diff --git a/STATUS.md b/STATUS.md index 60df929..6950748 100644 --- a/STATUS.md +++ b/STATUS.md @@ -17,22 +17,33 @@ 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 (2026-08-04): maxwell-equilibria onboarded -(new problem, E&M round per docs/PLAN-em-problems.md) and its first -attempt closed the marquee target: an explicit rational five-charge -configuration — units at e₁,e₂,e₃, charges 4367/1000000 at -(1/3,1/3,1/3) ± (17/200)(1,1,1) — is CERTIFIED to have exactly 24 -nondegenerate equilibria, > (5−1)² = 16: a concrete machine-checkable -instance of the six-day-old arXiv:2607.27197 refutation of Maxwell's -1873 conjecture, which the asymptotic preprint itself does not supply. -Complete census (472k boxes, all-positive localization, index sum −4), -independently re-verified leaf-by-leaf by the second engine (different -arithmetic + fixed-point theorem), both PASS. Escalation per contract: -skeptic review of the shared subdivision driver queued (new queue 16) -before this counts as settled. Float side-finding: the 24-count window -needs ε ≲ 0.18 and is ~0.5% wide in q. Previous headline (2026-07-31): -the two-track billiards plan (old queue 11+12); all four records are -skeptic-confirmed, six corrections total, none load-bearing. +Current standing. HEADLINE (2026-08-04, second cycle): the +maxwell-equilibria escalation is DISCHARGED — skeptic review (002) +confirms 001's certified 24-equilibria witness: a shifted+enlarged-region +re-census whose tree shares no box with 001's again certifies exactly 24 +(and the independent verifier PASSes it), an 80-bit re-run reproduces +every leaf verdict, and the fold brackets 001 predicted are now certified +counts — 12 at q = 4360/10⁶, 16 at q = 4400/10⁶. The centroid Hessian +closes exactly: signature decided by 6561t⁶ vs 128q², degeneracy at +q\* = 81√2·t³/16 ≈ 4.3968·10⁻³, certified index flip across it. Real +correction: verify_equilibria.py's tiling check (prefix-freeness + Kraft) +is UNSOUND for axis-labelled paths — coverage independence silently +rested on driver honesty; closed for 001's certificate by an +axis-consistency reconstruction (0 violations / 235,993 paths), hardening +queued. The explicit refutation witness is now settled library fact. +Union-closed: the i-AGGREGATED odds-ratio control SURVIVES the mandated +large-n probe (014, skeptic-confirmed 015) — certified positive in exact +rationals to n = 32 (15 certificates, softest +5.99e-7; independent +re-certification of 5, zero disagreement), float trends to n = 64, +minimum margin GROWING in n (the 012-style reversal was hunted for and +not found, including on 012's own genre), ~1900 instances with zero +in-regime violations. Products are the exact equality set (λ > 0, +inclusion proven; converse open), so the gap is a boundary-positivity +question — proof effort is now justified by the queue's own criterion. +Previous headlines: 2026-08-04 first cycle — maxwell-equilibria onboarded, +001 certified the explicit rational five-charge witness (24 > 16) for the +six-day-old arXiv:2607.27197 refutation; 2026-07-31 — the two-track +billiards plan, all four records skeptic-confirmed. 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 @@ -67,7 +78,7 @@ attempts (queue; run blind). | Problem | Status | Budget | Active line | |---|---|---|---| | Erdős–Gyárfás | n=24 done | high | next: 2-connected C16-free test (lead 2); n=26 is a ~16h run | -| Union-closed (Frankl) | ROUTE LIVE (verified) | high | gap (a′) refuted (007/013) → live gaps: aggregated OR control (probe n ≳ 20 BEFORE proof effort), δ-linear tax budget, corrected assembly budgets (012); plus three sweep leads (010) | +| Union-closed (Frankl) | ROUTE LIVE (verified) | high | aggregated OR control survived the mandated probe: certified to n = 32, margin growing (014/015) → proof effort now justified (queue 1: Gram/Frobenius + product-boundary Hessian program); δ-linear tax budget, corrected assembly budgets (012); three sweep leads (010) | | Erdős–Straus | 601 resolved | medium | next: prove the identity-poor mechanism (why QR classes force low f) | | Singmaster | census done | medium | next: Diophantine curve table (search-deeper is now low value) | | Lonely runner | k=8 done | medium | next: k=9 scan (k=8 likely settled by Rosenfeld preprint) | @@ -76,11 +87,11 @@ attempts (queue; run blind). | 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) | -| Maxwell equilibria | explicit 24-equilibria witness VERIFIED (001, both engines) | high | next: skeptic review of the driver (queue 16, escalation), certified window/fold brackets + exact centroid Hessian (001 leads 2, 5); n=3 census (queue 17) untouched | +| Maxwell equilibria | 24-equilibria witness SETTLED: skeptic-confirmed, escalation discharged, fold brackets 12/16 certified, centroid degeneracy exact (001+002) | high | next: harden verifier tiling check (queue 16, tier-0 fix); blind 3-charge strata map (queue 15); n=3 census hunting 4-vs-6 (queue 17); certified window edges + q\* sliver (002 leads 3-4) | ## Attempt queue (next cycles pull from the top) -1. [union-closed] The restated gap: i-AGGREGATED odds-ratio control (007's replacement after the per-i averaged form died — see 007/013). FIRST extend the certified probe to larger n: 012's orbit-symmetrized engine reaches n = 32, and its budget-growth reversal past n ≈ 22 is a standing warning against small-n optimism — the aggregate is certified positive only at n ≤ 7. If it survives n ≳ 20, attack the proof through 007's Gram/Frobenius machinery (PSD mass matrix; first-order-in-λ perfect square) and the margin-modulated variant. Any claimed bound → skeptic review before ledger entry. +1. [union-closed] ~~Probe the aggregated control at n ≳ 20~~ **DONE in 014/015** (survives, certified to n = 32, margin growing). Replacement — the proof program the probe now licenses: products are the exact equality set (λ > 0), so attack **boundary positivity**: (i) the product-boundary Hessian of A — minimize its quadratic form over perturbation directions (eigenproblem per (n, p, λ) via the orbit engine); a negative in-regime eigendirection kills the gap, a PSD proof settles the boundary layer. Mind 015's R1: fitted orders dip to d^1.39 at λ ≥ 0.5, so a Hessian-PSD result alone gives no finite-d positivity radius — a cubic-remainder bound is part of the job. (ii) Settle the equality-set converse for λ > 0. (iii) Map the near-zero slice-direction dip (+1.4e-7 at n ≈ 48, recovers by 64) over p × layer-ratio × n ≤ 96, tracking the λ→0 first-order coefficient. (iv) Run the margin-modulated variant (007 lead 2) through 014's battery — one-line weight change. Any claimed bound → skeptic review before ledger entry. 2. [union-closed] Rebuild the assembly budgets per 012's corrections: the tax is δ-LINEAR (restate B2), the τ_half step needs s₀ ≤ 0.0843, corrected δ₀ ≈ 0.004. The open question is whether ANY n-uniform budget object exists — 008's conditional theorem survives structurally; its constants need the corrected inputs, and the budget census machinery (uc_pert.py + uc_pert_skeptic.py's orbit engine) is ready. 3. [union-closed] Sweep leads from 010, one per cycle: (i) pairwise-closure LP/degree-2 SOS certification on the n ≤ 4 census (all 4958 families) — kill: certified bound converges to ≈ 0.382; (ii) union-transfer-operator eigenvalue field — test log-supermodularity of λ_C on the n ≤ 5 census (pre-check the total-positivity records first); (iii) bipartite-MIS decomposition — geng census to n = 12, connectivity of extremals, compositionality across 1- and 2-cuts. Kill conditions recorded in 010. 4. [erdos-straus] Prove the identity-poverty mechanism: why does QR-class membership mod 840 force fewer Type I covering congruences? Start from 001's obstruction analysis + 002's rate data; target a theorem "f(p) ≥ g(N_typeI(p))" or a disproof. @@ -95,7 +106,7 @@ attempts (queue; run blind). 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] 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. 15. [maxwell-equilibria] **Run blind.** First attempt: certified counts for structured 3-charge families beyond the harness self-test knowns — collinear with unequal charges (does the count stay 2 or drop?), isoceles families, a coarse (shape × charge-ratio) sweep. Deliverable is the count strata map, `EVIDENCE` scoped by grid and region. Every complete count must pass the index-sum identity; treat a violation as a harness bug, not a finding. -16. [maxwell-equilibria] ~~Explicit certified witness for the arXiv:2607.27197 refutation~~ **DONE in 001** (witness certified, both engines PASS). Replacement item — the escalation the contract demands: **skeptic review of 001**, fresh eyes on the shared subdivision driver (the residual common-mode risk: both engines re-check the same leaf tree, so a driver bug that produced a wrong tree AND self-consistent leaves would slip through). Concrete: re-run the census at different precision/min-width/box offsets (tree changes wholesale; certified count must not), verify the 24 enclosures pairwise disjoint exactly, audit the Krawczyk-verdict and leaf-emission code paths by hand. Cheap add-ons that bracket the structure: certified counts at q = 4360/1000000 (expect 12) and 4400/1000000 (expect 16) — 001 lead 2 — and the exact rational centroid-Hessian fold analysis (001 lead 5). +16. [maxwell-equilibria] ~~Skeptic review of 001~~ **DONE in 002** (confirmed-with-corrections; escalation discharged; brackets 12/16 certified; centroid Hessian exact). Replacement — harden the verification chain per 002's findings: (i) fix `verify_equilibria.py`'s tiling check with the axis-consistency reconstruction (prefix-freeness + Kraft is provably insufficient for axis-labelled paths — 002 R1) and add the two tamper-demo certificates from `explore/skeptic_verifier_gap_demo.py` as regression tests. Tier-0 change: keep it generic (a checker fix, no findings in prose), tests must pass. (ii) Guard `krawczyk()`'s rounded midpoint at low precision / deep min-width (002's latent hazard: margin 2⁻³¹ at 001's params, untriggered). (iii) Optional certified add-ons: binary-search the window edges q ∈ (4.389, 4.400)·10⁻³ and the sliver census q ∈ (4.389, 4.3968)·10⁻³ separating the 24→16 edge from the centroid degeneracy q\* = 81√2·t³/16. 17. [maxwell-equilibria] Three-positive-charge census hunting the open 4-vs-6 gap: after 15's strata map, target the strata boundaries (where counts jump) with unequal charges. Win condition kept in view by every run: any configuration certified with ≥ 5 isolated equilibria refutes the conjectured max 4 and is a result people have sought since 1873 — it must survive verify_equilibria.py and be reported as requiring escalation. Equal-magnitude configurations are settled (Tsai 2015, max 4) — spend no effort there. ## Verified results @@ -345,6 +356,53 @@ attempts (queue; run blind). only for ε ≲ 0.18 and spans ~0.5% in q. Escalation open: skeptic review of the shared driver (queue 16) before this is settled library fact. +- **[maxwell-equilibria] Witness settled: 001 skeptic-confirmed, fold + brackets certified, verifier gap found and closed for this certificate** + (2026-08-04, attempt 002): the escalation is discharged. A + shifted+enlarged-region re-census (472,775 boxes, tree sharing no box + with 001's) certifies exactly 24 again and PASSes the independent + verifier; an 80-bit re-run reproduces 001's tree verdict-by-verdict + (splits are grid-independent — precision perturbs nothing, the region + shift carries the independence weight). 001's float predictions are now + certified counts: 12 at q = 4360/10⁶ (index +4/−8), 16 at q = 4400/10⁶ + (+6/−10; 2.2× tree, near-degenerate). Fresh-code structural checks on + 001's certificate all pass in exact ℚ (tiling proof, 24 enclosures + pairwise disjoint with min gap 1.22e-3, index sum −4, region ⊇ ball). + Correction of record (R1): the checker's prefix-freeness + Kraft tiling + test is unsound for axis-labelled paths — a tampered certificate hiding + a real equilibrium PASSes it; coverage independence rested on the + (audited, correct) driver. Closed here by a strictly stronger + axis-consistency tree reconstruction: 0 violations over all 235,993 + paths. Exact centroid Hessian (001 lead 5): signature decided by + 6561t⁶ vs 128q², degeneracy q\* = 81√2·t³/16 ≈ 4.3968·10⁻³, certified + index flip +1 → −1 across it (matches C/D certs). Hardening leads in + queue 16. + +- **[union-closed] Aggregated odds-ratio control survives the mandated + large-n probe (skeptic-confirmed)** (2026-08-04, attempts 014+015): the + restated gap A(μ,λ) ≥ 0 (013's i-aggregated control) holds on every + instance tried — ~1900 total: 840 orbit-census (14 families × λ grids + incl. the 4.847/(n−3) window law, n ∈ {10,…,32}), 432 perturbative, 426 + sparse structured (the 007/013 witness genre re-laid-out per n, + multi-gadget stacks, crash/mirror), 82 aggregate-objective hill-climbs, + float trends to n = 64. Fifteen exact-rational certificates (Fraction + end-to-end, directed log₂ enclosures, widths ≤ 5.9e-12), softest + +5.988e-7 at n = 32. Minimum census margin GROWS monotonically with n + (crashmix at small λ: +4.9e-4 at n = 10 → +1.2e-3 at n = 32) — the + 012-style reversal was hunted for and not found, including on 012's own + genre. Structure: product measures are the exact equality set for + λ > 0 (inclusion proven; converse open — and false at λ = 0), so + perturbative departures rise ~quadratically and the tightest census + points are boundary noise. Skeptic (015): independent certifier on a + different algorithm re-certified 5/15 with zero disagreement (~200× + tighter), orbit formulas re-derived and cross-checked at n = 8, 9, dip + row (+1.4e-7 at n ≈ 48, recovers by 64) reproduced digit-for-digit and + shown not to be a grid artifact, ~116 new adversarial instances in + regions 014 could not see — all positive. Four reporting-level + corrections (fitted-order range d^1.39–d^2.06, converse overstatement, + docstring runtimes, instance count 1820). Finite-n `EVIDENCE`; proof + program in queue 1. + ## Insights / cross-problem notes - Infrastructure (2026-07-27): cross-pollination layer added. `mechanisms.json` @@ -492,6 +550,30 @@ attempts (queue; run blind). (always half-even, per spec) — directed sqrt bounds must be established by hand; caught by the verifier's own validation suite. +- Prefix-freeness + Kraft equality does NOT imply a tiling for + axis-labelled split paths (2026-08-04, maxwell 002): {"01","11"} is + prefix-free with Kraft sum 1 and tiles nothing — so a checker that + verifies coverage combinatorially from split paths accepts non-covering + trees, and a tampered certificate that hides a real equilibrium PASSes. + The lesson generalizes: an "independent" verifier must re-establish + coverage from leaf geometry (axis-consistency reconstruction), not from + combinatorial summaries the driver emitted; otherwise coverage + independence silently reduces to trusting the driver. Same cycle, same + family: changing working precision does not perturb a width-driven + subdivision tree at all (identical 471,985 boxes at 64 and 80 bits) — + region offsets, not precision, are the real tree-independence test. +- Probe-before-proof paid for itself twice (2026-08-04, union-closed + 014/015): the mandated large-n probe both cleared the proof gate AND + reshaped the proof target — products are the exact equality set of the + aggregated control (λ > 0), so the conjecture is boundary positivity of + a functional vanishing identically on products, and the census minima + are boundary noise rather than adversaries. Two engine-methodology + notes that made the certificates cheap: integer-scaled rows make the + aggregate denominator exact, and a from-scratch certifier on a + different enclosure algorithm (binary digit-extraction log₂ vs atanh + series) is a fast, decisive skeptic instrument (5 re-certifications, + nested intervals, zero disagreement). + ## Dead ends - **[union-closed] Pointwise Plackett odds-ratio control (gap (a) as diff --git a/problems/maxwell-equilibria/attempts/002-skeptic-review-of-001.md b/problems/maxwell-equilibria/attempts/002-skeptic-review-of-001.md new file mode 100644 index 0000000..a718053 --- /dev/null +++ b/problems/maxwell-equilibria/attempts/002-skeptic-review-of-001.md @@ -0,0 +1,416 @@ +# 002 — Skeptic review of 001 (abk explicit witness): adversarial verification + +- **Problem:** maxwell-equilibria, `problems/maxwell-equilibria/PROBLEM.md` +- **Date:** 2026-08-04 +- **Mode:** informed +- **Type:** adversarial verification of `001-abk-explicit-witness.md` (default + stance: refute). Target per STATUS queue item 16 / 001 lead 1: both engines + share the subdivision tree, so the residual common-mode risk is a bug in the + subdivision DRIVER producing a wrong tree with self-consistent leaves. +- **Outcome in one line:** the certified count of 24 nondegenerate equilibria + is REAL and survives every attack tried — including a re-census over a + wholesale-different subdivision tree and a finer-grid rerun of the original + one — but the review found a genuine + soundness gap in the *independent checker's* tiling check (prefix-freeness + + Kraft equality do NOT imply a tiling for axis-labelled paths; demonstrated + by a tampered certificate that hides a real equilibrium and still PASSes + verify_equilibria.py), so 001's "established by two independent + implementations" overstated the independence of the coverage claim. The gap + is closed for 001's actual certificate by a strictly stronger + axis-consistency tree check written fresh for this review. +- **Tools:** `explore/skeptic_cert_checks.py` (independent structural checks, + fresh code, exact rational arithmetic, no harness imports; ~4 s on the + 34 MB certificate); `explore/skeptic_verifier_gap_demo.py` (the tiling-gap + demonstrations, ~1 min); `explore/skeptic_centroid_hessian.py` (exact + centroid eigenvalue signature, instant); `harness/maxwell-equilibria/ + equilibria.py` re-runs at perturbed parameters (configs in + `explore/skeptic_*_config.json`, runtimes below); + `harness/maxwell-equilibria/verify_equilibria.py` on a perturbed + certificate. All deterministic, standard library only. +- **Sources:** attempt 001 and its certificate + `data/abk-witness-cert.json.gz`; the two harness files (read line by line); + no external papers consulted beyond what 001 cites. + +## Claims attacked + +1. **Coverage**: the driver's subdivision loop cannot drop a box, emit + overlapping leaves, or leave a hole; the certificate's 235,993 leaves tile + the search region exactly. +2. **The 24 isolation certificates**: pairwise disjoint enclosures, boxes + consistent with their split paths, enclosures inside their boxes. +3. **Nondegeneracy and index data**: all det signs ±1, index sum −4 = 1 − 5. +4. **Completeness scope**: the localization lemma and its instantiation + (the region really contains every equilibrium); the domination radii. +5. **Independence of engine 2**: every place `verify_equilibria.py` trusts + driver metadata rather than re-deriving it. +6. **Robustness of the count to the tree**: re-censuses at different + precision and over a geometrically different box decomposition — the tree + must change wholesale, the certified count must not. +7. **001's float side-predictions** (add-on): q = 4360/1000000 gives a + complete count of 12; q = 4400/1000000 gives 16. +8. **001 lead 5** (add-on): the centroid (1/3,1/3,1/3) is a rational + equilibrium of the whole family; its Hessian signature is decidable in + exact arithmetic. + +## Refutations found + +### R1. The independent checker's tiling check is not sound by itself: +### prefix-freeness + Kraft equality do not imply a tiling + +`verify_equilibria.py` validates the leaf decomposition by (i) prefix-freeness +of the split paths and (ii) the Kraft equality Σ 2^−depth = 1, and its +docstring claims this "holds iff the leaves tile the root". **That equivalence +is false for paths carrying a per-node axis choice.** The leaf set +`{"01", "11"}` (axis-0 upper half and axis-1 upper half of the root) is +prefix-free with Kraft sum exactly 1, yet the two boxes overlap on the +(hi, hi) x-y quadrant and jointly miss the (lo, lo) quadrant. Kraft counting +is blind to the axis digits; the claimed equivalence only holds *given* that +all paths through each internal node agree on that node's split axis — which +is exactly the driver-honesty assumption the checker was supposed to +discharge. + +Demonstrations (`python problems/maxwell-equilibria/explore/ +skeptic_verifier_gap_demo.py`, deterministic, ~1 min): + +- **DEMO 1**: a synthetic certificate with leaves `{"01", "11"}` over a + region where the field is sign-definite (the checker's own validation + region), every leaf genuinely sign-excluded. `verify()` returns **PASS** + on a non-tiling leaf list. +- **DEMO 2 (the load-bearing one)**: the driver's honest certificate for two + unit charges at (±1,0,0) — exactly one equilibrium, at the origin — is + tampered with: the isolated leaf is replaced by two mixed-axis exclusion + leaves that each avoid the origin but jointly leave a hole containing it, + and the count is edited from 1 to 0. `verify_equilibria.py` returns + **PASS with count 0**: the checker endorses a "complete census" that + misses a real equilibrium. (The same leaf lists are rejected by this + review's tree reconstruction, at the mixed-axis node.) + +Consequences, stated precisely: + +- What is actually wrong: 001's sentence "established by two independent + implementations ..." and the harness docstring's "exact combinatorial check + (Kraft equality + prefix-freeness) that the leaf boxes tile the search + region". The *coverage* half of the two-engine claim was never + independently verified; it rested on the driver constructing + axis-consistent trees. +- What is **not** wrong: the driver (`equilibria.py`) demonstrably emits both + children of every split with the same axis digit (step 5 of + `count_equilibria` pushes `path + f"{axis}{side}"` for both sides of one + chosen `axis`), so certificates it actually produces do tile. The gap is in + what the checker *catches*, not in any existing certificate. +- The fix applied here (checker hardening left as lead 1 — no harness edit in + this review): the tiling claim for 001's actual certificate is + re-established by a strictly stronger check written fresh for this review + (`skeptic_cert_checks.py`, TREE block): reconstruct the subdivision tree + from all 235,993 paths and verify at every internal node that (a) all + paths agree on the split axis and (b) both children are present. By + induction this proves the leaves partition the root box. **001's + certificate passes with zero violations** (max depth 58 splits; Kraft and + prefix-freeness re-confirmed as corollaries). + +### R2. Reporting-level: the "independent" det-sign verification leans on an +### unstated (but valid) argument + +The checker computes det signs over the driver's recorded `enclosure` without +re-deriving that the zero lies in it (the driver knows this from its own +Krawczyk image; the checker never recomputes an enclosure). This turns out to +be sound anyway, by an argument neither file states: success of the checker's +interval Gauss elimination on Y·J(B) forces every matrix in the interval +Jacobian J(B) to be nonsingular; J(B) is an entrywise-interval matrix set, +hence convex and connected, so det has one sign on all of J(B); DF at the +true zero and DF over any sub-box of B all lie in J(B); hence the sign +computed over the enclosure equals the index of the zero regardless of where +in B the zero sits. Recorded so the soundness of the index data does not +silently rest on an unstated lemma. No numerical conclusion changes. + +## Claims that survive (and what was thrown at them) + +### S1. Driver audit: no box can be dropped, no leaves overlap, the region +### is as claimed (attacks 1, 4 — hand audit, line by line) + +- **No dropped boxes**: every box popped in `count_equilibria` either appends + exactly one leaf (`continue` paths: outside / ball / sign / newton / + isolated / unresolved) or pushes both children of one split; the only other + exit is the `max_boxes` RuntimeError, which aborts the entire run with no + certificate written. There is no code path that discards a box silently. +- **No overlap**: splits are exact Fraction midpoints; children share only a + face; leaves are interior-disjoint by construction — and now also verified + combinatorially from the emitted paths (R1 fix). A zero exactly on a shared + face cannot be double-counted: a Krawczyk-unique verdict puts the leaf's + single zero in the leaf's *interior*, so a face zero would be a second zero + in the closed box and the verdict could not have been issued; face zeros + can only surface as unresolved leaves (001 has none), and the offset region + makes them unreachable for symmetric rational points. +- **Unresolved is honest**: a box at minimum width that certifies nothing is + emitted as `unresolved` and counted; 001's certificate has zero. +- **Localization region cannot be smaller than claimed**: `auto_region` uses + half-width r_up + 1/16 with offsets 1/97, 1/101, 1/103 < 1/16, so the + region contains the closed ball |x| ≤ R0 for any outward rounding of + r_up ≥ R0. The lemma itself re-derived: for |x| ≥ R0, x·F(x) = + Σ qᵢ x·(x−xᵢ)/rᵢ³ has every term ≥ 0 (Cauchy–Schwarz plus |x| ≥ |xᵢ|), and + simultaneous equality would force x = xᵢ for every i — impossible for ≥ 2 + distinct charges. The exclusion test (`box_min_norm_sq(box) >= r0_sq`, + exact) therefore correctly excludes the closed outside including the + boundary sphere. Re-checked exactly for all 15 excluded-outside leaves, + with R0² = 1 recomputed from the config (REGION, BALL blocks — pass). +- **Domination lemma**: radii re-instantiated exactly with this review's own + isqrt bounds (certified *lower* bounds for d_ij make the inequality + conservative); all 18 excluded-ball leaves re-checked exactly. Pass. +- **Krawczyk verdict line**: operator formula, strict `inside_open`, strict + `disjoint`, enclosure = K ∩ box, det-sign fallback to the full box — all + match the standard statements; the mean-value emptiness argument is valid + for any fixed rational Y, so the float-computed preconditioner is outside + the trust chain on both engines. One latent hazard found, not triggered: + see "what survived" below. +- **Shared-assumption inventory** (attack 5), beyond R1/R2: both engines use + the same F/DF formulas (unavoidable; anchored by the classical validation + knowns), both use a float preconditioner Y (certificates Y-independent by + construction), the checker's `excluded-newton` fallback reuses the + Krawczyk *theorem* (different arithmetic and code, same mean-value form — + independence there is implementational, not mathematical), and both parse + the same config JSON (checked by hand: it is the configuration 001 + states). The checker re-derives both lemmas and the region containment; + domination radii values are taken from the certificate but only as + candidates whose defining inequality is re-proved. The checker also PASSes + certificates with unresolved leaves (count = lower bound, by design) and + silently skips the index-sum identity if any det sign cannot be + re-established — harmless for 001 (0 unresolved, all 24 signs + re-established) but worth knowing when reading future PASS verdicts. + +### S2. Structural checks on the actual certificate (attacks 1–3), fresh code + +`python problems/maxwell-equilibria/explore/skeptic_cert_checks.py + --centroid --expect-count 24` — all checks pass: + +- TREE: full axis-consistency reconstruction of all 235,993 leaf paths + (the R1 fix); Kraft = 1 and prefix-freeness re-confirmed independently. +- BOX: all 24 recorded isolated boxes equal the boxes rebuilt from their + paths by this review's own midpoint code; every enclosure sits inside its + leaf box; isolated records ↔ isolated leaves are 1:1. +- DISJ: the 24 isolation enclosures are pairwise disjoint by exact rational + comparison (276 pairs; minimum separating gap ≈ 1.220·10⁻³ — consistent + with 001's float minimum position separation 1.51·10⁻³ less enclosure + widths). +- INDEX: all det signs ±1; sum = −4 = 1 − 5 (10 positive, 14 negative), + recomputed from the leaf data. +- TALLY: leaf-kind tallies match the certificate summary exactly + (7262 sign / 228674 newton / 18 ball / 15 outside / 24 isolated, + 0 unresolved). +- CENTROID: (1/3,1/3,1/3) lies in exactly one enclosure. + +### S3. The count is robust to the subdivision tree (attack 6) — +### four full re-censuses, every tree different, every count as predicted + +All runs single-core from the repo root, `--max-boxes 3000000`, +deterministic (wall times below are under 4-way CPU contention; 001's +baseline 1984 s was uncontended). Baseline (001): 471,985 boxes / 235,993 +leaves, count 24, index sum −4. Certificates gzipped in `data/` as +`skeptic-002-cert{A,B,C,D}.json.gz`. + +| run | perturbation | config / flags | boxes / leaves | secs | count | index sum | +|---|---|---|---|---|---|---| +| A | arithmetic grid: `--prec-bits 80`, same auto region | `data/abk-witness-config.json` | 471,985 / 235,993 | 2609 | **24** | −4 | +| B | wholesale different geometry: explicit region, offsets 1/89, 1/91, 1/93, half-width 5/4 (vs auto 1/97, 1/101, 1/103, 17/16) | `explore/skeptic_region_config.json`, `--prec-bits 64` | 472,775 / 236,388 | 2539 | **24** | −4 (from leaf data) | +| C | charge q = 4360/1000000 (window bracket, low) | `explore/skeptic_q4360_config.json`, `--prec-bits 64` | 359,577 / 179,789 | 1988 | **12** | −4 (+4/−8) | +| D | charge q = 4400/1000000 (window bracket, high) | `explore/skeptic_q4400_config.json`, `--prec-bits 64` | 1,022,511 / 511,256 | 5307 | **16** | −4 (+6/−10) | + +Reproduce with (outputs land wherever `--out` points; the gzipped copies in +`data/` are byte-identical up to the `seconds` field): + + python harness/maxwell-equilibria/equilibria.py \ + --config problems/maxwell-equilibria/data/abk-witness-config.json \ + --out certA-prec80.json --prec-bits 80 --max-boxes 3000000 + python harness/maxwell-equilibria/equilibria.py \ + --config problems/maxwell-equilibria/explore/skeptic_region_config.json \ + --out certB-region.json --prec-bits 64 --max-boxes 3000000 + python harness/maxwell-equilibria/equilibria.py \ + --config problems/maxwell-equilibria/explore/skeptic_q4360_config.json \ + --out certC-q4360.json --prec-bits 64 --max-boxes 3000000 + python harness/maxwell-equilibria/equilibria.py \ + --config problems/maxwell-equilibria/explore/skeptic_q4400_config.json \ + --out certD-q4400.json --prec-bits 64 --max-boxes 3000000 + +Notes: + +- **Run A came back with the *identical* tree** — the same 471,985 boxes, + leaf-by-leaf identical (path, kind) lists and identical isolated boxes and + det signs; only the isolation enclosures tightened (Krawczyk images on the + finer 2⁻⁸⁰ grid). Checked by direct list comparison against 001's + certificate. This is *expected on reflection* — split axis (max width) and + midpoints are exact-arithmetic decisions independent of the grid, so the + tree can only change where a certification test flips outcome, and every + one of the ~472k test outcomes has margin far above the 2⁻⁶⁴ → 2⁻⁸⁰ + rounding difference. So A is evidence that no leaf verdict in 001 is a + rounding-margin artifact, but it is a *weak* independence check for the + tree itself; run B is the one that carries the tree-perturbation weight + (different region ⇒ no box in common with 001's tree; leaf count 236,388 + vs 235,993, max depth 65 vs 58, 26 ball-exclusions vs 18, no + outside-localization leaves — and the same 24 zeros, +10/−14). +- Run B supplies the region explicitly, so the driver does not itself claim + completeness (`complete: false`) and skips the outside-localization + shortcut and the index-sum field (that is why its log reports + `index_sum=None`); the census is nevertheless complete because the region + contains the closed ball |x| ≤ R0 = 1 (checked exactly on the run-B + certificate) and the localization lemma was re-derived in S1. Its index + sum, recomputed from the leaf data, is −4 (10 positive, 14 negative). +- Every run was pushed through `skeptic_cert_checks.py` — all checks pass on + all four certificates (C with `--expect-count 12`, D with + `--expect-count 16`; the centroid lies in exactly one enclosure in every + run). Run B — the tree-perturbation run — was additionally re-verified + end-to-end by the second engine: + `python harness/maxwell-equilibria/verify_equilibria.py --cert + certB-region.json` → **PASS, zero failures, 1098 s**: all 236,388 leaves + re-established (the 24 isolations by preconditioned interval Newton, the + 26 ball-exclusions and both lemmas re-derived). Its report shows + `index_sum: null` because the verifier runs the Poincaré–Hopf identity + only for certificates claiming `complete: true` — a benign, scope-honest + limitation (the checker refuses to infer a completeness the certificate + did not claim), *distinct* from the R1 gap; the leaf det signs it did + re-establish sum to −4 (INDEX check above). +- **Run D doubles as a conditioning stress test**, and the exact centroid + Hessian below explains why: q = 4400/1000000 sits only 3.2·10⁻⁶ above the + exact centroid degeneracy q\* (S4), where det DF(centroid) ≈ −1.6·10⁻⁸, + a thousand times smaller than 001's minimum |det DF| ≈ 1.3·10⁻⁵. The + driver needed 1.02M boxes (2.2× the witness run) and isolated the + centroid at depth 64 (enclosure width ≈ 2.8·10⁻⁷) — with zero unresolved + leaves. A useful calibration: the harness still resolves a census this + close to a degeneracy at 64 bits. + +### S4. The add-on predictions (attacks 7, 8) + +- **Window brackets**: certified complete counts **12** at q = 4360/1000000 + and **16** at q = 4400/1000000 — exactly 001's float predictions. The + 12 → 24 → 16 sequence is now bracketed by certificates, not floats; both + index sums are −4 (+4/−8 and +6/−10), as the identity demands. +- **Exact centroid Hessian** (001 lead 5, closed): by the D₃ symmetry the + Jacobian at the centroid is A·I + B·J (J = all-ones; the rank-one sums + Σ_tri d dᵀ = I − J/3 and axial d dᵀ = t²J are verified exactly), and + trace DF = 0 forces B = −A, so the full signature reduces to the sign of + the axial eigenvalue λ_ax = 3/r_t³ − 4q/r_a³ (r_t² = 2/3, r_a² = 3t²) — + i.e. to the single exact rational comparison **6561·t⁶ vs 128·q²** + (`explore/skeptic_centroid_hessian.py`; also verifies F(c) = 0 for the + whole family). At t = 17/200: + - q = 4360/1000000 and q = **4367/1000000 (the witness)**: λ_ax > 0, + signature (+,−,−), det DF(c) > 0 — **matching the certified det_sign +1 + of the centroid enclosure in 001's certificate**: an independent exact + cross-check of one certified index by pure symmetry algebra, no + intervals involved. + - q = 4400/1000000: λ_ax < 0, signature (−,+,+), det DF(c) < 0. + - The predicted index flip is **confirmed in the certificates**: the + enclosure containing (1/3,1/3,1/3) carries certified det_sign +1 in the + B and C runs (and in 001's own certificate) and −1 in the D run — three + interval-arithmetic censuses agreeing with the pure symmetry algebra on + both sides of q\*. + - The centroid is **exactly degenerate at q\*(t) = 81√2·t³/16**; at + t = 17/200, q\* = 4.396801013961…·10⁻³ (certified enclosure of width + 2⁻¹²⁰ in the script output). Note q\* is *above* 001's float window edge + ≈ 4.389·10⁻³. SPECULATION: the 24→16 transition at the window's high + edge is NOT the centroid degeneracy; the centroid degenerates slightly + later, already inside the 16-count regime, flipping its index +1 → −1 + with a compensating event elsewhere. A certified census inside + q ∈ (4.389, 4.3968)·10⁻³ would separate the two events — lead 3. + +## Outcome + +**VERIFIED_WITH_CORRECTIONS** (scope: attempt 001's certificate and its +verification chain; the four perturbed configurations listed above; nothing +about any other (t, q), and nothing about ABK's asymptotic theorem). + +The headline stands: the configuration of 001 (units at e₁,e₂,e₃, charges +4367/1000000 at (251/600,251/600,251/600) and (149/600,149/600,149/600)) has +**exactly 24 nondegenerate equilibria**, now resting on: the original two +engines, this review's independent structural checks in exact arithmetic +(including a tiling proof strictly stronger than the checker's), and four +re-censuses — 24 again on a subdivision tree sharing no box with 001's +(run B) and on a finer-grid rerun that reproduced 001's tree verdict-by- +verdict (run A), with exactly the predicted bracket counts off the witness +(runs C, D). +24 > 16 = (5−1)²: the explicit refutation instance of Maxwell's bound at +n = 5 is confirmed, and the escalation required by the verification contract +is discharged. + +Corrections to 001 (none touches the count): + +1. "Established by two independent implementations" overstates the coverage + half: the tiling check in `verify_equilibria.py` is not sound by itself + (R1); coverage independence is only established as of this review. +2. "The Kraft-equality tiling check exact" (001 step 3) — the arithmetic is + exact, but the check is not equivalent to tiling; same correction. +3. R2: the det-sign location argument is sound but was never stated; recorded + here. + +**Not claimed:** that the harness driver is bug-free in regimes not exercised +(see the latent midpoint hazard below); anything about the window edges +beyond the two certified bracket points; the fold interpretation of the +window (still SPECULATION, now sharpened by the exact q\*). + +## Why it failed / what survived + +Nothing in the review kills 001. Found and worth keeping: + +- **R1 (the real finding)**: the two-engine protocol had a common-mode hole + on coverage — the checker's tiling check assumed the very driver behavior + (per-node axis consistency) it was meant to verify. A driver bug emitting + mixed-axis children would have produced certificates that PASS both + engines while missing equilibria. The demonstrated tamper (DEMO 2) is the + failure mode queue item 16 hypothesized, one level up: not a wrong tree + with self-consistent leaves, but a wrong *tiling check* that + self-consistent leaves slip through. +- **Latent driver hazard, not triggered**: `IV.mid()` rounds the midpoint + down to the 2^−prec grid and `krawczyk()` uses it with J(B) taken over the + box alone; if the rounding error 2^−prec ever reached half the box width + (roughly prec_bits ≲ min_width_bits + 1 at unit scale), m could exit B and + the mean-value form would be invalid. At 001's parameters the margin is + 2⁻³¹ vs 2⁻⁶⁴ — safe by 10 decimal orders — and the checker (exact + midpoints) would catch resulting false leaves; but a future run at low + `--prec-bits` with deep `--min-width-bits` should be refused rather than + trusted. Lead 2. + +Reusable: + +- `explore/skeptic_cert_checks.py`: certificate-level structural audit with + the axis-consistency tiling proof — cheap (~4 s) and engine-independent; + run it on every future census certificate alongside the checker. +- `explore/skeptic_centroid_hessian.py`: the exact symmetry reduction of the + centroid Hessian for the whole (t, q) family, and the closed-form + degeneracy locus q\*(t) = 81√2·t³/16. +- The perturbed-region trick (explicit region in the config with fresh + non-dyadic offsets) as the standard way to force a wholesale-different + tree without touching the harness. +- Certified hard-instance data points for the family: counts 12 / 24 / 16 at + q = 4360, 4367, 4400 (·10⁻⁶) — the two folds of 001's window are now + bracketed by certificates, not floats. + +## Leads generated + +1. **Harden the checker's tiling check** (harness change, tier 0, generic): + replace prefix-freeness + Kraft with the axis-consistency reconstruction + (or add it), and port the same case to both `--validate` suites as a + tamper test. Falsifiable: DEMO 1 and DEMO 2 certificates must flip to + FAIL; all existing certificates must still PASS. +2. **Guard the midpoint rounding** (harness change, tier 0, generic): make + `krawczyk()` return "unknown" when the box width is within a safety + factor of the 2^−prec grid (or intersect m into B). Falsifiable: a + synthetic low-precision deep-subdivision run must stop issuing + certificates instead of mis-centering m. +3. **The sliver q ∈ (4.389, 4.3968)·10⁻³**: certified censuses on both sides + of q\* = 81√2·t³/16 inside the 16-count regime — does the total count + change at q\* (the centroid index flip +1 → −1 must be compensated by ±2 + somewhere), and is the 24→16 window edge a separate fold as 001 + speculated? Two ~35-min runs decide. +4. **Certified window edges** (001 lead 2, sharpened): the brackets are now + certified (12 at 4360, 16 at 4400, ·10⁻⁶); binary-search both edges to + certified 10⁻⁵-wide q-intervals with the same pipeline. + +## References + +- Attempt 001: `problems/maxwell-equilibria/attempts/001-abk-explicit-witness.md` + and its data in `problems/maxwell-equilibria/data/`. +- Harness under review: `harness/maxwell-equilibria/equilibria.py`, + `harness/maxwell-equilibria/verify_equilibria.py` (both read in full). +- A. Neumaier, *Interval Methods for Systems of Equations*, CUP 1990, §5.2 — + as cited by the harness docstrings for the Krawczyk and interval-Gauss + certificates; statements checked against the implementations, book not + re-consulted. +- P. Arathoon, G. Ball, M. D. Kvalheim, *The Maxwell Conjecture is False*, + arXiv:2607.27197 (July 2026) [T, via 001 — not re-read here]. diff --git a/problems/maxwell-equilibria/data/skeptic-002-certA-prec80.json.gz b/problems/maxwell-equilibria/data/skeptic-002-certA-prec80.json.gz new file mode 100644 index 0000000..15bc6e9 Binary files /dev/null and b/problems/maxwell-equilibria/data/skeptic-002-certA-prec80.json.gz differ diff --git a/problems/maxwell-equilibria/data/skeptic-002-certB-region.json.gz b/problems/maxwell-equilibria/data/skeptic-002-certB-region.json.gz new file mode 100644 index 0000000..19a13e0 Binary files /dev/null and b/problems/maxwell-equilibria/data/skeptic-002-certB-region.json.gz differ diff --git a/problems/maxwell-equilibria/data/skeptic-002-certC-q4360.json.gz b/problems/maxwell-equilibria/data/skeptic-002-certC-q4360.json.gz new file mode 100644 index 0000000..aa9cbe3 Binary files /dev/null and b/problems/maxwell-equilibria/data/skeptic-002-certC-q4360.json.gz differ diff --git a/problems/maxwell-equilibria/data/skeptic-002-certD-q4400.json.gz b/problems/maxwell-equilibria/data/skeptic-002-certD-q4400.json.gz new file mode 100644 index 0000000..589b461 Binary files /dev/null and b/problems/maxwell-equilibria/data/skeptic-002-certD-q4400.json.gz differ diff --git a/problems/maxwell-equilibria/explore/skeptic_centroid_hessian.py b/problems/maxwell-equilibria/explore/skeptic_centroid_hessian.py new file mode 100644 index 0000000..cf6f8f8 --- /dev/null +++ b/problems/maxwell-equilibria/explore/skeptic_centroid_hessian.py @@ -0,0 +1,110 @@ +#!/usr/bin/env python3 +"""Exact eigenvalue signature of DF at the centroid for the abk family. + +Configuration family (attempt 001): unit charges at e1, e2, e3, and two +charges q > 0 at c +- t*(1,1,1), where c = (1/3,1/3,1/3) is the centroid. + +Facts derived here (proofs in attempt 002; every load-bearing inequality is +instantiated below in exact rational arithmetic): + + 1. F(c) = 0 for every (t, q): the three triangle terms have equal radii + r_t = |c - e_i| and directions summing to 3c - (e1+e2+e3) = 0; the two + axial terms are opposite with equal radii r_a = t*sqrt(3). + + 2. DF(c) = A*I + B*J with J the all-ones matrix (D3 symmetry + the + rank-one structure of each charge term q*(I/r^3 - 3 d d^T/r^5)): + sum_tri d_i d_i^T = I - J/3 (exact, verified below), + axial d d^T = t^2 J. + Since trace DF = -laplacian V = 0, B = -A; equivalently the identity + 3/r_t^3 - 3/r_t^5 = -(3/2)/r_t^3 holds exactly because r_t^2 = 2/3 + (verified below). Eigenvalues: lambda_ax = A + 3B = -2A on (1,1,1), + lambda_perp = A (multiplicity 2). + + 3. lambda_ax = 3/r_t^3 - 4q/r_a^3. Both terms positive, so + sign(lambda_ax) = sign(9*r_a^6 - 16 q^2 r_t^6) + = sign(6561 t^6 - 128 q^2) (exact rational), + using r_a^6 = 27 t^6 and r_t^6 = 8/27. + 6561 t^6 > 128 q^2: signature (+,-,-), det DF > 0 (nondegenerate saddle) + 6561 t^6 < 128 q^2: signature (-,+,+), det DF < 0 + equality: DEGENERATE, at q*(t) = 81*sqrt(2)*t^3/16. + +Usage: skeptic_centroid_hessian.py [--t 17/200] [--q 4367/1000000 ...] +Prints the exact comparison for each q, and q*(t) exactly and as a decimal. +Standard library only, exact except the labelled float cross-check. +""" + +from __future__ import annotations + +import argparse +from fractions import Fraction +from math import isqrt + + +def main() -> None: + ap = argparse.ArgumentParser() + ap.add_argument("--t", default="17/200") + ap.add_argument("--q", nargs="*", + default=["4360/1000000", "4367/1000000", "4400/1000000"]) + args = ap.parse_args() + t = Fraction(args.t) + + # -- fact 1: F(c) = 0, exact pieces ------------------------------------- + c = (Fraction(1, 3),) * 3 + tri = [(Fraction(1), Fraction(0), Fraction(0)), + (Fraction(0), Fraction(1), Fraction(0)), + (Fraction(0), Fraction(0), Fraction(1))] + dirs = [tuple(c[k] - p[k] for k in range(3)) for p in tri] + assert all(sum(d[k] for d in dirs) == 0 for k in range(3)), "triangle dirs" + r2s = {sum(d[k] ** 2 for k in range(3)) for d in dirs} + assert r2s == {Fraction(2, 3)}, "equal triangle radii, r_t^2 = 2/3" + print("F(c) = 0: triangle directions sum to 0 at equal radii (r_t^2 = 2/3),") + print(" axial terms opposite at equal radii (r_a^2 = 3 t^2) [exact]") + + # -- fact 2: sum_tri d d^T = I - J/3, and the traceless identity -------- + S = [[sum(d[a] * d[b] for d in dirs) for b in range(3)] for a in range(3)] + expect = [[Fraction(2, 3) if a == b else Fraction(-1, 3) for b in range(3)] + for a in range(3)] + assert S == expect, "sum_tri d d^T != I - J/3" + # identity 3/r^3 - 3/r^5 = -(3/2)/r^3 <=> r^2 = 2/3 (with r = r_t): + assert Fraction(2, 3) * (3 + Fraction(3, 2)) == 3, "traceless identity" + print("DF(c) = A*I + B*J with B = -A: rank-one sums and the r_t^2 = 2/3") + print(" traceless identity verified exactly") + + # -- fact 3: the sign of lambda_ax -------------------------------------- + lhs = 6561 * t ** 6 + print(f"\nt = {t}: 6561 t^6 = {lhs} ~ {float(lhs):.6e}") + for qs in args.q: + q = Fraction(qs) + rhs = 128 * q ** 2 + if lhs > rhs: + verdict = "lambda_ax > 0: signature (+,-,-), det DF(c) > 0 (index +1)" + elif lhs < rhs: + verdict = "lambda_ax < 0: signature (-,+,+), det DF(c) < 0 (index -1)" + else: + verdict = "DEGENERATE (lambda_ax = 0)" + print(f"q = {qs}: 128 q^2 = {rhs} ~ {float(rhs):.6e}\n" + f" -> {verdict} [exact rational comparison]") + + # -- the degeneracy locus ------------------------------------------------ + print(f"\nq*(t) = 81*sqrt(2)/16 * t^3 (exact algebraic form), t^3 = {t**3}") + # certified decimal enclosure of q* via integer sqrt at 2^-120 + v = 2 * (Fraction(81, 16) * t ** 3) ** 2 # q*^2 = 2*(81 t^3/16)^2 + s = 1 << 120 + r = isqrt(v.numerator * v.denominator * s * s) + lo = Fraction(r, v.denominator * s) + hi = Fraction(r + 1, v.denominator * s) + assert lo ** 2 <= v <= hi ** 2 + print(f"q*({t}) in [{float(lo):.12e}, {float(hi):.12e}] (certified)") + + # -- float cross-check (EVIDENCE only, never certifies) ------------------ + tf, r_t3, = float(t), (2.0 / 3.0) ** 1.5 + r_a3 = (3.0 * tf * tf) ** 1.5 + for qs in args.q: + qf = float(Fraction(qs)) + lam_ax = 3.0 / r_t3 - 4.0 * qf / r_a3 + print(f"float check q={qs}: lambda_ax ~ {lam_ax:+.6e}, " + f"lambda_perp ~ {-lam_ax / 2:+.6e} (x2)") + + +if __name__ == "__main__": + main() diff --git a/problems/maxwell-equilibria/explore/skeptic_cert_checks.py b/problems/maxwell-equilibria/explore/skeptic_cert_checks.py new file mode 100644 index 0000000..fc4fd24 --- /dev/null +++ b/problems/maxwell-equilibria/explore/skeptic_cert_checks.py @@ -0,0 +1,346 @@ +#!/usr/bin/env python3 +"""Independent structural checks on an equilibria.py certificate (skeptic pass). + +Written for the skeptic review of attempt 001 (abk-witness). Shares no code +with harness/maxwell-equilibria/equilibria.py or verify_equilibria.py; every +check below is re-implemented from scratch in exact rational arithmetic +(fractions.Fraction + math.isqrt only). + +Checks, and why each is stronger than (or independent of) what the two +harness engines already did: + + TREE The verifier's tiling check is prefix-freeness + the Kraft equality + sum 2^-depth = 1. That is NOT sufficient for a leaf set whose paths + carry per-node axis choices: the set {"01", "11"} (axis-0 upper + half, axis-1 upper half) is prefix-free with Kraft sum 1 and tiles + nothing -- the two boxes overlap and jointly miss a quadrant. The + check here reconstructs the subdivision tree from the paths and + verifies AXIS CONSISTENCY at every internal node: each node is + either a leaf (path ends there, no continuations) or ALL paths + through it agree on the split axis and both children subtrees are + present and well-formed. By induction this proves the leaf boxes + partition the root box exactly. Kraft and prefix-freeness are then + theorems, but are recomputed anyway as cross-checks. + + DISJ The 24 isolation enclosures are pairwise disjoint, by exact + rational comparison (some coordinate strictly separates each pair). + + BOX Every isolated leaf's recorded box equals the box rebuilt from its + split path (own midpoint-splitting code), its enclosure is contained + in it, and count == number of isolated leaves. + + INDEX All det signs are +-1 and sum to 1 - n (Poincare-Hopf identity for + all-positive configurations), recomputed from the leaf data. + + REGION R0^2 = max |x_i|^2 recomputed from the config; the region must + contain the closed localization ball (certified rational sqrt upper + bound, own isqrt code). + + RHO The domination inequality |q_i|/rho_i^2 > sum_j |q_j|/(d_ij-rho_i)^2 + is re-instantiated exactly (own certified lower bounds for d_ij). + + BALL Every excluded-ball leaf's box lies in the closed ball of its + domination radius (exact); every excluded-outside leaf's box has + min |x|^2 >= R0^2 (exact). + + TALLY Leaf-kind tallies match the certificate's summary fields. + +Usage: skeptic_cert_checks.py CERT.json [--centroid] [--expect-count N] + --centroid additionally checks that (1/3,1/3,1/3) lies in exactly + one isolation enclosure (the abk-witness family has a symmetric + rational equilibrium there). + +Exit status 0 iff every check passes. Standard library only. +""" + +from __future__ import annotations + +import argparse +import json +import sys +from fractions import Fraction +from math import isqrt + +FAILURES = [] + + +def check(name: str, cond: bool, detail: str = "") -> None: + line = ("ok " if cond else "FAIL ") + name + (f" [{detail}]" if detail else "") + print(line) + if not cond: + FAILURES.append(name) + + +def frac(s) -> Fraction: + return Fraction(s) + + +def sqrt_up(f: Fraction, bits: int = 128) -> Fraction: + """Certified upper bound for sqrt(f), f >= 0. Own implementation.""" + s = 1 << bits + t = f.numerator * f.denominator * s * s + r = isqrt(t) + if r * r != t: + r += 1 + return Fraction(r, f.denominator * s) + + +def sqrt_dn(f: Fraction, bits: int = 128) -> Fraction: + s = 1 << bits + return Fraction(isqrt(f.numerator * f.denominator * s * s), f.denominator * s) + + +# -------------------------------------------------------------------------- +# TREE: full subdivision-tree reconstruction with axis consistency +# -------------------------------------------------------------------------- + +def check_tree(paths: list[str]) -> None: + """Prove the leaf paths are exactly the leaves of one well-formed binary + subdivision tree (every internal node splits along a single axis into + both halves). This implies the leaf boxes tile the root box.""" + TERM = "$" + root: dict = {} + dup = badchar = 0 + for p in paths: + if len(p) % 2: + badchar += 1 + continue + node = root + ok = True + for i in range(0, len(p), 2): + pair = p[i:i + 2] + if pair[0] not in "012" or pair[1] not in "01": + ok = False + break + node = node.setdefault(pair, {}) + if not ok: + badchar += 1 + continue + if TERM in node: + dup += 1 + node[TERM] = True + check("TREE: all paths well-formed (even length, axis in 0-2, side in 0-1)", + badchar == 0, f"{badchar} malformed") + check("TREE: no duplicate leaf paths", dup == 0, f"{dup} duplicates") + + bad_mixed = bad_onechild = bad_termint = empty = 0 + n_leaves = 0 + max_depth = 0 + stack = [(root, 0)] + while stack: + node, depth = stack.pop() + kids = [k for k in node if k != TERM] + term = TERM in node + if term and kids: + bad_termint += 1 # a leaf path is a strict prefix of another + if not term and not kids: + empty += 1 # cannot happen structurally, defensive + if term and not kids: + n_leaves += 1 + max_depth = max(max_depth, depth) + continue + axes = {k[0] for k in kids} + if len(axes) != 1: + bad_mixed += 1 # two different split axes at one node + else: + a = axes.pop() + if set(kids) != {a + "0", a + "1"}: + bad_onechild += 1 # only one half present -> a hole + for k in kids: + stack.append((node[k], depth + 1)) + check("TREE: no leaf path is a prefix of another", bad_termint == 0, + f"{bad_termint} violations") + check("TREE: single split axis at every internal node", bad_mixed == 0, + f"{bad_mixed} mixed-axis nodes") + check("TREE: both children present at every internal node", bad_onechild == 0, + f"{bad_onechild} one-child nodes") + check("TREE: leaf count from tree equals path count", + n_leaves == len(paths), f"{n_leaves} vs {len(paths)}") + # cross-checks (implied by the above, but cheap) + kraft = sum(Fraction(1, 1 << (len(p) // 2)) for p in paths) + check("TREE: Kraft equality sum 2^-depth == 1", kraft == 1, str(kraft)) + print(f" leaves={n_leaves} max_depth={max_depth} splits") + + +# -------------------------------------------------------------------------- +# box rebuild (own code) +# -------------------------------------------------------------------------- + +def rebuild(root_box, path: str): + box = [list(iv) for iv in root_box] + for i in range(0, len(path), 2): + a, s = int(path[i]), int(path[i + 1]) + lo, hi = box[a] + mid = (lo + hi) / 2 + box[a] = [lo, mid] if s == 0 else [mid, hi] + return [tuple(iv) for iv in box] + + +def main() -> int: + ap = argparse.ArgumentParser() + ap.add_argument("cert") + ap.add_argument("--centroid", action="store_true") + ap.add_argument("--expect-count", type=int, default=None) + args = ap.parse_args() + + with open(args.cert, encoding="utf-8") as fh: + cert = json.load(fh) + + charges = [(frac(c["q"]), tuple(frac(v) for v in c["pos"])) + for c in cert["config"]["charges"]] + n = len(charges) + root_box = [(frac(lo), frac(hi)) for lo, hi in cert["region"]] + leaves = cert["leaves"] + isolated = cert["isolated"] + print(f"cert: n={n} charges, {len(leaves)} leaves, count={cert['count']}, " + f"unresolved={cert['unresolved']}, complete={cert.get('complete')}") + + # ---- TREE ---- + check_tree([lf["path"] for lf in leaves]) + + # ---- TALLY ---- + tally: dict = {} + for lf in leaves: + tally[lf["kind"]] = tally.get(lf["kind"], 0) + 1 + check("TALLY: isolated leaves == count field", + tally.get("isolated", 0) == cert["count"], + f"{tally.get('isolated', 0)} vs {cert['count']}") + check("TALLY: unresolved leaves == unresolved field", + tally.get("unresolved", 0) == cert["unresolved"], + f"{tally.get('unresolved', 0)} vs {cert['unresolved']}") + for kind, cnt in cert["excluded"].items(): + check(f"TALLY: {kind} == summary", tally.get(kind, 0) == cnt, + f"{tally.get(kind, 0)} vs {cnt}") + known = {"isolated", "unresolved"} | set(cert["excluded"]) + check("TALLY: no unknown leaf kinds", set(tally) <= known, str(set(tally) - known)) + if args.expect_count is not None: + check(f"TALLY: count == expected {args.expect_count}", + cert["count"] == args.expect_count, str(cert["count"])) + + # ---- BOX ---- + iso_paths = {lf["path"] for lf in leaves if lf["kind"] == "isolated"} + check("BOX: isolated records match isolated leaves 1:1", + iso_paths == {it["path"] for it in isolated} + and len(isolated) == len(iso_paths)) + bad_box = bad_enc = 0 + encs = [] + for it in isolated: + want = rebuild(root_box, it["path"]) + got = [tuple(frac(v) for v in iv) for iv in it["box"]] + if want != got: + bad_box += 1 + enc = [tuple(frac(v) for v in iv) for iv in it["enclosure"]] + for k in range(3): + if not (got[k][0] <= enc[k][0] <= enc[k][1] <= got[k][1]): + bad_enc += 1 + break + encs.append(enc) + check("BOX: every recorded isolated box == box rebuilt from its path", + bad_box == 0, f"{bad_box} mismatches") + check("BOX: every enclosure inside its leaf box", bad_enc == 0, + f"{bad_enc} violations") + + # ---- DISJ ---- + overlap = 0 + min_gap = None + for i in range(len(encs)): + for j in range(i + 1, len(encs)): + seps = [] + for k in range(3): + if encs[i][k][1] < encs[j][k][0]: + seps.append(encs[j][k][0] - encs[i][k][1]) + elif encs[j][k][1] < encs[i][k][0]: + seps.append(encs[i][k][0] - encs[j][k][1]) + if not seps: + overlap += 1 + else: + g = max(seps) + min_gap = g if min_gap is None else min(min_gap, g) + check("DISJ: isolation enclosures pairwise disjoint (exact)", overlap == 0, + f"{overlap} overlapping pairs of {len(encs) * (len(encs) - 1) // 2}") + if min_gap is not None: + print(f" minimum pairwise separating gap ~ {float(min_gap):.3e}") + + # ---- INDEX ---- + signs = [it["det_sign"] for it in isolated] + check("INDEX: every det sign is +1 or -1", all(s in (1, -1) for s in signs)) + check(f"INDEX: sum of det signs == 1 - n == {1 - n}", + sum(signs) == 1 - n, + f"sum={sum(signs)}, +1s={signs.count(1)}, -1s={signs.count(-1)}") + + # ---- REGION ---- + r0_sq = max(p[0] ** 2 + p[1] ** 2 + p[2] ** 2 for _, p in charges) + if all(q > 0 for q, _ in charges): + r_up = sqrt_up(r0_sq) + ok = all(root_box[k][0] <= -r_up and r_up <= root_box[k][1] for k in range(3)) + check("REGION: region contains the closed localization ball |x| <= R0", ok, + f"R0^2 = {r0_sq}") + else: + print("REGION: charges not all positive; localization not applicable") + + # ---- RHO ---- + rhos = [frac(r) for r in cert["domination_rhos"]] + ok = len(rhos) == n + if ok: + for i, (qi, pi) in enumerate(charges): + rho = rhos[i] + if rho <= 0: + ok = False + break + rhs = Fraction(0) + for j, (qj, pj) in enumerate(charges): + if j == i: + continue + d_lo = sqrt_dn(sum((pi[k] - pj[k]) ** 2 for k in range(3))) + if d_lo <= rho: + ok = False + break + rhs += abs(qj) / (d_lo - rho) ** 2 + if not ok or not abs(qi) / rho ** 2 > rhs: + ok = False + break + check("RHO: domination inequality re-instantiated exactly for every charge", ok) + + # ---- BALL / OUTSIDE ---- + bad_ball = bad_out = 0 + for lf in leaves: + if lf["kind"] == "excluded-ball": + box = rebuild(root_box, lf["path"]) + i = lf["ball"] + if not (isinstance(i, int) and 0 <= i < n): + bad_ball += 1 + continue + c = charges[i][1] + s = sum(max(abs(box[k][0] - c[k]), abs(box[k][1] - c[k])) ** 2 + for k in range(3)) + if s > rhos[i] ** 2: + bad_ball += 1 + elif lf["kind"] == "excluded-outside": + box = rebuild(root_box, lf["path"]) + s = Fraction(0) + for k in range(3): + if box[k][0] > 0: + s += box[k][0] ** 2 + elif box[k][1] < 0: + s += box[k][1] ** 2 + if s < r0_sq: + bad_out += 1 + check("BALL: every excluded-ball box inside its domination ball (exact)", + bad_ball == 0, f"{bad_ball} violations") + check("BALL: every excluded-outside box outside the localization ball (exact)", + bad_out == 0, f"{bad_out} violations") + + # ---- CENTROID (optional) ---- + if args.centroid: + c = Fraction(1, 3) + hits = sum(1 for enc in encs + if all(enc[k][0] <= c <= enc[k][1] for k in range(3))) + check("CENTROID: (1/3,1/3,1/3) lies in exactly one enclosure", hits == 1, + f"{hits} hits") + + print("FAILURES:", len(FAILURES)) + return 1 if FAILURES else 0 + + +if __name__ == "__main__": + sys.exit(main()) diff --git a/problems/maxwell-equilibria/explore/skeptic_q4360_config.json b/problems/maxwell-equilibria/explore/skeptic_q4360_config.json new file mode 100644 index 0000000..b6e816e --- /dev/null +++ b/problems/maxwell-equilibria/explore/skeptic_q4360_config.json @@ -0,0 +1,10 @@ +{ + "_comment": "Fold-window bracket, low side (attempt 001 lead 2): same geometry as the abk witness but q = 4360/1000000, below the float-observed 24-window (~(4.366, 4.389)e-3). 001's float scan predicts a complete count of 12 here.", + "charges": [ + {"q": "1", "pos": ["1", "0", "0"]}, + {"q": "1", "pos": ["0", "1", "0"]}, + {"q": "1", "pos": ["0", "0", "1"]}, + {"q": "4360/1000000", "pos": ["251/600", "251/600", "251/600"]}, + {"q": "4360/1000000", "pos": ["149/600", "149/600", "149/600"]} + ] +} diff --git a/problems/maxwell-equilibria/explore/skeptic_q4400_config.json b/problems/maxwell-equilibria/explore/skeptic_q4400_config.json new file mode 100644 index 0000000..1003dfa --- /dev/null +++ b/problems/maxwell-equilibria/explore/skeptic_q4400_config.json @@ -0,0 +1,10 @@ +{ + "_comment": "Fold-window bracket, high side (attempt 001 lead 2): same geometry as the abk witness but q = 4400/1000000, above the float-observed 24-window (~(4.366, 4.389)e-3). 001's float scan predicts a complete count of 16 here.", + "charges": [ + {"q": "1", "pos": ["1", "0", "0"]}, + {"q": "1", "pos": ["0", "1", "0"]}, + {"q": "1", "pos": ["0", "0", "1"]}, + {"q": "4400/1000000", "pos": ["251/600", "251/600", "251/600"]}, + {"q": "4400/1000000", "pos": ["149/600", "149/600", "149/600"]} + ] +} diff --git a/problems/maxwell-equilibria/explore/skeptic_region_config.json b/problems/maxwell-equilibria/explore/skeptic_region_config.json new file mode 100644 index 0000000..24b6f79 --- /dev/null +++ b/problems/maxwell-equilibria/explore/skeptic_region_config.json @@ -0,0 +1,11 @@ +{ + "_comment": "Skeptic re-run of the abk-witness census (attempt 001) over a geometrically different search region: offsets 1/89,1/91,1/93 (vs the auto-region 1/97,1/101,1/103), half-width 5/4 (vs 17/16). Region strictly contains the localization ball |x| <= R0 = 1, so by the all-positive localization lemma a census over it is complete even though the tool, given an explicit region, does not claim completeness itself. Every subdivision face is (non-dyadic offset + dyadic), so no symmetric rational point of small height lies on a face. The subdivision tree this induces shares no box with the 001 tree; the certified count must still be 24.", + "charges": [ + {"q": "1", "pos": ["1", "0", "0"]}, + {"q": "1", "pos": ["0", "1", "0"]}, + {"q": "1", "pos": ["0", "0", "1"]}, + {"q": "4367/1000000", "pos": ["251/600", "251/600", "251/600"]}, + {"q": "4367/1000000", "pos": ["149/600", "149/600", "149/600"]} + ], + "region": [["-441/356", "449/356"], ["-451/364", "459/364"], ["-461/372", "469/372"]] +} diff --git a/problems/maxwell-equilibria/explore/skeptic_verifier_gap_demo.py b/problems/maxwell-equilibria/explore/skeptic_verifier_gap_demo.py new file mode 100644 index 0000000..d2a6e34 --- /dev/null +++ b/problems/maxwell-equilibria/explore/skeptic_verifier_gap_demo.py @@ -0,0 +1,156 @@ +#!/usr/bin/env python3 +"""Demonstration: verify_equilibria.py's tiling check is not sound by itself. + +The independent checker validates the leaf decomposition of a certificate by +(i) prefix-freeness of the split paths and (ii) the Kraft equality +sum 2^-depth = 1. For paths that carry a PER-NODE AXIS CHOICE those two +conditions do not imply a tiling: {"01", "11"} (axis-0 upper half and axis-1 +upper half of the root) is prefix-free with Kraft sum 1, yet the two boxes +overlap and jointly miss the (lo, lo) quadrant of the root. + +This script shows the consequence at both severities: + + DEMO 1 A synthetic certificate whose two leaves overlap and leave a hole + (no equilibrium in the region, every leaf genuinely sign-excluded): + verify() returns PASS on a non-tiling leaf list. + + DEMO 2 The real driver's certificate for two unit charges at (+-1, 0, 0) + (exactly one equilibrium, at the origin) is tampered with: the + isolated leaf is replaced by two mixed-axis exclusion leaves that + each avoid the origin but jointly leave a hole containing it, and + the count is edited from 1 to 0. If verify() PASSes, the checker + has endorsed a "complete census" that misses a real equilibrium. + + Both demos are also run through the skeptic's own tree reconstruction + (skeptic_cert_checks.check_tree), which rejects them at the mixed-axis + node -- i.e. axis consistency is exactly the missing condition. + +The driver itself (equilibria.py) always emits both children of a split with +the same axis digit, so certificates it actually produces are axis-consistent +and DO tile; the gap is in what the independent checker is able to catch, +not a wrong count in any existing certificate. See attempt 002. + +Standard library only. Deterministic. Runtime ~1 minute (DEMO 2 runs the +two-charge census at 64 bits). +""" + +from __future__ import annotations + +import json +import os +import sys +from fractions import Fraction + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, os.path.join(HERE, "..", "..", "..", "harness", "maxwell-equilibria")) +sys.path.insert(0, HERE) + +import equilibria as eq # noqa: E402 (the driver) +import verify_equilibria as ve # noqa: E402 (the checker under test) +import skeptic_cert_checks as scc # noqa: E402 (this review's tree check) + + +def own_tree_check(paths) -> bool: + """Run the skeptic tree reconstruction; True iff it accepts.""" + scc.FAILURES.clear() + scc.check_tree(list(paths)) + ok = not scc.FAILURES + scc.FAILURES.clear() + return ok + + +def demo1() -> bool: + print("=" * 72) + print("DEMO 1: overlapping leaves + hole, no zero involved") + print("=" * 72) + cfg2 = {"charges": [{"q": "1", "pos": ["1", "0", "0"]}, + {"q": "1", "pos": ["-1", "0", "0"]}]} + # Region from the checker's own validation suite: F is sign-definite on + # all of it, so ANY sub-box is genuinely sign-excludable. + bad = { + "config": cfg2, + "region": [["2", "9/4"], ["0", "1/4"], ["0", "1/4"]], + "complete": False, + "domination_rhos": ["1/4", "1/4"], + "count": 0, "unresolved": 0, "isolated": [], + "leaves": [{"path": "01", "kind": "excluded-sign", "ball": None}, + {"path": "11", "kind": "excluded-sign", "ball": None}], + } + # the two boxes: axis-0 upper half and axis-1 upper half of the region -- + # they overlap on (x hi, y hi) and miss (x lo, y lo) entirely. + res = ve.verify(bad) + print("verify_equilibria.verify() verdict:", res["verdict"], res["failures"]) + mine = own_tree_check(["01", "11"]) + print("skeptic tree reconstruction accepts:", mine) + exploited = res["verdict"] == "PASS" and not mine + print("GAP DEMONSTRATED:" if exploited else "gap NOT demonstrated:", + "checker PASSes a non-tiling leaf list that the axis-consistency " + "check rejects" if exploited else res) + return exploited + + +def demo2() -> bool: + print("=" * 72) + print("DEMO 2: hiding the real origin equilibrium of two unit charges") + print("=" * 72) + eq.set_precision(64) + cfg = eq.Config([(Fraction(1), (Fraction(1), Fraction(0), Fraction(0))), + (Fraction(1), (Fraction(-1), Fraction(0), Fraction(0)))]) + cert = eq.count_equilibria(cfg) + print(f"honest driver run: count={cert['count']} " + f"unresolved={cert['unresolved']} leaves={len(cert['leaves'])}") + assert cert["count"] == 1 and cert["unresolved"] == 0 + iso = cert["isolated"][0] + path = iso["path"] + box = [(Fraction(lo), Fraction(hi)) for lo, hi in iso["box"]] + # the unique equilibrium is the origin (by symmetry); it is interior here + assert all(lo < 0 < hi for lo, hi in box) + + # Replace the isolated leaf with two mixed-axis half-leaves that each + # avoid the origin; their union misses the quadrant of the origin. + # Use the two axes whose midpoints are farthest from 0 (best exclusion + # margin), and on each keep the half NOT containing the origin. + mids = [(box[a][0] + box[a][1]) / 2 for a in range(3)] + assert all(m != 0 for m in mids), "face through the origin" + ax1, ax2 = sorted(range(3), key=lambda a: abs(mids[a]), reverse=True)[:2] + fakes = [] + for axis in sorted((ax1, ax2)): + mid = mids[axis] + side = "1" if mid > 0 else "0" # origin is in the other half + fakes.append(f"{path}{axis}{side}") + print("isolated leaf", repr(path), "replaced by mixed-axis leaves", + [repr(f) for f in fakes]) + + tampered = dict(cert) + tampered["count"] = 0 + tampered["isolated"] = [] + tampered["index_sum"] = None + tampered["index_sum_ok"] = None + tampered["leaves"] = ( + [lf for lf in cert["leaves"] if lf["path"] != path] + + [{"path": f, "kind": "excluded-newton", "ball": None} for f in fakes]) + tampered = json.loads(json.dumps(tampered)) # same round-trip as a file + + res = ve.verify(tampered) + print("verify_equilibria.verify() verdict:", res["verdict"], + "count:", res["count"], res["failures"][:3]) + mine = own_tree_check([lf["path"] for lf in tampered["leaves"]]) + print("skeptic tree reconstruction accepts:", mine) + exploited = res["verdict"] == "PASS" and res["count"] == 0 and not mine + print("GAP DEMONSTRATED:" if exploited else "gap NOT demonstrated:", + "checker endorses a census claiming 0 equilibria for a region " + "that provably contains one (the origin)" if exploited else res) + return exploited + + +def main() -> int: + d1 = demo1() + d2 = demo2() + print("=" * 72) + print(f"DEMO 1 (overlap+hole PASSes): {d1}; " + f"DEMO 2 (hidden real zero PASSes): {d2}") + return 0 if d1 else 1 # demo 1 is the load-bearing demonstration + + +if __name__ == "__main__": + sys.exit(main()) diff --git a/problems/maxwell-equilibria/prior-art.json b/problems/maxwell-equilibria/prior-art.json index 168c69d..602d84e 100644 --- a/problems/maxwell-equilibria/prior-art.json +++ b/problems/maxwell-equilibria/prior-art.json @@ -1,7 +1,7 @@ { "problem": "maxwell-equilibria", "route_status": "VERIFIED", - "route_summary": "Explicit rational five-charge witness certified with exactly 24 nondegenerate equilibria (> 16 = (5-1)^2) by two independent engines - a concrete, machine-checkable instance of the arXiv:2607.27197 refutation. Escalation: a skeptic review of the harness driver is queued before this is treated as settled. The n=3 gap (4 vs 6) is untouched.", + "route_summary": "Explicit rational five-charge witness certified with exactly 24 nondegenerate equilibria (> 16 = (5-1)^2) - a concrete, machine-checkable instance of the arXiv:2607.27197 refutation. The escalation is discharged: the skeptic review (002) confirmed the count against a driver audit, fresh exact structural checks and four perturbed re-censuses, while finding and closing a real common-mode gap in the checker's tiling test. The n=3 gap (4 vs 6) is untouched.", "attempts": [ { "id": "001", @@ -20,6 +20,33 @@ "abk-witness", "24-count window" ] + }, + { + "id": "002", + "file": "attempts/002-skeptic-review-of-001.md", + "date": "2026-08-04", + "mode": "informed", + "status": "VERIFIED_WITH_CORRECTIONS", + "verifies": "001", + "mechanism": [ + "adversarial-verification", + "exact-rational-arithmetic", + "independent-reimplementation" + ], + "one_line": "Skeptic pass on 001: the 24-count survives a driver audit, fresh exact structural checks, and four re-censuses over different subdivision trees (also certifying the 12/16 window brackets and the exact centroid Hessian) - but the checker's Kraft+prefix tiling test is proven non-sound (a tampered mixed-axis certificate hiding a real zero PASSes), so coverage independence dates from this review's axis-consistency check.", + "range": "001's certificate re-audited leaf-by-leaf structurally; re-censuses at prec 80, a shifted/enlarged region, q=4360/1000000 and q=4400/1000000; centroid Hessian exact for the (t,q) family", + "corrections": [ + "'established by two independent implementations' overstated: verify_equilibria.py's tiling check (prefix-freeness + Kraft equality) does not imply a tiling for axis-labelled paths, so coverage rested on driver honesty until this review's stronger tree check", + "the checker's det-sign evaluation over the driver's enclosure is sound only via an unstated convexity argument on the interval Jacobian, now recorded", + "latent (untriggered) driver hazard: rounded midpoint can exit the box when prec_bits is small relative to min_width_bits; safe by 10 decimal orders at 001's parameters" + ], + "leak_terms": [ + "mixed-axis", + "axis-consistency", + "81*sqrt(2)*t^3/16", + "4.3968", + "12/24/16 brackets" + ] } ] } diff --git a/problems/union-closed/attempts/014-aggregated-or-probe-large-n.md b/problems/union-closed/attempts/014-aggregated-or-probe-large-n.md new file mode 100644 index 0000000..65c7fec --- /dev/null +++ b/problems/union-closed/attempts/014-aggregated-or-probe-large-n.md @@ -0,0 +1,397 @@ +# 014 — Aggregated OR control probed to n = 32 (trend runs to n = 64): survives, certified + +- **Problem:** union-closed, `problems/union-closed/PROBLEM.md` +- **Date:** 2026-08-04 +- **Mode:** informed +- **Type:** computational probe (queue item 1, the mandated first step: + extend the certified probe of the restated gap to n ≳ 20 BEFORE any + proof effort). Deliberately NOT a proof attempt. +- **Tools:** `explore/uc_or_agg_probe.py` (written and run here; standard + library only; deterministic, fixed seeds; parts X/O/P/T/S/H/E, runnable + independently). Checkpoints `data/or_agg_probe_part[XOPTSHE]*.json`, log + `data/or_agg_probe_run.log`. Commands reproducing every number below: + + python problems/union-closed/explore/uc_or_agg_probe.py 2>&1 | \ + tee -a problems/union-closed/data/or_agg_probe_run.log + # or per part, in the order run here: + python .../uc_or_agg_probe.py X # engine cross-checks (~15 s) + python .../uc_or_agg_probe.py O # orbit family census (~3 min) + python .../uc_or_agg_probe.py P # perturbative d-sweep (~80 s) + python .../uc_or_agg_probe.py T # trend runs to n = 64 (~2 min) + python .../uc_or_agg_probe.py S # sparse structured census (~5 s) + python .../uc_or_agg_probe.py H # aggregate hill-climbs (~3.5 min) + python .../uc_or_agg_probe.py E # exact certification (~2 min) + +- **Sources:** 005/006/007/008/009/011/012/013 (this repo); no external + fetches. Engines consulted for conventions and the orbit idea: + `explore/uc_or_avg.py` (007), `explore/uc_or_avg_skeptic.py` (013), + `explore/uc_pert_skeptic.py` (012's orbit engine). All code here is + fresh; nothing is imported from them. + +## The exact object probed + +Notation as in 005/006/007: tilt coupling `pi_lambda(A,B) prop +u(A) u(B) 2^{lambda|A&B|}` with both marginals mu; `t = 2^lambda`; +histories `a = A_{= lambda` with the 10-atom +witness; certified positive only at n <= 7 before this attempt, 013 part C: ++1.844669) is the claim whose sign this attempt probes: + + A(mu, lambda) = sum_{i < n, N_i != empty} w_i (M_i - lambda) + / sum_{i < n, N_i != empty} w_i >= 0 + for all mu with every elementwise marginal < 0.38271, all lambda >= 0, + +with `w_i` = the pi-mass of `N_i x N_i` ("defined mass"; a common +normalization cancels in the ratio). `i = n` is excluded because +`M_n = lambda` identically (005 Prop 1); coordinates with `N_i` empty +contribute nothing. + +**Convention robustness of the sign (new observation, two lines).** Under +013's conjecture-friendliest alternative bookkeeping — score every +margin-degenerate history at exactly lambda instead of conditioning it +out — every degenerate history adds `w * (lambda - lambda) = 0` to the +numerator and positive mass to the denominator. The numerator of A is +unchanged and the denominator only grows, so **the SIGN of A is identical +under either convention**. A sign probe of the conditioning-out aggregate +is therefore convention-robust for free (unlike the per-i magnitude +statements 013 had to re-check separately). + +## Approach + +Two engines, then exact certification of the tightest points: + +- **Sparse engine** (any n, m atoms, O(m² n)): direct atom-pair census + from a potential, using 005's trick (any positive u is its own + marginal's Sinkhorn potential). This is what makes the witness genre, + crash/mirror, gadget stacks and adversarial searches reachable at + n = 32 with no approximation — the cost of the obvious alternative + (full-support Sinkhorn + census) is 4^n and dies at n ≈ 13. +- **Orbit engine** (n to 64+): for measures invariant under permutations + of coords {3..n} — every structured adversary genre on file (Bernoulli + cells, Sawin geometric mixtures, delta_0+Bern, Chase–Lovett slice and + smoothed slice, eps-mixed crash, 012's crash-mixture cells). Histories + collapse to orbit classes `(oa, ob, ja, jb)` (special-coordinate + patterns, tail-prefix weights) with the prefix-overlap j summed into a + mass factor `JW[ja][jb] = sum_j Ncl(ja,jb,j) t^j`; cells per class are + exact orbit sums through tail pair-kernels + `PK[s][l] = C(f,s) sum_w C(s,w) C(f-s,l-w) t^w`. The OR of a class does + not depend on j (it cancels), which is what keeps the census O(n^4) + per instance. Same S_{n-2} symmetry as 012's budget engine; fresh code, + cross-checked against the direct engine at n = 6, 7 (part X2). +- **Exact rational certifier** (the 2026-07-31 standard: float agreement + is NOT certification): both engines re-run end-to-end in + `fractions.Fraction` at exact rational tilts (181/16, 4, 271/256), all + potentials dyadic rationals, every log2 replaced by a certified + enclosure — atanh series with explicit tail bound and **directed dyadic + fixed-point compaction** (every intermediate rounded toward the safe + side onto 120-bit dyadics, so long certified sums stay gcd-free and + cheap; this is what makes exact certification affordable at n = 32). + +Why these adversaries: the queue names them. The 10-atom witness genre +(near-proportional light slices, anti-aligned cross-ratios) is the only +known killer of the per-i form; Sawin mixtures were 007's closest census +approach; crash/eps-crash are 005/006's genres; the crash-mixture (p,d) +cells are exactly the genre whose budget growth reversed past n ≈ 22 in +012 (the standing warning this probe exists to answer); the lambda grid +includes the 011 window law `lam_max ~ 4.847/(n-3)` (half/at/twice) plus +the tilt-recipe cells log2(1.06), log2(1.20) and O(1) values up to 5. + +## What was done + +### X — engine cross-checks (all PASS) + +- Sparse engine reproduces 007's witness at lambda 3.5: `M_5 - lambda = + -0.122033`, aggregate +1.844754, max marginal 0.3178. +- Own exact path reproduces 013 part C identically: aggregate at + t = 181/16 in [+1.844669005341 ± 1.4e-15] — same number 013 certified. +- Orbit vs direct engine on full supports (slice, Sawin, eps-crash, + d0+Bern at n = 6, 7): max |M_i or aggregate diff| = 8.0e-13. +- Exact orbit rows vs float orbit rows: diff 7.9e-15. + +### O — orbit family census: 840 instances, 0 violations + +14 families × 10 lambdas × n in {10, 14, 20, 24, 28, 32} (660 of 840 +in-regime; Sinkhorn residual <= 1e-11 on all; `|M_n - lambda| <= 1.4e-13` +sanity on all). Key structure in the result: + +- **Product measures are the exact equality set.** Every Bernoulli cell + gives aggregate = 0 to float noise (|A| <= 9.3e-15 across all n, + lambda): product mu has a product potential, hence proportional + response slices at every history, hence `OR = 2^lambda` pointwise + (005 Prop 2's equality case), hence `M_i = lambda` for every i and + A = 0 **identically**. The aggregated control, if true, is tight on + the entire product family — so any proof must be second-order at that + boundary, and any violation only needs a downward curvature direction. +- Tightest non-product in-regime point per n, always at the smallest + grid lambda (~0.083–0.084) and always the 012 genre `crashmix_383_05` + = 0.95·Bern(0.383)^n + 0.05·crash: + + n 10 14 20 24 28 32 + min aggregate +4.88e-4 +6.49e-4 +8.50e-4 +9.76e-4 +1.10e-3 +1.21e-3 + + The minimum margin **grows monotonically with n** — no 012-style + reversal in the aggregate on this genre up to n = 32. +- Family ordering at the tight lambda (n = 32): crashmix_383_05 + +1.2e-3 < crashmix_30_05 +1.4e-3 < partE_mix +3.5e-3 < sawin_geo_a + +4.3e-3 < smoothed slice +6.2e-3; slices, d0bern, eps-crash all >= 1e-2. + +### P — perturbative sweep around the product equality boundary: 432 instances, 0 negative + +Since products are the equality set, the kill direction is perturbative: +`mu_d = (1-d) Bern(p)^n + d nu`, nu in {crash, empty, full, slice at the +record layer}, p in {0.3823, 0.30}, d in [0.005, 0.2], lambda in +{log2 1.06, 0.5, 4.847/(n-3)}, n in {10, 20, 32}. Result: the aggregate +leaves the boundary **quadratically upward in every direction tested** +(fitted order d^1.85–d^2.06 at small d; positive at every point). +Contrast with 012 recorded explicitly: on the same crash-mixture genre +whose per-coordinate averaged DOWNWARD budget grows with n (killing +008's B1), the aggregate's margin grows with n too — the aggregate +absorbs the growing downward budget because the compensating upward +deviations grow faster in the mass-weighted sum. + +### T — trend extension to n = 64 on the softest directions: 0 negative + +The one decaying trend in part P — the slice-direction curvature at +p = 0.30 (aggregate at d = 0.05, lambda 0.084: 1.35e-5 at n = 10 → +6.2e-7 at n = 32) — chased to n = 64 with the orbit engine: + + n 32 40 48 56 64 + slice p=.30 +6.15e-7 +2.22e-7 +1.41e-7 +2.33e-7 +4.31e-7 + +The margin **dips to +1.4e-7 near n = 48 and then recovers** — a genuine +non-monotonicity (the 012 lesson about small-n reading is live), but no +zero crossing to n = 64. All other tracked directions (matched-layer +slice, both p; crash direction at fixed lambda and at the moving window +lambda; crashmix_383_05) grow or plateau positive. + +### S — sparse structured census at n in {10..32}: 426 instances, 0 violations + +- Pure crash / mirror (4 atoms, Theorem-B-safe: aggregate >= 0 is a + theorem there; run for the margin trend). +- **Witness genre re-laid-out at every n** (marker coord 1, fresh-prefix + dilution singletons 2..n-3, response n-2, futures n-1, n; by 007's + dilution invariance the response coordinate keeps `M_i - lambda = + -0.122` exactly at lambda 3.5, independent of n), with the dilution + weight swept 0.25..40 and to 1e6 for the limit: the in-regime aggregate + decays like Theta(1/n) — at lambda 0.5: +0.163 (n=10) → +0.0549 (n=32), + with n·A → ~1.76 — because a bounded numerator (the gadget's fixed + deficit plus fixed surpluses) is spread over Theta(n) coordinates' + defined mass. Positive at every n, lambda, wdil; the wdil → infinity + limit converges to a positive constant (dilution coordinates are + single-diagonal-history coordinates, `M_i >= lambda` by 005 Prop 2). +- **Multi-gadget stacks** (007 lead 1 generalized): K = floor((n-2)/5) + copies of the witness's light-slice gadget on disjoint coordinate + blocks, each atom carrying its gadget's marker coordinate so the + blocks do not pollute each other's active prefixes. All positive + (tightest +1.83, at n = 20). + +### H — adversarial hill-climbs on the aggregate: 82 endpoints, 0 violations + +Marginal-penalized (target 0.375), dynamic-range-clamped (007's underflow +guard) multiplicative climbs with atom add/drop moves, minimizing the +aggregate itself: free sparse supports (8–18 atoms) at n = 10, 14; +witness-seeded at n = 10, 14, 20; multi-gadget-seeded at n = 14, 20, 24; +small-lambda climbs (lambda in {0.084, 0.15, 0.3, 4.847/(n-3)}) at +n = 10, 14. Lowest in-regime endpoint +0.005602 (n = 10, lambda 0.084 — +consistent with the O(lambda) scaling of everything at small lambda; on +the lambda >= 0.5 climbs the lowest endpoint is +0.20). + +### E — exact rational certification: 15 certificates, all positive + +Every certificate is a sign theorem about a stated exact rational measure +(dyadic potential, exact rational tilt), computed in Fraction end-to-end +with certified log2 enclosures; enclosure widths <= 5.9e-12. Scope +nuance, as in 013 R2: the orbit certificates are for quantized refit +measures mu-tilde (36-bit dyadic potentials, Sinkhorn residual <= 1e-11 +from the named family), which suffices for the universally-quantified +claim; the sparse witness-genre certificates are for the exactly-stated +atom weights themselves. + + point n tilt t aggregate (certified) + witness genre, tightest in-regime wdil 10 181/16 +1.709542464 + (same, per n) 14 181/16 +1.474040147 + 20 181/16 +1.162947590 + 24 181/16 +1.013509091 + 28 181/16 +0.896763710 + 32 181/16 +0.803594797 + witness genre, lambda = 2 EXACTLY 20 4 +0.483059569 + crashmix_383_05 (tightest census fam) 10 271/256 +0.000476924 + (same, per n; exact max marginal 14 271/256 +0.000634269 + 0.382350 < 0.38271 checked as an 20 271/256 +0.000829968 + exact rational at every n) 24 271/256 +0.000952638 + 28 271/256 +0.001075588 + 32 271/256 +0.001201865 + softest direction (slice-dir of 20 271/256 +0.000004637 + Bern(.30), d = 0.05) 32 271/256 +0.000000599 + +At t = 4 the n = 20 statement is fully rational with lambda = 2 exactly — +no irrational number appears anywhere in it. The Bernoulli equality +boundary is deliberately NOT in this table: an enclosure there straddles +0 by construction and certifies nothing (recorded as boundary, not as a +near-violation). + +## Outcome + +**EVIDENCE — the i-aggregated odds-ratio control survives everything this +probe threw at it, now far beyond its previous n <= 7 support.** Scope: +0 in-regime violations across ~1780 float instances (840 orbit-census at +n in {10, 14, 20, 24, 28, 32} × 14 families × 10-point lambda grids in +[0.079, 5]; 432 perturbative instances around the product boundary; 40 +trend instances to n = 64; 426 sparse structured instances including the +witness genre and gadget stacks at every n; 82 adversarial climb +endpoints at n <= 24), and 15 exact-rational certificates (all positive, +widths <= 5.9e-12) at n up to 32 including the tightest point found +anywhere (+5.99e-7, certified). The minimum margin over the census GROWS +with n on every structured family; the single decaying direction turns +around at n ≈ 48 without crossing zero. + +**Not claimed:** no claim that the aggregated control is true — this is +finite-n, finite-grid, finite-family EVIDENCE, and part T's dip-and- +recover is a reminder that trends here are not asymptotics; no claim +about lambda outside [0.079, 5] or off-grid; no claim for adversaries +outside the tested genres (the climbs are hill-climbs, not exhaustive — +in particular the space of K-gadget interactions is sampled thinly); the +orbit certificates certify quantized refit measures, not the ideal +family members (013 R2 scope); the equality-boundary observation is a +corollary of known facts (005 Props 1–2), not new mathematics; per repo +rules nothing here is a result until an independent skeptic pass. + +**Queue-item verdict:** the mandated probe condition is met — the +aggregate is now certified positive at n = 32 (vs n <= 7 before), the +margin trend is growing-in-n on every known adversary genre, and the +012-style reversal was looked for and not found (including on 012's own +genre). Proof effort on the aggregated control is now justified by the +queue's own criterion. + +## Why it failed / what survived + +Nothing failed — the probed statement survived; what the probe exposed is +WHERE it is hard and where it is soft: + +- **The obstruction to any easy proof is the product equality boundary.** + A = 0 identically on all product measures (proportional slices force + `OR = 2^lambda` pointwise), so the aggregated control has an + (n+1)-parameter equality set inside the regime, and any proof must + show the aggregate is a NONNEGATIVE-CURVATURE functional at every + product point in every admissible direction, plus something global. + The probe measured that curvature to be positive (d^1.85–d^2.06 + departure, part P) in crash/empty/full/slice directions — but the + slice-direction curvature at p = 0.30 decays to +1.4e-7 by n = 48 + before recovering. **The quantity a proof must control is the Hessian + of A at product measures; the quantity an adversary should attack is + its smallest eigenvalue over directions, which this probe only sampled + along four rays.** +- The witness genre's per-i deficit is REAL at every n (M_{n-2} - lambda + = -0.122 exactly, dilution-invariant) but the aggregate absorbs it with + Theta(1/n) room to spare, and no stacking of gadgets found here beats + the bookkeeping: each gadget brings its own positive coordinates. +- 012's warning did not materialize for the aggregate: the growing + downward budget on crash mixtures is outrun by the growing upward + deviations in the mass-weighted sum. (Measured, not explained — see + lead 3.) + +Survived / reusable: + +- `explore/uc_or_agg_probe.py`: the tail-exchangeable ORBIT CENSUS for + M_i/aggregate (n = 64 reachable in seconds, arithmetic-generic so the + same code path runs float and Fraction), the sparse any-n census, and + the **directed dyadic fixed-point enclosure kit** (`_dround`, + `_ln_1_to_2_fp`, compacted `agg_enclosure`) that makes exact + certification ~60x cheaper than 013's Fraction-naive path — reusable + for any future certified census in this route. +- The equality-boundary framing: aggregated control = "second-order + nonnegativity at products + global". This is the statement to prove or + kill, and it connects directly to 007's Theorem C (first order in + lambda is a perfect square) and to 008/012's perturbative machinery. +- The witness-at-n layout and multi-gadget builder as hard-instance + generators for any future aggregate claim. +- Trend tables (parts O/P/T checkpoints) for margin-vs-n on every genre. + +## Leads generated + +1. **Prove or kill the product-boundary curvature.** Compute the Hessian + quadratic form of `A(mu_d)` at d = 0 for product mu analytically (sum + 007's Theorem-A pairing over i; the first order vanishes on the + boundary, so the d² coefficient is a computable quadratic form in the + direction nu). Falsifiable both ways: numerically minimize the form + over nu (an eigenproblem per (n, p, lambda); the orbit engine gives + the matrix for tail-exchangeable nu) — a negative eigenvalue with an + in-regime eigendirection kills the aggregated control; a proof that + the form is PSD for all p, lambda settles the boundary layer and + names the remaining global gap. +2. **Map the slice-direction dip.** The +1.4e-7 minimum at (p = 0.30, + record-layer slice, d = 0.05, lambda ~ 0.084, n ~ 48) is the softest + point known. Sweep p in [0.25, 0.38] × layer ratio in [p, 0.383] × + n to 96 (orbit engine, hours at most) and fit where the minimum moves: + if some cell drives it below 0 the gap dies at EVIDENCE level; if the + dip is bounded away from 0 uniformly, record the floor. +3. **Explain the 012 contrast.** On crash mixtures the per-coordinate + downward budget grows with n (012) yet the aggregate margin grows + too. Decompose the aggregate numerator per coordinate into down/up + parts (the engine already has the rows): is the compensation local + (same i) or cross-i? A local identity would be a provable lemma; + cross-i compensation would say the aggregate is the WRONG restatement + and 007 lead 2 (margin-modulated, h-sensitivity-weighted) is the one + to probe next. +4. **Margin-modulated control (007 lead 2) is still untested** and is the + assembly-relevant statement; the census infrastructure here evaluates + it with a one-line weight change. Run the same battery. +5. **(proof side, now unblocked by the queue's own criterion)** Attack + `sum_i` of 007's Theorem-A pairing directly: the diagonal funding + `sum_i sum_a m_aa e_a` is 005 part B's strict-excess budget; the probe + says the inequality has growing room at every tested family, so a + lossy bound may suffice away from products, with lead 1 handling the + boundary layer. + +## References + +- This repo: `attempts/005-odds-ratio-control-refuted.md` (Props 1–2, 5–6, + crash family, slice reduction); `attempts/006-skeptic-review-of-005.md` + (S8, lambda > 0 equality correction); `attempts/007-averaged-or-control.md` + (well-posed M_i, the witness, dilution invariance, aggregated form = + §5 item 1, leads 1–2); `attempts/008-perturbative-assembly.md` + + `attempts/012-skeptic-review-of-008.md` (orbit-engine idea, budget-growth + reversal warning, crash-mixture cells); `attempts/009-mutual-information-tax.md` + + `attempts/011-skeptic-review-of-009.md` (lambda-window law 4.847/(n-3), + tilt-recipe cells); `attempts/013-skeptic-review-of-007.md` (exact + certification standard, t = 181/16, R2 refit-scope convention, + surrogate-lambda bookkeeping). +- Tools/data: `explore/uc_or_agg_probe.py`; + `data/or_agg_probe_part[XOPTSHE]*.json`; `data/or_agg_probe_run.log`; + conventions cross-checked against `explore/uc_or_avg.py`, + `explore/uc_or_avg_skeptic.py`, `explore/uc_pert_skeptic.py`. +- No external sources; the atanh-series ln bound is standard calculus, + re-derived in the tool's docstrings (as in 013). + +## For the skeptic (load-bearing claims, in attack order) + +1. The 15 exact certificates (part E), especially crashmix_383_05 at + n = 32 (+0.001201865435) and the softest point (+5.988e-7 at n = 32): + they rest on my ~60-line directed-rounding enclosure kit + (`_dround` / `_ln_1_to_2_fp` / compacted `agg_enclosure`), which is + NOT independently reviewed. Re-derive the directed-rounding soundness + and re-run at least one certificate with an independent exact path + (013's slower enclosure code is on disk and takes ~1 min at n = 10). +2. The orbit-engine formulas: class count `Ncl`, mass factor `JW`, cell + `= t^{oa.ob + j + alpha.beta} S(ja+alpha, jb+beta)`, the j-independence + of the class OR, and the structural N_i reach test. Hand-derivable; + the direct-engine cross-check (part X2) is only at n = 6, 7. +3. The claim that Bernoulli products give A = 0 identically (equality + boundary) — two-line corollary of 005 Props 1–2 but it carries the + whole "census minima at bern rows are noise" reading of part O. +4. The aggregate's sign convention-robustness argument (numerator + unchanged, denominator grows under surrogate-lambda scoring). +5. The trend claims (growing margin in n; dip-and-recover at n ~ 48) are + float-only and grid-limited — checkpoints carry every number. diff --git a/problems/union-closed/attempts/015-skeptic-review-of-014.md b/problems/union-closed/attempts/015-skeptic-review-of-014.md new file mode 100644 index 0000000..2bdcb53 --- /dev/null +++ b/problems/union-closed/attempts/015-skeptic-review-of-014.md @@ -0,0 +1,358 @@ +# 015 — Skeptic review of 014 (aggregated OR probe to n = 32): independent re-certification + +- **Problem:** union-closed, `problems/union-closed/PROBLEM.md` +- **Date:** 2026-08-04 +- **Mode:** informed +- **Type:** adversarial verification of `014-aggregated-or-probe-large-n.md` + (default stance: refute). 014's highest-value confessed vulnerability is + that its 15 exact certificates rest on its own ~60-line directed-rounding + enclosure kit, reviewed by nobody. The decisive check here is therefore an + **independent exact certifier built on a different algorithm**: binary + digit-extraction (square-and-compare with directed integer shifts, sound- + ness proved in the docstring) instead of 014's atanh series + dyadic + compaction, integer-scaled row builders with an *exact* aggregate + denominator, own orbit-formula re-derivation, own Sinkhorn (0.4-damped, + not 014's sqrt damping), own direct full-support evaluator. Nothing on + the critical path is shared with `uc_or_agg_probe.py`; it is imported + only to reproduce the exact rational refit measures its orbit + certificates are *about* (the measure spec is data, not verification + logic) and as a comparison target. +- **Outcome in one line:** 014 holds: all five re-certified certificates + (including the softest, +5.99e-7 at n = 32, and the fully rational t = 4 + point) reproduce inside 014's enclosures at my fixed width ~1.1e-14 + (~200x tighter on the n = 32 orbit certificates), + the enclosure kit survives hand re-derivation and 437-case adversarial + cross-bracketing, the orbit formulas re-derive and cross-check at n = 8, 9 + with exact within-class OR agreement, the dip row reproduces digit-for- + digit and is not a grid artifact, and ~116 new instances in regions 014 + could not see (witness-direction off the product boundary, marginals + pushed to 0.3827, lambda below the grid floor) found 0 violations. + Corrections are reporting-level: the part-P "d^1.85–d^2.06" order range + contradicts its own checkpoint (true range d^1.39–d^2.06, grid-edge + artifact), and the "products are the exact equality set" phrase claims an + unproven converse (false outright at lambda = 0). +- **Tools:** `explore/uc_or_agg_skeptic2.py` (parts K/X/C/T/V; stdlib only; + deterministic; runtime ~3 min plus a ~1 min T-addendum inline script; + checkpoints `data/oragg_sk2_part[KXCTV].json`, log + `data/oragg_sk2_run.log`). Commands reproducing every number: + + python problems/union-closed/explore/uc_or_agg_skeptic2.py 2>&1 | \ + tee -a problems/union-closed/data/oragg_sk2_run.log + # per part: K (kit attack), X (cross-checks), C (re-certification), + # T (trend attack), V (new adversaries); the small-d order study is + # the inline script logged at the end of oragg_sk2_run.log + python problems/union-closed/explore/uc_or_agg_probe.py X # re-run + +- **Sources:** 005/006/007/012/013/014 (this repo). No external fetches. + +Notation as in 007/013/014: tilt coupling `pi ∝ u(A)u(B) t^{|A∩B|}`, +`t = 2^lambda`; `M_i` = 007 §1's well-posed per-coordinate average; +`A(mu, lambda) = sum_{i 0 proviso 006 had to add to 005 Prop 2's equality + case). Downstream uses are unaffected: part E's exclusion of bern rows + from certification and the "proof must be second-order at the boundary" + framing need only the forward inclusion. Converse probe: the natural + non-product candidate (the mirror family, which 006 S8 saw at per-i + equality) fails by hand computation — its support is block-product + {∅,{1}} × {∅,{2..n}}, the block target is product, so Sinkhorn gives a + block-product potential, M_1 = lambda exactly but M_2 = (n−1) lambda + > lambda, hence A > 0. The honest statement: products ⊆ equality set + (all lambda); equality set = products is open for lambda > 0. +- **R3 (cosmetic — docstring drift, the 006-R3/013-R1 genre).** + `uc_or_agg_probe.py`'s docstring says "O ~ 40-80 min … E ~ 10-30 min, + X+S+H ~ 15 min"; the actual run (log + checkpoint `seconds` fields) is + O ≈ 146 s, E ≈ 113 s, H ≈ 208 s. The record's *own* per-part times + (~3 min, ~2 min) are correct; the docstring is stale. +- **R4 (cosmetic — instance-count slip).** The Outcome's "~1780 float + instances" understates its own listed components: 840 + 432 + 40 + 426 + + 82 = 1820. + +## Claims that survive (and what was done to break them) + +### 1. The certification kit and the 15 certificates — CONFIRMED; five re-certified independently, exact agreement + +*Hand re-derivation of the kit (part K docstrings carry the proofs):* +`_dround` floors/ceils onto ≤120-bit dyadics via integer floor division — +directionally sound including negatives (2000-case fuzz: 0 failures). +`_ln_1_to_2_fp`: the atanh partial sum is a lower bound (positive terms); +the tail bound `2 z^{2J+1}/((2J+1)(1−z²))` is the correct geometric +majorant; for the upper pass every intermediate is rounded up and the tail +uses the up-rounded z², which only enlarges it — outward everywhere. In +`log2_enclosure` the range reduction is exact (Fraction halving), and the +division by ln 2 picks the correct bound (LN2D_LO under a positive +numerator for the upper bound, etc.; the negative-numerator branches are +dead code but also correct). `agg_enclosure`'s compaction rounds every +running sum outward; the num/den interval ratio takes worst cases +independently, which is conservative (valid, slightly loose). +**One-sided verdict: sound.** + +*Independent adversarial test (part K1):* my own enclosure — +digit-extraction `log2_bracket`, working mantissa 80 bits, 48 extracted +bits, ±(1,2)-ulp safety margins, soundness proof in the docstring, no +shared code or algorithm — against their `log2_enclosure` on 437 +rationals: 400 random with numerators/denominators up to 200 bits, plus +2^k(1 ± 1e-25), 2^k(1 ± 1/3) for k ∈ {−100..100}, 181/16, 271/256. +Since my bracket provably contains the truth at width 1.1e-14, any +their-interval disjoint from mine convicts them: **0 failures** (their +max width 1.5e-13). + +*Independent re-certification (part C):* five of the fifteen, on the +**identical exact rational measures** — sparse witnesses rebuilt from the +stated atom weights and layout; orbit refit measures mu-tilde rebuilt by +re-running 014's deterministic float Sinkhorn + 36-bit quantization as an +input spec, then everything downstream (row builder, integer scaling, +log2, marginals) mine: + + certificate mine [lo, hi] 014's [lo, hi] overlap + witness_n10_wdil16 (t=181/16) [+1.70954246371, +1.70954246371] [+1.70954246371, +1.70954246371] yes + witness_n20_t4 (lambda=2 exact) [+0.483059568557, +0.483059568557] same to all printed digits yes + witness_n32_wdil20 (t=181/16) [+0.803594797389, +0.803594797389] same to all printed digits yes + crashmix_383_05_n32 (t=271/256) [+1.20186543655e-3, +1.20186543656e-3] [+1.2018654345e-3, +1.2018654366e-3] yes + softest: mixdir_slice p=.3 n=32 [+5.98823635652e-7, +5.9882364631e-7] [+5.9881776522e-7, +5.9882370424e-7] yes + + My width is 1.1e-14 everywhere (their orbit widths 2–6e-12; their + sparse widths ~1e-15 are finer still); on the orbit certificates my + intervals sit *inside* theirs and on the sparse ones the intervals + agree to all printed digits; all signs positive, exact in-regime + marginals confirmed + (crashmix max marginal exactly 0.38235 < 0.38271 as a rational + comparison). **No exact disagreement anywhere.** In addition a fresh + certificate at the softest point on **my own** mu-tilde-prime (my + Sinkhorn damping, my 40-bit quantization — a different exact rational + measure near the same family point): [+5.9882e-7, +5.9882e-7], + positive. The aggregate's insensitivity (10 digits) to which ~1e-11 + refit is used says the certified value is a property of the family + point, not of the quantization. + +### 2. The orbit-engine formulas — re-derived by hand; CONFIRMED at n = 8, 9 + +- *Re-derivation:* for tail-exchangeable mu and i ≥ 3, conditioning on + prefix classes gives cell(α,β) = `t^{oa·ob + j} · t^{αβ} · + S[ja+α][jb+β]` with S the future pair-kernel contraction; the `t^j` and + `t^{oa·ob}` factors cancel in the cross-ratio, so **OR = t · S00·S11 / + (S01·S10) is j-independent** and the j-sum collapses into + `JW[ja][jb] = Σ_j Ncl t^j` — 014's formulas are forced. `Ncl(q,ja,jb,j) + = C(q,j)C(q−j,ja−j)C(q−ja,jb−j)` and `PK_f[s][l] = C(f,s) Σ_w + C(s,w)C(f−s,l−w) t^w` were validated by brute-force subset enumeration + (part K4: all q ≤ 6, f ≤ 5, rational t = 7/3 exact — 0 failures). +- *Cross-checks at n = 8 AND 9* (014 stopped at 6, 7), part X2: my orbit + engine vs my own direct full-support evaluator vs 014's orbit engine on + the same fitted potential, four families (crashmix, sawin_geo_a, + slice-direction mixture, d0bern) × lambda ∈ {0.5, 1}: max disagreement + across all 16 instances **1.1e-13**. +- *j-independence tested directly* (part X3): at n = 9, i = 6, per-history + ORs from the direct evaluator grouped into (oa,ob,ja,jb) classes: 256 + classes, within-class relative spread **exactly 0.0**. +- *N_i reach test:* implied by the direct-engine agreement (my evaluator + derives N_i from the actual support) and by the exact dichotomy asserts + in both exact paths never firing. + +### 3. The equality boundary — forward direction CONFIRMED (re-proved and re-measured); converse is R2 + +Two-line forward proof re-derived: a product potential factorizes the +kernel, so at every history both response slices are proportional, OR = +2^lambda pointwise (005 Prop 2 equality case), M_i ≡ lambda, A ≡ 0 — at +every lambda, every n. Verified with my independent evaluator (numbers in +R2). The *census-noise reading* of part O (bern rows at ±1e-14 are +boundary noise, not near-violations) is therefore right, and part E's +refusal to "certify" a boundary point is the correct move. The +biconditional phrasing is R2. + +### 4. Surrogate-lambda sign-robustness — CONFIRMED (re-derived) + +Under 013's alternative bookkeeping every margin-degenerate history +contributes `w·(lambda − lambda) = 0` to the numerator and `w > 0` to the +denominator; this also covers coordinates whose N_i is empty under +conditioning-out (they enter the surrogate denominator with zero +numerator). So the numerator of A is literally unchanged, the denominator +only grows: sign preserved, magnitude shrinks toward 0. One boundary case +014 skates over, harmless: if *no* coordinate has defined mass the +conditioning-out A is undefined while the surrogate A is 0 — no sign to +disagree about. Sound as stated. + +### 5. The float trends — CONFIRMED; the dip and the minima are real features, not grid artifacts + +- *Dip row reproduced digit-for-digit* by my engine (part T1): +6.150e-7 / + +2.217e-7 / +1.408e-7 / +2.330e-7 / +4.305e-7 at n = 32..64 — matches + 014's checkpoints to 4 digits at every n, with independent Sinkhorn. +- *Not a lambda-grid artifact* (part T2): at the dip point (n = 48) lambda + swept 0.005–0.5 including below 014's grid floor 0.079: A > 0 + throughout; A/lambda decreases monotonically to **+6.4e-7 as + lambda → 0** — the first-order-in-lambda coefficient (007 Theorem C's + perfect square) stays bounded away from 0. Not a d artifact (d ∈ + {0.01..0.2} positive), and p = 0.30 is the local minimizer over p ∈ + {0.26..0.34} at n = 48 (so 014's softest point is a genuine local + extremum of the sampled family, not where its grid stopped). +- *Census-minimum growth in n confirmed off-grid* (part T3): crashmix_383_05 + at ten lambdas spanning 0.005–4.4 (including values between 014's big + grid points): A monotone increasing in lambda at fixed n, and the + margin **grows with n at every lambda tested** — the growth claim is + not a property of the lambda ≈ 0.083 grid floor. The census minimum + sitting at the smallest grid lambda is the O(lambda) scaling of A, as + 014 itself says, not a hidden interior extremum. +- Spot-verified from checkpoints against the record: per-n census minima + (+4.88e-4 … +1.21e-3 at crashmix, smallest grid lambda), family + ordering at n = 32, 660/840 in-regime, Sinkhorn residuals ≤ 1e-11 on + all 840, max |M_n − lambda| = 1.39e-13, enclosure widths ≤ 5.94e-12, + H best endpoint +0.005602, witness n·A → 1.7555 (my evaluator, + independent). 014's part X re-run reproduces its checkpoint. + +### New-adversary sweep (part V) — 0 in-regime violations in ~116 new instances + +Chosen to hit exactly what 014's search could not see: + +- **Witness-direction departure from the product boundary** (n = 10): + `u = x^{|A|} + d · u_witness` in potential space — a non-tail-symmetric + direction; 014's part P sampled only 4 symmetric rays and its record + itself names the Hessian's direction space as the attack surface. + 24 cells (bulk marginal targets 0.30 / 0.375 × d ∈ {0.03, 0.3, 3} × + lambda ∈ {0.084, 0.5, 2, 3.5}): all positive, min +3.2e-8 at the + smallest departure — consistent with quadratic upward curvature in this + direction too. +- **Boundary-of-regime marginals** (max marginal pushed to 0.38264–0.38270 + by tuning crashmix p, d0bern components, and a scaled Sawin mixture, at + n = 20, 32): all positive, and with *larger* margins than the interior + census — the regime boundary is not where the aggregate is tight. +- **Tiny lambda** (0.002–0.04, below 014's grid floor) at the two tightest + families, n = 20, 32: positive, cleanly linear in lambda. + +## Verdict + +| # | 014 claim | Verdict | +|---|-----------|---------| +| 1 | 15 exact certificates + enclosure kit | **CONFIRMED** — kit sound by hand re-derivation + 437-case adversarial cross-bracketing (0 failures); 5/15 re-certified on identical exact inputs by an independent algorithm, exact agreement, my tighter intervals nested inside theirs; fresh independent certificate at the softest point also positive | +| 2 | Orbit-engine formulas, j-independence, N_i test | **CONFIRMED** — hand re-derivation; combinatorics brute-forced; 3-way engine agreement ≤ 1.1e-13 at n = 8, 9; within-class OR spread exactly 0 | +| 3 | Products are the exact equality set | **CONFIRMED as an inclusion, corrected as stated** (R2): forward re-proved and re-measured; converse unproven and false at lambda = 0; mirror-candidate fails to break it | +| 4 | Surrogate-lambda sign-robustness | **CONFIRMED** (re-derived, one harmless undefined-A boundary case noted) | +| 5 | Trend claims (margin growth, dip-and-recover) | **CONFIRMED** — dip reproduced digit-for-digit; shown robust off-grid in lambda, d, p; growth-in-n holds at every off-grid lambda tested; part-P order range corrected (R1) | +| — | 0 violations in ~1780 instances | **CONFIRMED and extended** — count is 1820 (R4); +~116 new instances in unseen regions, 0 violations | + +**Net assessment:** 014's headline — the i-aggregated OR control survives +to n = 32 certified and to n = 64 on trends, with growing margin and no +012-style reversal — should be treated as **VERIFIED at its stated scope** +(finite n, finite grids, tested genres; certificates are sign theorems +about the stated exact rational measures). The queue-item verdict (proof +effort now justified, lead 1 = product-boundary Hessian) stands, with one +sharpening from R1: at lambda ≳ 0.5 the third-order term in d is large +enough to halve the fitted order at d = 0.005, so **a Hessian-PSD result +alone will not control the boundary layer at finite d — lead 1 needs a +quantitative cubic remainder bound**, or a restriction to the small-d +regime it certifies. + +## Why it failed / what survived + +Nothing failed in the reviewed record beyond reporting-level slips; the +review's own kill attempts failed for identifiable reasons worth keeping: + +- The kit attack failed because every rounding in 014's kit really is + outward; the compaction idea (directed dyadic fixed-point) is sound and + is *the* reason exact certification is cheap at n = 32. My digit- + extraction bracket holds a fixed 1.1e-14 width at similar cost (~200x + tighter on the n = 32 orbit certificates) — worth adopting for future + certificates where margins are ~1e-7. +- The witness-direction boundary attack failed with clean quadratic + positivity: even the one genre known to produce per-i deficits + (M_i − lambda = −0.122) enters the aggregate quadratically upward when + diluted into a product bulk. The bulk floods every prefix, so the + gadget's deficit rows carry vanishing relative mass — same mechanism as + 014's Theta(1/n) witness-genre decay, seen from the boundary side. +- The regime-boundary attack failed because pushing marginals toward + 0.38271 *raises* the aggregate on every family tried: the tight + direction is small lambda near products, not large marginals. + +Residual risk, labelled: all engines (005–014 and this one) formalize the +same census definition from 003; exact arithmetic and re-derivation remove +numerical and algebraic error, not a shared conceptual error in that +definition (already flagged in 013; still true). The orbit certificates +remain statements about quantized refit measures (013 R2 scope), which is +the right scope for the universally-quantified claim. + +Reusable: `explore/uc_or_agg_skeptic2.py` — the digit-extraction log2 +bracket (`log2_bracket`, soundness proof inline, width 1.1e-14), the +integer-scaled exact orbit/sparse certifiers with exact denominators, the +from-scratch direct evaluator, and the boundary/witness-direction +instance builders. Checkpoints carry every number +(`data/oragg_sk2_part[KXCTV].json`). + +## Leads generated + +1. **(sharpens 014 lead 1)** The Hessian program at the product boundary + needs a cubic remainder bound: measured fitted orders at d = 0.005 drop + to 1.39 (full direction, lambda = 0.5, n = 32) before recovering + toward 2 as d → 0. Compute the third derivative's growth in (n, + lambda) along the full/crash rays with the orbit engine before + trusting any finite-d positivity radius from the Hessian alone. +2. **Settle the equality-set converse for lambda > 0** (from R2): is + A(mu, lambda) = 0 only at (block-degenerate) products? For ≤ 4-atom + supports Theorem B reduces it to per-i equality, where 005 Prop 2's + proportional-slices characterization applies; the full-support case + looks like a rank-1-slices induction. A non-product equality point + would be a second tangency of the conjectured inequality that the + Hessian program would silently miss; a proof closes the boundary + description in 014's proof plan. +3. The first-order coefficient at the dip point (A/lambda → +6.4e-7 at + n = 48) is the smallest Theorem-C perfect-square value seen anywhere; + 014's lead 2 (map the dip in (p, layer, n)) should track this + lambda → 0 coefficient rather than A at a fixed small lambda — it is + the quantity that would actually cross zero first if the control fails + along this family. + +## References + +- Reviewed record: `problems/union-closed/attempts/014-aggregated-or-probe-large-n.md`; + its tool `explore/uc_or_agg_probe.py` and checkpoints + `data/or_agg_probe_part[XOPTSHE]*.json`, `data/or_agg_probe_run.log`. +- Context: `attempts/007-averaged-or-control.md` (M_i definition, witness, + dilution invariance); `attempts/013-skeptic-review-of-007.md` (exact + certification standard, refit scope, surrogate-lambda); + `attempts/012-skeptic-review-of-008.md` (orbit symmetry, reversal + warning); `attempts/005-*`/`006-*` (Props 1–2 and the lambda > 0 + equality proviso reused in R2). +- This review's tool/data: `explore/uc_or_agg_skeptic2.py`; + `data/oragg_sk2_part[KXCTV].json`; `data/oragg_sk2_run.log`. +- No external sources; the digit-extraction log2 soundness argument is + elementary and written out in the tool's docstrings. diff --git a/problems/union-closed/data/or_agg_probe_partE.json b/problems/union-closed/data/or_agg_probe_partE.json new file mode 100644 index 0000000..1652c7f --- /dev/null +++ b/problems/union-closed/data/or_agg_probe_partE.json @@ -0,0 +1,217 @@ +{ + "orbit": [ + { + "agg_hi": 0.00047692397516781366, + "agg_lo": 0.00047692397155503225, + "certified_sign": "positive", + "enclosure_width": 3.612781410386039e-12, + "engine": "orbit", + "lambda": 0.08214904135387156, + "max_marginal": 0.38235000000080877, + "max_marginal_exact_lt_record": true, + "n": 10, + "name": "crashmix_383_05_n10", + "seconds": 0.6, + "sinkhorn_resid": 5.628608690244619e-12, + "t": "271/256" + }, + { + "agg_hi": 0.0006342686849517653, + "agg_lo": 0.0006342686819540854, + "certified_sign": "positive", + "enclosure_width": 2.9976799035232697e-12, + "engine": "orbit", + "lambda": 0.08214904135387156, + "max_marginal": 0.38235000000014047, + "max_marginal_exact_lt_record": true, + "n": 14, + "name": "crashmix_383_05_n14", + "seconds": 2.1, + "sinkhorn_resid": 5.294764626739834e-12, + "t": "271/256" + }, + { + "agg_hi": 0.0008299678041200834, + "agg_lo": 0.000829967801625059, + "certified_sign": "positive", + "enclosure_width": 2.4950243794809437e-12, + "engine": "orbit", + "lambda": 0.08214904135387156, + "max_marginal": 0.3823500000005936, + "max_marginal_exact_lt_record": true, + "n": 20, + "name": "crashmix_383_05_n20", + "seconds": 7.6, + "sinkhorn_resid": 8.719136523893667e-12, + "t": "271/256" + }, + { + "agg_hi": 0.0009526375609496559, + "agg_lo": 0.0009526375586484367, + "certified_sign": "positive", + "enclosure_width": 2.3012191391591883e-12, + "engine": "orbit", + "lambda": 0.08214904135387156, + "max_marginal": 0.38234999999998837, + "max_marginal_exact_lt_record": true, + "n": 24, + "name": "crashmix_383_05_n24", + "seconds": 14.3, + "sinkhorn_resid": 5.373368416883295e-12, + "t": "271/256" + }, + { + "agg_hi": 0.0010755884907999072, + "agg_lo": 0.001075588488635752, + "certified_sign": "positive", + "enclosure_width": 2.164155144909046e-12, + "engine": "orbit", + "lambda": 0.08214904135387156, + "max_marginal": 0.3823500000000963, + "max_marginal_exact_lt_record": true, + "n": 28, + "name": "crashmix_383_05_n28", + "seconds": 25.0, + "sinkhorn_resid": 6.2553295876455195e-12, + "t": "271/256" + }, + { + "agg_hi": 0.0012018654365805032, + "agg_lo": 0.0012018654345185519, + "certified_sign": "positive", + "enclosure_width": 2.061951300960488e-12, + "engine": "orbit", + "lambda": 0.08214904135387156, + "max_marginal": 0.38235000000005875, + "max_marginal_exact_lt_record": true, + "n": 32, + "name": "crashmix_383_05_n32", + "seconds": 39.4, + "sinkhorn_resid": 6.996958568095124e-12, + "t": "271/256" + }, + { + "agg_hi": 4.637006726853692e-06, + "agg_lo": 4.637001982432064e-06, + "certified_sign": "positive", + "enclosure_width": 4.744421627743592e-12, + "engine": "orbit", + "lambda": 0.08214904135387156, + "max_marginal": 0.3025000000002532, + "max_marginal_exact_lt_record": true, + "n": 20, + "name": "mixdir_slice_p0.3_d0.05_n20", + "seconds": 3.4, + "sinkhorn_resid": 7.300160476120254e-12, + "t": "271/256" + }, + { + "agg_hi": 5.988237042421268e-07, + "agg_lo": 5.988177652162741e-07, + "certified_sign": "positive", + "enclosure_width": 5.939025852692808e-12, + "engine": "orbit", + "lambda": 0.08214904135387156, + "max_marginal": 0.303749999999851, + "max_marginal_exact_lt_record": true, + "n": 32, + "name": "mixdir_slice_p0.3_d0.05_n32", + "seconds": 20.0, + "sinkhorn_resid": 5.720979245893432e-12, + "t": "271/256" + } + ], + "sparse": [ + { + "agg_hi": 1.7095424637138141, + "agg_lo": 1.7095424637138115, + "certified_sign": "positive", + "enclosure_width": 2.7020851819931803e-15, + "engine": "sparse", + "max_marginal": 0.27049832331955115, + "max_marginal_exact_lt_record": true, + "n": 10, + "name": "witness_n10_wdil16", + "seconds": 0.0, + "t": "181/16" + }, + { + "agg_hi": 1.4740401471251452, + "agg_lo": 1.4740401471251292, + "certified_sign": "positive", + "enclosure_width": 1.5890349997392047e-14, + "engine": "sparse", + "max_marginal": 0.19672031651417546, + "max_marginal_exact_lt_record": true, + "n": 14, + "name": "witness_n14_wdil16", + "seconds": 0.0, + "t": "181/16" + }, + { + "agg_hi": 1.1629475903583624, + "agg_lo": 1.162947590358356, + "certified_sign": "positive", + "enclosure_width": 6.285520433445028e-15, + "engine": "sparse", + "max_marginal": 0.10440100446929078, + "max_marginal_exact_lt_record": true, + "n": 20, + "name": "witness_n20_wdil20", + "seconds": 0.0, + "t": "181/16" + }, + { + "agg_hi": 1.0135090911747144, + "agg_lo": 1.0135090911747104, + "certified_sign": "positive", + "enclosure_width": 3.933652884563209e-15, + "engine": "sparse", 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1.3739009929736312e-15, + "aggregate": 0.002205353552242407, + "argmin_i": 55, + "in_regime": true, + "lambda": 0.084064, + "max_marginal": 0.38235000000002184, + "min_excess_lt_n": 6.386556615914363e-05, + "n": 56, + "name": "crashmix_383_05", + "seconds": 2.85, + "sinkhorn_resid": 7.049028027950044e-12 + }, + { + "Mn_minus_lam_abs": 5.412337245047638e-16, + "aggregate": 4.305224233247369e-07, + "argmin_i": 54, + "d": 0.05, + "in_regime": true, + "lambda": 0.084, + "max_marginal": 0.3037500000000252, + "min_excess_lt_n": 3.9702401574692736e-08, + "n": 64, + "name": "mixdir_slice_p0.3_d0.05", + "seconds": 5.48, + "sinkhorn_resid": 5.5146998079180776e-12 + }, + { + "Mn_minus_lam_abs": 5.551115123125783e-17, + "aggregate": 4.767854242005066e-06, + "argmin_i": 55, + "d": 0.05, + "in_regime": true, + "lambda": 0.5, + "max_marginal": 0.3037500000000218, + "min_excess_lt_n": 1.5270855864280009e-06, + "n": 64, + "name": "mixdir_slice_p0.3_d0.05", + "seconds": 5.35, + "sinkhorn_resid": 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0.38235000000002617, + "min_excess_lt_n": 6.973246466125682e-05, + "n": 64, + "name": "crashmix_383_05", + "seconds": 4.88, + "sinkhorn_resid": 9.11382080914791e-12 + } + ] +} diff --git a/problems/union-closed/data/or_agg_probe_partX.json b/problems/union-closed/data/or_agg_probe_partX.json new file mode 100644 index 0000000..7b1f261 --- /dev/null +++ b/problems/union-closed/data/or_agg_probe_partX.json @@ -0,0 +1,17 @@ +{ + "x1": { + "M5_minus_lam": -0.1220334102236671, + "aggregate": 1.844754315091965, + "max_marginal": 0.31776857435310957, + "pass": true + }, + "x1_exact": { + "hi": 1.8446690053408297, + "lo": 1.8446690053408268, + "pass": true + }, + "x2_max_diff": 7.949196856316121e-13, + "x2_pass": true, + "x3_diff": 7.889522368742519e-15, + "x3_pass": true +} diff --git a/problems/union-closed/data/or_agg_probe_run.log b/problems/union-closed/data/or_agg_probe_run.log new file mode 100644 index 0000000..53df261 --- /dev/null +++ b/problems/union-closed/data/or_agg_probe_run.log @@ -0,0 +1,137 @@ +== part O: orbit family census == +[O] n=10: 140 instances (110 in-regime), aggregate violations: 0; min in-regime aggregate = -0.000000 at bern_p3823 lam=3.5 (i* = 9, min per-i excess -0.0000); max resid 1.0e-11, max |M_n - lam| 2.1e-14; 1s +[O] n=14: 140 instances (110 in-regime), aggregate violations: 0; min in-regime aggregate = -0.000000 at bern_p3823 lam=3.5 (i* = 12, min per-i excess -0.0000); max resid 1.0e-11, max |M_n - lam| 4.9e-14; 4s +[O] n=20: 140 instances (110 in-regime), aggregate violations: 0; min in-regime aggregate = -0.000000 at bern_p3823 lam=5 (i* = 16, min per-i excess -0.0000); max resid 1.0e-11, max |M_n - lam| 6.0e-14; 14s +[O] n=24: 140 instances (110 in-regime), aggregate violations: 0; min in-regime aggregate = -0.000000 at bern_p3823 lam=3.5 (i* = 23, min per-i excess -0.0000); max resid 9.7e-12, max |M_n - lam| 5.2e-14; 21s +[O] n=28: 140 instances (110 in-regime), aggregate violations: 0; min in-regime aggregate = -0.000000 at bern_p3823 lam=3.5 (i* = 27, min per-i excess -0.0000); max resid 9.9e-12, max |M_n - lam| 6.2e-14; 39s +[O] n=32: 140 instances (110 in-regime), aggregate violations: 0; min in-regime aggregate = -0.000000 at bern_p30 lam=5 (i* = 30, min per-i excess -0.0000); max resid 9.9e-12, max |M_n - lam| 1.4e-13; 67s +done in 146s; checkpoints in /home/user/math-lab/problems/union-closed/data +== part P: perturbative d-sweep around product boundary == +[P] n=10 p=0.3823 dir=crash lam=0.084064: agg(d) = 0.005:+6.40e-06 0.01:+2.42e-05 0.02:+8.99e-05 0.05:+4.89e-04 0.1:+1.69e-03 0.2:+5.64e-03 order~d^1.92 +[P] n=10 p=0.3823 dir=crash lam=0.5: agg(d) = 0.005:+4.42e-05 0.01:+1.66e-04 0.02:+6.12e-04 0.05:+3.28e-03 0.1:+1.12e-02 0.2:+3.66e-02 order~d^1.91 +[P] n=10 p=0.3823 dir=crash lam=0.692429: agg(d) = 0.005:+6.72e-05 0.01:+2.52e-04 0.02:+9.20e-04 0.05:+4.87e-03 0.1:+1.63e-02 0.2:+5.23e-02 order~d^1.90 +[P] n=10 p=0.3823 dir=empty lam=0.084064: agg(d) = 0.005:+4.06e-06 0.01:+1.58e-05 0.02:+6.00e-05 0.05:+3.38e-04 0.1:+1.20e-03 0.2:+4.10e-03 order~d^1.96 +[P] n=10 p=0.3823 dir=empty lam=0.5: agg(d) = 0.005:+2.99e-05 0.01:+1.16e-04 0.02:+4.36e-04 0.05:+2.41e-03 0.1:+8.33e-03 0.2:+2.74e-02 order~d^1.95 +[P] n=10 p=0.3823 dir=empty lam=0.692429: agg(d) = 0.005:+4.55e-05 0.01:+1.75e-04 0.02:+6.58e-04 0.05:+3.59e-03 0.1:+1.23e-02 0.2:+3.97e-02 order~d^1.94 +[P] n=10 p=0.3823 dir=full lam=0.084064: agg(d) = 0.005:+2.26e-05 0.01:+8.15e-05 0.02:+2.86e-04 0.05:+1.42e-03 0.1:+4.55e-03 0.2:+1.37e-02 order~d^1.85 +[P] n=10 p=0.3823 dir=full lam=0.5: agg(d) = 0.005:+2.28e-04 0.01:+8.04e-04 0.02:+2.73e-03 0.05:+1.27e-02 0.1:+3.75e-02 0.2:+1.03e-01 order~d^1.82 +[P] n=10 p=0.3823 dir=full lam=0.692429: agg(d) = 0.005:+3.98e-04 0.01:+1.39e-03 0.02:+4.61e-03 0.05:+2.05e-02 0.1:+5.80e-02 0.2:+1.52e-01 order~d^1.80 +[P] n=10 p=0.3823 dir=slice lam=0.084064: agg(d) = 0.005:+5.56e-08 0.01:+2.24e-07 0.02:+9.15e-07 0.05:+6.05e-06 0.1:+2.65e-05 0.2:+1.26e-04 order~d^2.01 +[P] n=10 p=0.3823 dir=slice lam=0.5: agg(d) = 0.005:+4.29e-07 0.01:+1.73e-06 0.02:+7.01e-06 0.05:+4.57e-05 0.1:+1.96e-04 0.2:+9.01e-04 order~d^2.01 +[P] n=10 p=0.3823 dir=slice lam=0.692429: agg(d) = 0.005:+6.56e-07 0.01:+2.64e-06 0.02:+1.07e-05 0.05:+6.94e-05 0.1:+2.96e-04 0.2:+1.34e-03 order~d^2.01 +[P] n=10 p=0.3 dir=crash lam=0.084064: agg(d) = 0.005:+7.74e-06 0.01:+2.87e-05 0.02:+1.04e-04 0.05:+5.49e-04 0.1:+1.85e-03 0.2:+5.96e-03 order~d^1.89 +[P] n=10 p=0.3 dir=crash lam=0.5: agg(d) = 0.005:+6.27e-05 0.01:+2.29e-04 0.02:+8.15e-04 0.05:+4.15e-03 0.1:+1.35e-02 0.2:+4.14e-02 order~d^1.87 +[P] n=10 p=0.3 dir=crash lam=0.692429: agg(d) = 0.005:+1.03e-04 0.01:+3.72e-04 0.02:+1.31e-03 0.05:+6.48e-03 0.1:+2.04e-02 0.2:+6.05e-02 order~d^1.85 +[P] n=10 p=0.3 dir=empty lam=0.084064: agg(d) = 0.005:+1.96e-06 0.01:+7.74e-06 0.02:+3.02e-05 0.05:+1.77e-04 0.1:+6.53e-04 0.2:+2.32e-03 order~d^1.98 +[P] n=10 p=0.3 dir=empty lam=0.5: agg(d) = 0.005:+1.36e-05 0.01:+5.33e-05 0.02:+2.07e-04 0.05:+1.20e-03 0.1:+4.37e-03 0.2:+1.52e-02 order~d^1.98 +[P] n=10 p=0.3 dir=empty lam=0.692429: agg(d) = 0.005:+2.00e-05 0.01:+7.87e-05 0.02:+3.05e-04 0.05:+1.76e-03 0.1:+6.34e-03 0.2:+2.19e-02 order~d^1.97 +[P] n=10 p=0.3 dir=full lam=0.084064: agg(d) = 0.005:+4.02e-05 0.01:+1.41e-04 0.02:+4.82e-04 0.05:+2.29e-03 0.1:+6.96e-03 0.2:+1.98e-02 order~d^1.81 +[P] n=10 p=0.3 dir=full lam=0.5: agg(d) = 0.005:+4.70e-04 0.01:+1.61e-03 0.02:+5.22e-03 0.05:+2.24e-02 0.1:+6.13e-02 0.2:+1.54e-01 order~d^1.77 +[P] n=10 p=0.3 dir=full lam=0.692429: agg(d) = 0.005:+8.79e-04 0.01:+2.93e-03 0.02:+9.18e-03 0.05:+3.69e-02 0.1:+9.55e-02 0.2:+2.29e-01 order~d^1.74 +[P] n=10 p=0.3 dir=slice lam=0.084064: agg(d) = 0.005:+1.31e-07 0.01:+5.26e-07 0.02:+2.12e-06 0.05:+1.35e-05 0.1:+5.61e-05 0.2:+2.41e-04 order~d^2.00 +[P] n=10 p=0.3 dir=slice lam=0.5: agg(d) = 0.005:+8.07e-07 0.01:+3.24e-06 0.02:+1.30e-05 0.05:+8.33e-05 0.1:+3.45e-04 0.2:+1.48e-03 order~d^2.00 +[P] n=10 p=0.3 dir=slice lam=0.692429: agg(d) = 0.005:+1.13e-06 0.01:+4.55e-06 0.02:+1.83e-05 0.05:+1.17e-04 0.1:+4.85e-04 0.2:+2.08e-03 order~d^2.00 +[P] n=20 p=0.3823 dir=crash lam=0.084064: agg(d) = 0.005:+1.17e-05 0.01:+4.31e-05 0.02:+1.57e-04 0.05:+8.51e-04 0.1:+2.98e-03 0.2:+1.03e-02 order~d^1.89 +[P] n=20 p=0.3823 dir=crash lam=0.285118: agg(d) = 0.005:+4.89e-05 0.01:+1.80e-04 0.02:+6.55e-04 0.05:+3.49e-03 0.1:+1.20e-02 0.2:+3.97e-02 order~d^1.88 +[P] n=20 p=0.3823 dir=crash lam=0.5: agg(d) = 0.005:+1.26e-04 0.01:+4.59e-04 0.02:+1.64e-03 0.05:+8.32e-03 0.1:+2.69e-02 0.2:+8.23e-02 order~d^1.87 +[P] n=20 p=0.3823 dir=empty lam=0.084064: agg(d) = 0.005:+7.40e-06 0.01:+2.79e-05 0.02:+1.04e-04 0.05:+5.76e-04 0.1:+2.06e-03 0.2:+7.18e-03 order~d^1.91 +[P] n=20 p=0.3823 dir=empty lam=0.285118: agg(d) = 0.005:+3.18e-05 0.01:+1.20e-04 0.02:+4.43e-04 0.05:+2.42e-03 0.1:+8.42e-03 0.2:+2.82e-02 order~d^1.91 +[P] n=20 p=0.3823 dir=empty lam=0.5: agg(d) = 0.005:+7.21e-05 0.01:+2.70e-04 0.02:+9.89e-04 0.05:+5.27e-03 0.1:+1.77e-02 0.2:+5.67e-02 order~d^1.90 +[P] n=20 p=0.3823 dir=full lam=0.084064: agg(d) = 0.005:+3.45e-05 0.01:+1.24e-04 0.02:+4.38e-04 0.05:+2.23e-03 0.1:+7.34e-03 0.2:+2.32e-02 order~d^1.85 +[P] n=20 p=0.3823 dir=full lam=0.285118: agg(d) = 0.005:+2.17e-04 0.01:+7.71e-04 0.02:+2.66e-03 0.05:+1.27e-02 0.1:+3.82e-02 0.2:+1.08e-01 order~d^1.83 +[P] n=20 p=0.3823 dir=full lam=0.5: agg(d) = 0.005:+7.37e-04 0.01:+2.53e-03 0.02:+8.17e-03 0.05:+3.43e-02 0.1:+9.24e-02 0.2:+2.33e-01 order~d^1.78 +[P] n=20 p=0.3823 dir=slice lam=0.084064: agg(d) = 0.005:+4.61e-08 0.01:+1.86e-07 0.02:+7.54e-07 0.05:+4.94e-06 0.1:+2.14e-05 0.2:+9.99e-05 order~d^2.01 +[P] n=20 p=0.3823 dir=slice lam=0.285118: agg(d) = 0.005:+1.62e-07 0.01:+6.51e-07 0.02:+2.64e-06 0.05:+1.73e-05 0.1:+7.46e-05 0.2:+3.47e-04 order~d^2.01 +[P] n=20 p=0.3823 dir=slice lam=0.5: agg(d) = 0.005:+2.93e-07 0.01:+1.18e-06 0.02:+4.79e-06 0.05:+3.12e-05 0.1:+1.35e-04 0.2:+6.25e-04 order~d^2.01 +[P] n=20 p=0.3 dir=crash lam=0.084064: agg(d) = 0.005:+1.34e-05 0.01:+4.90e-05 0.02:+1.77e-04 0.05:+9.34e-04 0.1:+3.20e-03 0.2:+1.07e-02 order~d^1.87 +[P] n=20 p=0.3 dir=crash lam=0.285118: agg(d) = 0.005:+6.98e-05 0.01:+2.53e-04 0.02:+8.95e-04 0.05:+4.56e-03 0.1:+1.49e-02 0.2:+4.61e-02 order~d^1.86 +[P] n=20 p=0.3 dir=crash lam=0.5: agg(d) = 0.005:+2.32e-04 0.01:+8.18e-04 0.02:+2.76e-03 0.05:+1.26e-02 0.1:+3.69e-02 0.2:+1.01e-01 order~d^1.82 +[P] n=20 p=0.3 dir=empty lam=0.084064: agg(d) = 0.005:+3.82e-06 0.01:+1.47e-05 0.02:+5.58e-05 0.05:+3.18e-04 0.1:+1.17e-03 0.2:+4.18e-03 order~d^1.94 +[P] n=20 p=0.3 dir=empty lam=0.285118: agg(d) = 0.005:+1.52e-05 0.01:+5.82e-05 0.02:+2.20e-04 0.05:+1.24e-03 0.1:+4.48e-03 0.2:+1.57e-02 order~d^1.94 +[P] n=20 p=0.3 dir=empty lam=0.5: agg(d) = 0.005:+3.15e-05 0.01:+1.20e-04 0.02:+4.53e-04 0.05:+2.52e-03 0.1:+8.93e-03 0.2:+3.04e-02 order~d^1.93 +[P] n=20 p=0.3 dir=full lam=0.084064: agg(d) = 0.005:+5.94e-05 0.01:+2.09e-04 0.02:+7.21e-04 0.05:+3.51e-03 0.1:+1.10e-02 0.2:+3.31e-02 order~d^1.82 +[P] n=20 p=0.3 dir=full lam=0.285118: agg(d) = 0.005:+4.45e-04 0.01:+1.53e-03 0.02:+5.06e-03 0.05:+2.22e-02 0.1:+6.20e-02 0.2:+1.61e-01 order~d^1.79 +[P] n=20 p=0.3 dir=full lam=0.5: agg(d) = 0.005:+1.79e-03 0.01:+5.70e-03 0.02:+1.68e-02 0.05:+6.14e-02 0.1:+1.50e-01 0.2:+3.49e-01 order~d^1.67 +[P] n=20 p=0.3 dir=slice lam=0.084064: agg(d) = 0.005:+4.39e-08 0.01:+1.77e-07 0.02:+7.21e-07 0.05:+4.75e-06 0.1:+2.07e-05 0.2:+9.81e-05 order~d^2.01 +[P] n=20 p=0.3 dir=slice lam=0.285118: agg(d) = 0.005:+1.55e-07 0.01:+6.25e-07 0.02:+2.54e-06 0.05:+1.67e-05 0.1:+7.25e-05 0.2:+3.42e-04 order~d^2.01 +[P] n=20 p=0.3 dir=slice lam=0.5: agg(d) = 0.005:+2.84e-07 0.01:+1.14e-06 0.02:+4.64e-06 0.05:+3.04e-05 0.1:+1.32e-04 0.2:+6.16e-04 order~d^2.01 +[P] n=32 p=0.3823 dir=crash lam=0.084064: agg(d) = 0.005:+1.63e-05 0.01:+6.08e-05 0.02:+2.24e-04 0.05:+1.23e-03 0.1:+4.42e-03 0.2:+1.56e-02 order~d^1.89 +[P] n=32 p=0.3823 dir=crash lam=0.167138: agg(d) = 0.005:+3.90e-05 0.01:+1.45e-04 0.02:+5.34e-04 0.05:+2.92e-03 0.1:+1.03e-02 0.2:+3.52e-02 order~d^1.90 +[P] n=32 p=0.3823 dir=crash lam=0.5: agg(d) = 0.005:+4.59e-04 0.01:+1.60e-03 0.02:+5.29e-03 0.05:+2.30e-02 0.1:+6.49e-02 0.2:+1.74e-01 order~d^1.81 +[P] n=32 p=0.3823 dir=empty lam=0.084064: agg(d) = 0.005:+1.08e-05 0.01:+4.06e-05 0.02:+1.52e-04 0.05:+8.57e-04 0.1:+3.11e-03 0.2:+1.10e-02 order~d^1.92 +[P] n=32 p=0.3823 dir=empty lam=0.167138: agg(d) = 0.005:+2.54e-05 0.01:+9.60e-05 0.02:+3.59e-04 0.05:+2.00e-03 0.1:+7.13e-03 0.2:+2.45e-02 order~d^1.92 +[P] n=32 p=0.3823 dir=empty lam=0.5: agg(d) = 0.005:+1.56e-04 0.01:+5.83e-04 0.02:+2.12e-03 0.05:+1.09e-02 0.1:+3.52e-02 0.2:+1.05e-01 order~d^1.91 +[P] n=32 p=0.3823 dir=full lam=0.084064: agg(d) = 0.005:+5.10e-05 0.01:+1.86e-04 0.02:+6.67e-04 0.05:+3.47e-03 0.1:+1.16e-02 0.2:+3.71e-02 order~d^1.87 +[P] n=32 p=0.3823 dir=full lam=0.167138: agg(d) = 0.005:+1.61e-04 0.01:+5.85e-04 0.02:+2.06e-03 0.05:+1.02e-02 0.1:+3.20e-02 0.2:+9.35e-02 order~d^1.86 +[P] n=32 p=0.3823 dir=full lam=0.5: agg(d) = 0.005:+2.86e-03 0.01:+8.67e-03 0.02:+2.42e-02 0.05:+8.31e-02 0.1:+1.97e-01 0.2:+4.50e-01 order~d^1.60 +[P] n=32 p=0.3823 dir=slice lam=0.084064: agg(d) = 0.005:+3.43e-08 0.01:+1.38e-07 0.02:+5.62e-07 0.05:+3.68e-06 0.1:+1.59e-05 0.2:+7.46e-05 order~d^2.01 +[P] n=32 p=0.3823 dir=slice lam=0.167138: agg(d) = 0.005:+6.85e-08 0.01:+2.76e-07 0.02:+1.12e-06 0.05:+7.33e-06 0.1:+3.17e-05 0.2:+1.49e-04 order~d^2.01 +[P] n=32 p=0.3823 dir=slice lam=0.5: agg(d) = 0.005:+2.07e-07 0.01:+8.35e-07 0.02:+3.39e-06 0.05:+2.22e-05 0.1:+9.60e-05 0.2:+4.51e-04 order~d^2.01 +[P] n=32 p=0.3 dir=crash lam=0.084064: agg(d) = 0.005:+1.93e-05 0.01:+7.10e-05 0.02:+2.58e-04 0.05:+1.39e-03 0.1:+4.86e-03 0.2:+1.66e-02 order~d^1.88 +[P] n=32 p=0.3 dir=crash lam=0.167138: agg(d) = 0.005:+5.45e-05 0.01:+2.00e-04 0.02:+7.21e-04 0.05:+3.78e-03 0.1:+1.27e-02 0.2:+4.09e-02 order~d^1.87 +[P] n=32 p=0.3 dir=crash lam=0.5: agg(d) = 0.005:+1.19e-03 0.01:+3.67e-03 0.02:+1.04e-02 0.05:+3.69e-02 0.1:+9.03e-02 0.2:+2.15e-01 order~d^1.62 +[P] n=32 p=0.3 dir=empty lam=0.084064: agg(d) = 0.005:+5.69e-06 0.01:+2.18e-05 0.02:+8.25e-05 0.05:+4.74e-04 0.1:+1.75e-03 0.2:+6.36e-03 order~d^1.93 +[P] n=32 p=0.3 dir=empty lam=0.167138: agg(d) = 0.005:+1.26e-05 0.01:+4.82e-05 0.02:+1.83e-04 0.05:+1.04e-03 0.1:+3.82e-03 0.2:+1.36e-02 order~d^1.93 +[P] n=32 p=0.3 dir=empty lam=0.5: agg(d) = 0.005:+5.89e-05 0.01:+2.25e-04 0.02:+8.43e-04 0.05:+4.65e-03 0.1:+1.62e-02 0.2:+5.32e-02 order~d^1.93 +[P] n=32 p=0.3 dir=full lam=0.084064: agg(d) = 0.005:+8.90e-05 0.01:+3.18e-04 0.02:+1.11e-03 0.05:+5.54e-03 0.1:+1.77e-02 0.2:+5.34e-02 order~d^1.84 +[P] n=32 p=0.3 dir=full lam=0.167138: agg(d) = 0.005:+3.21e-04 0.01:+1.13e-03 0.02:+3.84e-03 0.05:+1.77e-02 0.1:+5.15e-02 0.2:+1.40e-01 order~d^1.82 +[P] n=32 p=0.3 dir=full lam=0.5: agg(d) = 0.005:+7.22e-03 0.01:+1.89e-02 0.02:+4.60e-02 0.05:+1.37e-01 0.1:+3.01e-01 0.2:+6.45e-01 order~d^1.39 +[P] n=32 p=0.3 dir=slice lam=0.084064: agg(d) = 0.005:+4.23e-09 0.01:+1.76e-08 0.02:+7.66e-08 0.05:+6.16e-07 0.1:+3.61e-06 0.2:+2.64e-05 order~d^2.06 +[P] n=32 p=0.3 dir=slice lam=0.167138: agg(d) = 0.005:+1.08e-08 0.01:+4.47e-08 0.02:+1.91e-07 0.05:+1.46e-06 0.1:+8.14e-06 0.2:+5.61e-05 order~d^2.05 +[P] n=32 p=0.3 dir=slice lam=0.5: agg(d) = 0.005:+6.24e-08 0.01:+2.54e-07 0.02:+1.05e-06 0.05:+7.37e-06 0.1:+3.62e-05 0.2:+2.14e-04 order~d^2.02 +[P] 432 instances (369 in-regime); negative aggregates: 0; min in-regime aggregate = +4.229e-09 at n=32 mixdir_slice_p0.3 d=0.005 lam=0.084064 +done in 76s; checkpoints in /home/user/math-lab/problems/union-closed/data +== part T: margin trend past n = 32 == +[T] n=32: mixdir_slice_p0.3_d0.05@0.084:+6.150e-07 mixdir_slice_p0.3_d0.05@0.5:+7.370e-06 mixdir_slice_matched_p0.3_d0.05@0.084:+3.763e-06 mixdir_slice_p0.3823_d0.05@0.084:+3.674e-06 mixdir_slice_matched_p0.3823_d0.05@0.084:+3.674e-06 mixdir_crash_p0.3823_d0.05@0.084:+1.233e-03 mixdir_crash_p0.3823_d0.05@0.167138:+2.916e-03 crashmix_383_05@0.084064:+1.234e-03 +[T] n=40: mixdir_slice_p0.3_d0.05@0.084:+2.217e-07 mixdir_slice_p0.3_d0.05@0.5:+4.621e-06 mixdir_slice_matched_p0.3_d0.05@0.084:+2.982e-06 mixdir_slice_p0.3823_d0.05@0.084:+2.908e-06 mixdir_slice_matched_p0.3823_d0.05@0.084:+2.908e-06 mixdir_crash_p0.3823_d0.05@0.084:+1.514e-03 mixdir_crash_p0.3823_d0.05@0.131:+2.721e-03 crashmix_383_05@0.084064:+1.514e-03 +[T] n=48: mixdir_slice_p0.3_d0.05@0.084:+1.408e-07 mixdir_slice_p0.3_d0.05@0.5:+3.747e-06 mixdir_slice_matched_p0.3_d0.05@0.084:+2.334e-06 mixdir_slice_p0.3823_d0.05@0.084:+2.406e-06 mixdir_slice_matched_p0.3823_d0.05@0.084:+2.406e-06 mixdir_crash_p0.3823_d0.05@0.084:+1.834e-03 mixdir_crash_p0.3823_d0.05@0.107711:+2.591e-03 crashmix_383_05@0.084064:+1.834e-03 +[T] n=56: mixdir_slice_p0.3_d0.05@0.084:+2.330e-07 mixdir_slice_p0.3_d0.05@0.5:+3.929e-06 mixdir_slice_matched_p0.3_d0.05@0.084:+2.129e-06 mixdir_slice_p0.3823_d0.05@0.084:+2.051e-06 mixdir_slice_matched_p0.3823_d0.05@0.084:+2.051e-06 mixdir_crash_p0.3823_d0.05@0.084:+2.205e-03 mixdir_crash_p0.3823_d0.05@0.091453:+2.497e-03 crashmix_383_05@0.084064:+2.205e-03 +[T] n=64: mixdir_slice_p0.3_d0.05@0.084:+4.305e-07 mixdir_slice_p0.3_d0.05@0.5:+4.768e-06 mixdir_slice_matched_p0.3_d0.05@0.084:+1.818e-06 mixdir_slice_p0.3823_d0.05@0.084:+1.787e-06 mixdir_slice_matched_p0.3823_d0.05@0.084:+1.787e-06 mixdir_crash_p0.3823_d0.05@0.084:+2.642e-03 mixdir_crash_p0.3823_d0.05@0.079459:+2.427e-03 crashmix_383_05@0.084064:+2.642e-03 +[T] negative in-regime aggregates: 0 +== part H: adversarial hill-climbs on the aggregate == +[H] 82 climb endpoints; in-regime aggregate violations: 0; five lowest in-regime endpoints: +0.00560 +0.00700 +0.00882 +0.00932 +0.00977 +[H] lowest in-regime endpoint: +0.005602 (small_lam n=10 lam=0.084, 25 atoms) +done in 208s; checkpoints in /home/user/math-lab/problems/union-closed/data +== part X: engine cross-checks == +[X1] witness n=7 lam=3.5: M_5 - lam = -0.122033 (007: -0.122033), aggregate = +1.844754 (007: +1.84), max marginal = 0.3178 (007: 0.3178) +[X1] exact aggregate at t=181/16 in [+1.844669005, +1.844669005] (013: +1.844669), width 2.7e-15 +[X2] orbit vs direct, slice_w2 n=6: max |diff| = 1.55e-15 (resids 2.2e-16/2.8e-17) +[X2] orbit vs direct, sawin_geo_a n=6: max |diff| = 2.77e-13 (resids 7.7e-12/9.9e-14) +[X2] orbit vs direct, crash_eps2 n=6: max |diff| = 7.95e-13 (resids 6.9e-12/5.9e-14) +[X2] orbit vs direct, d0bern_30_51 n=7: max |diff| = 3.43e-13 (resids 7.1e-12/6.7e-14) +[X3] exact vs float orbit rows (partE_mix n=8): float +0.021228885, exact [+0.021228885, +0.021228885], diff 7.9e-15 +[X] ALL PASS +== part E: exact rational certification == +[E] witness_n10_wdil16 (sparse, n=10, t=181/16): aggregate in [+1.709542464, +1.709542464] -> positive; exact max marginal 0.270498 < 0.38271; 0.0s +[E] witness_n14_wdil16 (sparse, n=14, t=181/16): aggregate in [+1.474040147, +1.474040147] -> positive; exact max marginal 0.196720 < 0.38271; 0.0s +[E] witness_n20_wdil20 (sparse, n=20, t=181/16): aggregate in [+1.162947590, +1.162947590] -> positive; exact max marginal 0.104401 < 0.38271; 0.0s +[E] witness_n24_wdil20 (sparse, n=24, t=181/16): aggregate in [+1.013509091, +1.013509091] -> positive; exact max marginal 0.084848 < 0.38271; 0.1s +[E] witness_n28_wdil20 (sparse, n=28, t=181/16): aggregate in [+0.896763710, +0.896763710] -> positive; exact max marginal 0.071167 < 0.38271; 0.1s +[E] witness_n32_wdil20 (sparse, n=32, t=181/16): aggregate in [+0.803594797, +0.803594797] -> positive; exact max marginal 0.061118 < 0.38271; 0.1s +[E] witness_n20_t4_lam2exact (sparse, n=20, t=4/1): aggregate in [+0.904942086, +0.904942086] -> positive; exact max marginal 0.463729 >= 0.38271; 0.0s +[E] crashmix_383_05_n10 (orbit, n=10, lam~0.082149): aggregate in [+0.000476924, +0.000476924] -> positive; exact max marginal 0.382350; resid 5.6e-12; 0.6s +[E] crashmix_383_05_n14 (orbit, n=14, lam~0.082149): aggregate in [+0.000634269, +0.000634269] -> positive; exact max marginal 0.382350; resid 5.3e-12; 2.1s +[E] crashmix_383_05_n20 (orbit, n=20, lam~0.082149): aggregate in [+0.000829968, +0.000829968] -> positive; exact max marginal 0.382350; resid 8.7e-12; 7.7s +[E] crashmix_383_05_n24 (orbit, n=24, lam~0.082149): aggregate in [+0.000952638, +0.000952638] -> positive; exact max marginal 0.382350; resid 5.4e-12; 15.5s +[E] crashmix_383_05_n28 (orbit, n=28, lam~0.082149): aggregate in [+0.001075588, +0.001075588] -> positive; exact max marginal 0.382350; resid 6.3e-12; 25.3s +[E] crashmix_383_05_n32 (orbit, n=32, lam~0.082149): aggregate in [+0.001201865, +0.001201865] -> positive; exact max marginal 0.382350; resid 7.0e-12; 41.3s +[E] mixdir_slice_p0.3_d0.05_n20 (orbit, n=20, lam~0.082149): aggregate in [+0.000004637, +0.000004637] -> positive; exact max marginal 0.302500; resid 7.3e-12; 3.4s +[E] mixdir_slice_p0.3_d0.05_n32 (orbit, n=32, lam~0.082149): aggregate in [+0.000000599, +0.000000599] -> positive; exact max marginal 0.303750; resid 5.7e-12; 18.8s +done in 116s; checkpoints in /home/user/math-lab/problems/union-closed/data +== part E: exact rational certification == +[E] witness_n10_wdil16 (sparse, n=10, t=181/16): aggregate in [+1.709542464, +1.709542464] -> positive; exact max marginal 0.270498 < 0.38271; 0.0s +[E] witness_n14_wdil16 (sparse, n=14, t=181/16): aggregate in [+1.474040147, +1.474040147] -> positive; exact max marginal 0.196720 < 0.38271; 0.0s +[E] witness_n20_wdil20 (sparse, n=20, t=181/16): aggregate in [+1.162947590, +1.162947590] -> positive; exact max marginal 0.104401 < 0.38271; 0.0s +[E] witness_n24_wdil20 (sparse, n=24, t=181/16): aggregate in [+1.013509091, +1.013509091] -> positive; exact max marginal 0.084848 < 0.38271; 0.1s +[E] witness_n28_wdil20 (sparse, n=28, t=181/16): aggregate in [+0.896763710, +0.896763710] -> positive; exact max marginal 0.071167 < 0.38271; 0.1s +[E] witness_n32_wdil20 (sparse, n=32, t=181/16): aggregate in [+0.803594797, +0.803594797] -> positive; exact max marginal 0.061118 < 0.38271; 0.1s +[E] witness_n20_t4_lam2exact (sparse, n=20, t=4/1): aggregate in [+0.483059569, +0.483059569] -> positive; exact max marginal 0.091560 < 0.38271; 0.0s +[E] crashmix_383_05_n10 (orbit, n=10, lam~0.082149): aggregate in [+0.000476924, +0.000476924] -> positive; exact max marginal 0.382350; resid 5.6e-12; 0.6s +[E] crashmix_383_05_n14 (orbit, n=14, lam~0.082149): aggregate in [+0.000634269, +0.000634269] -> positive; exact max marginal 0.382350; resid 5.3e-12; 2.1s +[E] crashmix_383_05_n20 (orbit, n=20, lam~0.082149): aggregate in [+0.000829968, +0.000829968] -> positive; exact max marginal 0.382350; resid 8.7e-12; 7.6s +[E] crashmix_383_05_n24 (orbit, n=24, lam~0.082149): aggregate in [+0.000952638, +0.000952638] -> positive; exact max marginal 0.382350; resid 5.4e-12; 14.3s +[E] crashmix_383_05_n28 (orbit, n=28, lam~0.082149): aggregate in [+0.001075588, +0.001075588] -> positive; exact max marginal 0.382350; resid 6.3e-12; 25.0s +[E] crashmix_383_05_n32 (orbit, n=32, lam~0.082149): aggregate in [+0.001201865, +0.001201865] -> positive; exact max marginal 0.382350; resid 7.0e-12; 39.4s +[E] mixdir_slice_p0.3_d0.05_n20 (orbit, n=20, lam~0.082149): aggregate in [+0.000004637, +0.000004637] -> positive; exact max marginal 0.302500; resid 7.3e-12; 3.4s +[E] mixdir_slice_p0.3_d0.05_n32 (orbit, n=32, lam~0.082149): aggregate in [+0.000000599, +0.000000599] -> positive; exact max marginal 0.303750; resid 5.7e-12; 20.0s +done in 113s; checkpoints in /home/user/math-lab/problems/union-closed/data diff --git a/problems/union-closed/data/oragg_sk2_partC.json b/problems/union-closed/data/oragg_sk2_partC.json new file mode 100644 index 0000000..52f61ef --- /dev/null +++ b/problems/union-closed/data/oragg_sk2_partC.json @@ -0,0 +1,78 @@ +{ + "certs": [ + { + "in_regime_exact": true, + "intervals_overlap": true, + "max_marginal": 0.27049832331955115, + "my_hi": 1.7095424637138186, + "my_lo": 1.709542463713808, + "my_sign": "positive", + "my_width": 1.0658141036401503e-14, + "name": "witness_n10_wdil16", + "sign_agrees": true, + "their_hi": 1.7095424637138141, + "their_lo": 1.7095424637138115 + }, + { + "in_regime_exact": true, + "intervals_overlap": true, + "max_marginal": 0.09156022350625077, + "my_hi": 0.48305956855679133, + "my_lo": 0.48305956855678067, + "my_sign": "positive", + "my_width": 1.0658141036401503e-14, + "name": "witness_n20_t4_lam2exact", + "sign_agrees": true, + "their_hi": 0.48305956855678606, + "their_lo": 0.48305956855678567 + }, + { + "in_regime_exact": true, + "intervals_overlap": true, + "max_marginal": 0.06111843723141829, + "my_hi": 0.8035947973888391, + "my_lo": 0.8035947973888284, + "my_sign": "positive", + "my_width": 1.0658141036401503e-14, + "name": "witness_n32_wdil20", + "sign_agrees": true, + "their_hi": 0.8035947973888339, + "their_lo": 0.803594797388832 + }, + { + "in_regime_exact": true, + "intervals_overlap": true, + "max_marginal": 0.38235000000005875, + "my_hi": 0.001201865436563904, + "my_lo": 0.001201865436553246, + "my_sign": "positive", + "my_width": 1.0658141036401503e-14, + "name": "crashmix_383_05_n32", + "sign_agrees": true, + "their_hi": 0.0012018654365805032, + "their_lo": 0.0012018654345185519 + }, + { + "in_regime_exact": true, + "intervals_overlap": true, + "max_marginal": 0.303749999999851, + "my_hi": 5.988236463096674e-07, + "my_lo": 5.988236356515264e-07, + "my_sign": "positive", + "my_width": 1.0658141036401503e-14, + "name": "mixdir_slice_p0.3_d0.05_n32", + "sign_agrees": true, + "their_hi": 5.988237042421268e-07, + "their_lo": 5.988177652162741e-07 + }, + { + "in_regime_exact": true, + "max_marginal": 0.30375000000005453, + "my_hi": 5.98823646304681e-07, + "my_lo": 5.9882363564654e-07, + "my_sign": "positive", + "name": "mixdir_slice_p0.3_d0.05_n32_MY_mu", + "sinkhorn_resid": 6.902478588699523e-12 + } + ] +} diff --git a/problems/union-closed/data/oragg_sk2_partK.json b/problems/union-closed/data/oragg_sk2_partK.json new file mode 100644 index 0000000..0c90428 --- /dev/null +++ b/problems/union-closed/data/oragg_sk2_partK.json @@ -0,0 +1,10 @@ +{ + "k1_cases": 437, + "k1_failures": [], + "k1_pass": true, + "k2_failures": 0, + "k2_pass": true, + "k3_failures": 0, + "k3_pass": true, + "k4_pass": true +} diff --git a/problems/union-closed/data/oragg_sk2_partT.json b/problems/union-closed/data/oragg_sk2_partT.json new file mode 100644 index 0000000..9b35bc3 --- /dev/null +++ b/problems/union-closed/data/oragg_sk2_partT.json @@ -0,0 +1,599 @@ +{ + "n_negative": 0, + "results": [ + { + "aggregate": 6.149830063855562e-07, + "argmin_i": 31, + "in_regime": true, + "lambda": 0.084, + "max_marginal": 0.30375000000006114, + "min_excess": 3.7111347908624914e-07, + "n": 32, + "name": "dip_slice_p.30_d.05", + "resid": 6.846523348258415e-12 + }, + { + "aggregate": 2.2170250623519912e-07, + "argmin_i": 26, + "in_regime": true, + "lambda": 0.084, + "max_marginal": 0.30375000000006147, + "min_excess": 1.7886266100164416e-07, + "n": 40, + "name": "dip_slice_p.30_d.05", + 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"name": "slice_dir_tiny_lambda", + "resid": 8.313794097603022e-12 + } + ] +} diff --git a/problems/union-closed/data/oragg_sk2_partX.json b/problems/union-closed/data/oragg_sk2_partX.json new file mode 100644 index 0000000..6e7a659 --- /dev/null +++ b/problems/union-closed/data/oragg_sk2_partX.json @@ -0,0 +1,142 @@ +{ + "checks": [ + { + "M5_minus_lam": -0.1220334102236662, + "agg": 1.844754315091965, + "name": "witness_n7", + "pass": true + }, + { + "agg_014_orbit": 0.0025237958196272663, + "agg_mine_direct": 0.002523795819619652, + "agg_mine_orbit": 0.002523795819627282, + "maxdiff": 7.629747528214992e-15, + "name": "x2_crashmix_383_05_n8_lam0.5", + "resid": 7.504996624163596e-12 + }, + { + "agg_014_orbit": 0.016367617438402798, + "agg_mine_direct": 0.016367617438380368, + "agg_mine_orbit": 0.016367617438402784, + "maxdiff": 2.24160967565723e-14, + "name": "x2_sawin_geo_a_n8_lam0.5", + "resid": 8.473666213149045e-12 + }, + { + "agg_014_orbit": 9.766680110388583e-05, + 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"agg_mine_direct": 0.000198759624678639, + "agg_mine_orbit": 0.0001987596246984524, + "maxdiff": 1.9813415231412224e-14, + "name": "x2_mixdir_slice_p.30_d.05_n9_lam1.0", + "resid": 9.072409490329392e-12 + }, + { + "agg_014_orbit": 0.21093845061141692, + "agg_mine_direct": 0.21093845061138916, + "agg_mine_orbit": 0.21093845061141692, + "maxdiff": 2.7755575615628914e-14, + "name": "x2_d0bern_30_51_n9_lam1.0", + "resid": 7.620792885632e-12 + } + ], + "x2_maxdiff": 1.0455525334407412e-13, + "x2_pass": true, + "x3_class_or_rel_spread": 0.0, + "x3_pass": true +} diff --git a/problems/union-closed/data/oragg_sk2_run.log b/problems/union-closed/data/oragg_sk2_run.log new file mode 100644 index 0000000..080a6ff --- /dev/null +++ b/problems/union-closed/data/oragg_sk2_run.log @@ -0,0 +1,132 @@ +== part K: enclosure-kit attack == +[K1] 437 rationals: disjoint-enclosure failures = 0; max widths theirs 1.5e-13 / mine 1.1e-14 +[K2] _dround directionality: 0 failures / 2000 +[K3] _ln_1_to_2_fp anchor vs my bracket*ln2 (slack 1e-12): 0 failures / 300 +[K4] combinatorics brute force: Ncl failures 0, PK failures 0 +done in 0s +== part X: formula cross-checks == +[X1] my direct on witness n=7 lam=3.5: M_5-lam = -0.122033 (007: -0.122033), agg = +1.844754 (014: +1.844754), marg = 0.3178 (0.3178) +[X2] crashmix_383_05 n=8 lam=0.5: mine-orbit +0.002523796 mine-direct +0.002523796 014-orbit +0.002523796 (maxdiff 7.63e-15) +[X2] sawin_geo_a n=8 lam=0.5: mine-orbit +0.016367617 mine-direct +0.016367617 014-orbit +0.016367617 (maxdiff 2.24e-14) +[X2] mixdir_slice_p.30_d.05 n=8 lam=0.5: mine-orbit +0.000097667 mine-direct +0.000097667 014-orbit +0.000097667 (maxdiff 4.61e-14) +[X2] d0bern_30_51 n=8 lam=0.5: mine-orbit +0.088665065 mine-direct +0.088665065 014-orbit +0.088665065 (maxdiff 1.24e-14) +[X2] crashmix_383_05 n=8 lam=1.0: mine-orbit +0.005641590 mine-direct +0.005641590 014-orbit +0.005641590 (maxdiff 1.25e-14) +[X2] sawin_geo_a n=8 lam=1.0: mine-orbit +0.032687699 mine-direct +0.032687699 014-orbit +0.032687699 (maxdiff 1.83e-14) +[X2] mixdir_slice_p.30_d.05 n=8 lam=1.0: mine-orbit +0.000209476 mine-direct +0.000209476 014-orbit +0.000209476 (maxdiff 7.58e-15) +[X2] d0bern_30_51 n=8 lam=1.0: mine-orbit +0.187274148 mine-direct +0.187274148 014-orbit +0.187274148 (maxdiff 2.40e-14) +[X2] crashmix_383_05 n=9 lam=0.5: mine-orbit +0.002905299 mine-direct +0.002905299 014-orbit +0.002905299 (maxdiff 3.05e-14) +[X2] sawin_geo_a n=9 lam=0.5: mine-orbit +0.017510437 mine-direct +0.017510437 014-orbit +0.017510437 (maxdiff 3.59e-14) +[X2] mixdir_slice_p.30_d.05 n=9 lam=0.5: mine-orbit +0.000096401 mine-direct +0.000096401 014-orbit +0.000096401 (maxdiff 3.32e-14) +[X2] d0bern_30_51 n=9 lam=0.5: mine-orbit +0.098394651 mine-direct +0.098394651 014-orbit +0.098394651 (maxdiff 1.05e-13) +[X2] crashmix_383_05 n=9 lam=1.0: mine-orbit +0.006716601 mine-direct +0.006716601 014-orbit +0.006716601 (maxdiff 6.45e-14) +[X2] sawin_geo_a n=9 lam=1.0: mine-orbit +0.034888993 mine-direct +0.034888993 014-orbit +0.034888993 (maxdiff 5.00e-15) +[X2] mixdir_slice_p.30_d.05 n=9 lam=1.0: mine-orbit +0.000198760 mine-direct +0.000198760 014-orbit +0.000198760 (maxdiff 1.98e-14) +[X2] d0bern_30_51 n=9 lam=1.0: mine-orbit +0.210938451 mine-direct +0.210938451 014-orbit +0.210938451 (maxdiff 2.78e-14) +[X3] j-independence at n=9 i=6: 256 classes, max rel spread of within-class ORs = 0.00e+00 +[X] ALL PASS +done in 3s +== part C: independent re-certification == +[C] witness_n10_wdil16: mine [+1.70954246371, +1.70954246371] (w 1.1e-14) vs theirs [+1.70954246371, +1.70954246371] -> overlap=True sign=positive marg=0.270498 +[C] witness_n20_t4_lam2exact: mine [+0.483059568557, +0.483059568557] (w 1.1e-14) vs theirs [+0.483059568557, +0.483059568557] -> overlap=True sign=positive marg=0.091560 +[C] witness_n32_wdil20: mine [+0.803594797389, +0.803594797389] (w 1.1e-14) vs theirs [+0.803594797389, +0.803594797389] -> overlap=True sign=positive marg=0.061118 + (sparse certs 0s) +[C] crashmix_383_05_n32: mine [+0.00120186543655, +0.00120186543656] (w 1.1e-14) vs theirs [+0.00120186543452, +0.00120186543658] -> overlap=True sign=positive marg=0.382350 + (3s, their-resid 7.0e-12) +[C] mixdir_slice_p0.3_d0.05_n32: mine [+5.98823635652e-07, +5.9882364631e-07] (w 1.1e-14) vs theirs [+5.98817765216e-07, +5.98823704242e-07] -> overlap=True sign=positive marg=0.303750 + (2s) +[C] MY OWN mu-tilde' at softest point: [+5.988236356e-07, +5.988236463e-07] -> positive, marg 0.303750, resid 6.9e-12 (2s) +done in 7s +== part T: trend attack == +[T1] n=32: agg +6.150e-07 (014: 6.15e-7 / 2.22e-7 / 1.41e-7 / 2.33e-7 / 4.31e-7) resid 6.8e-12 +[T1] n=40: agg +2.217e-07 (014: 6.15e-7 / 2.22e-7 / 1.41e-7 / 2.33e-7 / 4.31e-7) resid 8.5e-12 +[T1] n=48: agg +1.408e-07 (014: 6.15e-7 / 2.22e-7 / 1.41e-7 / 2.33e-7 / 4.31e-7) resid 6.1e-12 +[T1] n=56: agg +2.330e-07 (014: 6.15e-7 / 2.22e-7 / 1.41e-7 / 2.33e-7 / 4.31e-7) resid 7.1e-12 +[T1] n=64: agg +4.305e-07 (014: 6.15e-7 / 2.22e-7 / 1.41e-7 / 2.33e-7 / 4.31e-7) resid 8.0e-12 +[T2] n=48 lam=0.005: agg +3.223e-09 agg/lam +6.446e-07 +[T2] n=48 lam=0.01: agg +7.092e-09 agg/lam +7.092e-07 +[T2] n=48 lam=0.02: agg +1.677e-08 agg/lam +8.386e-07 +[T2] n=48 lam=0.04: agg +4.395e-08 agg/lam +1.099e-06 +[T2] n=48 lam=0.06: agg +8.161e-08 agg/lam +1.360e-06 +[T2] n=48 lam=0.12: agg +2.584e-07 agg/lam +2.154e-06 +[T2] n=48 lam=0.25: agg +9.798e-07 agg/lam +3.919e-06 +[T2] n=48 lam=0.5: agg +3.747e-06 agg/lam +7.494e-06 +[T2] n=48 d=0.01: agg +4.886e-09 +[T2] n=48 d=0.02: agg +1.890e-08 +[T2] n=48 d=0.1: agg +1.065e-06 +[T2] n=48 d=0.2: agg +1.168e-05 +[T2] n=48 p=0.26 (record layer): agg +6.720e-06 +[T2] n=48 p=0.28 (record layer): agg +1.457e-06 +[T2] n=48 p=0.32 (record layer): agg +6.644e-07 +[T2] n=48 p=0.34 (record layer): agg +1.663e-06 +[T2] n=48 p=0.28 (matched layer): agg +2.294e-06 +[T2] n=48 p=0.3 (matched layer): agg +2.334e-06 +[T2] n=48 p=0.32 (matched layer): agg +2.367e-06 +[T3] crashmix n=10: 0.005:+2.85e-05 0.02:+1.14e-04 0.05:+2.88e-04 0.084:+4.88e-04 0.15:+8.86e-04 0.3:+1.85e-03 0.7:+4.93e-03 1.5:+1.42e-02 2.75:+3.42e-02 4.4:+5.55e-02 +[T3] crashmix n=20: 0.005:+4.88e-05 0.02:+1.96e-04 0.05:+4.97e-04 0.084:+8.50e-04 0.15:+1.59e-03 0.3:+3.74e-03 0.7:+1.56e-02 1.5:+6.26e-02 2.75:+1.29e-01 4.4:+1.80e-01 +[T3] crashmix n=32: 0.005:+6.77e-05 0.02:+2.73e-04 0.05:+7.01e-04 0.084:+1.23e-03 0.15:+2.51e-03 0.3:+7.73e-03 0.7:+4.65e-02 1.5:+1.49e-01 2.75:+2.67e-01 4.4:+3.56e-01 +[T] negative in-regime aggregates: 0 +done in 53s +== part V: new-adversary hunt == +[V1] lam=0.084 p0=0.3 d=0.03: agg +1.570515e-07 marg 0.3003 minexc +0.0000 +[V1] lam=0.084 p0=0.3 d=0.3: agg +1.435734e-05 marg 0.3027 minexc +0.0000 +[V1] lam=0.084 p0=0.3 d=3: agg +7.901415e-04 marg 0.3222 minexc +0.0000 +[V1] lam=0.084 p0=0.375 d=0.03: agg +3.216775e-08 marg 0.3750 minexc +0.0000 +[V1] lam=0.084 p0=0.375 d=0.3: agg +3.096566e-06 marg 0.3753 minexc +0.0000 +[V1] lam=0.084 p0=0.375 d=3: agg +2.310343e-04 marg 0.3777 minexc +0.0000 +[V1] lam=0.5 p0=0.3 d=0.03: agg +1.363987e-06 marg 0.3003 minexc +0.0000 +[V1] lam=0.5 p0=0.3 d=0.3: agg +1.236621e-04 marg 0.3031 minexc +0.0000 +[V1] lam=0.5 p0=0.3 d=3: agg +6.566013e-03 marg 0.3268 minexc +0.0005 +[V1] lam=0.5 p0=0.375 d=0.03: agg +2.899298e-07 marg 0.3750 minexc +0.0000 +[V1] lam=0.5 p0=0.375 d=0.3: agg +2.774709e-05 marg 0.3753 minexc +0.0000 +[V1] lam=0.5 p0=0.375 d=3: agg +2.006391e-03 marg 0.3784 minexc +0.0001 +[V1] lam=2 p0=0.3 d=0.03: agg +2.614547e-05 marg 0.3006 minexc +0.0000 +[V1] lam=2 p0=0.3 d=0.3: agg +2.279550e-03 marg 0.3060 minexc +0.0007 +[V1] lam=2 p0=0.3 d=3: agg +9.941058e-02 marg 0.3586 minexc +0.0205 +[V1] lam=2 p0=0.375 d=0.03: agg +6.230744e-06 marg 0.3751 minexc +0.0000 +[V1] lam=2 p0=0.375 d=0.3: agg +5.798669e-04 marg 0.3756 minexc +0.0001 +[V1] lam=2 p0=0.375 d=3: agg +3.676327e-02 marg 0.3850 minexc +0.0058 OUT +[V1] lam=3.5 p0=0.3 d=0.03: agg +2.727813e-04 marg 0.3011 minexc +0.0001 +[V1] lam=3.5 p0=0.3 d=0.3: agg +2.218938e-02 marg 0.3125 minexc +0.0087 +[V1] lam=3.5 p0=0.3 d=3: agg +5.761402e-01 marg 0.4016 minexc +0.1570 OUT +[V1] lam=3.5 p0=0.375 d=0.03: agg +7.225921e-05 marg 0.3751 minexc +0.0000 +[V1] lam=3.5 p0=0.375 d=0.3: agg +6.435270e-03 marg 0.3763 minexc +0.0022 +[V1] lam=3.5 p0=0.375 d=3: agg +3.115537e-01 marg 0.4014 minexc +0.0762 OUT +[V2] n=20 lam=0.02 p=0.383363: agg +1.962e-04 marg 0.382695 +[V2] n=20 lam=0.084 p=0.383363: agg +8.495e-04 marg 0.382695 +[V2] n=20 lam=0.5 p=0.383363: agg +8.286e-03 marg 0.382695 +[V2] n=20 d0bern(0.304145,0.55) lam=0.084: agg +3.447e-02 marg 0.382720 +[V2] n=20 d0bern(0.304145,0.55) lam=0.5: agg +2.893e-01 marg 0.382720 +[V2] n=20 d0bern(0.48976,0.75) lam=0.084: agg +1.531e-01 marg 0.382680 +[V2] n=20 d0bern(0.48976,0.75) lam=0.5: agg +1.190e+00 marg 0.382680 +[V2] n=20 sawin-boundary lam=0.084: agg +4.620e-03 marg 0.382643 +[V2] n=20 sawin-boundary lam=0.5: agg +3.152e-02 marg 0.382643 +[V2] n=32 lam=0.02 p=0.383363: agg +2.733e-04 marg 0.382695 +[V2] n=32 lam=0.084 p=0.383363: agg +1.232e-03 marg 0.382695 +[V2] n=32 lam=0.5 p=0.383363: agg +2.290e-02 marg 0.382695 +[V2] n=32 d0bern(0.304145,0.55) lam=0.084: agg +5.462e-02 marg 0.382720 +[V2] n=32 d0bern(0.304145,0.55) lam=0.5: agg +5.350e-01 marg 0.382720 +[V2] n=32 d0bern(0.48976,0.75) lam=0.084: agg +2.480e-01 marg 0.382680 +[V2] n=32 d0bern(0.48976,0.75) lam=0.5: agg +2.005e+00 marg 0.382680 +[V2] n=32 sawin-boundary lam=0.084: agg +5.292e-03 marg 0.382643 +[V2] n=32 sawin-boundary lam=0.5: agg +3.503e-02 marg 0.382643 +[V3] n=20 lam=0.002: crashmix +1.951e-05 slice-dir +1.113e-07 +[V3] n=20 lam=0.01: crashmix +9.777e-05 slice-dir +5.572e-07 +[V3] n=20 lam=0.04: crashmix +3.955e-04 slice-dir +2.241e-06 +[V3] n=32 lam=0.002: crashmix +2.706e-05 slice-dir +1.185e-08 +[V3] n=32 lam=0.01: crashmix +1.358e-04 slice-dir +6.060e-08 +[V3] n=32 lam=0.04: crashmix +5.554e-04 slice-dir +2.628e-07 +[V] in-regime negatives: 0 +done in 74s +== part T addendum: order-vs-d for the worst part-P cell == +d=0.02: agg=4.603581e-02 order(2d->d)=nan resid=9.5e-12 +d=0.01: agg=1.890016e-02 order(2d->d)=1.284 resid=9.1e-12 +d=0.005: agg=7.222218e-03 order(2d->d)=1.388 resid=8.7e-12 +d=0.0025: agg=2.535775e-03 order(2d->d)=1.510 resid=8.2e-12 +d=0.00125: agg=8.154163e-04 order(2d->d)=1.637 resid=7.7e-12 +d=0.000625: agg=2.426045e-04 order(2d->d)=1.749 resid=6.9e-12 +crash dir: +d=0.02: agg=1.041370e-02 order(2d->d)=nan resid=9.7e-12 +d=0.01: agg=3.671295e-03 order(2d->d)=1.504 resid=9.3e-12 +d=0.005: agg=1.191232e-03 order(2d->d)=1.624 resid=8.8e-12 +d=0.0025: agg=3.579897e-04 order(2d->d)=1.734 resid=8.0e-12 +d=0.00125: agg=1.013996e-04 order(2d->d)=1.820 resid=6.4e-12 diff --git a/problems/union-closed/explore/uc_or_agg_probe.py b/problems/union-closed/explore/uc_or_agg_probe.py new file mode 100644 index 0000000..63d4d0d --- /dev/null +++ b/problems/union-closed/explore/uc_or_agg_probe.py @@ -0,0 +1,1614 @@ +#!/usr/bin/env python3 +"""Attempt 014: large-n certified probe of the i-AGGREGATED odds-ratio control. + +Object under test (007 SS5 "what survives" item 1, exactly as 013 part C +certified it at n = 7): for the tilt coupling pi_lambda(A,B) prop +u(A) u(B) 2^{lambda |A&B|} with both marginals mu, and M_i the well-posed +per-coordinate average of 007 SS1 (mass-weighted mean of log2 OR_i over +N_i x N_i histories; N_i = reachable (i-1)-bit prefixes with both +continuations reachable in supp(mu); i = n excluded since M_n = lambda +identically), + + A(mu, lambda) = sum_{i= 0 + with w_i = pi-mass of N_i x N_i (the "defined mass"), + +for all mu in the record regime (every elementwise marginal < 0.38271) and +all lambda >= 0? Before this attempt the aggregate was certified positive +only at n <= 7 (013 part C: +1.844669 on the 10-atom witness). Per the +standing warning from 012 (small-n flattening reversed past n ~ 22) this +probe extends the census to n in {10, 14, 20, 24, 28, 32} BEFORE any proof +effort. + +Sign convention note: under the conjecture-friendliest alternative +bookkeeping (score every margin-degenerate history at exactly lambda, 013's +"surrogate-lambda" convention) the aggregate's NUMERATOR is unchanged and +its denominator only grows, so the SIGN of A is the same under either +convention. A sign probe of the conditioning-out aggregate is therefore +convention-robust for free. + +Engines: + * sparse engine -- direct atom-pair census from a potential (005's trick: + any positive u is its own marginal's Sinkhorn potential). Cost + O(m^2 n) for m atoms, so it reaches ANY n on sparse supports. Fresh + code, conventions copied from 007 SS1 / 013 (cross-checked in part X). + * orbit engine -- for measures invariant under permutations of coords + {3..n} (all the structured families: Bernoulli/Sawin mixtures, slices, + delta_0+Bern, crash mixtures). Histories are collapsed to orbit + classes ((a1,a2) special-coordinate pattern, tail prefix weights, + overlap), the same S_{n-2} symmetry 012's budget engine used to reach + n = 32. Cells per class are exact orbit sums via tail pair-kernels + PK[s][l] = C(f,s) * sum_j C(s,j) C(f-s,l-j) t^j. + * exact rational certifier -- both engines re-run end-to-end in + fractions.Fraction at rational tilt t, with every log2 replaced by a + certified rational enclosure (atanh series, directed rounding, explicit + tail bound; own implementation following the 2026-07-31 lab standard: + float agreement is NOT certification). Potentials are dyadic + rationals (Fraction(float)), so all table entries stay dyadic and no + denominator blow-up occurs. + +Parts (run `python uc_or_agg_probe.py X O S H E` or any subset; default all): + X cross-checks: (X1) sparse engine reproduces 007's witness numbers + (M_5 - lambda = -0.122033, aggregate +1.8447) and 013's exact + aggregate enclosure at t = 181/16; (X2) orbit engine vs direct + sparse+Sinkhorn on full supports at n = 6, 7 (slice, Sawin mixture, + eps-crash, d0+Bern); (X3) exact orbit rows vs float orbit rows. + O orbit family census over n in {10,14,20,24,28,32} x lambda grid + (incl. the 011 window law 4.847/(n-3)): Bernoulli cells, Sawin + geometric mixtures, delta0+Bern, Chase-Lovett slice + smoothed slice, + eps-mixed crash, 012 crash-mixture cells (p,d). + S sparse structured census at the same n: pure crash / mirror, the + 10-atom witness genre re-laid-out at each n with its dilution-weight + sweep (tightest in-regime point per n), multi-gadget stacks (K + light-slice gadgets, 007 lead 1 generalized). + H seeded adversarial hill-climbs on the AGGREGATE objective (marginal- + penalized, dynamic-range-clamped): free sparse supports, witness- + seeded, multi-gadget-parametrized, at n = 10, 14, 20 (+ spot 24). + E exact rational certification of the tightest points found (per-n + minimum and the global minimum), plus exact in-regime marginals. + +Standard library only. Deterministic (fixed seeds). Runtime: X+S+H ~ 15 +min, O ~ 40-80 min (n = 28, 32 dominate), E ~ 10-30 min. +Checkpoints: problems/union-closed/data/or_agg_probe_part[XOSHE]*.json +Log: problems/union-closed/data/or_agg_probe_run.log (tee by hand) + +Usage: + python problems/union-closed/explore/uc_or_agg_probe.py [parts] 2>&1 | + tee -a problems/union-closed/data/or_agg_probe_run.log +""" + +from __future__ import annotations + +import json +import math +import random +import sys +import time +from fractions import Fraction +from pathlib import Path + +ROOT = Path(__file__).resolve().parent.parent +DATA = ROOT / "data" + +RECORD = 0.38271 +F0 = Fraction(0) +NS_GRID = (10, 14, 20, 24, 28, 32) + + +def popc(x: int) -> int: + return bin(x).count("1") + + +def save(name: str, obj) -> None: + DATA.mkdir(exist_ok=True) + (DATA / name).write_text( + json.dumps(obj, indent=1, sort_keys=True, default=str) + "\n", + encoding="utf-8") + + +def lam_grid(n: int) -> list[float]: + """lambda grid per n: tilt-recipe cells log2(1.06), log2(1.20) from + 008/012's calibration, O(1) values, and the 011 window law + lam_max ~ 4.847/(n-3) (half / at / twice).""" + base = [math.log2(1.06), math.log2(1.20), 0.5, 1.0, 2.0, 3.5, 5.0] + win = [2.4 / (n - 3), 4.847 / (n - 3), 9.7 / (n - 3)] + return sorted(set(round(v, 6) for v in base + win)) + + +# ====================================================================== +# certified log2 enclosures (rational in, rational bounds out). +# Own implementation of the 013 standard: ln s = 2 atanh((s-1)/(s+1)), +# partial sum is a lower bound (all terms positive), tail after J terms +# <= 2 z^{2J+1} / ((2J+1)(1-z^2)). Inputs are directed-rounded to +# bounded precision first so Fraction sizes stay tame. +# ====================================================================== + +def _ln_1_to_2(s: Fraction, terms: int) -> tuple[Fraction, Fraction]: + if s == 1: + return F0, F0 + z = (s - 1) / (s + 1) + z2 = z * z + term = z + acc = F0 + for j in range(terms): + acc += term / (2 * j + 1) + term *= z2 + lo = 2 * acc + tail = 2 * term / ((2 * terms + 1) * (1 - z2)) + return lo, lo + tail + + +LN2_LO, LN2_HI = _ln_1_to_2(Fraction(2), 48) # width ~ 1e-47 + + +def _dround(x: Fraction, up: bool, bits: int = 120) -> Fraction: + """Directed rounding of x to a dyadic rational with a <= bits-bit + numerator: result <= x if not up, result >= x if up. Keeping every + intermediate dyadic avoids the gcd churn that makes long certified + sums explode.""" + if x == 0: + return F0 + n, d = x.numerator, x.denominator + shift = bits - (abs(n).bit_length() - d.bit_length()) + if shift >= 0: + num, den = n << shift, d + else: + num, den = n, d << (-shift) + q = num // den # floor (rounds down, also for <0) + if up and q * den != num: + q += 1 + if shift >= 0: + return Fraction(q, 1 << shift) + return Fraction(q << (-shift)) + + +def _ln_1_to_2_fp(s: Fraction, up: bool, terms: int, + bits: int = 120) -> Fraction: + """Directed bound on ln s for 1 <= s < 2, dyadic fixed-point: every + operation is rounded in the safe direction, and the atanh-series tail + 2 z^{2J+1}/((2J+1)(1-z^2)) is added on the upper side.""" + if s == 1: + return F0 + z = _dround((s - 1) / (s + 1), up, bits) + z2 = _dround(z * z, up, bits) + term = z + acc = F0 + for j in range(terms): + acc = _dround(acc + term / (2 * j + 1), up, bits) + term = _dround(term * z2, up, bits) + val = 2 * acc + if up: + tail = _dround(2 * term / ((2 * terms + 1) * (1 - z2)), True, bits) + val = _dround(val + tail, True, bits) + return val + + +LN2D_LO = _dround(LN2_LO, False) +LN2D_HI = _dround(LN2_HI, True) + + +def log2_enclosure(r: Fraction, terms: int = 12, + bits: int = 120) -> tuple[Fraction, Fraction]: + """[lo, hi] with lo <= log2 r <= hi, dyadic rationals; r > 0. + terms=12 gives width ~ 1e-13 (z <= 1/3 after range reduction).""" + assert r > 0 + out = [] + for up in (False, True): + rr = _dround(r, up, bits) + if rr <= 0: + rr = r + k = 0 + while rr >= 2: + rr /= 2 + k += 1 + while rr < 1: + rr *= 2 + k -= 1 + ln_dir = _ln_1_to_2_fp(rr, up, terms, bits) + if up: + out.append(_dround(k + ln_dir / LN2D_LO, True, bits) + if ln_dir >= 0 else + _dround(k + ln_dir / LN2D_HI, True, bits)) + else: + out.append(_dround(k + ln_dir / LN2D_HI, False, bits) + if ln_dir >= 0 else + _dround(k + ln_dir / LN2D_LO, False, bits)) + return out[0], out[1] + + +# ====================================================================== +# sparse engine (float): census + aggregate from a potential +# ====================================================================== + +def sparse_rows(u: dict[int, float], lam: float, n: int): + """Per i = 1..n: dict(rows=[(mass, log2 OR)], def_mass, n_active, + cell_floor). Masses unnormalized (one common Z for all i, which + cancels in every ratio reported). N_i membership is structural + (from supp(u)); a structurally nondegenerate history with a zero + float cell is an underflow and is flagged.""" + t = 2.0 ** lam + supp = sorted(u) + kern = {} + for a in supp: + ua = u[a] + for b in supp: + kern[(a, b)] = ua * u[b] * t ** popc(a & b) + out = [] + for i in range(1, n + 1): + pmask = (1 << (i - 1)) - 1 + bit = 1 << (i - 1) + prefixes = {A & pmask for A in supp} + active = {c for c in prefixes + if any(A & pmask == c and A & bit for A in supp) + and any(A & pmask == c and not A & bit for A in supp)} + tables: dict[tuple[int, int], list[float]] = {} + for (A, B), w in kern.items(): + key = (A & pmask, B & pmask) + tab = tables.setdefault(key, [0.0] * 4) + tab[(2 if A & bit else 0) + (1 if B & bit else 0)] += w + rows = [] + dm = 0.0 + floor = None + underflow = False + for (a, b), (p00, p01, p10, p11) in tables.items(): + if a in active and b in active: + if min(p00, p01, p10, p11) <= 0.0: + underflow = True # structural nondeg, float zero + continue + mass = p00 + p01 + p10 + p11 + v = (math.log2(p11) + math.log2(p00) + - math.log2(p10) - math.log2(p01)) + rows.append((a, b, mass, v)) + dm += mass + fl = min(p00, p01, p10, p11) / mass + floor = fl if floor is None else min(floor, fl) + out.append({"i": i, "rows": rows, "def_mass": dm, + "n_active": len(active), "cell_floor": floor, + "underflow": underflow}) + return out + + +def agg_of_rows(per_i, lam: float, n: int): + """(aggregate, min per-i excess over i= n or not c["rows"]: + continue + w = c["def_mass"] + mi = sum(m * v for (_a, _b, m, v) in c["rows"]) / w + num += w * (mi - lam) + den += w + if worst is None or mi - lam < worst: + worst, worst_i = mi - lam, c["i"] + if den == 0.0: + return None, None, None + return num / den, worst, worst_i + + +def sparse_marginals(u: dict[int, float], lam: float, n: int): + t = 2.0 ** lam + supp = sorted(u) + z = 0.0 + mu = {} + for a in supp: + row = sum(u[a] * u[b] * t ** popc(a & b) for b in supp) + mu[a] = row + z += row + mu = {a: v / z for a, v in mu.items()} + return mu, [sum(v for a, v in mu.items() if a >> j & 1) for j in range(n)] + + +def sparse_eval(u: dict[int, float], lam: float, n: int): + """(agg, worst per-i excess, worst i, max marginal, underflow flag).""" + per_i = sparse_rows(u, lam, n) + agg, worst, wi = agg_of_rows(per_i, lam, n) + _mu, marg = sparse_marginals(u, lam, n) + return agg, worst, wi, max(marg), any(c["underflow"] for c in per_i) + + +def sinkhorn_atoms(mu: dict[int, float], lam: float, iters: int = 30000, + tol: float = 1e-13): + """Symmetric geometric-damped scaling on atom space (013's iteration + style); for mu-specified sparse instances (crash) and small-n direct + cross-checks of the orbit engine.""" + supp = sorted(mu) + N = len(supp) + K = [[2.0 ** (lam * popc(supp[a] & supp[b])) for b in range(N)] + for a in range(N)] + tgt = [mu[s] for s in supp] + u = [math.sqrt(max(v, 1e-300)) for v in tgt] + resid = float("inf") + for _ in range(iters): + cur = [u[a] * sum(u[b] * K[a][b] for b in range(N)) for a in range(N)] + z = sum(cur) + resid = max(abs(c / z - g) for c, g in zip(cur, tgt)) + if resid < tol: + break + u = [ua * math.sqrt(g * z / c) for ua, c, g in zip(u, cur, tgt)] + return {supp[a]: u[a] for a in range(N)}, resid + + +# ====================================================================== +# sparse engine (exact): same census with Fractions; certified aggregate +# ====================================================================== + +def sparse_rows_exact(u: dict[int, Fraction], t: Fraction, n: int): + supp = sorted(u) + kern = {} + for a in supp: + ua = u[a] + for b in supp: + kern[(a, b)] = ua * u[b] * t ** popc(a & b) + out = [] + for i in range(1, n + 1): + pmask = (1 << (i - 1)) - 1 + bit = 1 << (i - 1) + prefixes = {A & pmask for A in supp} + active = {c for c in prefixes + if any(A & pmask == c and A & bit for A in supp) + and any(A & pmask == c and not A & bit for A in supp)} + tables: dict[tuple[int, int], list[Fraction]] = {} + for (A, B), w in kern.items(): + key = (A & pmask, B & pmask) + tab = tables.setdefault(key, [F0] * 4) + tab[(2 if A & bit else 0) + (1 if B & bit else 0)] += w + rows = [] + for (a, b), (p00, p01, p10, p11) in tables.items(): + nondeg = min(p00, p01, p10, p11) > 0 + assert nondeg == (a in active and b in active), \ + f"dichotomy failure at i={i}" # exact: must never fire + if nondeg: + rows.append((p00 + p01 + p10 + p11, + (p11 * p00) / (p10 * p01))) + out.append(rows) + return out + + +def sparse_marginals_exact(u: dict[int, Fraction], t: Fraction, n: int): + supp = sorted(u) + z = F0 + mu = {} + for a in supp: + row = sum(u[a] * u[b] * t ** popc(a & b) for b in supp) + mu[a] = row + z += row + mu = {a: v / z for a, v in mu.items()} + return [sum(v for a, v in mu.items() if a >> j & 1) for j in range(n)] + + +def agg_enclosure(rows_by_i, t: Fraction, n: int, terms: int = 12): + """Certified [lo, hi] for A = sum_{i= n or not rows: + per_i.append(None) + continue + slo = shi = w_lo = w_hi = F0 + for (m, orr) in rows: + q = orr / t + enc = memo.get(q) + if enc is None: + enc = log2_enclosure(q, terms=terms) + memo[q] = enc + mlo = _dround(m, False) + mhi = _dround(m, True) + # m > 0: directed products depend on the sign of the bound + slo = _dround(slo + (mlo if enc[0] >= 0 else mhi) * enc[0], + False) + shi = _dround(shi + (mhi if enc[1] >= 0 else mlo) * enc[1], + True) + w_lo = _dround(w_lo + mlo, False) + w_hi = _dround(w_hi + mhi, True) + per_i.append((slo / w_hi if slo >= 0 else slo / w_lo, + shi / w_lo if shi >= 0 else shi / w_hi, + (w_lo + w_hi) / 2)) + num_lo = _dround(num_lo + slo, False) + num_hi = _dround(num_hi + shi, True) + den_lo = _dround(den_lo + w_lo, False) + den_hi = _dround(den_hi + w_hi, True) + a_lo = num_lo / den_hi if num_lo >= 0 else num_lo / den_lo + a_hi = num_hi / den_lo if num_hi >= 0 else num_hi / den_hi + return a_lo, a_hi, per_i + + +# ====================================================================== +# orbit engine: measures invariant under permutations of coords {3..n}. +# States s = (a1, a2, k): membership of coords 1, 2 and tail count +# k = |A & {3..n}|. Same symmetry class as 012's engine; fresh code. +# ====================================================================== + +def orbit_states(n: int): + r = n - 2 + return [(a1, a2, k) for a1 in (0, 1) for a2 in (0, 1) + for k in range(r + 1)] + + +def wtail_gen(r: int, t): + """Wt[k][l] = sum over tail sets T_B of size l of t^{|T_A & T_B|}, + T_A fixed of size k. Arithmetic-generic (float or Fraction t).""" + tp = [t ** j for j in range(r + 1)] + Wt = [[None] * (r + 1) for _ in range(r + 1)] + for k in range(r + 1): + for l in range(r + 1): + s = 0 + for j in range(max(0, k + l - r), min(k, l) + 1): + s += math.comb(k, j) * math.comb(r - k, l - j) * tp[j] + Wt[k][l] = s + return Wt + + +def pair_kernel(f: int, t): + """PK[s][l] = C(f,s) * Wt_f[s][l] = sum over PAIRS of future sets + (|X_A|=s, |X_B|=l) of t^{|X_A & X_B|}.""" + Wt = wtail_gen(f, t) + return [[math.comb(f, s) * Wt[s][l] for l in range(f + 1)] + for s in range(f + 1)] + + +def orbit_kernel_matrix(n: int, t, states): + """M[s][t2] = t^{s1 t1 + s2 t2} * Wt[k_s][k_t]: for a FIXED atom in + state s, the total kernel weight against the whole orbit t2.""" + r = n - 2 + Wt = wtail_gen(r, t) + return [[(t ** (s[0] * t2[0] + s[1] * t2[1])) * Wt[s[2]][t2[2]] + for t2 in states] for s in states] + + +def orbit_sinkhorn(n: int, tgt: dict, lam: float, iters: int = 12000, + tol: float = 1e-11): + """Symmetric scaling g <- g * sqrt(tgt/cur) on orbit space; tgt is the + PER-ATOM target mass by state (zero-mass states excluded from the + support). Returns (g, relative residual).""" + states = [s for s in orbit_states(n) if tgt.get(s, 0.0) > 0.0] + cnt = [math.comb(n - 2, s[2]) for s in states] + M = orbit_kernel_matrix(n, 2.0 ** lam, states) + NS = len(states) + tv = [tgt[s] for s in states] + g = [math.sqrt(tv[a]) for a in range(NS)] + resid = float("inf") + for it in range(iters): + cur = [g[a] * sum(g[b] * M[a][b] for b in range(NS)) + for a in range(NS)] + z = sum(cnt[a] * cur[a] for a in range(NS)) + resid = max(abs(cur[a] / (z * tv[a]) - 1.0) for a in range(NS)) + if resid < tol: + break + g = [g[a] * math.sqrt(tv[a] * z / cur[a]) for a in range(NS)] + return {s: g[a] for a, s in enumerate(states)}, resid + + +def orbit_rows(n: int, t, g: dict): + """Per i = 1..n: list of (mass, OR) over nondegenerate history orbit + classes, masses unnormalized. Arithmetic-generic: t and the values of + g may be floats or Fractions (math.comb counts are ints). + + i = 1: response coord 1; prefix empty. + i = 2: response coord 2; prefix = coord-1 value. + i >= 3: response = a tail coord; prefix = (a1, a2) + q = i-3 tail + coords; future f = n-i tail coords. History classes are + (oa, ob, ja, jb) with the prefix-overlap j summed into the mass factor + JW[ja][jb] = sum_j Ncl(ja,jb,j) t^j; the OR does not depend on j.""" + r = n - 2 + zero = 0 * t + + def gv(o, c): + return g.get((o[0], o[1], c), zero) + + orb4 = [(0, 0), (0, 1), (1, 0), (1, 1)] + out = [] + + # ---- i = 1 + PK = pair_kernel(r, t) + + def quad(o_a, ca, o_b, cb, PKm, f): + acc = zero + for s in range(f + 1): + ga = gv(o_a, ca + s) + if ga == 0: + continue + row = PKm[s] + sub = zero + for l in range(f + 1): + gb = gv(o_b, cb + l) + if gb != 0: + sub += row[l] * gb + acc += ga * sub + return acc + + # i=1: cells over (alpha, beta) = coord-1 values; sum over coord-2 + reach1 = {al: any(g.get((al, a2, k), zero) != 0 + for a2 in (0, 1) for k in range(r + 1)) + for al in (0, 1)} + rows1 = [] + if reach1[0] and reach1[1]: + cell = {} + for al in (0, 1): + for be in (0, 1): + acc = zero + for a2 in (0, 1): + for b2 in (0, 1): + acc += (t ** (a2 * b2)) * quad((al, a2), 0, + (be, b2), 0, PK, r) + cell[(al, be)] = (t ** (al * be)) * acc + mass = cell[(0, 0)] + cell[(0, 1)] + cell[(1, 0)] + cell[(1, 1)] + orr = (cell[(1, 1)] * cell[(0, 0)]) / (cell[(1, 0)] * cell[(0, 1)]) + rows1.append((mass, orr)) + out.append(rows1) + + # i=2: classes (a, b) = coord-1 prefix values + reach2 = {(a, al): any(g.get((a, al, k), zero) != 0 + for k in range(r + 1)) + for a in (0, 1) for al in (0, 1)} + act2 = {a for a in (0, 1) if reach2[(a, 0)] and reach2[(a, 1)]} + rows2 = [] + for a in act2: + for b in act2: + cell = {} + for al in (0, 1): + for be in (0, 1): + cell[(al, be)] = (t ** (a * b + al * be)) * \ + quad((a, al), 0, (b, be), 0, PK, r) + mass = (cell[(0, 0)] + cell[(0, 1)] + cell[(1, 0)] + + cell[(1, 1)]) + orr = (cell[(1, 1)] * cell[(0, 0)]) / (cell[(1, 0)] + * cell[(0, 1)]) + rows2.append((mass, orr)) + out.append(rows2) + + # i >= 3 + for i in range(3, n + 1): + q = i - 3 + f = n - i + PKf = pair_kernel(f, t) + # S matrices per (oa, ob): S[x][y], x,y in 0..q+1 + # via Gmat_o[x][s] = g(o, x+s) + Gm = {} + for o in orb4: + Gm[o] = [[gv(o, x + s) if x + s <= r else zero + for s in range(f + 1)] for x in range(q + 2)] + # Half_o[l][y] = sum_s PKf[s][l] ... compute H_o = PKf^T Gm_o^T: + # H_o[l][y] = sum_s PKf[s][l] * Gm_o[y][s] (PK symmetric would + # allow PKf[l][s]; keep general) + Ho = {} + for o in orb4: + Go = Gm[o] + H = [[zero] * (q + 2) for _ in range(f + 1)] + for l in range(f + 1): + Hl = H[l] + for s in range(f + 1): + ps = PKf[s][l] + if ps == 0: + continue + for y in range(q + 2): + gy = Go[y][s] + if gy != 0: + Hl[y] += ps * gy + Ho[o] = H + # JW[ja][jb] = sum_j Ncl * t^j + tp = [t ** j for j in range(q + 1)] + JW = [[zero] * (q + 1) for _ in range(q + 1)] + for ja in range(q + 1): + for jb in range(q + 1): + acc = zero + for j in range(max(0, ja + jb - q), min(ja, jb) + 1): + acc += (math.comb(q, j) * math.comb(q - j, ja - j) + * math.comb(q - ja, jb - j)) * tp[j] + JW[ja][jb] = acc + # reachability of continuations c = 0..q+1 per orbit pattern + reach = {o: [any(gv(o, c + s) != 0 for s in range(f + 1)) + for c in range(q + 3)] for o in orb4} + rows = [] + for oa in orb4: + Ra = reach[oa] + Ga = Gm[oa] + for ob in orb4: + Rb = reach[ob] + Hb = Ho[ob] + base = t ** (oa[0] * ob[0] + oa[1] * ob[1]) + # S[x][y] = sum_s Ga[x][s] * sum_l PKf[s][l] Gb[y][l] + # = sum_l ... use Hb: S[x][y] = sum_s Ga[x][s]*.. + # Hb[l][y] = sum_s PKf[s][l] Gb[y][s] -- careful: Hb was + # built from Gm[ob], i.e. Hb[l][y] = sum_s PK[s][l] Gb[y][s] + # We need S[x][y] = sum_{s,l} Ga[x][s] PK[s][l] Gb[y][l] + # = sum_l (sum_s Ga[x][s] PK[s][l]) Gb[y][l] + # PK symmetric (pair count), so sum_s Ga[x][s] PK[s][l] + # = (H built from Ga)[l][x]. Use Ho[oa]. + Ha = Ho[oa] + S = [[zero] * (q + 2) for _ in range(q + 2)] + Gb = Gm[ob] + for x in range(q + 2): + Sx = S[x] + for l in range(f + 1): + hax = Ha[l][x] + if hax == 0: + continue + for y in range(q + 2): + gby = Gb[y][l] + if gby != 0: + Sx[y] += hax * gby + for ja in range(q + 1): + if not (Ra[ja] and Ra[ja + 1]): + continue + for jb in range(q + 1): + if not (Rb[jb] and Rb[jb + 1]): + continue + s00 = S[ja][jb] + s01 = S[ja][jb + 1] + s10 = S[ja + 1][jb] + s11 = S[ja + 1][jb + 1] + if 0 in (s00, s01, s10, s11): + continue # unreachable class (zero count) + jw = JW[ja][jb] + if jw == 0: + continue + orr = t * (s00 * s11) / (s01 * s10) + mass = base * jw * (s00 + s01 + s10 + t * s11) + rows.append((mass, orr)) + out.append(rows) + return out + + +def orbit_agg(rows_by_i, lam: float, n: int): + """Float aggregate + per-i excess from orbit rows.""" + num = den = 0.0 + worst = worst_i = None + per_i = [] + for i, rows in enumerate(rows_by_i, start=1): + if not rows: + per_i.append(None) + continue + W = sum(m for (m, _o) in rows) + mi = sum(m * math.log2(o) for (m, o) in rows) / W + per_i.append({"i": i, "M_i": mi, "def_mass": W}) + if i < n: + num += W * (mi - lam) + den += W + if worst is None or mi - lam < worst: + worst, worst_i = mi - lam, i + if den == 0.0: + return None, None, None, per_i + return num / den, worst, worst_i, per_i + + +def orbit_marginals(n: int, t, g: dict): + """Elementwise marginals [coord1, coord2, tail coord] for the tilt + coupling of orbit potential g. Arithmetic-generic.""" + r = n - 2 + zero = 0 * t + states = [s for s in orbit_states(n) if g.get(s, zero) != zero] + M = orbit_kernel_matrix(n, t, states) + z = zero + m1 = m2 = mt = zero + for a, s in enumerate(states): + row = zero + for b, s2 in enumerate(states): + row += g[s2] * M[a][b] + w = math.comb(r, s[2]) * g[s] * row + z += w + if s[0]: + m1 += w + if s[1]: + m2 += w + mt += w * Fraction(s[2], r) if isinstance(t, Fraction) \ + else w * (s[2] / r) + return m1 / z, m2 / z, mt / z + + +# ---------------------------------------------------------------------- +# orbit family targets: PER-ATOM mass by state (a1, a2, k) +# ---------------------------------------------------------------------- + +def st_bern(n: int, p: float) -> dict: + return {(a1, a2, k): p ** (a1 + a2 + k) * (1 - p) ** (n - a1 - a2 - k) + for (a1, a2, k) in orbit_states(n)} + + +def st_mixture(n: int, comps) -> dict: + out = {s: 0.0 for s in orbit_states(n)} + for w, p in comps: + for s, v in st_bern(n, p).items(): + out[s] += w * v + return {s: v for s, v in out.items() if v > 0} + + +def st_slice(n: int, w0: int) -> dict: + tot = math.comb(n, w0) + return {(a1, a2, k): 1.0 / tot + for (a1, a2, k) in orbit_states(n) if a1 + a2 + k == w0} + + +def st_smslice(n: int, w0: int, tau: float) -> dict: + out = {s: (1 - tau) * v for s, v in st_slice(n, w0).items()} + for s, v in st_bern(n, w0 / n).items(): + out[s] = out.get(s, 0.0) + tau * v + return out + + +def st_d0bern(n: int, pl: float, ph: float) -> dict: + out = {s: (1 - pl) * v for s, v in st_bern(n, ph).items()} + out[(0, 0, 0)] = out.get((0, 0, 0), 0.0) + pl + return out + + +def st_crashmix(n: int, p: float, d: float) -> dict: + """(1-d) Bern(p)^n + d * crash (005's 4-atom family), 012's cells.""" + r = n - 2 + out = {s: (1 - d) * v for s, v in st_bern(n, p).items()} + out[(0, 1, 0)] = out.get((0, 1, 0), 0.0) + d * 0.33 + out[(0, 0, r)] = out.get((0, 0, r), 0.0) + d * 0.33 / math.comb(r, r) + out[(1, 1, r)] = out.get((1, 1, r), 0.0) + d * 0.04 + out[(1, 0, 0)] = out.get((1, 0, 0), 0.0) + d * 0.30 + return out + + +def st_epscrash(n: int, eps: float) -> dict: + """(1-eps) crash + eps * uniform (007 part K2 genre).""" + out = {s: eps * 2.0 ** (-n) for s in orbit_states(n)} + r = n - 2 + out[(0, 1, 0)] += (1 - eps) * 0.33 + out[(0, 0, r)] += (1 - eps) * 0.33 + out[(1, 1, r)] += (1 - eps) * 0.04 + out[(1, 0, 0)] += (1 - eps) * 0.30 + return out + + +def st_mix_dir(n: int, p: float, nu: str, d: float) -> dict: + """(1-d) Bern(p)^n + d * direction, for the perturbative sweep around + the product EQUALITY BOUNDARY (aggregate == 0 identically at d = 0). + Directions: crash (005), empty (delta_0), full (delta_[n]), + slice (uniform on the in-regime layer).""" + out = {s: (1 - d) * v for s, v in st_bern(n, p).items()} + r = n - 2 + if nu == "crash": + out[(0, 1, 0)] += d * 0.33 + out[(0, 0, r)] += d * 0.33 + out[(1, 1, r)] += d * 0.04 + out[(1, 0, 0)] += d * 0.30 + elif nu == "empty": + out[(0, 0, 0)] += d + elif nu == "full": + out[(1, 1, r)] += d + elif nu == "slice": + w0 = int(RECORD * n) + for s, v in st_slice(n, w0).items(): + out[s] = out.get(s, 0.0) + d * v + elif nu == "slice_matched": + w0 = round(p * n) # the layer under the Bern bulk + for s, v in st_slice(n, w0).items(): + out[s] = out.get(s, 0.0) + d * v + else: + raise ValueError(nu) + return out + + +def orbit_families(n: int) -> dict: + """The structured adversary genres, per-atom orbit targets.""" + w0 = int(RECORD * n) # slice weight, in-regime by construction + fams = { + "bern_p3823": st_bern(n, 0.3823), + "bern_p30": st_bern(n, 0.30), + "sawin_geo_a": st_mixture(n, [(0.80, 0.28), (0.12, 0.55), + (0.06, 0.75), (0.02, 0.92)]), + "sawin_geo_b": st_mixture(n, [(0.70, 0.32), (0.20, 0.50), + (0.08, 0.70), (0.02, 0.90)]), + "sawin_geo_c": st_mixture(n, [(0.60, 0.30), (0.25, 0.45), + (0.10, 0.60), (0.05, 0.80)]), + "partE_mix": st_mixture(n, [(0.7, 0.30), (0.3, 0.05)]), + "d0bern_30_51": st_d0bern(n, 0.30, 0.51), + "d0bern_30_55": st_d0bern(n, 0.30, 0.55), + f"slice_w{w0}": st_slice(n, w0), + f"smslice_w{w0}": st_smslice(n, w0, 2.0 ** -8), + "crash_eps2": st_epscrash(n, 1e-2), + "crash_eps3": st_epscrash(n, 1e-3), + "crashmix_383_05": st_crashmix(n, 0.383, 0.05), + "crashmix_30_05": st_crashmix(n, 0.30, 0.05), + } + return fams + + +def orbit_instance(name: str, n: int, tgt: dict, lam: float): + """Sinkhorn-fit + census + marginals; returns result dict + g.""" + t0 = time.time() + g, resid = orbit_sinkhorn(n, tgt, lam) + rows = orbit_rows(n, 2.0 ** lam, g) + agg, worst, wi, per_i = orbit_agg(rows, lam, n) + m1, m2, mt = orbit_marginals(n, 2.0 ** lam, g) + mm = max(m1, m2, mt) + # sanity: M_n = lambda identically (Prop 1) + mn_err = (abs(per_i[n - 1]["M_i"] - lam) + if per_i[n - 1] is not None else None) + return { + "name": name, "n": n, "lambda": lam, "aggregate": agg, + "min_excess_lt_n": worst, "argmin_i": wi, "max_marginal": mm, + "in_regime": mm < RECORD, "sinkhorn_resid": resid, + "Mn_minus_lam_abs": mn_err, "seconds": round(time.time() - t0, 2), + }, g + + +# ====================================================================== +# sparse structured builders: witness genre at any n, multi-gadget stacks +# ====================================================================== + +# 007's 10-atom witness, original layout (n=7: marker 1, dilution 2-4, +# response 5, futures 6-7), bit j-1 = coord j. +WITNESS_U7 = { + 0b0100001: 3.802705744653898, # {1,6} + 0b1000001: 8.094727225702503, # {1,7} + 0b1010001: 0.004792968160237178, # {1,5,7} (light slice) + 0b0000000: 0.2526660836353666, # {} + 0b0100000: 27.28641199464732, # {6} + 0b1000000: 12.529780963546393, # {7} + 0b0110000: 0.3609234435898425, # {5,6} + 0b0000010: 20.0, 0b0000100: 20.0, 0b0001000: 20.0, +} + + +def witness_at_n(n: int, wdil: float = 20.0) -> tuple[dict, int]: + """The witness genre re-laid-out at size n: coord 1 = block marker, + coords 2..n-3 = fresh-prefix dilution singletons (weight wdil), + coord n-2 = response, coords n-1, n = futures. By 007's dilution + invariance M_response = -0.122 independent of the number of dilution + atoms; what changes with n is the AGGREGATE bookkeeping (new + coordinates bring their own w_i and M_i). Returns (u, response_i).""" + i_resp = n - 2 + b_resp = 1 << (i_resp - 1) + b_f1 = 1 << (n - 2) + b_f2 = 1 << (n - 1) + b_mark = 1 + u = { + b_mark | b_f1: 3.802705744653898, # {1, f1} + b_mark | b_f2: 8.094727225702503, # {1, f2} + b_mark | b_resp | b_f2: 0.004792968160237178, # light slice + 0: 0.2526660836353666, # {} + b_f1: 27.28641199464732, + b_f2: 12.529780963546393, + b_resp | b_f1: 0.3609234435898425, + } + for j in range(2, n - 2): # dilution singletons + u[1 << (j - 1)] = wdil + return u, i_resp + + +def multi_gadget(n: int, K: int, m: int, slices, wdil: float, + sub_w: float = 1.0): + """K light-slice gadgets stacked: gadget k occupies a marker coord, a + sub-marker coord, a response coord and m future coords; remaining + coords are fresh-prefix dilution singletons. Every atom of gadget k + carries marker_k, so at gadget k's response coordinate all other + gadgets' atoms sit on fresh prefixes (deterministic response) and do + not pollute the active blocks -- the multi-coordinate generalization + 007 lead 1 asked for. slices = list of K tuples (F0, F1, G0, G1), + each a vector over 2^m future patterns: + a-block atoms: marker + sub-marker + response alpha + future x, + b-block atoms: marker + response alpha + future x. + Returns (u, [response coords]) or None if it does not fit in n.""" + span = K * (3 + m) + ndil = n - span + if ndil < 0: + return None + u: dict[int, float] = {} + resp_coords = [] + pos = 0 + for k in range(K): + b_mark = 1 << pos + b_sub = 1 << (pos + 1) + i_resp = pos + 3 + b_resp = 1 << (pos + 2) + fut0 = pos + 3 + F0, F1, G0, G1 = slices[k] + for al, (FA, GA) in enumerate(((F0, G0), (F1, G1))): + for x in range(1 << m): + fbits = 0 + for jj in range(m): + if x >> jj & 1: + fbits |= 1 << (fut0 + jj) + if FA[x] > 0: + key = b_mark | b_sub | (b_resp if al else 0) | fbits + u[key] = u.get(key, 0.0) + FA[x] * sub_w + if GA[x] > 0: + key = b_mark | (b_resp if al else 0) | fbits + u[key] = u.get(key, 0.0) + GA[x] + resp_coords.append(i_resp) + pos += 3 + m + for j in range(ndil): + u[1 << (pos + j)] = wdil + return u, resp_coords + + +# The witness's slices in (F0, F1, G0, G1) form over 2 future coords +# (x bit0 = first future, bit1 = second future): +WITNESS_SLICES = ( + [0.0, 3.802705744653898, 8.094727225702503, 0.0], # F0: {f1}, {f2} + [0.0, 0.0, 0.004792968160237178, 0.0], # F1: {f2} light + [0.2526660836353666, 27.28641199464732, 12.529780963546393, 0.0], + [0.0, 0.3609234435898425, 0.0, 0.0], # G1: {f1} +) + + +def rand_slice(rng, dim: int): + if rng.random() < 0.5: + v = [0.0] * dim + for _ in range(rng.randrange(1, 3)): + v[rng.randrange(dim)] = math.exp(rng.uniform(-2, 2)) + if sum(v) == 0: + v[rng.randrange(dim)] = 1.0 + return v + return [math.exp(rng.uniform(-3, 3)) if rng.random() < 0.7 else 0.0 + for _ in range(dim)] + + +def clamp_pot(u: dict[int, float], rel: float = 1e-9) -> dict[int, float]: + """Dynamic-range clamp (007's underflow guard): drop atoms below + rel * max weight. Honest support changes, handled by N_i.""" + mx = max(u.values()) + return {k: v for k, v in u.items() if v >= mx * rel} + + +# ====================================================================== +# part X -- cross-checks +# ====================================================================== + +def part_x() -> dict: + out: dict = {} + print("== part X: engine cross-checks ==") + + # X1a: sparse engine reproduces 007's witness numbers at lambda = 3.5 + lam, n = 3.5, 7 + per_i = sparse_rows(WITNESS_U7, lam, n) + agg, worst, wi, per = agg_of_rows(per_i, lam, n), None, None, None + a, w, wi2 = agg + _mu, marg = sparse_marginals(WITNESS_U7, lam, n) + w5 = per_i[4] + m5 = sum(m * v for (_a, _b, m, v) in w5["rows"]) / w5["def_mass"] + print(f"[X1] witness n=7 lam=3.5: M_5 - lam = {m5 - lam:+.6f} " + f"(007: -0.122033), aggregate = {a:+.6f} (007: +1.84), " + f"max marginal = {max(marg):.4f} (007: 0.3178)") + ok1 = abs(m5 - lam + 0.122033) < 5e-6 and abs(a - 1.8447) < 5e-4 + out["x1"] = {"M5_minus_lam": m5 - lam, "aggregate": a, + "max_marginal": max(marg), "pass": ok1} + + # X1b: exact aggregate at t = 181/16 vs 013 part C (+1.844669) + tF = Fraction(181, 16) + uF = {k: Fraction(v) for k, v in WITNESS_U7.items()} + rows = sparse_rows_exact(uF, tF, n) + lo, hi, _pi = agg_enclosure(rows, tF, n) + print(f"[X1] exact aggregate at t=181/16 in " + f"[{float(lo):+.9f}, {float(hi):+.9f}] (013: +1.844669), " + f"width {float(hi - lo):.1e}") + ok1b = lo > 0 and abs(float(lo) - 1.844669) < 1e-5 + out["x1_exact"] = {"lo": float(lo), "hi": float(hi), "pass": ok1b} + + # X2: orbit engine vs direct sparse+Sinkhorn on full supports, n=6,7. + # Direct path: atom-space Sinkhorn on the full 2^n support, then the + # sparse census; orbit path: orbit Sinkhorn + orbit rows. + max_dv = 0.0 + for (name, n2, tgt_orbit, mu_full) in x2_instances(): + lam2 = 1.0 + gg, res_o = orbit_sinkhorn(n2, tgt_orbit, lam2) + rows_o = orbit_rows(n2, 2.0 ** lam2, gg) + agg_o, w_o, _wi, per_o = orbit_agg(rows_o, lam2, n2) + u_d, res_d = sinkhorn_atoms(mu_full, lam2) + per_d = sparse_rows(u_d, lam2, n2) + agg_d, w_d, _x = agg_of_rows(per_d, lam2, n2) + dv = abs(agg_o - agg_d) + # per-i comparison + for i in range(1, n2 + 1): + po = per_o[i - 1] + rows_i = per_d[i - 1]["rows"] + if po is None or not rows_i: + assert po is None and not rows_i, (name, i) + continue + mid = sum(m * v for (_a, _b, m, v) in rows_i) \ + / per_d[i - 1]["def_mass"] + dv = max(dv, abs(po["M_i"] - mid)) + max_dv = max(max_dv, dv) + print(f"[X2] orbit vs direct, {name} n={n2}: max |diff| = {dv:.2e} " + f"(resids {res_o:.1e}/{res_d:.1e})") + out["x2_max_diff"] = max_dv + out["x2_pass"] = max_dv < 5e-7 + + # X3: exact orbit rows vs float orbit rows on one instance + n3, lam3 = 8, 1.0 + tgt = st_mixture(n3, [(0.7, 0.30), (0.3, 0.05)]) + g3, _res = orbit_sinkhorn(n3, tgt, lam3) + rows_f = orbit_rows(n3, 2.0 ** lam3, g3) + agg_f, _w, _i, _p = orbit_agg(rows_f, lam3, n3) + g3x = {s: Fraction(v) for s, v in g3.items()} + t3x = Fraction(2.0 ** lam3) + rows_x = orbit_rows(n3, t3x, g3x) + lo3, hi3, _pp = agg_enclosure(rows_x, t3x, n3) + d3 = max(abs(agg_f - float(lo3)), abs(agg_f - float(hi3))) + print(f"[X3] exact vs float orbit rows (partE_mix n=8): float " + f"{agg_f:+.9f}, exact [{float(lo3):+.9f}, {float(hi3):+.9f}], " + f"diff {d3:.1e}") + out["x3_diff"] = d3 + out["x3_pass"] = d3 < 1e-7 + + save("or_agg_probe_partX.json", out) + print(f"[X] ALL {'PASS' if all(v for k, v in out.items() if str(k).endswith('pass')) else 'FAIL'}") + return out + + +def x2_instances(): + """(name, n, orbit target, full-support mu) for the X2 cross-check.""" + def bern_full(n, p): + return {m: p ** popc(m) * (1 - p) ** (n - popc(m)) + for m in range(1 << n)} + + def mix_full(n, comps): + mu = {m: 0.0 for m in range(1 << n)} + for w, p in comps: + for m2, v in bern_full(n, p).items(): + mu[m2] += w * v + return {m: v for m, v in mu.items() if v > 0} + + insts = [] + n = 6 + w0 = 2 + mu = {m: 1.0 / math.comb(n, w0) for m in range(1 << n) + if popc(m) == w0} + insts.append((f"slice_w{w0}", n, st_slice(n, w0), mu)) + insts.append(("sawin_geo_a", n, + st_mixture(n, [(0.80, 0.28), (0.12, 0.55), + (0.06, 0.75), (0.02, 0.92)]), + mix_full(n, [(0.80, 0.28), (0.12, 0.55), + (0.06, 0.75), (0.02, 0.92)]))) + # eps-crash n=6 + eps = 1e-2 + mu_ec = {m: eps / (1 << n) for m in range(1 << n)} + full = (1 << n) - 1 + for a, wgt in ((0b10, 0.33), (full ^ 0b11, 0.33), (full, 0.04), + (0b01, 0.30)): + mu_ec[a] += (1 - eps) * wgt + insts.append(("crash_eps2", n, st_epscrash(n, eps), mu_ec)) + n = 7 + insts.append(("d0bern_30_51", n, st_d0bern(n, 0.30, 0.51), + mix_full(n, [(0.30, 0.0), (0.70, 0.51)]))) + return insts + + +# ====================================================================== +# part O -- orbit family census over the n grid +# ====================================================================== + +def part_o(ns=NS_GRID) -> dict: + print("== part O: orbit family census ==") + all_res = {} + tightest = {} + for n in ns: + res = [] + fams = orbit_families(n) + lams = lam_grid(n) + t_n = time.time() + for name, tgt in fams.items(): + for lam in lams: + rec, _g = orbit_instance(name, n, tgt, lam) + res.append(rec) + # per-n summary over IN-REGIME instances + inreg = [r for r in res if r["in_regime"] and r["aggregate"] is not None] + n_viol = sum(1 for r in inreg if r["aggregate"] < -1e-9) + worst = min(inreg, key=lambda r: r["aggregate"]) if inreg else None + max_resid = max(r["sinkhorn_resid"] for r in res) + max_mn = max(r["Mn_minus_lam_abs"] for r in res + if r["Mn_minus_lam_abs"] is not None) + print(f"[O] n={n}: {len(res)} instances ({len(inreg)} in-regime), " + f"aggregate violations: {n_viol}; min in-regime aggregate = " + f"{worst['aggregate']:+.6f} at {worst['name']} " + f"lam={worst['lambda']:g} (i* = {worst['argmin_i']}, " + f"min per-i excess {worst['min_excess_lt_n']:+.4f}); " + f"max resid {max_resid:.1e}, max |M_n - lam| {max_mn:.1e}; " + f"{time.time() - t_n:.0f}s") + all_res[n] = res + tightest[n] = worst + save(f"or_agg_probe_partO_n{n}.json", + {"n": n, "results": res, "n_in_regime_violations": n_viol, + "min_in_regime_aggregate": worst}) + save("or_agg_probe_partO_summary.json", + {str(n): tightest[n] for n in all_res}) + return {"results": all_res, "tightest": tightest} + + +# ====================================================================== +# part P -- perturbative sweep around the product equality boundary. +# Product measures have proportional slices, hence OR == 2^lambda +# pointwise and aggregate == 0 IDENTICALLY: they are the equality set of +# the aggregated control. A violation therefore only needs the +# d-derivative (or curvature) of the aggregate to be negative along SOME +# direction at SOME (n, p, lambda) -- and 012 showed the perturbative +# downward-budget grows with n on exactly this genre. Sweep d and fit +# the small-d scaling. +# ====================================================================== + +def part_p(ns=(10, 20, 32)) -> dict: + print("== part P: perturbative d-sweep around product boundary ==") + res = [] + dirs = ("crash", "empty", "full", "slice") + dgrid = (0.005, 0.01, 0.02, 0.05, 0.1, 0.2) + for n in ns: + lams = sorted({round(math.log2(1.06), 6), 0.5, + round(4.847 / (n - 3), 6)}) + for p in (0.3823, 0.30): + for nu in dirs: + for lam in lams: + aggs = [] + for d in dgrid: + tgt = st_mix_dir(n, p, nu, d) + rec, _g = orbit_instance( + f"mixdir_{nu}_p{p:g}", n, tgt, lam) + rec["d"] = d + res.append(rec) + aggs.append((d, rec["aggregate"], + rec["in_regime"])) + # small-d scaling: ratio agg(2d)/agg(d) ~ 2^order + a1, a2 = aggs[0][1], aggs[1][1] + order = (math.log2(a2 / a1) + if a1 and a2 and a1 > 0 and a2 > 0 else None) + neg = [a for a in aggs if a[1] is not None + and a[1] < -1e-12 and a[2]] + print(f"[P] n={n} p={p:g} dir={nu:6s} lam={lam:g}: " + f"agg(d) = " + + " ".join(f"{d:g}:{a:+.2e}" for d, a, _r in aggs) + + (f" order~d^{order:.2f}" if order else "") + + (" <-- NEGATIVE" if neg else "")) + inreg = [r for r in res if r["in_regime"] and r["aggregate"] is not None] + nv = sum(1 for r in inreg if r["aggregate"] < -1e-12) + worst = min(inreg, key=lambda r: r["aggregate"]) + print(f"[P] {len(res)} instances ({len(inreg)} in-regime); negative " + f"aggregates: {nv}; min in-regime aggregate = " + f"{worst['aggregate']:+.3e} at n={worst['n']} {worst['name']} " + f"d={worst['d']:g} lam={worst['lambda']:g}") + save("or_agg_probe_partP.json", + {"results": res, "n_negative": nv, "min_in_regime": worst}) + return {"results": res, "min_in_regime": worst} + + +# ====================================================================== +# part T -- trend extension past n = 32 on the softest observed +# directions (part P's slice-direction curvature at p = 0.30 decays with +# n: 1.35e-5 -> 6.2e-7 over n = 10 -> 32). The orbit engine is cheap +# enough to chase the trend to n = 64: does the margin cross zero, decay +# to 0 polynomially, or plateau? Also tracks the tightest census family +# (crashmix_383_05) and the crash direction at the window lambda. +# ====================================================================== + +def part_t(ns=(32, 40, 48, 56, 64)) -> dict: + print("== part T: margin trend past n = 32 ==") + res = [] + for n in ns: + for (p, nu, lam, d) in ( + (0.30, "slice", 0.084, 0.05), + (0.30, "slice", 0.5, 0.05), + (0.30, "slice_matched", 0.084, 0.05), + (0.3823, "slice", 0.084, 0.05), + (0.3823, "slice_matched", 0.084, 0.05), + (0.3823, "crash", 0.084, 0.05), + (0.3823, "crash", round(4.847 / (n - 3), 6), 0.05), + ): + tgt = st_mix_dir(n, p, nu, d) + rec, _g = orbit_instance(f"mixdir_{nu}_p{p:g}_d{d:g}", n, tgt, + lam) + rec["d"] = d + res.append(rec) + # the tightest census family, at its tightest lambda + tgt = st_crashmix(n, 0.383, 0.05) + lamw = round(math.log2(1.06), 6) + rec, _g = orbit_instance("crashmix_383_05", n, tgt, lamw) + res.append(rec) + row = [r for r in res if r["n"] == n] + print(f"[T] n={n}: " + + " ".join(f"{r['name']}@{r['lambda']:g}:" + f"{r['aggregate']:+.3e}" for r in row)) + neg = [r for r in res if r["in_regime"] and r["aggregate"] is not None + and r["aggregate"] < -1e-12] + print(f"[T] negative in-regime aggregates: {len(neg)}") + save("or_agg_probe_partT.json", {"results": res, "n_negative": len(neg)}) + return {"results": res} + + +# ====================================================================== +# part S -- sparse structured census: crash/mirror, witness genre, +# multi-gadget stacks +# ====================================================================== + +def part_s(ns=NS_GRID) -> dict: + print("== part S: sparse structured census ==") + res = [] + + def run_pot(name, u, n, lam): + agg, worst, wi, mm, uf = sparse_eval(u, lam, n) + rec = {"name": name, "n": n, "lambda": lam, "aggregate": agg, + "min_excess_lt_n": worst, "argmin_i": wi, + "max_marginal": mm, "in_regime": mm < RECORD, + "underflow": uf} + res.append(rec) + return rec + + full_atoms = lambda n: (1 << n) - 1 + + for n in ns: + lams = lam_grid(n) + # crash / mirror (4 atoms; Theorem B territory -- margin trend) + full = full_atoms(n) + crash = {0b10: 0.33, full ^ 0b11: 0.33, full: 0.04, 0b01: 0.30} + mirror = {0b10 | (full ^ 0b11): 0.3, 0b00: 0.2, full: 0.3, + 0b01: 0.2} + for lam in lams: + for fname, mu in (("crash", crash), ("mirror", mirror)): + u, resid = sinkhorn_atoms(mu, lam) + rec = run_pot(f"{fname}", u, n, lam) + rec["sinkhorn_resid"] = resid + + # witness genre re-laid-out at n, dilution-weight sweep: the + # tightest in-regime aggregate the genre can reach at this n + for lam in (0.5, 1.0, 2.0, 3.5, 5.0): + best = None + for wdil in (0.25, 0.5, 1.0, 2.0, 4.0, 8.0, 16.0, 20.0, 40.0): + u, i_resp = witness_at_n(n, wdil) + agg, worst, wi, mm, uf = sparse_eval(u, lam, n) + rec = {"name": f"witness_wdil{wdil:g}", "n": n, + "lambda": lam, "aggregate": agg, + "min_excess_lt_n": worst, "argmin_i": wi, + "max_marginal": mm, "in_regime": mm < RECORD, + "underflow": uf} + res.append(rec) + if rec["in_regime"] and (best is None + or agg < best["aggregate"]): + best = rec + if best is not None: + print(f"[S] n={n} lam={lam:g}: witness-genre tightest " + f"in-regime aggregate {best['aggregate']:+.6f} " + f"({best['name']}, min per-i excess " + f"{best['min_excess_lt_n']:+.4f} at i=" + f"{best['argmin_i']}, marg {best['max_marginal']:.4f})") + + # multi-gadget stacks seeded with the witness slices: K gadgets, + # m = 2 futures each (K as large as fits), sub-marker weight and + # dilution weight swept coarsely + for lam in (2.0, 3.5): + K = (n - 2) // 5 + if K < 1: + continue + for wdil in (1.0, 4.0, 16.0): + built = multi_gadget(n, K, 2, [WITNESS_SLICES] * K, wdil) + if built is None: + continue + u, resp = built + u = clamp_pot(u) + rec = run_pot(f"stack_K{K}_wdil{wdil:g}", u, n, lam) + stack = [r for r in res if r["n"] == n and r["name"].startswith("stack")] + if stack: + b = min((r for r in stack if r["in_regime"]), + key=lambda r: r["aggregate"], default=None) + if b: + print(f"[S] n={n}: tightest in-regime witness-slice stack " + f"aggregate {b['aggregate']:+.6f} ({b['name']} " + f"lam={b['lambda']:g})") + + inreg = [r for r in res if r["in_regime"] and r["aggregate"] is not None] + n_viol = sum(1 for r in inreg if r["aggregate"] < -1e-9) + worst = min(inreg, key=lambda r: r["aggregate"]) + print(f"[S] total {len(res)} instances ({len(inreg)} in-regime); " + f"aggregate violations: {n_viol}; min in-regime aggregate = " + f"{worst['aggregate']:+.6f} at n={worst['n']} {worst['name']} " + f"lam={worst['lambda']:g}") + save("or_agg_probe_partS.json", + {"results": res, "n_in_regime_violations": n_viol, + "min_in_regime": worst}) + return {"results": res, "min_in_regime": worst} + + +# ====================================================================== +# part H -- adversarial hill-climbs on the aggregate objective +# ====================================================================== + +MARG_TARGET = 0.375 + + +def climb(u0: dict[int, float], n: int, lam: float, rng, steps: int = 400): + """Multiplicative hill-climb minimizing aggregate + marginal penalty, + dynamic-range-clamped. Returns (best value, best u, max marginal).""" + def obj(u): + u = clamp_pot(u) + agg, _w, _i, mm, uf = sparse_eval(u, lam, n) + if agg is None or uf: + return float("inf"), None, None + return agg + 25.0 * max(0.0, mm - MARG_TARGET), agg, mm + + cur, cagg, cmm = obj(u0) + u = dict(u0) + if not math.isfinite(cur): + return None, None, None + for step in range(steps): + sigma = 1.0 * (1 - step / steps) + 0.05 + cand = dict(u) + r = rng.random() + if r < 0.12 and len(cand) > 5: # drop an atom + cand.pop(rng.choice(sorted(cand))) + elif r < 0.24: # add a random atom + a = rng.randrange(1 << n) + cand[a] = math.exp(rng.uniform(-2, 2)) + else: # perturb one weight + a = rng.choice(sorted(cand)) + cand[a] = max(cand[a], 1e-300) * math.exp(rng.gauss(0, sigma)) + v, agg, mm = obj(cand) + if v < cur: + u, cur, cagg, cmm = clamp_pot(cand), v, agg, mm + return cagg, u, cmm + + +def part_h() -> dict: + print("== part H: adversarial hill-climbs on the aggregate ==") + rng = random.Random(140001) + endpoints = [] + best_overall = None + + def record(tag, n, lam, cagg, u, cmm): + nonlocal best_overall + e = {"tag": tag, "n": n, "lambda": lam, "aggregate": cagg, + "max_marginal": cmm, "n_atoms": len(u) if u else None} + endpoints.append(e) + if (cagg is not None and cmm is not None and cmm < RECORD + and (best_overall is None + or cagg < best_overall["aggregate"])): + best_overall = dict(e) + best_overall["u"] = {format(k, "b"): v for k, v in u.items()} + + # H-a: free sparse supports at n = 10, 14 (8..18 atoms) + for n in (10, 14): + for restart in range(10): + k = rng.randrange(8, 19) + atoms = rng.sample(range(1 << n), k) + u0 = {a: math.exp(rng.uniform(-2, 2)) for a in atoms} + lam = rng.choice([0.5, 1.0, 2.0, 3.5]) + cagg, u, cmm = climb(u0, n, lam, rng, steps=400) + record("free", n, lam, cagg, u, cmm) + + # H-b: witness-seeded at n = 10, 14, 20 + for n in (10, 14, 20): + for lam in (2.0, 3.5): + for restart in range(3): + u0, _i = witness_at_n(n, wdil=rng.choice([1.0, 4.0, 20.0])) + if restart: # jiggle the seed + u0 = {k: v * math.exp(rng.gauss(0, 0.1)) + for k, v in u0.items()} + cagg, u, cmm = climb(u0, n, lam, rng, steps=350) + record("witness_seed", n, lam, cagg, u, cmm) + + # H-c: multi-gadget-parametrized at n = 14, 20, 24: witness slices + + # random slices, K gadgets, climbing atom weights freely + for n in (14, 20, 24): + K = (n - 2) // 5 + for restart in range(4): + m = 2 + slices = [] + for k in range(K): + if rng.random() < 0.5: + slices.append(WITNESS_SLICES) + else: + slices.append(tuple(rand_slice(rng, 1 << m) + for _ in range(4))) + built = multi_gadget(n, K, m, slices, wdil=4.0) + if built is None: + continue + u0, _resp = built + lam = rng.choice([2.0, 3.5]) + cagg, u, cmm = climb(clamp_pot(u0), n, lam, rng, steps=350) + record("stack_seed", n, lam, cagg, u, cmm) + + # H-d: small-lambda climbs (Theorem C makes the first order in lambda + # a perfect square per i, so a small-lambda kill needs all D-sums ~ 0 + # with negative curvature -- let the search try, near the window law) + for n in (10, 14): + for lam in (0.084, 0.15, 0.3, 4.847 / (n - 3)): + for restart in range(4): + if restart % 2 == 0: + k = rng.randrange(8, 16) + u0 = {a: math.exp(rng.uniform(-2, 2)) + for a in rng.sample(range(1 << n), k)} + else: + u0, _i = witness_at_n(n, wdil=4.0) + u0 = {k2: v * math.exp(rng.gauss(0, 0.1)) + for k2, v in u0.items()} + cagg, u, cmm = climb(u0, n, lam, rng, steps=350) + record("small_lam", n, lam, cagg, u, cmm) + + ok = [e for e in endpoints if e["aggregate"] is not None] + nv = sum(1 for e in ok if e["aggregate"] < -1e-9 + and e["max_marginal"] < RECORD) + lows = sorted(e["aggregate"] for e in ok + if e["max_marginal"] < RECORD)[:5] + print(f"[H] {len(ok)} climb endpoints; in-regime aggregate violations: " + f"{nv}; five lowest in-regime endpoints: " + + " ".join(f"{v:+.5f}" for v in lows)) + if best_overall: + print(f"[H] lowest in-regime endpoint: {best_overall['aggregate']:+.6f} " + f"({best_overall['tag']} n={best_overall['n']} " + f"lam={best_overall['lambda']:g}, " + f"{best_overall['n_atoms']} atoms)") + save("or_agg_probe_partH.json", + {"endpoints": endpoints, "n_in_regime_violations": nv, + "best": best_overall}) + return {"endpoints": endpoints, "best": best_overall} + + +# ====================================================================== +# part E -- exact rational certification of the tightest points +# ====================================================================== + +def certify_sparse(name: str, u: dict[int, float], lam_num: int, + lam_den: int, n: int, use_t: Fraction | None = None): + """Certify the aggregate sign for the exact dyadic-rational potential + Fraction(u) at exact rational tilt. If use_t is given it IS the tilt + t (lambda = log2 t enclosed); else t = 2^{lam_num/lam_den} is + float-rationalized (dyadic). Returns dict.""" + t = use_t if use_t is not None else Fraction(2.0 ** (lam_num / lam_den)) + uF = {k: Fraction(v) for k, v in u.items()} + t0 = time.time() + rows = sparse_rows_exact(uF, t, n) + lo, hi, per_i = agg_enclosure(rows, t, n) + marg = sparse_marginals_exact(uF, t, n) + mm = max(marg) + rec = {"name": name, "n": n, "t": str(t), "engine": "sparse", + "agg_lo": float(lo), "agg_hi": float(hi), + "enclosure_width": float(hi - lo), + "certified_sign": ("positive" if lo > 0 else + "NEGATIVE" if hi < 0 else "indeterminate"), + "max_marginal_exact_lt_record": bool(mm < Fraction(38271, 100000)), + "max_marginal": float(mm), "seconds": round(time.time() - t0, 1)} + print(f"[E] {name} (sparse, n={n}, t={t.numerator}/{t.denominator}" + f"{'' if t.denominator < 10**6 else ' ~ ' + format(float(t), '.6g')}):" + f" aggregate in [{float(lo):+.9f}, {float(hi):+.9f}] -> " + f"{rec['certified_sign']}; exact max marginal " + f"{float(mm):.6f} {'<' if rec['max_marginal_exact_lt_record'] else '>='} 0.38271; {rec['seconds']}s") + return rec + + +def _quant(v: float, bits: int = 36) -> Fraction: + """Nearest dyadic with a bits-bit mantissa (exact, sign-preserving).""" + if v == 0.0: + return F0 + m, e = math.frexp(v) + return Fraction(round(m * (1 << bits)), 1 << bits) * Fraction(2) ** e + + +def certify_orbit(name: str, n: int, tgt: dict, lam: float, + terms: int = 10, use_t: Fraction | None = None, + quant_bits: int = 36): + """Sinkhorn-fit at float lambda, rationalize the potential (dyadic, + quantized to quant_bits-bit mantissas to keep exact arithmetic fast) + and the tilt (use_t if given -- prefer a short rational like 271/256 + -- else the dyadic float-rationalization of 2^lambda), then certify + the aggregate sign for that EXACT rational measure (013's + rationalized-refit scope: the certificate is about a mu-tilde within + ~Sinkhorn-residual + 2^-quant_bits relative of the named family, + which suffices for the universally-quantified claim).""" + t0 = time.time() + if use_t is not None: + t = use_t + lam = math.log2(float(t.numerator) / float(t.denominator)) + else: + t = Fraction(2.0 ** lam) + g, resid = orbit_sinkhorn(n, tgt, lam) + gF = {s: _quant(v, quant_bits) for s, v in g.items()} + rows = orbit_rows(n, t, gF) + lo, hi, _per = agg_enclosure(rows, t, n, terms=terms) + m1, m2, mt = orbit_marginals(n, t, gF) + mm = max(m1, m2, mt) + rec = {"name": name, "n": n, "lambda": lam, "t": str(t), + "engine": "orbit", "sinkhorn_resid": resid, + "agg_lo": float(lo), "agg_hi": float(hi), + "enclosure_width": float(hi - lo), + "certified_sign": ("positive" if lo > 0 else + "NEGATIVE" if hi < 0 else "indeterminate"), + "max_marginal_exact_lt_record": bool(mm < Fraction(38271, 100000)), + "max_marginal": float(mm), "seconds": round(time.time() - t0, 1)} + print(f"[E] {name} (orbit, n={n}, lam~{lam:g}): aggregate in " + f"[{float(lo):+.9f}, {float(hi):+.9f}] -> {rec['certified_sign']};" + f" exact max marginal {float(mm):.6f}; resid {resid:.1e}; " + f"{rec['seconds']}s") + return rec + + +def part_e(o_tightest=None) -> dict: + """Certify: (a) the witness-genre tightest in-regime point per n + (sparse, cheap, all n up to 32); (b) the tightest orbit point per n + (from part O's summary if available on disk); (c) rational-lambda + witness-genre points (t = 4 exactly, lambda = 2).""" + print("== part E: exact rational certification ==") + out = {"sparse": [], "orbit": []} + + # (a) witness genre at every n: the wdil giving the tightest in-regime + # aggregate at lam = 3.5 (float pre-scan), certified at t = 181/16 + for n in NS_GRID: + best = None + for wdil in (0.25, 0.5, 1.0, 2.0, 4.0, 8.0, 16.0, 20.0): + u, _i = witness_at_n(n, wdil) + agg, _w, _wi, mm, _uf = sparse_eval(u, 3.5, n) + if mm < RECORD and (best is None or agg < best[0]): + best = (agg, wdil, u) + if best is None: + continue + _agg, wdil, u = best + rec = certify_sparse(f"witness_n{n}_wdil{wdil:g}", u, 7, 2, n, + use_t=Fraction(181, 16)) + out["sparse"].append(rec) + + # (c) fully rational statement at lambda = 2 exactly (t = 4): the + # witness genre at n = 20 (wdil = 20 keeps it in-regime at t = 4) + u20, _i = witness_at_n(20, 20.0) + rec = certify_sparse("witness_n20_t4_lam2exact", u20, 2, 1, 20, + use_t=Fraction(4)) + out["sparse"].append(rec) + + # (b) tightest NON-PRODUCT orbit point per n, read from part O + # checkpoints. Product (bern) cells are the exact equality boundary + # of the aggregated control (proportional slices => OR == 2^lambda + # pointwise => aggregate == 0 identically); an enclosure there + # straddles 0 by construction and certifies nothing, so they are + # excluded from the "tightest point" and recorded as boundary. + for n in NS_GRID: + p = DATA / f"or_agg_probe_partO_n{n}.json" + if not p.exists(): + continue + allres = json.loads(p.read_text())["results"] + cand = [r for r in allres if r["in_regime"] + and r["aggregate"] is not None + and not r["name"].startswith("bern")] + if not cand: + continue + worst = min(cand, key=lambda r: r["aggregate"]) + name = worst["name"] + fams = orbit_families(n) + if name not in fams: + continue + # census min sits at the smallest grid lambda (~0.083-0.084); + # certify at the nearby SHORT exact rational tilt t = 271/256 + # (lambda = log2(271/256) ~ 0.08215), 013's clean-rational move + rec = certify_orbit(f"{name}_n{n}", n, fams[name], 0.0, + terms=10, use_t=Fraction(271, 256)) + out["orbit"].append(rec) + + # (d) the softest direction found (part P/T): slice-direction + # perturbation of Bern(0.30) at d = 0.05, small lambda -- the + # smallest nonzero margin in the whole probe (~6e-7 at n = 32). + # Certify at n = 20 and 32 (enclosure width ~1e-10 << margin). + for n_soft in (20, 32): + tgt = st_mix_dir(n_soft, 0.30, "slice", 0.05) + rec = certify_orbit(f"mixdir_slice_p0.3_d0.05_n{n_soft}", n_soft, + tgt, 0.0, terms=10, use_t=Fraction(271, 256)) + out["orbit"].append(rec) + + save("or_agg_probe_partE.json", out) + return out + + +def main() -> None: + parts = [p.upper() for p in sys.argv[1:] if p.upper() in "XOPTSHE"] \ + or list("XOPTSHE") + t0 = time.time() + if "X" in parts: + part_x() + if "O" in parts: + part_o() + if "P" in parts: + part_p() + if "T" in parts: + part_t() + if "S" in parts: + part_s() + if "H" in parts: + part_h() + if "E" in parts: + part_e() + print(f"done in {time.time() - t0:.0f}s; checkpoints in {DATA}") + + +if __name__ == "__main__": + main() diff --git a/problems/union-closed/explore/uc_or_agg_skeptic2.py b/problems/union-closed/explore/uc_or_agg_skeptic2.py new file mode 100644 index 0000000..4c7e5fa --- /dev/null +++ b/problems/union-closed/explore/uc_or_agg_skeptic2.py @@ -0,0 +1,1294 @@ +#!/usr/bin/env python3 +"""Attempt 015: adversarial skeptic review of 014 (aggregated OR probe). + +Independent re-implementation, from scratch, of everything 014's headline +numbers rest on. NOTHING on the critical path is shared with +uc_or_agg_probe.py: + + * log2 certification: binary DIGIT-EXTRACTION (square-and-compare) with + directed integer shifts -- a different algorithm from 014's atanh + series + dyadic compaction. Soundness proof in log2_bracket's + docstring. + * exact aggregate: integer-scaled row builders (all masses exact + integers on one common scale; only the log2 values are intervals), so + the denominator of A is EXACT and only the numerator is an interval. + * own direct full-support census evaluator (float + the exact sparse + one), own tail-exchangeable orbit evaluator re-derived by hand, own + Sinkhorn (exponent-0.4-damped scaling, not 014's sqrt damping). + +uc_or_agg_probe.py is imported ONLY (a) to reproduce the exact rational +refit measures mu-tilde its certificates are ABOUT (part C re-certifies +the same stated inputs -- the measure spec is data, not verification +logic), and (b) to compare its float orbit rows against mine (part X). + +Parts: + K attack 014's enclosure kit: directional unit tests of _dround / + _ln_1_to_2_fp / log2_enclosure against my bracket enclosure on + adversarial inputs (near powers of 2, tiny/huge, tight rationals); + brute-force validation of the orbit combinatorics (Ncl, PK) that my + hand re-derivation predicts. + X formula cross-checks: my direct evaluator vs my orbit evaluator at + n = 8 AND 9 (014 only did 6, 7), on crashmix / sawin / slice-dir / + d0bern; my direct vs 014's sparse engine on the witness; explicit + j-independence check of class ORs at n = 9. + C independent re-certification of five of 014's fifteen certificates + (incl. the softest, +5.99e-7 at n = 32, and the fully rational + t = 4 point), on the identical exact rational inputs, plus one + fresh certificate at the softest point from MY OWN Sinkhorn+quant. + T trend attack: the n ~ 48 dip re-computed with my engine, off-grid + lambda (below 014's grid floor 0.079 and between its big-grid + points), and the census-minimum's lambda-scaling. + V new-adversary hunt in regions 014's search could not see: + witness-DIRECTION perturbation of a dense product bulk (non-tail- + symmetric direction off the equality boundary, n = 10), + boundary-of-regime marginals (max marginal pushed to ~0.3827), + tiny lambda. + +Standard library only. Deterministic. Runtime ~10-25 min total. +Checkpoints: problems/union-closed/data/oragg_sk2_part[KXCTV].json +Usage: python problems/union-closed/explore/uc_or_agg_skeptic2.py [parts] +""" + +from __future__ import annotations + +import json +import math +import sys +import time +from fractions import Fraction +from pathlib import Path + +ROOT = Path(__file__).resolve().parent.parent +DATA = ROOT / "data" +sys.path.insert(0, str(Path(__file__).resolve().parent)) + +RECORD = Fraction(38271, 100000) +RECORD_F = 0.38271 + + +def popc(x: int) -> int: + return bin(x).count("1") + + +def save(name: str, obj) -> None: + DATA.mkdir(exist_ok=True) + (DATA / name).write_text( + json.dumps(obj, indent=1, sort_keys=True, default=str) + "\n", + encoding="utf-8") + + +# ====================================================================== +# my certified log2: binary digit extraction (square-and-compare). +# +# For y in [1, 2) kept as an integer mantissa m with y = m / 2^WB: +# repeat B times: y <- y^2; if y >= 2 emit digit 1 and halve else emit 0. +# Then log2(y_0) = sum_j d_j 2^{-j} + 2^{-B} log2(y_B) with y_B in [1,2). +# +# Soundness with directed integer shifts (proved, used below): +# DOWN pass: m <- (m*m) >> (WB + d) (floor). By induction the tracked +# value y_j = m_j/2^WB satisfies y_j <= v_j, where v_j is the EXACT +# process that uses the algorithm's emitted digits d_j +# (v_{j+1} = v_j^2/2^{d_j}, v_0 = true input). Digits stay in sync by +# construction (they are defined by the tracked m). Unrolling +# log2 v_{j+1} = 2 log2 v_j - d_j gives +# log2 v_0 = sum d_j 2^{-(j+1)} + 2^{-B} log2 v_B, and v_B >= y_B >= 1 +# so log2 v_B >= 0: acc/2^B <= log2(input). +# UP pass: ceil shifts give y_j >= v_j and y stays <= 2 (squaring a +# value <= 2 then halving on digit 1 keeps it <= 2; ceil adds < 1 ulp), +# so v_B <= y_B <= 2 and log2 v_B <= 1: log2(input) <= (acc+1)/2^B. +# The initial mantissa is floor/ceil of q * 2^(WB-e), which only widens +# the bracket in the safe direction (log2 is increasing). +# ====================================================================== + +WB = 80 # working mantissa bits +BB = 48 # extracted fraction bits + + +def _digits_dir(m: int, up: bool, wb: int = WB, bits: int = BB) -> int: + """Truncated binary expansion of log2(m/2^wb), m/2^wb in [1,2]; + floor variant if not up (acc/2^bits <= log2), ceil variant if up + (log2 <= (acc+1)/2^bits).""" + acc = 0 + two = 1 << (2 * wb + 1) # value 2.0 at the squared scale + for _ in range(bits): + m *= m + acc <<= 1 + if m >= two: + acc |= 1 + sh = wb + 1 + else: + sh = wb + if up: + m = -((-m) >> sh) # ceil + else: + m >>= sh # floor + return acc + + +def log2_bracket(q: Fraction) -> tuple[Fraction, Fraction]: + """[lo, hi] dyadic with lo <= log2 q <= hi, width ~ 3*2^-BB; q > 0. + + Safety margins: the tracked mantissa can drift below 1 (floor pass) + or above 2 (ceil pass) by < 2^-(WB-2), so log2 v_B lies in + [-2^-(WB-1), 1 + 2^-(WB-1)] instead of [0, 1]; the -1/+2 ulps on the + extracted digits absorb that with room to spare.""" + assert q > 0 + num, den = q.numerator, q.denominator + e = num.bit_length() - den.bit_length() + # ensure 2^e <= q < 2^(e+1) + ge = num >= (den << e) if e >= 0 else (num << (-e)) >= den + if not ge: + e -= 1 + # mantissa q/2^e in [1,2): integer bracket at WB bits + sh = WB - e + if sh >= 0: + m_lo = (num << sh) // den + m_hi = -((-(num << sh)) // den) + else: + m_lo = num // (den << (-sh)) + m_hi = -((-num) // (den << (-sh))) + lo = Fraction(_digits_dir(m_lo, False) - 1, 1 << BB) + e + hi = Fraction(_digits_dir(m_hi, True) + 2, 1 << BB) + e + return lo, hi + + +# ====================================================================== +# my direct full-support / sparse census evaluator (float), from the +# 007 SS1 definition, written from scratch: N_i = reachable prefixes with +# both continuations reachable in supp(u); M_i = mass-weighted mean of +# log2 OR over N_i x N_i; A = sum_{i> j & 1) / Z + for j in range(n)] + per_i = [] + num = den = 0.0 + worst = worst_i = None + for i in range(1, n + 1): + mask = (1 << (i - 1)) - 1 + bit = 1 << (i - 1) + byp: dict[int, list[int]] = {} + for x in range(m): + byp.setdefault(atoms[x] & mask, []).append(x) + act = [p for p, xs in byp.items() + if any(atoms[x] & bit for x in xs) + and any(not atoms[x] & bit for x in xs)] + rows_w = 0.0 + rows_s = 0.0 + any_row = False + for pa in act: + for pb in act: + c = [0.0, 0.0, 0.0, 0.0] + for x in byp[pa]: + ax = 2 if atoms[x] & bit else 0 + kx = kern[x] + for y in byp[pb]: + c[ax + (1 if atoms[y] & bit else 0)] += kx[y] + if min(c) <= 0.0: + raise RuntimeError(f"underflow/degeneracy i={i}") + mass = c[0] + c[1] + c[2] + c[3] + v = math.log2(c[3] * c[0] / (c[2] * c[1])) + rows_w += mass + rows_s += mass * v + any_row = True + if not any_row: + per_i.append(None) + continue + mi = rows_s / rows_w + per_i.append((mi, rows_w)) + if i < n: + num += rows_w * (mi - lam) + den += rows_w + if worst is None or mi - lam < worst: + worst, worst_i = mi - lam, i + agg = num / den if den > 0 else None + return agg, per_i, max(marg), worst, worst_i + + +# ====================================================================== +# my exact sparse certifier: integer-scaled tables + log2_bracket. +# t = tn/td rational; u values Fractions -> common-denominator integers. +# All kernel entries are scaled by td^n and the u common denominator: +# one global scale, so masses are exact ints and A's denominator exact. +# ====================================================================== + +def my_certify_sparse(u: dict[int, Fraction], tn: int, td: int, n: int): + atoms = sorted(u) + m = len(atoms) + cd = 1 + for a in atoms: + cd = cd * u[a].denominator // math.gcd(cd, u[a].denominator) + ui = [int(u[a] * cd) for a in atoms] + tp = [tn ** k * td ** (n - k) for k in range(n + 1)] + kern = [[ui[x] * ui[y] * tp[popc(atoms[x] & atoms[y])] + for y in range(m)] for x in range(m)] + # exact marginals vs RECORD + rowsum = [sum(kern[x]) for x in range(m)] + Z = sum(rowsum) + margs = [Fraction(sum(rowsum[x] for x in range(m) + if atoms[x] >> j & 1), Z) for j in range(n)] + mm = max(margs) + num_lo = num_hi = Fraction(0) + den = 0 + memo: dict[tuple[int, int], tuple[Fraction, Fraction]] = {} + for i in range(1, n): # i = n excluded + mask = (1 << (i - 1)) - 1 + bit = 1 << (i - 1) + byp: dict[int, list[int]] = {} + for x in range(m): + byp.setdefault(atoms[x] & mask, []).append(x) + act = [p for p, xs in byp.items() + if any(atoms[x] & bit for x in xs) + and any(not atoms[x] & bit for x in xs)] + for pa in act: + for pb in act: + c = [0, 0, 0, 0] + for x in byp[pa]: + ax = 2 if atoms[x] & bit else 0 + kx = kern[x] + for y in byp[pb]: + c[ax + (1 if atoms[y] & bit else 0)] += kx[y] + assert min(c) > 0, f"degenerate cell in active pair i={i}" + mass = c[0] + c[1] + c[2] + c[3] + # log2(OR/t) = log2( (c11 c00 td) / (c01 c10 tn) ) + qn, qd = c[3] * c[0] * td, c[2] * c[1] * tn + g = math.gcd(qn, qd) + key = (qn // g, qd // g) + enc = memo.get(key) + if enc is None: + enc = log2_bracket(Fraction(key[0], key[1])) + memo[key] = enc + num_lo += mass * enc[0] + num_hi += mass * enc[1] + den += mass + assert den > 0 + return num_lo / den, num_hi / den, mm + + +# ====================================================================== +# my orbit evaluator for tail-exchangeable measures (own re-derivation). +# +# Atom state s = (a1, a2, k), k = |A cap {3..n}|, r = n - 2 tail coords, +# potential g by state. For coordinate i >= 3 (response = tail coord i): +# q = i - 3 prefix tail coords, f = n - i future tail coords. History +# class of a prefix pair: (oa, ob) in {0,1}^2 x {0,1}^2 (special coords), +# (ja, jb) prefix tail counts, j = |prefix tails of a cap of b|. The +# 2x2 table of (a, b) is +# cell(al, be) = t^{oa.ob + j} * t^{al.be} * S[ja+al][jb+be], +# S[x][y] = sum_{s,l} g(oa, x+s) PK_f[s][l] g(ob, y+l), +# PK_f[s][l] = C(f,s) sum_w C(s,w) C(f-s,l-w) t^w +# (= sum over ordered pairs of future sets, |XA| = s, |XB| = l, of +# t^{|XA cap XB|}), +# so OR = t * S[ja][jb] S[ja+1][jb+1] / (S[ja][jb+1] S[ja+1][jb]) is +# j-independent, and the class mass sums the j-factor into +# JW[ja][jb] = sum_j Ncl(q, ja, jb, j) t^j, +# Ncl = C(q,j) C(q-j, ja-j) C(q-ja, jb-j) (ordered prefix pairs). +# i = 1, 2 are handled directly over the <= 4 special-coordinate +# histories with the full r-tail pair kernel. Membership in N_i is +# structural: pattern (o, c) is active iff both continuations c, c+1 +# reach supp(g), i.e. exists s <= f with g(o, c + al + s) > 0. +# +# Exact mode: g values as integers over a common denominator 2^G (or cd), +# t-powers as tn^w td^(f-w): every S entry, JW and mass is an exact +# INTEGER on a per-i scale cd^2 * td^(n-3+extras); the extras are aligned +# across i so the aggregate denominator is one exact integer. +# ====================================================================== + +def _pairkernel_int(f: int, tn: int, td: int): + """PK_f[s][l] scaled by td^f (exact ints).""" + tp = [tn ** w * td ** (f - w) for w in range(f + 1)] + PK = [[0] * (f + 1) for _ in range(f + 1)] + for s in range(f + 1): + cfs = math.comb(f, s) + for l in range(f + 1): + acc = 0 + for w in range(max(0, s + l - f), min(s, l) + 1): + acc += math.comb(s, w) * math.comb(f - s, l - w) * tp[w] + PK[s][l] = cfs * acc + return PK + + +def _jw_int(q: int, tn: int, td: int): + """JW[ja][jb] scaled by td^q (exact ints).""" + tp = [tn ** j * td ** (q - j) for j in range(q + 1)] + JW = [[0] * (q + 1) for _ in range(q + 1)] + for ja in range(q + 1): + for jb in range(q + 1): + acc = 0 + for j in range(max(0, ja + jb - q), min(ja, jb) + 1): + acc += (math.comb(q, j) * math.comb(q - j, ja - j) + * math.comb(q - ja, jb - j)) * tp[j] + JW[ja][jb] = acc + return JW + + +ORB4 = ((0, 0), (0, 1), (1, 0), (1, 1)) + + +def my_orbit_rows_int(n: int, gi: dict, tn: int, td: int): + """Exact integer orbit rows. gi: state -> int (common denominator + folded out; it cancels in OR and is a common mass factor). Returns + per i in 1..n a list of (mass_int, or_num, or_den) with mass_int on + ONE common scale across i and or_num/or_den = OR/t exactly.""" + r = n - 2 + + def g(o, c): + return gi.get((o[0], o[1], c), 0) + + out = [] + # ---- i = 1, 2: special coords. Q[(oa, ob)][x][y] over continuation + # counts is the full-tail pair kernel contraction. + PKr = _pairkernel_int(r, tn, td) + + def quad(oa, ob): + acc = 0 + for s in range(r + 1): + ga = g(oa, s) + if ga: + row = PKr[s] + acc += ga * sum(row[l] * gb for l, gb in + ((l, g(ob, l)) for l in range(r + 1)) if gb) + return acc + + QQ = {(oa, ob): quad(oa, ob) for oa in ORB4 for ob in ORB4} + # i = 1: response coord 1; history (empty, empty) if both values reach + reach_c1 = {v: any(g((v, a2), k) for a2 in (0, 1) for k in range(r + 1)) + for v in (0, 1)} + rows1 = [] + if reach_c1[0] and reach_c1[1]: + # cell(al, be) = t^{al be} sum_{a2 b2} t^{a2 b2} QQ[(al,a2),(be,b2)] + # scale: td^r (QQ) * td^2 (two explicit t powers) -> mult by td^2 + cell = {} + for al in (0, 1): + for be in (0, 1): + acc = 0 + for a2 in (0, 1): + for b2 in (0, 1): + acc += (tn if a2 * b2 else td) * QQ[((al, a2), + (be, b2))] + cell[(al, be)] = (tn if al * be else td) * acc + mass = cell[(0, 0)] + cell[(0, 1)] + cell[(1, 0)] + cell[(1, 1)] + rows1.append((mass, cell[(1, 1)] * cell[(0, 0)] * td, + cell[(1, 0)] * cell[(0, 1)] * tn)) + out.append(rows1) + # i = 2: response coord 2; prefixes = coord-1 values + reach_12 = {(v, w): any(g((v, w), k) for k in range(r + 1)) + for v in (0, 1) for w in (0, 1)} + act2 = [v for v in (0, 1) if reach_12[(v, 0)] and reach_12[(v, 1)]] + rows2 = [] + for a in act2: + for b in act2: + cell = {} + for al in (0, 1): + for be in (0, 1): + cell[(al, be)] = ((tn if a * b else td) + * (tn if al * be else td) + * QQ[((a, al), (b, be))]) + mass = (cell[(0, 0)] + cell[(0, 1)] + cell[(1, 0)] + + cell[(1, 1)]) + rows2.append((mass, cell[(1, 1)] * cell[(0, 0)] * td, + cell[(1, 0)] * cell[(0, 1)] * tn)) + out.append(rows2) + # scale so far: i=1: td^{r+2}; i=2: td^{r+2}. For i >= 3 the natural + # scale is td^{q + f + 2} = td^{n - 3} (JW) * (cells: td^{f} * td^2) + # ... one td short of the i<=2 scale (r + 2 = n). Track exponents + # and align at the end. + scale12 = r + 2 # td exponent used above + per_i_scale = [scale12, scale12] + for i in range(3, n + 1): + q, f = i - 3, n - i + PK = _pairkernel_int(f, tn, td) + JW = _jw_int(q, tn, td) + rows = [] + # T[o][x][s] = g(o, x+s); S per (oa, ob) + for oa in ORB4: + ga = [[g(oa, x + s) if x + s <= r else 0 + for s in range(f + 1)] for x in range(q + 2)] + if not any(any(row) for row in ga): + continue + # A[x][l] = sum_s ga[x][s] PK[s][l] + Amat = [[sum(ga[x][s] * PK[s][l] for s in range(f + 1) + if ga[x][s]) for l in range(f + 1)] + for x in range(q + 2)] + reach_a = [any(ga[c]) for c in range(q + 2)] + for ob in ORB4: + gb = [[g(ob, y + l) if y + l <= r else 0 + for l in range(f + 1)] for y in range(q + 2)] + if not any(any(row) for row in gb): + continue + reach_b = [any(gb[c]) for c in range(q + 2)] + S = [[sum(Amat[x][l] * gb[y][l] for l in range(f + 1) + if gb[y][l]) for y in range(q + 2)] + for x in range(q + 2)] + e = oa[0] * ob[0] + oa[1] * ob[1] # in {0, 1, 2} + base = tn ** e * td ** (2 - e) + for ja in range(q + 1): + if not (reach_a[ja] and reach_a[ja + 1]): + continue + for jb in range(q + 1): + if not (reach_b[jb] and reach_b[jb + 1]): + continue + s00, s01 = S[ja][jb], S[ja][jb + 1] + s10, s11 = S[ja + 1][jb], S[ja + 1][jb + 1] + if not (s00 and s01 and s10 and s11): + continue + mass = base * JW[ja][jb] * ( + td * (s00 + s01 + s10) + tn * s11) + rows.append((mass, s00 * s11, s01 * s10)) + out.append(rows) + per_i_scale.append(q + f + 3) # td^{2-e}*tn^e (2) + JW q + PK f + 1 + # align all masses to the max scale + mx = max(per_i_scale) + aligned = [] + for sc, rows in zip(per_i_scale, out): + mult = td ** (mx - sc) + aligned.append([(m * mult, on, od) for (m, on, od) in rows] + if mult != 1 else rows) + return aligned + + +def my_orbit_agg_float(n: int, gf: dict, lam: float): + """Float aggregate via the integer-mode row builder run on floats is + unsafe (overflow); instead run the same formulas in float directly.""" + t = 2.0 ** lam + rows_by_i = my_orbit_rows_generic(n, gf, t) + num = den = 0.0 + worst = worst_i = None + for i, rows in enumerate(rows_by_i, start=1): + if i >= n or not rows: + continue + W = sum(m for m, _o in rows) + mi = sum(m * math.log2(o) for m, o in rows) / W + num += W * (mi - lam) + den += W + if worst is None or mi - lam < worst: + worst, worst_i = mi - lam, i + return (num / den if den else None), worst, worst_i + + +def my_orbit_rows_generic(n: int, g: dict, t: float): + """Float rows (mass, OR). Same hand-derived formulas, float t.""" + r = n - 2 + + def gv(o, c): + return g.get((o[0], o[1], c), 0.0) + + def pk(f): + return [[math.comb(f, s) * sum( + math.comb(s, w) * math.comb(f - s, l - w) * t ** w + for w in range(max(0, s + l - f), min(s, l) + 1)) + for l in range(f + 1)] for s in range(f + 1)] + + out = [] + PKr = pk(r) + + def quad(oa, ob): + acc = 0.0 + for s in range(r + 1): + ga = gv(oa, s) + if ga: + acc += ga * sum(PKr[s][l] * gv(ob, l) + for l in range(r + 1) if gv(ob, l)) + return acc + + QQ = {(oa, ob): quad(oa, ob) for oa in ORB4 for ob in ORB4} + reach_c1 = {v: any(gv((v, a2), k) for a2 in (0, 1) + for k in range(r + 1)) for v in (0, 1)} + rows1 = [] + if reach_c1[0] and reach_c1[1]: + cell = {} + for al in (0, 1): + for be in (0, 1): + cell[(al, be)] = t ** (al * be) * sum( + t ** (a2 * b2) * QQ[((al, a2), (be, b2))] + for a2 in (0, 1) for b2 in (0, 1)) + mass = sum(cell.values()) + rows1.append((mass, cell[(1, 1)] * cell[(0, 0)] + / (cell[(1, 0)] * cell[(0, 1)]))) + out.append(rows1) + reach_12 = {(v, w): any(gv((v, w), k) for k in range(r + 1)) + for v in (0, 1) for w in (0, 1)} + act2 = [v for v in (0, 1) if reach_12[(v, 0)] and reach_12[(v, 1)]] + rows2 = [] + for a in act2: + for b in act2: + cell = {(al, be): t ** (a * b + al * be) * QQ[((a, al), (b, be))] + for al in (0, 1) for be in (0, 1)} + rows2.append((sum(cell.values()), + cell[(1, 1)] * cell[(0, 0)] + / (cell[(1, 0)] * cell[(0, 1)]))) + out.append(rows2) + for i in range(3, n + 1): + q, f = i - 3, n - i + PK = pk(f) + tpj = [t ** j for j in range(q + 1)] + JW = [[sum(math.comb(q, j) * math.comb(q - j, ja - j) + * math.comb(q - ja, jb - j) * tpj[j] + for j in range(max(0, ja + jb - q), min(ja, jb) + 1)) + for jb in range(q + 1)] for ja in range(q + 1)] + rows = [] + for oa in ORB4: + ga = [[gv(oa, x + s) if x + s <= r else 0.0 + for s in range(f + 1)] for x in range(q + 2)] + if not any(any(row) for row in ga): + continue + Amat = [[sum(ga[x][s] * PK[s][l] for s in range(f + 1) + if ga[x][s]) for l in range(f + 1)] + for x in range(q + 2)] + reach_a = [any(ga[c]) for c in range(q + 2)] + for ob in ORB4: + gb = [[gv(ob, y + l) if y + l <= r else 0.0 + for l in range(f + 1)] for y in range(q + 2)] + if not any(any(row) for row in gb): + continue + reach_b = [any(gb[c]) for c in range(q + 2)] + S = [[sum(Amat[x][l] * gb[y][l] for l in range(f + 1) + if gb[y][l]) for y in range(q + 2)] + for x in range(q + 2)] + base = t ** (oa[0] * ob[0] + oa[1] * ob[1]) + for ja in range(q + 1): + if not (reach_a[ja] and reach_a[ja + 1]): + continue + for jb in range(q + 1): + if not (reach_b[jb] and reach_b[jb + 1]): + continue + s00, s01 = S[ja][jb], S[ja][jb + 1] + s10, s11 = S[ja + 1][jb], S[ja + 1][jb + 1] + if 0.0 in (s00, s01, s10, s11): + continue + rows.append((base * JW[ja][jb] + * (s00 + s01 + s10 + t * s11), + t * s00 * s11 / (s01 * s10))) + out.append(rows) + return out + + +def my_orbit_marginals(n: int, g: dict, t: float): + r = n - 2 + states = [s for s in g if g[s] > 0] + + def wt(k, l): + return sum(math.comb(k, j) * math.comb(r - k, l - j) * t ** j + for j in range(max(0, k + l - r), min(k, l) + 1)) + + z = m1 = m2 = mt = 0.0 + for s in states: + row = sum(g[s2] * t ** (s[0] * s2[0] + s[1] * s2[1]) + * wt(s[2], s2[2]) for s2 in states) + w = math.comb(r, s[2]) * g[s] * row + z += w + m1 += w * s[0] + m2 += w * s[1] + mt += w * s[2] / r + return m1 / z, m2 / z, mt / z + + +def my_orbit_sinkhorn(n: int, tgt: dict, lam: float, iters: int = 40000, + tol: float = 1e-11): + """My scaling iteration: g <- g * (tgt*z/cur)^0.4 (different damping + exponent from 014's 0.5).""" + r = n - 2 + t = 2.0 ** lam + states = [s for s in sorted(tgt) if tgt[s] > 0] + + def wt(k, l): + return sum(math.comb(k, j) * math.comb(r - k, l - j) * t ** j + for j in range(max(0, k + l - r), min(k, l) + 1)) + + M = [[t ** (a[0] * b[0] + a[1] * b[1]) * wt(a[2], b[2]) + for b in states] for a in states] + cnt = [math.comb(r, s[2]) for s in states] + tv = [tgt[s] for s in states] + N = len(states) + g = [tv[a] ** 0.5 for a in range(N)] + resid = float("inf") + for _ in range(iters): + cur = [g[a] * sum(g[b] * M[a][b] for b in range(N)) + for a in range(N)] + z = sum(cnt[a] * cur[a] for a in range(N)) + resid = max(abs(cur[a] / (z * tv[a]) - 1.0) for a in range(N)) + if resid < tol: + break + g = [g[a] * (tv[a] * z / cur[a]) ** 0.4 for a in range(N)] + return {s: g[a] for a, s in enumerate(states)}, resid + + +def my_certify_orbit_rows(n: int, gF: dict, tn: int, td: int): + """Exact aggregate enclosure from integer orbit rows + log2_bracket. + gF: state -> Fraction (dyadic). Returns (lo, hi, den>0).""" + cd = 1 + for v in gF.values(): + cd = cd * v.denominator // math.gcd(cd, v.denominator) + gi = {s: int(v * cd) for s, v in gF.items() if v != 0} + rows_by_i = my_orbit_rows_int(n, gi, tn, td) + num_lo = num_hi = Fraction(0) + den = 0 + memo: dict[tuple[int, int], tuple[Fraction, Fraction]] = {} + for i, rows in enumerate(rows_by_i, start=1): + if i >= n: + continue + for (mass, on, od) in rows: + # OR/t = (on*td.. already exact: on/od * ... ) -- rows carry + # or_num/or_den = OR/t exactly (see builder) + g2 = math.gcd(on, od) + key = (on // g2, od // g2) + enc = memo.get(key) + if enc is None: + enc = log2_bracket(Fraction(key[0], key[1])) + memo[key] = enc + num_lo += mass * enc[0] + num_hi += mass * enc[1] + den += mass + assert den > 0 + return num_lo / den, num_hi / den + + +def my_orbit_marginals_exact(n: int, gF: dict, tn: int, td: int): + r = n - 2 + states = [s for s in gF if gF[s] > 0] + cd = 1 + for v in gF.values(): + cd = cd * v.denominator // math.gcd(cd, v.denominator) + gi = {s: int(gF[s] * cd) for s in states} + + def wt(k, l): + return sum(math.comb(k, j) * math.comb(r - k, l - j) + * tn ** j * td ** (r - j) + for j in range(max(0, k + l - r), min(k, l) + 1)) + + z = m1 = m2 = Fraction(0) + mt = Fraction(0) + for a in states: + row = sum(gi[b] * tn ** (a[0] * b[0] + a[1] * b[1]) + * td ** (2 - a[0] * b[0] - a[1] * b[1]) + * wt(a[2], b[2]) for b in states) + w = math.comb(r, a[2]) * gi[a] * row + z += w + m1 += w * a[0] + m2 += w * a[1] + mt += Fraction(w * a[2], r) + return m1 / z, m2 / z, mt / z + + +# ====================================================================== +# family targets (own code, same definitions as the record states them) +# ====================================================================== + +def orbit_states(n: int): + return [(a1, a2, k) for a1 in (0, 1) for a2 in (0, 1) + for k in range(n - 1)] + + +def st_bern(n: int, p: float) -> dict: + return {(a1, a2, k): p ** (a1 + a2 + k) * (1 - p) ** (n - a1 - a2 - k) + for a1 in (0, 1) for a2 in (0, 1) for k in range(n - 1)} + + +def st_mixture(n: int, comps) -> dict: + out = {} + for w, p in comps: + for s, v in st_bern(n, p).items(): + out[s] = out.get(s, 0.0) + w * v + return {s: v for s, v in out.items() if v > 0} + + +def st_slice(n: int, w0: int) -> dict: + tot = math.comb(n, w0) + return {(a1, a2, k): 1.0 / tot for a1 in (0, 1) for a2 in (0, 1) + for k in range(n - 1) if a1 + a2 + k == w0} + + +def st_crashmix(n: int, p: float, d: float) -> dict: + r = n - 2 + out = {s: (1 - d) * v for s, v in st_bern(n, p).items()} + for s, w in (((0, 1, 0), 0.33), ((0, 0, r), 0.33), ((1, 1, r), 0.04), + ((1, 0, 0), 0.30)): + out[s] = out.get(s, 0.0) + d * w + return out + + +def st_mix_dir(n: int, p: float, nu: str, d: float) -> dict: + out = {s: (1 - d) * v for s, v in st_bern(n, p).items()} + r = n - 2 + if nu == "crash": + for s, w in (((0, 1, 0), 0.33), ((0, 0, r), 0.33), + ((1, 1, r), 0.04), ((1, 0, 0), 0.30)): + out[s] = out.get(s, 0.0) + d * w + elif nu == "slice": + for s, v in st_slice(n, int(RECORD_F * n)).items(): + out[s] = out.get(s, 0.0) + d * v + elif nu == "slice_matched": + for s, v in st_slice(n, round(p * n)).items(): + out[s] = out.get(s, 0.0) + d * v + else: + raise ValueError(nu) + return out + + +def st_d0bern(n: int, pl: float, ph: float) -> dict: + out = {s: (1 - pl) * v for s, v in st_bern(n, ph).items()} + out[(0, 0, 0)] = out.get((0, 0, 0), 0.0) + pl + return out + + +def orbit_to_full(n: int, g: dict) -> dict[int, float]: + """Expand an orbit potential to an atom potential on 2^[n].""" + u = {} + for A in range(1 << n): + s = (A & 1, (A >> 1) & 1, popc(A >> 2)) + v = g.get(s, 0.0) + if v > 0: + u[A] = v + return u + + +# 007's 10-atom witness weights (the measure definition, data): +W_LIGHT = 0.004792968160237178 +W_ATOMS = (3.802705744653898, 8.094727225702503, W_LIGHT, + 0.2526660836353666, 27.28641199464732, 12.529780963546393, + 0.3609234435898425) + + +def my_witness_at_n(n: int, wdil: float) -> dict[int, float]: + """Own implementation of the record's witness-at-n layout: marker + coord 1, dilution singletons 2..n-3, response n-2, futures n-1, n.""" + bm, br = 1, 1 << (n - 3) + f1, f2 = 1 << (n - 2), 1 << (n - 1) + a = W_ATOMS + u = {bm | f1: a[0], bm | f2: a[1], bm | br | f2: a[2], 0: a[3], + f1: a[4], f2: a[5], br | f1: a[6]} + for j in range(2, n - 2): + u[1 << (j - 1)] = wdil + return u + + +# ====================================================================== +# part K -- attack the enclosure kit + combinatorics +# ====================================================================== + +def part_k() -> dict: + print("== part K: enclosure-kit attack ==") + import uc_or_agg_probe as probe + out = {} + import random + rng = random.Random(20260804) + + # K1: their log2_enclosure vs my log2_bracket on adversarial rationals + cases = [] + for _ in range(400): + num = rng.randrange(1, 1 << rng.randrange(4, 200)) + den = rng.randrange(1, 1 << rng.randrange(4, 200)) + cases.append(Fraction(num, den)) + # near powers of two (both sides), tiny, huge, exact powers + for k in (-100, -7, -1, 0, 1, 7, 100): + for eps_num, eps_den in ((0, 1), (1, 10 ** 25), (-1, 10 ** 25), + (1, 3), (-1, 3)): + q = Fraction(2) ** k * (1 + Fraction(eps_num, eps_den)) + if q > 0: + cases.append(q) + cases.append(Fraction(181, 16)) + cases.append(Fraction(271, 256)) + bad = [] + maxw_theirs = maxw_mine = 0.0 + for q in cases: + tlo, thi = probe.log2_enclosure(q) + mlo, mhi = log2_bracket(q) + # both must contain the true value: cross-containment demands + # overlap, and each interval must contain the other's midpoint + # up to widths; the decisive check: tlo <= mhi and mlo <= thi + if not (tlo <= mhi and mlo <= thi): + bad.append((str(q), float(tlo), float(thi), float(mlo), + float(mhi))) + maxw_theirs = max(maxw_theirs, float(thi - tlo)) + maxw_mine = max(maxw_mine, float(mhi - mlo)) + # sanity anchor: floor consistency with float log2 where safe + fl = math.log2(float(q)) if 1e-300 < float(q) < 1e300 else None + if fl is not None and (fl < float(mlo) - 1e-6 + or fl > float(mhi) + 1e-6): + bad.append(("float-anchor", str(q), fl, float(mlo), float(mhi))) + print(f"[K1] {len(cases)} rationals: disjoint-enclosure failures = " + f"{len(bad)}; max widths theirs {maxw_theirs:.1e} / mine " + f"{maxw_mine:.1e}") + out["k1_cases"] = len(cases) + out["k1_failures"] = bad + out["k1_pass"] = not bad + + # K2: their _dround directional soundness, random Fractions inc. negatives + dbad = 0 + for _ in range(2000): + x = Fraction(rng.randrange(-10 ** 30, 10 ** 30), + rng.randrange(1, 10 ** 30)) + lo = probe._dround(x, False) + hi = probe._dround(x, True) + if not (lo <= x <= hi): + dbad += 1 + print(f"[K2] _dround directionality: {dbad} failures / 2000") + out["k2_failures"] = dbad + out["k2_pass"] = dbad == 0 + + # K3: their _ln_1_to_2_fp directed bounds -- sanity anchor against my + # bracket via ln s = log2(s) * ln 2 (float ln 2; the RIGOROUS + # end-to-end test of their composed log2_enclosure is K1) + lbad = 0 + for _ in range(300): + s = 1 + Fraction(rng.randrange(0, 10 ** 18), 10 ** 18) + if s >= 2 or s == 1: + continue + lo = probe._ln_1_to_2_fp(s, False, 12) + hi = probe._ln_1_to_2_fp(s, True, 12) + l2lo, l2hi = log2_bracket(s) + mid = (float(l2lo) + float(l2hi)) / 2 * math.log(2.0) + if not (float(lo) - 1e-12 <= mid <= float(hi) + 1e-12): + lbad += 1 + print(f"[K3] _ln_1_to_2_fp anchor vs my bracket*ln2 (slack 1e-12): " + f"{lbad} failures / 300") + out["k3_failures"] = lbad + out["k3_pass"] = lbad == 0 + + # K4: brute-force the orbit combinatorics my derivation predicts + # Ncl(q,ja,jb,j) counts ordered prefix pairs; PK_f pair kernel + ncl_bad = pk_bad = 0 + for q in range(0, 7): + for ja in range(q + 1): + for jb in range(q + 1): + cnt = {} + for a in range(1 << q): + if popc(a) != ja: + continue + for b in range(1 << q): + if popc(b) != jb: + continue + cnt[popc(a & b)] = cnt.get(popc(a & b), 0) + 1 + for j in range(q + 1): + pred = (math.comb(q, j) * math.comb(q - j, ja - j) + * math.comb(q - ja, jb - j) + if j <= min(ja, jb) and ja + jb - j <= q else 0) + if cnt.get(j, 0) != pred: + ncl_bad += 1 + t_test = Fraction(7, 3) + for f in range(0, 6): + PK = _pairkernel_int(f, 7, 3) + for s in range(f + 1): + for l in range(f + 1): + acc = Fraction(0) + for XA in range(1 << f): + if popc(XA) != s: + continue + for XB in range(1 << f): + if popc(XB) != l: + continue + acc += t_test ** popc(XA & XB) + if acc * 3 ** f != PK[s][l]: + pk_bad += 1 + print(f"[K4] combinatorics brute force: Ncl failures {ncl_bad}, " + f"PK failures {pk_bad}") + out["k4_pass"] = ncl_bad == 0 and pk_bad == 0 + + save("oragg_sk2_partK.json", out) + return out + + +# ====================================================================== +# part X -- formula cross-checks (n = 8, 9; j-independence; witness) +# ====================================================================== + +def part_x() -> dict: + print("== part X: formula cross-checks ==") + import uc_or_agg_probe as probe + out = {"checks": []} + + # X1: my direct evaluator on the witness at n = 7 (007 numbers) + u7 = my_witness_at_n(7, 20.0) + agg, per_i, mm, worst, wi = my_eval_float(u7, 3.5, 7) + print(f"[X1] my direct on witness n=7 lam=3.5: M_5-lam = " + f"{per_i[4][0] - 3.5:+.6f} (007: -0.122033), agg = {agg:+.6f} " + f"(014: +1.844754), marg = {mm:.4f} (0.3178)") + ok = (abs(per_i[4][0] - 3.5 + 0.122033) < 5e-6 + and abs(agg - 1.844754) < 5e-5 and abs(mm - 0.317769) < 5e-6) + out["checks"].append({"name": "witness_n7", "pass": bool(ok), + "agg": agg, "M5_minus_lam": per_i[4][0] - 3.5}) + + # X2: my orbit vs my direct vs 014's orbit at n = 8, 9 + fams = [("crashmix_383_05", lambda n: st_crashmix(n, 0.383, 0.05)), + ("sawin_geo_a", lambda n: st_mixture( + n, [(0.80, 0.28), (0.12, 0.55), (0.06, 0.75), + (0.02, 0.92)])), + ("mixdir_slice_p.30_d.05", lambda n: st_mix_dir( + n, 0.30, "slice", 0.05)), + ("d0bern_30_51", lambda n: st_d0bern(n, 0.30, 0.51))] + worst_dd = 0.0 + for n in (8, 9): + for lam in (0.5, 1.0): + for name, mk in fams: + tgt = mk(n) + g, res = my_orbit_sinkhorn(n, tgt, lam) + agg_o, w_o, _ = my_orbit_agg_float(n, g, lam) + # direct on the SAME potential expanded to atoms + u_full = orbit_to_full(n, g) + agg_d, per_d, mm_d, w_d, _ = my_eval_float(u_full, lam, n) + # 014's orbit rows on the same g + rows_p = probe.orbit_rows(n, 2.0 ** lam, g) + agg_p, w_p, _wi, _pp = probe.orbit_agg(rows_p, lam, n) + dd = max(abs(agg_o - agg_d), abs(agg_o - agg_p)) + worst_dd = max(worst_dd, dd) + out["checks"].append( + {"name": f"x2_{name}_n{n}_lam{lam}", "agg_mine_orbit": + agg_o, "agg_mine_direct": agg_d, "agg_014_orbit": + agg_p, "maxdiff": dd, "resid": res}) + print(f"[X2] {name} n={n} lam={lam}: mine-orbit " + f"{agg_o:+.9f} mine-direct {agg_d:+.9f} 014-orbit " + f"{agg_p:+.9f} (maxdiff {dd:.2e})") + out["x2_maxdiff"] = worst_dd + out["x2_pass"] = worst_dd < 1e-8 + + # X3: explicit j-independence of class ORs at n = 9, i = 6: + # per-history ORs from my direct evaluator grouped by orbit class + n, lam = 9, 1.0 + g, _ = my_orbit_sinkhorn(n, st_crashmix(n, 0.383, 0.05), lam) + u = orbit_to_full(n, g) + t = 2.0 ** lam + atoms = sorted(u) + i = 6 + mask, bit = (1 << (i - 1)) - 1, 1 << (i - 1) + byp: dict[int, list[int]] = {} + for a in atoms: + byp.setdefault(a & mask, []).append(a) + act = [p for p, xs in byp.items() + if any(x & bit for x in xs) and any(not x & bit for x in xs)] + classes: dict[tuple, list[float]] = {} + for pa in act: + for pb in act: + c = [0.0] * 4 + for A in byp[pa]: + for B in byp[pb]: + c[(2 if A & bit else 0) + (1 if B & bit else 0)] += \ + u[A] * u[B] * t ** popc(A & B) + orr = c[3] * c[0] / (c[2] * c[1]) + key = (pa & 1, (pa >> 1) & 1, pb & 1, (pb >> 1) & 1, + popc(pa >> 2), popc(pb >> 2)) + classes.setdefault(key, []).append(orr) + spread = max((max(v) - min(v)) / max(v) for v in classes.values()) + print(f"[X3] j-independence at n=9 i=6: {len(classes)} classes, max " + f"rel spread of within-class ORs = {spread:.2e}") + out["x3_class_or_rel_spread"] = spread + out["x3_pass"] = spread < 1e-9 + + ok_all = all(c.get("pass", True) for c in out["checks"]) and \ + out["x2_pass"] and out["x3_pass"] + print(f"[X] {'ALL PASS' if ok_all else 'FAILURES PRESENT'}") + save("oragg_sk2_partX.json", out) + return out + + +# ====================================================================== +# part C -- independent re-certification +# ====================================================================== + +def part_c() -> dict: + print("== part C: independent re-certification ==") + import uc_or_agg_probe as probe + out = {"certs": []} + E = json.loads((DATA / "or_agg_probe_partE.json").read_text()) + theirs = {r["name"]: r for r in E["sparse"] + E["orbit"]} + + def report(name, lo, hi, mm, ref): + rec = {"name": name, "my_lo": float(lo), "my_hi": float(hi), + "my_width": float(hi - lo), + "my_sign": "positive" if lo > 0 else + ("NEGATIVE" if hi < 0 else "indeterminate"), + "max_marginal": float(mm), + "in_regime_exact": bool(mm < RECORD), + "their_lo": ref["agg_lo"], "their_hi": ref["agg_hi"]} + # decisive comparison: intervals must overlap (mine is proven + # sound with width ~1e-14, so non-overlap convicts theirs) + overlap = not (float(hi) < ref["agg_lo"] - 1e-15 + or float(lo) > ref["agg_hi"] + 1e-15) + rec["intervals_overlap"] = bool(overlap) + rec["sign_agrees"] = rec["my_sign"] == ref["certified_sign"] + out["certs"].append(rec) + print(f"[C] {name}: mine [{float(lo):+.12g}, {float(hi):+.12g}] " + f"(w {float(hi - lo):.1e}) vs theirs [{ref['agg_lo']:+.12g}," + f" {ref['agg_hi']:+.12g}] -> overlap={overlap} " + f"sign={rec['my_sign']} marg={float(mm):.6f}") + return rec + + # C1: witness n=10 wdil=16 at t = 181/16 (sparse; my own builder) + t0 = time.time() + u = {k: Fraction(v) for k, v in my_witness_at_n(10, 16.0).items()} + lo, hi, mm = my_certify_sparse(u, 181, 16, 10) + report("witness_n10_wdil16", lo, hi, mm, theirs["witness_n10_wdil16"]) + + # C2: the fully rational t = 4 point (n = 20, wdil = 20, lambda = 2) + u = {k: Fraction(v) for k, v in my_witness_at_n(20, 20.0).items()} + lo, hi, mm = my_certify_sparse(u, 4, 1, 20) + report("witness_n20_t4_lam2exact", lo, hi, mm, + theirs["witness_n20_t4_lam2exact"]) + + # C3: witness n=32 at t = 181/16 + u = {k: Fraction(v) for k, v in my_witness_at_n(32, 20.0).items()} + lo, hi, mm = my_certify_sparse(u, 181, 16, 32) + report("witness_n32_wdil20", lo, hi, mm, theirs["witness_n32_wdil20"]) + print(f" (sparse certs {time.time() - t0:.0f}s)") + + # C4: crashmix_383_05 n=32 at t = 271/256 -- SAME mu-tilde as 014: + # their float Sinkhorn + their 36-bit quantization (input spec), + # my exact rows + my log2 from there on. + t0 = time.time() + n = 32 + lamf = math.log2(271 / 256) + g, resid = probe.orbit_sinkhorn(n, probe.st_crashmix(n, 0.383, 0.05), + lamf) + gF = {s: probe._quant(v, 36) for s, v in g.items()} + lo, hi = my_certify_orbit_rows(n, gF, 271, 256) + m1, m2, mt = my_orbit_marginals_exact(n, gF, 271, 256) + report("crashmix_383_05_n32", lo, hi, max(m1, m2, mt), + theirs["crashmix_383_05_n32"]) + print(f" ({time.time() - t0:.0f}s, their-resid {resid:.1e})") + + # C5: the softest point, mixdir_slice p=0.30 d=0.05 n=32, same mu-tilde + t0 = time.time() + tgt = probe.st_mix_dir(32, 0.30, "slice", 0.05) + g, resid = probe.orbit_sinkhorn(32, tgt, lamf) + gF = {s: probe._quant(v, 36) for s, v in g.items()} + lo, hi = my_certify_orbit_rows(32, gF, 271, 256) + m1, m2, mt = my_orbit_marginals_exact(32, gF, 271, 256) + report("mixdir_slice_p0.3_d0.05_n32", lo, hi, max(m1, m2, mt), + theirs["mixdir_slice_p0.3_d0.05_n32"]) + print(f" ({time.time() - t0:.0f}s)") + + # C6: fresh certificate at the softest point from MY OWN pipeline + # (my Sinkhorn damping, my 40-bit quantization): an independent + # mu-tilde' near the same family point -- new evidence, not a rerun. + t0 = time.time() + tgt = st_mix_dir(32, 0.30, "slice", 0.05) + g, resid = my_orbit_sinkhorn(32, tgt, lamf) + + def quant40(v: float) -> Fraction: + mant, e = math.frexp(v) + return Fraction(round(mant * (1 << 40)), 1 << 40) * Fraction(2) ** e + + gF2 = {s: quant40(v) for s, v in g.items()} + lo, hi = my_certify_orbit_rows(32, gF2, 271, 256) + m1, m2, mt = my_orbit_marginals_exact(32, gF2, 271, 256) + mm = max(m1, m2, mt) + rec = {"name": "mixdir_slice_p0.3_d0.05_n32_MY_mu", "my_lo": float(lo), + "my_hi": float(hi), "my_sign": "positive" if lo > 0 else + ("NEGATIVE" if hi < 0 else "indeterminate"), + "max_marginal": float(mm), "in_regime_exact": bool(mm < RECORD), + "sinkhorn_resid": resid} + out["certs"].append(rec) + print(f"[C] MY OWN mu-tilde' at softest point: [{float(lo):+.9e}, " + f"{float(hi):+.9e}] -> {rec['my_sign']}, marg {float(mm):.6f}, " + f"resid {resid:.1e} ({time.time() - t0:.0f}s)") + + save("oragg_sk2_partC.json", out) + return out + + +# ====================================================================== +# part T -- trend attack: the dip, off-grid lambda, lambda-scaling +# ====================================================================== + +def run_orbit_float(n, tgt, lam): + g, resid = my_orbit_sinkhorn(n, tgt, lam) + agg, worst, wi = my_orbit_agg_float(n, g, lam) + m1, m2, mt = my_orbit_marginals(n, g, 2.0 ** lam) + return {"n": n, "lambda": lam, "aggregate": agg, "min_excess": worst, + "argmin_i": wi, "max_marginal": max(m1, m2, mt), + "in_regime": max(m1, m2, mt) < RECORD_F, "resid": resid} + + +def part_t() -> dict: + print("== part T: trend attack ==") + res = [] + + # T1: reproduce the dip row with my engine + for n in (32, 40, 48, 56, 64): + r = run_orbit_float(n, st_mix_dir(n, 0.30, "slice", 0.05), 0.084) + r["name"] = "dip_slice_p.30_d.05" + res.append(r) + print(f"[T1] n={n}: agg {r['aggregate']:+.3e} (014: 6.15e-7 / " + f"2.22e-7 / 1.41e-7 / 2.33e-7 / 4.31e-7) resid {r['resid']:.1e}") + + # T2: the dip point at off-grid lambda (below 014's grid floor and + # around it) and other d -- does anything cross zero? + for lam in (0.005, 0.01, 0.02, 0.04, 0.06, 0.12, 0.25, 0.5): + r = run_orbit_float(48, st_mix_dir(48, 0.30, "slice", 0.05), lam) + r["name"] = "dip_lambda_scan" + res.append(r) + print(f"[T2] n=48 lam={lam:g}: agg {r['aggregate']:+.3e} " + f"agg/lam {r['aggregate'] / lam:+.3e}") + for d in (0.01, 0.02, 0.1, 0.2): + r = run_orbit_float(48, st_mix_dir(48, 0.30, "slice", d), 0.084) + r["name"] = f"dip_d{d:g}" + res.append(r) + print(f"[T2] n=48 d={d:g}: agg {r['aggregate']:+.3e}") + # neighborhood in p and layer (lead-2 spot sample) + for p in (0.26, 0.28, 0.32, 0.34): + r = run_orbit_float(48, st_mix_dir(48, p, "slice", 0.05), 0.084) + r["name"] = f"dip_p{p:g}" + res.append(r) + print(f"[T2] n=48 p={p:g} (record layer): agg {r['aggregate']:+.3e}") + for p in (0.28, 0.30, 0.32): + r = run_orbit_float(48, st_mix_dir(48, p, "slice_matched", 0.05), + 0.084) + r["name"] = f"dipm_p{p:g}" + res.append(r) + print(f"[T2] n=48 p={p:g} (matched layer): agg {r['aggregate']:+.3e}") + + # T3: census-minimum family at off-grid lambda: crashmix_383_05, + # lambda scan incl. tiny and between 014's big grid points + for n in (10, 20, 32): + for lam in (0.005, 0.02, 0.05, 0.084, 0.15, 0.3, 0.7, 1.5, + 2.75, 4.4): + r = run_orbit_float(n, st_crashmix(n, 0.383, 0.05), lam) + r["name"] = "crashmix_383_05_scan" + res.append(r) + row = [x for x in res if x["name"] == "crashmix_383_05_scan" + and x["n"] == n] + print(f"[T3] crashmix n={n}: " + " ".join( + f"{x['lambda']:g}:{x['aggregate']:+.2e}" for x in row)) + + neg = [r for r in res if r["in_regime"] and r["aggregate"] is not None + and r["aggregate"] < -1e-12] + print(f"[T] negative in-regime aggregates: {len(neg)}") + for r in neg: + print(" NEG:", r) + save("oragg_sk2_partT.json", {"results": res, "n_negative": len(neg)}) + return {"results": res, "n_negative": len(neg)} + + +# ====================================================================== +# part V -- new adversaries 014 could not see +# ====================================================================== + +def part_v() -> dict: + print("== part V: new-adversary hunt ==") + res = [] + + # V1: witness-DIRECTION perturbation of a dense product bulk at + # n = 10: u = x^{|A|} (product potential, an equality point) + d * + # (witness atoms). Non-tail-symmetric direction; 014's part P had + # only 4 symmetric rays. x is set so the pure-bulk marginal is ~ p0. + n = 10 + + def bulk_x(p0: float, t: float) -> float: + # per-coordinate: p = (x + x^2 t)/(1 + 2x + x^2 t); solve for x + lo_, hi_ = 1e-6, 10.0 + for _ in range(200): + mid = (lo_ + hi_) / 2 + p = (mid + mid * mid * t) / (1 + 2 * mid + mid * mid * t) + if p < p0: + lo_ = mid + else: + hi_ = mid + return (lo_ + hi_) / 2 + + for lam in (0.084, 0.5, 2.0, 3.5): + t = 2.0 ** lam + for p0 in (0.30, 0.375): + x = bulk_x(p0, t) + base = {A: x ** popc(A) for A in range(1 << n)} + uwit = my_witness_at_n(n, 4.0) + scale = max(base.values()) + for d in (0.03, 0.3, 3.0): + u = dict(base) + for a, v in uwit.items(): + u[a] = u.get(a, 0.0) + d * scale * v / max(uwit.values()) + agg, per_i, mm, worst, wi = my_eval_float(u, lam, n) + rec = {"name": f"dense_bulk_wit_p{p0:g}_d{d:g}", + "n": n, "lambda": lam, "aggregate": agg, + "max_marginal": mm, "in_regime": mm < RECORD_F, + "min_excess": worst, "argmin_i": wi} + res.append(rec) + print(f"[V1] lam={lam:g} p0={p0:g} d={d:g}: agg " + f"{agg:+.6e} marg {mm:.4f} minexc {worst:+.4f}" + f"{' OUT' if not rec['in_regime'] else ''}") + + # V2: boundary-of-regime marginals: crashmix with p tuned so the max + # marginal sits just under 0.38271, at small/window lambda + for n in (20, 32): + for lam in (0.02, 0.084, 0.5): + p_lo, p_hi = 0.383, 0.3845 + for _ in range(20): + p = (p_lo + p_hi) / 2 + r = run_orbit_float(n, st_crashmix(n, p, 0.05), lam) + if r["max_marginal"] < 0.382695: + p_lo = p + else: + p_hi = p + r = run_orbit_float(n, st_crashmix(n, p_lo, 0.05), lam) + r["name"] = f"crashmix_boundary_p{p_lo:.6f}" + res.append(r) + print(f"[V2] n={n} lam={lam:g} p={p_lo:.6f}: agg " + f"{r['aggregate']:+.3e} marg {r['max_marginal']:.6f}") + # d0bern pushed to the boundary: (1-pl) ph ~ 0.38268 + for (pl, ph) in ((0.304145, 0.55), (0.48976, 0.75)): + for lam in (0.084, 0.5): + r = run_orbit_float(n, st_d0bern(n, pl, ph), lam) + r["name"] = f"d0bern_boundary_{pl:g}_{ph:g}" + res.append(r) + print(f"[V2] n={n} d0bern({pl:g},{ph:g}) lam={lam:g}: agg " + f"{r['aggregate']:+.3e} marg {r['max_marginal']:.6f}") + # sawin mixture scaled to the boundary + comps = [(0.80, 0.30317), (0.12, 0.59551), (0.06, 0.81206), + (0.02, 0.99612)] + for lam in (0.084, 0.5): + r = run_orbit_float(n, st_mixture(n, comps), lam) + r["name"] = "sawin_boundary" + res.append(r) + print(f"[V2] n={n} sawin-boundary lam={lam:g}: agg " + f"{r['aggregate']:+.3e} marg {r['max_marginal']:.6f}") + + # V3: tiny lambda at the tight census points (below 014's grid floor) + for n in (20, 32): + for lam in (0.002, 0.01, 0.04): + r = run_orbit_float(n, st_crashmix(n, 0.383, 0.05), lam) + r["name"] = "crashmix_tiny_lambda" + res.append(r) + r2 = run_orbit_float(n, st_mix_dir(n, 0.30, "slice", 0.05), lam) + r2["name"] = "slice_dir_tiny_lambda" + res.append(r2) + print(f"[V3] n={n} lam={lam:g}: crashmix {r['aggregate']:+.3e} " + f"slice-dir {r2['aggregate']:+.3e}") + + neg = [r for r in res if r["in_regime"] and r["aggregate"] is not None + and r["aggregate"] < -1e-12] + print(f"[V] in-regime negatives: {len(neg)}") + for r in neg: + print(" NEG:", r) + save("oragg_sk2_partV.json", {"results": res, "n_negative": len(neg)}) + return {"results": res, "n_negative": len(neg)} + + +def main() -> None: + parts = [p.upper() for p in sys.argv[1:] if p.upper() in "KXCTV"] \ + or list("KXCTV") + t0 = time.time() + if "K" in parts: + part_k() + if "X" in parts: + part_x() + if "C" in parts: + part_c() + if "T" in parts: + part_t() + if "V" in parts: + part_v() + print(f"done in {time.time() - t0:.0f}s") + + +if __name__ == "__main__": + main() diff --git a/problems/union-closed/prior-art.json b/problems/union-closed/prior-art.json index ebd532d..f550a36 100644 --- a/problems/union-closed/prior-art.json +++ b/problems/union-closed/prior-art.json @@ -439,6 +439,87 @@ "minor ratio 6.68" ], "note": "Decisive check: 007's confessed vulnerability (both engines share IEEE floats; its first witnesses were shared-underflow artifacts) is closed by re-verifying the witness in exact rational arithmetic with directed-rounding log2 enclosures. At t = 4 and t = 16 the refutation of M_i >= lambda is a fully rational statement with lambda = 2, 4 exactly. New strengthening beyond 007: the violation does not hinge on the conditioning-out convention (degenerate histories scored at exactly lambda still give M_5 - lambda = -0.067, certified). Nuance: exact certificates at lambda = 1, 2, 3 are for rationalized-refit measures within 9.2e-16 of the witness mu, not the identical rational mu. Theorem C's smoothness step is analogous to but not literally discharged by 008's P6-prime (P6-prime is d/d-mu at fixed lambda; Theorem C needs d/d-lambda at fixed mu)." + }, + { + "id": "014", + "file": "attempts/014-aggregated-or-probe-large-n.md", + "date": "2026-08-04", + "mode": "informed", + "status": "EVIDENCE", + "mechanism": [ + "coupling", + "overlap-tilt", + "sinkhorn", + "exact-rational-arithmetic", + "group-orbit-enumeration" + ], + "one_line": "The queue-mandated large-n probe of the restated gap: the i-aggregated OR control survives to n = 32 certified (15 exact-rational certificates, tightest +5.99e-7) and to n = 64 on float trends -- 0 violations in ~1780 instances spanning every known adversary genre plus aggregate-objective hill-climbs; product measures are its exact equality set, all perturbative departures are quadratically upward, the minimum margin GROWS with n on every structured family (no 012-style reversal), and the one decaying direction dips to +1.4e-7 at n ~ 48 then recovers.", + "range": "n in {10,14,20,24,28,32} across 14 orbit families x 10-point lambda grids in [0.079, 5] (840 instances) + 432 perturbative instances around the product boundary + trend runs to n = 64 + 426 sparse structured instances (witness genre re-laid-out per n, multi-gadget stacks, crash/mirror) + 82 climb endpoints at n <= 24; 15 certificates in exact rational arithmetic at tilts 181/16, 4 (lambda = 2 exactly), 271/256, enclosure widths <= 5.9e-12", + "refuted_by": null, + "gaps": [ + "aggregated-or-control", + "margin-modulated-or-control", + "mutual-information-tax", + "recipe-totality" + ], + "leak_terms": [ + "uc_or_agg_probe", + "or_agg_probe", + "aggregate survives n=32", + "equality boundary at products", + "product equality set", + "slice-direction dip", + "271/256", + "0.001201865", + "5.99e-7", + "dyadic compaction", + "witness-at-n", + "gadget stack" + ], + "note": "Probe only, per the queue's ordering (012's budget-growth reversal was the standing warning; it was looked for and not found in the aggregate, including on 012's own crash-mixture genre). Sign of the aggregate shown convention-robust under 013's surrogate-lambda bookkeeping (numerator unchanged, denominator grows). Certificates are for quantized refit measures at exact rational tilts (013 R2 scope). Proof effort on the aggregated control is now justified by the queue's own criterion; lead 1 (product-boundary Hessian) is the entry point." + }, + { + "id": "015", + "file": "attempts/015-skeptic-review-of-014.md", + "date": "2026-08-04", + "mode": "informed", + "status": "VERIFIED_WITH_CORRECTIONS", + "verifies": "014", + "mechanism": [ + "adversarial-verification", + "independent-reimplementation", + "exact-rational-arithmetic", + "coupling", + "overlap-tilt", + "sinkhorn", + "group-orbit-enumeration" + ], + "one_line": "Independent re-implementation (digit-extraction log2 bracket instead of the atanh kit, integer-scaled exact rows, own orbit derivation and Sinkhorn) confirms 014 in full: five certificates incl. the softest +5.99e-7 and the fully rational t = 4 point re-certified on identical exact inputs with no disagreement, kit sound by hand and by 437-case cross-bracketing, orbit formulas cross-checked at n = 8-9 with exactly zero within-class OR spread, the n ~ 48 dip reproduced digit-for-digit and shown robust off-grid, and ~116 new adversary instances (witness-direction off the product boundary, marginals pushed to 0.3827, lambda below the grid floor) all positive; corrections are reporting-level (part-P order range is d^1.39-d^2.06 on its own checkpoint, not d^1.85-d^2.06; 'products are the exact equality set' claims an unproven converse that is false at lambda = 0; stale docstring runtimes; 1820 not ~1780 instances).", + "corrections": [ + "part P 'fitted order d^1.85-d^2.06 in every direction' misreports its own checkpoint: 19 of 72 fitted cells lie below 1.85, minimum d^1.39 (n=32, full direction, lambda=0.5, in-regime); successive-halving orders climb back toward 2 as d -> 0, so the qualitative quadratic-upward reading survives but the quoted range is a grid-edge artifact, and the cubic term is large at lambda >= 0.5 (lead-1 consequence recorded)", + "'product measures are the exact equality set' proves only the forward inclusion; the converse is untested and false outright at lambda = 0 (A = 0 for every mu), needing the same lambda > 0 proviso 006 added to 005 Prop 2; no downstream use is affected", + "uc_or_agg_probe.py docstring runtimes (O ~ 40-80 min, E ~ 10-30 min) are stale; actual O ~ 146 s, E ~ 113 s (the record's own times are right)", + "the Outcome's '~1780 float instances' should be 1820 (840+432+40+426+82)" + ], + "refuted_by": null, + "gaps": [ + "aggregated-or-control", + "margin-modulated-or-control", + "mutual-information-tax", + "recipe-totality" + ], + "leak_terms": [ + "oragg_sk2", + "uc_or_agg_skeptic2", + "digit-extraction bracket", + "log2_bracket", + "d^1.39", + "equality-set converse", + "cubic remainder", + "witness-direction boundary attack", + "6.4e-7 first-order floor" + ], + "note": "Decisive check: 014's unreviewed enclosure kit is re-derived by hand and beaten against a from-scratch certifier using a different algorithm (binary digit extraction with directed integer shifts; exact integer masses so the aggregate denominator is exact); the five re-certifications agree with 014's enclosures at width 1.1e-14, including on the identical quantized refit measures. New floor observed: the lambda -> 0 first-order coefficient at the dip point is +6.4e-7 (n = 48), the smallest Theorem-C square seen anywhere; lead 2 should track that coefficient, not A at fixed lambda." } ] }