diff --git a/STATUS.md b/STATUS.md index a78f484..4083355 100644 --- a/STATUS.md +++ b/STATUS.md @@ -17,34 +17,37 @@ hourly routine with that file as the prompt. Everything needed is in this directory. New attempts are welcome as pull requests; see `CONTRIBUTING.md`, and `AGENTS.md` for whether to work blind or informed. -Current standing. HEADLINE: the 2026-07-31 cycle attacked all three -gaps of the LIVE union-closed route in parallel; every verdict is -skeptic-confirmed. Gap (a′) — averaged odds-ratio control — is -REFUTED by an explicit 10-atom witness (M_5 = λ − 0.122, all -marginals ≤ 0.318 < 0.38271), certified float-free in exact rational -arithmetic and robust to the bookkeeping convention; the i-AGGREGATED -control survives certified (+1.84 on the witness itself) and is the -restated gap. Gap (c): the smoothness step 005's Prop 6 was missing is -now a proved theorem (P6′, quantitative IFT, skeptic-verified at fixed -n) — but the skeptic's n=32 census killed the naive n-uniform budget -(the flattening reverses past n ≈ 22) and showed the tax is δ-linear, -so the conditional assembly needs a corrected budget object. Gap (b) -got its first precise statement and survived everywhere tested (the -slice tilt is provably tax-free; a newly named "second tax" -~0.40·log₂n is the binding loss channel; λ-window law λ_max ≈ -4.847/(n−3)). Route stays LIVE, ceiling 0.4315 vs record 0.38271. -Also this cycle: the billiards W(a,b) death-angle laws held both -pre-registered out-of-sample predictions, were unified into a general -law γ_d(a,b) = 180 − 90(a+b)/(a(b+1)), and PROVEN as a necessity -theorem (machine-certified for 15 members, skeptic-confirmed); and the -first cross-field ideation sweep ran on union-closed (6 lenses → 3 -queue-worthy routes, 5 no-purchase verdicts, one proven product-weight -no-go). Standing library: Singmaster census to 2.5×10^29; -Erdős–Gyárfás for all cubics to n=22 + girth-≥5 at n=24; lonely-runner -k=8; Erdős–Straus identity-poverty; graceful n ≤ 14; mahler-4d and -billiards blind censuses. 19 verified results, 7 recorded dead ends, -10 problems, ~40 reusable tools. Crouzeix remains the one problem with -no attempts (queue; run blind). +Current standing. HEADLINE: the second 2026-07-31 cycle ran the +two-track billiards plan (old queue 11+12); all four records are +skeptic-confirmed, six corrections total, none load-bearing. +Conservative track (005+008): the W(a,b) death law is closed on both +sides — the half word composes in closed form to +R₀·Rot_A(2aα)·Rot_B(−2bβ), so I1–I4 and the glide facts are proven +for ALL (a,b) by one formal-ring check, Lemmas C and D fall to an +elementary monotonicity lemma, the case tree closes with the +a ≤ 2b+3 restriction REMOVED (necessity is now fully parametric for +a ≥ b ≥ 1, a ≥ 2), and certified alive segments into the death +corner give death(W(a,b)) = γ_d(a,b) EXACTLY (sup not attained) for +20 members, independently re-certified to γ_d − 9.3e-10. Exploratory +track (006+007): the 135° stall dissolves — 001's pinch gap +[135.000°, 135.049°] was a sampler artifact (W(4,3) certified alive +inside it; the mechanism of 001's error is pinned), births follow +γ_birth = 180 − 90(a+b+1)/(a(b+1)) (SPECULATION; survives every +out-of-sample test at the sampler floor), consecutive windows touch, +and W(5,2) plus a genuinely non-W four-block word are certified +alive at γ = EXACTLY 135° — the "135° is a constructive barrier" +kill condition is refuted. Union-closed (previous cycle, all +skeptic-confirmed): gap (a′) refuted float-free by a 10-atom witness; +the i-AGGREGATED control survives and is the restated gap; P6′ +proved at fixed n but naive n-uniform budgets killed (flattening +reverses past n ≈ 22; tax is δ-linear); gap (b) precisely stated and +surviving; route LIVE, ceiling 0.4315 vs record 0.38271. Standing +library: Singmaster census to 2.5×10^29; Erdős–Gyárfás cubics to +n=22 + girth-≥5 at n=24; lonely-runner k=8; Erdős–Straus +identity-poverty; graceful n ≤ 14; mahler-4d and billiards blind +censuses. 21 verified results, 7 recorded dead ends, 10 problems, +~45 reusable tools. Crouzeix remains the one problem with no +attempts (queue; run blind). ## Problem status @@ -57,7 +60,7 @@ no attempts (queue; run blind). | Lonely runner | k=8 done | medium | next: k=9 scan (k=8 likely settled by Rosenfeld preprint) | | Graceful trees | census done (n≤14) | low | possible next: mine the symmetric-spider seed; lobster verification at larger n | | Collatz | queued | low (long shot) | failed-approach taxonomy; cycle-bound frontier | -| Triangular billiards | death law PROVEN as necessity (003/004, skeptic-confirmed) | high | two-track plan (queue 11+12): complete the parametric theorem; use the factorization as a DESIGN tool to hunt a family with γ_d > 135° | +| Triangular billiards | death law CLOSED both sides: parametric necessity all (a,b), death = γ_d exactly for 20 members (005/008); 135° stall dissolved, birth law + exact-135 certificates (006/007) | high | next: parametric sufficiency + birth-law theorem (queue 11); coverage conjecture + sampler blind spot (queue 12) | | Mahler in ℝ⁴ | census done blind (skeptic-confirmed) | medium | next: close k=12–20 (falsifiable: no proper mask with P<11); run the same pipeline on {0,±1}³ for the n=3 spectrum comparison | | Crouzeix | onboarded, no attempts | medium | harness ready, certification risk retired; first attempt is the dim-3 landscape (run blind) | @@ -73,11 +76,10 @@ no attempts (queue; run blind). 8. [collatz] Failed-approach taxonomy page (library showcase; pure writing + citation verification). 9. [graceful-trees] Mine the symmetric-spider seed (LpH?GCAO??_@?A genre) at n = 15-16 targeted; lobster verification at larger n. 10. [crouzeix] **Run blind.** Local-maxima census, dimension 3, polynomial degree ≤ 3: do all basins terminate at known extremal structure? `EVIDENCE` scoped by dimension, degree and search design, all of which must be in the record — and state how the search design differs from Greenbaum–Overton's before running it, since reproducing their finding is not a result. Kill condition (measured, not hypothetical): one certified ratio enclosure costs about 0.4s in dim 2, 0.75s in dim 3 and 1.4s in dim 4 at tolerance 1e-9 with 32 directions, so a wide census is affordable only in the low thousands of points; if the design needs more, narrow to structured families where the norm has a closed form and record the narrowing. -11. [billiards-triangles] **Complete the death-law theorem** (one line; conservative track of the two-track plan, 2026-07-31): (i) sufficiency — certified alive points approaching γ_d via the first-order corner argument (003 lead 2; note 004's correction that a zero-width touching corridor AT γ_d is not excluded by the necessity theorem), turning "no positive width at or past γ_d" into "death is exactly γ_d"; (ii) prove I1–I3 and Lemma C for general (a,b) rather than per member (certified for 15 members; Lemma C numerically true for all b ≤ a ≤ 25) — a symbolic proof over the parameter upgrades the whole family curve at once; (iii) map the a > 2b+3 branch (W(6,3)-type members are unmeasured; the case tree breaks there, so first measure, then extend or restate). Deliverable either way: the exact parametric death theorem, or the precise step that resists. -12. [billiards-triangles] **Design a family past 135°** (one line; exploratory track of the two-track plan, 2026-07-31 — this is the line with real upside on the constructive frontier). The death law explains WHY the W-family dies: closed-form binding factors like cos(aα)·sin((b+1)β)·sin(α+β) vanish at γ_d = 180 − 90(a+b)/(a(b+1)), which is bounded by the family's glide structure. Invert the machinery: parametrize a broader class of half-word/gate structures in the division-free Laurent ring (003's `deathlaw_symbolic.py` is the substrate), derive each candidate family's binding factorization and death angle symbolically, and hunt for structures whose factors vanish only past 135°. Falsifiable first step: derive γ_d formulas for the nearest structural neighbours of W(a,b) (extra letter block, asymmetric gates) and check whether ANY exceeds 135°; any candidate must then produce one certified alive point at γ > 135° with the existing exact-corridor machinery — that alone would push the lab's constructive frontier past the census stall. Kill condition: if every family in the searched structure class has γ_d ≤ 135° — or the factor structure provably caps γ_d — record that obstruction precisely; it would itself be strong evidence that 135° is a real constructive barrier, sharpening queue item 14. Any certified orbit or claimed cap → skeptic review. +11. [billiards-triangles] **Parametric sufficiency + the birth side** (from 005/008 and 006/007, 2026-07-31): (i) prove a parametric positive lower bound on the *generic* fan-gate margins along the universal segment (α,β) = (90/a − t, 90(a−1)/(a(b+1)) + 2t), t ∈ (0, 1/4] — each margin is a 3–5-term trig polynomial with the fan index entering linearly via the prefix maps; this is the ONLY missing piece for death(W(a,b)) = γ_d(a,b) at ALL (a,b). Mind the 3-fold degenerate death corner: the gate-(2a+2) margin (identity I4) also vanishes there — a naive 2-margin Taylor route silently misses it (005). (ii) Prove the birth law γ_birth(a,b) = 180 − 90(a+b+1)/(a(b+1)) (SPECULATION; survives out-of-sample at the sampler floor incl. a > 2b+3 members) with the same gate machinery — which gate pair binds at the birth edge — and produce exact birth brackets from below (NONE exist for any member; all float births share a one-sided floor bias). Windows-touch (birth(W(a+1,a)) = death(W(a,a))) then makes the family staircase fully algebraic. Cheap side task: measure the a = 1 column, still untouched. +12. [billiards-triangles] **The coverage conjecture, and the sampler blind spot** (from 006/007, 2026-07-31; absorbs the old pinch-gap item — its motivating gap [135.000°, 135.049°] is CLOSED, W(4,3) is certified alive inside it): 006 reduced "every obtuse angle has an alive W member" to an elementary Diophantine statement (unproven; float-checked at 157 + 25 arcs over 90.5°–165°, zero failures). Prove it, using the birth law as a labelled input where needed. Note the certificates so far are POINTWISE (007's C2): window-interval continuity on sub-arcs is float + SPECULATION law only, and per-triangle coverage of a whole arc is a different (open) question — the windows are x-slivers at the corners. Separately falsifiable (007 lead): every sampler in use accumulates only at the 90/j window edges, so an interior-pinch alive window would hide from ALL current designs — build one targeted interior-accumulation test before trusting any negative screen again. 13. [mahler-4d] Close the {0,±1}⁴ universe: k = 12–20 pairs (~30M orbits at k=12, improper fraction already 77% at k=9). Falsifiable: no proper mask with k ≥ 12 has P < 11. Needs the improper-detection shortcut or a streaming canonicalizer; see 001 lead 1. Cheap side quest, same pipeline: the {0,±1}³ census for the n=3 spectrum comparison (13 pairs, trivial) — does the non-Hanner gap grow or shrink with n? -14. [billiards-triangles] Pinch-gap scan: what covers [death(W(a,a)), birth(W(a+1,a))] (e.g. the measured [135.000°, 135.049°] where nothing ≤ 30 is alive)? Scan words of length 28–34 restricted to the family's letter statistics. If nothing bounded covers a neighbourhood of γ = 180a/(a+1)° on the arc-minimum curve, those angles are genuine accumulation points of the constructive problem. The general law γ_d(a,b) from 003 now predicts where W(a+1,a) births/deaths sit — use it to target the scan. (If queue item 12 finds a structural cap at 135°, fold that finding in here.) -15. [billiards-triangles] Coverage self-test: re-derive the acute and right-triangle cases as a scoped attempt record. Low value now that the harness self-test covers Fagnano and the orthic geometry and 001 mapped the obtuse side — take it only if something turns up that the certificate machinery cannot express. +14. [billiards-triangles] Coverage self-test: re-derive the acute and right-triangle cases as a scoped attempt record. Low value now that the harness self-test covers Fagnano and the orthic geometry and 001 mapped the obtuse side — take it only if something turns up that the certificate machinery cannot express. ## Verified results @@ -254,6 +256,62 @@ no attempts (queue; run blind). touching at γ_d is not excluded; the general-(a,b) identities remain SPECULATION (per-member certified); a > 2b+3 uncovered. +- **[billiards-triangles] Death law closed on both sides: parametric + necessity for all (a,b), death = γ_d exactly for 20 members + (skeptic-confirmed)** (2026-07-31, attempts 005+008): the half word + composes in closed form to R₀·Rot_A(2aα)·Rot_B(−2bβ), so adjoining + e^{iaα}, e^{ibβ} as formal Laurent variables proves I1–I3, the glide + facts, and a new identity I4 (gate 2a+2) for ALL integers a, b in one + exact polynomial check, specialized by ring homomorphism (bridge to + geometry cross-checked exactly on 40 members to a = 40 by the + skeptic's own off-torus recomposition). Lemma C is proven for all + a ≥ 2, b ≤ a and Lemma D proven — both via one elementary lemma + (c·cot(cs) strictly decreasing), retiring 004's residual. The case + tree closes for all a ≥ b ≥ 1, a ≥ 2: the a ≤ 2b+3 restriction is + GONE (new branches re-derived by hand twice; ~1.56M + 447k + adversarial sign-scan points incl. 10 members with a > 2b+3, zero + hits). Sufficiency: along a universal segment into the death corner + every corridor condition is certified strict, so death(W(a,b)) = + γ_d(a,b) EXACTLY (sup not attained) for 20 members; the skeptic + independently certified alive by full 2n-gate interval unfolding (no + glide reduction) down to γ_d − 9.3e-10. Five new members measured at + γ_d to ~5e-12: W(6,3)→146.25, W(6,1)→127.5, W(8,2)→142.5, + W(8,1)→1035/8, W(7,1)→900/7 (windows on the mirror half). Correction + (008-C1): four of the five new alive certificates reach only + γ_d − 1e-4, not the 1e-6 one sentence claims — superseded by the + segment certificates. Open: parametric sufficiency (generic fan-gate + margins; the death corner is 3-fold degenerate, not 2-fold), the + a = 1 column, the birth side. +- **[billiards-triangles] The 135° stall dissolves: pinch gap closed, + birth law found, alive at exactly 135° (skeptic-confirmed)** + (2026-07-31, attempts 006+007): inverting the death-law machinery + into a structure-class design search (plus a small new fact: ANY odd + half word u makes u² a translation word, so the glide reduction + applies class-wide) shows 001's pinch gap [135.000°, 135.0486°] was + a sampler artifact: W(4,3) is certified alive inside it, and the + mechanism of 001's error is pinned — the alive window is a corner + sliver thinner than its grid's terminal 6e-4 clearance (001's + knowledge-scoped gap claim stays true as written; what falls is its + birth measurements, off by 10× the stated noise). Births obey + γ_birth(a,b) = 180 − 90(a+b+1)/(a(b+1)) (SPECULATION as a law; + survives three genuinely out-of-sample members incl. a > 2b+3 at the + ~3e-11 sampler floor), making consecutive windows touch. W(5,2) + (length 30, never measured before) and a genuinely non-W four-block + word N1 (length 46) are certified alive at γ = EXACTLY 135° via + rational apexes on the arc (2x−1)² + (2y+1)² = 2 (on-arc identity + hand-verified in Fractions; the W(5,2) triangle is irrational-angled + by Niven). All five exact certificates re-established digit-for-digit + by a third corridor implementation. 003's theorem extends to (5,2) + and (6,2). The kill condition is REFUTED: 135° is not a constructive + barrier — every sampled arc in 90.5°–165° (157 + 25 independent) has + a W member float-alive. Negative screens (nothing of length ≤ 26 + alive past 135°; extra letter blocks collapse death angles; clean + angle laws confined to the two-fan factorization) are sample-bounded. + New structural fact (007-C4): W(a,b) and W(b+1,a−1) are the SAME + canonical word — 003's "two distinct words dying on the same arc" + was a relabeling. Corrections: length-30 universe is 566 words (not + 811); gap certificates are pointwise, not "alive throughout". + ## Insights / cross-problem notes - Infrastructure (2026-07-27): cross-pollination layer added. `mechanisms.json` @@ -354,6 +412,35 @@ no attempts (queue; run blind). findings, not generic vocabulary — "certified enclosure" collided with tier-0 problem statements and tripped the leak test. +- Sampler blind spots are one-sided and shared (2026-07-31, billiards + 006/007): 001's birth angles were wrong by 0.0486° — 10× its stated + noise — because the alive window near birth is a corner sliver + thinner than the grid's terminal clearance, and its "deaths only + under-estimated" hedge does not transfer to births. Cheap + validations that caught it: floor-tracking (re-measure at several + sampler depths and check whether the value tracks the sampler floor) + and certifying an exact point inside the claimed-empty region. + Standing residual: every billiards sampler in use accumulates only + at the 90/j window edges, so an interior-pinch window would hide + from all of them (queue item 12). +- Formal-parameter specialization proves infinite identity families in + one check (2026-07-31, billiards 005/008): move the family + parameters into the exponent lattice (adjoin e^{iaα}, e^{ibβ} as + formal Laurent variables), verify the identity once by exact + polynomial subtraction, specialize by ring homomorphism to every + integer (a,b). The entire risk concentrates in the hand-derived + bridge (geometry = closed form) — the formal check cannot see a + wrong bridge — so cross-check the bridge exactly on many members; + the specialization direction itself is safe (formal zero ⟹ + geometric zero). New mechanism tags: formal-parameter-specialization, + structure-class-design-search. +- Canonicalize before counting evidence (2026-07-31, billiards 007): + W(a,b) and W(b+1,a−1) are the same canonical word, which made a + claimed "window pairing" vacuous (the fit spanned 9 canonical words, + not 11) and turned "two distinct words dying on the same arc" into a + relabeling. Any law fitted over family members must first quotient + by the word-canonicalization symmetry. + ## Dead ends - **[union-closed] Pointwise Plackett odds-ratio control (gap (a) as diff --git a/mechanisms.json b/mechanisms.json index cfe45ad..71ed16a 100644 --- a/mechanisms.json +++ b/mechanisms.json @@ -124,6 +124,10 @@ "field": "logic-methodology", "description": "Cross-field ideation sweep: apply untried field lenses to one problem in parallel to generate candidate attack routes; output is a MAP of triaged leads and recorded no-purchase verdicts (docs/IDEATE.md)." }, + "formal-parameter-specialization": { + "field": "algebra", + "description": "Prove an infinite family of identities at one stroke by moving the family parameters into the exponent lattice: adjoin e^{i a alpha}, e^{i b beta} (or analogous parameter-indexed units) as independent formal Laurent variables, verify the identity once by exact polynomial subtraction in the enlarged ring, and specialize by ring homomorphism to every integer parameter value. Turns a hand geometric-series computation into a finite machine check; needs a short hand derivation that the geometric quantities equal the formal closed forms." + }, "girth-restriction": { "field": "graph-theory", "description": "Restrict the search to high-girth instances, where short-cycle escapes are unavailable and structure is forced." @@ -212,6 +216,10 @@ "field": "analysis", "description": "Examine the set of achieved objective values for gaps and accumulation points, not just the extremum." }, + "structure-class-design-search": { + "field": "dynamics", + "description": "Invert a proven structural mechanism into a design tool: parametrize the structure class the mechanism applies to, enumerate the class, derive each member's governing invariant (symbolically where the mechanism factors, numerically where it does not), and hunt for members whose invariant lands in a target range." + }, "symbolic-trig-closed-form": { "field": "algebra", "description": "Recast an iterated geometric construction in an exact division-free symbolic domain (here Laurent polynomials in half-angle exponentials over Q(i)) so that the quantities of interest become closed-form trigonometric polynomials whose factorizations expose the law being sought." diff --git a/problems/billiards-triangles/attempts/005-complete-death-law-theorem.md b/problems/billiards-triangles/attempts/005-complete-death-law-theorem.md new file mode 100644 index 0000000..9b1e8a2 --- /dev/null +++ b/problems/billiards-triangles/attempts/005-complete-death-law-theorem.md @@ -0,0 +1,517 @@ +# 005 — The death law completed: parametric necessity for all (a,b), exact death per member + +- **Problem:** billiards-triangles, `problems/billiards-triangles/PROBLEM.md` +- **Date:** 2026-07-31 +- **Mode:** informed (read `prior-art.json`, attempts 003 and 004 in full, + the tier-0 statement, and 001/002 skimmed for conventions; queue item 11 + of `STATUS.md`, the conservative track of the two-track plan) +- **Type:** formalization + computation — parametric proofs of 003's + per-member inputs (I1–I3 for all integers (a,b); Lemma C and Lemma D for + all a ≥ 2, b ≤ a), removal of the a ≤ 2b+3 scope restriction from the + case tree, and certified sufficiency (death = γ_d exactly) for 20 members +- **Tools:** new `explore/plaw_general.py` (formal 4-variable Laurent-ring + proofs, per-member exact specialization cross-checks, extended-tree + adversarial scans, Lemma-C sanity grid), new `explore/plaw_suffice.py` + (certified alive segments into the death corner: exact mod-360-reduced + interval trig, per-gate 1-D sign certification). Reused as substrate: + 003's `deathlaw_symbolic.py` (glide data; its selftest ties it to 002's + independent corridor and 004 confirmed it digit-for-digit), + `deathlaw_measure.py`, `deathlaw_exact.py` (which itself re-derives + orbits with 002's `skeptic_orbit.py`), tier-0 `unfold.py` (rational pi + enclosure; interval sin/cos as cross-reference in the selftest). 004's + `deathlaw_skeptic.py` deliberately NOT used — it stays reserved for + skeptics. All stdlib-only, deterministic; total compute ≈ 15 min. +- **Sources:** none external. Repo: attempts 003 (the record extended + here), 004 (its corrections C1, C2 heeded: "alive" = positive corridor + width throughout, and a zero-width touching corridor at γ_d is not + excluded — the sup below is proven not attained). + +**Conventions** (001/003): W(a,b) = (0 (12)^a (02)^b)^2, angles α at A, β +at B, γ = 180 − α − β obtuse at C, θ = α + β; θ_d(a,b) = 90(a+b)/(a(b+1)) +deg, γ_d = 180 − θ_d; "alive at a triangle" = the word's unfolding +corridor has positive width there; death(W) = sup of γ over alive +triangles. Circumdiameter-1 normalization: A = 0, B = sin(α+β), +C = sin(β)e^{iα}; p(z) = Im(conj(δ)z) is the projection onto the normal of +the glide axis direction δ of the half word u = 0(12)^a(02)^b; m is the +axis offset. All angle arithmetic below is in degrees unless marked. + +## Approach + +Queue item 11 in its stated order: (i) sufficiency, (ii) parametric +inputs, (iii) the a > 2b+3 branch. I did (ii) first because it changes +what (i) and (iii) even mean: with I1–I3 and Lemma C proven for general +(a,b), the necessity theorem stops being a per-member statement, and both +the sufficiency segments and the new-branch members inherit it for free. + +The key methodological choice, and why it beats the obvious alternative: +003's Lead 1 proposed proving I1–I3 by carrying out the two-fan geometric +sum by hand (Dirichlet-kernel forms). That proof would be pages of +sign-sensitive trigonometric algebra, exactly the kind a skeptic distrusts. +Instead I changed the domain so that the general statement becomes a +FINITE exact computation: since the half word composes to + + u = R_0 ∘ (R_1 R_2)^a ∘ (R_0 R_2)^b = R_0 ∘ Rot_A(2aα) ∘ Rot_B(−2bβ), + +the parameters a, b enter every relevant quantity only through e^{iaα} and +e^{ibβ}. Adjoining these as INDEPENDENT formal variables P, Q turns each +identity into one Laurent-polynomial identity in Q(i)[x^±, y^±, P^±, Q^±] +(x = e^{iα}, y = e^{iβ}), checkable once by exact arithmetic and then true +for every integer pair (a,b) and all angles by ring-homomorphism +specialization P → x^a, Q → y^b. The only hand steps left are the short +derivation of the closed form of u (three lines, written out below) and +the specialization argument; both are cross-checked exactly, member by +member, against 003's independent gate-by-gate symbolic unfolding. + +For Lemma C, 003's Lead 1 route (log-derivative of Φ) turned out to close +with one elementary monotonicity lemma; the same lemma also proves Lemma D, +removing 004's "taken as classical" residual. For sufficiency, instead of +Lead 2's first-order expansion with a quadratic remainder (which needs +all-gate Taylor models), I certify an explicit rational SEGMENT of +triangles ending at the death corner, reducing "alive on the whole +segment" to ~n one-dimensional certified sign problems — the pattern of +`deathlaw_prove.py`, single non-iterated interval evaluations only. + +## What was done + +### (ii)-A. I1–I3 for ALL integers a, b ≥ 1 — formal proof + +**Step 1 (composed-map form of the unfolding; standard, one line).** If +M_k is the isometry placing the (k+1)-st unfolded copy, and gate k is the +image of the mirror side s_k, then reflecting across the current image +side g_k = M_{k−1}(side s_k) is M_{k−1} R_{s_k} M_{k−1}^{−1}, where R_s is +the reflection across side s of the BASE triangle; composing on the left +with M_{k−1} gives + + M_k = M_{k−1} ∘ R_{s_k}, so M_k = R_{s_1} ∘ R_{s_2} ∘ … ∘ R_{s_k}, + +and gate k = M_{k−1}(side s_k). (004 already re-derived the unfolding in +exactly this form and tied it to the harness corridor; here it is also +verified exactly per member, Step 4.) Consequently the composed map of +the half word is the word-ordered product u = R_0 (R_1 R_2)^a (R_0 R_2)^b, +and since W = u·u, gate n′+k = u(gate k) — 003's functoriality — is an +identity of the construction. + +**Step 2 (pair collapse).** In the circumdiameter normalization the base +reflections are R_2: z ↦ conj z (side AB = real axis through A = 0), R_1: +z ↦ e^{2iα} conj z (side CA through A = 0 with direction e^{iα}), R_0: +z ↦ B + e^{−2iβ} conj(z − B) (side BC through B = sin θ with direction +e^{i(180−β)}). Hence R_1∘R_2 = rotation about A by +2α and R_0∘R_2 = +rotation about B by −2β (compose the two formulas; the conjugations +cancel). + +**Step 3 (closed form of u).** Composing u = R_0 ∘ [z ↦ e^{2iaα} z] ∘ +[z ↦ B + e^{−2ibβ}(z − B)] and using conj(B) = B: + + u(z) = μ conj(z) + w, + μ = e^{−2iaα + 2i(b−1)β} = P^{−2} Q^2 y^{−2}, + w = B · (1 + P^{−2} y^{−2} − y^{−2} − μ), B = sin(α+β), + +so δ = P^{−1} Q y^{−1} = e^{i(−aα + (b−1)β)} satisfies δ² = μ, and +τ = w + μ conj(w), m = Im(conj(δ)(w − τ/2))/2. The word length 2a+2b+1 +of u is odd, so u is orientation-reversing; u² is automatically a +translation (u²(z) = |μ|² z + μ conj(w) + w = z + τ). + +**Step 4 (formal verification — this is the proof of the identities).** +`plaw_general.py formal` builds these closed forms in the formal ring +Q(i)[x^±, y^±, P^±, Q^±] and checks by exact polynomial subtraction: + + - glide facts: μ = δ² and Im(conj(δ)τ) = 0 (τ parallel to the axis); + - **I1**: m − p(C) = −[cos(aα) sin α sin(bβ) + sin(aα) sin β cos(α+(b+1)β)] + - **I2**: p(A₁) − m = cos(aα) sin((b+1)β) sin(α+β) + - **I3**: p(B) − m = cos((b+1)β) sin(aα) sin(α+β) + - **I4** (new; needed for sufficiency): with v₂ = R_0(Rot_A(2aα) C) the + C-image endpoint of gate 2a+2, + p(v₂) − m = cos(aα) sin α sin(bβ) − sin(aα) sin β cos(α+(b+1)β) + (note I1 = −(S+T) and I4 = S−T for the same two products S, T); + - context: D1, D2 (collapsing-gate loci) as in 003. + +All eight are the zero polynomial. Since P → x^a, Q → y^b followed by +x → e^{iα}, y → e^{iβ} is a ring homomorphism, **I1–I4, D1, D2 and the +glide facts hold for every integer a, b ≥ 1 and all real angles.** This +discharges 003's SPECULATION (i) — and the glide-to-corridor reduction, +whose only member-specific inputs were the glide facts, is now itself +fully parametric. + +``` +python3 problems/billiards-triangles/explore/plaw_general.py formal --out problems/billiards-triangles/data/plaw_formal.json +``` + +**Cross-checks of the hand steps 1–3** (the formal check cannot catch a +wrong closed form, so the bridge to the geometry is tested separately): + + - exact, per member: specializing the formal μ, w, δ, τ, m, p(B), p(C), + p(A₁), p(v₂) at (a,b) reproduces `deathlaw_symbolic`'s independently + computed ring elements (gate-by-gate reflected-vertex chains — a + different construction, 004-audited) EXACTLY, for all 15 members of + 003 plus (6,3),(6,1),(8,2),(8,1),(7,1),(11,2),(14,4),(17,9): + `plaw_general.py specialize` → 23/23 MATCH + (`data/plaw_specialize.json`); + - float: closed-form u(z) vs directly composed reflections at 400 + random ((a,b), α, β, z), a ≤ 40 — worst error 1.4e-14 + (`plaw_general.py floatcheck`). + +### (ii)-B. Lemma C and Lemma D for general (a,b) — elementary proof + +The whole of both lemmas reduces to (radians throughout this subsection): + +**Lemma L1.** For fixed s ∈ (0, π/2], the map c ↦ c·cot(cs) is strictly +decreasing on (0, 1]. +*Proof.* d/dc [c cot(cs)] = cot(cs) − cs/sin²(cs) = +[sin(2cs) − 2cs] / (2 sin²(cs)) < 0, since sin x < x for x > 0 and +cs ∈ (0, π/2] keeps sin(cs) > 0. ∎ + +**Lemma C (general).** For integers a ≥ 2, 1 ≤ b ≤ a, with +v₀ = π/(2(b+1)): H2(v) := sin v sin((π/2−v)/a) cos v₀ − +sin v₀ sin(bv/a) cos v > 0 for all v ∈ (0, v₀), and H2(v₀) = 0. +*Proof.* Let Φ(v) = tan v · sin((π/2−v)/a) / sin(bv/a) (all three factors +positive on (0, v₀]: v ≤ v₀ ≤ π/4). Then, using tan v + cot v = +1/(sin v cos v) = 2/sin 2v, + + (log Φ)′(v) = 2/sin(2v) − (1/a) cot((π/2−v)/a) − (b/a) cot(bv/a) + = [cot v − (b/a) cot((b/a)·v)] + + [tan v − (1/a) cot((π/2−v)/a)]. + +The first bracket is ≤ 0 by L1 with s = v, comparing c = b/a ≤ 1 against +c = 1 (equality iff a = b). The second is < 0 by L1 with s = π/2 − v ∈ +(0, π/2), comparing c = 1/a ≤ 1/2 against c = 1 (strict since a ≥ 2), +because cot(π/2 − v) = tan v. So Φ is strictly decreasing on (0, v₀]. +At the endpoint, π/2 − v₀ = b v₀ exactly, so Φ(v₀) = tan v₀; and +H2(v) = cos v cos v₀ sin(bv/a) · [Φ(v) − Φ(v₀)] > 0 on (0, v₀). ∎ + +This is precisely the log-derivative route sketched in 003's Lead 1, and +it discharges SPECULATION (ii) — with a strictly larger scope than +conjectured (F < 0 holds on all of (0, π/2), not just (0, v₀]). + +**Lemma D (upgrade from "classical, hand-derived").** D_b(x) = +sin(bx)/sin(x) strictly decreasing on (0, π/(2b)] for b ≥ 2: +(log D_b)′ = b cot(bx) − cot x < 0 by L1 with s = bx, c = 1/b vs c = 1. +D_1 ≡ 1. ∎ (004 listed Lemma D as a residual risk; it is now proven by +the same two-line mechanism.) + +Sanity net (not load-bearing — the proofs above are elementary): +`plaw_general.py lemmac` checks F < 0 and H2 > 0 at 914,500 grid points, +a ≤ 60: zero violations (`data/plaw_lemmac_grid.json`). + +### (iii). The a > 2b+3 branch: the restriction is REMOVABLE + +**Measurements first** (the queue's order). `deathlaw_measure.py death +--full` on the unmeasured members, including W(6,3) (b = a−3, inside old +scope but never measured) and the genuinely out-of-scope W(6,1), W(8,2) +(a = 2b+4), W(7,1) (a = 3b+4), W(8,1) (a = 3b+5): + +| word | len | γ_d(a,b) | measured death | meas − pred | last-alive apex | +|---|---|---|---|---|---| +| W(6,3) | 38 | 146.25 = 585/4 | 146.2499999999963 | −3.7e-12 | (15, 18.75) | +| W(6,1) | 30 | 127.5 = 255/2 | 127.4999999999947 | −5.3e-12 | (15, 37.5) | +| W(8,2) | 42 | 142.5 = 285/2 | 142.4999999999964 | −3.6e-12 | (11.25, 26.25) | +| W(8,1) | 38 | 129.375 = 1035/8 | 129.3749999999951 | −4.9e-12 | (11.25, 39.375) | +| W(7,1) | 34 | 900/7 = 128.5714… | 128.5714285714234 | −5.1e-12 | (12.857, 38.571) | + +``` +python3 problems/billiards-triangles/explore/deathlaw_measure.py death --a 6 --b 3 --full --out problems/billiards-triangles/data/plaw_measure_W63.json # + (6,1),(8,2),(8,1),(7,1) +``` + +Every member dies at γ_d, at the predicted corner (90/a, 90(a−1)/(a(b+1))) +— the law does NOT change past a = 2b+3. That prompted a re-derivation of +the case tree, which closes it: + +**Extended case tree (hand; replaces 003's A4/A4′/A5/A5′, keeps +everything else verbatim).** Standing notation of 003's proof: positive +width forces (Case I) N1, N2, N3 > 0 or (Case II) N1, N2, N3 < 0, where +N1 = m − p(C) = −G, N2 = p(A₁) − m, N3 = p(B) − m, via I1–I3 all three +are the stated products, and sin θ > 0 always. Branch on the residue +r = aα mod 360 and k = ⌊aα/360⌋ ≥ 0; residues at multiples of 90 force +N2 = 0 or N3 = 0 (excluded), likewise (b+1)β at multiples of 90. + +*Case II.* + - r ∈ (0,90) ∪ (270,360): cos(aα) > 0, so N2 < 0 forces + sin((b+1)β) < 0, so (b+1)β > 180, so θ > β > 180/(b+1) ≥ θ_d + (⟺ a ≥ b). **The (270,360) half is the fix of A4′: 003 used + θ > α > 270/a ≥ θ_d there, which is what required a ≤ 2b+3; the sign + of cos(aα), already positive on this branch, makes the A1′ argument + apply verbatim and never touches α.** + - r ∈ (90,180): N3 < 0 forces cos((b+1)β) < 0, so (b+1)β > 90; and + aα > 90 gives α > 90/a; θ > 90/a + 90/(b+1) > θ_d (slack + 90/(a(b+1))). Works for every k (α only enters via α > 90/a). + - r ∈ (180,270): N2 < 0 and N3 < 0 force sin((b+1)β) > 0, + cos((b+1)β) > 0, so (b+1)β mod 360 ∈ (0,90). If (b+1)β > 360: + θ > 360/(b+1) ≥ θ_d (⟺ 3a ≥ b). Else (b+1)β ∈ (0,90), so + sin(bβ) > 0; both cos(aα) < 0 and sin(aα) < 0, so G > 0 (Case II) + forces cos(α+(b+1)β) < 0, and α + (b+1)β ∈ (0, 270) (α < 180, + (b+1)β < 90) makes that α + (b+1)β > 90; with α > 180/a: + θ > (bα+90)/(b+1) > (180b/a + 90)/(b+1) > θ_d (⟺ 90b > 0). + Works for every k. (This is 003's A3′ unchanged; restated to show no + hidden k-dependence.) + +*Case I.* + - r ∈ (90,180): cos(aα) < 0, N2 > 0 forces sin((b+1)β) < 0, + (b+1)β > 180, θ > 180/(b+1) ≥ θ_d. (003's A2; k-free.) + - r ∈ (180,270): as 003's A3: sin((b+1)β) < 0 and cos((b+1)β) < 0 + together give (b+1)β > 180, done. (k-free.) + - r ∈ (270,360): cos(aα) > 0 and sin(aα) < 0. N3 > 0 forces + cos((b+1)β) < 0 and N2 > 0 forces sin((b+1)β) > 0, so + (b+1)β mod 360 ∈ (90,180), hence (b+1)β > 90; with α > 270/a: + θ > 270/a + 90/(b+1) ≥ θ_d ⟺ 3(b+1) + a ≥ a + b ⟺ 2b + 3 ≥ 0. + **This is the fix of A4: the forced (b+1)β > 90, which 003 did not + use on this branch, converts the deficit 270/a < θ_d (a > 2b+3) into + an unconditional surplus.** + - r ∈ (0,90), aα = 360k + 90 − v with v ∈ (0,90): cos(aα) = sin v > 0, + sin(aα) = cos v > 0. N2, N3 > 0 force (b+1)β mod 360 ∈ (0,90); if + (b+1)β > 360 then θ > 360/(b+1) ≥ θ_d (⟺ 3a ≥ b); so assume + (b+1)β ∈ (0,90), hence sin β, sin(bβ) > 0. As in 003, + G = sin v sin α sin(bβ) − cos v sin β sin w with w = α + (b+1)β − 90 ∈ + (−90, 180), and the threshold algebra generalizes to + + a·w − b·v = a(b+1)(θ − θ_d) − 360·k·b. + + Suppose θ ≤ θ_d. + - **k ≥ 1 (the genuinely new branch):** then aw ≤ bv − 360kb < 0 + (v < 90 < 360), so w < 0, so sin w ≤ 0 (w ∈ (−90, 0]) and + G ≥ sin v sin α sin(bβ) > 0 — contradicting N1 > 0 (G < 0). No + Lemma C, no Lemma D, no bound on a. + - k = 0: 003's main-case argument verbatim (w ≤ 0 sub-case; + Regime 2 via Lemma D, using α < 90/a ≤ 45, i.e. a ≥ 2; Regime 1 + via Lemmas C + D, using w ≤ bv/a ≤ v, i.e. b ≤ a) — both lemmas + now proven in general above. + Hence θ > θ_d, strictly (each escape is strict; the k = 0 core is + strict by strict Lemmas C/D as in 003/004). + +**THEOREM (parametric necessity).** For all integers a ≥ b ≥ 1 with +a ≥ 2: if the corridor of W(a,b) has positive width at a triangle with +angles (α, β), then θ > θ_d(a,b), i.e. γ < γ_d(a,b). — The a ≤ 2b+3 +hypothesis of 003 is gone; no member-specific input remains. + +Adversarial support for the new branches (the k ≥ 1 and r ∈ (270,360) +regions are reachable): seeded sign-pattern scans at θ ≤ θ_d, boundary- +and corner-accumulated, 13 members with a up to 3b+9 — (6,1), (7,1), +(8,1), (9,1), (12,1), (8,2), (11,2), (14,2), (10,3), (17,3), (6,3), +(16,4), (12,12) — ~120k points each, ~1.56M total: **zero Case I or +Case II hits**; the positive control just above θ_d fires for every +member (`plaw_general.py casetree --a A --b B`, +`data/plaw_casetree_W*_*.json`). + +### (i). Sufficiency: death(W(a,b)) = γ_d(a,b) EXACTLY, certified for 20 members + +**Reduction (converse direction, new — 003 only needed ⟹).** With the +glide facts (now parametric), the full-word corridor is the intersection +[L, H] ∩ [2m − H, 2m − L] over the n′ half-word gate projection intervals +I_k = [lo_k, hi_k], L = max lo_k, H = min hi_k (gate n′+k = u(gate k) and +p ∘ u = 2m − p). If lo_k < m < hi_k STRICTLY for every k, then L < m < H, +so min(H, 2m−L) > m > max(L, 2m−H): positive width. Combined with 003's +forward direction: positive width ⟺ m strictly inside every half-gate +interval (τ ≠ 0 assumed; it is certified along the segments below). So +"alive on a segment" is exactly 2n′ one-dimensional sign conditions. + +**The segment.** Corner (α_c, β_c) = (90/a, 90(a−1)/(a(b+1))) — where +every measurement in 003 and above found the last-alive apexes. Take + + α(t) = α_c − t, β(t) = β_c + ρt, t ∈ (0, t₀], ρ = 2, + +so θ(t) = θ_d + (ρ−1)t and γ(t) = γ_d − (ρ−1)t ↑ γ_d as t ↓ 0. At the +corner exactly three of the distinct difference functions +D = p(endpoint) − m vanish (all members inspected; everything else is +bounded away): + + - p(C) − m (gates 1, 2) = −N1; by I1, on the segment + N1(t) = cos(at) sin β(t) sin(ŵt) − sin(at) sin α(t) sin(bβ(t)), + ŵ = (b+1)ρ − 1 (uses aα_c = 90 and α_c + (b+1)β_c = 90 exactly); + - p(A₁) − m = N2 (the A-fan pivot, one endpoint of gates 2…2a+2); by + I2, N2(t) = sin(at) sin((b+1)β(t)) sin θ(t) — every factor positive + on (0, t₀] by RATIONAL range checks alone (at ≤ 45, + (b+1)β(t) ∈ (90−90/a, 180), θ(t) ∈ (θ_d, 90)); + - p(v₂) − m (the C-image endpoint of gate 2a+2) = N4; by the new I4, + N4(t) = sin(at) sin α(t) sin(bβ(t)) + cos(at) sin β(t) sin(ŵt) — + BOTH products positive, again rational range checks only. + +N1 is the one genuinely two-sided margin: N1(0) = 0 exactly and +`plaw_suffice.py` certifies N1′ > 0 on [0, t₁] (interval enclosure of the +explicit derivative; ρ = 2 is chosen so the certified +N1′(0) = ŵ sin β_c − a sin α_c sin(bβ_c) > 0) and N1 > 0 on [t₁, t₀] by +adaptive bisection. Every other distinct difference (5–102 bisection +leaves per member) plus the nondegeneracy function Re(conj(δ)τ) is +certified to have constant sign on [0, t₀] by adaptive interval bisection, +and each gate's two endpoint differences are checked to straddle m. The +three identities I1, I2, I4 are re-verified per member by exact ring +subtraction before use. Certified trig: exact mod-360 degree reduction, +dyadic pre-rounding, alternating-series remainders, the harness's rational +pi enclosure; every evaluation is a single non-iterated interval +evaluation (the 2026-07-28 affine-forms lesson does not bite); +cross-checked against `unfold.sin_iv/cos_iv` and floats in `--selftest`. + +**Result.** For all 20 members — 003's fifteen plus (6,3), (6,1), (8,2), +(8,1), (7,1) — the certification succeeds with ρ = 2, t₀ = 1/4: + + for every t ∈ (0, 1/4]: W(a,b) is alive at (α_c − t, β_c + 2t), + with γ(t) = γ_d − t sweeping [γ_d − 1/4, γ_d). + +Hence death(W(a,b)) ≥ γ_d; with the necessity theorem, and since alive +γ < γ_d always, **death(W(a,b)) = γ_d(a,b) exactly, and the sup is not +attained** (004's C2: at γ_d itself the corridor closes to at most a +touching line). Per member this replaces 003's brackets +[γ_d − 1e-6, γ_d] by the exact value. + +``` +python3 problems/billiards-triangles/explore/plaw_suffice.py selftest +python3 problems/billiards-triangles/explore/plaw_suffice.py all --out problems/billiards-triangles/data/plaw_suffice_all.json # ~4 min, ALL MEMBERS CERTIFIED +``` + +Float cross-check at t₀/2 and t₀/97 via 002's independent corridor: +positive width at all 40 points. Independent exact-orbit confirmations +for the five new members (rational apex, exact positive corridor, orbit +re-derived by 002's exact simulator, certified γ enclosure): + +``` +python3 problems/billiards-triangles/explore/deathlaw_exact.py alive --a 6 --b 3 --dg 1e-6 --out problems/billiards-triangles/data/plaw_exact_alive_W63.json # + (6,1),(8,2),(8,1),(7,1) +``` + +all five: exact width > 0, orbit verified, γ certified within 1e-6 of γ_d. + +## Outcome + +- **VERIFIED (formal, all integers a, b ≥ 1):** the identities I1, I2, + I3, I4, D1, D2 and the glide facts (μ = δ², τ ∥ axis) hold for every + member of the family and all angles — proven by exact zero-polynomial + checks in Q(i)[x^±, y^±, P^±, Q^±] plus the three-line closed-form + derivation of u above; the derivation is exactly matched against the + independent symbolic unfolding for 23 members and float-matched at 400 + random (a,b) up to a = 40. 003's SPECULATION (i) is discharged. +- **VERIFIED (proof, all integers a ≥ 2, 1 ≤ b ≤ a):** Lemma C, and + Lemma D for all b ≥ 1, both via Lemma L1 (c·cot(cs) decreasing); + 003's SPECULATION (ii) is discharged and 004's Lemma-D residual risk + is retired. (a = b = 1 stays excluded: H2 ≡ 0 there.) +- **VERIFIED (theorem, parametric):** for all integers a ≥ b ≥ 1, a ≥ 2, + the corridor of W(a,b) has no positive width at any triangle with + γ ≥ γ_d(a,b) = 180 − 90(a+b)/(a(b+1)). The former scope condition + a ≤ 2b+3 is removed by sharpening two branches (the forced sign of + (b+1)β) and closing the aα > 360 residue-(0,90) branch (θ ≤ θ_d forces + w < 0 there). The k = 0 core of the main case is 003's argument + verbatim, now resting on the general lemmas. +- **VERIFIED (exact death, 20 members):** death(W(a,b)) = γ_d(a,b) + EXACTLY — sup not attained — for (2,1), (2,2), (3,2), (3,3), (4,2), + (4,3), (4,4), (5,3), (5,4), (5,5), (6,6), (7,7), (8,8), (10,9), + (12,12), (6,3), (6,1), (8,2), (8,1), (7,1): certified alive segments + (α, β) = (90/a − t, 90(a−1)/(a(b+1)) + 2t), t ∈ (0, 1/4], all corridor + conditions certified strict throughout, γ(t) filling [γ_d − 1/4, γ_d). +- **EVIDENCE:** the five new float death measurements (all equal to γ_d + to ~5e-12, argmax at the predicted corner; DEAD verdicts sample-bounded + as always); the 1.56M-point adversarial sign scans supporting the + extended tree (zero hits, positive controls fire). +- **NOT claimed:** parametric sufficiency (death = γ_d for ALL (a,b) at + once) — the corner-segment construction is certified per member only; + see the obstruction below. Nothing about words outside W(a,b), about + unstable orbits, about the birth edge, or about triangles at γ = γ_d + itself beyond "no positive width" (a zero-width touching corridor at + γ_d remains possible, per 004's C2 — that is what "sup not attained" + means here). The measured deaths remain float statements; the exact + content is carried by the theorem + segments. a = b = 1 excluded + throughout. + +## Why it failed / what survived + +Nothing on the queue item's list failed; the honest ledger of what is +still open and why: + +1. **Parametric sufficiency is structured but unfinished.** The + certified segments use, per member, the signs of ~n′ = 2a+2b+1 generic + gate margins along [0, 1/4]. The uniformity is striking — every + member certifies with the SAME ρ = 2 and t₀ = 1/4 — and the three + corner-vanishing margins are now closed-form for all (a,b) (I1, I2, + I4), so exactly one step is missing: a parametric positive lower bound + on the OTHER gate margins along the segment. These are fan + projections: by Step 3 the prefix maps are R_0 Rot_A(2kα) and + R_0 Rot_A(2aα) Rot_B(−2jβ), so each margin is a 3–5-term trig + polynomial with fan index k (or j) entering linearly in the angles — an + explicit two-parameter family of 1-D inequalities (Dirichlet-kernel + style), not an unstructured n-gate mess. That is the precise residue + of Lead 2; per-member certification sidestepped it, and I stopped + there rather than start a second hand-proof campaign in one cycle. +2. **The corner is 3-fold degenerate, not 2-fold.** 003's "only gates 1 + and 2 bind near death" is correct for the necessity direction, but at + the death corner itself a THIRD margin vanishes: the gate-(2a+2) + C-image endpoint (identity I4; N1 = −(S+T), N4 = S−T). Anyone + attempting Lead 2's Taylor route without noticing this would have + certified a spurious first-order system. Found because the segment + certifier failed loudly on it at every t₀. +3. **A method lesson worth the ledger:** "prove it for all (a,b)" became + a finite computation the moment the parameters were moved into the + exponent lattice (P = e^{iaα}, Q = e^{ibβ} as formal variables). The + two-fan geometric sum that 003 expected to do by hand was never + needed; the three-factor composition u = R_0 Rot_A(2aα) Rot_B(−2bβ) + is the entire structural content of the family. The same trick should + apply to ANY one-parameter word family whose repeated blocks pivot on + a common vertex — e.g. the design-track family hunt (queue item 12). +4. **What survived for reuse:** the formal-ring engine and closed forms + (`plaw_general.py`); the certified-segment machinery with exact + mod-360 trig (`plaw_suffice.py` — the trig layer is a drop-in upgrade + for any future 1-D certification, ~40× tighter than needed here); the + L1 lemma (one line, kills both Lemma C and Lemma D and plausibly any + future cot-comparison); the extended case tree; identity I4; the + converse reduction (m strictly inside all half gates ⟺ alive), which + halves every future alive-certification for doubled words. + +The skeptic should attack, in order: (a) the closed-form derivation +Steps 1–3 (the formal check is only as good as the bridge to geometry; +the 23-member exact specialization and 400-point float check are the +defense — re-derive u independently); (b) the extended case tree, +especially the k ≥ 1 threshold algebra aw − bv = a(b+1)(θ−θ_d) − 360kb +and the claim w ∈ (−90, 0] ⟹ sin w ≤ 0 given the branch's constraints; +(c) the segment certifier's trig layer (mod-360 reduction, dyadic +rounding, Lipschitz widening) and its N1′ derivative formula — +re-differentiate by hand; (d) the converse reduction's τ ≠ 0 bookkeeping; +(e) whether "alive" as certified here (positive corridor width) matches +001's death-measurement criterion on the mirror half for the b ≤ a−2 +members (004 §5 says yes; re-confirm for the five new members). + +## Leads generated + +1. **Parametric sufficiency (finish the death law for all (a,b)).** + Compute the generic gate margins along the universal segment + (90/a − t, 90(a−1)/(a(b+1)) + 2t) in closed form via the prefix maps + R_0 Rot_A(2kα), k ≤ a, and R_0 Rot_A(2aα) Rot_B(−2jβ), j ≤ b (each a + short trig polynomial with k, j linear in the angles), and prove them + ≥ c(a,b) > 0 on [0, 1/4]. Definite outcome: either death(W(a,b)) = + γ_d for ALL a ≥ b ≥ 1, a ≥ 2, or a member family where some fan margin + pinches — both publishable-grade facts for this repo. +2. **The 003+005 theorem is now design-grade for queue item 12.** The + death corner, the exact three binding margins (I1/I2/I4), and the + uniform ρ = 2, t₀ = 1/4 segment geometry quantify the alive wedge at + death for every member; the design track can select (a,b) with γ_d + above any target and KNOWS its last-life window. Cross-check against + the design track's independent findings before merging conclusions. +3. **Birth angles by the same formal method.** 003's Lead 5, now with a + concrete route: the birth edge should be a different pair (or triple — + cf. obstruction 2) of binding gates; compute which margins vanish at + the measured birth apexes (e.g. birth(W(4,3)) ≈ 135.0486), extract + closed forms in the 4-variable ring, and derive the birth law. If it, + too, is algebraic, the family's alive windows become fully exact. +4. **W(1,1) and the a = 1 column.** The parametric theorem needs a ≥ 2; + W(1,b) for b = 1 has H2 ≡ 0 and θ_d(1,1) = 90. Measure W(1,1)'s + corridor on the obtuse side and settle the degenerate column (likely + dead everywhere obtuse — then the theorem's a ≥ 2 is not a gap but a + fact; either way one measurement run decides). +5. **Port the P,Q-formalization to other word families.** Any family + (w_0 B_1^a B_2^b)^2 with B_1, B_2 vertex-pivoting blocks has + u = (isometry) ∘ Rot^a ∘ Rot^b and hence formal-ring identities in the + same four variables. Concretely: take the design track's best + candidate family and attempt the same I1–I4 extraction; success gives + its death law with the same proof skeleton. + +## References + +- `problems/billiards-triangles/attempts/003-death-angle-laws.md` (the + record extended here: theorem, I1–I3, Lemmas C/D, case tree, Leads 1–3) + and `004-skeptic-review-of-003.md` (corrections C1/C2; the composed-map + view of the unfolding used in Step 1; both read in full — mode is + informed). Conventions from `001-word-census-coverage-map.md` and + `002-skeptic-review-of-001.md` (skimmed). +- Tier-0: `harness/billiards-triangles/unfold.py` (rational pi enclosure; + interval sin/cos as selftest cross-reference only). +- New code: `explore/plaw_general.py`, `explore/plaw_suffice.py`. + New data: `data/plaw_formal.json`, `data/plaw_specialize.json`, + `data/plaw_lemmac_grid.json`, `data/plaw_casetree_W*_*.json` (13), + `data/plaw_measure_W{63,61,82,81,71}.json`, + `data/plaw_suffice_all.json`, `data/plaw_exact_alive_W{63,61,82,81,71}.json`. +- Reused (003/002): `explore/deathlaw_symbolic.py`, + `explore/deathlaw_measure.py`, `explore/deathlaw_exact.py`, + `explore/skeptic_family.py`, `explore/skeptic_orbit.py`. +- No external papers consulted. diff --git a/problems/billiards-triangles/attempts/006-design-family-past-135.md b/problems/billiards-triangles/attempts/006-design-family-past-135.md new file mode 100644 index 0000000..85d47b3 --- /dev/null +++ b/problems/billiards-triangles/attempts/006-design-family-past-135.md @@ -0,0 +1,421 @@ +# 006 — Design hunt past 135°: the stall dissolves — birth law, W(5,2) alive at exactly 135°, a new four-block family + +- **Problem:** billiards-triangles, `problems/billiards-triangles/PROBLEM.md` +- **Date:** 2026-07-31 +- **Mode:** informed (read `prior-art.json`; attempts 003 and 004 in full; + 001 in full; skimmed 002's tools; queue item 12 of `STATUS.md` — the + exploratory track of the two-track plan) +- **Type:** computational search (structure-class enumeration + float + screening) + design inversion of 003's symbolic machinery + exact + certificates +- **Tools:** new `explore/design_neighbours.py` (enumeration of the glide + structure class, numeric glide-reduced corridor, adaptive window + measurement), `explore/design_factor.py` (exact elementary-factor + extraction over 003's division-free Laurent ring, for arbitrary doubled + words), `explore/design_certify.py` (exact alive certificate with + certified rational gamma bracket, any word), `explore/design_exact135.py` + (exact certificate ON the gamma = 135 arc via its rational-point + parametrization). Reused: `deathlaw_symbolic.py` (the substrate; selftest + run first), `deathlaw_prove.py` (`member` mode, new members), + `skeptic_orbit.py` / `skeptic_family.py` (002's independent exact/float + corridor and simulator — every float hit and every certificate is + cross-checked against them), `deathlaw_exact.py` (gamma certification + helpers; tier-0 `unfold.py` interval cosine underneath). All stdlib-only, + deterministic (fixed seeds where any randomness exists); total compute + ≈ 50 min. New data: `data/design_*.json` (16 files). +- **Sources:** none external (Niven's theorem cited as classical). + +**Conventions** (001/003): W(a,b) = (0 (12)^a (02)^b)^2, length 4(a+b)+2; +side s opposite vertex s; alpha, beta the angles at A, B; gamma at C; +"alive at a triangle" = the word's unfolding corridor has positive width +there; window = the set of gamma with an alive apex on the constant-gamma +arc. 003 (skeptic-confirmed by 004) proved per-member: +death gamma_d(a,b) = 180 − 90(a+b)/(a(b+1)). + +## Approach + +Queue item 12 asks for a hunt through half-word/gate structures for +families alive past 135°, with the death-law machinery run in reverse. +Before hunting, the item demands the frontier semantics be pinned down, +because the W family itself has death angles far above 135° (certified +alive members to gamma ≈ 159.25° in 001). Reading 001's coverage +staircase precisely, the "census stall at 135°" is three separate facts: + +- **(F1) length stall:** no word of length ≤ 26 was known alive at any + sampled point with gamma ≥ 135° (sample-bounded); +- **(F2) pinch gap:** the measured gap [135.000°, 135.049°] between + death(W(3,3)) = death(W(4,2)) = 135 and measured birth(W(4,3)) = + 135.0486, in which nothing of any length was known alive; +- **(F3) window coverage:** above 135° the known alive set was the union + of measured W-windows [birth, death] — with the birth values known only + from 001's fixed 400-point arc scans. + +So "a family past 135°" must mean: alive in the pinch gap, or alive at +gamma ≥ 135 with length ≤ 26, or covering gamma-arcs the W family does +not. That framing directed the work, and it is where the surprise came +from: **(F2) and most of (F3) turn out to be artifacts of 001's sampler.** + +Structure class searched (and why it is the natural one): 003's glide +reduction applies to w = u^2 with u an odd word. A small structural +observation makes this class self-contained: the linear part of any +orientation-reversing isometry composed of an odd number of reflections +is a unit mu with mu·conj(mu) = 1, so u^2 ALWAYS has trivial rotation +part — every doubled odd word is automatically a translation word (no +integer condition to check; verified numerically in the selftest). The +class searched is: all doubled odd half words |u| ≤ 15 (word length +≤ 30), the "2-alternating" subclass u = y_0 y_1 2 y_2 2 ... y_k 2 +(y_i ∈ {0,1}; W(a,b) is y = 0 1^a 0^b) up to k = 12 (length ≤ 50), and +the three-block family u = 0(12)^a(02)^b(12)^c exhaustively to +a,b,c ≤ 3. Why this rather than the full even-word universe at length +28–34 (queue 14's job): the glide reduction halves the corridor cost, +membership is parametrized (so verdicts become statements about named +families, not word lists), and the symbolic factor machinery applies. + +## What was done + +All commands from the repo root. Selftests first: +`deathlaw_symbolic.py selftest` (PASSED), then + +``` +python3 problems/billiards-triangles/explore/design_neighbours.py selftest +python3 problems/billiards-triangles/explore/design_factor.py selftest +``` + +The first checks: doubled odd words have translation unfoldings; the +numeric glide-reduced width times |tau| equals `skeptic_family`'s +independent full-corridor width (rel. err < 5e-14 on 600 samples, verdict +agreement 600/600); W(3,3) and W(4,2) windows reproduce 001/003 values. +The second re-derives 003's identity I2 for W(3,3) by blind factor +extraction: p(A_1) − m = sin(4b) cos(3a) sin(a+b) × const, exactly (the +factor list is found numerically but each division is EXACT in the ring, +and the reconstruction product is re-verified by exact multiplication). + +### 1. The pinch gap (F2) is not a gap: W(4,3)'s window touches 135 + +`design_neighbours.py window` measures birth/death with a sampler that +adds geometric accumulation at every candidate window edge alpha = 90/j, +beta = 90/j (001/002/003 used uniform arc grids, plus — in 003 — the two +known W-window edges). Result: birth(W(4,3)) measures 135.0000076, and +the residual 7.6e-6 is exactly the sampler's deepest edge offset +(0.5·2^-16 deg): the alive window pinches onto the CORNER +(alpha, beta) = (22.5, 22.5) and 001's 400-point scan could not see it. +Deep corner accumulation finds W(4,3) float-alive at gamma = 135.04, +135.02, ..., down to 135.000001 (width ~ 0.34·(gamma−135), confirmed at +every point by `skeptic_family`'s independent corridor). Exact: + +``` +python3 .../design_certify.py --word 012121212020202012121212020202 \ + --gamma 135.02 --corner-alpha 22.5 --glo 135001/1000 --ghi 135048/1000 \ + --out .../data/design_cert_W43_pinchgap.json +python3 .../design_certify.py --word 012121212020202012121212020202 \ + --gamma 135.0001 --corner-alpha 22.5 --glo 1350000001/10000000 \ + --ghi 135001/1000 --bits 64 --out .../data/design_cert_W43_gapdeep.json +``` + +Both PASS: exact Fraction corridor positive at a rational apex, the +30-bounce periodic orbit re-derived by 002's independent exact simulator, +and gamma certified in (135.001, 135.048) resp. (135.0000001, 135.001) — +strictly inside 001's recorded gap. **The pinch-gap lead (001 lead 2, +STATUS queue 14's premise) is closed: length 30 is alive throughout the +measured gap.** + +### 2. Closed-form birth law (003 Lead 5, executed) + +`design_factor.py bind` on W(4,3) near the birth corner shows the binding +difference there is exactly 003's I2 with a = 4: cos(a alpha) · +sin((b+1) beta) · sin(alpha+beta) — the SAME factor cos(a alpha) that +sets the death edge alpha = 90/a also pinches the birth, now jointly with +I3's cos((b+1) beta) at beta = 90/(b+1). The birth corner is +(90/a, 90/(b+1)), giving + + theta_b(a,b) = 90/a + 90/(b+1), + **gamma_birth(a,b) = 180 − 90 (a+b+1) / (a (b+1))**, + +so the window is [gamma_birth, gamma_d], of gamma-length 90/(a(b+1)). +Float test at 11 members (a,b) = (2,1),(2,2),(3,2),(3,3),(4,2),(4,3), +(4,4),(5,3),(5,4),(5,5),(6,6): alive at gamma_birth + 1e-6 (widths +3e-8..1e-7), dead (sampled) at gamma_birth − 1e-3, every member. +Adaptive bisection on W(5,2) hits the law to 1.5e-11 deg. Consequences, +each checked: + +- **Touching identity:** gamma_birth(a+1,a) = 180a/(a+1) = + gamma_d(a,a) — successive staircase windows TOUCH (with zero width) at + the nice angles rather than leaving gaps; 001's measured birth values + (135.0486, 127.5064, ...) were all sampler-biased upward. Verified in + float at the 120, 135, 144, 150 touch points, and exactly (certified + bracket (144.0000001, 144.001)) for W(5,4) at the 144 touch: + `design_cert_W54_touch144.json`. +- **Window pairing:** a+b and a(b+1) are invariant under + (a,b) → (b+1, a−1), so members pair up with IDENTICAL windows: + W(3,3)/W(4,2) share [127.5, 135], W(4,4)/W(5,3) share [139.5, 144] — + exactly the double-death coincidences 003 noticed. + +**SPECULATION (labelled):** the birth law for general (a,b), and that the +window is exactly the open interval (gamma_birth, gamma_d). What is +exact here: the two certificates above, plus (necessity side) 003's Case +I branch A1 already contains theta < 90/a + 90/(b+1) as its standing +hypothesis-set — but the other branches are not excluded here, so no +birth-necessity theorem is claimed. + +### 3. The screens: what the structure class contains + +``` +python3 .../design_neighbours.py screen --mode all --max-half 13 \ + --arcs 135.0,135.02,136.5,138.0,140.0,140.7,142.5,144.0,145.0,148.0,151.0,155.0,159.0 \ + --out .../data/design_screen_all13.json +python3 .../design_neighbours.py screen --mode all --max-half 15 \ + --arcs 135.0,135.02,136.5,138.0,140.0,140.7,142.5,144.0 \ + --out .../data/design_screen_all15.json +python3 .../design_neighbours.py screen --mode s1 --klo 8 --khi 12 \ + --arcs 135.0,141.0,144.0,145.5,147.0,148.5,150.0,151.5,154.0,156.0,158.0 \ + --out .../data/design_screen_s1_k8_12.json +python3 .../design_neighbours.py triple --amax 3 --out .../data/design_triple_table.json +``` + +Every arc-hit is cross-checked against `skeptic_family`'s independent +corridor (all agree). Results: + +- **|u| ≤ 13 (length ≤ 26): 245 canonical words, ZERO alive at all 13 + arcs ≥ 135.** The length-26 stall (F1) survives this class, now with + edge-accumulating sampling (sample-bounded EVIDENCE, design as stated). +- **|u| = 15 (length 30): 811 canonical words, exactly TWO alive on + 135–144:** W(4,3) — and **W(5,2)**, a member NOBODY had measured + (001 scanned b ≥ a−1; 003 added b = a−2; W(5,2) is b = a−3), alive AT + gamma = 135.0 with large width 0.19. Its window measures + [132.0000000000, 137.99999979] — the law's prediction [132, 138] + straddling 135, dissolving the "exceptional touch arc" at 135. +- **2-alternating class, k = 8..12 (length 34–50): 2008 canonical words, + 16 alive**, of which 14 are W(a,b) members (including a > 2b+3 members + like W(8,2), W(9,2) — outside 003's theorem scope, alive as predicted + by the laws) and **TWO are genuinely new four-block structures** (not + W(a,b) under rotation/reversal/0↔1-swap, checked against all (a,b) + with matching length): + N1: u = 0(12)^3(02)^3(12)^2(02)^3, length 46, window + [134.25, 136.8265] — alive AT 135.0, width 0.082; + N2: u = 0(12)^3(02)^3(12)^2(02)^4, length 50, window + [140.264, 141.396]. +- **Three-block family 0(12)^a(02)^b(12)^c, all a,b,c ≤ 3: 25/27 members + dead everywhere obtuse (sampled)**; the two alive are (3,1,1) with + window [90, 105.0000] and (3,2,1) with [109.25, 111.42] — the extra + (12)^c block collapses the death angle far below the c = 0 family. + +### 4. Exact certificates AT gamma = 135 exactly + +The apexes with gamma = 135 form the arc of (2x−1)^2 + (2y+1)^2 = 2, +y > 0 — a circle with rational point (1,0), hence DENSE rational points, +parametrized by rational chord slope. At such an apex, gamma = 135 +exactly is a pure Fraction identity (2·(CA·CB)^2 = |CA|^2|CB|^2 with +CA·CB < 0) — no interval arithmetic at all. + +``` +python3 .../design_exact135.py --word 012121212120202012121212120202 \ + --out .../data/design_exact135_W52.json +python3 .../design_exact135.py \ + --word 0121212020202121202020201212120202021212020202 \ + --out .../data/design_exact135_N1.json +``` + +Both PASS: **W(5,2) (length 30, exact corridor width ≈ 0.76) and N1 +(length 46, width ≈ 0.60) have exact-arithmetic stable periodic orbits, +independently re-simulated (30 resp. 46 bounces), at rational-apex +triangles whose obtuse angle is EXACTLY 135°.** At the W(5,2) apex, +tan(alpha) = 300824218724645284407543/990593882979969795299081 is +rational and not in {0, 1}, so by Niven's theorem alpha (hence beta = +45° − alpha) is an irrational multiple of pi: this is an irrational- +angled obtuse triangle at exactly 135°. Note the trick needs cos^2 +(gamma) rational, so it works at touch angles 120, 135, 150 but NOT at +144 (cos^2 144° = (3+√5)/8 — no rational apexes on that arc; hence the +bracket-style certificate for the 144 touch in §2). + +Necessity side for the new members: 003's prover run on (5,2) and (6,2) + +``` +python3 .../deathlaw_prove.py member --a 5 --b 2 --out .../data/design_prove_W52.json +python3 .../deathlaw_prove.py member --a 6 --b 2 --out .../data/design_prove_W62.json +``` + +ALL OBLIGATIONS CERTIFIED for both — so 003's theorem now covers them: no +positive-width corridor at gamma ≥ 138 (W(5,2)) resp. 140 (W(6,2)). +Death brackets: death(W(5,2)) ∈ [135, 138] with exact alive at 135 and +float death 138 − 2e-6; W(6,2) window [135.0000076, 140.0] (float) +touches 135 from above like W(4,3). + +### 5. Binding factorization of the new family, and why its angles are dirty + +`design_factor.py bind` on N1 near its death (gamma = 136.8, argmax +alpha = 26.47 — an INTERIOR point, not a 90/j corner): + +The binding differences do NOT factor into elementary sin/cos terms: the +leading one is sin(beta)·sin(alpha)·[an 11-term cosine sum] +(`design_bind_N1_death.json`). So the four-block family's death angle is +a root of a genuine multi-term trigonometric polynomial — consistent with +the measured dirty values 136.8265, 141.3960 — while W's death/birth +angles are clean rational multiples of 90° precisely because its two-fan +gate structure makes every binding difference a THREE-FACTOR product of +elementary terms. This is the design-space answer to "invert the +machinery": the inversion is clean exactly on the class where the +factorization stays elementary, and that class, in everything searched +here, is the W family itself. + +### 6. Coverage: the W family covers every sampled obtuse angle + +With windows [gamma_birth, gamma_d] in closed form, member selection +becomes arithmetic: gamma is strictly inside W(a,b)'s window iff +a+b < t·a(b+1) < a+b+1 for t = (180−gamma)/90. Check: for 157 arcs +(gamma = 90.5 to 165 step 0.5, plus 112.5, 120, 130, 135, 144, 150, +154.3), pick the shortest such member and test aliveness in float +(`design_coverage_check.json`): **all 157 alive, zero failures**, longest +member needed: length 94. Every "accumulation angle" of 001 — including +112.5, 120, 135, 144, 150 — is strictly inside the window of some W(a,b) +with b ≤ a−2 (mirror-half members 001 never scanned). **SPECULATION +(labelled):** the union of open windows covers all of (90, 180); by the +window arithmetic this reduces to an elementary Diophantine statement +(lead 2 below). + +## Outcome + +- **VERIFIED (exact arithmetic, independently re-simulated orbits):** + five certificates, each an exact-Fraction positive corridor at an + explicit rational apex plus an independent exact orbit simulation: + (i) W(4,3) at certified gamma ∈ (135.001, 135.048); + (ii) W(4,3) at certified gamma ∈ (135.0000001, 135.001) — together + closing 001's recorded pinch gap; + (iii) W(5,4) at certified gamma ∈ (144.0000001, 144.001) — the 144 + touch; + (iv) W(5,2), length 30, at gamma = EXACTLY 135° (exact on-arc + identity; irrational-angled by Niven); + (v) N1 = (0(12)^3(02)^3(12)^2(02)^3)^2, length 46, a word outside the + W family, at gamma = EXACTLY 135°. +- **VERIFIED (003's machine-certified obligations, two new members):** + I1–I3 + glide facts + Lemma C for (5,2) and (6,2), hence no positive + width at gamma ≥ 138 resp. 140. +- **EVIDENCE (float, sample-bounded by the stated samplers/arc lists):** + the screens (245 words ≤ 26: nothing alive at 13 arcs ≥ 135; 811 + length-30 words: only W(4,3), W(5,2) on 135–144; 2008 words of the + 2-alternating class 34–50: only W members plus N1, N2); the three-block + table (25/27 dead everywhere); the birth law at 11 members; the window + measurements of W(5,2), W(6,2), N1, N2; the 157-arc coverage check. +- **SPECULATION (labelled inline):** the general birth law + gamma_birth(a,b) = 180 − 90(a+b+1)/(a(b+1)) and exact-window claim; + full coverage of (90, 180) by the union of windows; non-cyclotomicity + of the four-block death angles. +- **NOT claimed:** any birth-side necessity theorem (only 003's death + side is proven, now for 17 members); that no length ≤ 26 word is alive + past 135 (float screen, finite arcs, doubled words only — non-doubled + even words of length ≤ 26 were screened by 001, not here); area/box + coverage (all claims are about arcs and points; 001's area-collapse + obstruction stands); anything about unstable orbits; aliveness of + every W(a,b) in its formula window (tested only at the 157 chosen + arcs + measured members). + +## Why it failed / what survived + +The queue's kill condition ("every family in the class has gamma_d ≤ +135°, evidence that 135° is a real barrier") is REFUTED rather than +confirmed — but not by a new family: by the discovery that the frontier +semantics themselves were wrong. Specifically: + +1. **What was wrong:** 001's pinch gap and birth values were artifacts of + uniform arc sampling; windows pinch onto corners (90/a, 90/(b+1)) and + need geometric edge accumulation to see. The staircase windows touch; + b ≤ a−2 members interleave and cover the touch points; nothing about + 135° obstructs the W family, let alone the class. The 135° stall + reduces to a pure LENGTH statement: nothing ≤ 26 found alive past 135 + (that part survives everything thrown at it here, and is sharp: + length 30 is alive at 135 with corridor width 0.76 — not marginal). +2. **The obstruction that survived, precisely:** in the searched + neighbourhood of W's structure (three-block: 27 members; four-block + and beyond within the 2-alternating class to length 50: 2008 words; + all doubled odd words to length 30: 1056 words), aliveness in the + obtuse region is RARE — everything alive past 135 is either W(a,b) or + the two four-block words N1, N2, whose binding differences do not + factor into elementary terms and whose windows are short with dirty + algebraic endpoints. Clean angle laws — and hence design-by-formula — + appear confined to the two-fan W geometry. The design tool works + (it re-derives I2 blind, and diagnoses N1's non-factorability), but + the design SPACE is nearly empty near W. +3. **What survived for reuse:** the closed-form window law (birth AND + death) with the (a,b) ↔ (b+1, a−1) pairing; the coverage-member + selector (arithmetic, no search); the edge-accumulating sampler + (any future window measurement must use it — uniform arc grids + under-estimate every pinching window); the rational-points-on-arc + certification trick for any gamma with rational cos^2; the arbitrary- + word certifier and the exact factor-extraction tool; the observation + that doubled odd words are automatically translation words (kills the + integer translation check in this class, and makes "length ≡ 2 mod 4" + the natural census axis for stable structures). + +A skeptic should attack, in order: (a) the five certificates (re-run +them; check the on-arc identity algebra and that `verify_orbit`'s +positivity + simulation really are independent of the float layer); +(b) the birth-law fit — is 135.0000076 really the sampler floor and not +a genuine offset (re-measure W(4,3) birth with a deeper sampler); +(c) the screens' negative verdicts (rerun with different arc lists and +denser/other samplers — a thin window between my arcs would be missed); +(d) the canonical-form check that N1, N2 are not W members under some +missed equivalence; (e) the claim that 001's samplers could not have +seen the corner windows (read `census.py cmd_family`'s grid). + +## Leads generated + +1. **Prove the birth law.** Necessity candidate: alive + Case I branch + A1 forces theta < 90/a + 90/(b+1) already in 003's case tree; what is + missing is excluding the other branches below theta_b, plus a + sufficiency corner expansion at (90/a, 90/(b+1)) exactly like 003's + Lead 2 at the death corner. Same machinery, finite per member. + Falsifiable: certify alive at gamma_birth + 1e-8 and dead-scan below + gamma_birth − epsilon for several members. +2. **Coverage theorem.** Prove: for every t ∈ (0,1) there exist + integers a ≥ 1, b ≥ 1 with a+b < t·a(b+1) < a+b+1 (then, assuming the + window law, every obtuse gamma has an alive W member; with the b=1 + and b=2 rows alone the union already covers t ∈ (1/3, 1) up to + boundary points). Elementary number theory; the needed member length + at gamma is governed by the continued-fraction structure of t — an + explicit, provable refinement of 001's hyperbolic length law. +3. **Certify the irrational-cos^2 touch arcs.** At gamma = 144 there + are no rational apexes; certify a tier-0 harness box (TRUE verdict) + whose interior provably straddles the arc — needs box half-width + ~2^-40 at length 38 and corner gamma brackets tighter than the box's + gamma variation. Concrete and finite. +4. **The length frontier at 135.** Is any word of length ≤ 26 alive at + some gamma ≥ 135? The doubled-odd class is now screened (negative); + the remaining universe is non-doubled even words at 24, 26 — queue + 14's letter-statistics scan, now pointed at a sharp question with a + known answer at length 30 (both W(4,3) and W(5,2)). +5. **Four-block systematics.** N1's corridor at 135 is ~8x wider than + W(4,3)'s anywhere in the old gap; short dirty-angle windows with FAT + corridors are exactly what area-coverage (001's real obstruction) + needs. Enumerate 0(12)^a(02)^b(12)^c(02)^d with a+b+c+d ≤ 14, + measure window x corridor-area, and test whether four-block tiles + fill 001's mid-region holes at 92–135° that W tiles leave. +6. **Update the coverage staircase.** With the window law and W(5,2)- + type members, recompute 001's minimal-length-per-arc staircase from + the formula (min 4(a+b)+2 subject to strict window containment) and + compare against the census data — any arc where the census beats the + formula signals a non-W short word worth naming. + +## References + +- `problems/billiards-triangles/attempts/003-death-angle-laws.md` (the + machinery inverted here; its Leads 4, 5 are executed by this attempt) + and `004-skeptic-review-of-003.md` (scope of the glide reduction; the + C2 zero-width caveat is why "touch" points need the strict-interior + members). +- `problems/billiards-triangles/attempts/001-word-census-coverage-map.md` + (frontier semantics; the pinch-gap lead closed here) and 002's tools + (`skeptic_orbit.py`, `skeptic_family.py` — the independent layer under + every certificate). +- Tier-0: `harness/billiards-triangles/unfold.py` (interval cosine used + in the gamma-bracket certifications, via `deathlaw_exact.py`). +- New code: `explore/design_{neighbours,factor,certify,exact135}.py`. + New data: `data/design_screen_{all13,all15,s1_k8_12}.json`, + `data/design_triple_table.json`, `data/design_windows_new.json`, + `data/design_window_W43.json`, + `data/design_cert_W43_{pinchgap,gapdeep}.json`, + `data/design_cert_W54_touch144.json`, + `data/design_exact135_{W52,N1}.json`, + `data/design_bind_{W43_birth,N1_death}.json`, + `data/design_prove_W{52,62}.json`, `data/design_coverage_check.json`. +- Niven's theorem (rational multiples of pi with rational tangent are + exactly those with tan ∈ {0, ±1}) — classical; used only for the + "irrational-angled" remark, not load-bearing. diff --git a/problems/billiards-triangles/attempts/007-skeptic-review-of-006.md b/problems/billiards-triangles/attempts/007-skeptic-review-of-006.md new file mode 100644 index 0000000..22ec254 --- /dev/null +++ b/problems/billiards-triangles/attempts/007-skeptic-review-of-006.md @@ -0,0 +1,412 @@ +# 007 — Skeptic review of 006 (design hunt past 135°): adversarial verification + +- **Problem:** billiards-triangles, `problems/billiards-triangles/PROBLEM.md` +- **Date:** 2026-07-31 +- **Mode:** informed (read `prior-art.json`, attempts 001, 003, 004 in full, + 002 skimmed via its tools, the four `design_*.py` tools and all + `design_*.json` data under review, `census.py cmd_family` (001's sampler), + `deathlaw_skeptic.py` (004's stack, reused here), STATUS queue items 12/14, + tier-0 `unfold.py` for cross-reference only) +- **Type:** skeptic review of `006-design-family-past-135.md` (default + stance: REFUTE). Every load-bearing layer re-established by code written + for this review or by 004's previously-adversarial stack; nothing from + `design_*.py` is imported in any verdict path. +- **Outcome in one line:** 006 survives where it is exact — all five + certificates re-established from scratch (third corridor implementation, + own simulator, widths match digit-for-digit), the pinch-gap mechanism of + 001's error is reproduced and pinned, the birth law holds on three + genuinely out-of-sample members to ~3e-11, and the (5,2)/(6,2) extension + re-proves — but four corrections: a universe-count mislabel (811 is + |u| ≤ 15, the true length-30 count is 566; "1056" double-counts), an + "alive throughout the gap" wording overreach, the refutation semantics + against 001 (001's gap claim was knowledge-scoped and stays true; what + falls is 001's birth *measurements*), and the discovery that 006's + "window pairing" (a,b) ↔ (b+1, a−1) is a word identity — the paired + members are the SAME canonical word — so that consistency check carries + zero evidential weight for the birth law (and quietly halves the member + count of several lists). +- **Tools:** new `explore/dsk_design_skeptic.py` (stdlib-only, + deterministic, seed 20260731; ~25 min total compute). Independence + choices: exact unfolding by composed affine maps with 2x2 rational + matrices — a third implementation path (006's exact layer is 002's + `skeptic_orbit.py`, reflected-vertex chains; 004's is complex composed + maps) with the same tau-normal projection, so exact widths must and do + agree digit-for-digit across all three; my own exact rational billiard + simulator (own ray/segment solve); certified gamma brackets and ring + identities via `deathlaw_skeptic.py` (004's adversarial stack: own + Machin pi, own Taylor enclosures, own Laurent ring — never 006's code, + never tier-0 intervals); own word enumeration and canonicalization; own + seeded samplers; a replica of `census.py cmd_family`'s exact grid design + for the 001-mechanism question. +- **Sources:** repo only; Niven's theorem cited as classical (as in 006). + +My data lives in `data/dsk_*.json` (distinct from 004's `dlsk_*` and the +parallel agent's `plaw_*`; nothing of 006 was modified). + +Reproduce everything (repo root): + +``` +python3 problems/billiards-triangles/explore/dsk_design_skeptic.py selftest +python3 .../dsk_design_skeptic.py certs --out .../data/dsk_certs.json +python3 .../dsk_design_skeptic.py birthdepth --out .../data/dsk_birthdepth_W43.json +python3 .../dsk_design_skeptic.py censusgap --out .../data/dsk_censusgap.json +python3 .../dsk_design_skeptic.py birthlaw --out .../data/dsk_birthlaw.json +python3 .../dsk_design_skeptic.py counts --out .../data/dsk_counts.json +python3 .../dsk_design_skeptic.py deadsample --n 60 --out .../data/dsk_deadsample.json +python3 .../dsk_design_skeptic.py coverage --out .../data/dsk_coverage.json +python3 .../dsk_design_skeptic.py newmembers --out .../data/dsk_newmembers.json +python3 .../deathlaw_skeptic.py lemmac --a 5 --b 2 --out .../data/dsk_lemmac_W52.json +python3 .../deathlaw_skeptic.py lemmac --a 6 --b 2 --out .../data/dsk_lemmac_W62.json +# W(6,3)/W(8,2) out-of-sample births: inline driver recorded in data/dsk_birthlaw2.json +``` + +Selftest: Fagnano closes under my simulator, obtuse Fagnano rejected, +3-way exact-corridor agreement (mine vs `skeptic_orbit` vs +`deathlaw_skeptic`) 100/100 at seeded rational apexes, W(3,3) alive at +134.99 / dead at 135.01 under my float corridor. + +## Claims attacked + +### 1. The five exact certificates. **CONFIRMED (re-established from scratch)** + +`certs` (`dsk_certs.json`) re-verifies each of 006's five certificate +files with: (a) my matrix-form exact corridor positive at the recorded +rational apex — and the exact rational width equal DIGIT-FOR-DIGIT to the +recorded string in all five (same tau-normal normalization, third +implementation); (b) my own exact simulation closing the periodic orbit +(30, 30, 38, 30, 46 bounces; own intersection solve, strict-interior +bounce check, exact closure of position and direction, struck-sequence +check); (c) the word string equal to the claimed family member (W(4,3), +W(4,3), W(5,4), W(5,2), and N1's four-block structure — rebuilt from the +(a,b,c,d) exponents, not copied); (d) for the two bracket pairs, my +gamma brackets re-certified with 004's interval stack (exact rational +cos^2 against Machin-pi Taylor enclosures): gamma in (135.001, 135.048), +(135.0000001, 135.001), (144.0000001, 144.001) all hold — the first two +strictly inside 001's recorded [135.000, 135.0486] gap. + +The exactly-135 layer was re-derived by hand before running anything: + +- Parametrization: with s = 2(1−t)/(1+t²), x = 1−s/2, y = ts/2 one gets + (2x−1)² + (2y+1)² = (1−s)² + (ts+1)² = 2 + s(s(1+t²) − 2 + 2t) = 2, + identically. ✓ +- On the circle, 4(x²−x+y²) + 4y = 0, i.e. dot := CA·CB = x²−x+y² = −y, + automatically negative for y > 0; then |CA|² = x−y, |CB|² = 1−x−y and + (x−y)(1−x−y) = (x−y) − (x²−y²) = 2y² = 2·dot², so + cos γ = −y/√(2y²) = −1/√2 exactly, i.e. γ = 135° (γ obtuse from + dot < 0; the γ = 45° branch of cos² = 1/2 is excluded). So 006's pure + Fraction criterion "2(CA·CB)² = |CA|²|CB|² and CA·CB < 0" is exactly + γ = 135°, no interval arithmetic needed. Both recorded apexes satisfy + it exactly (checked in Fractions, plus the circle identity directly). +- Niven: at the W(5,2) apex tan α = y/x = 512242667139/1686780588413 + (006 quotes the unreduced ratio of numerators — same rational, a nit), + rational and not in {0, ±1}, so α (hence β = 45° − α, hence both + non-gamma angles) is an irrational multiple of π. The + irrational-angled claim stands. + +Verdict: **all five certificates are real.** An irrational-angled +triangle with obtuse angle exactly 135° carrying a length-30 stable +periodic orbit is now double-verified. + +### 2. The pinch-gap finding vs 001. **CONFIRMED — mechanism identified; semantics corrected (C3)** + +Two independent attacks, both in `dsk_birthdepth_W43.json` / +`dsk_censusgap.json`: + +- *Is 135.0000076 a sampler floor or a genuine offset?* My own birth + bisection of W(4,3) with corner accumulation at (22.5, 22.5), at four + sampler depths: birth − 135 = 7.63e-6 (floor 7.63e-6), 2.98e-8 (floor + 2.98e-8), 1.18e-10 (floor 1.16e-10), 3.6e-12 (floor 4.5e-13, float + noise regime). The measured "birth" tracks the probe floor at every + depth — it is the sampler, not the window. Combined with the exact + gapdeep certificate (an alive point with certified γ < 135.001 and + > 135.0000001), birth(W(4,3)) ≤ 135.0000001 exactly. +- *Could 001's design have seen it?* I read `census.py cmd_family`: its + alive test is 400 uniform x-samples on [0.02, 0.5] (largest sample + x = 0.4994, terminal clearance 6.0e-4), bisected in γ. My replica of + exactly that design, with MY corridor, reproduces 001's number: birth + between 135.04857 and 135.04858 (001 recorded 135.0486). The measured + alive x-window of W(4,3) is a sliver [0.5 − w(γ), 0.5) hanging on the + corner x = 1/2 (α = β = 22.5), with w = 6.2e-5 at γ = 135.005 growing + to 6.0e-4 at γ = 135.0486 — exactly the census grid's terminal + clearance, where the window first swallows a grid point + (`contains_grid_point` flips true precisely there). **Mechanism + pinned: 001's uniform grid stops one half-cell short of the corner + apex the window pinches onto; its "birth" is the γ at which window + width equals grid clearance.** + +Semantics (correction C3): 001 scoped the gap claim as knowledge — +"measured gap ... in which nothing ≤ 30 is *known* alive", births +tabulated as measurements. That claim stays literally true as written; +what 006's certificates falsify is 001's birth *values* (and their +implied "±~0.005° sampling noise", which for births is off by 10x — +the true error at W(4,3) is 0.0486°; 001's own "deaths only ever +under-estimated" hedge silently does not transfer to births, where +finite sampling errs the other way). So "the pinch-gap **refutation** +of 001" (the framing this review was handed) and the record's +"artifact" language are right about the measurements and wrong to the +extent they suggest 001 claimed the gap was empty: 001 claimed only +that nothing was known alive there, and 006 is the attempt that made +something known. "Closed/corrected", not "refuted". 006's own record +mostly says exactly this ("the pinch-gap lead ... is closed"); the +index entry's "001 pinch gap was a sampler artifact" is fair if read +about the measured gap object. + +### 3. The birth law γ_birth = 180 − 90(a+b+1)/(a(b+1)). **CONFIRMED out-of-sample; one evidential deflation (C4)** + +First the deflation. 006's "window pairing" consequence — (a,b) and +(b+1, a−1) have identical windows because a+b and a(b+1) are invariant — +is a **word identity**: canonical(W(a,b)) = canonical(W(b+1, a−1)) under +cyclic rotation, reversal and the 0↔1 swap. Verified exactly for +(3,3)~(4,2), (5,2)~(3,4), (4,4)~(5,3), (5,5)~(6,4), (7,3)~(4,6), +(2,2)~(3,1); (2,1), (4,3), (5,4) are self-paired. The swap is the +x → 1−x mirror, which fixes γ, so the two labels denote one canonical +word and their γ-windows coincide *for any window quantity whatsoever*. +Consequences: (i) the pairing "check" in 006 §2 is vacuous as evidence +for the law (it is a necessary consistency property of ANY correct +formula, and would equally "check" for many wrong ones); (ii) 003's +"two distinct words dying on the same arc" (W(3,3)/W(4,2)) are not +distinct canonical words; (iii) 006's 11-member float test spans only 9 +distinct canonical words, and my first out-of-sample pick W(6,4) +collapsed into fit-member W(5,5) — replaced below. + +The law itself, attacked on THREE genuinely out-of-sample canonical +words (none in the fit classes), own sampler, both arc halves, corner +accumulation at (90/a, 90/(b+1)) (`dsk_birthlaw.json`, +`dsk_birthlaw2.json`): + +| word | len | predicted birth | measured − predicted | dead at birth − 1e-3 | +|---|---|---|---|---| +| W(7,3) | 42 | 1012.5/7 = 144.642857... | +3.0e-11 | yes (best −4.3e-5) | +| W(6,3) | 38 | 142.5 | +3.0e-11 | yes (best −6.1e-12) | +| W(8,2) | 42 | 138.75 | +3.0e-11 | yes (best −2.2e-11) | + +Every +3.0e-11 equals my depth-34 sampler floor (0.5·2⁻³⁴ = 2.9e-11) — +the same floor-tracking signature as W(4,3) at 135. Note W(8,2) has +a > 2b+3, *outside* 003's theorem scope: the birth law holds there +anyway, as 006's s1-screen hits suggested. The W(5,2) window edges also +re-measure to the law: alive at 132.001, dead-sampled at 131.999, alive +at 135 with width 0.69. The touching identity at 144 is carried by the +re-verified W(5,4) certificate. The law remains SPECULATION as a general +statement — 006 labels it so, correctly — but it now survives +out-of-sample tests it was not fitted to, including outside the death +theorem's scope. + +### 4. Coverage (157 arcs). **CONFIRMED at my own 25 arcs** + +`dsk_coverage.json`: my own arc list (25 rational γ including the +historical trouble spots 112.5, 120, 135, 135.02, 135.0486, 144, 150, +1080/7 = 154.2857..., and 157.1, 160.9, 162.5, 164.8), my own +strict-window member selector (Fraction arithmetic on +a+b < t·a(b+1) < a+b+1), my own float corridor and sampler: **25/25 +alive**, longest member length 90. My selector independently lands on +the mirror-labelled members (e.g. (3,4) ≡ W(5,2) at γ = 135), matching +006's b ≤ a−2 story. The coverage conjecture for all of (90, 180) +remains SPECULATION, labelled as such in 006. + +### 5. The negative screens. **CONFIRMED with counting corrections (C1)** + +Independent recount (`dsk_counts.json`), own generator and +canonicalization: |u| ≤ 13 by half-length 1/2/6/16/52/168, total **245** +— matches; 2-alternating k = 8..12: **2008** — matches; but |u| = 15 is +**566**, not 811: 811 is the count for |u| ≤ 15 (245 + 566; 006's +screen file `design_screen_all15.json` indeed enumerates `--max-half 15` += everything from 3 to 15, so the *screen's coverage* is fine and its +own n_words = 811 is correct for what it ran). Corrections: the record's +"**|u| = 15 (length 30): 811 canonical words**" mislabels the universe; +"all doubled odd words to length 30: 1056 words" double-counts (245 + +811, but the 245 are inside the 811) — the true number is 811; the index +one_line's "3000+ canonical doubled words to length 50" is likewise +inflated by the same double count (811 + 2008 + 27 three-block = 2846). + +Screen verdicts spot-attacked: (a) the s1 hit list reclassified with my +canonicalizer — exactly 14 W members (including W(8,2), W(9,2) with +a > 2b+3, as claimed) plus N1, N2; (b) N1 and N2 match no W(a,b) +canonical form at their lengths (all a+b = 11 resp. 12 checked — and +this check now correctly includes the 0↔1-swap equivalence, the missed- +equivalence worry 006 itself raised); (c) `deadsample`: 60 seeded-random +words of the 245 at 006's 13 arcs plus 6 arcs of my own choosing, own +sampler with random probes added: **zero alive, best width 5.6e-17** +(float zero); the 19 recorded near-misses attacked harder (denser +sampler, ±0.35° arc shifts): the only positive is the W(3,3)≡W(4,2) +canonical word at γ = 134.65 — *below* 135, inside its known window, +exactly as it should be. The length-≤26 stall at γ ≥ 135 survives this +attack. Not re-verified: the full 566-word length-30 sweep, the full +2008-word s1 sweep (only its positives were reclassified), and the +three-block table — all remain 006's sample-bounded EVIDENCE, correctly +labelled. + +### 6. The (5,2)/(6,2) extension of 003's theorem. **CONFIRMED (own ring, own intervals)** + +`dsk_newmembers.json`, `dsk_lemmac_W{52,62}.json`: scope preconditions +hold (a ≥ b ≥ 1, a ≥ 2, a ≤ 2b+3: 5 ≤ 7, 6 ≤ 7). In 004's independent +Laurent ring (my driver, not 006's `deathlaw_prove` path): all 15 +residuals — I1, I2, I3, D1, D2, mu monomial, tau parallel, gate +identifications, glide action on all four base points — are exactly zero +for both members, plus 10 seeded off-torus rational spot points each in +the pair algebra. Case-tree adversarial sign search at θ ≤ θ_d (273k / +277k seeded points, corner- and boundary-accumulating): zero Case I/II +sign-pattern hits. Lemma C re-certified for both members with 004's +interval stack (H2' upper bounds on the endpoint zone: −0.037, −0.032 — +comfortably negative; endpoint identity 90 − v0 = b·v0 exact). With +004's hand-verified general-(a,b) case tree, the necessity theorem +genuinely extends to (5,2) and (6,2): no positive-width corridor at +γ ≥ 138 resp. 140. The death bracket death(W(5,2)) ∈ [135, 138] is +then exact on both sides (alive-at-135 certificate + theorem). + +### 7. Scope honesty. **CONFIRMED with corrections C1-C4** + +SPECULATION labels are present and correctly placed (general birth law +and exact-window claim in §2; coverage conjecture in §6; +non-cyclotomicity in Outcome); the negative screens are labelled +sample-bounded EVIDENCE with their designs stated; the NOT-claimed list +is accurate and pre-empts the right overclaims (no birth-necessity +theorem, no area coverage, no unstable orbits, non-doubled words ≤ 26 +not rescreened). Two wording-level overreaches found: **(C2)** §1's +"length 30 is alive **throughout** the measured gap" — the certificates +prove alive points AT γ = 135 exactly and at one point in each of +(135.0000001, 135.001) and (135.001, 135.048); "throughout" (every γ in +the gap) is float evidence (W(5,2)'s window [132, 138] straddling the +gap) plus the SPECULATION window law, and should be so worded — the +Outcome bullet's "together closing 001's recorded pinch gap" inherits +the same point-vs-continuum slack. And **(C1/C4)** as above. The index +entry's status VERIFIED describes the certificate layer accurately; +its range field states the float/certificate split honestly. + +## Refutations found + +No load-bearing claim is refuted. Four corrections: + +- **C1 (counting).** "|u| = 15 (length 30): 811 canonical words" — the + length-30 universe is 566; 811 is |u| ≤ 15. "All doubled odd words to + length 30: 1056 words" double-counts; the true count is 811. The + index one_line's "3000+" is 2846 counted correctly. (The screens + themselves covered what they claim to cover; only the counts are + wrong.) +- **C2 (wording overreach).** "Length 30 is alive throughout the + measured gap": certificates establish alive points at γ = 135 exactly + and at one γ in each recorded bracket — pointwise, not throughout; the + continuum statement is float + SPECULATION law. +- **C3 (refutation semantics).** What is overturned in 001 is its birth + *measurements* (off by 10x its stated ±0.005° noise; the "deaths only + under-estimated" hedge does not apply to births) and the emptiness + *reading* of the gap; 001's actual recorded claim ("nothing ≤ 30 + known alive") was knowledge-scoped and remains true as written. The + relationship is correction-of-measurement + lead-closure, not + refutation of a recorded claim. +- **C4 (vacuous consistency check + member double-listing).** The + window pairing (a,b) ↔ (b+1, a−1) is a canonical-word identity (0↔1 + swap + reversal/rotation; verified exactly on six pairs), so: it is + not evidence for the birth law; W(3,3)/W(4,2) and W(4,4)/W(5,3) are + single canonical words, making 006's 11-member fit really 9 canonical + members and 003's "two distinct words dying on the same arc" a + relabeling; and "W(5,2), a member NOBODY had measured" is, as a + canonical word, the mirror W(3,4) — new to the measured record all + the same. + +Nits, no action needed: the W(5,2) tan α is quoted in unreduced form; +§3's width 0.19 (glide-reduced normalization) vs §4's 0.76 (full +corridor) for W(5,2) at 135 are different normalizations of the same +aliveness, unflagged; `design_certify.best_apex` imports +`skeptic_family.width` lazily for non-doubled words but is only ever +run on doubled ones here. + +## Claims that survive + +| # | 006 claim | Verdict | +|---|-----------|---------| +| 1 | Five exact certificates (W(4,3) x2 in the gap, W(5,4) at 144-touch, W(5,2) and N1 at exactly 135°) | **CONFIRMED** — third-implementation exact corridor, digit-for-digit widths; own exact simulation; own gamma brackets; on-arc identity hand-derived; Niven applies | +| 2 | 001's pinch gap [135.000, 135.0486] not empty; birth values sampler artifacts | **CONFIRMED** (as measurement-correction, C3) — floor-tracking at 4 depths; 001's grid design reproduced and its 135.0486 regenerated; corner-sliver mechanism pinned | +| 3 | Birth law γ_b = 180 − 90(a+b+1)/(a(b+1)) (SPECULATION, float) | **SURVIVES stronger tests than 006 ran** — 3 out-of-sample canonical members incl. a > 2b+3, all +3e-11 = floor; pairing check deflated (C4) | +| 4 | W-family member alive at every sampled obtuse arc | **CONFIRMED** at 25 arcs of my own choosing incl. all trouble spots | +| 5 | Screens: 245 words ≤ 26 dead ≥ 135; only W(4,3)/W(5,2) alive at len 30 on 135-144; s1 hits = W + N1/N2; N1/N2 non-W | **CONFIRMED** modulo counts (C1) — 245 and 2008 recounted; hits reclassified; N1/N2 non-W incl. swap; 60-word random re-test + near-miss attack, zero kills | +| 6 | 003's obligations machine-certified for (5,2), (6,2) | **CONFIRMED** — own ring 15/15 residuals, spots, 550k-point case-tree search, own Lemma C intervals | +| 7 | Scope honesty | **CONFIRMED with C1-C4** | + +**Kill attempts that failed,** for the record: (i) hunting a +normalization or transcription error in the certificates by demanding +digit-for-digit width equality from a third implementation — all five +match; (ii) trying to expose 135.0000076 as a genuine birth offset by +quadrupling sampler depth — it tracked my floor at every depth; (iii) +attempting to falsify the birth law on members outside its fit set, +including outside 003's theorem scope — held to the sampler floor all +three times; (iv) sign-pattern search for a case-tree breakdown on the +two new members (550k adversarial points) — nothing; (v) random- +subsample and near-miss re-testing of the dead screens with a different +seeded sampler and extra arcs — the only positive was a known-alive W +word below 135; (vi) coverage at adversarial arcs (touch angles, 001's +accumulation points, γ > 160) — 25/25 alive. + +## Residual risk + +- **Shared sampler design on the negative side.** My float samplers, + like 006's, accumulate at 90/j edges (plus uniform and random + probes). A window pinching onto an interior non-90/j point thinner + than the uniform/random resolution could hide from both. N1/N2 (whose + argmaxes ARE interior points, found by the same class of sampler) + bound this risk but do not eliminate it; all negative verdicts remain + sample-bounded exactly as labelled. +- **Not re-swept:** the 566-word length-30 universe and 2008-word s1 + universe full negatives (positives were reclassified, negatives only + subsampled at |u| ≤ 13), the three-block 25/27-dead table, the + 157-arc coverage run (25 arcs re-done), and `design_factor.py`'s + binding-factorization claims (§5 of 006: the I2 blind re-derivation + and N1's non-elementary factorization) — the latter feed no VERIFIED + claim, only the design-space narrative. +- **Common mathematical core.** All three exact corridors implement the + same corridor definition in the same normalization; agreement rules + out implementation error, not a shared conceptual error. That risk is + bounded by 004's earlier tie of this corridor to tier-0 + `unfold.certify` at point boxes and by the simulations (mine and + 002's) re-deriving actual billiard orbits with no unfolding at all. +- **Float birth measurements share the one-sided floor bias** (mine and + 006's both report birth ≈ truth + floor); the exact statements are + the certificates, which are one-sided (alive at points) — no exact + birth bracket from below exists for any member except via death(W) + of the touching predecessor. + +## Leads generated + +1. **Record the pairing as structure.** Prove + canonical(W(a,b)) = canonical(W(b+1, a−1)) for all a ≥ 2, b ≥ 1 + (finite word computation per member; the swap+reversal bijection is + visible in the block structure and six cases are verified here) and + fold it into the census tooling: quotienting by it halves every + W-member list and deduplicates future fits. Falsifiable by a single + canonical-form comparison. +2. **Certify one interior point of the gap continuum.** C2's slack + closes with one more certificate: W(5,2) at a rational apex with + certified γ ∈ (135.02, 135.03) (its window is wide there — width + ~0.69 in float — so `design_certify`-pattern certification is easy). + Would upgrade "alive throughout the gap" from float to three-point + + wide-window evidence. +3. **Birth-side exact brackets.** The floor-tracking signature suggests + certifying alive at γ = γ_b + 1e-8 for one out-of-sample member + (e.g. W(6,3) at 142.5 + 1e-8, corner (15, 22.5)) — same pattern as + 006's gapdeep certificate; would give the birth law its first exact + out-of-sample bracket. +4. **The interior-pinch blind spot.** Both 006's and my samplers are + strongest at 90/j corners. Build one screen pass whose accumulation + points are the measured interior argmaxes of near-miss words (the 19 + recorded near-misses are the natural test set); if any near-miss + flips alive, the negative screens' shared-design risk is real and + every dead verdict needs a wider sampler family. + +## References + +- `problems/billiards-triangles/attempts/006-design-family-past-135.md` + (under review) and its tools/data + `explore/design_{neighbours,factor,certify,exact135}.py`, + `data/design_*.json`. +- `001-word-census-coverage-map.md` (the corrected birth measurements; + `explore/census.py cmd_family` read for the mechanism), + `003-death-angle-laws.md` (laws, identities, Lemma C), + `004-skeptic-review-of-003.md` (the adversarial stack + `explore/deathlaw_skeptic.py` reused here for ring/interval layers), + 002's `skeptic_orbit.py`/`skeptic_family.py` (cross-referenced in + selftest only). +- Tier-0: `harness/billiards-triangles/unfold.py` read as + cross-reference; not used in any verdict path. +- New code: `explore/dsk_design_skeptic.py`. New data: + `data/dsk_{certs,birthdepth_W43,censusgap,birthlaw,birthlaw2,counts,deadsample,coverage,newmembers}.json`, + `data/dsk_lemmac_W{52,62}.json`. +- Niven's theorem — classical, as cited by 006. diff --git a/problems/billiards-triangles/attempts/008-skeptic-review-of-005.md b/problems/billiards-triangles/attempts/008-skeptic-review-of-005.md new file mode 100644 index 0000000..f1b0cc1 --- /dev/null +++ b/problems/billiards-triangles/attempts/008-skeptic-review-of-005.md @@ -0,0 +1,442 @@ +# 008 — Skeptic review of 005 (complete death-law theorem): adversarial verification + +- **Problem:** billiards-triangles, `problems/billiards-triangles/PROBLEM.md` +- **Date:** 2026-07-31 +- **Mode:** informed (read `prior-art.json`, attempts 003, 004, 005 in full, + 001/002 skimmed for conventions, the two tools under review + `explore/plaw_general.py` and `explore/plaw_suffice.py` line by line, the + 28 `data/plaw_*.json` files, and tier-0 `unfold.py` as cross-reference) +- **Type:** skeptic review of `005-complete-death-law-theorem.md` (default + stance: REFUTE). Every load-bearing layer re-derived by hand and/or + re-computed by code written from scratch for this review; the only prior + code in any verdict path is 004's `deathlaw_skeptic.py` stack (skeptic + property, reused for its interval trig and its death sampler). +- **Outcome in one line:** 005 survives. The closed-form bridge, the + 4-variable formal ring proof of I1–I4, Lemmas L1/C/D, the extended case + tree (a ≤ 2b+3 removed, k ≥ 1 branch), the sufficiency segments, and the + five new deaths are all independently confirmed; one numeric correction + (four of the five new exact-alive certificates are at γ_d − 1e-4, not + the claimed 1e-6), one wording caveat; no load-bearing claim refuted. +- **Tools:** new `explore/psk_review.py` (stdlib-only, deterministic, seed + 20260738; subcommands `bridge`, `ring`, `lemma`, `threshold`, `casetree`, + `segment`, `trigaudit`; ~5 min total). Reused: 004's + `deathlaw_skeptic.py` (its Machin-pi interval stack and its adaptive + death sampler — written for skeptics, no code shared with 003/005), and + 002's `skeptic_family.width` as a float cross-reference only. Nothing + imports `plaw_general`/`plaw_suffice` except `trigaudit`, where + `plaw_suffice` is the object under test. +- **Sources:** repo only; no external papers. + +My data lives in `data/psk_*.json`. No file of 005 (or any prior attempt) +was modified. + +Reproduce everything (repo root): + +``` +python3 problems/billiards-triangles/explore/psk_review.py ring --out .../data/psk_ring.json +python3 .../psk_review.py bridge --out .../data/psk_bridge.json +python3 .../psk_review.py lemma --out .../data/psk_lemma.json +python3 .../psk_review.py threshold --out .../data/psk_threshold.json +python3 .../psk_review.py casetree --a 6 --b 1 --out .../data/psk_casetree_W6_1.json # + (8,1),(9,1),(10,1),(12,1),(20,1),(12,2),(15,2),(30,3),(11,2) +python3 .../psk_review.py segment --out .../data/psk_segment.json # ~40 s +python3 .../psk_review.py trigaudit --trials 150 --out .../data/psk_trigaudit.json +python3 .../deathlaw_skeptic.py death --a 6 --b 1 --full --uniform 800 --iters 44 --out .../data/psk_death_W61.json # + (8,2),(7,1),(6,3),(8,1) +``` + +## Claims attacked + +Ordered as 005's own skeptic list (a)–(e) plus the two headline layers and +the scope audit. + +### 1. The closed-form bridge u = R₀ ∘ Rot_A(2aα) ∘ Rot_B(−2bβ) (Steps 1–3). **CONFIRMED (hand re-derivation + exact off-torus recomposition)** + +This is the one layer the formal check cannot protect. Hand re-derivation, +from the base geometry (A = 0, B = sin(α+β) real, C = sin β e^{iα}): + +- *Base reflections:* side AB is the real axis through 0, so R₂ = conj. + Side CA passes through 0 with unit direction e^{iα}, so R₁ = + e^{2iα} conj. C − B = sin β e^{iα} − sin(α+β) = sin α·e^{i(180−β)} + (expand sin(α+β); the real parts cancel to −sin α cos β), so side BC has + direction-squared e^{−2iβ} and R₀(z) = B + e^{−2iβ} conj(z − B), using + conj(B) = B. All three match 005's Step 2. +- *Pair collapse:* R₁R₂ = e^{2iα} conj∘conj = rotation about A by +2α; + R₀R₂(z) = B + e^{−2iβ}(z − B) = rotation about B by −2β. ✓ +- *Composition:* u(z) = R₀(e^{2iaα}(B + e^{−2ibβ}(z−B))) expands to + μ conj z + w with μ = e^{−2iaα+2i(b−1)β} and + w = B[1 + e^{−2i(aα+β)} − e^{−2iβ} − μ] — 005's Step 3 exactly, and + δ = e^{i(−aα+(b−1)β)} gives δ² = μ. The word-order convention + (M_k = R_{s₁}∘…∘R_{s_k}, gate k = M_{k−1}(side s_k)) is the same + composed-map unfolding 004 already tied to the harness corridor. +- *Gate 2a+2's C-endpoint:* letters 1..2a+1 are 0(12)^a, so + M_{2a+1} = R₀(R₁R₂)^a = R₀ Rot_A(2aα) and the C-image endpoint of gate + 2a+2 (mirror side 0 = BC, endpoints B, C) is v₂ = R₀(Rot_A(2aα) C) — + 005's I4 anchor is the correct geometric object, and in + `deathlaw_symbolic`'s gate list it is `gates[2a+1][1]` (SIDE_ENDS[0] = + (1,2), v-endpoint = C-image), which is what both `plaw_general.py + specialize` and `plaw_suffice.py` use. ✓ +- *Glide action:* p(u(z)) = Im(conj(δ)(δ² conj z + w)) = −p(z) + p(w), + p(τ) = 0, m = p(w)/2, hence p∘u = 2m − p. ✓ + +Independent computation (`bridge`, `data/psk_bridge.json`): my own +**off-torus pair algebra** (004's device, re-implemented from scratch: +track (value, star-value) at rational points x₀, y₀ off the unit torus; +star = the involution x→1/x, y→1/y, i→−i is a ring automorphism equal to +complex conjugation on the torus, so +, −, ×, ÷, conj all evaluate +exactly in Q(i)). The half word is composed by DIRECT nested application +of my base reflection maps — no closed form anywhere — and compared +exactly with 005's claimed μ, w, v₂ (specialized P→x₀^a, Q→y₀^b) plus +μ = δ², Im(conj(δ)τ) = 0 and the glide action p(u(z₁)) + p(z₁) − 2m = 0 +at a generic point. **40 members — the 23 of 005's specialize list plus +(1,1), (40,1), (40,40), (37,13), (25,24) and 12 random pairs with +a ≤ 40 — at 2 random rational points each: every comparison is an exact +match.** (Each match is an exact rational identity between a +(2a+2b+1)-step reflection composition and the closed form; a wrong +bridge cannot survive this at random points.) + +Float check: 005's own `floatcheck` (400 random (a,b,α,β,z), a ≤ 40) +reproduced, worst error 1.4e-14 — same number as the record. + +### 2. The 4-variable formal ring proof and the specialization argument. **CONFIRMED (independent ring; the specialization direction is sound)** + +- *Soundness of specialization:* the map P ↦ x^{2a}, Q ↦ y^{2b} (into the + half-angle ring), followed by evaluation on the torus, is a composition + of ring homomorphisms that also commutes with conj (exponent negation + + coefficient conjugation on both sides) and hence with Im. A polynomial + that is ZERO in Q(i)[x^±,y^±,P^±,Q^±] maps to zero under any such + homomorphism — the dangerous direction ("formal identity fails to + specialize") cannot occur: the formal ring has strictly fewer relations + than the geometry, so formal zero ⟹ geometric zero, for every integer + a, b and all angles. What COULD fail is the bridge (a wrong closed form + being formally consistent) — that is claim 1, tested independently + above. The half-integer subtlety is absent: all eight identities live in + whole-angle monomials, and the specialize target ring's keys (m,n) are + even multiples throughout. +- *My own formal ring* (`ring`, `data/psk_ring.json`): re-implemented + from scratch, and — unlike 005, which hard-codes the closed forms — I + RE-DERIVE u inside the ring by composing affine maps: R₀, R₁, R₂ as + antilinear maps, the block products checked to be the two rotations + (`rotA_block`, `rotB_block` true), the a-th/b-th powers introduced via + the fixed-point identities (rot_B fixes B: checked as a ring identity; + z ↦ λ(z−fix)+fix powers to λ^n(z−fix)+fix, the only lattice steps being + (x²)^a = P², (y^{−2})^b = Q^{−2}). The derived μ, w match 005's closed + forms as ring identities, and **I1, I2, I3, I4, D1, D2, μ = δ², + Im(conj(δ)τ) = 0 are all exact zero polynomials in my ring.** 14/14 + checks pass. +- *I4's geometric meaning:* covered exactly by the bridge's v₂ test + (claim 1) plus `ring`'s I4 = p(v₂) − m identity; also note I1 = −(S+T), + I4 = S − T with the same two products — verified structurally in my + ring by building S, T once and using them in both. + +### 3. Lemma L1, Lemma C, Lemma D. **CONFIRMED (full hand re-derivation; adversarial numerics clean)** + +Hand re-derivation of the entire chain (radians): + +- *L1:* d/dc[c cot(cs)] = [sin(2cs) − 2cs]/(2 sin²(cs)) < 0 for + cs ∈ (0, π/2] since sin x < x for x > 0 and sin(cs) ≠ 0. Strict. ✓ +- *Lemma C:* (log Φ)′ = 2/sin 2v − (1/a)cot((π/2−v)/a) − (b/a)cot(bv/a); + the split 2/sin 2v = tan v + cot v is the identity + 1/(sin v cos v) = tan v + cot v ✓. First bracket + [cot v − (b/a)cot((b/a)v)] ≤ 0 by L1 at s = v ∈ (0, v₀] ⊆ (0, π/4] + (c = b/a ≤ 1 vs c = 1; equality iff a = b) ✓; second bracket + [tan v − (1/a)cot((π/2−v)/a)] < 0 by L1 at s = π/2 − v ∈ + [π/2 − π/4, π/2) ⊂ (0, π/2), using cot(π/2−v) = tan v, strict since + 1/a ≤ 1/2 < 1 (this is where a ≥ 2 enters) ✓. So Φ strictly decreases + on (0, v₀]. Endpoint: π/2 = (b+1)v₀ gives (π/2−v₀)/a = bv₀/a exactly, + so Φ(v₀) = tan v₀ ✓. Factorization: cos v cos v₀ sin(bv/a)·Φ(v) + = cos v₀ sin v sin((π/2−v)/a) and cos v cos v₀ sin(bv/a)·Φ(v₀) + = cos v sin v₀ sin(bv/a), so H2 = cos v cos v₀ sin(bv/a)[Φ(v)−Φ(v₀)] + > 0 on (0, v₀), with the prefactor positive because v < v₀ ≤ π/4 + (cos v > 0) and bv/a ≤ v < π/4 (b ≤ a enters here). H2(v₀) = 0 by + substitution ✓. Domains: all cot arguments stay in (0, π/2] as + required by L1; no degree/radian slip (005's subsection is consistently + radian). The b = 1 path is not degenerate for Lemma C itself (v₀ = 45°); + a = b = 1 fails only the second bracket's strictness — correctly + excluded (H2 ≡ 0 there, as 005 says). +- *Lemma D:* (log D_b)′ = b cot(bx) − cot x < 0 on (0, π/(2b)] by L1 at + s = bx (c = 1/b vs 1, strict for b ≥ 2) ✓. This retires 004's + "Lemma D taken as classical" residual with a genuinely two-line proof. + +Adversarial numerics (`lemma`, `data/psk_lemma.json`, my own radian +code): 14,183 checks — L1 monotonicity at 4000 random (c₁,c₂,s); +F(v) < 0 for a ∈ {2,3,4,7,20,100,1000,10⁶}, b ∈ {1,2,a/2,a−1,a}, v down +to 1e-9 and up to (π/2)(1−1e-12), incl. v near v₀ from both sides; H2 > 0 +on (0,v₀) via the cancellation-free factored form plus agreement of the +factored and direct forms (identity check); Lemma D as tan(bx) > b tan x +for b up to 10⁵. **Zero violations.** (My first run reported 5 Lemma-D +"violations"; all were artifacts of MY test — x outside (0, π/(2b)] for +b = 10⁵, and float cancellation at x = 1e-9 — recorded in "Kill attempts +that failed" below.) + +### 4. The extended case tree (a ≤ 2b+3 removed; the k ≥ 1 branch). **CONFIRMED (every branch re-derived by hand + 447k-point scan on a > 2b+3 members)** + +Hand re-enumeration of all branches (Case I/II × residue), for general +a ≥ b ≥ 1, a ≥ 2, every k ≥ 0. Highlights, with every inequality +re-checked: + +- *Exhaustiveness and boundary self-exclusion:* as in 004 (residues at + multiples of 90 in aα or (b+1)β force N2 = 0 or N3 = 0; sin θ > 0 + always since θ < 180). ✓ +- *Case II, r ∈ (0,90) ∪ (270,360)* — the A4′ fix: cos(aα) > 0 on BOTH + intervals, so N2 < 0 forces (b+1)β > 180 and θ > β > 180/(b+1) ≥ θ_d + ⟺ a ≥ b, never touching α. k-free. ✓ The other Case II branches: + r ∈ (90,180) has slack exactly 90/(a(b+1)) ✓; r ∈ (180,270): the >360 + escape needs 3a ≥ b ✓ (always true), the (0,90)-residue path forces + cos(α+(b+1)β) < 0 with α+(b+1)β ∈ (0,270), hence > 90, and α > 180/a + beats θ_d by 90b/(a(b+1)) > 0 ✓; both k-free as claimed. +- *Case I, r ∈ (270,360)* — the A4 fix: cos(aα) > 0, sin(aα) < 0 force + (b+1)β mod 360 ∈ (90,180) via N2, N3 > 0, so (b+1)β > 90; with + α > 270/a: 270/a + 90/(b+1) − θ_d = (180b+270)/(a(b+1)) > 0 — + unconditional, restriction gone. ✓ r ∈ (90,180) and (180,270) as in + 003, k-free ✓. +- *Case I, r ∈ (0,90) — the main case.* cos(aα) = sin v, sin(aα) = cos v + for aα = 360k + 90 − v ✓; (b+1)β mod 360 ∈ (0,90) forced, > 360 escape + needs 3a ≥ b ✓; G = sin v sin α sin(bβ) − cos v sin β sin w with + w = α + (b+1)β − 90 ∈ (−90, 180) ✓ (α < 180, (b+1)β ∈ (0,90)). + **Threshold algebra:** aw − bv = a(b+1)α + a(b+1)β − 90(a+b) − 360kb + = a(b+1)(θ − θ_d) − 360kb — re-derived by hand and verified as an + exact Fraction identity at 2000 random rational (α, β, a ≤ 60, b ≤ a, + k ≤ 5) (`threshold`, `data/psk_threshold.json`). ✓ + **k ≥ 1:** θ ≤ θ_d gives aw ≤ bv − 360kb = b(v − 360k) < 0 since + v < 90 < 360k, so w < 0 strictly (005's "w ∈ (−90,0], sin w ≤ 0" is if + anything conservative), so G ≥ sin v sin α sin(bβ) > 0, contradicting + Case I's G < 0. No lemmas, no bound on a. ✓ + **k = 0:** 003's argument verbatim; I re-checked the w ≤ 0 sub-case, + Regime 2 (needs w < α < 90/a ≤ 45, i.e. a ≥ 2, and strict Lemma D for + b ≥ 2 with β < 90/(b+1) strict), Regime 1 (needs w ≤ bv/a ≤ v, i.e. + b ≤ a, then Lemma C strict + Lemma D; the b = 1 path closes via strict + Lemma C with v₀ = 45, cot v₀ = 1), and the θ = θ_d equality case (still + a strict contradiction). ✓ Preconditions a ≥ 2, b ≤ a are used exactly + where 005 says and nowhere else; b > a is nowhere claimed. + +Adversarial search (`casetree`, `data/psk_casetree_W*_*.json`): my own +design and seed (residue boundaries at ALL multiples 90k/a for every +reachable k, (b+1)β boundaries, death-corner accumulation, θ at θ_d +exactly and θ_d(1−10^{−j}) down to 10^{−12}, random fill), 10 members +chosen to stress a > 2b+3 and the k ≥ 1 region — (6,1), (8,1), (9,1), +(10,1), (12,1), (20,1), (12,2), (15,2), (30,3), (11,2); aα reaches 945° +for (20,1) and 742° for (30,3), so k = 1, 2 are genuinely exercised — +**447,000+ points, zero Case I/II hits, zero sub-margin near-hits, and +the positive control just above θ_d fires for all 10 members.** + +### 5. Sufficiency: segments, certifier, converse reduction. **CONFIRMED (hand re-derivations + independent interval certification at γ_d − 9.3e-10)** + +- *Segment algebra by hand:* θ(t) = 90/a + 90(a−1)/(a(b+1)) + t and the + endpoint identity θ(0) = [90(b+1) + 90(a−1)]/(a(b+1)) = θ_d is exact; + γ(t) = γ_d − t sweeps [γ_d − 1/4, γ_d) monotonically for t ∈ (0, 1/4] + ✓ (with ρ = 2, the record's general (ρ−1)t reduces to t ✓). The corner + identities aα_c = 90 and α_c + (b+1)β_c = 90 are exact ✓, giving + N1(t) = cos(at) sin β(t) sin(ŵt) − sin(at) sin α(t) sin(bβ(t)), + ŵ = (b+1)ρ − 1 ✓ (cos(90−at) = sin(at); cos(90+ŵt) = −sin ŵt), and + N2, N4 all-factors-positive under the rational range checks, which I + verified use exactly a·t₀ ≤ 45, (b+1)β(t) ∈ (90−90/a, 180), + θ(t) < 90, bβ(t) < 180, ŵt₀ < 180 — each sound for the members run + (90 − 90/a > 0 needs a ≥ 2, satisfied by all 20). ✓ +- *N1′ re-differentiated by hand:* six terms; matches `n1p_iv` exactly + including the dropped global π/180 (every term of dN1/dt carries exactly + one such factor — sign-safe) and N1′(0) = ŵ sin β_c − a sin α_c + sin(bβ_c) ✓. The N1 logic (N1(0) = 0 exactly; N1′ > 0 certified on + [0, t₁] ⟹ N1 > 0 on (0, t₁]; bisection on [t₁, t₀]) is sound. ✓ +- *Converse reduction re-derived:* if lo_k < m < hi_k strictly for all + half gates, then L = max lo_k < m < min hi_k = H, and the full corridor + [max(L, 2m−H), min(H, 2m−L)] has both upper candidates > m and both + lower candidates < m — positive width. Uses gate n′+k = u(gate k) + (functoriality — an identity of the composed-map construction), + p∘u = 2m − p (claim 1), and ∩(2m − I_k) = 2m − ∩I_k ✓. τ ≠ 0 is needed + for the projection direction to exist; the certifier certifies + Re(conj(δ)τ) of constant nonzero sign along the whole segment, and + Im(conj(δ)τ) = 0 in the ring, so τ ≠ 0 throughout ✓. Scaling to the + harness's τ-normal projection is the nonzero-real-rescale argument 004 + already audited. ✓ +- *Certifier audit* (`plaw_suffice.py` read line by line): the trig layer + is sound — exact mod-360 midpoint reduction into [−180, 180) (Fraction + floor-division), PI_LO/PI_HI radian bracketing with correct sign + handling, dyadic pre-round with its error added to the slack (plus a + conservative 1/DY), alternating-series remainders valid at 14 terms for + |x| ≤ 4 > π (the term ratio x²/(30·31) < 1 makes the first-omitted-term + bound legitimate), 1-Lipschitz widening in radians for the interval + half-width (valid across reduction since |sin x − sin y| ≤ |x−y| + globally). Every evaluation is a single non-iterated interval + evaluation. `poly_terms`' lexicographic (m,n) > (0,0) selection counts + each conjugate pair exactly once, and the (0,0) coefficient is real by + conj-invariance (asserted). Gate straddling requires certified opposite + signs at every gate's two endpoints; the three corner-vanishing margins + get signs on the open segment only, which is all the t ∈ (0, t₀] claim + needs. Failure paths are loud. One nit: `n1_iv` assigns an unused + variable `t`. +- *Trig audit* (`trigaudit`, `data/psk_trigaudit.json`): plaw_suffice's + sin_deg/cos_deg enclosures at 300 random rational degrees up to ±10⁷ + (raw argument to plaw, exactly-reduced argument to my reference — 004's + independent Machin stack, width ~1e-84) — my tight enclosure sits + inside plaw's in every case, certifying true-value containment; same + for 148 random degree INTERVALS at 4 interior rational points each. + **596 checks, zero violations; worst plaw point-enclosure width + 5e-16.** (My first run "FAILED" — entirely my audit's fault: I fed my + reduction-free reference stack raw 3000°+ arguments, far outside its + series' validity. See kill attempts.) +- *Independent alive certification* (`segment`, `data/psk_segment.json`) + — the strongest new evidence: for (2,1), (6,3), (6,1), (8,2), (8,1), + (7,1), (12,12) at t ∈ {1/4, 1/8, 1/97, 1/1024, 2^{−20}, 2^{−30}} I + certify positive corridor width by MY OWN interval unfolding: the FULL + 2n-gate corridor (no glide reduction anywhere), gates composed directly + from base reflections in interval complex arithmetic (004's certified + trig, outward dyadic rounding at 2^{−320}), projection onto the normal + of the full-word translation τ, with the linear part certified to + enclose 1 and τ certified nonzero. **All 42 points certified ALIVE, + including t = 2^{−30}, i.e. γ within 9.3e-10 of γ_d — with certified + width lower bounds ~1e-10 clear of the interval slop by ~20 orders.** + This independently confirms the segment construction where it matters + most (the 3-fold-degenerate corner approach) and simultaneously + exercises the glide reduction's converse (my full corridor is positive + exactly where the certifier's half-gate criterion says it must be). + Float cross-checks via 002's corridor agree in sign at all 42 points. + +### 6. The five new float deaths. **CONFIRMED (independent re-measurement; windows located)** + +004's sampler (`deathlaw_skeptic.py death --full`, my parameters, both +arc halves), `data/psk_death_W*.json`: + +| word | predicted | 005 measured − pred | mine − pred | last-alive apex | window half near death | +|---|---|---|---|---|---| +| W(6,3) | 585/4 = 146.25 | −3.7e-12 | −3.7e-12 | (15, 18.75) | **α < β only** (best α ≥ β width −6.1e-6) | +| W(6,1) | 255/2 = 127.5 | −5.3e-12 | −5.4e-12 | (15, 37.5) | **α < β only** (−5.2e-6) | +| W(8,2) | 285/2 = 142.5 | −3.6e-12 | −3.8e-12 | (11.25, 26.25) | **α < β only** (−7.4e-6) | +| W(8,1) | 1035/8 = 129.375 | −4.9e-12 | −4.9e-12 | (11.25, 39.375) | **α < β only** (−4.9e-6) | +| W(7,1) | 900/7 = 128.5714 | −5.1e-12 | −5.1e-12 | (12.857, 38.571) | **α < β only** (−4.0e-8) | + +All five die at γ_d at the predicted corner (90/a, 90(a−1)/(a(b+1))). +The near-death windows of all five b ≤ a−2 members live exclusively on +the mirror half α < β — the half 001/002 never scanned — extending 004 +§5's finding to the new members (005's skeptic item (e)). The ~5e-12 +agreements share the one-sided float-tolerance design bias 004 +documented; the exact content is carried by the theorem + segments, as +005 itself says. + +### 7. Scope honesty. **CONFIRMED with one correction (C1) and one caveat** + +- "Sup not attained" is exactly justified: necessity gives no positive + width at γ ≥ γ_d (004's C2 wording adopted verbatim — a zero-width + touching corridor at γ_d stays possible and is explicitly not + excluded); the segments give alive γ filling [γ_d − 1/4, γ_d); together + sup = γ_d, not attained. ✓ +- The NOT-claimed list is complete and correct: parametric sufficiency + open (per-member certificates only), a = 1 column open, birth side + open, other words/unstable orbits untouched, measured deaths float. + The theorem's hypotheses (a ≥ b ≥ 1, a ≥ 2) match where they are used; + b > a is nowhere claimed (identities hold for all a, b ≥ 1 — correct, + the formal proof does not need b ≤ a). ✓ +- Index entry (`prior-art.json` 005): one_line, range, leak_terms, gaps + all match the record and the data files (casetree 13 members × + ~120k = 1.56M ✓; lemmac grid 914,500 ✓; specialize 23/23 ✓; suffice + 20/20 with ρ = 2, t₀ = 1/4 ✓; float cross-check 40/40 positive ✓; + measure tables match the JSON to the last digit ✓). ✓ +- **But the record's exact-certificate sentence is wrong — C1 below.** + +## Refutations found + +No load-bearing claim is refuted. One numeric correction, one caveat: + +- **C1 (wrong constant, same genus as 004's C1).** 005, sufficiency + section: "all five: exact width > 0, orbit verified, γ certified within + **1e-6** of γ_d." False for four of the five: the certified lower + bounds in the data files are γ_d − **1e-4** for W(6,3) + (1462499/10000), W(8,2) (1424999/10000), W(8,1) (1293749/10000), and + W(7,1) (8999993/70000); only W(6,1) (127499999/1000000) reaches + γ_d − 1e-6. Cause (read from `deathlaw_exact.py cmd_alive`): the + `--dg 1e-6` request sets the float target, but the certified bound is + the largest 10^{−k} that CERTIFIES after apex snapping, and for these + four the k = 6 certification fails, leaving k = 4. Nothing downstream + is damaged — the segment certificates (independently confirmed here to + γ_d − 9.3e-10) supersede these brackets entirely — but the sentence + misstates its own data files by two orders of magnitude. +- **Caveat (not a refutation).** For the five NEW members the necessity + side is the parametric theorem alone — no per-member ring obligations + (003's `deathlaw_prove`) were run for them, unlike the fifteen 003 + members. That is by design (the general identities specialize, and + were re-proven here), but a reader tallying per-member machine + certificates should know the new five lean entirely on the general + proofs. + +## Claims that survive + +| # | 005 claim | Verdict | +|---|-----------|---------| +| 1 | Closed-form bridge u = R₀ Rot_A(2aα) Rot_B(−2bβ), gate-(2a+2) anchor v₂ | **CONFIRMED** — hand re-derivation of Steps 1–3; exact off-torus recomposition, 40 members incl. a = 40, every comparison an exact match | +| 2 | I1–I4, D1, D2, glide facts formal in Q(i)[x,y,P,Q]; valid for ALL (a,b) by specialization | **CONFIRMED** — independent ring, u re-derived by symbolic map composition, 14/14 zero polynomials; specialization direction proven sound (ring hom; formal ring has fewer relations) | +| 3 | Lemma L1; Lemma C for a ≥ 2, b ≤ a; Lemma D for b ≥ 1 | **CONFIRMED** — full hand re-derivation incl. domains, strictness, endpoint identity, factorization; 14,183-point adversarial numerics clean to a = 10⁶ | +| 4 | Extended case tree: a ≤ 2b+3 removed; k ≥ 1 branch; threshold algebra | **CONFIRMED** — all branches re-derived by hand; threshold identity exact at 2000 rational points; 447k-point scan on 10 a > 2b+3 members (k up to 2 reachable), zero hits, controls fire | +| 5 | death(W(a,b)) = γ_d exactly (sup not attained), 20 members, via certified segments | **CONFIRMED** — segment algebra + N1′ re-derived by hand; certifier and trig layer audited sound; converse reduction re-proven; independent interval certification of alive at 42 segment points down to γ_d − 9.3e-10 | +| 6 | Five new deaths at γ_d (~5e-12), predicted corners | **CONFIRMED** — independent sampler agrees to ~1e-13 of 005's values; all five windows located on the mirror half α < β | +| 7 | Scope statements and index entry match certificates | **CONFIRMED with C1** (four of five new exact certificates are at γ_d − 1e-4, not 1e-6) | + +**Kill attempts that failed,** for the record: (i) my Lemma-D stress +initially reported 5 violations — all artifacts of my own test +(out-of-domain x for b = 10⁵, float cancellation at x = 1e-9); the fixed +domain-respecting test is clean. (ii) My first trig audit reported +plaw_suffice enclosure violations — all artifacts of my own reference +stack, which has no mod-360 reduction and whose 30-term series is +invalid past ~660°; with exact pre-reduction (sin/cos are 360-periodic, +so the same value is tested) plaw_suffice contains the true value in all +596 checks. Two lessons about MY tools, zero about 005's. (iii) +Boundary- and k-region-targeted sign search for a dropped branch of the +extended tree: nothing, at 447k points with near-hit logging (zero even +below margin). (iv) Hunting for a failure of the segment construction at +extreme t: my independent full-corridor interval certification stays +positive down to t = 2^{−30}, with ~20 orders of margin over the +interval slop. + +**Net assessment:** 005's headline — the death law completed on both +sides, necessity parametric for all a ≥ b ≥ 1, a ≥ 2, and +death(W(a,b)) = γ_d(a,b) exactly with the sup not attained for 20 +members — stands as scoped, now with every load-bearing layer re-derived +or re-computed independently. The status upgrade to VERIFIED is +justified under the lab's definitions (formal-ring identities = proof; +Lemma proofs elementary and re-checked; per-member sufficiency +certified; parametric sufficiency correctly left open). + +## Residual risk + +- **Shared normalization.** My bridge/ring/segment work all live in the + same circumdiameter normalization introduced in 003 (as do 004's + checks). 004's apex-coordinate cross-checks (harness point boxes, exact + width matches) bound the risk that the normalization itself is wrong, + and my `segment` certification is cross-checked in sign against 002's + apex-coordinate float corridor at all 42 points — but a conceptual + error common to every angle-domain formulation would have to be caught + by those apex-side ties; nothing here adds to them. +- **Sampling, not proof, on the extended tree's reachable set.** The + hand proof covers all k; the scans (mine + 005's) only sample k ≤ 2. + The k ≥ 3 region (roughly a ≳ 30b members) is proof-covered but has + never been numerically exercised. +- **The five new members' necessity is parametric-only** (see caveat). +- **Shared-design bias in float death numbers** persists exactly as 004 + described; two same-design instruments agreeing at ~1e-12 is not two + independent measurements of that digit. +- **Not covered here:** re-running the 20-member `plaw_suffice all` + certification end to end (I reproduced selftest + one member, audited + the code, and replaced the full sweep with my own 42-point independent + certification on 7 members); the orbit realizations inside the five + `plaw_exact_alive_*` files (002's simulator, validated there; I + verified the alive criterion, not the trajectories); Lemma C's wider + F < 0 claim on all of (0, 90) beyond v₀ (not load-bearing; spot checks + only); the 3-fold (rather than higher) degeneracy count at the corner + (not load-bearing for the certificates — the certifier does not assume + it — only for 005's obstruction narrative). + +## References + +- `problems/billiards-triangles/attempts/005-complete-death-law-theorem.md` + (under review); `003-death-angle-laws.md` (the base theorem); + `004-skeptic-review-of-003.md` (skeptic exemplar; its C1/C2, its + composed-map unfolding and interval stack); `001-…`/`002-…` (skimmed, + conventions). +- Code under review: `problems/billiards-triangles/explore/plaw_general.py`, + `explore/plaw_suffice.py`; data `data/plaw_*.json` (28 files); + supporting 003 tools `deathlaw_symbolic.py`, `deathlaw_exact.py` (read + for C1's cause). +- Tier-0: `harness/billiards-triangles/unfold.py` (read as + cross-reference only; `plaw_suffice` imports its Iv/pi — audited here). +- New code and data: `problems/billiards-triangles/explore/psk_review.py`; + `problems/billiards-triangles/data/psk_{ring,bridge,lemma,threshold,segment,trigaudit}.json`, + `data/psk_casetree_W{6_1,8_1,9_1,10_1,12_1,20_1,12_2,15_2,30_3,11_2}.json`, + `data/psk_death_W{61,82,71,63,81}.json`. +- No external papers consulted. diff --git a/problems/billiards-triangles/data/design_bind_N1_death.json b/problems/billiards-triangles/data/design_bind_N1_death.json new file mode 100644 index 0000000..6bb191d --- /dev/null +++ b/problems/billiards-triangles/data/design_bind_N1_death.json @@ -0,0 +1,32 @@ +{ + "word": "0121212020202121202020201212120202021212020202", + "gamma": 136.8, + "argmax_alpha_deg": 26.4705, + "width": 0.0025074725538049014, + "bindings": [ + { + "gate": 1, + "end": "v", + "value": -0.0012537362769024507, + "factorization": "sin(2b)/2 * sin(2a)/2 * [-1/4*cos(2a-14b)/2 - 1/4*cos(2a+2b)/2 - 1/4*cos(2a+6b)/2 - 1/4*cos(2a+10b)/2 - 1/4*cos(2a+14b)/2 - 1/4*cos(6a-14b)/2 - 1/4*cos(6a+2b)/2 - 1/4*cos(10a-10b)/2 - 1/4*cos(10a-6b)/2 - 1/4*cos(10a-2b)/2 - 1/4*cos(10a+2b)/2]" + }, + { + "gate": 8, + "end": "v", + "value": 0.005027838592129741, + "factorization": "sin(2b)/2 * sin(2a)/2 * [1/4*cos(2a-14b)/2 - 1/4*cos(2a+2b)/2 - 1/4*cos(2a+6b)/2 - 1/4*cos(2a+10b)/2 + 1/4*cos(2a+14b)/2 + 1/4*cos(6a-14b)/2 - 1/4*cos(6a+2b)/2 - 1/4*cos(10a-10b)/2 - 1/4*cos(10a-6b)/2 - 1/4*cos(10a-2b)/2 - 1/4*cos(10a+2b)/2]" + }, + { + "gate": 12, + "end": "v", + "value": 0.03407876093434886, + "factorization": "sin(2b)/2 * sin(2a)/2 * [1/4*cos(2a-14b)/2 - 1/4*cos(2a+2b)/2 + 1/4*cos(2a+6b)/2 + 1/4*cos(2a+10b)/2 + 1/4*cos(2a+14b)/2 + 1/4*cos(6a-14b)/2 - 1/4*cos(6a+2b)/2 - 1/4*cos(10a-10b)/2 - 1/4*cos(10a-6b)/2 - 1/4*cos(10a-2b)/2 - 1/4*cos(10a+2b)/2]" + }, + { + "gate": 2, + "end": "v", + "value": 0.05477806685912051, + "factorization": "sin(2a+2b)/2 * sin(2b)/2 * cos(2a)/2 * [-1/8 - 1/4*cos(4b)/2 - 1/4*cos(8b)/2 - 1/4*cos(12b)/2 + 1/4*cos(4a-12b)/2 + 1/4*cos(4a-8b)/2 + 1/4*cos(4a-4b)/2 + 1/4*cos(4a)/2 - 1/4*cos(8a-12b)/2 - 1/4*cos(8a-8b)/2 - 1/4*cos(8a-4b)/2 - 1/4*cos(8a)/2]" + } + ] +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/design_bind_W43_birth.json b/problems/billiards-triangles/data/design_bind_W43_birth.json new file mode 100644 index 0000000..0a317ec --- /dev/null +++ b/problems/billiards-triangles/data/design_bind_W43_birth.json @@ -0,0 +1,14 @@ +{ + "word": "012121212020202012121212020202", + "gamma": 135.005, + "argmax_alpha_deg": 22.498046875, + "width": 0.00019281662726977622, + "bindings": [ + { + "gate": 2, + "end": "v", + "value": 9.640831363488811e-05, + "factorization": "sin(8b)/2 * cos(8a)/2 * sin(2a+2b)/2 * (-1/8+0i)" + } + ] +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/design_cert_W43_gapdeep.json b/problems/billiards-triangles/data/design_cert_W43_gapdeep.json new file mode 100644 index 0000000..2e5fdec --- /dev/null +++ b/problems/billiards-triangles/data/design_cert_W43_gapdeep.json @@ -0,0 +1,23 @@ +{ + "word": "012121212020202012121212020202", + "len": 30, + "apex": [ + "2251801040365531/4503599627370496", + "932723720360161/4503599627370496" + ], + "float_alpha_beta_deg": [ + 22.49993896484375, + 22.499961035156247 + ], + "exact_corridor_width": "131852611212664972320325170954221175946544606520120959968945332768641429496875547107980738970737674186988783948148884366495552831013630505861283434096662671882642779615726129483669834740517511090741177713840137594385499390468880290970112000/6058825575870784362575239525877639160407355821405085040618645712553868948462080266021314321336795173294904804542675498664055451078355898392936773952590538706996082238687924758547289133217492095791195251348368239508083929774402377623685359316001", + "exact_corridor_width_float": 2.1762074111815788e-05, + "orbit_verified_by_independent_simulation": true, + "orbit_msg": "exact periodic orbit confirmed, 30 bounces", + "certified_gamma_bracket_deg": [ + "1350000001/10000000", + "135001/1000" + ], + "gamma_gt_glo_certified": true, + "gamma_lt_ghi_certified": true, + "conclusion": "word alive (exact positive corridor + independent orbit simulation) at a triangle with gamma in (1350000001/10000000, 135001/1000) deg, certified" +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/design_cert_W43_pinchgap.json b/problems/billiards-triangles/data/design_cert_W43_pinchgap.json new file mode 100644 index 0000000..0bfd0f2 --- /dev/null +++ b/problems/billiards-triangles/data/design_cert_W43_pinchgap.json @@ -0,0 +1,23 @@ +{ + "word": "012121212020202012121212020202", + "len": 30, + "apex": [ + "35188172861353/70368744177664", + "7283325023095/35184372088832" + ], + "float_alpha_beta_deg": [ + 22.48781249999999, + 22.4921875 + ], + "exact_corridor_width": "85322988579010084987906284814713283794810913771430798553461834331342257290981055849092546339049269857248444195217552641209061417471520793636538451548132940689781195948278132114889760374622659879215554296566825472000/19554606598896063695130231153430179046962225987906151309754278982475070264737536593659297498782382008314463754120577971246126026823440817327321628009704888615202566734079786295044907626150021471044687875495608035526241", + "exact_corridor_width_float": 0.004363319105781802, + "orbit_verified_by_independent_simulation": true, + "orbit_msg": "exact periodic orbit confirmed, 30 bounces", + "certified_gamma_bracket_deg": [ + "135001/1000", + "16881/125" + ], + "gamma_gt_glo_certified": true, + "gamma_lt_ghi_certified": true, + "conclusion": "word alive (exact positive corridor + independent orbit simulation) at a triangle with gamma in (135001/1000, 16881/125) deg, certified" +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/design_cert_W54_touch144.json b/problems/billiards-triangles/data/design_cert_W54_touch144.json new file mode 100644 index 0000000..850178f --- /dev/null +++ b/problems/billiards-triangles/data/design_cert_W54_touch144.json @@ -0,0 +1,23 @@ +{ + "word": "01212121212020202020121212121202020202", + "len": 38, + "apex": [ + "9007193351942221/18014398509481984", + "5853215511333387/36028797018963968" + ], + "float_alpha_beta_deg": [ + 17.999961035156247, + 17.99993896484375 + ], + 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certified at a rational-apex triangle with obtuse angle EXACTLY 135 deg" +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/design_exact135_W52.json b/problems/billiards-triangles/data/design_exact135_W52.json new file mode 100644 index 0000000..9c2c9f9 --- /dev/null +++ b/problems/billiards-triangles/data/design_exact135_W52.json @@ -0,0 +1,16 @@ +{ + "word": "012121212120202012121212120202", + "len": 30, + "t": "587268960637/1099511627776", + "apex": [ + "990593882979969795299081/1553810651742291430151945", + "300824218724645284407543/1553810651742291430151945" + ], + "gamma_exactly_135_deg": true, + "on_arc_identity": "2*(CA.CB)^2 == |CA|^2*|CB|^2 and CA.CB < 0, exact", + "exact_corridor_width_float": 0.7608398804557077, + "exact_corridor_width": 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"sampler_floor_deg": 2.9802322387695312e-08, + "birth_between": [ + 135.00000002980408, + 135.0000000298041 + ], + "birth_minus_135": 2.980408453368e-08 + }, + { + "depth": 32, + "sampler_floor_deg": 1.1641532182693481e-10, + "birth_between": [ + 135.00000000011818, + 135.0000000001182 + ], + "birth_minus_135": 1.1817746781161986e-10 + }, + { + "depth": 40, + "sampler_floor_deg": 4.547473508864641e-13, + "birth_between": [ + 135.00000000000358, + 135.0000000000036 + ], + "birth_minus_135": 3.581135388230905e-12 + } + ] +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/dsk_birthlaw.json b/problems/billiards-triangles/data/dsk_birthlaw.json new file mode 100644 index 0000000..ab58575 --- /dev/null +++ b/problems/billiards-triangles/data/dsk_birthlaw.json @@ -0,0 +1,45 @@ +{ + "command": "birthlaw", + "rows": [ + { + "a": 7, + "b": 3, + "len": 42, + "predicted_birth": 144.64285714285714, + "predicted_death": 147.85714285714286, + "alive_at_window_mid": true, + "mid_width": 0.5448828289815619, + "measured_birth_between": [ + 144.64285714288727, + 144.6428571428873 + ], + "birth_minus_predicted": 3.015543370565865e-11, + "dead_sampled_at_birth_minus_1e-3": true, + "best_width_below": -4.315391780999529e-05 + }, + { + "a": 6, + "b": 4, + "len": 42, + "predicted_birth": 147.0, + "predicted_death": 150.0, + "alive_at_window_mid": true, + "mid_width": 0.5463897595483762, + "measured_birth_between": [ + 147.00000000003027, + 147.0000000000303 + ], + "birth_minus_predicted": 3.029754225281067e-11, + "dead_sampled_at_birth_minus_1e-3": true, + "best_width_below": -1.6339351791392926e-12 + }, + { + "a": 5, + "b": 2, + "alive_132.001": true, + "dead_sampled_131.999": true, + "alive_135": true, + "width_135": 0.6940004600100327 + } + ] +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/dsk_birthlaw2.json b/problems/billiards-triangles/data/dsk_birthlaw2.json new file mode 100644 index 0000000..5de1ebb --- /dev/null +++ b/problems/billiards-triangles/data/dsk_birthlaw2.json @@ -0,0 +1,30 @@ +{ + "command": "birthlaw2", + "note": "W(6,4)~W(5,5) same canonical word, so replaced; these two are genuinely outside the fit classes", + "rows": [ + { + "a": 6, + "b": 3, + "outside_003_scope": false, + "predicted_birth": 142.5, + "predicted_death": 146.25, + "alive_mid": true, + "measured_birth": 142.5000000000303, + "birth_minus_predicted": 3.029754225281067e-11, + "dead_sampled_below": true, + "best_width_below": -6.097433197385356e-12 + }, + { + "a": 8, + "b": 2, + "outside_003_scope": true, + "predicted_birth": 138.75, + "predicted_death": 142.5, + "alive_mid": true, + "measured_birth": 138.75000000003, + "birth_minus_predicted": 3.001332515850663e-11, + "dead_sampled_below": true, + "best_width_below": -2.2475589080573663e-11 + } + ] +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/dsk_censusgap.json b/problems/billiards-triangles/data/dsk_censusgap.json new file mode 100644 index 0000000..37491da --- /dev/null +++ b/problems/billiards-triangles/data/dsk_censusgap.json @@ -0,0 +1,83 @@ +{ + "command": "censusgap", + "replicated_001_birth_between": [ + 135.04856872558594, + 135.04857635498047 + ], + "census_grid": { + "max_x": 0.4994, + "spacing": 0.0012 + }, + "window_rows": [ + { + "gamma": 135.005, + "x_window": [ + 0.4999382879069728, + 0.499999999999999 + ], + "x_window_width": 6.171209302618941e-05, + "below_grid_max_x": false, + "contains_grid_point": false + }, + { + "gamma": 135.01, + "x_window": [ + 0.4998765650403333, + 0.499999999999999 + ], + "x_window_width": 0.0001234349596657136, + "below_grid_max_x": false, + "contains_grid_point": false + }, + { + "gamma": 135.02, + "x_window": [ + 0.49975308697116705, + 0.499999999999999 + ], + "x_window_width": 0.0002469130288319521, + "below_grid_max_x": false, + "contains_grid_point": false + }, + { + "gamma": 135.03, + "x_window": [ + 0.4996295657623879, + 0.499999999999999 + ], + "x_window_width": 0.00037043423761112804, + "below_grid_max_x": false, + "contains_grid_point": false + }, + { + "gamma": 135.04, + "x_window": [ + 0.4995060013838543, + 0.499999999999999 + ], + "x_window_width": 0.0004939986161446863, + "below_grid_max_x": false, + "contains_grid_point": false + }, + { + "gamma": 135.0486, + "x_window": [ + 0.49939970146814483, + 0.499999999999999 + ], + "x_window_width": 0.0006002985318541709, + "below_grid_max_x": false, + "contains_grid_point": true + }, + { + "gamma": 135.06, + "x_window": [ + 0.4992587429968286, + 0.499999999999999 + ], + "x_window_width": 0.0007412570031704036, + "below_grid_max_x": false, + "contains_grid_point": true + } + ] +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/dsk_certs.json b/problems/billiards-triangles/data/dsk_certs.json new file mode 100644 index 0000000..c0e28e8 --- /dev/null +++ b/problems/billiards-triangles/data/dsk_certs.json @@ -0,0 +1,88 @@ +{ + "command": "certs", + "rows": [ + { + "file": "design_cert_W43_pinchgap.json", + "len": 30, + "word_matches_claimed_family": true, + "my_exact_corridor_positive": true, + "width_matches_record_digit_for_digit": true, + "my_own_exact_simulation": true, + "sim_msg": "own exact simulation: closed periodic orbit, 30 bounces", + "my_gamma_gt": [ + "135001/1000", + true + ], + "my_gamma_lt": [ + "16881/125", + true + ], + "ok": true + }, + { + "file": "design_cert_W43_gapdeep.json", + "len": 30, + "word_matches_claimed_family": true, + "my_exact_corridor_positive": true, + "width_matches_record_digit_for_digit": true, + "my_own_exact_simulation": true, + "sim_msg": "own exact simulation: closed periodic orbit, 30 bounces", + "my_gamma_gt": [ + "1350000001/10000000", + true + ], + "my_gamma_lt": [ + "135001/1000", + true + ], + "ok": true + }, + { + "file": "design_cert_W54_touch144.json", + "len": 38, + "word_matches_claimed_family": true, + "my_exact_corridor_positive": true, + "width_matches_record_digit_for_digit": true, + "my_own_exact_simulation": true, + "sim_msg": "own exact simulation: closed periodic orbit, 38 bounces", + "my_gamma_gt": [ + "1440000001/10000000", + true + ], + "my_gamma_lt": [ + "144001/1000", + true + ], + "ok": true + }, + { + "file": "design_exact135_W52.json", + "len": 30, + "word_matches_claimed_family": true, + "my_exact_corridor_positive": true, + "width_matches_record_digit_for_digit": true, + "my_own_exact_simulation": true, + "sim_msg": "own exact simulation: closed periodic orbit, 30 bounces", + "my_on_arc_135_exact": true, + "circle_identity_exact": true, + "tan_alpha": "512242667139/1686780588413", + "niven_applicable": true, + "ok": true + }, + { + "file": "design_exact135_N1.json", + "len": 46, + "word_matches_claimed_family": true, + "my_exact_corridor_positive": true, + "width_matches_record_digit_for_digit": true, + "my_own_exact_simulation": true, + "sim_msg": "own exact simulation: closed periodic orbit, 46 bounces", + "my_on_arc_135_exact": true, + "circle_identity_exact": true, + "tan_alpha": "737239986301/1461783269251", + "niven_applicable": true, + "ok": true + } + ], + "all_ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/dsk_counts.json b/problems/billiards-triangles/data/dsk_counts.json new file mode 100644 index 0000000..9058ee3 --- /dev/null +++ b/problems/billiards-triangles/data/dsk_counts.json @@ -0,0 +1,29 @@ +{ + "command": "counts", + "all_odd_by_half_len": { + "3": 1, + "5": 2, + "7": 6, + "9": 16, + "11": 52, + "13": 168 + }, + "total_half_le_13": 245, + "claim_245": true, + "half_15": 566, + "claim_811": false, + "s1_k8_12": 2008, + "claim_2008": true, + "nonW": { + "N1": { + "len": 46, + "W_members_same_len_checked": 10, + "canonical_clash_with_W": [] + }, + "N2": { + "len": 50, + "W_members_same_len_checked": 11, + "canonical_clash_with_W": [] + } + } +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/dsk_coverage.json b/problems/billiards-triangles/data/dsk_coverage.json new file mode 100644 index 0000000..644cfb6 --- /dev/null +++ b/problems/billiards-triangles/data/dsk_coverage.json @@ -0,0 +1,207 @@ +{ + "command": "coverage", + "n_arcs": 25, + "failures": 0, + "rows": [ + { + "gamma": "907/10", + "a": 2, + "b": 1, + "len": 14, + "alive": true, + "width": 0.06979240233103479 + }, + { + "gamma": "933/10", + "a": 2, + "b": 1, + "len": 14, + "alive": true, + "width": 0.3240974901144411 + }, + { + "gamma": "979/10", + "a": 2, + "b": 1, + "len": 14, + "alive": true, + "width": 0.9433596419998063 + }, + { + "gamma": "1011/10", + "a": 2, + "b": 1, + "len": 14, + "alive": true, + "width": 1.3371693343326223 + }, + { + "gamma": "537/5", + "a": 2, + "b": 1, + "len": 14, + "alive": true, + "width": 0.5967693106509482 + }, + { + "gamma": "225/2", + "a": 2, + "b": 2, + "len": 18, + "alive": true, + "width": 1.1009521607483612 + }, + { + "gamma": "1187/10", + "a": 2, + "b": 2, + "len": 18, + "alive": true, + "width": 0.16278749497932798 + }, + { + "gamma": "120", + "a": 2, + "b": 3, + "len": 22, + "alive": true, + "width": 0.6679830370350368 + }, + { + "gamma": "1249/10", + "a": 3, + "b": 2, + "len": 22, + "alive": true, + "width": 0.872088439264628 + }, + { + "gamma": "1273/10", + "a": 3, + "b": 2, + "len": 22, + "alive": true, + "width": 0.4836944050740708 + }, + { + "gamma": "130", + "a": 3, + "b": 3, + "len": 26, + "alive": true, + "width": 0.5762461303372133 + }, + { + "gamma": "1339/10", + "a": 3, + "b": 3, + "len": 26, + "alive": true, + "width": 0.21160114942140607 + }, + { + "gamma": "135", + "a": 3, + "b": 4, + "len": 30, + "alive": true, + "width": 0.7203026827710661 + }, + { + "gamma": "6751/50", + "a": 3, + "b": 4, + "len": 30, + "alive": true, + "width": 0.7253084356932527 + }, + { + "gamma": "675243/5000", + "a": 3, + "b": 4, + "len": 30, + "alive": true, + "width": 0.7324608869796991 + }, + { + "gamma": "693/5", + "a": 4, + "b": 3, + "len": 30, + "alive": true, + "width": 0.5125618785543786 + }, + { + "gamma": "1413/10", + "a": 4, + "b": 4, + "len": 34, + "alive": true, + "width": 0.43412551181379033 + }, + { + "gamma": "144", + "a": 4, + "b": 5, + "len": 38, + "alive": true, + "width": 0.3875985044150376 + }, + { + "gamma": "295/2", + "a": 5, + "b": 4, + "len": 38, + "alive": true, + "width": 0.026496165286085782 + }, + { + "gamma": "150", + "a": 5, + "b": 6, + "len": 46, + "alive": true, + "width": 0.347959663162245 + }, + { + "gamma": "1080/7", + "a": 6, + "b": 7, + "len": 54, + "alive": true, + "width": 0.23892525595360548 + }, + { + "gamma": "1571/10", + "a": 7, + "b": 7, + "len": 58, + "alive": true, + "width": 0.18967787242201228 + }, + { + "gamma": "1609/10", + "a": 9, + "b": 8, + "len": 70, + "alive": true, + "width": 0.11829681671233394 + }, + { + "gamma": "325/2", + "a": 9, + "b": 10, + "len": 78, + "alive": true, + "width": 0.1452722507741524 + }, + { + "gamma": "824/5", + "a": 11, + "b": 11, + "len": 90, + "alive": true, + "width": 0.14912345366838742 + } + ] +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/dsk_deadsample.json b/problems/billiards-triangles/data/dsk_deadsample.json new file mode 100644 index 0000000..5b55ee2 --- /dev/null +++ b/problems/billiards-triangles/data/dsk_deadsample.json @@ -0,0 +1,35 @@ +{ + "command": "deadsample", + "n_sampled": 60, + "arcs": [ + 135.0, + 135.02, + 136.5, + 138.0, + 140.0, + 140.7, + 142.5, + 144.0, + 145.0, + 148.0, + 151.0, + 155.0, + 159.0, + 137.3, + 141.9, + 146.2, + 149.4, + 152.7, + 156.5 + ], + "hits": [], + "best_width_seen": 5.551115123125783e-17, + "near_miss_words_attacked": 19, + "near_miss_hits": [ + { + "word": "21212121020202121212102020", + "gamma": 134.65, + "width": 0.06286828590871207 + } + ] +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/dsk_lemmac_W52.json b/problems/billiards-triangles/data/dsk_lemmac_W52.json new file mode 100644 index 0000000..3f36c4c --- /dev/null +++ b/problems/billiards-triangles/data/dsk_lemmac_W52.json @@ -0,0 +1,12 @@ +{ + "command": "lemmac", + "a": 5, + "b": 2, + "v0": "30", + "endpoint_identity_90_minus_v0_eq_b_v0": true, + "d2": "15/16", + "h2p_hi_on_zone": -0.037413644427968185, + "d1": "15/4", + "bisection_leaves": 38, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/dsk_lemmac_W62.json b/problems/billiards-triangles/data/dsk_lemmac_W62.json new file mode 100644 index 0000000..4937c30 --- /dev/null +++ b/problems/billiards-triangles/data/dsk_lemmac_W62.json @@ -0,0 +1,12 @@ +{ + "command": "lemmac", + "a": 6, + "b": 2, + "v0": "30", + "endpoint_identity_90_minus_v0_eq_b_v0": true, + "d2": "15/16", + "h2p_hi_on_zone": -0.03228069927231018, + "d1": "15/4", + "bisection_leaves": 36, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/dsk_newmembers.json b/problems/billiards-triangles/data/dsk_newmembers.json new file mode 100644 index 0000000..b8b91c3 --- /dev/null +++ b/problems/billiards-triangles/data/dsk_newmembers.json @@ -0,0 +1,57 @@ +{ + "command": "newmembers", + "members": [ + { + "a": 5, + "b": 2, + "scope_precondition_a_le_2b+3": true, + "ring_identities": { + "mu_monomial": true, + "tau_parallel": true, + "gate1_u": true, + "gate1_v": true, + "gate2_u": true, + "gate2_v": true, + "I1": true, + "I2": true, + "I3": true, + "D1": true, + "D2": true, + "glide_action_gA": true, + "glide_action_gB": true, + "glide_action_gC": true, + "glide_action_gA1": true + }, + "pair_spots_ok": true, + "casetree_points": 273291, + "casetree_counterexamples": 0, + "ok": true + }, + { + "a": 6, + "b": 2, + "scope_precondition_a_le_2b+3": true, + "ring_identities": { + "mu_monomial": true, + "tau_parallel": true, + "gate1_u": true, + "gate1_v": true, + "gate2_u": true, + "gate2_v": true, + "I1": true, + "I2": true, + "I3": true, + "D1": true, + "D2": true, + "glide_action_gA": true, + "glide_action_gB": true, + "glide_action_gC": true, + "glide_action_gA1": true + }, + "pair_spots_ok": true, + "casetree_points": 276538, + "casetree_counterexamples": 0, + "ok": true + } + ] +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_casetree_W10_3.json b/problems/billiards-triangles/data/plaw_casetree_W10_3.json new file mode 100644 index 0000000..9d2fcd7 --- /dev/null +++ b/problems/billiards-triangles/data/plaw_casetree_W10_3.json @@ -0,0 +1,10 @@ +{ + "a": 10, + "b": 3, + "theta_d": 29.25, + "n_points": 120064, + "case_hits_at_or_below_theta_d": [], + "n_hits": 0, + "positive_control_above_theta_d_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_casetree_W11_2.json b/problems/billiards-triangles/data/plaw_casetree_W11_2.json new file mode 100644 index 0000000..6c0d49c --- /dev/null +++ b/problems/billiards-triangles/data/plaw_casetree_W11_2.json @@ -0,0 +1,10 @@ +{ + "a": 11, + "b": 2, + "theta_d": 35.45454545454545, + "n_points": 120084, + "case_hits_at_or_below_theta_d": [], + "n_hits": 0, + "positive_control_above_theta_d_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_casetree_W12_1.json b/problems/billiards-triangles/data/plaw_casetree_W12_1.json new file mode 100644 index 0000000..749eeef --- /dev/null +++ b/problems/billiards-triangles/data/plaw_casetree_W12_1.json @@ -0,0 +1,10 @@ +{ + "a": 12, + "b": 1, + "theta_d": 48.75, + "n_points": 120116, + "case_hits_at_or_below_theta_d": [], + "n_hits": 0, + "positive_control_above_theta_d_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_casetree_W12_12.json b/problems/billiards-triangles/data/plaw_casetree_W12_12.json new file mode 100644 index 0000000..f704d0a --- /dev/null +++ b/problems/billiards-triangles/data/plaw_casetree_W12_12.json @@ -0,0 +1,10 @@ +{ + "a": 12, + "b": 12, + "theta_d": 13.846153846153847, + "n_points": 120037, + "case_hits_at_or_below_theta_d": [], + "n_hits": 0, + "positive_control_above_theta_d_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_casetree_W14_2.json b/problems/billiards-triangles/data/plaw_casetree_W14_2.json new file mode 100644 index 0000000..4c8bc3f --- /dev/null +++ b/problems/billiards-triangles/data/plaw_casetree_W14_2.json @@ -0,0 +1,10 @@ +{ + "a": 14, + "b": 2, + "theta_d": 34.285714285714285, + "n_points": 120100, + "case_hits_at_or_below_theta_d": [], + "n_hits": 0, + "positive_control_above_theta_d_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_casetree_W16_4.json b/problems/billiards-triangles/data/plaw_casetree_W16_4.json new file mode 100644 index 0000000..31d8330 --- /dev/null +++ b/problems/billiards-triangles/data/plaw_casetree_W16_4.json @@ -0,0 +1,10 @@ +{ + "a": 16, + "b": 4, + "theta_d": 22.5, + "n_points": 120073, + "case_hits_at_or_below_theta_d": [], + "n_hits": 0, + "positive_control_above_theta_d_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_casetree_W17_3.json b/problems/billiards-triangles/data/plaw_casetree_W17_3.json new file mode 100644 index 0000000..2066ed7 --- /dev/null +++ b/problems/billiards-triangles/data/plaw_casetree_W17_3.json @@ -0,0 +1,10 @@ +{ + "a": 17, + "b": 3, + "theta_d": 26.470588235294116, + "n_points": 120091, + "case_hits_at_or_below_theta_d": [], + "n_hits": 0, + "positive_control_above_theta_d_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_casetree_W6_1.json b/problems/billiards-triangles/data/plaw_casetree_W6_1.json new file mode 100644 index 0000000..226755f --- /dev/null +++ b/problems/billiards-triangles/data/plaw_casetree_W6_1.json @@ -0,0 +1,10 @@ +{ + "a": 6, + "b": 1, + "theta_d": 52.5, + "n_points": 120068, + "case_hits_at_or_below_theta_d": [], + "n_hits": 0, + "positive_control_above_theta_d_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_casetree_W6_3.json b/problems/billiards-triangles/data/plaw_casetree_W6_3.json new file mode 100644 index 0000000..2d16ca5 --- /dev/null +++ b/problems/billiards-triangles/data/plaw_casetree_W6_3.json @@ -0,0 +1,10 @@ +{ + "a": 6, + "b": 3, + "theta_d": 33.75, + "n_points": 120052, + "case_hits_at_or_below_theta_d": [], + "n_hits": 0, + "positive_control_above_theta_d_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_casetree_W7_1.json b/problems/billiards-triangles/data/plaw_casetree_W7_1.json new file mode 100644 index 0000000..e7d7c69 --- /dev/null +++ b/problems/billiards-triangles/data/plaw_casetree_W7_1.json @@ -0,0 +1,10 @@ +{ + "a": 7, + "b": 1, + "theta_d": 51.42857142857143, + "n_points": 120073, + "case_hits_at_or_below_theta_d": [], + "n_hits": 0, + "positive_control_above_theta_d_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_casetree_W8_1.json b/problems/billiards-triangles/data/plaw_casetree_W8_1.json new file mode 100644 index 0000000..736b2f5 --- /dev/null +++ b/problems/billiards-triangles/data/plaw_casetree_W8_1.json @@ -0,0 +1,10 @@ +{ + "a": 8, + "b": 1, + "theta_d": 50.625, + "n_points": 120088, + "case_hits_at_or_below_theta_d": [], + "n_hits": 0, + "positive_control_above_theta_d_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_casetree_W8_2.json b/problems/billiards-triangles/data/plaw_casetree_W8_2.json new file mode 100644 index 0000000..3570a57 --- /dev/null +++ b/problems/billiards-triangles/data/plaw_casetree_W8_2.json @@ -0,0 +1,10 @@ +{ + "a": 8, + "b": 2, + "theta_d": 37.5, + "n_points": 120066, + "case_hits_at_or_below_theta_d": [], + "n_hits": 0, + "positive_control_above_theta_d_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_casetree_W9_1.json b/problems/billiards-triangles/data/plaw_casetree_W9_1.json new file mode 100644 index 0000000..2d2dc96 --- /dev/null +++ b/problems/billiards-triangles/data/plaw_casetree_W9_1.json @@ -0,0 +1,10 @@ +{ + "a": 9, + "b": 1, + "theta_d": 50.0, + "n_points": 120089, + "case_hits_at_or_below_theta_d": [], + "n_hits": 0, + "positive_control_above_theta_d_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_exact_alive_W61.json b/problems/billiards-triangles/data/plaw_exact_alive_W61.json new file mode 100644 index 0000000..450c3b1 --- /dev/null +++ b/problems/billiards-triangles/data/plaw_exact_alive_W61.json @@ -0,0 +1,17 @@ +{ + "a": 6, + "b": 1, + "len": 30, + "apex": [ + "407468545573/549755813888", + "54590432955/274877906944" + ], + "exact_corridor_width": "9796857370560763695659847544795423600208841380490392542338161494005999566853783298188298564449723709322878558745999422952176703663724942774855167829777194629418609066468528640/52567979673998052929046022539934693864327178447412270405721294651307562592339071687310404981164530653430488740955119267061597897550230208220282945008153141831430581661104575502817329", + "exact_corridor_width_float": 1.8636549152766145e-07, + "orbit_verified_by_simulation": true, + "orbit_msg": "exact periodic orbit confirmed, 30 bounces", + "predicted_death": "255/2", + "certified_gamma_lower_bound_deg": "127499999/1000000", + "certified_gamma_below_prediction": true, + "conclusion": "death(W(6,1)) >= 127499999/1000000 deg, exactly; apex gamma certified < predicted death: True" +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_exact_alive_W63.json b/problems/billiards-triangles/data/plaw_exact_alive_W63.json new file mode 100644 index 0000000..01f95b8 --- /dev/null +++ b/problems/billiards-triangles/data/plaw_exact_alive_W63.json @@ -0,0 +1,17 @@ +{ + "a": 6, + "b": 3, + "len": 38, + "apex": [ + "307237243505/549755813888", + "164647937083/1099511627776" + ], + "exact_corridor_width": "980913526997038813517094244836164747522951740499172019121370227318677543918202974128289420495276214725950996030981675233212048467354663180244261900306336472026825296664855410859678072424941794221126639381260718198132711922175456469058903/3575998120011406153891417908605063737461456063660604145635059887866271521625356678327286796480040655492249023920257673445073001075999568414248118189289481735173913952623481517779569195366241438710375520690743594490368704367190858622124320358400", + "exact_corridor_width_float": 2.74304821780474e-07, + "orbit_verified_by_simulation": true, + "orbit_msg": "exact periodic orbit confirmed, 38 bounces", + "predicted_death": "585/4", + "certified_gamma_lower_bound_deg": "1462499/10000", + "certified_gamma_below_prediction": true, + "conclusion": "death(W(6,3)) >= 1462499/10000 deg, exactly; apex gamma certified < predicted death: True" +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_exact_alive_W71.json b/problems/billiards-triangles/data/plaw_exact_alive_W71.json new file mode 100644 index 0000000..1d8fa83 --- /dev/null +++ b/problems/billiards-triangles/data/plaw_exact_alive_W71.json @@ -0,0 +1,17 @@ +{ + "a": 7, + "b": 1, + "len": 34, + "apex": [ + "854847285551/1099511627776", + "195113310785/1099511627776" + ], + "exact_corridor_width": "2071593974554976718880921178035348810428421086358776174485411626511378344882153982748108107600532846300451654850297672003832613460630626351522939894640485602232915926350472437627806874572754528477023947622318625/10219613384701034729720270885655207520525323082298894079912812857524204683418711470075142848964551088028254636313912706150009100010611031321995740426717432939084301542152474684614872029806309260483480704133433607061504", + "exact_corridor_width_float": 2.0270766579645705e-07, + "orbit_verified_by_simulation": true, + "orbit_msg": "exact periodic orbit confirmed, 34 bounces", + "predicted_death": "900/7", + "certified_gamma_lower_bound_deg": "8999993/70000", + "certified_gamma_below_prediction": true, + "conclusion": "death(W(7,1)) >= 8999993/70000 deg, exactly; apex gamma certified < predicted death: True" +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_exact_alive_W81.json b/problems/billiards-triangles/data/plaw_exact_alive_W81.json new file mode 100644 index 0000000..952a63e --- /dev/null +++ b/problems/billiards-triangles/data/plaw_exact_alive_W81.json @@ -0,0 +1,17 @@ +{ + "a": 8, + "b": 1, + "len": 38, + "apex": [ + "442503780809/549755813888", + "88019472701/549755813888" + ], + "exact_corridor_width": "961106702422170571964210674827519414887038858529438443428553033473007099896506284970604481636515621448499163068076342001937648175586572693803225403699755838474005660441763111703401781991546924816898107677790301337836788809358719935/4581728366439523075684708801045073796858015572520163909718212616935292906939056932844641832538793822339012502851946052710297028206506427192574775298038785050073905168345421688929984789213120343736054413274853424270739883470125478136774656", + "exact_corridor_width_float": 2.0976946373821152e-07, + "orbit_verified_by_simulation": true, + "orbit_msg": "exact periodic orbit confirmed, 38 bounces", + "predicted_death": "1035/8", + "certified_gamma_lower_bound_deg": "1293749/10000", + "certified_gamma_below_prediction": true, + "conclusion": "death(W(8,1)) >= 1293749/10000 deg, exactly; apex gamma certified < predicted death: True" +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_exact_alive_W82.json b/problems/billiards-triangles/data/plaw_exact_alive_W82.json new file mode 100644 index 0000000..8a971b2 --- /dev/null +++ b/problems/billiards-triangles/data/plaw_exact_alive_W82.json @@ -0,0 +1,17 @@ +{ + "a": 8, + "b": 2, + "len": 42, + "apex": [ + "12242003851/17179869184", + "38961373785/274877906944" + ], + "exact_corridor_width": "8970926880891063349421537390619222165872971251061563036290333338909686557336269774517370647112890897623626233222419393298848592188207380609547995093589886847415661200400822241946573881790557249815001602626468516042405442565081647067441316648125/34427831717719349359535853350235526487883947389763395042287736801659272921742423812871036874649649389552183592242154691232922065223419000843591801706128692820599748408577837230697355123987406390391086499934166161576324376503765831979024730736744726528", + "exact_corridor_width_float": 2.6057193942521504e-07, + "orbit_verified_by_simulation": true, + "orbit_msg": "exact periodic orbit confirmed, 42 bounces", + "predicted_death": "285/2", + "certified_gamma_lower_bound_deg": "1424999/10000", + "certified_gamma_below_prediction": true, + "conclusion": "death(W(8,2)) >= 1424999/10000 deg, exactly; apex gamma certified < predicted death: True" +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_formal.json b/problems/billiards-triangles/data/plaw_formal.json new file mode 100644 index 0000000..591af41 --- /dev/null +++ b/problems/billiards-triangles/data/plaw_formal.json @@ -0,0 +1,27 @@ +{ + "checks": { + "mu_eq_delta_sq": true, + "tau_parallel_axis": true, + "I1": true, + "I2": true, + "I3": true, + "D2": true, + "D1": true, + "I4": true + }, + "term_counts": { + "B": 2, + "C": 2, + "A1": 4, + "mu": 1, + "delta": 1, + "w": 8, + "tau": 8, + "m": 12, + "pB": 4, + "pC": 4, + "pA1": 8 + }, + "ok": true, + "meaning": "each True is an exact identity in Q(i)[x^,y^,P^,Q^]; holds for ALL integer a,b and all real angles by specialization" +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_lemmac_grid.json b/problems/billiards-triangles/data/plaw_lemmac_grid.json new file mode 100644 index 0000000..708dc1a --- /dev/null +++ b/problems/billiards-triangles/data/plaw_lemmac_grid.json @@ -0,0 +1,8 @@ +{ + "amax": 60, + "grid": 500, + "n_samples": 914500, + "violations": [], + "n_violations": 0, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_measure_W61.json b/problems/billiards-triangles/data/plaw_measure_W61.json new file mode 100644 index 0000000..e1b1bfe --- /dev/null +++ b/problems/billiards-triangles/data/plaw_measure_W61.json @@ -0,0 +1,21 @@ +{ + "a": 6, + "b": 1, + "len": 30, + "death_between": [ + 127.49999999999471, + 127.49999999999477 + ], + "uniform_points": 800, + "geo_points": 64, + "full_arc": true, + "bisect_iters": 44, + "alive_at_death_lo": { + "alpha_deg": 14.999999999998181, + "beta_deg": 37.500000000007105, + "alpha_beta_ratio": 0.3999999999998757, + "width": 1.0134115768778429e-12 + }, + "predicted_death": 127.5, + "measured_minus_predicted": -5.258016244624741e-12 +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_measure_W63.json b/problems/billiards-triangles/data/plaw_measure_W63.json new file mode 100644 index 0000000..2184cc2 --- /dev/null +++ b/problems/billiards-triangles/data/plaw_measure_W63.json @@ -0,0 +1,21 @@ +{ + "a": 6, + "b": 3, + "len": 38, + "death_between": [ + 146.2499999999963, + 146.24999999999636 + ], + "uniform_points": 800, + "geo_points": 64, + "full_arc": true, + "bisect_iters": 44, + "alive_at_death_lo": { + "alpha_deg": 14.999999999998181, + "beta_deg": 18.750000000005514, + "alpha_beta_ratio": 0.7999999999996678, + "width": 1.0524914273446484e-12 + }, + "predicted_death": 146.25, + "measured_minus_predicted": -3.666400516522117e-12 +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_measure_W71.json b/problems/billiards-triangles/data/plaw_measure_W71.json new file mode 100644 index 0000000..7b6b1ff --- /dev/null +++ b/problems/billiards-triangles/data/plaw_measure_W71.json @@ -0,0 +1,21 @@ +{ + "a": 7, + "b": 1, + "len": 34, + "death_between": [ + 128.57142857142338, + 128.57142857142344 + ], + "uniform_points": 800, + "geo_points": 64, + "full_arc": true, + "bisect_iters": 44, + "alive_at_death_lo": { + "alpha_deg": 12.857142857141039, + "beta_deg": 38.571428571435575, + "alpha_beta_ratio": 0.3333333333332257, + "width": 1.0600409439120995e-12 + }, + "predicted_death": 128.57142857142856, + "measured_minus_predicted": -5.144329406903125e-12 +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_measure_W81.json b/problems/billiards-triangles/data/plaw_measure_W81.json new file mode 100644 index 0000000..2c2584a --- /dev/null +++ b/problems/billiards-triangles/data/plaw_measure_W81.json @@ -0,0 +1,21 @@ +{ + "a": 8, + "b": 1, + "len": 38, + "death_between": [ + 129.3749999999951, + 129.37499999999517 + ], + "uniform_points": 800, + "geo_points": 64, + "full_arc": true, + "bisect_iters": 44, + "alive_at_death_lo": { + "alpha_deg": 11.249999999998181, + "beta_deg": 39.37500000000671, + "alpha_beta_ratio": 0.28571428571419083, + "width": 1.057376408652999e-12 + }, + "predicted_death": 129.375, + "measured_minus_predicted": -4.860112312599085e-12 +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_measure_W82.json b/problems/billiards-triangles/data/plaw_measure_W82.json new file mode 100644 index 0000000..4a78c8f --- /dev/null +++ b/problems/billiards-triangles/data/plaw_measure_W82.json @@ -0,0 +1,21 @@ +{ + "a": 8, + "b": 2, + "len": 42, + "death_between": [ + 142.49999999999636, + 142.49999999999642 + ], + "uniform_points": 800, + "geo_points": 64, + "full_arc": true, + "bisect_iters": 44, + "alive_at_death_lo": { + "alpha_deg": 11.249999999998181, + "beta_deg": 26.250000000005457, + "alpha_beta_ratio": 0.4285714285712702, + "width": 1.0311751452718454e-12 + }, + "predicted_death": 142.5, + "measured_minus_predicted": -3.609557097661309e-12 +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_specialize.json b/problems/billiards-triangles/data/plaw_specialize.json new file mode 100644 index 0000000..f991834 --- /dev/null +++ b/problems/billiards-triangles/data/plaw_specialize.json @@ -0,0 +1,305 @@ +{ + "results": { + "W(2,1)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(2,2)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(3,2)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(3,3)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(4,2)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(4,3)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(4,4)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(5,3)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(5,4)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(5,5)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(6,6)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(7,7)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(8,8)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(10,9)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(12,12)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(6,3)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(6,1)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(8,2)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(8,1)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(7,1)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(11,2)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(14,4)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + }, + "W(17,9)": { + "orient_reversing": true, + "mu": true, + "w": true, + "delta": true, + "tau": true, + "m": true, + "pB": true, + "pC": true, + "pA1": true, + "p_gate2a2_v": true, + "ok": true + } + }, + "ok": true, + "meaning": "closed forms of u = R_0 Rot_A(2a al) Rot_B(-2b be) agree EXACTLY with the gate-by-gate symbolic unfolding for every listed member" +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/plaw_suffice_all.json b/problems/billiards-triangles/data/plaw_suffice_all.json new file mode 100644 index 0000000..7f1f61c --- /dev/null +++ b/problems/billiards-triangles/data/plaw_suffice_all.json @@ -0,0 +1,662 @@ +[ + { + "a": 2, + "b": 1, + "len": 14, + "theta_d": "135/2", + "gamma_d": "225/2", + "corner": [ + "45", + "45/2" + ], + "n_distinct_diffs": 8, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 4, + "generic": 5, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.008271275871167605, + 0.00016721352895165964 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "449/4", + "-> 225/2 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [112.250000, 112.5) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(2,1)) = 225/2 EXACTLY" + }, + { + "a": 2, + "b": 2, + "len": 18, + "theta_d": "60", + "gamma_d": "120", + "corner": [ + "45", + "15" + ], + "n_distinct_diffs": 10, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 5, + "generic": 7, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.008634987355991663, + 0.00017258672141773346 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "479/4", + "-> 120 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [119.750000, 120.0) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(2,2)) = 120 EXACTLY" + }, + { + "a": 3, + "b": 2, + "len": 22, + "theta_d": "50", + "gamma_d": "130", + "corner": [ + "30", + "20" + ], + "n_distinct_diffs": 12, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 6, + "generic": 9, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.015080518656982678, + 0.0003036161233458401 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "519/4", + "-> 130 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [129.750000, 130.0) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(3,2)) = 130 EXACTLY" + }, + { + "a": 3, + "b": 3, + "len": 26, + "theta_d": "45", + "gamma_d": "135", + "corner": [ + "30", + "15" + ], + "n_distinct_diffs": 14, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 6, + "generic": 11, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.01660184191556402, + 0.00033125930304978013 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "539/4", + "-> 135 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [134.750000, 135.0) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(3,3)) = 135 EXACTLY" + }, + { + "a": 4, + "b": 2, + "len": 26, + "theta_d": "45", + "gamma_d": "135", + "corner": [ + "45/2", + "45/2" + ], + "n_distinct_diffs": 14, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 6, + "generic": 11, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.019384397213460858, + 0.00039091814364589084 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "539/4", + "-> 135 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [134.750000, 135.0) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(4,2)) = 135 EXACTLY" + }, + { + "a": 4, + "b": 3, + "len": 30, + "theta_d": "315/8", + "gamma_d": "1125/8", + "corner": [ + "45/2", + "135/8" + ], + "n_distinct_diffs": 16, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 6, + "generic": 13, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.02224763681635422, + 0.00044507936132665016 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "1123/8", + "-> 1125/8 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [140.375000, 140.625) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(4,3)) = 1125/8 EXACTLY" + }, + { + "a": 4, + "b": 4, + "len": 34, + "theta_d": "36", + "gamma_d": "144", + "corner": [ + "45/2", + "27/2" + ], + "n_distinct_diffs": 18, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 6, + "generic": 15, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.02460177897663729, + 0.0004883217924147587 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "575/4", + "-> 144 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [143.750000, 144.0) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(4,4)) = 144 EXACTLY" + }, + { + "a": 5, + "b": 3, + "len": 34, + "theta_d": "36", + "gamma_d": "144", + "corner": [ + "18", + "18" + ], + "n_distinct_diffs": 18, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 6, + "generic": 15, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.026572470962002548, + 0.0005320163696000968 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "575/4", + "-> 144 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [143.750000, 144.0) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(5,3)) = 144 EXACTLY" + }, + { + "a": 5, + "b": 4, + "len": 38, + "theta_d": "162/5", + "gamma_d": "738/5", + "corner": [ + "18", + "72/5" + ], + "n_distinct_diffs": 20, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 6, + "generic": 17, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.03002575610360081, + 0.0005968217762120087 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "2947/20", + "-> 738/5 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [147.350000, 147.6) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(5,4)) = 738/5 EXACTLY" + }, + { + "a": 5, + "b": 5, + "len": 42, + "theta_d": "30", + "gamma_d": "150", + "corner": [ + "18", + "12" + ], + "n_distinct_diffs": 22, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 6, + "generic": 19, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.03294444957364662, + 0.0006501659274507077 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "599/4", + "-> 150 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [149.750000, 150.0) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(5,5)) = 150 EXACTLY" + }, + { + "a": 6, + "b": 6, + "len": 50, + "theta_d": "180/7", + "gamma_d": "1080/7", + "corner": [ + "15", + "75/7" + ], + "n_distinct_diffs": 26, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 6, + "generic": 23, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.04159840724181363, + 0.00081621327276582 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "4313/28", + "-> 1080/7 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [154.035714, 154.28571428571428) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(6,6)) = 1080/7 EXACTLY" + }, + { + "a": 7, + "b": 7, + "len": 58, + "theta_d": "45/2", + "gamma_d": "315/2", + "corner": [ + "90/7", + "135/14" + ], + "n_distinct_diffs": 30, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 6, + "generic": 29, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.05050831632093855, + 0.0009853753738819915 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "629/4", + "-> 315/2 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [157.250000, 157.5) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(7,7)) = 315/2 EXACTLY" + }, + { + "a": 8, + "b": 8, + "len": 66, + "theta_d": "20", + "gamma_d": "160", + "corner": [ + "45/4", + "35/4" + ], + "n_distinct_diffs": 34, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 6, + "generic": 35, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.059631514908079275, + 0.001156812645383809 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "639/4", + "-> 160 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [159.750000, 160.0) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(8,8)) = 160 EXACTLY" + }, + { + "a": 10, + "b": 9, + "len": 78, + "theta_d": "171/10", + "gamma_d": "1629/10", + "corner": [ + "9", + "81/10" + ], + "n_distinct_diffs": 40, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 5, + "generic": 52, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.074153501392493, + 0.0014308051446241254 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "3253/20", + "-> 1629/10 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [162.650000, 162.9) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(10,9)) = 1629/10 EXACTLY" + }, + { + "a": 12, + "b": 12, + "len": 98, + "theta_d": "180/13", + "gamma_d": "2160/13", + "corner": [ + "15/2", + "165/26" + ], + "n_distinct_diffs": 50, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 5, + "generic": 102, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.09777230517601831, + 0.0018559367472987809 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "8627/52", + "-> 2160/13 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [165.903846, 166.15384615384616) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(12,12)) = 2160/13 EXACTLY" + }, + { + "a": 6, + "b": 3, + "len": 38, + "theta_d": "135/4", + "gamma_d": "585/4", + "corner": [ + "15", + "75/4" + ], + "n_distinct_diffs": 20, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 6, + "generic": 17, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.02998385610623533, + 0.0006003131386775884 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "146", + "-> 585/4 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [146.000000, 146.25) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(6,3)) = 585/4 EXACTLY" + }, + { + "a": 6, + "b": 1, + "len": 30, + "theta_d": "105/2", + "gamma_d": "255/2", + "corner": [ + "15", + "75/2" + ], + "n_distinct_diffs": 16, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 4, + "generic": 13, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.018997614198509538, + 0.0003860905611792198 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "509/4", + "-> 255/2 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [127.250000, 127.5) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(6,1)) = 255/2 EXACTLY" + }, + { + "a": 8, + "b": 2, + "len": 42, + "theta_d": "75/2", + "gamma_d": "285/2", + "corner": [ + "45/4", + "105/4" + ], + "n_distinct_diffs": 22, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 5, + "generic": 19, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.028046729753725685, + 0.0005645577132145085 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "569/4", + "-> 285/2 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [142.250000, 142.5) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(8,2)) = 285/2 EXACTLY" + }, + { + "a": 8, + "b": 1, + "len": 38, + "theta_d": "405/8", + "gamma_d": "1035/8", + "corner": [ + "45/4", + "315/8" + ], + "n_distinct_diffs": 20, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 4, + "generic": 17, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.020557916642400365, + 0.0004170429733560965 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "1033/8", + "-> 1035/8 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [129.125000, 129.375) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(8,1)) = 1035/8 EXACTLY" + }, + { + "a": 7, + "b": 1, + "len": 34, + "theta_d": "360/7", + "gamma_d": "900/7", + "corner": [ + "90/7", + "270/7" + ], + "n_distinct_diffs": 18, + "rho": 2, + "leaves": { + "N2": 0, + "N4": 0, + "N1_deriv": 1, + "N1_bisect": 4, + "generic": 15, + "tau_r": 1 + }, + "t0": "1/4", + "ok": true, + "float_widths_at_t0/2_t0/97": [ + 0.01987993020830281, + 0.00040368223407805104 + ], + "float_alive_crosscheck": true, + "gamma_range_alive": [ + "3593/28", + "-> 900/7 (open)" + ], + "conclusion": "alive for all t in (0, 1/4]: gamma in [128.321429, 128.57142857142858) all realized; with the parametric necessity theorem (003 + 005 extended case tree): death(W(7,1)) = 900/7 EXACTLY" + } +] \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_bridge.json b/problems/billiards-triangles/data/psk_bridge.json new file mode 100644 index 0000000..bf0843a --- /dev/null +++ b/problems/billiards-triangles/data/psk_bridge.json @@ -0,0 +1,528 @@ +{ + "results": { + "W(2,1)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(2,2)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(3,2)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(3,3)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(4,2)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(4,3)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(4,4)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(5,3)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(5,4)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(5,5)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(6,6)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(7,7)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(8,8)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(10,9)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(12,12)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(6,3)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(6,1)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(8,2)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(8,1)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(7,1)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(11,2)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(14,4)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(17,9)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(1,1)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(40,1)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(40,40)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(37,13)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(25,24)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(15,15)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(23,23)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(28,1)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(19,2)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(24,23)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(19,17)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(10,4)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(38,29)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(22,14)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(9,6)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(6,2)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + }, + "W(13,8)": { + "points": 2, + "ok": true, + "detail": { + "mu": true, + "w": true, + "v2": true, + "delta_sq": true, + "tau_axis": true, + "glide_action": true, + "ok": true + } + } + }, + "seed": 20260738, + "n_members": 40, + "ok": true, + "meaning": "direct reflection-composition of the half word agrees EXACTLY (off-torus pair algebra, my own maps) with 005's closed forms mu, w, v2 and glide facts at random rational points" +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_casetree_W10_1.json b/problems/billiards-triangles/data/psk_casetree_W10_1.json new file mode 100644 index 0000000..5c9b1c6 --- /dev/null +++ b/problems/billiards-triangles/data/psk_casetree_W10_1.json @@ -0,0 +1,13 @@ +{ + "a": 10, + "b": 1, + "theta_d": 49.5, + "seed": 20260738, + "n_points": 44660, + "n_hits": 0, + "hits": [], + "n_nearhits_below_margin": 0, + "nearhits": [], + "positive_control_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_casetree_W11_2.json b/problems/billiards-triangles/data/psk_casetree_W11_2.json new file mode 100644 index 0000000..8d04be3 --- /dev/null +++ b/problems/billiards-triangles/data/psk_casetree_W11_2.json @@ -0,0 +1,13 @@ +{ + "a": 11, + "b": 2, + "theta_d": 35.45454545454545, + "seed": 20260738, + "n_points": 44550, + "n_hits": 0, + "hits": [], + "n_nearhits_below_margin": 0, + "nearhits": [], + "positive_control_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_casetree_W12_1.json b/problems/billiards-triangles/data/psk_casetree_W12_1.json new file mode 100644 index 0000000..9c79cac --- /dev/null +++ b/problems/billiards-triangles/data/psk_casetree_W12_1.json @@ -0,0 +1,13 @@ +{ + "a": 12, + "b": 1, + "theta_d": 48.75, + "seed": 20260738, + "n_points": 44760, + "n_hits": 0, + "hits": [], + "n_nearhits_below_margin": 0, + "nearhits": [], + "positive_control_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_casetree_W12_2.json b/problems/billiards-triangles/data/psk_casetree_W12_2.json new file mode 100644 index 0000000..1239a9d --- /dev/null +++ b/problems/billiards-triangles/data/psk_casetree_W12_2.json @@ -0,0 +1,13 @@ +{ + "a": 12, + "b": 2, + "theta_d": 35.0, + "seed": 20260738, + "n_points": 44550, + "n_hits": 0, + "hits": [], + "n_nearhits_below_margin": 0, + "nearhits": [], + "positive_control_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_casetree_W15_2.json b/problems/billiards-triangles/data/psk_casetree_W15_2.json new file mode 100644 index 0000000..9491764 --- /dev/null +++ b/problems/billiards-triangles/data/psk_casetree_W15_2.json @@ -0,0 +1,13 @@ +{ + "a": 15, + "b": 2, + "theta_d": 34.0, + "seed": 20260738, + "n_points": 44660, + "n_hits": 0, + "hits": [], + "n_nearhits_below_margin": 0, + "nearhits": [], + "positive_control_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_casetree_W20_1.json b/problems/billiards-triangles/data/psk_casetree_W20_1.json new file mode 100644 index 0000000..13a3396 --- /dev/null +++ b/problems/billiards-triangles/data/psk_casetree_W20_1.json @@ -0,0 +1,13 @@ +{ + "a": 20, + "b": 1, + "theta_d": 47.25, + "seed": 20260738, + "n_points": 45121, + "n_hits": 0, + "hits": [], + "n_nearhits_below_margin": 0, + "nearhits": [], + "positive_control_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_casetree_W30_3.json b/problems/billiards-triangles/data/psk_casetree_W30_3.json new file mode 100644 index 0000000..c0f5e98 --- /dev/null +++ b/problems/billiards-triangles/data/psk_casetree_W30_3.json @@ -0,0 +1,13 @@ +{ + "a": 30, + "b": 3, + "theta_d": 24.75, + "seed": 20260738, + "n_points": 44930, + "n_hits": 0, + "hits": [], + "n_nearhits_below_margin": 0, + "nearhits": [], + "positive_control_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_casetree_W6_1.json b/problems/billiards-triangles/data/psk_casetree_W6_1.json new file mode 100644 index 0000000..b012aea --- /dev/null +++ b/problems/billiards-triangles/data/psk_casetree_W6_1.json @@ -0,0 +1,13 @@ +{ + "a": 6, + "b": 1, + "theta_d": 52.5, + "seed": 20260738, + "n_points": 44450, + "n_hits": 0, + "hits": [], + "n_nearhits_below_margin": 0, + "nearhits": [], + "positive_control_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_casetree_W8_1.json b/problems/billiards-triangles/data/psk_casetree_W8_1.json new file mode 100644 index 0000000..1246780 --- /dev/null +++ b/problems/billiards-triangles/data/psk_casetree_W8_1.json @@ -0,0 +1,13 @@ +{ + "a": 8, + "b": 1, + "theta_d": 50.625, + "seed": 20260738, + "n_points": 44550, + "n_hits": 0, + "hits": [], + "n_nearhits_below_margin": 0, + "nearhits": [], + "positive_control_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_casetree_W9_1.json b/problems/billiards-triangles/data/psk_casetree_W9_1.json new file mode 100644 index 0000000..4c957f2 --- /dev/null +++ b/problems/billiards-triangles/data/psk_casetree_W9_1.json @@ -0,0 +1,13 @@ +{ + "a": 9, + "b": 1, + "theta_d": 50.0, + "seed": 20260738, + "n_points": 44584, + "n_hits": 0, + "hits": [], + "n_nearhits_below_margin": 0, + "nearhits": [], + "positive_control_fires": true, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_death_W61.json b/problems/billiards-triangles/data/psk_death_W61.json new file mode 100644 index 0000000..9a771ab --- /dev/null +++ b/problems/billiards-triangles/data/psk_death_W61.json @@ -0,0 +1,22 @@ +{ + "command": "death", + "a": 6, + "b": 1, + "len": 30, + "uniform": 800, + "iters": 44, + "full": true, + "death_between": [ + 127.4999999999946, + 127.49999999999466 + ], + "predicted": 127.5, + "measured_minus_predicted": -5.3717030823463574e-12, + "last_alive": { + "alpha": 14.999999999998181, + "beta": 37.50000000000722, + "width": 1.0196288258157438e-12, + "best_width_alpha_ge_beta": -5.247384812717352e-06, + "best_width_alpha_lt_beta": 1.0196288258157438e-12 + } +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_death_W63.json b/problems/billiards-triangles/data/psk_death_W63.json new file mode 100644 index 0000000..1c88ff5 --- /dev/null +++ b/problems/billiards-triangles/data/psk_death_W63.json @@ -0,0 +1,22 @@ +{ + "command": "death", + "a": 6, + "b": 3, + "len": 38, + "uniform": 800, + "iters": 44, + "full": true, + "death_between": [ + 146.2499999999963, + 146.24999999999636 + ], + "predicted": 146.25, + "measured_minus_predicted": -3.666400516522117e-12, + "last_alive": { + "alpha": 14.999999999998181, + "beta": 18.750000000005514, + "width": 1.0125233984581428e-12, + "best_width_alpha_ge_beta": -6.106605024812764e-06, + "best_width_alpha_lt_beta": 1.0125233984581428e-12 + } +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_death_W71.json b/problems/billiards-triangles/data/psk_death_W71.json new file mode 100644 index 0000000..ba4d465 --- /dev/null +++ b/problems/billiards-triangles/data/psk_death_W71.json @@ -0,0 +1,22 @@ +{ + "command": "death", + "a": 7, + "b": 1, + "len": 34, + "uniform": 800, + "iters": 44, + "full": true, + "death_between": [ + 128.57142857142347, + 128.57142857142352 + ], + "predicted": 128.57142857142858, + "measured_minus_predicted": -5.087485988042317e-12, + "last_alive": { + "alpha": 12.857142857141039, + "beta": 38.57142857143549, + "width": 1.0063061495202419e-12, + "best_width_alpha_ge_beta": -3.954839910162811e-08, + "best_width_alpha_lt_beta": 1.0063061495202419e-12 + } +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_death_W81.json b/problems/billiards-triangles/data/psk_death_W81.json new file mode 100644 index 0000000..5bdf17f --- /dev/null +++ b/problems/billiards-triangles/data/psk_death_W81.json @@ -0,0 +1,22 @@ +{ + "command": "death", + "a": 8, + "b": 1, + "len": 38, + "uniform": 800, + "iters": 44, + "full": true, + "death_between": [ + 129.37499999999505, + 129.3749999999951 + ], + "predicted": 129.375, + "measured_minus_predicted": -4.916955731459893e-12, + "last_alive": { + "alpha": 11.249999999998181, + "beta": 39.375000000006764, + 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b/problems/billiards-triangles/data/psk_lemma.json new file mode 100644 index 0000000..f14001e --- /dev/null +++ b/problems/billiards-triangles/data/psk_lemma.json @@ -0,0 +1,7 @@ +{ + "seed": 20260738, + "n_checks": 14183, + "violations": [], + "n_violations": 0, + "ok": true +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_ring.json b/problems/billiards-triangles/data/psk_ring.json new file mode 100644 index 0000000..1dc4bb5 --- /dev/null +++ b/problems/billiards-triangles/data/psk_ring.json @@ -0,0 +1,20 @@ +{ + "checks": { + "rotA_block": true, + "rotB_block": true, + "rotB_fixes_B": true, + "u_antilinear": true, + "mu_matches_005": true, + "w_matches_005": true, + "mu_eq_delta_sq": true, + "tau_parallel_axis": true, + "I1": true, + "I2": true, + "I3": true, + "I4": true, + "D1": true, + "D2": true + }, + "ok": true, + "seed": 20260738 +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_segment.json 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"width_iv": [ + 4.023422923578153e-05, + 4.023422923578153e-05 + ], + "float_width": 0.0007024143447091902 + }, + { + "a": 12, + "b": 12, + "t": "1/1048576", + "gamma_minus_gamma_d": -9.5367431640625e-07, + "certified_alive": true, + "width_iv": [ + 3.9258131186314934e-08, + 3.9258131186314934e-08 + ], + "float_width": 6.854683793910965e-07 + }, + { + "a": 12, + "b": 12, + "t": "1/1073741824", + "gamma_minus_gamma_d": -9.313225746154785e-10, + "certified_alive": true, + "width_iv": [ + 3.833798716164971e-11, + 3.833798716164971e-11 + ], + "float_width": 6.695413112822735e-10 + } + ], + "ok": true, + "seed": 20260738, + "meaning": "full 2n-gate corridor built by my own interval unfolding (no glide reduction), certified positive width at every sampled t on 005's universal segment incl. t = 2^-30 (gamma within 1e-9 of gamma_d)" +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_threshold.json b/problems/billiards-triangles/data/psk_threshold.json new file mode 100644 index 0000000..7d917e0 --- /dev/null +++ b/problems/billiards-triangles/data/psk_threshold.json @@ -0,0 +1,5 @@ +{ + "ok": true, + "n": 2000, + "seed": 20260738 +} \ No newline at end of file diff --git a/problems/billiards-triangles/data/psk_trigaudit.json b/problems/billiards-triangles/data/psk_trigaudit.json new file mode 100644 index 0000000..f8b5870 --- /dev/null +++ b/problems/billiards-triangles/data/psk_trigaudit.json @@ -0,0 +1,7 @@ +{ + "n_checks": 596, + "worst_point_width": 4.990496198076935e-16, + "ok": true, + "seed": 20260738, + "note": "earlier FAILED run used an UNREDUCED reference argument (my stack has no mod-360 reduction and its series is invalid past ~660 deg) -- an artifact of my audit, not of plaw_suffice; see record" +} \ No newline at end of file diff --git a/problems/billiards-triangles/explore/design_certify.py b/problems/billiards-triangles/explore/design_certify.py new file mode 100644 index 0000000..ac23a6c --- /dev/null +++ b/problems/billiards-triangles/explore/design_certify.py @@ -0,0 +1,125 @@ +#!/usr/bin/env python3 +"""Attempt 006: exact alive certificate for an arbitrary word at a target +gamma, with a certified rational gamma bracket. + +Same certification pattern as deathlaw_exact.py (003), generalized from the +W(a,b) family to any even bounce word and any target arc: + + 1. float search for the widest apex on the arc gamma = --gamma, using + design_neighbours' whole-arc sampler PLUS deep geometric accumulation + at the caller-supplied corner --corner-alpha (the windows this attempt + hunts pinch onto corners like alpha = beta = 22.5); + 2. snap the apex to denominator 2^--bits; corridor positivity is then + checked in EXACT Fraction arithmetic (skeptic_orbit.corridor, 002's + independent implementation); + 3. the periodic orbit is re-derived by exact rational simulation + (skeptic_orbit.verify_orbit -- independent of any unfolding); + 4. gamma is certified strictly inside the rational bracket + (--glo, --ghi) via exact rational cos^2 against interval enclosures + (deathlaw_exact.gamma_bounds_certified; single non-iterated interval + evaluations, tier-0 unfold.py interval cosine + rational pi). + +The result "word alive at a triangle with gamma in (glo, ghi), exactly" is +what this prints and writes. + +Usage example (the pinch-gap certificate of attempt 006): + python3 problems/billiards-triangles/explore/design_certify.py \ + --word 012121212020202012121212020202 \ + --gamma 135.02 --corner-alpha 22.5 \ + --glo 135000001/1000000 --ghi 135048/1000 \ + --out problems/billiards-triangles/data/design_cert_W43_gap.json + +Stdlib only, deterministic. +""" + +import argparse +import json +import math +import os +import sys +from fractions import Fraction as F + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +from design_neighbours import (apex_from_angles, reduced_width, # noqa: E402 + arc_alphas) +from deathlaw_exact import gamma_bounds_certified, snap # noqa: E402 +import skeptic_orbit # noqa: E402 + + +def best_apex(word, g, corner_alpha, uniform=400, depth=48): + th = 180.0 - g + als = arc_alphas(g, uniform=uniform, jmax=12, depth=14) + if corner_alpha is not None: + for d in range(1, depth + 1): + eps = 0.5 * 2.0 ** -d + for al in (corner_alpha - eps, corner_alpha + eps, + th - corner_alpha - eps, th - corner_alpha + eps): + if 0 < al < th: + als.append(al) + best, arg = -math.inf, None + doubled = (len(word) % 2 == 0 and (len(word) // 2) % 2 == 1 + and word == word[:len(word) // 2] * 2) + for al in als: + be = th - al + x, y = apex_from_angles(al, be) + if doubled: + v, _ = reduced_width(word, x, y) + else: + from skeptic_family import width as sk_w + v = sk_w(word, x, y) + if v > best: + best, arg = v, (al, be) + return best, arg + + +def main(): + ap = argparse.ArgumentParser() + ap.add_argument("--word", required=True) + ap.add_argument("--gamma", type=float, required=True) + ap.add_argument("--corner-alpha", type=float, default=None) + ap.add_argument("--glo", required=True, + help="rational lower gamma bound to certify (deg)") + ap.add_argument("--ghi", required=True, + help="rational upper gamma bound to certify (deg)") + ap.add_argument("--bits", type=int, default=48) + ap.add_argument("--out") + args = ap.parse_args() + w = tuple(int(c) for c in args.word) + glo, ghi = F(args.glo), F(args.ghi) + + bw, arg = best_apex(w, args.gamma, args.corner_alpha) + if not bw > 0: + raise SystemExit(f"no alive float apex on arc gamma={args.gamma}") + x, y = apex_from_angles(*arg) + xq, yq = snap(x, args.bits), snap(y, args.bits) + c = skeptic_orbit.corridor(w, (xq, yq)) + if c is None or not c[1] > c[0]: + raise SystemExit("exact corridor not positive after snap; " + "raise --bits or move --gamma") + ok, msg = skeptic_orbit.verify_orbit(w, (xq, yq)) + lo_ok, hi_ok = gamma_bounds_certified(xq, yq, glo, ghi) + res = { + "word": args.word, "len": len(w), + "apex": [str(xq), str(yq)], + "float_alpha_beta_deg": [arg[0], arg[1]], + "exact_corridor_width": str(c[1] - c[0]), + "exact_corridor_width_float": float(c[1] - c[0]), + "orbit_verified_by_independent_simulation": bool(ok), + "orbit_msg": str(msg), + "certified_gamma_bracket_deg": [str(glo), str(ghi)], + "gamma_gt_glo_certified": bool(lo_ok), + "gamma_lt_ghi_certified": bool(hi_ok), + "conclusion": (f"word alive (exact positive corridor + independent " + f"orbit simulation) at a triangle with gamma in " + f"({glo}, {ghi}) deg, certified" + if (ok and lo_ok and hi_ok) else "INCOMPLETE"), + } + if args.out: + json.dump(res, open(args.out, "w"), indent=1) + print(json.dumps(res, indent=1)) + sys.exit(0 if (ok and lo_ok and hi_ok) else 1) + + +if __name__ == "__main__": + main() diff --git a/problems/billiards-triangles/explore/design_exact135.py b/problems/billiards-triangles/explore/design_exact135.py new file mode 100644 index 0000000..4962e82 --- /dev/null +++ b/problems/billiards-triangles/explore/design_exact135.py @@ -0,0 +1,131 @@ +#!/usr/bin/env python3 +"""Attempt 006: exact certificate of a stable periodic orbit at a triangle +whose obtuse angle is EXACTLY 135 degrees (gamma = 3 pi / 4). + +The apexes seeing AB = [(0,0),(1,0)] at angle 135 deg form the open arc of +the circle (2x-1)^2 + (2y+1)^2 = 2 with y > 0 (inscribed-angle theorem; +proven here directly from the dot product, no trigonometry: for C = (x,y), +cos(gamma) = -1/sqrt(2) <=> CA.CB < 0 and 2 (CA.CB)^2 = |CA|^2 |CB|^2). +The circle has the rational point (x,y) = (1,0) (vertex B), hence a full +rational parametrization by the slope t of chords through it: + + s = 2(1-t)/(1+t^2), x = 1 - s/2, y = t s / 2. + +So rational apexes are DENSE on the 135-arc. For a word alive on that arc +we pick t rational near the float argmax, and verify EXACTLY (Fractions): + + (E1) the apex lies on the arc: y > 0 and CA.CB < 0 and + 2 (CA.CB)^2 == |CA|^2 |CB|^2 -- so gamma = 135 deg exactly; + (E2) the word's corridor at that apex has positive width + (skeptic_orbit.corridor, 002's independent exact implementation); + (E3) the periodic orbit is re-derived by exact rational billiard + simulation (skeptic_orbit.verify_orbit -- no unfolding). + +Positive corridor width is an open condition, so the orbit is stable under +perturbation of the triangle; and a rational apex on this arc generically +has alpha, beta irrational multiples of pi (not checked, not claimed). + +Usage: + python3 .../design_exact135.py --word [--denom-bits B] [--out F] + +Stdlib only, deterministic. +""" + +import argparse +import json +import math +import os +import sys +from fractions import Fraction as F + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +from design_neighbours import reduced_width # noqa: E402 +from skeptic_family import width as sk_width # noqa: E402 +import skeptic_orbit # noqa: E402 + + +def apex_from_t(t): + """Rational apex on the gamma = 135 deg arc, from rational chord slope + t in (0, 1) (t -> 0+ degenerates to vertex A, t -> 1- to vertex B).""" + s = 2 * (1 - t) / (1 + t * t) + return 1 - s / 2, t * s / 2 + + +def on_arc_exact(x, y): + """(E1): gamma = 135 deg exactly at apex (x, y). Pure Fraction check.""" + if not y > 0: + return False + dot = (-x) * (1 - x) + y * y + if not dot < 0: + return False + ca2 = x * x + y * y + cb2 = (1 - x) ** 2 + y * y + return 2 * dot * dot == ca2 * cb2 + + +def main(): + ap = argparse.ArgumentParser() + ap.add_argument("--word", required=True) + ap.add_argument("--denom-bits", type=int, default=40, + help="t is snapped to denominator 2^bits") + ap.add_argument("--out") + args = ap.parse_args() + w = tuple(int(c) for c in args.word) + + # float hunt for the widest apex ON the arc, parametrized by t + doubled = (len(w) % 2 == 0 and (len(w) // 2) % 2 == 1 + and w == w[:len(w) // 2] * 2) + def fwidth(x, y): + return reduced_width(w, x, y)[0] if doubled else sk_width(w, x, y) + best_t, best_v = None, -math.inf + N = 4000 + for k in range(1, N): + t = k / N + x, y = apex_from_t(t) + v = fwidth(x, y) + if v > best_v: + best_v, best_t = v, t + # golden-section-ish refinement by trisection + lo, hi = best_t - 1.0 / N, best_t + 1.0 / N + for _ in range(60): + t1, t2 = lo + (hi - lo) / 3, hi - (hi - lo) / 3 + v1 = fwidth(*apex_from_t(t1)) + v2 = fwidth(*apex_from_t(t2)) + if v1 < v2: + lo = t1 + else: + hi = t2 + best_t = 0.5 * (lo + hi) + if not best_v > 0: + raise SystemExit("word has no positive float corridor on the arc") + + tq = F(round(best_t * 2 ** args.denom_bits), 2 ** args.denom_bits) + xq, yq = apex_from_t(tq) + assert on_arc_exact(xq, yq), "exact on-arc check failed" + c = skeptic_orbit.corridor(w, (xq, yq)) + if c is None or not c[1] > c[0]: + raise SystemExit("exact corridor not positive at the snapped apex") + ok, msg = skeptic_orbit.verify_orbit(w, (xq, yq)) + res = { + "word": args.word, "len": len(w), + "t": str(tq), + "apex": [str(xq), str(yq)], + "gamma_exactly_135_deg": True, + "on_arc_identity": "2*(CA.CB)^2 == |CA|^2*|CB|^2 and CA.CB < 0, exact", + "exact_corridor_width_float": float(c[1] - c[0]), + "exact_corridor_width": str(c[1] - c[0]), + "orbit_verified_by_independent_simulation": bool(ok), + "orbit_msg": str(msg), + "conclusion": (f"stable periodic orbit (word length {len(w)}) " + f"certified at a rational-apex triangle with obtuse " + f"angle EXACTLY 135 deg" if ok else "INCOMPLETE"), + } + if args.out: + json.dump(res, open(args.out, "w"), indent=1) + print(json.dumps(res, indent=1)) + sys.exit(0 if ok else 1) + + +if __name__ == "__main__": + main() diff --git a/problems/billiards-triangles/explore/design_factor.py b/problems/billiards-triangles/explore/design_factor.py new file mode 100644 index 0000000..5e12e06 --- /dev/null +++ b/problems/billiards-triangles/explore/design_factor.py @@ -0,0 +1,331 @@ +#!/usr/bin/env python3 +"""Attempt 006: symbolic binding-factor extraction for arbitrary doubled +words (the design inversion of 003's death-law machinery). + +Built on deathlaw_symbolic's division-free Laurent ring +Q(i)[e^{i alpha/2}, e^{i beta/2}]. For any doubled odd word w = u^2 this +tool computes the glide data (axis direction delta, offset m, half-gate +endpoint projections p(z) = Im(conj(delta) z)) exactly, then FACTORS the +binding differences p(z) - m and p(z) - p(z') into elementary factors + + s(p,q) = sin((p alpha + q beta)/2), c(p,q) = cos((p alpha + q beta)/2) + +by numeric candidate detection followed by EXACT division in the ring +(division by the binomial M -+ M^{-1}, M = A^p B^q, with a termination +cap; a reported factorization is exact: quotient times factors times a +monomial unit reproduces the polynomial, re-verified by multiplication). + +The death/birth angle of a family then reads off the factor list: an alive +window edge sits where a binding factor vanishes, e.g. c(2a,0) = cos(a +alpha) vanishing at alpha = 90/a is W(a,b)'s death edge (003's I2). + +Subcommands: + selftest re-derive 003's I2/I3 factorizations for W(3,3) + bind --word W --gamma G binding gates at the float argmax on arc G, + and the exact factorization of each binding + difference + birth --a A --b B factor the differences binding at the BIRTH + corner of W(a,b) (003 Lead 5) + +Stdlib only, deterministic. +""" + +import argparse +import json +import math +import os +import sys +from fractions import Fraction as F + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +import deathlaw_symbolic as DS # noqa: E402 +from deathlaw_symbolic import (padd, psub, pmul, pscale, pconj, pim, # noqa + pclean, mono, peval, ONE, I, HALF_OVER_I) +from design_neighbours import apex_from_angles, arc_alphas # noqa: E402 + + +def glide_data_word(word): + """deathlaw_symbolic.glide_data generalized to any doubled odd word.""" + n = len(word) + half = word[:n // 2] + assert n % 2 == 0 and word == half * 2 and len(half) % 2 == 1 + Uh = DS.unfold_sym(half) + orient, mu, w = Uh["maps"][-1] + assert orient == -1 + assert len(mu) == 1, f"glide linear part not a monomial: {mu}" + (mm, nn), c = next(iter(mu.items())) + assert mm % 2 == 0 and nn % 2 == 0 + if c == (F(1), F(0)): + delta = mono(mm // 2, nn // 2) + elif c == (F(-1), F(0)): + delta = mono(mm // 2, nn // 2, I) + else: + raise AssertionError(f"unexpected glide coefficient {c}") + dconj = pconj(delta) + tau = padd(w, pmul(mu, pconj(w))) + assert pclean(pim(pmul(dconj, tau))) == {}, "tau not parallel to axis" + m = pscale(pim(pmul(dconj, psub(w, pscale(tau, (F(1, 2), F(0)))))), + (F(1, 2), F(0))) + endpoints = [] # (gate index, endpoint label, projection ring element) + for k, (u, v) in enumerate(Uh["gates"]): + endpoints.append((k + 1, "u", pim(pmul(dconj, u)))) + endpoints.append((k + 1, "v", pim(pmul(dconj, v)))) + return {"delta": delta, "m": m, "tau": tau, "endpoints": endpoints, + "gates": Uh["gates"], "half": half, "word": word} + + +# ----------------------------------------------------- exact factorization + +def divide_binomial(P, p, q, sign): + """Exact division of Laurent poly P by D = A^p B^q + sign * A^-p B^-q. + Returns quotient Q with P = Q*D, or None if not divisible (cap hit or + nonzero remainder).""" + if not P: + return None + D = padd(mono(p, q), mono(-p, -q, (F(sign), F(0)))) + Q = {} + R = dict(P) + cap = 4 * (len(P) + 8) * (abs(p) + abs(q) + 2) + for _ in range(cap): + if not R: + return Q + # lead term: max of phi = p*m + q*n, tie-break by exponent tuple + k = max(R, key=lambda mn: (p * mn[0] + q * mn[1], mn)) + c = R[k] + qk = (k[0] - p, k[1] - q) + # if the lead cannot be reduced further the division fails + if p * k[0] + q * k[1] <= p * qk[0] + q * qk[1]: + return None + Q[qk] = c + R = psub(R, pmul({qk: c}, D)) + return None + + +def elementary_candidates(P, pq_max=40, tol=1e-9, npts=4): + """Numeric candidates (p, q, sign) with D = M + sign*M^{-1} dividing P: + P must vanish on the locus (p alpha + q beta)/2 = 0 (mod pi) [sign -] + or = pi/2 (mod pi) [sign +]. Sampled at npts real points per locus.""" + # scale reference + import random + rng = random.Random(20260731) + ref = max(abs(peval(P, rng.uniform(0.3, 2.5), rng.uniform(0.3, 2.5))) + for _ in range(6)) + out = [] + seen = set() + for p in range(0, pq_max + 1): + for q in range(-pq_max, pq_max + 1): + if p == 0 and q <= 0: + continue + if math.gcd(p, abs(q)) != 1 and not (p == 0 or q == 0): + pass # non-primitive directions allowed: frequencies matter + for sign, off in ((-1, 0.0), (1, math.pi)): + key = (p, q, sign) + if key in seen: + continue + good = True + for t in range(npts): + al = 0.35 + 0.53 * t + # (p al + q be)/2 = off/2 (mod pi): pick k spread out + k = t - npts // 2 + if q != 0: + be = (off + 2 * math.pi * k - p * al) / q + else: + if p == 0: + good = False + break + al = (off + 2 * math.pi * k) / p + be = 0.7 + 0.31 * t + v = abs(peval(P, al, be)) + if v > tol * max(ref, 1e-30): + good = False + break + if good: + seen.add(key) + out.append((p, q, sign)) + return out + + +def factor_elementary(P, pq_max=40): + """Peel exact elementary factors off P greedily. Returns + (factors, cofactor) with P == unit-monomial-scaled product; each factor + reported as ('sin'|'cos', p, q) meaning sin/cos((p a + q b)/2). + Division is EXACT; the numeric step only proposes candidates. + The reconstruction P == c * prod * cofactor is re-verified exactly.""" + factors = [] + cur = dict(P) + changed = True + while changed and cur: + changed = False + cands = elementary_candidates(cur, pq_max) + # peel the largest factors first, so composite loci (e.g. sin(4b) + # = product of four half-angle factors) come out in maximal form + cands.sort(key=lambda t: -(t[0] * t[0] + t[1] * t[1])) + for (p, q, sign) in cands: + while True: + Q = divide_binomial(cur, p, q, sign) + if Q is None: + break + factors.append(("cos" if sign == 1 else "sin", p, q)) + cur = Q + changed = True + # exact reconstruction check + rec = {k: c for k, c in cur.items()} + for (kind, p, q) in factors: + sign = 1 if kind == "cos" else -1 + rec = pmul(rec, padd(mono(p, q), mono(-p, -q, (F(sign), F(0))))) + # rec should equal P (both are the raw binomial products; the sin/cos + # normalization constants (2i resp. 2) are carried by the caller) + assert pclean(psub(rec, P)) == {}, "reconstruction failed" + return factors, cur + + +def fmt_factors(factors, cofactor): + def ang(p, q): + parts = [] + if p: + parts.append(f"{p}a" if p != 1 else "a") + if q: + parts.append(f"{'+' if q > 0 else '-'}{abs(q)}b" + if abs(q) != 1 else ("+b" if q > 0 else "-b")) + return "(" + "".join(parts).lstrip("+") + ")/2" + s = " * ".join(f"{k}{ang(p, q)}" for (k, p, q) in factors) + if cofactor and pclean(cofactor) not in ({}, ): + if len(cofactor) == 1 and list(cofactor)[0] == (0, 0): + c = cofactor[(0, 0)] + s += f" * ({c[0]}{'+' if c[1] >= 0 else ''}{c[1]}i)" + else: + s += f" * [{DS.trig_str(pre(cofactor)) if is_real(cofactor) else 'cofactor(' + str(len(cofactor)) + ' terms)'}]" + return s or "1" + + +def is_real(p): + return pclean(psub(p, pconj(p))) == {} + + +def pre(p): + return DS.pre(p) + + +# ------------------------------------------------------------- subcommands + +def binding_report(word, gamma, corner=None, out=None): + gd = glide_data_word(tuple(word)) + th = math.radians(180.0 - gamma) + # float argmax on the arc + best, arg = -math.inf, None + als = [math.radians(a) for a in arc_alphas(gamma, uniform=200, depth=14)] + if corner is not None: + for d in range(1, 46): + eps = 0.5 * 2.0 ** -d + for a in (corner - eps, corner + eps): + aa = math.radians(a) + if 0 < aa < th: + als.append(aa) + mval_p = gd["m"] + dconj = pconj(gd["delta"]) + for al in als: + be = th - al + m = peval(mval_p, al, be).real + lo, hi = -math.inf, math.inf + for (u, v) in gd["gates"]: + pu = peval(pim(pmul(dconj, u)), al, be).real + pv = peval(pim(pmul(dconj, v)), al, be).real + if pu > pv: + pu, pv = pv, pu + lo, hi = max(lo, pu), min(hi, pv) + wdt = min(hi, 2 * m - lo) - max(lo, 2 * m - hi) + if wdt > best: + best, arg = wdt, (al, be) + al, be = arg + m = peval(mval_p, al, be).real + rows = [] + for (k, lbl, pz) in gd["endpoints"]: + rows.append((k, lbl, pz, peval(pz, al, be).real - m)) + rows.sort(key=lambda r: abs(r[3])) + print(f"word {''.join(map(str, word))} gamma={gamma} " + f"argmax alpha={math.degrees(al):.6f} width={best:.3e}") + res = {"word": "".join(map(str, word)), "gamma": gamma, + "argmax_alpha_deg": math.degrees(al), "width": best, + "bindings": []} + seen_forms = set() + for (k, lbl, pz, dv) in rows[:6]: + diff = pclean(psub(pz, mval_p)) + key = json.dumps(sorted([(mn, (str(c[0]), str(c[1]))) + for mn, c in diff.items()])) + if key in seen_forms: + continue + seen_forms.add(key) + if not diff: + desc = "0 (endpoint ON the axis offset identically)" + else: + fac, cof = factor_elementary(diff) + desc = fmt_factors(fac, cof) + print(f" gate {k}{lbl}: p - m = {dv:+.3e} factors: {desc}") + res["bindings"].append({"gate": k, "end": lbl, "value": dv, + "factorization": desc}) + if out: + json.dump(res, open(out, "w"), indent=1) + return res + + +def cmd_bind(args): + w = tuple(int(c) for c in args.word) + binding_report(w, args.gamma, corner=args.corner, out=args.out) + + +def cmd_selftest(_args): + ok = True + # I2/I3 for W(3,3): p(A1) - m = cos(a alpha) sin((b+1) beta) sin(a+b) + # in half-angle vocabulary: cos((2a alpha)/2) sin((2(b+1) beta)/2) + # sin((2 alpha + 2 beta)/2), each carrying its normalization constant. + a, b = 3, 3 + word = ((0,) + (1, 2) * a + (0, 2) * b) * 2 + gd = glide_data_word(word) + # gate 2 endpoint v is A1 (R0(A)); find its projection minus m + k, lbl, pz = [e for e in gd["endpoints"] if e[0] == 2 and e[1] == "v"][0] + diff = pclean(psub(pz, gd["m"])) + fac, cof = factor_elementary(diff) + want = sorted([("cos", 2 * a, 0), ("sin", 0, 2 * (b + 1)), + ("sin", 2, 2)]) + got = sorted(fac) + # cofactor must be the constant scale: sin*sin -> (1/2i)^2, cos -> 1/2 + const_ok = (len(cof) == 1 and (0, 0) in cof) + good = got == want and const_ok + ok &= good + print(f"[I2 factorization W(3,3): {fmt_factors(fac, cof)}]", + "OK" if good else f"FAIL (got {got}, want {want})") + # numeric agreement of the factored form at a random angle + al, be = 0.41, 0.35 + lhs = peval(diff, al, be) + rhs = (math.cos(a * al) * math.sin((b + 1) * be) * math.sin(al + be)) + c = cof[(0, 0)] + cval = complex(c[0], c[1]) + # each sin binomial carries 2i, cos carries 2: + scale = cval * (2j) * (2j) * 2 / (2j) ** 0 # sin,sin,cos -> (2i)(2i)(2) + good = abs(lhs - rhs * (scale / abs(scale)) * abs(scale)) < 1e-9 or \ + abs(abs(lhs) - abs(rhs)) < 1e-9 + ok &= good + print(f"[I2 numeric magnitude {abs(lhs):.6f} vs {abs(rhs):.6f}]", + "OK" if good else "FAIL") + print("SELFTEST", "PASSED" if ok else "FAILED") + sys.exit(0 if ok else 1) + + +def main(): + ap = argparse.ArgumentParser() + sub = ap.add_subparsers(dest="cmd", required=True) + sub.add_parser("selftest").set_defaults(fn=cmd_selftest) + c = sub.add_parser("bind") + c.add_argument("--word", required=True) + c.add_argument("--gamma", type=float, required=True) + c.add_argument("--corner", type=float, default=None, + help="corner alpha in deg for deep accumulation") + c.add_argument("--out") + c.set_defaults(fn=cmd_bind) + args = ap.parse_args() + args.fn(args) + + +if __name__ == "__main__": + main() diff --git a/problems/billiards-triangles/explore/design_neighbours.py b/problems/billiards-triangles/explore/design_neighbours.py new file mode 100644 index 0000000..2b9a376 --- /dev/null +++ b/problems/billiards-triangles/explore/design_neighbours.py @@ -0,0 +1,479 @@ +#!/usr/bin/env python3 +"""Attempt 006: design hunt for stable-orbit word structures past 135 deg. + +Queue item 12 (exploratory track). The W(a,b) = (0(12)^a(02)^b)^2 family's +death law gamma_d = 180 - 90(a+b)/(a(b+1)) is now a per-member theorem +(003/004). This tool INVERTS the machinery: it enumerates the structure +class that the glide-reflection reduction applies to -- ALL doubled odd +half-words w = u^2 (any odd word u composed of reflections is +orientation-reversing; u^2 always has trivial rotation part, so every such +w is automatically a translation word) -- and hunts for members whose alive +windows reach angle regions the W family does not cover. + +Frontier semantics (pinned; see attempt 006 record): the W family itself is +alive far past 135 (certified members to gamma ~ 159.25 in 001). The actual +135-deg stall is: (i) no word of length <= 26 is known alive at any +gamma >= 135 (001's census, sample-bounded); (ii) the pinch gap +(135.000, 135.0486) between death(W(3,3)) = death(W(4,2)) = 135 and +birth(W(4,3)) ~ 135.0486 has NOTHING known alive at any length; (iii) above +135 the known alive set is the union of the W-family windows +[birth(a,b), gamma_d(a,b)] -- thin isolated windows, not area. A new +structure advances the frontier iff it is alive (a) inside a pinch gap, +(b) at gamma >= 135 with length <= 26, or (c) at some gamma with a shorter +word than the W family provides. + +Float screening only -- nothing here is a certificate; exact claims go +through the deathlaw_exact.py pattern. Corridor evaluation is by the +numeric glide reduction (half gates + axis offset m, the 003 reduction +confirmed by 004), cross-checked in --selftest against skeptic_family's +independent full-corridor width (they agree up to the exact factor |tau|). + +Subcommands: + selftest reduction vs full corridor; translation + facts; W-family windows reproduced + screen --max-half L [--klo K --khi K2] [--arcs ...] --out F + enumerate + screen a class: + mode all: every odd half word |u| <= L + mode s1: 2-alternating class + u = y0 y1 2 y2 2 ... yk 2 + window --word W --out F adaptive birth/death bisection of one word + triple --amax A --out F death table of the three-block family + u = 0(12)^a(02)^b(12)^c + gapscan --words F --gamma G dense scan of one arc for listed words + +Deterministic (no randomness in enumeration/screen paths). Stdlib only. +""" + +import argparse +import cmath +import json +import math +import os +import sys + +sys.path.insert(0, os.path.dirname(os.path.abspath(__file__))) +from skeptic_family import width as sk_width # noqa: E402 + +ENDS = ((1, 2), (2, 0), (0, 1)) + + +# ---------------------------------------------------------------- geometry + +def apex_from_angles(al_deg, be_deg): + ta = math.tan(math.radians(al_deg)) + tb = math.tan(math.radians(be_deg)) + return tb / (ta + tb), ta * tb / (ta + tb) + + +def half_glide_numeric(half, ax, ay): + """Unfold the HALF word; return (gates, mu, t) with the composed + orientation-reversing map z -> mu*conj(z) + t, or None if half even.""" + A, B, C = 0j, 1 + 0j, complex(ax, ay) + tri = [A, B, C] + gates = [] + for s in half: + i, j = ENDS[s] + u, v = tri[i], tri[j] + gates.append((u, v)) + d = v - u + e = d * d / (d.real * d.real + d.imag * d.imag) + tri = [u + e * (z - u).conjugate() for z in tri] + # composed map from vertex images: z -> mu conj(z) + t, A = 0 -> tri[0] + t = tri[0] + mu = (tri[1] - tri[0]) # / conj(B - A) = /1 + return gates, mu, t + + +def reduced_width(word, ax, ay): + """Corridor width of the doubled word u^2 via the glide reduction, + normalized (unit axis direction). Returns (width, |tau|). + width = min(hi, 2m-lo) - max(lo, 2m-hi) over half-gate projections. + Requires word = u*2 with |u| odd (checked).""" + n = len(word) + half = word[:n // 2] + if n % 2 or (n // 2) % 2 == 0 or word != half * 2: + raise ValueError("reduced_width needs a doubled odd half word") + gates, mu, t = half_glide_numeric(half, ax, ay) + tau = t + mu * t.conjugate() + at = abs(tau) + if at < 1e-14: + return -math.inf, at + delta = tau / at + dc = delta.conjugate() + m = (dc * t).imag / 2.0 + lo, hi = -math.inf, math.inf + for u, v in gates: + pu = (dc * u).imag + pv = (dc * v).imag + if pu > pv: + pu, pv = pv, pu + if pu > lo: + lo = pu + if pv < hi: + hi = pv + a = lo if lo > 2 * m - hi else 2 * m - hi + b = hi if hi < 2 * m - lo else 2 * m - lo + return b - a, at + + +# ---------------------------------------------------------------- sampling + +def arc_alphas(g, uniform=60, jmax=12, depth=10): + """Sample alphas on the arc gamma = g: uniform grid over the WHOLE arc + (both halves) plus geometric accumulation at every candidate window edge + alpha = 90/j and beta = 90/j (the W family's window edges sit at such + values; a new family's may too -- the uniform grid is the fallback).""" + th = 180.0 - g + als = [th * (k + 0.5) / uniform for k in range(uniform)] + for j in range(1, jmax + 1): + e = 90.0 / j + if e >= th: + continue + for d in range(1, depth + 1): + eps = 0.5 * 2.0 ** -d + for al in (e - eps, e + eps, th - e + eps, th - e - eps): + if 0 < al < th: + als.append(al) + return als + + +def best_alive(word, g, uniform=60, jmax=12, depth=10): + """(best reduced width, (alpha, beta) argmax) on the arc gamma = g.""" + th = 180.0 - g + best, arg = -math.inf, None + for al in arc_alphas(g, uniform, jmax, depth): + be = th - al + x, y = apex_from_angles(al, be) + v, _ = reduced_width(word, x, y) + if v > best: + best, arg = v, (al, be) + return best, arg + + +ALIVE_TOL = 1e-12 + + +# ------------------------------------------------------------- enumeration + +def canonical(word): + """Canonical form of the doubled word under cyclic rotation, reversal, + and the 0<->1 letter swap (mirror; gamma-coverage-equivalent).""" + n = len(word) + best = None + for w in (word, word[::-1]): + for sw in (False, True): + ww = tuple((1 - c if c < 2 else 2) for c in w) if sw else w + for r in range(n): + cand = ww[r:] + ww[:r] + if best is None or cand < best: + best = cand + return best + + +def is_power(u): + """True if u = v^k for some k >= 2 (then u^2 duplicates v^2's corridor + when |v| odd, or is a repeat of an even translation word).""" + n = len(u) + for p in range(1, n): + if n % p == 0 and u == u[:p] * (n // p): + return True + return False + + +def gen_all_odd(L): + """All odd half words u, |u| = L, over {0,1,2}, no equal adjacent + letters in u^2 (i.e. within u and u[-1] != u[0]), u not a proper power; + yields deduped doubled words via canonical().""" + seen = set() + stack = [(c,) for c in (0, 1, 2)] + while stack: + u = stack.pop() + if len(u) == L: + if u[-1] == u[0] or is_power(u): + continue + w = u * 2 + key = canonical(w) + if key not in seen: + seen.add(key) + yield w + continue + for c in (0, 1, 2): + if c != u[-1]: + stack.append(u + (c,)) + + +def gen_s1(k): + """The 2-alternating class: u = y0 y1 2 y2 2 ... yk 2, y in {0,1}^{k+1}, + y0 != y1 (one double cell, side 2 struck every other bounce elsewhere). + W(a,b) is y = 0 1^a 0^b. |u| = 2k+1, |w| = 4k+2.""" + seen = set() + for bits in range(2 ** (k + 1)): + y = [(bits >> i) & 1 for i in range(k + 1)] + if y[0] == y[1]: + continue + u = (y[0], y[1], 2) + for i in range(2, k + 1): + u += (y[i], 2) + if is_power(u): + continue + w = u * 2 + key = canonical(w) + if key not in seen: + seen.add(key) + yield w + + +# ------------------------------------------------------------- subcommands + +DEFAULT_ARCS = [135.02, 135.5, 136.5, 138.0, 140.0, 142.5, 145.0, + 148.0, 151.0, 155.0, 159.0] + + +def cmd_screen(args): + arcs = [float(g) for g in args.arcs.split(",")] if args.arcs \ + else DEFAULT_ARCS + words = [] + if args.mode == "all": + for L in range(3, args.max_half + 1, 2): + words.extend(gen_all_odd(L)) + else: + for k in range(args.klo, args.khi + 1): + words.extend(gen_s1(k)) + print(f"# {len(words)} canonical doubled words, arcs {arcs}") + hits, near = [], [] + for idx, w in enumerate(words): + best_g, best_v, best_arg = None, -math.inf, None + for g in arcs: + v, arg = best_alive(w, g, args.uniform, depth=args.depth) + if v > best_v: + best_v, best_g, best_arg = v, g, arg + if v > ALIVE_TOL: + hits.append({"word": "".join(map(str, w)), "len": len(w), + "gamma": g, "width": v, + "alpha": arg[0], "beta": arg[1]}) + if ALIVE_TOL >= best_v > -args.near_tol: + near.append({"word": "".join(map(str, w)), "len": len(w), + "gamma": best_g, "width": best_v, + "alpha": best_arg[0], "beta": best_arg[1]}) + if (idx + 1) % 500 == 0: + print(f"# {idx + 1}/{len(words)} screened, " + f"{len(hits)} arc-hits so far", flush=True) + # cross-check every hit against the independent full corridor + for h in hits: + w = tuple(int(c) for c in h["word"]) + x, y = apex_from_angles(h["alpha"], h["beta"]) + h["sk_width"] = sk_width(w, x, y) + h["sk_agrees"] = bool(h["sk_width"] > 0) + res = {"mode": args.mode, "max_half": args.max_half, + "klo": args.klo, "khi": args.khi, + "n_words": len(words), "arcs": arcs, + "uniform": args.uniform, "depth": args.depth, + "hits": hits, "near_misses": near} + json.dump(res, open(args.out, "w"), indent=1) + print(f"# wrote {args.out}: {len(hits)} arc-hits " + f"({len(set(h['word'] for h in hits))} words), " + f"{len(near)} near misses") + + +def measure_window(w, g_seed=None, uniform=400, depth=16, iters=44): + """Adaptive (birth, death) bisection for word w. Coarse walk first.""" + alive = [] + g = 90.25 + while g < 179.6: + if best_alive(w, g, uniform=120, depth=12)[0] > ALIVE_TOL: + alive.append(g) + g += 0.25 + if not alive and g_seed is not None: + if best_alive(w, g_seed, uniform, depth=depth)[0] > ALIVE_TOL: + alive = [g_seed] + if not alive: + return None + # death: bisect above the last alive arc + lo, hi = alive[-1], alive[-1] + 0.25 + arg_d = None + for _ in range(iters): + mid = 0.5 * (lo + hi) + v, ar = best_alive(w, mid, uniform, depth=depth) + if v > ALIVE_TOL: + lo, arg_d = mid, ar + else: + hi = mid + death = (lo, hi) + # birth: bisect below the first alive arc + lo2, hi2 = alive[0] - 0.25, alive[0] + arg_b = None + for _ in range(iters): + mid = 0.5 * (lo2 + hi2) + v, ar = best_alive(w, mid, uniform, depth=depth) + if v > ALIVE_TOL: + hi2, arg_b = mid, ar + else: + lo2 = mid + birth = (lo2, hi2) + return {"birth_between": birth, "death_between": death, + "alive_coarse": [alive[0], alive[-1]], + "death_argmax": arg_d, "birth_argmin": arg_b} + + +def cmd_window(args): + w = tuple(int(c) for c in args.word) + r = measure_window(w, uniform=args.uniform, iters=args.iters) + res = {"word": args.word, "len": len(w), "window": r} + if args.out: + json.dump(res, open(args.out, "w"), indent=1) + print(json.dumps(res)) + + +def cmd_triple(args): + rows = [] + for a in range(1, args.amax + 1): + for b in range(1, args.amax + 1): + for c in range(1, args.amax + 1): + u = (0,) + (1, 2) * a + (0, 2) * b + (1, 2) * c + if is_power(u): + continue + w = u * 2 + r = measure_window(w, uniform=args.uniform, iters=30) + d = None if r is None else 0.5 * (r["death_between"][0] + + r["death_between"][1]) + rows.append({"a": a, "b": b, "c": c, "len": len(w), + "window": r, "death_mid": d}) + print(f"({a},{b},{c}) len {len(w)}: " + f"{'DEAD everywhere sampled' if r is None else r}", + flush=True) + json.dump({"family": "u=0(12)^a(02)^b(12)^c doubled", + "amax": args.amax, "rows": rows}, + open(args.out, "w"), indent=1) + print(f"# wrote {args.out}") + + +def cmd_gapscan(args): + words = [tuple(int(c) for c in s) for s in + json.load(open(args.words))] if args.words.endswith(".json") \ + else [tuple(int(c) for c in args.words)] + out = [] + for w in words: + v, arg = best_alive(w, args.gamma, uniform=args.uniform, + depth=args.depth) + out.append({"word": "".join(map(str, w)), "gamma": args.gamma, + "best_width": v, + "alpha": None if arg is None else arg[0]}) + if v > ALIVE_TOL: + print(f"ALIVE {out[-1]}") + print(json.dumps({"n": len(words), + "n_alive": sum(1 for o in out + if o["best_width"] > ALIVE_TOL), + "max_width": max(o["best_width"] for o in out)})) + if args.out: + json.dump(out, open(args.out, "w"), indent=1) + + +def cmd_selftest(_args): + ok = True + # 1. doubled odd words are translation words: full-word composed map has + # linear part 1 (checked numerically at a generic apex) + test_words = [((0,) + (1, 2) * 2 + (0, 2) * 1) * 2, + (0, 1, 2) * 2, + (0, 2, 1, 2, 0, 2, 1) * 2] + for w in test_words: + A, B = 0j, 1 + 0j + tri = [A, B, complex(0.41, 0.093)] + lin, orient = 1 + 0j, 1 + for s in w: + i, j = ENDS[s] + u, v = tri[i], tri[j] + d = v - u + e = d * d / abs(d) ** 2 + tri = [u + e * (z - u).conjugate() for z in tri] + lin = e * lin.conjugate() + orient = -orient + good = orient == 1 and abs(lin - 1) < 1e-9 + ok &= good + print(f"[translation {''.join(map(str, w))}]", + "OK" if good else "FAIL") + # 2. reduced width * |tau| == full corridor width (skeptic_family), on + # the W family and random-ish other doubled words at many apexes + words = [((0,) + (1, 2) * a + (0, 2) * b) * 2 + for a, b in ((2, 1), (3, 3), (4, 2))] + words += [(0, 1, 2, 1, 2, 1, 2, 0, 2, 1, 2) * 2, + (0, 2, 1, 2, 1, 2, 0, 2, 0, 2, 1) * 2] + import random + rng = random.Random(20260731) + worst = 0.0 + n_alive_agree = 0 + for w in words: + for _ in range(120): + g = rng.uniform(91.0, 178.0) + th = 180.0 - g + al = rng.uniform(0.03 * th, 0.97 * th) + x, y = apex_from_angles(al, th - al) + rw, at = reduced_width(w, x, y) + fw = sk_width(w, x, y) + if math.isinf(rw) or math.isinf(fw): + continue + err = abs(rw * at - fw) / max(1.0, abs(fw)) + worst = max(worst, err) + if (rw > 1e-12) == (fw > 1e-12): + n_alive_agree += 1 + good = worst < 1e-8 + ok &= good + print(f"[reduced vs full corridor: rel err {worst:.2e}, " + f"verdict agreement {n_alive_agree}/600]", "OK" if good else "FAIL") + # 3. reproduce two W-family windows (001/003 values) + w33 = ((0,) + (1, 2) * 3 + (0, 2) * 3) * 2 + r = measure_window(w33, uniform=200, iters=30) + good = r is not None and abs(r["death_between"][0] - 135.0) < 2e-3 \ + and abs(r["birth_between"][1] - 127.5064) < 2e-2 + ok &= good + print(f"[W(3,3) window {r['birth_between'][1]:.4f}.." + f"{r['death_between'][0]:.4f} vs 127.5064..135.0]", + "OK" if good else "FAIL") + w42 = ((0,) + (1, 2) * 4 + (0, 2) * 2) * 2 + r = measure_window(w42, uniform=200, iters=30) + good = r is not None and abs(r["death_between"][0] - 135.0) < 2e-3 + ok &= good + print(f"[W(4,2) death {r['death_between'][0]:.4f} vs 135.0 " + f"(mirror-half window)]", "OK" if good else "FAIL") + print("SELFTEST", "PASSED" if ok else "FAILED") + sys.exit(0 if ok else 1) + + +def main(): + ap = argparse.ArgumentParser() + sub = ap.add_subparsers(dest="cmd", required=True) + sub.add_parser("selftest").set_defaults(fn=cmd_selftest) + c = sub.add_parser("screen") + c.add_argument("--mode", choices=["all", "s1"], default="all") + c.add_argument("--max-half", type=int, default=13) + c.add_argument("--klo", type=int, default=1) + c.add_argument("--khi", type=int, default=7) + c.add_argument("--arcs") + c.add_argument("--uniform", type=int, default=60) + c.add_argument("--depth", type=int, default=10) + c.add_argument("--near-tol", type=float, default=1e-3) + c.add_argument("--out", required=True) + c.set_defaults(fn=cmd_screen) + c = sub.add_parser("window") + c.add_argument("--word", required=True) + c.add_argument("--uniform", type=int, default=400) + c.add_argument("--iters", type=int, default=44) + c.add_argument("--out") + c.set_defaults(fn=cmd_window) + c = sub.add_parser("triple") + c.add_argument("--amax", type=int, default=3) + c.add_argument("--uniform", type=int, default=200) + c.add_argument("--out", required=True) + c.set_defaults(fn=cmd_triple) + c = sub.add_parser("gapscan") + c.add_argument("--words", required=True) + c.add_argument("--gamma", type=float, required=True) + c.add_argument("--uniform", type=int, default=2000) + c.add_argument("--depth", type=int, default=20) + c.add_argument("--out") + c.set_defaults(fn=cmd_gapscan) + args = ap.parse_args() + args.fn(args) + + +if __name__ == "__main__": + main() diff --git a/problems/billiards-triangles/explore/dsk_design_skeptic.py b/problems/billiards-triangles/explore/dsk_design_skeptic.py new file mode 100644 index 0000000..485507b --- /dev/null +++ b/problems/billiards-triangles/explore/dsk_design_skeptic.py @@ -0,0 +1,773 @@ +#!/usr/bin/env python3 +"""Attempt 007: skeptic review of 006 (design hunt past 135). Default REFUTE. + +Own verification stack, written for this review: + + * Exact unfolding by COMPOSED AFFINE MAPS with 2x2 rational matrices + (z -> M z + v), a third implementation path: 006's exact layer is + skeptic_orbit.py (002; reflected-vertex chains, complex-free tuples) and + 004's is complex composed maps. Corridor = intersection of gate-endpoint + projections onto n = (-tau_y, tau_x); same normalization as both, so + exact widths must agree DIGIT-FOR-DIGIT with recorded certificates. + * My own exact rational billiard simulator (own ray/segment solve with + cross products, strict interior bounce check, closure of position and + direction, struck-sequence check). + * Certified gamma brackets via deathlaw_skeptic's interval stack (004's + own Machin pi + Taylor enclosures -- NOT unfold.py, NOT 006's path). + * Own word enumeration + canonicalization (independent recount of 006's + universes), own samplers (seeded), own census-grid replica for the 001 + pinch-gap mechanism. + +Nothing from design_*.py is imported in any verdict path. + +Subcommands (all deterministic, seed 20260731): + selftest Fagnano + cross-implementation corridor agreement + certs re-verify the 5 exact certificates of 006 from data files + birthdepth W(4,3) birth vs sampler depth (is 135.0000076 a floor?) + censusgap replicate 001's census.py family sampler; window x-extent + birthlaw out-of-sample birth-law tests W(7,3), W(6,4); W(5,2) edges + counts independent recount of the screened universes; N1/N2 non-W + deadsample re-test a random subsample of "dead" words, own sampler/arcs + coverage formula-member aliveness at my own arc list + newmembers identities + case-tree search for (5,2), (6,2) via my ring + driver (deathlaw_skeptic ring); Lemma C via its lemmac +""" + +import argparse +import json +import math +import os +import random +import subprocess +import sys +from fractions import Fraction as F + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) + +SIDE_ENDS = {0: (1, 2), 1: (2, 0), 2: (0, 1)} +DATA = os.path.join(HERE, "..", "data") +SEED = 20260731 + + +# ------------------------------------------------------------ affine maps +# isometry T = (M, v): z -> M z + v, M a 2x2 tuple ((a,b),(c,d)), z tuple. + +def mat_apply(M, z): + return (M[0][0] * z[0] + M[0][1] * z[1], + M[1][0] * z[0] + M[1][1] * z[1]) + + +def t_apply(T, z): + M, v = T + w = mat_apply(M, z) + return (w[0] + v[0], w[1] + v[1]) + + +def mat_mul(A, B): + return ((A[0][0] * B[0][0] + A[0][1] * B[1][0], + A[0][0] * B[0][1] + A[0][1] * B[1][1]), + (A[1][0] * B[0][0] + A[1][1] * B[1][0], + A[1][0] * B[0][1] + A[1][1] * B[1][1])) + + +def t_compose(T, S): + """T o S: z -> T(S(z)).""" + M, v = T + N, w = S + Mw = mat_apply(M, w) + return (mat_mul(M, N), (Mw[0] + v[0], Mw[1] + v[1])) + + +def reflection(P, Q): + """Isometry reflecting across the line through P, Q.""" + dx, dy = Q[0] - P[0], Q[1] - P[1] + n2 = dx * dx + dy * dy + S = (((dx * dx - dy * dy) / n2, 2 * dx * dy / n2), + (2 * dx * dy / n2, (dy * dy - dx * dx) / n2)) + SP = mat_apply(S, P) + return (S, (P[0] - SP[0], P[1] - SP[1])) + + +def unfold(word, apex, exact): + """(gates, tau) or (None, reason). gates = [(u, v)] images of base sides. + exact=True: Fraction arithmetic, exact identity test.""" + one, zero = (F(1), F(0)) if exact else (1.0, 0.0) + V = ((zero, zero), (one, zero), apex) + T = (((one, zero), (zero, one)), (zero, zero)) + gates = [] + for s in word: + i, j = SIDE_ENDS[s] + gates.append((t_apply(T, V[i]), t_apply(T, V[j]))) + T = t_compose(T, reflection(V[i], V[j])) + M, v = T + if exact: + ident = M == ((F(1), F(0)), (F(0), F(1))) + else: + ident = all(abs(M[r][c] - (1.0 if r == c else 0.0)) < 1e-9 + for r in range(2) for c in range(2)) + if not ident: + return None, "linear part is not the identity" + if v == (zero, zero) or (not exact and abs(v[0]) + abs(v[1]) < 1e-14): + return None, "zero translation" + return gates, v + + +def corridor(word, apex, exact=True): + """(lo, hi, tau, gates) with projections onto n = (-tau_y, tau_x).""" + gates, tau = unfold(word, apex, exact) + if gates is None: + return None + nx, ny = -tau[1], tau[0] + lo = hi = None + for u, v in gates: + a = u[0] * nx + u[1] * ny + b = v[0] * nx + v[1] * ny + if a > b: + a, b = b, a + lo = a if lo is None else (a if a > lo else lo) + hi = b if hi is None else (b if b < hi else hi) + return lo, hi, tau, gates + + +def fwidth(word, x, y): + c = corridor(word, (x, y), exact=False) + return -math.inf if c is None else c[1] - c[0] + + +# ------------------------------------------------------------- simulator + +def reflect_vec(d, e): + """Reflect direction d across direction e.""" + n2 = e[0] * e[0] + e[1] * e[1] + t = (d[0] * e[0] + d[1] * e[1]) / n2 + return (2 * t * e[0] - d[0], 2 * t * e[1] - d[1]) + + +def my_verify_orbit(word, apex): + """Own exact simulation. (ok, msg).""" + c = corridor(word, apex, exact=True) + if c is None: + return False, "not a translation word" + lo, hi, tau, gates = c + if not lo < hi: + return False, "corridor not positive" + m = (lo + hi) / 2 + nx, ny = -tau[1], tau[0] + V = ((F(0), F(0)), (F(1), F(0)), (F(apex[0]), F(apex[1]))) + i, j = SIDE_ENDS[word[0]] + P, Q = V[i], V[j] + den = (Q[0] - P[0]) * nx + (Q[1] - P[1]) * ny + if den == 0: + return False, "first gate parallel to axis" + s = (m - (P[0] * nx + P[1] * ny)) / den + if not (0 < s < 1): + return False, "midline misses first gate interior" + p0 = (P[0] + s * (Q[0] - P[0]), P[1] + s * (Q[1] - P[1])) + d0 = reflect_vec(tau, (Q[0] - P[0], Q[1] - P[1])) + p, d, cur = p0, d0, word[0] + struck = [] + for _ in range(len(word)): + best = None + for sd in (0, 1, 2): + if sd == cur: + continue + a, b = SIDE_ENDS[sd] + U, W = V[a], V[b] + ex, ey = W[0] - U[0], W[1] - U[1] + det = d[0] * ey - d[1] * ex + if det == 0: + continue + rx, ry = U[0] - p[0], U[1] - p[1] + t = (rx * ey - ry * ex) / det + r = (rx * d[1] - ry * d[0]) / det + if t <= 0: + continue + if not (0 < r < 1): + continue + if best is None or t < best[0]: + best = (t, sd, (p[0] + t * d[0], p[1] + t * d[1])) + if best is None: + return False, "escaped or vertex hit" + _, cur, p = best + a, b = SIDE_ENDS[cur] + d = reflect_vec(d, (V[b][0] - V[a][0], V[b][1] - V[a][1])) + struck.append(cur) + if struck != list(word[1:]) + [word[0]]: + return False, "wrong bounce sequence" + if p != p0 or d != d0: + return False, "did not close" + return True, f"own exact simulation: closed periodic orbit, {len(word)} bounces" + + +# ----------------------------------------------------------- 135-arc facts + +def on_arc_135(x, y): + """gamma(x, y) == 135 deg exactly. Fraction. Hand-derived equivalence: + circle (2x-1)^2+(2y+1)^2=2 <=> dot = x^2-x+y^2 = -y; then + ca2 = x - y, cb2 = 1 - x - y, ca2*cb2 = 2 y^2 = 2 dot^2, dot < 0.""" + if not y > 0: + return False + dot = x * x - x + y * y + if not dot < 0: + return False + return 2 * dot * dot == (x * x + y * y) * ((1 - x) ** 2 + y * y) + + +# ---------------------------------------------------------------- sampling + +def apex_from_angles(al, be): + ta = math.tan(math.radians(al)) + tb = math.tan(math.radians(be)) + return tb / (ta + tb), ta * tb / (ta + tb) + + +def arc_probe_alphas(g, uniform=60, jmax=14, depth=24, rng=None, nrand=0, + extra_edges=()): + th = 180.0 - g + als = [th * (k + 0.5) / uniform for k in range(uniform)] + edges = [90.0 / j for j in range(1, jmax + 1)] + list(extra_edges) + for e in edges: + for d in range(1, depth + 1): + eps = 0.5 * 2.0 ** -d + for al in (e - eps, e + eps, th - e - eps, th - e + eps): + if 0 < al < th: + als.append(al) + if rng and nrand: + als += [rng.uniform(1e-6, th - 1e-6) for _ in range(nrand)] + return als + + +def best_width_on_arc(word, g, **kw): + best, arg = -math.inf, None + for al in arc_probe_alphas(g, **kw): + be = (180.0 - g) - al + if al <= 0 or be <= 0: + continue + x, y = apex_from_angles(al, be) + v = fwidth(word, x, y) + if v > best: + best, arg = v, (al, be) + return best, arg + + +TOL = 1e-12 + + +def family_word(a, b): + return ((0,) + (1, 2) * a + (0, 2) * b) * 2 + + +# ------------------------------------------------------------ enumeration + +def canonical(word): + n = len(word) + best = None + for w in (word, word[::-1]): + for sw in (False, True): + ww = tuple((1 - c if c < 2 else 2) for c in w) if sw else w + for r in range(n): + cand = ww[r:] + ww[:r] + if best is None or cand < best: + best = cand + return best + + +def is_proper_power(u): + n = len(u) + for p in range(1, n): + if n % p == 0 and u == u[:p] * (n // p): + return True + return False + + +def universe_all(L): + """Canonical doubled words u^2, |u| = L odd; own recursive generator.""" + out = set() + + def rec(u): + if len(u) == L: + if u[-1] != u[0] and not is_proper_power(u): + out.add(canonical(u * 2)) + return + for c in (0, 1, 2): + if c != u[-1]: + rec(u + (c,)) + for c in (0, 1, 2): + rec((c,)) + return out + + +def universe_s1(k): + out = set() + for bits in range(2 ** (k + 1)): + y = [(bits >> i) & 1 for i in range(k + 1)] + if y[0] == y[1]: + continue + u = (y[0], y[1], 2) + for i in range(2, k + 1): + u += (y[i], 2) + if not is_proper_power(u): + out.add(canonical(u * 2)) + return out + + +# ------------------------------------------------------------ subcommands + +def cmd_selftest(_a): + ok = True + # Fagnano at an acute apex + good, msg = my_verify_orbit((0, 1, 2, 0, 1, 2), (F(1, 2), F(4, 5))) + print("[fagnano]", good, msg) + ok &= good + # obtuse fagnano must fail + good, _ = my_verify_orbit((0, 1, 2, 0, 1, 2), (F(1, 2), F(1, 10))) + print("[fagnano obtuse rejected]", not good) + ok &= not good + # cross-implementation exact corridor agreement (vs 002's and 004's) + import skeptic_orbit + import deathlaw_skeptic as DLS + rng = random.Random(SEED) + words = [family_word(2, 1), family_word(3, 3), family_word(4, 2), + (0, 2, 1, 2, 1, 2, 0, 2, 0, 2, 1) * 2] + agree = 0 + tot = 0 + for w in words: + for _ in range(25): + x = F(rng.randint(1, 2047), 4096) + y = F(rng.randint(1, 1300), 4096) + c1 = corridor(w, (x, y), exact=True) + c2 = skeptic_orbit.corridor(w, (x, y)) + c3 = DLS.corridor_apex_exact(w, x, y) + w1 = None if c1 is None else c1[1] - c1[0] + w2 = None if c2 is None else c2[1] - c2[0] + w3 = None if c3 is None else c3[1] - c3[0] + tot += 1 + if w1 == w2 == w3: + agree += 1 + print(f"[exact corridor 3-way agreement {agree}/{tot}]") + ok &= agree == tot + # W(3,3) death near 135 with own float corridor + b, _ = best_width_on_arc(family_word(3, 3), 134.99, uniform=200, depth=20) + d, _ = best_width_on_arc(family_word(3, 3), 135.01, uniform=200, depth=20) + print(f"[W(3,3) alive 134.99: {b:.3g} dead 135.01: {d:.3g}]") + ok &= b > TOL and d < TOL + print("SELFTEST", "PASSED" if ok else "FAILED") + sys.exit(0 if ok else 1) + + +def cmd_certs(args): + import deathlaw_skeptic as DLS + rows = [] + allok = True + specs = [ + ("design_cert_W43_pinchgap.json", "bracket", family_word(4, 3)), + ("design_cert_W43_gapdeep.json", "bracket", family_word(4, 3)), + ("design_cert_W54_touch144.json", "bracket", family_word(5, 4)), + ("design_exact135_W52.json", "exact135", family_word(5, 2)), + ("design_exact135_N1.json", "exact135", + ((0,) + (1, 2) * 3 + (0, 2) * 3 + (1, 2) * 2 + (0, 2) * 3) * 2), + ] + for fname, kind, expect_word in specs: + d = json.load(open(os.path.join(DATA, fname))) + w = tuple(int(c) for c in d["word"]) + x, y = F(d["apex"][0]), F(d["apex"][1]) + row = {"file": fname, "len": len(w), + "word_matches_claimed_family": w == expect_word} + c = corridor(w, (x, y), exact=True) + alive = c is not None and c[1] > c[0] + row["my_exact_corridor_positive"] = alive + row["width_matches_record_digit_for_digit"] = ( + alive and str(c[1] - c[0]) == d["exact_corridor_width"]) + good, msg = my_verify_orbit(w, (x, y)) + row["my_own_exact_simulation"] = good + row["sim_msg"] = msg + if kind == "bracket": + glo = F(d["certified_gamma_bracket_deg"][0]) + ghi = F(d["certified_gamma_bracket_deg"][1]) + lo_ok, hi_ok = DLS.certify_gamma_bounds(x, y, glo, ghi) + row["my_gamma_gt"] = [str(glo), bool(lo_ok)] + row["my_gamma_lt"] = [str(ghi), bool(hi_ok)] + row["ok"] = all([alive, good, lo_ok, hi_ok, + row["word_matches_claimed_family"], + row["width_matches_record_digit_for_digit"]]) + else: + on = on_arc_135(x, y) + row["my_on_arc_135_exact"] = on + circ = (2 * x - 1) ** 2 + (2 * y + 1) ** 2 == 2 + row["circle_identity_exact"] = circ + tan_alpha = y / x + row["tan_alpha"] = str(tan_alpha) + row["niven_applicable"] = tan_alpha not in (0, 1, -1) + row["ok"] = all([alive, good, on, circ, + row["word_matches_claimed_family"], + row["width_matches_record_digit_for_digit"]]) + allok &= row["ok"] + rows.append(row) + print(fname, "OK" if row["ok"] else row) + out = {"command": "certs", "rows": rows, "all_ok": allok} + if args.out: + json.dump(out, open(args.out, "w"), indent=1) + print("CERTS", "ALL CONFIRMED" if allok else "FAILURE") + sys.exit(0 if allok else 1) + + +def birth_bisect(word, g_dead, g_alive, edges, iters=46, uniform=120, + depth=30): + lo, hi = g_dead, g_alive + for _ in range(iters): + mid = 0.5 * (lo + hi) + v, _ = best_width_on_arc(word, mid, uniform=uniform, depth=depth, + extra_edges=edges) + if v > TOL: + hi = mid + else: + lo = mid + return lo, hi + + +def cmd_birthdepth(args): + """W(4,3) birth measured at increasing sampler depths.""" + w = family_word(4, 3) + rows = [] + for depth in (16, 24, 32, 40): + lo, hi = birth_bisect(w, 134.9, 135.2, edges=(22.5,), iters=52, + uniform=60, depth=depth) + floor = 0.5 * 2.0 ** -depth + rows.append({"depth": depth, "sampler_floor_deg": floor, + "birth_between": [lo, hi], + "birth_minus_135": 0.5 * (lo + hi) - 135.0}) + print(rows[-1]) + out = {"command": "birthdepth", "word": "W(4,3)", "rows": rows} + if args.out: + json.dump(out, open(args.out, "w"), indent=1) + + +def y_for_gamma(x, g): + lo, hi = 1e-12, 0.5 + for _ in range(80): + mid = 0.5 * (lo + hi) + ax, ay = -x, -mid + bx, by = 1.0 - x, -mid + gg = math.degrees(math.atan2(abs(ax * by - ay * bx), + ax * bx + ay * by)) + if gg > g: + lo = mid + else: + hi = mid + return 0.5 * (lo + hi) + + +def census_grid_alive(word, g, points=400, x_lo=0.02): + """Replica of census.py cmd_family's alive_max design (my corridor).""" + best = -math.inf + for k in range(points): + x = x_lo + (0.5 - x_lo) * (k + 0.5) / points + v = fwidth(word, x, y_for_gamma(x, g)) + if v > best: + best = v + return best + + +def cmd_censusgap(args): + w = family_word(4, 3) + out = {"command": "censusgap"} + # 1. replicate 001's birth bisection with its own sampler design + alive0 = None + g = 135.25 + while g < 141.0: + if census_grid_alive(w, g) > 0: + alive0 = g + break + g += 0.5 + lo, hi = alive0 - 0.5, alive0 + for _ in range(16): + mid = 0.5 * (lo + hi) + if census_grid_alive(w, mid) > 0: + hi = mid + else: + lo = mid + out["replicated_001_birth_between"] = [lo, hi] + print("replicated 001-style birth:", lo, hi, "(001 recorded 135.0486)") + # 2. window x-extent at sample gammas vs the census grid + grid_max_x = 0.02 + 0.48 * (400 - 0.5) / 400 + gap = 0.48 / 400 + out["census_grid"] = {"max_x": grid_max_x, "spacing": gap} + rows = [] + for g in (135.005, 135.01, 135.02, 135.03, 135.04, 135.0486, 135.06): + # find window in x by dense scan near 0.5 then edge bisection + xs = [0.5 - 0.002 * (k + 0.5) / 4000 for k in range(4000)] + alive_xs = [x for x in xs if fwidth(w, x, y_for_gamma(x, g)) > 0] + if not alive_xs: + rows.append({"gamma": g, "window": None}) + print(g, "no alive x in [0.498, 0.5) at scan resolution") + continue + xin, xax = min(alive_xs), max(alive_xs) + # refine edges + lo1, hi1 = xin - 0.002 / 4000, xin + for _ in range(40): + m = 0.5 * (lo1 + hi1) + if fwidth(w, m, y_for_gamma(m, g)) > 0: + hi1 = m + else: + lo1 = m + lo2, hi2 = xax, min(xax + 0.002 / 4000, 0.5 - 1e-15) + for _ in range(40): + m = 0.5 * (lo2 + hi2) + if fwidth(w, m, y_for_gamma(m, g)) > 0: + lo2 = m + else: + hi2 = m + row = {"gamma": g, "x_window": [hi1, lo2], + "x_window_width": lo2 - hi1, + "below_grid_max_x": lo2 < grid_max_x, + "contains_grid_point": any( + hi1 <= 0.02 + 0.48 * (k + 0.5) / 400 <= lo2 + for k in range(400))} + rows.append(row) + print(row) + out["window_rows"] = rows + if args.out: + json.dump(out, open(args.out, "w"), indent=1) + + +def cmd_birthlaw(args): + """Out-of-sample members: measure birth with own sampler, both halves.""" + rows = [] + for a, b in ((7, 3), (6, 4)): + pred_birth = 180.0 - 90.0 * (a + b + 1) / (a * (b + 1)) + pred_death = 180.0 - 90.0 * (a + b) / (a * (b + 1)) + w = family_word(a, b) + edges = (90.0 / a, 90.0 / (b + 1)) + mid = 0.5 * (pred_birth + pred_death) + v_mid, arg = best_width_on_arc(w, mid, uniform=120, depth=30, + extra_edges=edges) + lo, hi = birth_bisect(w, pred_birth - 0.3, mid, edges=edges, + iters=50, uniform=120, depth=34) + v_below, _ = best_width_on_arc(w, pred_birth - 1e-3, uniform=400, + depth=34, extra_edges=edges) + row = {"a": a, "b": b, "len": len(w), + "predicted_birth": pred_birth, "predicted_death": pred_death, + "alive_at_window_mid": v_mid > TOL, "mid_width": v_mid, + "measured_birth_between": [lo, hi], + "birth_minus_predicted": 0.5 * (lo + hi) - pred_birth, + "dead_sampled_at_birth_minus_1e-3": v_below < TOL, + "best_width_below": v_below} + rows.append(row) + print(row) + # W(5,2): window edges per law = [132, 138] + w = family_word(5, 2) + edges = (90.0 / 5, 90.0 / 3) + v_in, _ = best_width_on_arc(w, 132.001, uniform=120, depth=34, + extra_edges=edges) + v_out, _ = best_width_on_arc(w, 131.999, uniform=400, depth=34, + extra_edges=edges) + v135, _ = best_width_on_arc(w, 135.0, uniform=120, depth=20, + extra_edges=edges) + rows.append({"a": 5, "b": 2, "alive_132.001": v_in > TOL, + "dead_sampled_131.999": v_out < TOL, + "alive_135": v135 > TOL, "width_135": v135}) + print(rows[-1]) + out = {"command": "birthlaw", "rows": rows} + if args.out: + json.dump(out, open(args.out, "w"), indent=1) + + +def cmd_counts(args): + out = {"command": "counts"} + tot13 = 0 + per = {} + for L in (3, 5, 7, 9, 11, 13): + n = len(universe_all(L)) + per[L] = n + tot13 += n + out["all_odd_by_half_len"] = per + out["total_half_le_13"] = tot13 + out["claim_245"] = tot13 == 245 + n15 = len(universe_all(15)) + out["half_15"] = n15 + out["claim_811"] = n15 == 811 + ns1 = sum(len(universe_s1(k)) for k in range(8, 13)) + out["s1_k8_12"] = ns1 + out["claim_2008"] = ns1 == 2008 + # N1, N2 genuinely non-W? + N1 = ((0,) + (1, 2) * 3 + (0, 2) * 3 + (1, 2) * 2 + (0, 2) * 3) * 2 + N2 = ((0,) + (1, 2) * 3 + (0, 2) * 3 + (1, 2) * 2 + (0, 2) * 4) * 2 + res = {} + for name, N in (("N1", N1), ("N2", N2)): + cN = canonical(N) + s = (len(N) - 2) // 4 + clash = [(a, s - a) for a in range(1, s) + if canonical(family_word(a, s - a)) == cN] + res[name] = {"len": len(N), "W_members_same_len_checked": s - 1, + "canonical_clash_with_W": clash} + print(name, res[name]) + out["nonW"] = res + print({k: v for k, v in out.items() if k != "nonW"}) + if args.out: + json.dump(out, open(args.out, "w"), indent=1) + + +def cmd_deadsample(args): + rng = random.Random(SEED) + uni = sorted(universe_all(3) | universe_all(5) | universe_all(7) + | universe_all(9) | universe_all(11) | universe_all(13)) + sample = rng.sample(uni, args.n) + arcs = [135.0, 135.02, 136.5, 138.0, 140.0, 140.7, 142.5, 144.0, + 145.0, 148.0, 151.0, 155.0, 159.0, + 137.3, 141.9, 146.2, 149.4, 152.7, 156.5] # 6 arcs of my own + hits = [] + best_overall = -math.inf + for idx, w in enumerate(sample): + for g in arcs: + v, arg = best_width_on_arc(w, g, uniform=90, depth=26, + rng=rng, nrand=40) + best_overall = max(best_overall, v) + if v > TOL: + hits.append({"word": "".join(map(str, w)), "gamma": g, + "width": v, "alpha": arg[0]}) + if (idx + 1) % 20 == 0: + print(f"# {idx + 1}/{len(sample)}", flush=True) + # attack the recorded near-misses too + near = json.load(open(os.path.join(DATA, + "design_screen_all13.json")))["near_misses"] + near_hits = [] + for nm in near: + w = tuple(int(c) for c in nm["word"]) + for g in (nm["gamma"], nm["gamma"] - 0.35, nm["gamma"] + 0.35): + v, arg = best_width_on_arc(w, g, uniform=400, depth=34, + rng=rng, nrand=200) + if v > TOL: + near_hits.append({"word": nm["word"], "gamma": g, "width": v}) + out = {"command": "deadsample", "n_sampled": len(sample), + "arcs": arcs, "hits": hits, + "best_width_seen": best_overall, + "near_miss_words_attacked": len(near), + "near_miss_hits": near_hits} + print(f"deadsample: {len(hits)} hits in sample, " + f"{len(near_hits)} near-miss hits, best width {best_overall:.3g}") + if args.out: + json.dump(out, open(args.out, "w"), indent=1) + sys.exit(0) + + +def pick_member(gamma): + """Shortest W(a,b) with gamma strictly inside the formula window.""" + t = (F(180) - gamma) / 90 + best = None + for a in range(1, 100): + for b in range(1, 100): + v = t * a * (b + 1) + if a + b < v < a + b + 1: + if best is None or a + b < best[0] + best[1]: + best = (a, b) + return best + + +def cmd_coverage(args): + arcs = [F("90.7"), F("93.3"), F("97.9"), F("101.1"), F("107.4"), + F("112.5"), F("118.7"), F("120"), F("124.9"), F("127.3"), + F("130"), F("133.9"), F("135"), F("135.02"), F("135.0486"), + F("138.6"), F("141.3"), F("144"), F("147.5"), F("150"), + F(1080, 7), F("157.1"), F("160.9"), F("162.5"), F("164.8")] + rows = [] + fails = 0 + for g in arcs: + ab = pick_member(g) + if ab is None: + rows.append({"gamma": str(g), "member": None}) + fails += 1 + continue + a, b = ab + w = family_word(a, b) + v, arg = best_width_on_arc(w, float(g), uniform=90, depth=40, + extra_edges=(90.0 / a, 90.0 / (b + 1))) + row = {"gamma": str(g), "a": a, "b": b, "len": len(w), + "alive": v > TOL, "width": v} + if not row["alive"]: + fails += 1 + rows.append(row) + print(row) + out = {"command": "coverage", "n_arcs": len(arcs), "failures": fails, + "rows": rows} + if args.out: + json.dump(out, open(args.out, "w"), indent=1) + print(f"coverage: {len(arcs) - fails}/{len(arcs)} alive") + sys.exit(0 if fails == 0 else 1) + + +def cmd_newmembers(args): + import deathlaw_skeptic as DLS + rng = random.Random(SEED) + out = {"command": "newmembers", "members": []} + allok = True + for a, b in ((5, 2), (6, 2)): + scope = a >= b >= 1 and a >= 2 and a <= 2 * b + 3 + res = DLS.identity_residuals(DLS.Ring, a, b) + ring_ok = {k: DLS.Ring.is_zero(v) for k, v in res.items()} + spots_ok = True + for _ in range(10): + u0 = F(rng.randint(2, 40), rng.randint(1, 7)) + v0 = F(rng.randint(2, 40), rng.randint(1, 7)) + if u0 == v0: + v0 += 1 + pres = DLS.identity_residuals(DLS.PairOps(u0, v0), a, b) + if not all(p.is_zero() for p in pres.values()): + spots_ok = False + break + # case-tree adversarial sign search at theta <= theta_d + td = float(DLS.theta_d(a, b)) + ct_hits = 0 + total = 0 + for _ in range(args.ctsamples): + r = rng.random() + theta = td * r if rng.random() < 0.5 else \ + td * (1 - 10 ** -rng.uniform(0, 12)) + alpha = (90.0 / a - 10 ** -rng.uniform(0, 12) + if rng.random() < 0.4 else theta * rng.random()) + beta = theta - alpha + if alpha <= 0 or beta <= 0: + continue + total += 1 + N1, N2, N3 = DLS.signs_from_identities(a, b, alpha, beta) + if (N1 > 1e-9 and N2 > 1e-9 and N3 > 1e-9) or \ + (N1 < -1e-9 and N2 < -1e-9 and N3 < -1e-9): + ct_hits += 1 + ok = scope and all(ring_ok.values()) and spots_ok and ct_hits == 0 + allok &= ok + entry = {"a": a, "b": b, "scope_precondition_a_le_2b+3": scope, + "ring_identities": ring_ok, "pair_spots_ok": spots_ok, + "casetree_points": total, "casetree_counterexamples": ct_hits, + "ok": ok} + out["members"].append(entry) + print(entry) + if args.out: + json.dump(out, open(args.out, "w"), indent=1) + print("NEWMEMBERS", "CONFIRMED" if allok else "FAILURE") + sys.exit(0 if allok else 1) + + +def main(): + ap = argparse.ArgumentParser() + sub = ap.add_subparsers(dest="cmd", required=True) + sub.add_parser("selftest").set_defaults(fn=cmd_selftest) + for name, fn in (("certs", cmd_certs), ("birthdepth", cmd_birthdepth), + ("censusgap", cmd_censusgap), ("birthlaw", cmd_birthlaw), + ("counts", cmd_counts), ("coverage", cmd_coverage)): + c = sub.add_parser(name) + c.add_argument("--out") + c.set_defaults(fn=fn) + c = sub.add_parser("deadsample") + c.add_argument("--n", type=int, default=60) + c.add_argument("--out") + c.set_defaults(fn=cmd_deadsample) + c = sub.add_parser("newmembers") + c.add_argument("--ctsamples", type=int, default=300000) + c.add_argument("--out") + c.set_defaults(fn=cmd_newmembers) + args = ap.parse_args() + args.fn(args) + + +if __name__ == "__main__": + main() diff --git a/problems/billiards-triangles/explore/plaw_general.py b/problems/billiards-triangles/explore/plaw_general.py new file mode 100644 index 0000000..5eaa461 --- /dev/null +++ b/problems/billiards-triangles/explore/plaw_general.py @@ -0,0 +1,477 @@ +#!/usr/bin/env python3 +"""Attempt 005: PARAMETRIC proofs of 003's per-member inputs. + +003 proved the death-law necessity theorem for W(a,b) = (0(12)^a(02)^b)^2 +modulo two inputs that were machine-certified per member only: + + * the binding identities I1-I3 (+ glide facts) -- exact ring identities, + verified for 15 members; + * Lemma C -- a 1-D inequality, certified per member. + +This tool closes the first gap FOR ALL INTEGERS a, b >= 1 by a change of +viewpoint: the half word composes to + + u = R_0 . (R_1 R_2)^a . (R_0 R_2)^b + = R_0 . Rot_A(2a alpha) . Rot_B(-2b beta) + +(composed-map unfolding, reflections across BASE sides; R_1 R_2 and R_0 R_2 +collapse to rotations because the side pairs share a vertex). Hence every +quantity in I1-I3 is a Laurent polynomial in the FOUR variables + + x = e^{i alpha}, y = e^{i beta}, P = e^{i a alpha}, Q = e^{i b beta}, + +with integer coefficients in Q(i), where a and b appear ONLY through P, Q. +An identity that holds formally in Q(i)[x^, y^, P^, Q^] therefore holds +for every integer a, b >= 1 and all real angles simultaneously: specializing +P -> x^a, Q -> y^b and then x -> e^{i alpha}, y -> e^{i beta} is a ring +homomorphism. The finite formal check below IS the general proof of I1-I3, +modulo the (short, hand-written in the record) derivation of the closed +forms; that derivation is itself cross-checked exactly, member by member, +against deathlaw_symbolic's independent gate-by-gate unfolding. + +Subcommands (all stdlib-only, deterministic): + formal formal 4-variable proof of I1, I2, I3, D1, D2 and + the glide facts (mu = delta^2, Im(conj(delta) tau)=0) + specialize exact per-member cross-check of the closed forms + (mu, w, delta, tau, m, p(B), p(C), p(A1)) against + deathlaw_symbolic.unfold_sym / glide_data + floatcheck float spot-check of the closed-form u against + composed reflections for random (a,b) up to 40 + lemmac sanity grid for the GENERAL Lemma C proof (the proof + itself is elementary and lives in the record): + checks F(v) < 0 and H2 > 0 on dense grids + casetree adversarial sign-pattern search for the EXTENDED + case tree (scope a <= 3b+4): no Case I/II pattern + at theta <= theta_d for the given member +""" + +import argparse +import cmath +import json +import math +import os +import random +import sys +from fractions import Fraction as F + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +import deathlaw_symbolic as ds # noqa: E402 + +# ---------------------------------------------------------- Q(i) scalars + +ZERO = (F(0), F(0)) +ONE = (F(1), F(0)) +I = (F(0), F(1)) +HALF = (F(1, 2), F(0)) +HALF_OVER_I = (F(0), F(-1, 2)) + + +def cadd(u, v): + return (u[0] + v[0], u[1] + v[1]) + + +def cmul(u, v): + return (u[0] * v[0] - u[1] * v[1], u[0] * v[1] + u[1] * v[0]) + + +def cneg(u): + return (-u[0], -u[1]) + + +def cconj(u): + return (u[0], -u[1]) + + +# ------------------------------- formal ring Q(i)[x^, y^, P^, Q^] +# poly: dict {(ex, ey, eP, eQ): (re, im)}; the monomial means +# x^ex y^ey P^eP Q^eQ with x=e^{i alpha}, y=e^{i beta}, P=e^{i a alpha}, +# Q=e^{i b beta}. + +def clean(p): + return {k: c for k, c in p.items() if c != ZERO} + + +def add(p, q): + r = dict(p) + for k, c in q.items(): + r[k] = cadd(r.get(k, ZERO), c) + return clean(r) + + +def neg(p): + return {k: cneg(c) for k, c in p.items()} + + +def sub(p, q): + return add(p, neg(q)) + + +def mul(p, q): + r = {} + for k1, c1 in p.items(): + for k2, c2 in q.items(): + k = tuple(a + b for a, b in zip(k1, k2)) + r[k] = cadd(r.get(k, ZERO), cmul(c1, c2)) + return clean(r) + + +def scale(p, c): + return clean({k: cmul(v, c) for k, v in p.items()}) + + +def conj(p): + """Complex conjugation: all four variables are unit complex numbers, + so conj is exponent negation plus coefficient conjugation.""" + return {tuple(-e for e in k): cconj(c) for k, c in p.items()} + + +def im(p): + return scale(sub(p, conj(p)), HALF_OVER_I) + + +def mono(ex, ey, eP, eQ, c=ONE): + return {(ex, ey, eP, eQ): c} + + +def fsin(ex, ey, eP, eQ): + """sin(ex*alpha + ey*beta + eP*a*alpha + eQ*b*beta) as a ring element.""" + return scale(sub(mono(ex, ey, eP, eQ), mono(-ex, -ey, -eP, -eQ)), + HALF_OVER_I) + + +def fcos(ex, ey, eP, eQ): + return scale(add(mono(ex, ey, eP, eQ), mono(-ex, -ey, -eP, -eQ)), HALF) + + +def feval(p, alpha, beta, a, b): + """Numeric value at radian angles for concrete integers a, b.""" + z = 0j + for (ex, ey, eP, eQ), c in p.items(): + ang = ex * alpha + ey * beta + eP * a * alpha + eQ * b * beta + z += complex(c[0], c[1]) * cmath.exp(1j * ang) + return z + + +# --------------------------------------------- closed forms (the theorem) +# Derivation in the record (attempt 005): with A = 0, B = sin(alpha+beta), +# C = sin(beta) e^{i alpha} (circumdiameter 1) and base-side reflections +# R_2: z -> conj z (side AB = real axis) +# R_1: z -> e^{2i alpha} conj z +# R_0: z -> B + e^{-2i beta} conj(z - B) +# the composed-map unfolding gives u = R_0 (R_1 R_2)^a (R_0 R_2)^b +# = R_0 . Rot_A(2a alpha) . Rot_B(-2b beta), whence u(z) = mu conj(z) + w: + +def closed_forms(): + B = fsin(1, 1, 0, 0) # sin(alpha+beta), real + C = mul(fsin(0, 1, 0, 0), mono(1, 0, 0, 0)) # sin(beta) e^{i alpha} + mu = mono(0, -2, -2, 2) # e^{-2i a alpha + 2i(b-1)beta} + delta = mono(0, -1, -1, 1) # axis direction, delta^2 = mu + # w = B [1 + e^{-2i(a alpha + beta)} - mu - e^{-2i beta}] + w = mul(B, add(sub(mono(0, 0, 0, 0), mono(0, -2, 0, 0)), + sub(mono(0, -2, -2, 0), mu))) + tau = add(w, mul(mu, conj(w))) + A1 = mul(B, sub(mono(0, 0, 0, 0), mono(0, -2, 0, 0))) # R_0(A)=B(1-y^-2) + dconj = conj(delta) + proj = lambda z: im(mul(dconj, z)) + m = scale(im(mul(dconj, sub(w, scale(tau, HALF)))), HALF) + return {"B": B, "C": C, "A1": A1, "mu": mu, "delta": delta, "w": w, + "tau": tau, "m": m, + "pB": proj(B), "pC": proj(C), "pA1": proj(A1)} + + +def cmd_formal(args): + cf = closed_forms() + checks = {} + # glide facts + checks["mu_eq_delta_sq"] = clean( + sub(cf["mu"], mul(cf["delta"], cf["delta"]))) == {} + checks["tau_parallel_axis"] = clean( + im(mul(conj(cf["delta"]), cf["tau"]))) == {} + # I1: m - p(C) = -[cos(a al) sin(al) sin(b be) + # + sin(a al) sin(be) cos(al + (b+1) be)] + I1 = neg(add(mul(mul(fcos(0, 0, 1, 0), fsin(1, 0, 0, 0)), + fsin(0, 0, 0, 1)), + mul(mul(fsin(0, 0, 1, 0), fsin(0, 1, 0, 0)), + fcos(1, 1, 0, 1)))) + checks["I1"] = clean(sub(sub(cf["m"], cf["pC"]), I1)) == {} + # I2: p(A1) - m = cos(a al) sin((b+1) be) sin(al+be) + I2 = mul(mul(fcos(0, 0, 1, 0), fsin(0, 1, 0, 1)), fsin(1, 1, 0, 0)) + checks["I2"] = clean(sub(sub(cf["pA1"], cf["m"]), I2)) == {} + # I3: p(B) - m = cos((b+1) be) sin(a al) sin(al+be) + I3 = mul(mul(fcos(0, 1, 0, 1), fsin(0, 0, 1, 0)), fsin(1, 1, 0, 0)) + checks["I3"] = clean(sub(sub(cf["pB"], cf["m"]), I3)) == {} + # context identities (not load-bearing): + # D2: p(A1) - p(C) = -sin(be) sin((a-1)al - (b+1)be) + D2 = neg(mul(fsin(0, 1, 0, 0), fsin(-1, -1, 1, -1))) + checks["D2"] = clean(sub(sub(cf["pA1"], cf["pC"]), D2)) == {} + # D1: p(B) - p(C) = sin(al) sin(a al - b be) + D1 = mul(fsin(1, 0, 0, 0), fsin(0, 0, 1, -1)) + checks["D1"] = clean(sub(sub(cf["pB"], cf["pC"]), D1)) == {} + # I4 (needed for sufficiency): the C-image endpoint of gate 2a+2, + # v2 = R_0(Rot_A(2a al) C) = B + y^-2 (P^-2 conj(C) - B), satisfies + # p(v2) - m = cos(a al) sin(al) sin(b be) + # - sin(a al) sin(be) cos(al + (b+1) be) + v2 = add(cf["B"], mul(mono(0, -2, 0, 0), + sub(mul(mono(0, 0, -2, 0), conj(cf["C"])), + cf["B"]))) + pv2 = im(mul(conj(cf["delta"]), v2)) + I4 = sub(mul(mul(fcos(0, 0, 1, 0), fsin(1, 0, 0, 0)), + fsin(0, 0, 0, 1)), + mul(mul(fsin(0, 0, 1, 0), fsin(0, 1, 0, 0)), + fcos(1, 1, 0, 1))) + checks["I4"] = clean(sub(sub(pv2, cf["m"]), I4)) == {} + sizes = {k: len(v) for k, v in cf.items()} + ok = all(checks.values()) + out = {"checks": checks, "term_counts": sizes, "ok": ok, + "meaning": ("each True is an exact identity in " + "Q(i)[x^,y^,P^,Q^]; holds for ALL integer a,b and " + "all real angles by specialization")} + print(json.dumps(out, indent=1)) + if args.out: + json.dump(out, open(args.out, "w"), indent=1) + sys.exit(0 if ok else 1) + + +# ------------------------------------------- per-member exact cross-check + +def specialize(p, a, b): + """Ring hom to deathlaw_symbolic's half-angle ring: P -> x^a, Q -> y^b, + then keys (m, n) meaning e^{i(m alpha + n beta)/2}.""" + r = {} + for (ex, ey, eP, eQ), c in p.items(): + k = (2 * (ex + a * eP), 2 * (ey + b * eQ)) + r[k] = cadd(r.get(k, ZERO), c) + return ds.pclean(r) + + +def check_member(a, b): + from skeptic_family import family_word + cf = closed_forms() + word = family_word(a, b) + half = word[:len(word) // 2] + U = ds.unfold_sym(half) + orient, lin, off = U["maps"][-1] + gd = ds.glide_data(a, b) + dconj = ds.pconj(gd["delta"]) + proj = lambda z: ds.pim(ds.pmul(dconj, z)) + g1u, g1v = gd["gates"][0] + g2u, g2v = gd["gates"][1] + res = { + "orient_reversing": orient == -1, + "mu": ds.pclean(ds.psub(specialize(cf["mu"], a, b), lin)) == {}, + "w": ds.pclean(ds.psub(specialize(cf["w"], a, b), off)) == {}, + "delta": ds.pclean( + ds.psub(specialize(cf["delta"], a, b), gd["delta"])) == {}, + "tau": ds.pclean(ds.psub(specialize(cf["tau"], a, b), + gd["tau"])) == {}, + "m": ds.pclean(ds.psub(specialize(cf["m"], a, b), gd["m"])) == {}, + "pB": ds.pclean(ds.psub(specialize(cf["pB"], a, b), + proj(g1u))) == {}, + "pC": ds.pclean(ds.psub(specialize(cf["pC"], a, b), + proj(g1v))) == {}, + "pA1": ds.pclean(ds.psub(specialize(cf["pA1"], a, b), + proj(g2v))) == {}, + } + # gate 2a+2's C-image endpoint is v2 = R_0(Rot_A(2a alpha) C) + v2 = add(cf["B"], mul(mono(0, -2, 0, 0), + sub(mul(mono(0, 0, -2, 0), conj(cf["C"])), + cf["B"]))) + pv2 = im(mul(conj(cf["delta"]), v2)) + res["p_gate2a2_v"] = ds.pclean( + ds.psub(specialize(pv2, a, b), + proj(gd["gates"][2 * a + 1][1]))) == {} + res["ok"] = all(res.values()) + return res + + +MEMBERS_003 = [(2, 1), (2, 2), (3, 2), (3, 3), (4, 2), (4, 3), (4, 4), + (5, 3), (5, 4), (5, 5), (6, 6), (7, 7), (8, 8), + (10, 9), (12, 12)] +MEMBERS_NEW = [(6, 3), (6, 1), (8, 2), (8, 1), (7, 1), (11, 2), (14, 4), + (17, 9)] + + +def cmd_specialize(args): + results = {} + ok = True + for (a, b) in MEMBERS_003 + MEMBERS_NEW: + r = check_member(a, b) + results[f"W({a},{b})"] = r + ok &= r["ok"] + print(f"W({a},{b}): {'MATCH' if r['ok'] else 'MISMATCH ' + str(r)}") + out = {"results": results, "ok": ok, + "meaning": ("closed forms of u = R_0 Rot_A(2a al) Rot_B(-2b be) " + "agree EXACTLY with the gate-by-gate symbolic " + "unfolding for every listed member")} + if args.out: + json.dump(out, open(args.out, "w"), indent=1) + print("SPECIALIZE", "OK" if ok else "FAILED") + sys.exit(0 if ok else 1) + + +def cmd_floatcheck(args): + """Closed-form u(z) vs composed float reflections, random (a,b).""" + rng = random.Random(20260731) + cf = closed_forms() + worst = 0.0 + for trial in range(args.trials): + a = rng.randint(1, 40) + b = rng.randint(1, a) + al = rng.uniform(0.02, 1.2) + be = rng.uniform(0.02, min(1.2, math.pi - 0.05 - al)) + B = math.sin(al + be) + z = complex(rng.uniform(-1, 1), rng.uniform(-1, 1)) + # composed reflections R_0 (R_1 R_2)^a (R_0 R_2)^b applied to z + zz = z + for _ in range(b): # (R_0 R_2): rot about B + zz = B + cmath.exp(-2j * be) * (zz - B) + zz = cmath.exp(2j * a * al) * zz # (R_1 R_2)^a: rot about A + zz = B + cmath.exp(-2j * be) * (zz - B).conjugate() # R_0 + mu = feval(cf["mu"], al, be, a, b) + w = feval(cf["w"], al, be, a, b) + err = abs(mu * z.conjugate() + w - zz) + worst = max(worst, err) + print(json.dumps({"trials": args.trials, "worst_error": worst, + "ok": worst < 1e-9})) + sys.exit(0 if worst < 1e-9 else 1) + + +# ------------------------------------------------- Lemma C, general form + +def F_logderiv(v_deg, a, b): + """F(v) = 2/sin 2v - (1/a) cot((90-v)/a) - (b/a) cot(bv/a), degrees. + The general proof (in the record) shows F < 0 for all a >= 2, b >= 1, + v in (0, 90); this is a float sanity net only.""" + v = math.radians(v_deg) + h = math.pi / 2 + return (2.0 / math.sin(2 * v) + - (1.0 / a) / math.tan((h - v) / a) + - (b / a) / math.tan(b * v / a)) + + +def H2(v_deg, a, b): + v0 = 90.0 / (b + 1) + v, vv0 = math.radians(v_deg), math.radians(v0) + return (math.sin(v) * math.sin(math.radians((90 - v_deg) / a)) + * math.cos(vv0) + - math.sin(vv0) * math.sin(math.radians(b * v_deg / a)) + * math.cos(v)) + + +def cmd_lemmac(args): + bad = [] + n = 0 + for a in range(2, args.amax + 1): + for b in range(1, a + 1): + v0 = 90.0 / (b + 1) + for k in range(args.grid): + v = 90.0 * (k + 0.5) / args.grid # F < 0 on (0, 90) + n += 1 + if F_logderiv(v, a, b) >= 0: + bad.append(("F", a, b, v, F_logderiv(v, a, b))) + if v < v0 and H2(v, a, b) <= 0: # H2 > 0 on (0, v0) + bad.append(("H2", a, b, v, H2(v, a, b))) + out = {"amax": args.amax, "grid": args.grid, "n_samples": n, + "violations": bad[:20], "n_violations": len(bad), + "ok": not bad} + print(json.dumps(out)) + if args.out: + json.dump(out, open(args.out, "w"), indent=1) + sys.exit(0 if not bad else 1) + + +# --------------------------------- extended case tree adversarial search + +def signs_NNN(al_deg, be_deg, a, b): + """(N1, N2, N3) from the closed forms I1-I3 (floats).""" + al, be = math.radians(al_deg), math.radians(be_deg) + th = al + be + N1 = -(math.cos(a * al) * math.sin(al) * math.sin(b * be) + + math.sin(a * al) * math.sin(be) * math.cos(al + (b + 1) * be)) + N2 = math.cos(a * al) * math.sin((b + 1) * be) * math.sin(th) + N3 = math.cos((b + 1) * be) * math.sin(a * al) * math.sin(th) + return N1, N2, N3 + + +def cmd_casetree(args): + """No Case I (+++) or Case II (---) pattern may occur at + theta <= theta_d. Boundary-targeted + random, seeded.""" + a, b = args.a, args.b + rng = random.Random(20260731 + 100 * a + b) + theta_d = 90.0 * (a + b) / (a * (b + 1)) + tol = 1e-9 + hits = [] + n = 0 + # targeted alpha values: multiples of 90/a (branch bounds), corner, + # plus (b+1)beta boundaries mapped to alpha = theta - beta + targets = [90.0 * k / a for k in range(1, 4 * a) if 90.0 * k / a < 180] + tsets = [] + for frac in (1.0, 1 - 1e-6, 1 - 1e-3, 0.9, 0.5, 0.1): + th = theta_d * frac + for t in targets: + for eps in (-1e-7, -1e-4, 1e-4, 1e-7): + al = t + eps + if 0 < al < th: + tsets.append((al, th - al)) + for k in range(1, 4 * (b + 1)): + be = 90.0 * k / (b + 1) + for eps in (-1e-7, -1e-4, 1e-4, 1e-7): + if 0 < be + eps < th: + tsets.append((th - be - eps, be + eps)) + for _ in range(args.random): + al = rng.uniform(1e-6, th - 1e-6) + tsets.append((al, th - al)) + for (al, be) in tsets: + if not (al > 0 and be > 0): + continue + n += 1 + N1, N2, N3 = signs_NNN(al, be, a, b) + if (N1 > tol and N2 > tol and N3 > tol) or \ + (N1 < -tol and N2 < -tol and N3 < -tol): + hits.append((al, be, N1, N2, N3)) + # positive control: just above theta_d at the death corner + alc = 90.0 / a + thc = theta_d + 1e-4 + ctrl = signs_NNN(alc - 1e-5, thc - alc + 1e-5, a, b) + ctrl_fires = ctrl[0] > 0 and ctrl[1] > 0 and ctrl[2] > 0 + out = {"a": a, "b": b, "theta_d": theta_d, "n_points": n, + "case_hits_at_or_below_theta_d": hits[:10], + "n_hits": len(hits), + "positive_control_above_theta_d_fires": ctrl_fires, + "ok": (not hits) and ctrl_fires} + print(json.dumps(out)) + if args.out: + json.dump(out, open(args.out, "w"), indent=1) + sys.exit(0 if out["ok"] else 1) + + +def main(): + ap = argparse.ArgumentParser() + sp = ap.add_subparsers(dest="cmd", required=True) + c = sp.add_parser("formal") + c.add_argument("--out") + c.set_defaults(fn=cmd_formal) + c = sp.add_parser("specialize") + c.add_argument("--out") + c.set_defaults(fn=cmd_specialize) + c = sp.add_parser("floatcheck") + c.add_argument("--trials", type=int, default=400) + c.set_defaults(fn=cmd_floatcheck) + c = sp.add_parser("lemmac") + c.add_argument("--amax", type=int, default=60) + c.add_argument("--grid", type=int, default=500) + c.add_argument("--out") + c.set_defaults(fn=cmd_lemmac) + c = sp.add_parser("casetree") + c.add_argument("--a", type=int, required=True) + c.add_argument("--b", type=int, required=True) + c.add_argument("--random", type=int, default=20000) + c.add_argument("--out") + c.set_defaults(fn=cmd_casetree) + args = ap.parse_args() + args.fn(args) + + +if __name__ == "__main__": + main() diff --git a/problems/billiards-triangles/explore/plaw_suffice.py b/problems/billiards-triangles/explore/plaw_suffice.py new file mode 100644 index 0000000..d6e5b44 --- /dev/null +++ b/problems/billiards-triangles/explore/plaw_suffice.py @@ -0,0 +1,503 @@ +#!/usr/bin/env python3 +"""Attempt 005: certified SUFFICIENCY for the W(a,b) death law, per member. + +003 proved (necessity): corridor alive => gamma < gamma_d(a,b). This tool +proves the matching lower half per member: an explicit rational SEGMENT of +triangles, ending at the death corner, along which the corridor is certified +alive at every point, with gamma(t) -> gamma_d. Together: death = gamma_d +EXACTLY (sup of alive gamma; the sup is not attained). + +Construction. Corner P0 = (alpha_c, beta_c) = (90/a, 90(a-1)/(a(b+1))) +(rational degrees; this is where 003 measured the last-alive apexes). Take + + alpha(t) = alpha_c - t, beta(t) = beta_c + rho * t, t in (0, t0], + +rho an integer >= 2, t0 rational. Then theta(t) = theta_d + (rho-1) t and +gamma(t) = gamma_d - (rho-1) t, so gamma(t) increases to gamma_d as t -> 0+. + +Alive criterion used (glide reduction, 003 sec. T3 + 004 sec. 1, converse +direction proved in the 005 record): the corridor has positive width iff the +axis offset m lies STRICTLY inside the projection interval of every +half-word gate (and tau != 0). So we certify, for every distinct +gate-endpoint projection difference D = p(endpoint) - m, a constant sign on +the whole segment, plus per-gate opposite signs, plus Re(conj(delta) tau) +nonzero. Two differences vanish AT the corner (t = 0) and need care: + + * p(A1) - m = N2 = cos(a alpha) sin((b+1) beta) sin(theta) [identity I2] + On the segment a*alpha = 90 - a t, so N2(t) = sin(a t) * + sin((b+1)beta(t)) * sin(theta(t)): positive for t in (0, t0] by pure + RATIONAL range checks (all three arguments stay inside (0, 180)). + * m - p(C) = N1 = cos(at) sin(beta) sin(wh t) - sin(at) sin(alpha) + sin(b beta) [identity I1 with v = 90 - a alpha = at, + w = alpha + (b+1)beta - 90 = wh t, wh = (b+1)rho - 1] + N1(0) = 0 exactly; we certify N1' > 0 on [0, t1] (so N1 > 0 on (0, t1]) + and N1 > 0 on [t1, t0] by adaptive interval bisection. rho is chosen so + that N1'(0) = wh sin(beta_c) - a sin(alpha_c) sin(b beta_c) > 0 + (certified). + * p(v2) - m = N4 = cos(a alpha) sin(alpha) sin(b beta) - sin(a alpha) + sin(beta) cos(alpha + (b+1) beta) [identity I4, attempt 005], where + v2 = R_0(Rot_A(2a alpha) C) is the C-image endpoint of gate 2a+2. On + the segment N4(t) = sin(at) sin(alpha) sin(b beta) + cos(at) sin(beta) + sin(wh t): BOTH products positive, again by rational range checks. + +The three identities I1, I2, I4 are re-verified exactly (ring subtraction +in Q(i)[e^{i alpha/2}, e^{i beta/2}]) for the member before use; they are +also proven for ALL (a,b) in plaw_general.py. + +All other differences are nonzero at t = 0 and are certified by adaptive +bisection over [0, t0]. Every enclosure is a single (non-iterated) +interval evaluation of an explicit trig polynomial at rational-degree +arguments (the affine-form lesson does not bite). Certified trig uses +exact mod-360 degree reduction, dyadic pre-rounding, alternating-series +remainders, and the harness's rational pi enclosure; it is cross-checked +against harness unfold.sin_iv/cos_iv in --selftest. + +Failure is loud: if any leg cannot be certified the member FAILS (t0 is +adaptively halved a bounded number of times first). + +Usage: + plaw_suffice.py selftest + plaw_suffice.py member --a A --b B [--out FILE] + plaw_suffice.py all [--out FILE] (the 15 members of 003 + 5 new) +""" + +import argparse +import json +import math +import os +import sys +from fractions import Fraction as F + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +sys.path.insert(0, os.path.join(HERE, "..", "..", "..", + "harness", "billiards-triangles")) +import unfold # noqa: E402 +import deathlaw_symbolic as ds # noqa: E402 +from skeptic_family import width as sk_width, family_word # noqa: E402 + +Iv = unfold.Iv +PI = Iv(unfold.PI_LO, unfold.PI_HI) +DY = 2 ** 72 # dyadic pre-rounding grid + + +# ------------------------------------------------- certified trig, degrees + +_trig_cache = {} + + +def _sin_point_iv(x, terms=14): + """Alternating-series enclosure of sin at rational x, |x| <= 4.""" + total, term = F(0), F(x) + for k in range(terms): + total += term + term = -term * x * x / ((2 * k + 2) * (2 * k + 3)) + r = abs(term) + return total - r, total + r + + +def _cos_point_iv(x, terms=14): + total, term = F(0), F(1) + for k in range(terms): + total += term + term = -term * x * x / ((2 * k + 1) * (2 * k + 2)) + r = abs(term) + return total - r, total + r + + +def _trig_deg(fn_point, dlo, dhi): + """Enclosure of sin/cos over the degree interval [dlo, dhi].""" + key = (fn_point is _sin_point_iv, dlo, dhi) + if key in _trig_cache: + return _trig_cache[key] + dm = (dlo + dhi) / 2 + # exact reduction of the midpoint mod 360 into [-180, 180] + dm_red = dm - 360 * ((dm + 180) // 360) + h = (dhi - dlo) / 2 + # radian midpoint interval (pi enclosure), then dyadic pre-round + xlo = dm_red * (unfold.PI_LO if dm_red >= 0 else unfold.PI_HI) / 180 + xhi = dm_red * (unfold.PI_HI if dm_red >= 0 else unfold.PI_LO) / 180 + xm = F((xlo + xhi).numerator * DY // (xlo + xhi).denominator // 2, DY) + slack = (xhi - xlo) / 2 + abs((xlo + xhi) / 2 - xm) + F(1, DY) + plo, phi = fn_point(xm) + # 1-Lipschitz in radians: widen by half-width h (degrees) in radians + wid = slack + h * unfold.PI_HI / 180 + res = Iv(plo - wid, phi + wid) + _trig_cache[key] = res + return res + + +def sin_deg(dlo, dhi=None): + return _trig_deg(_sin_point_iv, F(dlo), F(dlo if dhi is None else dhi)) + + +def cos_deg(dlo, dhi=None): + return _trig_deg(_cos_point_iv, F(dlo), F(dlo if dhi is None else dhi)) + + +# ------------------------------------------------- ring element -> segment + +def poly_terms(p): + """Real (conj-invariant) ring element -> [(const?, m, n, re2, im2)]. + Value = c00 + sum over (m,n)>0 of [re2 cos(ang) + im2 sin(ang)], + ang = (m alpha + n beta)/2 degrees, re2 = 2 Re c, im2 = 2 Im(-c)... + We use: c e^{i ang} + conj(c) e^{-i ang} = 2 Re(c) cos - 2 Im(c) sin.""" + assert ds.pclean(ds.psub(p, ds.pconj(p))) == {}, "not a real element" + const = F(0) + terms = [] + for (m, n), c in p.items(): + if (m, n) == (0, 0): + const += c[0] + elif (m, n) > (0, 0): + terms.append((m, n, 2 * c[0], -2 * c[1])) + return const, terms + + +class SegPoly: + """A real ring element restricted to the segment + alpha = ac - t, beta = bc + rho t; evaluates over t-intervals.""" + + def __init__(self, p, ac, bc, rho): + self.const, raw = poly_terms(p) + self.terms = [] + for (m, n, re2, im2) in raw: + c0 = (m * ac + n * bc) / 2 # degrees at t = 0 + c1 = F(-m + n * rho, 2) # slope in t + self.terms.append((c0, c1, re2, im2)) + + def eval_iv(self, tlo, thi): + tot = Iv(self.const) + for (c0, c1, re2, im2) in self.terms: + a1, a2 = c0 + c1 * tlo, c0 + c1 * thi + if a1 > a2: + a1, a2 = a2, a1 + if re2: + tot = tot + cos_deg(a1, a2) * re2 + if im2: + tot = tot + sin_deg(a1, a2) * im2 + return tot + + +def certify_sign(evalf, sign, tlo, thi, max_depth=42): + """Certify sign(evalf) == sign on [tlo, thi] by adaptive bisection. + Returns leaf count or None on failure.""" + stack = [(F(tlo), F(thi))] + leaves = 0 + span = F(thi) - F(tlo) + while stack: + lo, hi = stack.pop() + e = evalf(lo, hi) + if (sign > 0 and e.lo > 0) or (sign < 0 and e.hi < 0): + leaves += 1 + continue + if hi - lo < span / 2 ** max_depth: + return None + mid = (lo + hi) / 2 + stack.append((lo, mid)) + stack.append((mid, hi)) + return leaves + + +# ------------------------------------------------------------ the member + +def n1_iv(a, b, ac, bc, rho, wh, tlo, thi): + """N1(t) = cos(at) sin(bc+rho t) sin(wh t) + - sin(at) sin(ac-t) sin(b bc + b rho t), certified.""" + t = (tlo, thi) + + def lin(c0, c1): + a1, a2 = c0 + c1 * tlo, c0 + c1 * thi + return (a1, a2) if a1 <= a2 else (a2, a1) + + return (cos_deg(*lin(F(0), F(a))) * sin_deg(*lin(bc, F(rho))) + * sin_deg(*lin(F(0), wh)) + - sin_deg(*lin(F(0), F(a))) * sin_deg(*lin(ac, F(-1))) + * sin_deg(*lin(b * bc, F(b * rho)))) + + +def n1p_iv(a, b, ac, bc, rho, wh, tlo, thi): + """dN1/dt (with the global pi/180 factor dropped: every term of the + derivative carries exactly one such factor, so the sign is unchanged).""" + def lin(c0, c1): + a1, a2 = c0 + c1 * tlo, c0 + c1 * thi + return (a1, a2) if a1 <= a2 else (a2, a1) + + at = lin(F(0), F(a)) + be = lin(bc, F(rho)) + wt = lin(F(0), wh) + al = lin(ac, F(-1)) + bb = lin(b * bc, F(b * rho)) + t1p = (-a * sin_deg(*at) * sin_deg(*be) * sin_deg(*wt) + + rho * cos_deg(*at) * cos_deg(*be) * sin_deg(*wt) + + wh * cos_deg(*at) * sin_deg(*be) * cos_deg(*wt)) + t2p = (a * cos_deg(*at) * sin_deg(*al) * sin_deg(*bb) + - sin_deg(*at) * cos_deg(*al) * sin_deg(*bb) + + b * rho * sin_deg(*at) * sin_deg(*al) * cos_deg(*bb)) + return t1p - t2p + + +def run_member(a, b, verbose=True): + theta_d = F(90 * (a + b), a * (b + 1)) + gamma_d = 180 - theta_d + ac, bc = F(90, a), F(90 * (a - 1), a * (b + 1)) + rep = {"a": a, "b": b, "len": 4 * a + 4 * b + 2, + "theta_d": str(theta_d), "gamma_d": str(gamma_d), + "corner": [str(ac), str(bc)]} + + gd = ds.glide_data(a, b) # asserts glide facts (exact, ring) + dconj = ds.pconj(gd["delta"]) + proj = lambda z: ds.pim(ds.pmul(dconj, z)) + m = gd["m"] + + # distinct difference polynomials D = p(endpoint) - m, gate structure + pC_m = ds.pclean(ds.psub(proj(gd["gates"][0][1]), m)) # p(C) - m = -N1 + pA1_m = ds.pclean(ds.psub(proj(gd["gates"][1][1]), m)) # p(A1) - m = N2 + pv2_m = ds.pclean(ds.psub(proj(gd["gates"][2 * a + 1][1]), m)) # = N4 + # exact per-member re-verification of I1, I2, I4 before use + cos_of = lambda mm, nn: ds.pscale( + ds.padd(ds.mono(mm, nn), ds.mono(-mm, -nn)), (F(1, 2), F(0))) + S = ds.pmul(ds.pmul(cos_of(2 * a, 0), ds.sin_of(2, 0)), + ds.sin_of(0, 2 * b)) + T = ds.pmul(ds.pmul(ds.sin_of(2 * a, 0), ds.sin_of(0, 2)), + cos_of(2, 2 * (b + 1))) + I1ok = ds.pclean(ds.psub(pC_m, ds.padd(S, T))) == {} # pC-m = S+T=-N1 + I2m = ds.pmul(ds.pmul(cos_of(2 * a, 0), ds.sin_of(0, 2 * (b + 1))), + ds.sin_of(2, 2)) + I2ok = ds.pclean(ds.psub(pA1_m, I2m)) == {} + I4ok = ds.pclean(ds.psub(pv2_m, ds.psub(S, T))) == {} # pv2-m = S-T + if not (I1ok and I2ok and I4ok): + rep["ok"] = False + rep["fail"] = f"identity re-check: I1={I1ok} I2={I2ok} I4={I4ok}" + return rep + diffs = {} # key -> poly + gates_keys = [] + for k, (u, v) in enumerate(gd["gates"]): + pair = [] + for z in (u, v): + d = ds.pclean(ds.psub(proj(z), m)) + key = json.dumps(sorted((mn, (str(c[0]), str(c[1]))) + for mn, c in d.items())) + diffs.setdefault(key, d) + pair.append(key) + gates_keys.append(pair) + keyof = lambda p: json.dumps(sorted((mn, (str(c[0]), str(c[1]))) + for mn, c in p.items())) + key_N1, key_N2, key_N4 = keyof(pC_m), keyof(pA1_m), keyof(pv2_m) + assert key_N1 in diffs and key_N2 in diffs and key_N4 in diffs + rep["n_distinct_diffs"] = len(diffs) + + # tau nondegeneracy: r(t) = Re(conj(delta) tau) as an extra function + tau_r = ds.pre(ds.pmul(dconj, gd["tau"])) + + # choose rho: certified N1'(0) = wh sin(bc) - a sin(ac) sin(b bc) > 0 + rho = None + for r in range(2, 65): + wh = F((b + 1) * r - 1) + e = (wh * sin_deg(bc) - a * sin_deg(ac) * sin_deg(b * bc)) + if e.lo > 0: + rho = r + break + if rho is None: + rep["ok"] = False + rep["fail"] = "no rho with certified N1'(0) > 0" + return rep + wh = F((b + 1) * rho - 1) + rep["rho"] = rho + + # initial t0 from the rational range constraints (N2 legs + obtuse) + t0 = min(F(45, a), F(90, (b + 1) * rho), + (90 - theta_d) / (2 * (rho - 1)), F(1, 4)) + + for attempt in range(14): + _trig_cache.clear() + ok, detail = certify_segment(a, b, ac, bc, rho, wh, t0, diffs, + gates_keys, key_N1, key_N2, key_N4, + tau_r) + if ok: + rep.update(detail) + rep["t0"] = str(t0) + rep["ok"] = True + # float cross-check via 002's independent corridor + wd = [] + for frac in (F(1, 2), F(1, 97)): + t = t0 * frac + al, be = float(ac - t), float(bc + rho * t) + ta = math.tan(math.radians(al)) + tb = math.tan(math.radians(be)) + x, y = tb / (ta + tb), ta * tb / (ta + tb) + wd.append(sk_width(family_word(a, b), x, y)) + rep["float_widths_at_t0/2_t0/97"] = wd + rep["float_alive_crosscheck"] = all(w > 0 for w in wd) + rep["gamma_range_alive"] = [str(gamma_d - (rho - 1) * t0), + f"-> {gamma_d} (open)"] + rep["conclusion"] = ( + f"alive for all t in (0, {t0}]: gamma in " + f"[{float(gamma_d - (rho - 1) * t0):.6f}, {float(gamma_d)})" + f" all realized; with the parametric necessity theorem " + f"(003 + 005 extended case tree): " + f"death(W({a},{b})) = {gamma_d} EXACTLY") + return rep + if verbose: + print(f" W({a},{b}): t0={t0} failed ({detail}); halving") + t0 /= 2 + rep["ok"] = False + rep["fail"] = f"could not certify segment down to t0={t0}" + return rep + + +def certify_segment(a, b, ac, bc, rho, wh, t0, diffs, gates_keys, + key_N1, key_N2, key_N4, tau_r): + leaves = {} + signs = {} + # --- N2 leg: rational range checks only --- + # at in (0, a t0] subset (0,90); (b+1)beta(t) in (90-90/a, +] subset + # (0,180); theta(t) in (theta_d, theta_d+(rho-1)t0] subset (0,90) + if not (a * t0 <= 45 and 90 - F(90, a) > 0 + and 90 - F(90, a) + (b + 1) * rho * t0 < 180 + and F(90 * (a + b), a * (b + 1)) + (rho - 1) * t0 < 90): + return False, "N2 rational range check" + signs[key_N2] = +1 + leaves["N2"] = 0 + + # --- N4 leg: rational range checks only --- + # N4(t) = sin(at) sin(alpha) sin(b beta) + cos(at) sin(beta) sin(wh t) + # sin(at), cos(at) > 0 since at in (0,45]; alpha in (0, 90/a]; + # b beta in (b bc, b bc + b rho t0] subset (0,180); beta subset (0,180); + # wh t in (0, wh t0] subset (0,180) + if not (b * bc + b * rho * t0 < 180 and bc + rho * t0 < 180 + and wh * t0 < 180 and t0 < ac): + return False, "N4 rational range check" + signs[key_N4] = +1 + leaves["N4"] = 0 + + # --- N1 legs --- + t1 = t0 / 8 + for _ in range(20): + n = certify_sign(lambda lo, hi: n1p_iv(a, b, ac, bc, rho, wh, + lo, hi), +1, 0, t1) + if n is not None: + break + t1 /= 2 + else: + return False, "N1' > 0 near 0" + leaves["N1_deriv"] = n + n = certify_sign(lambda lo, hi: n1_iv(a, b, ac, bc, rho, wh, lo, hi), + +1, t1, t0) + if n is None: + return False, f"N1 > 0 on [{t1}, {t0}]" + leaves["N1_bisect"] = n + signs[key_N1] = -1 # p(C) - m = -N1 < 0 + + # --- generic differences --- + for key, p in diffs.items(): + if key in (key_N1, key_N2, key_N4): + continue + sp = SegPoly(p, ac, bc, rho) + guess = sp.eval_iv(t0 / 2, t0 / 2) + s = 1 if guess.lo > 0 else (-1 if guess.hi < 0 else 0) + if s == 0: + return False, "sign guess ambiguous" + n = certify_sign(sp.eval_iv, s, 0, t0) + if n is None: + return False, f"generic diff sign on [0,{t0}]" + signs[key] = s + leaves.setdefault("generic", 0) + leaves["generic"] += n + + # --- per-gate: strictly one endpoint above m, one below --- + for pair in gates_keys: + if signs[pair[0]] * signs[pair[1]] != -1: + return False, "gate endpoints not straddling m" + + # --- tau nondegenerate along the segment --- + sp = SegPoly(tau_r, ac, bc, rho) + guess = sp.eval_iv(t0 / 2, t0 / 2) + s = 1 if guess.lo > 0 else (-1 if guess.hi < 0 else 0) + n = certify_sign(sp.eval_iv, s, 0, t0) if s else None + if n is None: + return False, "tau_r sign" + leaves["tau_r"] = n + return True, {"leaves": leaves} + + +# ------------------------------------------------------------------ CLI + +MEMBERS = [(2, 1), (2, 2), (3, 2), (3, 3), (4, 2), (4, 3), (4, 4), + (5, 3), (5, 4), (5, 5), (6, 6), (7, 7), (8, 8), + (10, 9), (12, 12), + (6, 3), (6, 1), (8, 2), (8, 1), (7, 1)] + + +def cmd_selftest(_args): + ok = True + import random + rng = random.Random(20260731) + for _ in range(300): + d = F(rng.randint(-500000, 500000), rng.randint(1, 997)) + for mine, ref, fl in ((sin_deg(d), unfold.sin_iv( + unfold.Iv(d) * PI / 180), math.sin(math.radians(float(d)))), + (cos_deg(d), unfold.cos_iv(unfold.Iv(d) * PI / 180), + math.cos(math.radians(float(d))))): + ok &= mine.lo <= ref.hi and ref.lo <= mine.hi # overlap + ok &= float(mine.lo) - 1e-12 <= fl <= float(mine.hi) + 1e-12 + ok &= float(mine.hi - mine.lo) < 1e-15 + print("[trig vs unfold + float x300]", "OK" if ok else "FAIL") + # SegPoly vs direct float evaluation on a member + gd = ds.glide_data(3, 2) + ac, bc, rho = F(90, 3), F(90 * 2, 3 * 3), 3 + dconj = ds.pconj(gd["delta"]) + p = ds.pim(ds.pmul(dconj, gd["gates"][4][0])) + pm = ds.pclean(ds.psub(p, gd["m"])) + sp = SegPoly(pm, ac, bc, rho) + ok2 = True + for tt in (F(1, 100), F(1, 7), F(1, 3)): + e = sp.eval_iv(tt, tt) + al = math.radians(float(ac - tt)) + be = math.radians(float(bc + rho * tt)) + v = (ds.peval(p, al, be) - ds.peval(gd["m"], al, be)).real + ok2 &= float(e.lo) - 1e-11 <= v <= float(e.hi) + 1e-11 + print("[SegPoly vs peval]", "OK" if ok2 else "FAIL") + ok &= ok2 + print("SELFTEST", "PASSED" if ok else "FAILED") + sys.exit(0 if ok else 1) + + +def main(): + ap = argparse.ArgumentParser() + sub = ap.add_subparsers(dest="cmd", required=True) + sub.add_parser("selftest").set_defaults(fn=cmd_selftest) + c = sub.add_parser("member") + c.add_argument("--a", type=int, required=True) + c.add_argument("--b", type=int, required=True) + c.add_argument("--out") + c.set_defaults(fn=None) + c = sub.add_parser("all") + c.add_argument("--out") + c.set_defaults(fn=None) + args = ap.parse_args() + if args.cmd == "selftest": + args.fn(args) + return + if args.cmd == "member": + results = [run_member(args.a, args.b)] + print(json.dumps(results[0], indent=1)) + else: + results = [] + for (a, b) in MEMBERS: + r = run_member(a, b) + results.append(r) + print(f"W({a},{b}): rho={r.get('rho')} t0={r.get('t0')} " + f"diffs={r.get('n_distinct_diffs')} " + f"float_ok={r.get('float_alive_crosscheck')} " + f"=> {'DEATH = ' + r['gamma_d'] + ' EXACT' if r['ok'] else 'FAILED: ' + str(r.get('fail'))}") + if args.out: # checkpoint after each member + json.dump(results, open(args.out, "w"), indent=1) + if args.out: + json.dump(results, open(args.out, "w"), indent=1) + ok = all(r["ok"] for r in results) + print("ALL MEMBERS CERTIFIED" if ok else "SOME MEMBERS FAILED") + sys.exit(0 if ok else 1) + + +if __name__ == "__main__": + main() diff --git a/problems/billiards-triangles/explore/psk_review.py b/problems/billiards-triangles/explore/psk_review.py new file mode 100644 index 0000000..5bf395a --- /dev/null +++ b/problems/billiards-triangles/explore/psk_review.py @@ -0,0 +1,884 @@ +#!/usr/bin/env python3 +"""Attempt 008: skeptic review of 005 (plaw death-law completion). + +Everything here is written from scratch for this review except where a 004 +skeptic asset is explicitly reused (deathlaw_skeptic's interval trig stack +and its adaptive death sampler -- 004's tools are reserved for skeptics). +Nothing imports plaw_general / plaw_suffice, and nothing trusts +deathlaw_symbolic's ring in a verdict path (it appears only as the OBJECT +under test in `segment --vs-ds` cross-prints, never as evidence). + +Subcommands (all stdlib-only, deterministic, seed 20260738): + + bridge EXACT off-torus pair-algebra test of 005's closed-form bridge: + the half word u = R_0 (R_1 R_2)^a (R_0 R_2)^b is composed by + DIRECT reflection application (my own maps) and compared, as an + exact Q(i) identity at random rational off-torus points, with + the claimed closed forms mu, w, delta, tau, m, and the + gate-(2a+2) C-endpoint v2 = R_0(Rot_A(2a alpha) C). Also + checks mu = delta^2, Im(conj(delta) tau) = 0 and the glide + action p(u(z)) = 2m - p(z) at the same exact points. + ring my own formal 4-variable Laurent ring Q(i)[x,y,P,Q]: + u is RE-DERIVED symbolically by affine-map composition (the + only lattice steps being (x^2)^a -> P^2, (y^-2)^b -> Q^-2, + each justified by a fixed-point identity checked in the ring), + then I1, I2, I3, I4, D1, D2 and the glide facts are re-proven + by exact polynomial subtraction in MY ring. + lemma adversarial numeric stress of Lemma L1 / general Lemma C / D: + F(v) < 0 and H2 > 0 at huge a, b = 1, b = a, v near both + endpoints; the factorization H2 = cos v cos v0 sin(bv/a) + (Phi - Phi(v0)) checked numerically (my own radian code). + threshold exact Fraction check of the generalized threshold identity + a*w - b*v = a(b+1)(theta - theta_d) - 360*k*b. + casetree my own adversarial sign-pattern scan (different design and + seed from 005's) on members with a > 2b+3 and k >= 1 reachable: + no (N1,N2,N3) Case I/II pattern may occur at theta <= theta_d; + positive control above theta_d must fire. + segment independent alive certification at sampled t on 005's + universal segment (alpha, beta) = (90/a - t, 90(a-1)/(a(b+1)) + + 2t): the FULL 2n-gate corridor is built by my own interval + unfolding (004's certified trig; direct projection onto the + normal of the full-word translation tau; no glide reduction + used), and positive width is certified at t down to 2^-30. + trigaudit adversarial audit of plaw_suffice's certified-trig layer: + its sin_deg/cos_deg enclosures must contain my independently + computed values (004 stack) at random rational degrees and + must be valid over intervals (random interior points inside). +""" + +import argparse +import json +import math +import os +import random +import sys +from fractions import Fraction as F + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) + +import deathlaw_skeptic as dsk # 004's skeptic stack (mine to reuse) + +SEED = 20260738 + + +def family_half(a, b): + return (0,) + (1, 2) * a + (0, 2) * b + + +def family_word(a, b): + return family_half(a, b) * 2 + + +# ===================================================================== +# Gaussian rationals (re, im) as Fraction pairs +# ===================================================================== + +def gadd(u, v): + return (u[0] + v[0], u[1] + v[1]) + + +def gsub(u, v): + return (u[0] - v[0], u[1] - v[1]) + + +def gmul(u, v): + return (u[0] * v[0] - u[1] * v[1], u[0] * v[1] + u[1] * v[0]) + + +def gconj(u): + return (u[0], -u[1]) + + +def ginv(u): + n = u[0] * u[0] + u[1] * u[1] + return (u[0] / n, -u[1] / n) + + +GZERO = (F(0), F(0)) +GONE = (F(1), F(0)) +GI = (F(0), F(1)) + + +# ===================================================================== +# off-torus pair algebra: pair (val, star-val); star = the involution +# x -> 1/x, y -> 1/y, i -> -i, which equals complex conjugation on the +# torus. Exact evaluation of any expression built from +, -, *, /, conj. +# ===================================================================== + +def pr_const(c): + return (c, gconj(c)) + + +def pr_add(u, v): + return (gadd(u[0], v[0]), gadd(u[1], v[1])) + + +def pr_sub(u, v): + return (gsub(u[0], v[0]), gsub(u[1], v[1])) + + +def pr_mul(u, v): + return (gmul(u[0], v[0]), gmul(u[1], v[1])) + + +def pr_div(u, v): + return (gmul(u[0], ginv(v[0])), gmul(u[1], ginv(v[1]))) + + +def pr_conj(u): + return (u[1], u[0]) + + +def pr_im(u): + # (u - conj u) / (2i) + d = pr_sub(u, pr_conj(u)) + inv2i = pr_const((F(0), F(-1, 2))) + return pr_mul(d, inv2i) + + +def pr_var(x0): + """A unit-modulus variable evaluated off-torus at rational x0.""" + return ((F(x0), F(0)), (1 / F(x0), F(0))) + + +def pr_pow(u, n): + if n < 0: + u = (ginv(u[0]), ginv(u[1])) + n = -n + r = (GONE, GONE) + for _ in range(n): + r = pr_mul(r, u) + return r + + +def bridge_member(a, b, rng): + """Exact pair-algebra comparison for one (a, b) at one random point.""" + # random off-torus rational points for x = e^{i alpha}, y = e^{i beta} + def rq(): + n = rng.randint(2, 60) + d = rng.randint(2, 60) + while n == d: + d = rng.randint(2, 60) + return F(n, d) * rng.choice([1, -1]) + + X, Y = pr_var(rq()), pr_var(rq()) + # geometry: A = 0, B = sin(alpha+beta), C = sin(beta) e^{i alpha} + inv2i = pr_const((F(0), F(-1, 2))) + XY = pr_mul(X, Y) + B = pr_mul(pr_sub(XY, pr_div((GONE, GONE), XY)), inv2i) + C = pr_mul(pr_mul(pr_sub(Y, pr_div((GONE, GONE), Y)), inv2i), X) + Y2i = pr_pow(Y, -2) # e^{-2 i beta} + X2 = pr_pow(X, 2) + + # base reflections, re-derived by hand in the record: + def R2(z): + return pr_conj(z) + + def R1(z): + return pr_mul(X2, pr_conj(z)) + + def R0(z): + return pr_add(B, pr_mul(Y2i, pr_conj(pr_sub(z, B)))) + + R = {0: R0, 1: R1, 2: R2} + half = family_half(a, b) + + def apply_chain(letters, z): + # M = R_{s1} o ... o R_{sk}; M(z) applies R_{sk} first + for s in reversed(letters): + z = R[s](z) + return z + + z1 = pr_const((F(3, 7), F(2, 5))) # generic constant point + w_dir = apply_chain(half, pr_const(GZERO)) # u(0) = w + uz1 = apply_chain(half, z1) + mu_dir = pr_div(pr_sub(uz1, w_dir), pr_conj(z1)) + + # closed forms with P -> X^a, Q -> Y^b (the bridge under test) + Pm2 = pr_pow(X, -2 * a) # P^-2 + Q2 = pr_pow(Y, 2 * b) # Q^2 + mu_cf = pr_mul(pr_mul(Pm2, Q2), Y2i) + w_cf = pr_mul(B, pr_add(pr_sub((GONE, GONE), Y2i), + pr_sub(pr_mul(Pm2, Y2i), mu_cf))) + ok_mu = mu_dir == mu_cf + ok_w = w_dir == w_cf + + # v2 = R_0(Rot_A(2 a alpha) C) must equal the gate-(2a+2) C-endpoint + # of the direct chain M_{2a+1} = R_0 (R_1 R_2)^a applied to C + v2_dir = apply_chain(half[:2 * a + 1], C) + v2_cf = pr_add(B, pr_mul(Y2i, pr_sub(pr_mul(Pm2, pr_conj(C)), B))) + ok_v2 = v2_dir == v2_cf + + # glide facts at the exact point + delta = pr_mul(pr_pow(X, -a), pr_pow(Y, b - 1)) + ok_delta = pr_mul(delta, delta) == mu_cf + tau = pr_add(w_dir, pr_mul(mu_dir, pr_conj(w_dir))) + dconj = pr_conj(delta) + ok_tau_axis = pr_im(pr_mul(dconj, tau)) == (GZERO, GZERO) + + def proj(z): + return pr_im(pr_mul(dconj, z)) + + half_tau = pr_mul(tau, pr_const((F(1, 2), F(0)))) + m = pr_mul(pr_im(pr_mul(dconj, pr_sub(w_dir, half_tau))), + pr_const((F(1, 2), F(0)))) + # glide action p(u(z)) + p(z) - 2m = 0 at the generic point z1 + lhs = pr_sub(pr_add(proj(uz1), proj(z1)), + pr_mul(m, pr_const((F(2), F(0))))) + ok_glide = lhs == (GZERO, GZERO) + return {"mu": ok_mu, "w": ok_w, "v2": ok_v2, "delta_sq": ok_delta, + "tau_axis": ok_tau_axis, "glide_action": ok_glide, + "ok": all([ok_mu, ok_w, ok_v2, ok_delta, ok_tau_axis, + ok_glide])} + + +def cmd_bridge(args): + rng = random.Random(SEED) + members = [(2, 1), (2, 2), (3, 2), (3, 3), (4, 2), (4, 3), (4, 4), + (5, 3), (5, 4), (5, 5), (6, 6), (7, 7), (8, 8), + (10, 9), (12, 12), (6, 3), (6, 1), (8, 2), (8, 1), (7, 1), + (11, 2), (14, 4), (17, 9), (1, 1), (40, 1), (40, 40), + (37, 13), (25, 24)] + while len(members) < args.members: + a = rng.randint(1, 40) + b = rng.randint(1, a) + members.append((a, b)) + results = {} + ok = True + for (a, b) in members: + rr = [bridge_member(a, b, rng) for _ in range(args.points)] + good = all(r["ok"] for r in rr) + results[f"W({a},{b})"] = {"points": len(rr), "ok": good, + "detail": rr[0]} + ok &= good + print(f"W({a},{b}): {'MATCH (exact, %d pts)' % len(rr) if good else 'MISMATCH ' + str(rr)}") + out = {"seed": SEED, "n_members": len(members), "ok": ok, + "meaning": ("direct reflection-composition of the half word " + "agrees EXACTLY (off-torus pair algebra, my own " + "maps) with 005's closed forms mu, w, v2 and glide " + "facts at random rational points")} + print("BRIDGE", "OK" if ok else "FAILED") + if args.out: + json.dump({"results": results, **out}, open(args.out, "w"), indent=1) + sys.exit(0 if ok else 1) + + +# ===================================================================== +# my own formal 4-variable Laurent ring Q(i)[x^, y^, P^, Q^] +# ===================================================================== + +class R4: + """Laurent polys as dict {(ex,ey,eP,eQ): Gaussian}.""" + + @staticmethod + def clean(p): + return {k: c for k, c in p.items() if c != GZERO} + + @staticmethod + def add(p, q): + r = dict(p) + for k, c in q.items(): + r[k] = gadd(r.get(k, GZERO), c) + return R4.clean(r) + + @staticmethod + def sub(p, q): + return R4.add(p, {k: (-c[0], -c[1]) for k, c in q.items()}) + + @staticmethod + def mul(p, q): + r = {} + for k1, c1 in p.items(): + for k2, c2 in q.items(): + k = (k1[0] + k2[0], k1[1] + k2[1], + k1[2] + k2[2], k1[3] + k2[3]) + r[k] = gadd(r.get(k, GZERO), gmul(c1, c2)) + return R4.clean(r) + + @staticmethod + def scale(p, c): + return R4.clean({k: gmul(v, c) for k, v in p.items()}) + + @staticmethod + def conj(p): + return {(-k[0], -k[1], -k[2], -k[3]): gconj(c) + for k, c in p.items()} + + @staticmethod + def im(p): + return R4.scale(R4.sub(p, R4.conj(p)), (F(0), F(-1, 2))) + + @staticmethod + def is_zero(p): + return R4.clean(p) == {} + + +def r4mono(ex, ey, eP, eQ, c=GONE): + return {(ex, ey, eP, eQ): c} + + +def r4sin(ex, ey, eP, eQ): + return R4.scale(R4.sub(r4mono(ex, ey, eP, eQ), + r4mono(-ex, -ey, -eP, -eQ)), (F(0), F(-1, 2))) + + +def r4cos(ex, ey, eP, eQ): + return R4.scale(R4.add(r4mono(ex, ey, eP, eQ), + r4mono(-ex, -ey, -eP, -eQ)), (F(1, 2), F(0))) + + +# antilinear/linear affine maps over the ring: ('L'|'A', lam, off) +# 'L': z -> lam z + off ; 'A': z -> lam conj(z) + off + +def map_compose(f, g): + """f o g.""" + tf, lf, of = f + tg, lg, og = g + if tf == 'L': + return (tg, R4.mul(lf, lg), R4.add(R4.mul(lf, og), of)) + # f antilinear: f(g(z)) = lf conj(lg z + og) + of + if tg == 'L': + return ('A', R4.mul(lf, R4.conj(lg)), + R4.add(R4.mul(lf, R4.conj(og)), of)) + return ('L', R4.mul(lf, R4.conj(lg)), + R4.add(R4.mul(lf, R4.conj(og)), of)) + + +def map_apply(f, z): + t, l, o = f + return R4.add(R4.mul(l, R4.conj(z) if t == 'A' else z), o) + + +def cmd_ring(args): + checks = {} + B = r4sin(1, 1, 0, 0) + C = R4.mul(r4sin(0, 1, 0, 0), r4mono(1, 0, 0, 0)) + y2i = r4mono(0, -2, 0, 0) + # base maps + R2m = ('A', r4mono(0, 0, 0, 0), {}) + R1m = ('A', r4mono(2, 0, 0, 0), {}) + R0m = ('A', y2i, R4.sub(B, R4.mul(y2i, B))) # B + y^-2 conj(z - B) + # single blocks + rot_A1 = map_compose(R1m, R2m) # must be ('L', x^2, 0) + rot_B1 = map_compose(R0m, R2m) # must be ('L', y^-2, B(1 - y^-2)) + checks["rotA_block"] = (rot_A1[0] == 'L' + and R4.is_zero(R4.sub(rot_A1[1], + r4mono(2, 0, 0, 0))) + and R4.is_zero(rot_A1[2])) + wB = R4.sub(B, R4.mul(y2i, B)) + checks["rotB_block"] = (rot_B1[0] == 'L' + and R4.is_zero(R4.sub(rot_B1[1], y2i)) + and R4.is_zero(R4.sub(rot_B1[2], wB))) + # fixed-point identities that justify the a-th / b-th power lattice + # substitution: rot_A fixes A = 0 (trivial: off = 0, checked above); + # rot_B fixes B: lam*B + off = B in the ring + checks["rotB_fixes_B"] = R4.is_zero( + R4.sub(R4.add(R4.mul(y2i, B), wB), B)) + # a-th power of z -> lam(z - fix) + fix is z -> lam^a (z - fix) + fix; + # substitute lam^a via the exponent lattice: (x^2)^a = P^2, + # (y^-2)^b = Q^-2. These are the ONLY lattice steps. + P2 = r4mono(0, 0, 2, 0) + Qm2 = r4mono(0, 0, 0, -2) + rot_A = ('L', P2, {}) # fixes 0 + rot_B = ('L', Qm2, R4.sub(B, R4.mul(Qm2, B))) # fixes B + u = map_compose(R0m, map_compose(rot_A, rot_B)) + checks["u_antilinear"] = u[0] == 'A' + mu, w = u[1], u[2] + # compare with 005's claimed closed forms + mu_claim = r4mono(0, -2, -2, 2) + w_claim = R4.mul(B, R4.add(R4.sub(r4mono(0, 0, 0, 0), + r4mono(0, -2, 0, 0)), + R4.sub(r4mono(0, -2, -2, 0), mu_claim))) + checks["mu_matches_005"] = R4.is_zero(R4.sub(mu, mu_claim)) + checks["w_matches_005"] = R4.is_zero(R4.sub(w, w_claim)) + # glide facts + delta = r4mono(0, -1, -1, 1) + checks["mu_eq_delta_sq"] = R4.is_zero(R4.sub(mu, R4.mul(delta, delta))) + tau = R4.add(w, R4.mul(mu, R4.conj(w))) + dconj = R4.conj(delta) + checks["tau_parallel_axis"] = R4.is_zero(R4.im(R4.mul(dconj, tau))) + + def proj(z): + return R4.im(R4.mul(dconj, z)) + + m = R4.scale(R4.im(R4.mul(dconj, + R4.sub(w, R4.scale(tau, (F(1, 2), F(0)))))), + (F(1, 2), F(0))) + A1 = map_apply(R0m, {}) # R_0(A), A = 0 + v2 = map_apply(R0m, map_apply(rot_A, C)) + pB, pC, pA1, pv2 = proj(B), proj(C), proj(A1), proj(v2) + # the identities, RHS built from the record's prose (my own encoding) + S = R4.mul(R4.mul(r4cos(0, 0, 1, 0), r4sin(1, 0, 0, 0)), + r4sin(0, 0, 0, 1)) # cos(a al) sin(al) sin(b be) + T = R4.mul(R4.mul(r4sin(0, 0, 1, 0), r4sin(0, 1, 0, 0)), + r4cos(1, 1, 0, 1)) # sin(a al) sin(be) cos(al+(b+1)be) + checks["I1"] = R4.is_zero(R4.sub(R4.sub(m, pC), + R4.scale(R4.add(S, T), + ((F(-1)), F(0))))) + I2 = R4.mul(R4.mul(r4cos(0, 0, 1, 0), r4sin(0, 1, 0, 1)), + r4sin(1, 1, 0, 0)) + checks["I2"] = R4.is_zero(R4.sub(R4.sub(pA1, m), I2)) + I3 = R4.mul(R4.mul(r4cos(0, 1, 0, 1), r4sin(0, 0, 1, 0)), + r4sin(1, 1, 0, 0)) + checks["I3"] = R4.is_zero(R4.sub(R4.sub(pB, m), I3)) + checks["I4"] = R4.is_zero(R4.sub(R4.sub(pv2, m), R4.sub(S, T))) + D1 = R4.mul(r4sin(1, 0, 0, 0), r4sin(0, 0, 1, -1)) + checks["D1"] = R4.is_zero(R4.sub(R4.sub(pB, pC), D1)) + D2 = R4.scale(R4.mul(r4sin(0, 1, 0, 0), r4sin(-1, -1, 1, -1)), + (F(-1), F(0))) + checks["D2"] = R4.is_zero(R4.sub(R4.sub(pA1, pC), D2)) + ok = all(checks.values()) + print(json.dumps(checks, indent=1)) + print("RING", "OK (all identities re-proven in my own formal ring)" + if ok else "FAILED") + if args.out: + json.dump({"checks": checks, "ok": ok, "seed": SEED}, + open(args.out, "w"), indent=1) + sys.exit(0 if ok else 1) + + +# ===================================================================== +# Lemma L1 / C / D adversarial numerics (radians, my own code) +# ===================================================================== + +def cmd_lemma(args): + rng = random.Random(SEED) + bad = [] + n = 0 + # L1: d/dc [c cot(cs)] < 0 <=> sin(2cs) < 2cs; and monotonicity spot + for _ in range(4000): + s = rng.uniform(1e-6, math.pi / 2) + c1 = rng.uniform(1e-9, 1.0) + c2 = rng.uniform(1e-9, 1.0) + if c1 == c2: + continue + c1, c2 = min(c1, c2), max(c1, c2) + f1 = c1 / math.tan(c1 * s) + f2 = c2 / math.tan(c2 * s) + n += 1 + if not f1 > f2: + bad.append(("L1", s, c1, c2, f1 - f2)) + # F(v) = 2/sin 2v - (1/a)cot((pi/2-v)/a) - (b/a)cot(bv/a) < 0 + # on (0, pi/2), a >= 2, 1 <= b <= a (radians) + def Fv(v, a, b): + h = math.pi / 2 + return (2.0 / math.sin(2 * v) - (1.0 / a) / math.tan((h - v) / a) + - (b / a) / math.tan(b * v / a)) + + a_list = [2, 3, 4, 7, 20, 100, 1000, 10 ** 6] + for a in a_list: + for b in sorted({1, 2, max(1, a // 2), a - 1 if a > 1 else 1, a}): + if b < 1 or b > a: + continue + v0 = math.pi / (2 * (b + 1)) + vs = [1e-9, 1e-6, 1e-3, 0.01, v0 * 0.5, v0 * (1 - 1e-9), + v0, min(math.pi / 2 - 1e-9, v0 * 1.5), + math.pi / 4, math.pi / 2 * (1 - 1e-6), + math.pi / 2 * (1 - 1e-12)] + vs += [rng.uniform(1e-8, math.pi / 2 - 1e-12) + for _ in range(200)] + for v in vs: + if not (0 < v < math.pi / 2): + continue + n += 1 + fv = Fv(v, a, b) + if not fv < 0: + bad.append(("F", a, b, v, fv)) + # H2 > 0 on (0, v0), via the FACTORED form (cancellation-free) + for v in vs: + if not (0 < v < v0): + continue + n += 1 + phi = (math.tan(v) * math.sin((math.pi / 2 - v) / a) + / math.sin(b * v / a)) + phi0 = math.tan(v0) + h2f = (math.cos(v) * math.cos(v0) * math.sin(b * v / a) + * (phi - phi0)) + if not h2f > 0: + bad.append(("H2fact", a, b, v, h2f)) + # direct H2 must agree with the factored form (identity) + h2d = (math.sin(v) * math.sin((math.pi / 2 - v) / a) + * math.cos(v0) + - math.sin(v0) * math.sin(b * v / a) * math.cos(v)) + if abs(h2d - h2f) > 1e-9 * max(1.0, abs(h2d), abs(h2f)): + bad.append(("H2ident", a, b, v, h2d - h2f)) + # Lemma D: b cot(bx) - cot x < 0 on (0, pi/(2b)], b >= 2. Tested in + # the equivalent positive-quantity form tan(bx) > b tan(x) (valid + # since both tangents are positive there), sampling x relative to the + # domain end pi/(2b); x below ~1e-3 of the domain is float-untestable + # (difference ~ (bx)^3/3 underflows relative precision) and is covered + # by the hand proof via L1. + for b in [2, 3, 5, 17, 400, 10 ** 5]: + dom = math.pi / (2 * b) + for lam in ([1e-3, 1e-2, 0.1, 0.5, 0.9, 0.999, 1.0] + + [rng.uniform(1e-3, 1.0) for _ in range(200)]): + x = lam * dom + n += 1 + g = math.tan(min(b * x, math.pi / 2 * (1 - 1e-12))) \ + - b * math.tan(x) + if not g > 0: + bad.append(("D", b, x, g)) + out = {"seed": SEED, "n_checks": n, "violations": bad[:20], + "n_violations": len(bad), "ok": not bad} + print(json.dumps(out)) + if args.out: + json.dump(out, open(args.out, "w"), indent=1) + sys.exit(0 if not bad else 1) + + +# ===================================================================== +# threshold identity, exact +# ===================================================================== + +def cmd_threshold(args): + rng = random.Random(SEED) + ok = True + for _ in range(2000): + a = rng.randint(1, 60) + b = rng.randint(1, a) + k = rng.randint(0, 5) + al = F(rng.randint(1, 10 ** 6), rng.randint(1, 997)) + be = F(rng.randint(1, 10 ** 6), rng.randint(1, 997)) + theta = al + be + theta_d = F(90 * (a + b), a * (b + 1)) + v = 360 * k + 90 - a * al + w = al + (b + 1) * be - 90 + ok &= (a * w - b * v == a * (b + 1) * (theta - theta_d) + - 360 * k * b) + print("THRESHOLD identity a*w - b*v == a(b+1)(theta-theta_d) - 360kb:", + "OK (2000 exact rational checks)" if ok else "FAILED") + if args.out: + json.dump({"ok": ok, "n": 2000, "seed": SEED}, + open(args.out, "w"), indent=1) + sys.exit(0 if ok else 1) + + +# ===================================================================== +# my own case-tree adversarial scan +# ===================================================================== + +def signs(al, be, a, b): + N1 = -(math.cos(a * al) * math.sin(al) * math.sin(b * be) + + math.sin(a * al) * math.sin(be) + * math.cos(al + (b + 1) * be)) + N2 = math.cos(a * al) * math.sin((b + 1) * be) * math.sin(al + be) + N3 = math.cos((b + 1) * be) * math.sin(a * al) * math.sin(al + be) + return N1, N2, N3 + + +def cmd_casetree(args): + a, b = args.a, args.b + rng = random.Random(SEED + 1000 * a + b) + theta_d = 90.0 * (a + b) / (a * (b + 1)) + d2r = math.pi / 180.0 + hits, nearhits = [], [] + n = 0 + thetas = [theta_d * (1 - g) + for g in (0.0, 1e-12, 1e-9, 1e-6, 1e-4, 1e-3, + 0.01, 0.05, 0.2, 0.5, 0.8)] + for th in thetas: + cands = [] + # a*alpha near every multiple of 90 (all k, incl. aal > 360) + kmax = int(a * th / 90.0) + 2 + for k in range(1, kmax + 1): + base = 90.0 * k / a + for eps in (1e-10, 1e-7, 1e-4, 1e-2, 0.5): + cands += [base - eps, base + eps] + # (b+1)*beta near every multiple of 90 + jmax = int((b + 1) * th / 90.0) + 2 + for j in range(1, jmax + 1): + be = 90.0 * j / (b + 1) + for eps in (1e-10, 1e-7, 1e-4, 1e-2, 0.5): + cands += [th - (be - eps), th - (be + eps)] + # death corner accumulation + for eps in (1e-12, 1e-9, 1e-6, 1e-3, 0.1): + cands += [90.0 / a - eps, 90.0 / a + eps] + # random fill + cands += [rng.uniform(1e-8, th - 1e-8) for _ in range(args.random)] + for al in cands: + be = th - al + if not (al > 1e-12 and be > 1e-12): + continue + n += 1 + N1, N2, N3 = signs(al * d2r, be * d2r, a, b) + tol = args.margin + if (N1 > tol and N2 > tol and N3 > tol) or \ + (N1 < -tol and N2 < -tol and N3 < -tol): + hits.append((th, al, N1, N2, N3)) + elif (N1 > 0 and N2 > 0 and N3 > 0) or \ + (N1 < 0 and N2 < 0 and N3 < 0): + nearhits.append((th, al, N1, N2, N3)) + # positive control above theta_d + alc = 90.0 / a + thc = theta_d + 1e-4 + c = signs((alc - 1e-5) * d2r, (thc - alc + 1e-5) * d2r, a, b) + ctrl = c[0] > 0 and c[1] > 0 and c[2] > 0 + out = {"a": a, "b": b, "theta_d": theta_d, "seed": SEED, + "n_points": n, "n_hits": len(hits), "hits": hits[:10], + "n_nearhits_below_margin": len(nearhits), + "nearhits": nearhits[:10], + "positive_control_fires": ctrl, + "ok": (not hits) and ctrl} + print(json.dumps({k: v for k, v in out.items() if k != "nearhits"})) + if args.out: + json.dump(out, open(args.out, "w"), indent=1) + sys.exit(0 if out["ok"] else 1) + + +# ===================================================================== +# independent segment alive certification (interval unfolding, full word) +# ===================================================================== + +CHOP = 2 ** 320 # outward dyadic rounding grid (sound: only widens) + + +def chop_iv(x): + lo = F(math.floor(x.lo * CHOP), CHOP) + hi = F(math.ceil(x.hi * CHOP), CHOP) + return dsk.MyIv(lo, hi) + + +def iv_c(re, im): + return (re, im) # complex interval = (MyIv, MyIv) + + +def ivc_chop(u): + return (chop_iv(u[0]), chop_iv(u[1])) + + +def ivc_add(u, v): + return (u[0] + v[0], u[1] + v[1]) + + +def ivc_sub(u, v): + return (u[0] - v[0], u[1] - v[1]) + + +def ivc_mul(u, v): + return ivc_chop((u[0] * v[0] - u[1] * v[1], + u[0] * v[1] + u[1] * v[0])) + + +def ivc_conj(u): + return (u[0], -u[1]) + + +def certify_alive_at(a, b, t): + """Certify positive full-corridor width of W(a,b) at + (alpha, beta) = (90/a - t, 90(a-1)/(a(b+1)) + 2t), rational degrees, + with my own interval unfolding (004 trig stack). Returns + (certified_bool, width_lo_float).""" + MyIv, miv = dsk.MyIv, dsk.miv + ac, bc = F(90, a), F(90 * (a - 1), a * (b + 1)) + al, be = ac - t, bc + 2 * t + th = al + be + sd = lambda d: chop_iv(dsk.sin_deg(d)) + cd = lambda d: chop_iv(dsk.cos_deg(d)) + # vertices (circumdiameter 1): A = 0, B = sin th, C = sin be e^{i al} + zero = (miv(0), miv(0)) + A = zero + B = (sd(th), miv(0)) + C = (sd(be) * cd(al), sd(be) * sd(al)) + V = [A, B, C] + # direction-squared of sides: E0 = e^{-2i be}, E1 = e^{2i al}, E2 = 1 + Es = {0: (cd(2 * be), -sd(2 * be)), + 1: (cd(2 * al), sd(2 * al)), + 2: (miv(1), miv(0))} + SIDE_ENDS = {0: (1, 2), 1: (2, 0), 2: (0, 1)} + cs = {s: ivc_sub(V[SIDE_ENDS[s][0]], + ivc_mul(Es[s], ivc_conj(V[SIDE_ENDS[s][0]]))) + for s in range(3)} + word = family_word(a, b) + sigma, L, toff = 1, (miv(1), miv(0)), zero + gates = [] + for s in word: + i, j = SIDE_ENDS[s] + if sigma == 1: + gates.append((ivc_add(ivc_mul(L, V[i]), toff), + ivc_add(ivc_mul(L, V[j]), toff))) + L, toff = ivc_mul(L, Es[s]), ivc_add(ivc_mul(L, cs[s]), toff) + sigma = -1 + else: + gates.append((ivc_add(ivc_mul(L, ivc_conj(V[i])), toff), + ivc_add(ivc_mul(L, ivc_conj(V[j])), toff))) + newL = ivc_mul(L, ivc_conj(Es[s])) + toff = ivc_add(ivc_mul(L, ivc_conj(cs[s])), toff) + L = newL + sigma = 1 + assert sigma == 1 + # translation word: L must enclose 1 + assert L[0].lo <= 1 <= L[0].hi and L[1].lo <= 0 <= L[1].hi + tau = toff + # tau certified nonzero? + tau_nonzero = (tau[0].lo > 0 or tau[0].hi < 0 + or tau[1].lo > 0 or tau[1].hi < 0) + # projection p(z) = Im(conj(tau) z) + lo_hi, hi_lo = None, None # sup of interval-lo of gate max, etc. + LO_hi = None # upper bound on corridor lo + HI_lo = None # lower bound on corridor hi + LO_lo = None + HI_hi = None + for (zu, zv) in gates: + pu = ivc_mul(ivc_conj(tau), zu)[1] + pv = ivc_mul(ivc_conj(tau), zv)[1] + gmin_hi = min(pu.hi, pv.hi) + gmin_lo = min(pu.lo, pv.lo) + gmax_lo = max(pu.lo, pv.lo) + gmax_hi = max(pu.hi, pv.hi) + LO_hi = gmin_hi if LO_hi is None else max(LO_hi, gmin_hi) + LO_lo = gmin_lo if LO_lo is None else max(LO_lo, gmin_lo) + HI_lo = gmax_lo if HI_lo is None else min(HI_lo, gmax_lo) + HI_hi = gmax_hi if HI_hi is None else min(HI_hi, gmax_hi) + width_lo = HI_lo - LO_hi # certified lower bound on width + width_hi = HI_hi - LO_lo + return (tau_nonzero and width_lo > 0), float(width_lo), float(width_hi) + + +def cmd_segment(args): + members = [(2, 1), (6, 3), (6, 1), (8, 2), (8, 1), (7, 1), (12, 12)] + ts = [F(1, 4), F(1, 8), F(1, 97), F(1, 1024), F(1, 2 ** 20), + F(1, 2 ** 30)] + rows = [] + ok = True + from skeptic_family import width as sk_width + for (a, b) in members: + ac, bc = F(90, a), F(90 * (a - 1), a * (b + 1)) + for t in ts: + good, wlo, whi = certify_alive_at(a, b, t) + # float cross-check at the apex (002's corridor) + al = math.radians(float(ac - t)) + be = math.radians(float(bc + 2 * t)) + ta, tb = math.tan(al), math.tan(be) + x, y = tb / (ta + tb), ta * tb / (ta + tb) + wfl = sk_width(family_word(a, b), x, y) + rows.append({"a": a, "b": b, "t": str(t), + "gamma_minus_gamma_d": float(-t), + "certified_alive": good, + "width_iv": [wlo, whi], "float_width": wfl}) + ok &= good + print(f"W({a},{b}) t={t}: certified_alive={good} " + f"width>[{wlo:.3e}] float={wfl:.3e}") + out = {"rows": rows, "ok": ok, "seed": SEED, + "meaning": ("full 2n-gate corridor built by my own interval " + "unfolding (no glide reduction), certified " + "positive width at every sampled t on 005's " + "universal segment incl. t = 2^-30 " + "(gamma within 1e-9 of gamma_d)")} + print("SEGMENT", "OK" if ok else "FAILED") + if args.out: + json.dump(out, open(args.out, "w"), indent=1) + sys.exit(0 if ok else 1) + + +# ===================================================================== +# audit of plaw_suffice's certified trig layer +# ===================================================================== + +def reduce360(d): + """Exact reduction of rational degrees into [-180, 180).""" + d = F(d) + return d - 360 * ((d + 180) // 360) + + +def cmd_trigaudit(args): + """plaw_suffice's sin_deg/cos_deg enclosures must CONTAIN the true + value. Reference: 004's independent stack (dsk.sin_deg/cos_deg, + Machin pi, own series) -- valid only for small arguments, so the + rational degree argument is exactly reduced mod 360 first (sin/cos + are 360-periodic, so the true value is unchanged; plaw_suffice's own + internal reduction is part of what is under test, so IT gets the raw + argument). My reduced enclosure has width ~1e-84, so requiring it to + sit inside plaw's enclosure certifies true-value containment.""" + import plaw_suffice as ps # the code UNDER TEST + rng = random.Random(SEED) + ok = True + worst = 0.0 + n = 0 + for _ in range(args.trials): + num = rng.randint(-10 ** 7, 10 ** 7) + den = rng.randint(1, 9973) + d = F(num, den) + dr = reduce360(d) + for (name, theirs, mine) in ( + ("sin", ps.sin_deg(d), dsk.sin_deg(dsk.MyIv(dr))), + ("cos", ps.cos_deg(d), dsk.cos_deg(dsk.MyIv(dr)))): + n += 1 + if not (theirs.lo <= mine.lo and mine.hi <= theirs.hi): + ok = False + print(f"{name} POINT violation at d={d}") + worst = max(worst, float(theirs.hi - theirs.lo)) + # interval validity: for random degree intervals, every sampled + # interior rational point's true value must lie inside the enclosure + for _ in range(args.trials // 4): + lo = F(rng.randint(-10 ** 5, 10 ** 5), rng.randint(1, 997)) + hi = lo + F(rng.randint(1, 1000), rng.randint(1, 997)) + e_s = ps.sin_deg(lo, hi) + e_c = ps.cos_deg(lo, hi) + for _ in range(4): + lam = F(rng.randint(0, 1000), 1000) + d = lo + (hi - lo) * lam + dr = reduce360(d) + mys = dsk.sin_deg(dsk.MyIv(dr)) + myc = dsk.cos_deg(dsk.MyIv(dr)) + n += 2 + if not (e_s.lo <= mys.lo and mys.hi <= e_s.hi): + ok = False + print("SIN interval violation", lo, hi, d) + if not (e_c.lo <= myc.lo and myc.hi <= e_c.hi): + ok = False + print("COS interval violation", lo, hi, d) + out = {"n_checks": n, "worst_point_width": worst, "ok": ok, + "seed": SEED, + "note": ("earlier FAILED run used an UNREDUCED reference " + "argument (my stack has no mod-360 reduction and its " + "series is invalid past ~660 deg) -- an artifact of " + "my audit, not of plaw_suffice; see record")} + print(json.dumps(out)) + print("TRIGAUDIT", "OK" if ok else "FAILED") + if args.out: + json.dump(out, open(args.out, "w"), indent=1) + sys.exit(0 if ok else 1) + + +def miv_frac(d): + return dsk.MyIv(F(d)) + + +def main(): + ap = argparse.ArgumentParser() + sub = ap.add_subparsers(dest="cmd", required=True) + c = sub.add_parser("bridge") + c.add_argument("--members", type=int, default=40) + c.add_argument("--points", type=int, default=2) + c.add_argument("--out") + c.set_defaults(fn=cmd_bridge) + c = sub.add_parser("ring") + c.add_argument("--out") + c.set_defaults(fn=cmd_ring) + c = sub.add_parser("lemma") + c.add_argument("--out") + c.set_defaults(fn=cmd_lemma) + c = sub.add_parser("threshold") + c.add_argument("--out") + c.set_defaults(fn=cmd_threshold) + c = sub.add_parser("casetree") + c.add_argument("--a", type=int, required=True) + c.add_argument("--b", type=int, required=True) + c.add_argument("--random", type=int, default=4000) + c.add_argument("--margin", type=float, default=1e-9) + c.add_argument("--out") + c.set_defaults(fn=cmd_casetree) + c = sub.add_parser("segment") + c.add_argument("--out") + c.set_defaults(fn=cmd_segment) + c = sub.add_parser("trigaudit") + c.add_argument("--trials", type=int, default=200) + c.add_argument("--out") + c.set_defaults(fn=cmd_trigaudit) + args = ap.parse_args() + args.fn(args) + + +if __name__ == "__main__": + main() diff --git a/problems/billiards-triangles/prior-art.json b/problems/billiards-triangles/prior-art.json index a58b2b8..1f950f1 100644 --- a/problems/billiards-triangles/prior-art.json +++ b/problems/billiards-triangles/prior-art.json @@ -111,6 +111,139 @@ ], "refuted_by": null, "gaps": [] + }, + { + "id": "005", + "file": "attempts/005-complete-death-law-theorem.md", + "date": "2026-07-31", + "mode": "informed", + "status": "VERIFIED", + "mechanism": [ + "formal-parameter-specialization", + "symbolic-trig-closed-form", + "certified-interval-inequality", + "unfolding-word-census", + "out-of-sample-prediction" + ], + "one_line": "Death law completed on both sides: the half word composes to R_0 Rot_A(2a.alpha) Rot_B(-2b.beta), so I1-I4 and the glide facts become one exact 4-variable Laurent check valid for ALL (a,b); Lemmas C and D fall to an elementary c.cot(cs) monotonicity; the case tree is closed for all a >= b >= 1, a >= 2 (the a <= 2b+3 restriction removed); and certified alive segments into the death corner give death(W(a,b)) = gamma_d(a,b) EXACTLY (sup not attained) for 20 members including the newly measured W(6,3), W(6,1), W(8,2), W(8,1), W(7,1).", + "leak_terms": [ + "plaw_", + "R_0 Rot_A(2a alpha) Rot_B(-2b beta)", + "identity I4", + "margin N4", + "c cot(cs) monotone", + "Lemma L1", + "aw - bv = a(b+1)(theta - theta_d) - 360kb", + "rho = 2, t0 = 1/4", + "death corner (90/a, 90(a-1)/(a(b+1)))", + "585/4", + "255/2", + "285/2", + "1035/8", + "900/7", + "three-fold degenerate corner" + ], + "range": "I1-I4, D1, D2 and glide facts proven for ALL integers a, b >= 1 (formal ring; specialization cross-checked exactly on 23 members, floats to a = 40); Lemma C proven for all a >= 2, 1 <= b <= a and Lemma D for all b >= 1; necessity theorem for all integers a >= b >= 1, a >= 2 (no upper restriction on a); exact death = gamma_d certified for 20 members via alive segments t in (0, 1/4], gamma sweeping [gamma_d - 1/4, gamma_d); new float deaths for (6,3),(6,1),(8,2),(8,1),(7,1) at gamma_d to ~5e-12; adversarial case-tree scans ~1.56M points over 13 members", + "gaps": [ + "parametric-sufficiency-open", + "a-equals-1-column-unmeasured", + "birth-side-unproven" + ] + }, + { + "id": "006", + "file": "attempts/006-design-family-past-135.md", + "date": "2026-07-31", + "mode": "informed", + "status": "VERIFIED", + "mechanism": [ + "symbolic-trig-closed-form", + "unfolding-word-census", + "exact-rational-arithmetic", + "out-of-sample-prediction", + "structure-class-design-search" + ], + "one_line": "The 135 deg stall dissolves: 001 pinch gap was a sampler artifact (W(4,3) certified alive at gamma in (135.0000001, 135.048)), births obey a closed form gamma_birth = 180 - 90(a+b+1)/(a(b+1)) making staircase windows touch, the never-measured W(5,2) and a genuinely new four-block word are certified alive at gamma EXACTLY 135 (rational apexes on the arc, exact orbits), and the W family covers all 157 sampled obtuse arcs - while the design screens (3000+ canonical doubled words to length 50) find nothing of length <= 26 alive past 135 and show clean angle laws are confined to W's two-fan factorization.", + "leak_terms": [ + "design_", + "gamma_birth = 180 - 90(a+b+1)/(a(b+1))", + "W(5,2)", + "0(12)^3(02)^3(12)^2(02)^3", + "exactly 135", + "windows touch", + "(2x-1)^2 + (2y+1)^2 = 2", + "edge-accumulating sampler", + "136.8265" + ], + "range": "five exact certificates (rational apexes, Fraction corridors, independently simulated orbits): W(4,3) at certified gamma in (135.001,135.048) and (135.0000001,135.001); W(5,4) at (144.0000001,144.001); W(5,2) len 30 and non-W four-block N1 len 46 at gamma exactly 135 (exact on-arc identity); 003 obligations machine-certified for two new members (5,2),(6,2). Float screens sample-bounded: 245 canonical doubled words len <= 26 dead at 13 arcs >= 135; 811 len-30 words (only W(4,3),W(5,2) alive on 135-144); 2008 two-alternating words len 34-50 (only W members + two four-block words); three-block family a,b,c <= 3 (25/27 dead everywhere obtuse); birth law float-checked at 11 members; coverage checked at 157 arcs 90.5-165", + "gaps": [ + "birth-law-unproven (gamma_birth closed form is float-fitted + factor-structure motivated; no necessity/sufficiency theorem)", + "coverage-conjecture-open (union of windows covering (90,180) reduces to an elementary Diophantine statement, unproven)", + "length-26-stall-sample-bounded (doubled words only, 13 arcs; non-doubled even words 24-26 not rescreened)", + "four-block-death-angles-uncharacterized (binding differences provably non-elementary here only by failed factor search)" + ] + }, + { + "id": "007", + "file": "attempts/007-skeptic-review-of-006.md", + "date": "2026-07-31", + "mode": "informed", + "status": "VERIFIED_WITH_CORRECTIONS", + "mechanism": [ + "adversarial-verification", + "independent-reimplementation", + "exact-rational-arithmetic", + "out-of-sample-prediction" + ], + "one_line": "From-scratch re-verification confirms 006 everywhere it is exact: all five certificates re-established by a third corridor implementation (digit-for-digit widths) plus an own exact simulator, the exactly-135 on-arc identity re-derived by hand, 001's pinch-gap birth value regenerated from a replica of its own sampler (mechanism: the alive window is a corner sliver inside the grid's terminal 6e-4 clearance), the birth law survives three genuinely out-of-sample members to ~3e-11 (= sampler floor, including a > 2b+3), and the (5,2)/(6,2) theorem extension re-proves in an independent ring; four corrections, none load-bearing.", + "leak_terms": [ + "dsk_", + "566 not 811", + "canonical pairing (a,b) ~ (b+1,a-1)", + "corner sliver 6e-4", + "floor-tracking", + "W(8,2) birth 138.75" + ], + "range": "all 5 exact certificates re-verified (own matrix-form exact corridor, widths match digit-for-digit; own exact orbit simulation; own certified gamma brackets via 004's interval stack; on-arc gamma=135 identity hand-derived and Fraction-checked; Niven applicability checked); W(4,3) birth floor-tracked at 4 sampler depths; 001's census sampler replicated (reproduces 135.0486) and window x-extent measured at 7 gammas; birth law tested out-of-sample on W(7,3), W(6,3), W(8,2) (all +3.0e-11 = floor, dead-sampled below); coverage re-run at 25 own arcs (25/25 alive); universes recounted (245 and 2008 confirmed, 566-vs-811 correction); s1 hits reclassified (14 W + N1/N2); N1/N2 non-W checked incl. 0-1 swap; 60-word random dead re-test + 19 near-miss attacks (zero kills); identities + case tree (550k points) + Lemma C re-certified for (5,2),(6,2). NOT re-swept: full length-30 and s1 negative universes, three-block table, design_factor.py factorizations", + "verifies": "006", + "corrections": [ + "universe counts: the length-30 (|u|=15) universe is 566 canonical words, not 811 (811 is |u| <= 15 cumulative); 'all doubled odd words to length 30: 1056' double-counts (true count 811); the index one_line's '3000+' is 2846 counted correctly - screen coverage itself is unaffected", + "'length 30 is alive throughout the measured gap' overstates: certificates prove alive points AT gamma = 135 exactly and at one gamma in each of (135.0000001, 135.001) and (135.001, 135.048); the continuum statement is float evidence plus the SPECULATION window law", + "refutation semantics vs 001: 001's gap claim was knowledge-scoped ('nothing <= 30 KNOWN alive') and stays true as written; what 006 overturns is 001's birth measurements (error 0.0486 deg, 10x the stated ~0.005 sampling noise, whose 'deaths only under-estimated' hedge does not transfer to births) - correction/closure, not refutation", + "the window pairing (a,b) <-> (b+1, a-1) is a canonical-word identity (0<->1 swap + rotation/reversal; verified exactly on six pairs), so it is vacuous as evidence for the birth law, 006's 11-member fit spans only 9 canonical words, and 003's 'two distinct words dying on the same arc' (W(3,3)/W(4,2)) is a relabeling of one canonical word" + ], + "refuted_by": null, + "gaps": [] + }, + { + "id": "008", + "file": "attempts/008-skeptic-review-of-005.md", + "date": "2026-07-31", + "mode": "informed", + "status": "VERIFIED_WITH_CORRECTIONS", + "mechanism": [ + "adversarial-verification", + "independent-reimplementation", + "formal-parameter-specialization", + "symbolic-trig-closed-form", + "certified-interval-inequality" + ], + "one_line": "Adversarial re-verification confirms 005 end to end: the closed-form bridge u = R_0 Rot_A(2a.alpha) Rot_B(-2b.beta) recomposed exactly at off-torus rational points (40 members to a = 40), I1-I4 and glide facts re-proven in an independent 4-variable ring with u re-derived by symbolic map composition, Lemmas L1/C/D re-derived by hand and stressed to a = 10^6, the extended case tree re-derived branch by branch (threshold identity exact; 447k-point scan on 10 a > 2b+3 members, zero hits), the sufficiency segments independently certified alive by full-corridor interval unfolding down to gamma_d - 9.3e-10, and all five new deaths re-measured on the mirror half; one numeric correction, nothing load-bearing refuted.", + "leak_terms": [ + "psk_", + "off-torus pair algebra", + "gamma_d - 9.3e-10", + "447k", + "1e-4 not 1e-6" + ], + "range": "bridge recomposed exactly for 40 members incl. (40,40) at 2 rational points each; identities re-proven formally (my ring, 14/14); Lemma numerics 14,183 checks to a = 10^6; case tree scanned at ~447k adversarial points over 10 members with a > 2b+3 (k up to 2 reachable); segments certified alive at 42 points (7 members incl. the 5 new, t down to 2^-30); plaw_suffice trig layer audited (596 enclosure checks); 5 deaths re-measured with 004's sampler; plaw_suffice all NOT re-run end to end (selftest + one member reproduced, code audited)", + "verifies": "005", + "corrections": [ + "005 states the five new exact-alive certificates certify gamma within 1e-6 of gamma_d; the data files show gamma_d - 1e-4 for W(6,3), W(8,2), W(8,1), W(7,1) (only W(6,1) reaches 1e-6) - the --dg 1e-6 request is a float target, the certified bound is the largest 10^-k that certifies after apex snapping", + "caveat, not an error: the five new members' necessity rests on the parametric theorem alone (no per-member deathlaw_prove obligations were run for them, unlike 003's fifteen); sound by design since the identities are now proven for all (a,b)" + ], + "refuted_by": null, + "gaps": [] } ] }