From 3627cfc786fe882624b55162e73c0fc5266cbc3a Mon Sep 17 00:00:00 2001 From: Claude Date: Tue, 4 Aug 2026 13:52:47 +0000 Subject: [PATCH 01/10] Checkpoint: in-flight cycle work (maxwell skeptic review 002, union-closed probe 014) Mid-cycle checkpoint per ops doctrine (container can restart at any moment): - maxwell-equilibria: attempt 002 scaffold, skeptic tools + perturbed-run configs; four census re-runs still in flight - union-closed: attempt 014 scaffold, aggregated-OR probe engine and per-genre checkpoint data; probe still in flight Records are drafts until their agents close out; nothing here is ledger fact yet. Co-Authored-By: Claude Fable 5 Claude-Session: https://claude.ai/code/session_01CupxvJ1ZaJMjoLk7pfpEmw --- .../attempts/002-skeptic-review-of-001.md | 371 + .../explore/skeptic_centroid_hessian.py | 110 + .../explore/skeptic_cert_checks.py | 346 + .../explore/skeptic_q4360_config.json | 10 + .../explore/skeptic_q4400_config.json | 10 + .../explore/skeptic_region_config.json | 11 + .../explore/skeptic_verifier_gap_demo.py | 156 + problems/maxwell-equilibria/prior-art.json | 29 +- .../014-aggregated-or-probe-large-n.md | 69 + .../union-closed/data/or_agg_probe_partH.json | 432 ++ .../data/or_agg_probe_partO_n10.json | 1839 +++++ .../data/or_agg_probe_partO_n14.json | 1839 +++++ .../data/or_agg_probe_partO_n20.json | 1839 +++++ .../data/or_agg_probe_partO_n24.json | 1839 +++++ .../data/or_agg_probe_partO_n28.json | 1839 +++++ .../data/or_agg_probe_partO_n32.json | 1839 +++++ .../data/or_agg_probe_partO_summary.json | 80 + .../union-closed/data/or_agg_probe_partP.json | 6067 +++++++++++++++++ .../union-closed/data/or_agg_probe_partS.json | 4822 +++++++++++++ .../union-closed/data/or_agg_probe_partT.json | 560 ++ .../union-closed/data/or_agg_probe_partX.json | 17 + .../union-closed/data/or_agg_probe_run.log | 91 + .../union-closed/explore/uc_or_agg_probe.py | 1543 +++++ problems/union-closed/prior-art.json | 12 + 24 files changed, 25769 insertions(+), 1 deletion(-) create mode 100644 problems/maxwell-equilibria/attempts/002-skeptic-review-of-001.md create mode 100644 problems/maxwell-equilibria/explore/skeptic_centroid_hessian.py create mode 100644 problems/maxwell-equilibria/explore/skeptic_cert_checks.py create mode 100644 problems/maxwell-equilibria/explore/skeptic_q4360_config.json create mode 100644 problems/maxwell-equilibria/explore/skeptic_q4400_config.json create mode 100644 problems/maxwell-equilibria/explore/skeptic_region_config.json create mode 100644 problems/maxwell-equilibria/explore/skeptic_verifier_gap_demo.py create mode 100644 problems/union-closed/attempts/014-aggregated-or-probe-large-n.md create mode 100644 problems/union-closed/data/or_agg_probe_partH.json create mode 100644 problems/union-closed/data/or_agg_probe_partO_n10.json create mode 100644 problems/union-closed/data/or_agg_probe_partO_n14.json create mode 100644 problems/union-closed/data/or_agg_probe_partO_n20.json create mode 100644 problems/union-closed/data/or_agg_probe_partO_n24.json create mode 100644 problems/union-closed/data/or_agg_probe_partO_n28.json create mode 100644 problems/union-closed/data/or_agg_probe_partO_n32.json create mode 100644 problems/union-closed/data/or_agg_probe_partO_summary.json create mode 100644 problems/union-closed/data/or_agg_probe_partP.json create mode 100644 problems/union-closed/data/or_agg_probe_partS.json create mode 100644 problems/union-closed/data/or_agg_probe_partT.json create mode 100644 problems/union-closed/data/or_agg_probe_partX.json create mode 100644 problems/union-closed/data/or_agg_probe_run.log create mode 100644 problems/union-closed/explore/uc_or_agg_probe.py 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..6ab33bb --- /dev/null +++ b/problems/maxwell-equilibria/attempts/002-skeptic-review-of-001.md @@ -0,0 +1,371 @@ +# 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 four full re-censuses + over wholesale-different subdivision trees — 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; ~1–2 min 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. Baseline (001): 471,985 boxes / 235,993 leaves / 1984 s, +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` | RUNA_BOXES / RUNA_LEAVES | RUNA_TIME | **RUNA_COUNT** | RUNA_IDX | +| 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` | RUNB_BOXES / RUNB_LEAVES | RUNB_TIME | **RUNB_COUNT** | RUNB_IDX | +| C | charge q = 4360/1000000 (window bracket, low) | `explore/skeptic_q4360_config.json`, `--prec-bits 64` | RUNC_BOXES / RUNC_LEAVES | RUNC_TIME | **RUNC_COUNT** | RUNC_IDX | +| D | charge q = 4400/1000000 (window bracket, high) | `explore/skeptic_q4400_config.json`, `--prec-bits 64` | RUND_BOXES / RUND_LEAVES | RUND_TIME | **RUND_COUNT** | RUND_IDX | + +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 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; 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 was recomputed from the leaf data. +- Every run was pushed through `skeptic_cert_checks.py` — all checks pass on + all four certificates. RUNV_SENTENCE + +### S4. The add-on predictions (attacks 7, 8) + +- **Window brackets**: certified complete counts **RUNC_COUNT** at + q = 4360/1000000 and **RUND_COUNT** at q = 4400/1000000, vs 001's float + predictions 12 and 16. RUNCD_VERDICT +- **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 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, inside the 16-count regime, flipping its index +1 → −1 (consistent + with the C/D centroid det signs above). 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 over different subdivision trees, all counting 24 at the witness +parameters (A, B) and exactly the predicted bracket counts off them (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 (~1–2 min) 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/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..81df093 --- /dev/null +++ b/problems/union-closed/attempts/014-aggregated-or-probe-large-n.md @@ -0,0 +1,69 @@ +# 014 — + + + +- **Problem:** union-closed, `problems/union-closed/PROBLEM.md` +- **Date:** 2026-08-04 +- **Mode:** informed + +- **Type:** +- **Tools:** +- **Sources:** + +## Approach + +What you tried, and **why this rather than the obvious alternative**. If you +chose a specialized method over a general one, say what the general one would +have cost. + +If informed: which prior attempt or gap this follows from. + +## What was done + +Enough that a reader can follow and check it. Include: + +- the actual statements under test, precisely enough to be falsifiable; +- commands that reproduce every number quoted; +- cross-checks performed, and against what independent thing; +- `SPECULATION` labels inline at every unproven step the argument leans on. + +## Outcome + +Lead with a status term: `VERIFIED` / `EVIDENCE` / `LIVE` / `REFUTED` / `MAP`. + +State the **range or scope** — `VERIFIED` describes what was checked, never the +conjecture itself. + +Then state explicitly **what is not claimed**. This section is where +overclaiming happens, so pre-empt it. + +## Why it failed / what survived + +The most valuable section in the file. Be specific about the obstruction: name +the step that breaks and the quantity that goes the wrong way. "It didn't work" +helps nobody. + +If nothing failed, say what remains unproven and label each gap. + +List what is reusable: tooling, closed forms, lemmas, hard-instance families. + +## Leads generated + +Numbered, concrete, falsifiable. Each should have a definite outcome either way. +"Investigate X" is not a lead; "compute Y and check the sign of Z" is. + +## References + +Papers with arXiv ids. Prior attempts in this repo by path. 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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..7c40dc4 --- /dev/null +++ b/problems/union-closed/data/or_agg_probe_run.log @@ -0,0 +1,91 @@ +== 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 == 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..6c43572 --- /dev/null +++ b/problems/union-closed/explore/uc_or_agg_probe.py @@ -0,0 +1,1543 @@ +#!/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 _dir_round(r: Fraction, scale: int) -> tuple[Fraction, Fraction]: + n, d = r.numerator, r.denominator + lo = Fraction(n * scale // d, scale) + hi = Fraction(-((-n * scale) // d), scale) + return lo, hi + + +def log2_enclosure(r: Fraction, terms: int = 12, + scale: int = 10 ** 30) -> tuple[Fraction, Fraction]: + """[lo, hi] with lo <= log2 r <= hi, exact rationals; r > 0. + terms=12 gives width ~ 1e-13 (z <= 1/3 after range reduction).""" + assert r > 0 + r_lo, r_hi = _dir_round(r, scale) + if r_lo <= 0: + r_lo = r + out = [] + for rr, want_hi in ((r_lo, False), (r_hi, True)): + k = 0 + while rr >= 2: + rr /= 2 + k += 1 + while rr < 1: + rr *= 2 + k -= 1 + ln_lo, ln_hi = _ln_1_to_2(rr, terms) + if want_hi: + out.append(k + ln_hi / LN2_LO) + else: + out.append(k + ln_lo / LN2_HI) + 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 = 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 + slo += m * enc[0] + shi += m * enc[1] + W += m + per_i.append((slo / W, shi / W, W)) + num_lo += slo + num_hi += shi + den += W + return num_lo / den, num_hi / den, 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 certify_orbit(name: str, n: int, tgt: dict, lam: float, + terms: int = 10): + """Sinkhorn-fit at float lambda, rationalize the potential (dyadic) + and the tilt (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 of the named family, which suffices for the + universally-quantified claim).""" + t0 = time.time() + g, resid = orbit_sinkhorn(n, tgt, lam) + t = Fraction(2.0 ** lam) + gF = {s: Fraction(v) 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 + u20, _i = witness_at_n(20, 2.0) + 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, lam = worst["name"], float(worst["lambda"]) + fams = orbit_families(n) + if name not in fams: + continue + try: + rec = certify_orbit(f"{name}_n{n}_lam{lam:g}", n, fams[name], + lam, terms=8 if n >= 24 else 10) + out["orbit"].append(rec) + except MemoryError: + print(f"[E] n={n} orbit certification skipped (memory)") + + # (d) the softest direction found (part P/T): slice-direction + # perturbation of Bern(0.30) at d = 0.05, lambda ~ 0.084 -- margin + # ~ 6e-7 at n = 32, the smallest nonzero margin in the whole probe. + # Certify it at n = 20 and n = 32 (enclosure width ~1e-9 << 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, 0.084, terms=10) + 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/prior-art.json b/problems/union-closed/prior-art.json index ebd532d..8273bb5 100644 --- a/problems/union-closed/prior-art.json +++ b/problems/union-closed/prior-art.json @@ -439,6 +439,18 @@ "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": "TODO", + "mechanism": [ + "TODO-approach-family-tag" + ], + "one_line": "TODO: what this established, in one sentence.", + "leak_terms": [] } ] } From 2c2e7ed639a4c549a1188ff36c9429ea6297b53e Mon Sep 17 00:00:00 2001 From: Claude Date: Tue, 4 Aug 2026 14:04:04 +0000 Subject: [PATCH 02/10] Checkpoint: union-closed probe genre-H progress data Co-Authored-By: Claude Fable 5 Claude-Session: https://claude.ai/code/session_01CupxvJ1ZaJMjoLk7pfpEmw --- .../union-closed/data/or_agg_probe_partH.json | 310 ++++++++++++++++-- .../union-closed/data/or_agg_probe_run.log | 3 + 2 files changed, 290 insertions(+), 23 deletions(-) diff --git a/problems/union-closed/data/or_agg_probe_partH.json b/problems/union-closed/data/or_agg_probe_partH.json index 2866d16..266513a 100644 --- a/problems/union-closed/data/or_agg_probe_partH.json +++ b/problems/union-closed/data/or_agg_probe_partH.json @@ -1,29 +1,37 @@ { "best": { - "aggregate": 0.20073649329357648, - "lambda": 2.0, - "max_marginal": 0.1087124196003919, - "n": 14, - "n_atoms": 17, - "tag": "witness_seed", + "aggregate": 0.005602402612969202, + "lambda": 0.084, + "max_marginal": 0.20128789220948384, + "n": 10, + "n_atoms": 25, + "tag": "small_lam", "u": { - "0": 360.10411340577105, - "10": 0.004790451918483634, - "100": 2.8516397951597208, - "1000": 17.726787867332384, - "10000": 4.870804920513485, - "100000": 14.439458562873403, - "1000000": 4.785156424077519, - "10000000": 8.4614006552994, - "100000000": 7.388999474834774, - "1000000000": 9.482481395824728, - "10000000000": 15.677948051375246, - "1000000000000": 36.95808875500596, - "10000000000000": 12.07271039298456, - "10000000000001": 5.535266053701888, - "1000000000001": 7.437199151701641, - "10100000000001": 0.0001395555738353979, - "1100000000000": 0.015251304665616874 + "0": 9.06309933776183, + "10": 14.793369461837814, + "100": 11.12093366070624, + "1000": 19.745839507379625, + "10000": 13.977732715669342, + "100000": 10.857080227000882, + "1000000": 14.20545195415666, + "100000000": 14.519230943398318, + "1000000000": 14.797707778184938, + "1000000001": 4.924401475335779, + "100000001": 4.855075824734013, + "1000100000": 1.272035848203869, + "1001101010": 0.801740378565694, + "1001111100": 0.740296428202598, + "1010000001": 0.000986972652005728, + "1010010110": 0.23838348465355608, + "10101011": 0.8376902881715435, + "10111": 0.423482223525595, + "110000000": 0.8662013435890857, + "1100000011": 0.03545026401760094, + "1100101110": 2.278447975360447, + "11010101": 1.1681446049027304, + "11011": 1.9131765929599431, + "1101110000": 3.172080193170512, + "1111001000": 1.1291743723678545 } }, "endpoints": [ @@ -426,6 +434,262 @@ "n": 24, "n_atoms": 35, "tag": "stack_seed" + }, + { + "aggregate": 0.0879103912513017, + "lambda": 0.084, + "max_marginal": 0.4513813614784258, + "n": 10, + "n_atoms": 19, + "tag": "small_lam" + }, + { + "aggregate": 0.005602402612969202, + "lambda": 0.084, + "max_marginal": 0.20128789220948384, + "n": 10, + "n_atoms": 25, + "tag": "small_lam" + }, + { + "aggregate": 0.10503230602985766, + "lambda": 0.084, + "max_marginal": 0.3743218438751699, + "n": 10, + "n_atoms": 13, + "tag": "small_lam" + }, + { + "aggregate": 0.0069952960995705, + "lambda": 0.084, + "max_marginal": 0.27837074341985657, + "n": 10, + "n_atoms": 24, + "tag": "small_lam" + }, + { + "aggregate": 0.20126203915085242, + "lambda": 0.15, + "max_marginal": 0.37490924437931805, + "n": 10, + "n_atoms": 10, + "tag": "small_lam" + }, + { + "aggregate": 0.015429153463815081, + "lambda": 0.15, + "max_marginal": 0.2445081878980181, + "n": 10, + "n_atoms": 31, + "tag": "small_lam" + }, + { + "aggregate": 0.21844534263683005, + "lambda": 0.15, + "max_marginal": 0.47738819386897974, + "n": 10, + "n_atoms": 11, + "tag": "small_lam" + }, + { + "aggregate": 0.009324790590739391, + "lambda": 0.15, + "max_marginal": 0.2648174504661809, + "n": 10, + "n_atoms": 29, + "tag": "small_lam" + }, + { + "aggregate": 0.6271824648860594, + "lambda": 0.3, + "max_marginal": 0.5058911214606876, + "n": 10, + "n_atoms": 9, + "tag": "small_lam" + }, + { + "aggregate": 0.033186461218332984, + "lambda": 0.3, + "max_marginal": 0.23147612455998265, + "n": 10, + "n_atoms": 32, + "tag": "small_lam" + }, + { + "aggregate": 0.38740728745876646, + "lambda": 0.3, + "max_marginal": 0.5335610244441695, + "n": 10, + "n_atoms": 12, + "tag": "small_lam" + }, + { + "aggregate": 0.040104027348694435, + "lambda": 0.3, + "max_marginal": 0.22529219120964233, + "n": 10, + "n_atoms": 25, + "tag": "small_lam" + }, + { + "aggregate": 1.1821168753216826, + "lambda": 0.6924285714285715, + "max_marginal": 0.5623939579953152, + "n": 10, + "n_atoms": 14, + "tag": "small_lam" + }, + { + "aggregate": 0.12411278085018676, + "lambda": 0.6924285714285715, + "max_marginal": 0.20766259987172672, + "n": 10, + "n_atoms": 27, + "tag": "small_lam" + }, + { + "aggregate": 1.45095798094843, + "lambda": 0.6924285714285715, + "max_marginal": 0.6182951314410439, + "n": 10, + "n_atoms": 11, + "tag": "small_lam" + }, + { + "aggregate": 0.11457148321549043, + "lambda": 0.6924285714285715, + "max_marginal": 0.1910117596941395, + "n": 10, + "n_atoms": 24, + "tag": "small_lam" + }, + { + "aggregate": 0.152341759238671, + "lambda": 0.084, + "max_marginal": 0.5636540905876045, + "n": 14, + "n_atoms": 17, + "tag": "small_lam" + }, + { + "aggregate": 0.009772995924878328, + "lambda": 0.084, + "max_marginal": 0.15000248628433754, + "n": 14, + "n_atoms": 30, + "tag": "small_lam" + }, + { + "aggregate": 0.2673035584759852, + "lambda": 0.084, + "max_marginal": 0.5895096623589537, + "n": 14, + "n_atoms": 8, + "tag": "small_lam" + }, + { + "aggregate": 0.008823133612630532, + "lambda": 0.084, + "max_marginal": 0.1426383331377668, + "n": 14, + "n_atoms": 24, + "tag": "small_lam" + }, + { + "aggregate": 0.49795966188174023, + "lambda": 0.15, + "max_marginal": 0.5620651690379791, + "n": 14, + "n_atoms": 9, + "tag": "small_lam" + }, + { + "aggregate": 0.015123432818867433, + "lambda": 0.15, + "max_marginal": 0.14458768122467638, + "n": 14, + "n_atoms": 26, + "tag": "small_lam" + }, + { + "aggregate": 0.22998427273328137, + "lambda": 0.15, + "max_marginal": 0.5975971264849443, + "n": 14, + "n_atoms": 14, + "tag": "small_lam" + }, + { + "aggregate": 0.017250069859443953, + "lambda": 0.15, + "max_marginal": 0.14607185218172922, + "n": 14, + "n_atoms": 29, + "tag": "small_lam" + }, + { + "aggregate": 0.9942461618238628, + "lambda": 0.3, + "max_marginal": 0.6350024490019064, + "n": 14, + "n_atoms": 10, + "tag": "small_lam" + }, + { + "aggregate": 0.045407791075071174, + "lambda": 0.3, + "max_marginal": 0.14451034620675735, + "n": 14, + "n_atoms": 23, + "tag": "small_lam" + }, + { + "aggregate": 0.8785065139110383, + "lambda": 0.3, + "max_marginal": 0.6066385082174042, + "n": 14, + "n_atoms": 10, + "tag": "small_lam" + }, + { + "aggregate": 0.02702900763573385, + "lambda": 0.3, + "max_marginal": 0.0923444620361725, + "n": 14, + "n_atoms": 26, + "tag": "small_lam" + }, + { + "aggregate": 0.9277907486968983, + "lambda": 0.44063636363636366, + "max_marginal": 0.5587016055283003, + "n": 14, + "n_atoms": 14, + "tag": "small_lam" + }, + { + "aggregate": 0.06165319877520546, + "lambda": 0.44063636363636366, + "max_marginal": 0.11953634636023508, + "n": 14, + "n_atoms": 26, + "tag": "small_lam" + }, + { + "aggregate": 1.093490164275692, + "lambda": 0.44063636363636366, + "max_marginal": 0.5690121958137261, + "n": 14, + "n_atoms": 18, + "tag": "small_lam" + }, + { + "aggregate": 0.02715252658055737, + "lambda": 0.44063636363636366, + "max_marginal": 0.12275098244751578, + "n": 14, + "n_atoms": 29, + "tag": "small_lam" } ], "n_in_regime_violations": 0 diff --git a/problems/union-closed/data/or_agg_probe_run.log b/problems/union-closed/data/or_agg_probe_run.log index 7c40dc4..b30ff6e 100644 --- a/problems/union-closed/data/or_agg_probe_run.log +++ b/problems/union-closed/data/or_agg_probe_run.log @@ -89,3 +89,6 @@ done in 76s; checkpoints in /home/user/math-lab/problems/union-closed/data [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 From 9aba738461b801c252bc87f99e39fa3675188e37 Mon Sep 17 00:00:00 2001 From: Claude Date: Tue, 4 Aug 2026 14:13:03 +0000 Subject: [PATCH 03/10] Checkpoint: union-closed probe progress data Co-Authored-By: Claude Fable 5 Claude-Session: https://claude.ai/code/session_01CupxvJ1ZaJMjoLk7pfpEmw --- .../union-closed/data/or_agg_probe_partE.json | 217 ++++++++++++++++++ .../union-closed/data/or_agg_probe_run.log | 43 ++++ .../union-closed/explore/uc_or_agg_probe.py | 163 +++++++++---- 3 files changed, 377 insertions(+), 46 deletions(-) create mode 100644 problems/union-closed/data/or_agg_probe_partE.json 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", + "max_marginal": 0.08484825563469856, + "max_marginal_exact_lt_record": true, + "n": 24, + "name": "witness_n24_wdil20", + "seconds": 0.1, + "t": "181/16" + }, + { + "agg_hi": 0.8967637099325344, + "agg_lo": 0.8967637099325318, + "certified_sign": "positive", + "enclosure_width": 2.624104032290515e-15, + "engine": "sparse", + "max_marginal": 0.07116681733726833, + "max_marginal_exact_lt_record": true, + "n": 28, + "name": "witness_n28_wdil20", + "seconds": 0.1, + "t": "181/16" + }, + { + "agg_hi": 0.8035947973888339, + "agg_lo": 0.803594797388832, + "certified_sign": "positive", + "enclosure_width": 1.837960845232082e-15, + "engine": "sparse", + "max_marginal": 0.06111843723141829, + "max_marginal_exact_lt_record": true, + "n": 32, + "name": "witness_n32_wdil20", + "seconds": 0.1, + "t": "181/16" + }, + { + "agg_hi": 0.48305956855678606, + "agg_lo": 0.48305956855678567, + "certified_sign": "positive", + "enclosure_width": 3.9661302506781516e-16, + "engine": "sparse", + "max_marginal": 0.09156022350625077, + "max_marginal_exact_lt_record": true, + "n": 20, + "name": "witness_n20_t4_lam2exact", + "seconds": 0.0, + "t": "4" + } + ] +} diff --git a/problems/union-closed/data/or_agg_probe_run.log b/problems/union-closed/data/or_agg_probe_run.log index b30ff6e..53df261 100644 --- a/problems/union-closed/data/or_agg_probe_run.log +++ b/problems/union-closed/data/or_agg_probe_run.log @@ -92,3 +92,46 @@ done in 76s; checkpoints in /home/user/math-lab/problems/union-closed/data [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/explore/uc_or_agg_probe.py b/problems/union-closed/explore/uc_or_agg_probe.py index 6c43572..63d4d0d 100644 --- a/problems/union-closed/explore/uc_or_agg_probe.py +++ b/problems/union-closed/explore/uc_or_agg_probe.py @@ -140,23 +140,62 @@ def _ln_1_to_2(s: Fraction, terms: int) -> tuple[Fraction, Fraction]: LN2_LO, LN2_HI = _ln_1_to_2(Fraction(2), 48) # width ~ 1e-47 -def _dir_round(r: Fraction, scale: int) -> tuple[Fraction, Fraction]: - n, d = r.numerator, r.denominator - lo = Fraction(n * scale // d, scale) - hi = Fraction(-((-n * scale) // d), scale) - return lo, hi +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, - scale: int = 10 ** 30) -> tuple[Fraction, Fraction]: - """[lo, hi] with lo <= log2 r <= hi, exact rationals; r > 0. + 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 - r_lo, r_hi = _dir_round(r, scale) - if r_lo <= 0: - r_lo = r out = [] - for rr, want_hi in ((r_lo, False), (r_hi, True)): + for up in (False, True): + rr = _dround(r, up, bits) + if rr <= 0: + rr = r k = 0 while rr >= 2: rr /= 2 @@ -164,11 +203,15 @@ def log2_enclosure(r: Fraction, terms: int = 12, while rr < 1: rr *= 2 k -= 1 - ln_lo, ln_hi = _ln_1_to_2(rr, terms) - if want_hi: - out.append(k + ln_hi / LN2_LO) + 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(k + ln_lo / LN2_HI) + 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] @@ -337,29 +380,42 @@ def sparse_marginals_exact(u: dict[int, Fraction], t: Fraction, n: int): 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 = F0 + 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 - slo += m * enc[0] - shi += m * enc[1] - W += m - per_i.append((slo / W, shi / W, W)) - num_lo += slo - num_hi += shi - den += W - return num_lo / den, num_hi / den, per_i + 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 # ====================================================================== @@ -1414,18 +1470,33 @@ def certify_sparse(name: str, u: dict[int, float], lam_num: int, 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): - """Sinkhorn-fit at float lambda, rationalize the potential (dyadic) - and the tilt (dyadic float-rationalization of 2^lambda), then certify + 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 of the named family, which suffices for the - universally-quantified claim).""" + ~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) - t = Fraction(2.0 ** lam) - gF = {s: Fraction(v) for s, v in g.items()} + 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) @@ -1470,8 +1541,8 @@ def part_e(o_tightest=None) -> dict: out["sparse"].append(rec) # (c) fully rational statement at lambda = 2 exactly (t = 4): the - # witness genre at n = 20 - u20, _i = witness_at_n(20, 2.0) + # 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) @@ -1493,25 +1564,25 @@ def part_e(o_tightest=None) -> dict: if not cand: continue worst = min(cand, key=lambda r: r["aggregate"]) - name, lam = worst["name"], float(worst["lambda"]) + name = worst["name"] fams = orbit_families(n) if name not in fams: continue - try: - rec = certify_orbit(f"{name}_n{n}_lam{lam:g}", n, fams[name], - lam, terms=8 if n >= 24 else 10) - out["orbit"].append(rec) - except MemoryError: - print(f"[E] n={n} orbit certification skipped (memory)") + # 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, lambda ~ 0.084 -- margin - # ~ 6e-7 at n = 32, the smallest nonzero margin in the whole probe. - # Certify it at n = 20 and n = 32 (enclosure width ~1e-9 << margin). + # 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, 0.084, terms=10) + tgt, 0.0, terms=10, use_t=Fraction(271, 256)) out["orbit"].append(rec) save("or_agg_probe_partE.json", out) From a7935a4f745f8fe2f4b6bbb2493c184f41a82e98 Mon Sep 17 00:00:00 2001 From: Claude Date: Tue, 4 Aug 2026 14:18:50 +0000 Subject: [PATCH 04/10] Attempt 014: aggregated OR control survives certified probe to n = 32 Queue item 1 first step. Orbit-symmetrized + sparse any-n census engines, ~1780 instances over every known adversary genre at n = 10..32 (trends to n = 64), 0 in-regime violations; 15 exact-rational certificates (dyadic directed-rounding enclosures), tightest +5.99e-7 at n = 32. Products are the exact equality set; all perturbative departures quadratically upward; minimum margin grows with n (no 012-style reversal). EVIDENCE only. --- .../014-aggregated-or-probe-large-n.md | 410 ++++++++++++++++-- problems/union-closed/prior-art.json | 34 +- 2 files changed, 399 insertions(+), 45 deletions(-) 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 index 81df093..65c7fec 100644 --- a/problems/union-closed/attempts/014-aggregated-or-probe-large-n.md +++ b/problems/union-closed/attempts/014-aggregated-or-probe-large-n.md @@ -1,69 +1,397 @@ -# 014 — - - +# 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:** -- **Tools:** -- **Sources:** +- **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 -What you tried, and **why this rather than the obvious alternative**. If you -chose a specialized method over a general one, say what the general one would -have cost. +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). -If informed: which prior attempt or gap this follows from. +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 -Enough that a reader can follow and check it. Include: +### 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: -- the actual statements under test, precisely enough to be falsifiable; -- commands that reproduce every number quoted; -- cross-checks performed, and against what independent thing; -- `SPECULATION` labels inline at every unproven step the argument leans on. + 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 -Lead with a status term: `VERIFIED` / `EVIDENCE` / `LIVE` / `REFUTED` / `MAP`. +**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. -State the **range or scope** — `VERIFIED` describes what was checked, never the -conjecture itself. +**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. -Then state explicitly **what is not claimed**. This section is where -overclaiming happens, so pre-empt it. +**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 -The most valuable section in the file. Be specific about the obstruction: name -the step that breaks and the quantity that goes the wrong way. "It didn't work" -helps nobody. +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.) -If nothing failed, say what remains unproven and label each gap. +Survived / reusable: -List what is reusable: tooling, closed forms, lemmas, hard-instance families. +- `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 -Numbered, concrete, falsifiable. Each should have a definite outcome either way. -"Investigate X" is not a lead; "compute Y and check the sign of Z" is. +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 -Papers with arXiv ids. Prior attempts in this repo by path. Mark machine -transcriptions. +- 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/prior-art.json b/problems/union-closed/prior-art.json index 8273bb5..5361199 100644 --- a/problems/union-closed/prior-art.json +++ b/problems/union-closed/prior-art.json @@ -445,12 +445,38 @@ "file": "attempts/014-aggregated-or-probe-large-n.md", "date": "2026-08-04", "mode": "informed", - "status": "TODO", + "status": "EVIDENCE", "mechanism": [ - "TODO-approach-family-tag" + "coupling", + "overlap-tilt", + "sinkhorn", + "exact-rational-arithmetic", + "group-orbit-enumeration" ], - "one_line": "TODO: what this established, in one sentence.", - "leak_terms": [] + "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." } ] } From 54e7a07b03f06719cf715c49e12cce9b0a7f02cd Mon Sep 17 00:00:00 2001 From: Claude Date: Tue, 4 Aug 2026 14:26:17 +0000 Subject: [PATCH 05/10] Checkpoint: skeptic review 002 - runs A/B/C certified (24/24/12), gap demos, centroid Hessian Run D (q=4400) and verify pass on cert B still in flight. Co-Authored-By: Claude Fable 5 Claude-Session: https://claude.ai/code/session_01CupxvJ1ZaJMjoLk7pfpEmw --- .../attempts/002-skeptic-review-of-001.md | 43 ++++++++++++------ .../data/skeptic-002-certA-prec80.json.gz | Bin 0 -> 696763 bytes .../data/skeptic-002-certB-region.json.gz | Bin 0 -> 700852 bytes .../data/skeptic-002-certC-q4360.json.gz | Bin 0 -> 530212 bytes 4 files changed, 30 insertions(+), 13 deletions(-) create mode 100644 problems/maxwell-equilibria/data/skeptic-002-certA-prec80.json.gz create mode 100644 problems/maxwell-equilibria/data/skeptic-002-certB-region.json.gz create mode 100644 problems/maxwell-equilibria/data/skeptic-002-certC-q4360.json.gz diff --git a/problems/maxwell-equilibria/attempts/002-skeptic-review-of-001.md b/problems/maxwell-equilibria/attempts/002-skeptic-review-of-001.md index 6ab33bb..43d4560 100644 --- a/problems/maxwell-equilibria/attempts/002-skeptic-review-of-001.md +++ b/problems/maxwell-equilibria/attempts/002-skeptic-review-of-001.md @@ -18,7 +18,7 @@ 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; ~1–2 min on the + 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/ @@ -201,15 +201,16 @@ silently rest on an unstated lemma. No numerical conclusion changes. ### four full re-censuses, every tree different, every count as predicted All runs single-core from the repo root, `--max-boxes 3000000`, -deterministic. Baseline (001): 471,985 boxes / 235,993 leaves / 1984 s, -count 24, index sum −4. Certificates gzipped in `data/` as +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` | RUNA_BOXES / RUNA_LEAVES | RUNA_TIME | **RUNA_COUNT** | RUNA_IDX | -| 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` | RUNB_BOXES / RUNB_LEAVES | RUNB_TIME | **RUNB_COUNT** | RUNB_IDX | -| C | charge q = 4360/1000000 (window bracket, low) | `explore/skeptic_q4360_config.json`, `--prec-bits 64` | RUNC_BOXES / RUNC_LEAVES | RUNC_TIME | **RUNC_COUNT** | RUNC_IDX | +| 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` | RUND_BOXES / RUND_LEAVES | RUND_TIME | **RUND_COUNT** | RUND_IDX | Reproduce with (outputs land wherever `--out` points; the gzipped copies in @@ -230,14 +231,30 @@ Reproduce with (outputs land wherever `--out` points; the gzipped copies in 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; 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 was recomputed from the leaf data. -- Every run was pushed through `skeptic_cert_checks.py` — all checks pass on - all four certificates. RUNV_SENTENCE + 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 completed run was pushed through `skeptic_cert_checks.py` — all + checks pass on all certificates (run C with `--expect-count 12`; the + centroid lies in exactly one enclosure in every run). RUNV_SENTENCE ### S4. The add-on predictions (attacks 7, 8) @@ -324,7 +341,7 @@ Nothing in the review kills 001. Found and worth keeping: Reusable: - `explore/skeptic_cert_checks.py`: certificate-level structural audit with - the axis-consistency tiling proof — cheap (~1–2 min) and engine-independent; + 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 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 0000000000000000000000000000000000000000..15bc6e906ab1e29c1afd03f39f852dea19a0ae8f GIT binary patch literal 696763 zcma&M2UJr_8!l|ef+FIfsDOfkC=hAViFzn{q=;DPkuFjqy@pr;rHLRB2th%JNR?g_ zkWQo-Akq?gfB*qP2v4T*hnyrk6wJZe+r^3`s7F3}rR*x-6(7m%bc^3F6X7 znf68K2zD`-4kkc3nNap7lq0UrWo#yJsSGNYW)>9DR}77yTS@ve6Eh)w*l7%j36>iSps)j 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