Union-Closed Sets Conjecture
Combine any two sets in your collection and you stay inside it. Must some single item then appear in half of them?
In plain terms
Take a collection of sets that is union-closed: whenever two of its sets are combined, the result is also in the collection. The conjecture says some single element must appear in at least half of the sets.
A small example. Take {}, {1}, {1,2}. Combining any two gives you something already there, so it is union-closed. Element 1 appears in two of the three sets — comfortably at least half.
It sounds like it should fall to a clever counting argument. It has resisted for over forty years.
What is known
Nothing at all was known about a constant fraction until 2022, when Justin Gilmer proved that some element appears in at least ~1% of the sets, using an argument from information theory rather than combinatorics. That was a genuine surprise: entropy is not the tool anyone expected to crack this.
Within months, a series of papers sharpened the constant to (3−√5)/2 ≈ 0.38197 — and then showed that the entropy method cannot go past that number without a new idea. The current published record, 0.38271, comes from a refinement by Liu.
Why it is hard
The entropy argument uses union-closure in only one way: it assumes the combination of two typical sets stays inside the collection. That is an average-case fact, and Chase and Lovett built collections that are approximately union-closed, satisfy that average-case fact, and still fail the conjecture at 0.38197. So any advance has to use exact closure — the fact that even weird, atypical, heavily-overlapping pairs stay inside — and nobody knows how to charge for that.
What a breakthrough would mean
Honestly: nothing outside mathematics. There is no application waiting on this.
Inside mathematics the interest is real, and it is about method. Gilmer's proof was a demonstration that information-theoretic arguments can attack extremal set theory, and the barrier at 0.38197 is unusually well understood — we know precisely which step fails and why. A proof reaching 1/2 would almost certainly carry a new technique for exploiting closure conditions, and that technique would be the actual prize.
This problem is also unusual in being quantitative. Most famous conjectures are open or closed. This one has a number attached that has moved from 0.01 to 0.38 in three years, which makes partial progress measurable — and makes it a good target for a machine loop.
What this lab tried
Standing: Live Dependent-coupling interface survives every known counterexample genre; model ceiling 0.4315 vs published record 0.38271. No bound claimed; three labeled proof gaps.
Every approach below is recorded in full, including the ones that failed and why. Attempt titles link to the complete record.
Attempt 001informedMap
Mapped the entropy-method frontier and the (3-sqrt5)/2 barrier; established the n<=4 exhaustive baseline; produced candidate ideas A-D.
Attempt 002informedRefuted
The weighted-KL ladder is dead for every c >= 0, killed by Sawin geometric mixtures; generalizes to a no-go covering all smoothing-insensitive functionals.
Attempt 003informedLive
Overlap-biased couplings separate every adversary that killed the KL ladder; evades the 002 no-go by being smoothing-sensitive. No bound claimed.
- Superseded by
- attempt 004
- Open gaps
- plackett-odds-ratio-control
- mutual-information-tax
- recipe-totality
Attempt 004informedVerified, with corrections
Independent re-implementation confirms 003's interface and separations are real; corrects the recipe ceiling to 0.431496 and refutes 003's mini-theorem (e) as stated.
- Verifies
- attempt 003
- Corrections
- mini-theorem (e) bracket positivity is false as stated; restated, with the {0} u [1/2,1) genre covered by a half-mixing coupling
- single-parameter recipe ceiling is 0.431496 in closed form, not 0.445 (grid-floor artifact)
- one non-reproducible data file; several overstated sentences
- Open gaps
- plackett-odds-ratio-control
- mutual-information-tax
- recipe-totality
Attempt 005informedRefuted
The Gap-1 pointwise odds-ratio control [1, 2^lambda] is false both ways: Cauchy-Schwarz forces OR >= 2^lambda on diagonal histories, and an explicit 4-atom family crashes OR to 2^{lambda(3-n)}; sharp range and MTP2 one-sided partial result proved. Route stays LIVE with Gap 1 restated.
- Open gaps
- averaged-odds-ratio-control
- mutual-information-tax
- recipe-totality
Attempt 006informedVerified, with corrections
Independent re-implementation and hand re-derivation confirm all six propositions of 005 including the crash identity at lambdas 005 never ran; the Karlin-Rinott step is checked against the literature and re-proved via Ahlswede-Daykin; corrections are minor (Prop 2 equality-iff needs lambda > 0, two part-B reporting slips, docstring bugs); the restated averaged-control gap survives its own first falsifiable test.
- Verifies
- attempt 005
- Corrections
- Prop 2 equality characterization ('equality iff proportional slices') is false at lambda = 0, where OR = 1 identically for every mu; the iff needs lambda > 0
- part-B prose misreports its checkpoint: the 1.0000000011 minimum is the i<n figure, not 'all diagonal histories'; median per-trial minimum excess is 5.2e-4, not 5.4e-4
- uc_odds_ratio.py docstring errors: part F says 'OR >= 1' where code and record prove OR >= 2^lambda; part G says overall deviation scales like delta where the data show delta^2
- 'about 83% of history mass' is the lambda = 0.5 figure; it is 81% at lambda = 1.5
- Open gaps
- averaged-odds-ratio-control
- mutual-information-tax
- recipe-totality
Attempt 007informedRefuted
The restated Gap-1 averaged control M_i >= lambda is made well-posed (forced degeneracy bookkeeping), proved for four structured subclasses, then refuted in the record-relevant regime by an explicit perturbation-stable 10-atom mu (marginals <= 0.318) with M_5 = lambda - 0.122; the i-aggregated form survives and becomes the new gap.
- Range checked
- 444-instance census plus clamped adversarial searches at n <= 8, lambda in [0.05, 6]; witness verified by two independent engines to 1e-14, stable under 3% perturbation and 2-digit rounding, violating for every lambda >= 0.03 on the same mu
- Open gaps
- aggregated-or-control
- margin-modulated-or-control
- mutual-information-tax
- recipe-totality
- Methodological lesson
- Cross-engine agreement is not independence when both engines share IEEE arithmetic: a 1e-299 atom made u(A)u(B) underflow to exact zero in one diagonal cell, and both implementations plus the Sinkhorn round-trip agreed on the resulting fake violation; a parameter sweep of the suspect light atom toward its boundary was what exposed it. Guard adopted: cell-floor diagnostic plus dynamic-range clamping in searches.
- Note
- Settles 005 lead 1 negatively (005's SPECULATION label was accurate; nothing in 005/006 needs correction). Route stays LIVE: this kills a candidate bridge, not the interface. Partial positives now proved: M_i >= lambda for supports of <= 4 atoms, for potential-MTP2 mu, at i in {1, n}, and to first order in lambda (perfect-square coefficient), the last modulo the standard Sinkhorn-smoothness step.
Attempt 008informedLive
Closes the missing smoothness step in 005 Prop 6 (quantitative IFT, explicit fixed-n budget 924min(p,1-p)^{-3n}delta^2), runs the perturbative assembly end-to-end (positive on every marginal-capped instance to delta=0.2, n<=9), and shows the pointwise delta^2 OR budget is NOT n-uniform (deviation grows ~0.3 bits/coordinate at fixed delta) while the coordinate-averaged downward budget flattens; conditional n-uniform assembly theorem with every budget labelled by its gap.
- Range checked
- Exact couplings at n <= 9; delta <= 0.2; directions crash / zero-prefix embedded crash / delta_empty / Bernoulli mixtures / slice / pm-mixture; (p, rho) = (0.383, 1.06|1.20) and (0.30, 2.0); 78 assembly runs, 80 budget-census fits; proofs (Lemmas A-C, Theorem P6', conditional theorem) are for all n but pending skeptic pass
- Open gaps
- averaged-odds-ratio-control
- mutual-information-tax
- recipe-totality
- sinkhorn-tv-stability
- Note
- Sub-results at different confidence levels inside one record: the Prop-6 closure is a written-out proof (VERIFIED pending independent skeptic pass); the lead-3 assembly check and the non-uniformity of the pointwise budget are EVIDENCE at n <= 9; the assembly theorem is CONDITIONAL on budgets (B1)-(B4) with n-uniform constants, of which (B1) is the perturbative form of gap a-prime and (B2) of gap b. 005's leads 3 and 4 are distinct (the task line conflated them); both were run.
Attempt 009informedEvidence
Gap 2 made precise as a chain-rule assembly statement (TAX at p) and probed: it survives all four adversary genres, is smoothing-sensitive on 002's own certificates, the pure slice tilt is provably tax-free, the half-mixing coupling provably loses nothing to chain-rule certification, and a previously unnamed second tax (measured ~0.40 log2 n on slices) is identified as the binding loss channel.
- Range checked
- Crash family n<=11, lambda in [-1,3]; half-mixing genre n<=14 at two parameter points; Sawin certificate gadgets n<=60000 (block recipe, rigorous bound); CL pure slice n<=80, lambda in [0,3], p in {0.42, 0.3823}. Finite-n, finite-grid EVIDENCE only; mini-lemmas SL/HM proved but not yet skeptic-reviewed.
- Open gaps
- averaged-odds-ratio-control
- mutual-information-tax
- recipe-totality
- second-tax-scaling
- Note
- Deliberately the smallest-budget line of the cycle: a probe, not a campaign. The expected outcome was a precise obstruction; instead the candidate survived everywhere tested and the losses were localized (T_A vs ST, by genre).
Attempt 010informedMap
First ideation sweep (6 field lenses): 3 queue-worthy routes (pairwise-closure LP certification, union-transfer-operator eigenvalue field, bipartite MIS decomposition), 6 secondary, 5 no-purchase verdicts incl. a proven product-weight no-go killing the multiplicative toolbox.
Attempt 011informedVerified, with corrections
Independent re-implementation (recursion engine, natural-log Sinkhorn, direct-MI cross-checks) confirms 009 in full: mini-lemmas SL and HM re-derived by hand and now VERIFIED, all headline numbers reproduce to <= 1.4e-10, the crash lambda-window sharpens to the law lam_max*(n-3) -> 4.8475, the ST ~ 0.40 log2 n fit survives extension to n = 240 and two new p, and the untested tilt-on-Sawin cell closes positive at n <= 300; corrections are reporting-level (grid-edge suprema quoted as suprema, mixed-p 'increasing margin', slice-ratio scope, a measured 1e-5 lgamma floor behind part D's 'exactly', and an unstated m_k = 1/2 legality hypothesis p_k >= 1-1/sqrt(2) in the block bound, verified to hold on every instance used).
- Verifies
- attempt 009
- Corrections
- slice sup_lambda CR values at p ~ 0.42 are grid-edge values; true suprema sit at lambda ~ 4-5 and are larger (e.g. +1.4655 vs reported +1.0199 at n=80, w0=34); 'increasing margin in n' conflates p = 0.4167 and p = 0.425 rows -- at fixed p the margin is cleanly increasing
- Outcome scope 'p in {0.42, 0.3823}' overstates: slice ratios actually run are {0.4167, 0.425, 0.3833, 0.3875}; p = 0.3823 itself was never tested
- part D's 'H(mu) computed exactly' has a measured ~4e-6 absolute lgamma floor at n = 60000 (harmless at the +2418 scale)
- the part-D block bound's m_k = 1/2 choice needs the unstated hypothesis p_k >= 1-1/sqrt(2) ~ 0.29289; holds on all instances used but would fail silently on mixtures with a component marginal below it
- Open gaps
- averaged-odds-ratio-control
- mutual-information-tax
- recipe-totality
- second-tax-scaling
Attempt 012informedVerified, with corrections
Line-by-line re-derivation and two independent engines (incl. an orbit-symmetrized census that reaches n = 32) confirm 008's Theorem P6' closure of the Prop-6 smoothness gap, the 78-run assembly table, the calibration block, and the centered-kernel closed form; but the conditional theorem's constants fail (box-uniform tau_half is analytically false at 008's own s0 = 0.10, ceiling s0 <= 0.0843; delta_0 = 0.022 should be <= 0.0079 by 008's own formula, ~0.004 corrected), and the extended census reverses the flattening reading: the averaged downward OR budget's increments stop shrinking and start growing past n ~ 22 (claimed limits 0.07/0.3 passed by n = 16/10, delta^2-coefficient verified stable in delta at n = 32), killing budget (B1) as stated at evidence level by 008's own lead-1 criterion, while the tax budget (B2) is first-order in delta, not quadratic, in every observable window.
- Range checked
- Hand re-derivation of Lemmas A-C and Theorem P6' in full; P6' bound stress-tested on 120 instances (n <= 6, five directions incl. delta_full and half-empty, delta up to the theorem boundary 0.5, zero violations); Lemma B checked at interior delta; 78-run table spot-verified to 5e-6 and 104-run capped-negative hunt (rho in {1.045, 1.06, 1.20}, n <= 9, delta <= 0.2, incl. the half-empty direction with marginals exactly p) found none; budget census re-run and extended to n = 32 on the crash direction via an orbit-symmetrized engine (n <= 9 values reproduce 008 to all printed digits), with delta -> 0 stability verified at n = 32 down to delta = 0.003125 for the downward budget (clean delta^2) and shown NOT to hold for the tax (delta-linear); conditional-theorem constants recomputed on 008's grid plus tau < 0 and global domains
- Verifies
- attempt 008
- Corrections
- conditional theorem's 'G nondecreasing in tau up to tau_half on the box' is analytically false at (p, rho, s0) = (0.383, 1.2, 0.10): (x0+s0)^2 = 0.5141 > 1/2 and z > xy for rho > 1, so tau_half(corner) = 0; the claimed P2d numerical check does not exist in uc_pert.py; repair requires s0 <= x_c - x0 = 0.0843 with re-measured constants
- delta_0 ~ 0.022 does not follow from 008's stated formula: the code drops the ||G||_box/s0^2 c3 term (||G||_box = 0.1456, never computed by 008 -> delta_0 = 0.0079); the off-box charge needs the global sup(-G) = 0.487, not ||G||_box; M_tau must include the tau < 0 side (0.0838 vs 0.0592); c3 is ~4.7 not 2.5; fully corrected delta_0 ~ 0.004
- the averaged downward budget does NOT flatten: at (0.383, 1.2) the delta->0 coefficient rises 0.065 (n=9) -> 0.104 (n=32) with increasing increments; at (0.30, 1.0) it passes the claimed 0.3 limit at n = 10 and reaches >= 1.24 by n = 32 (the fixed-delta turnover is a delta-saturation artifact); (B1) with n-uniform constant is dead at EVIDENCE level per 008's own lead-1 falsification criterion
- the tax is first-order in delta (halving ratios -> 2), not O(delta^2), in every observable window at large n; fixed-delta tax/delta^2 exceeds the instantiated c2 = 0.01 by 10x at n = 32; (B2) must be restated as a linear-in-delta budget (absolute tax stays tiny, so end-to-end positivity is not threatened)
- 'worst in-class net = +0.0003 (slice n=5 delta=0.2)' is out of class by 008's own capped flag; true worst in-class min-net is ~ +0.00105 (rho=1.06 crash n=6 delta=0.02)
- P6' constant chase: the stated step 'using m0 <= 1/2' yields 831, not 924 (924 corresponds to m0 <= 1; bound valid either way); the 'measured c ~ 0.6' looseness illustration ignores 008's own census worst case c ~ 20.6
- Open gaps
- averaged-odds-ratio-control
- mutual-information-tax
- recipe-totality
- sinkhorn-tv-stability
Attempt 013informedVerified
Exact rational arithmetic (Fraction end-to-end, certified log2 enclosures) confirms 007 in full: the 10-atom witness's violation is certified float-free at rational tilts spanning lambda ~ 0.0144..5 including exactly rational lambda = 2, 3, 4 (positive at small t exactly where Theorem C forces it), both invariances hold as exact rational identities, the aggregate is certified positive (+1.844669) and survives witness-seeded two-coordinate attacks, all four partial theorems re-derive by hand, and the refutation even survives the conjecture-friendliest alternative bookkeeping (degenerate histories scored at lambda: still -0.067); no corrections, one cosmetic runtime discrepancy.
- Range checked
- Witness certified at 12 exact rational tilt parameters (fixed-u family t = 101/100 .. 32 plus rationalized-refit measures at lambda = 1, 2, 3 within 9.2e-16 of the witness mu); enclosure widths < 1e-40; own 96-instance battery + 21-endpoint aggregate hunt at n <= 7; Theorem B probed on 500 random <= 4-atom instances; hand re-derivations of section 1, Lemmas G/M, Theorems A/B/C and both invariances
- Verifies
- attempt 007
- Open gaps
- aggregated-or-control
- margin-modulated-or-control
- mutual-information-tax
- recipe-totality
- 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).
Attempt 014informedEvidence
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 checked
- 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
- Superseded by
- attempt 016
- Open gaps
- aggregated-or-control
- margin-modulated-or-control
- mutual-information-tax
- recipe-totality
- 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. Superseded where they differ by 016/017 (merged from a parallel cycle): the same control is certified NEGATIVE on a unit-replicated ladder at n = 96-160, so the n <= 32 positivity recorded here is a small-n fact, not a licence for proof effort on the forall-n statement. The two searches are consistent -- 016 also finds nothing negative at n <= 32.
Attempt 015informedVerified, with corrections
Independent re-implementation (digit-extraction log2 bracket instead of the atanh kit, integer-scaled exact rows, own orbit derivation and Sinkhorn) confirms 014 in full: five certificates incl. the softest +5.99e-7 and the fully rational t = 4 point re-certified on identical exact inputs with no disagreement, kit sound by hand and by 437-case cross-bracketing, orbit formulas cross-checked at n = 8-9 with exactly zero within-class OR spread, the n ~ 48 dip reproduced digit-for-digit and shown robust off-grid, and ~116 new adversary instances (witness-direction off the product boundary, marginals pushed to 0.3827, lambda below the grid floor) all positive; corrections are reporting-level (part-P order range is d^1.39-d^2.06 on its own checkpoint, not d^1.85-d^2.06; 'products are the exact equality set' claims an unproven converse that is false at lambda = 0; stale docstring runtimes; 1820 not ~1780 instances).
- Verifies
- attempt 014
- Corrections
- part P 'fitted order d^1.85-d^2.06 in every direction' misreports its own checkpoint: 19 of 72 fitted cells lie below 1.85, minimum d^1.39 (n=32, full direction, lambda=0.5, in-regime); successive-halving orders climb back toward 2 as d -> 0, so the qualitative quadratic-upward reading survives but the quoted range is a grid-edge artifact, and the cubic term is large at lambda >= 0.5 (lead-1 consequence recorded)
- 'product measures are the exact equality set' proves only the forward inclusion; the converse is untested and false outright at lambda = 0 (A = 0 for every mu), needing the same lambda > 0 proviso 006 added to 005 Prop 2; no downstream use is affected
- uc_or_agg_probe.py docstring runtimes (O ~ 40-80 min, E ~ 10-30 min) are stale; actual O ~ 146 s, E ~ 113 s (the record's own times are right)
- the Outcome's '~1780 float instances' should be 1820 (840+432+40+426+82)
- Open gaps
- aggregated-or-control
- margin-modulated-or-control
- mutual-information-tax
- recipe-totality
- Note
- Decisive check: 014's unreviewed enclosure kit is re-derived by hand and beaten against a from-scratch certifier using a different algorithm (binary digit extraction with directed integer shifts; exact integer masses so the aggregate denominator is exact); the five re-certifications agree with 014's enclosures at width 1.1e-14, including on the identical quantized refit measures. New floor observed: the lambda -> 0 first-order coefficient at the dip point is +6.4e-7 (n = 48), the smallest Theorem-C square seen anywhere; lead 2 should track that coefficient, not A at fixed lambda.
Attempt 016informedRefuted
Large-n probe of the restated Gap 1 kills it: unit-replicating the 10-atom witness genre (r crash units sharing one block marker) with re-optimized shared weights drives the i-aggregated control negative past n ~ 90, certified float-free at t = 4 (n = 96 and 160, marginals <= 0.309); the n <= 32 space stays positive everywhere searched.
- Range checked
- Free-support seeded climbs n <= 32 (58 climbs round 1, deep annealed round 2) plus permutation orbits: 0 violations; unit-symmetric ladder evaluated to n = 160 (raw witness weights: positive through n = 128, min +0.157; optimized theta*: float crossing in (88, 96], -0.0242 at n = 160, 20/20 perturbations stay negative); exact rational certificates at t = 181/16 and t = 4, enclosure widths < 1e-18
- Open gaps
- margin-modulated-or-control
- mutual-information-tax
- recipe-totality
- Note
- Settles queue item 1's probe-first branch negatively: the aggregated control is certified positive at n <= 32 on everything searched, so proof effort there was exactly the small-n trap 012 warned about. The kill is convention-robust (scoring degenerate histories at lambda only grows the denominator, never the numerator). Route stays LIVE; Gap 1 falls to the margin-modulated candidate (007 lead 2), which this record's ladder family must now be run against. Recorded as 016/017 (not 014/015) and on engines uc_agg_ctrl_probe*.py / uc_agg_ctrl_skeptic.py: this line ran in parallel with 014/015, which took the 014/015 slots and the uc_or_agg_probe.py name first. The two are independent and compatible -- 014 certifies the aggregate positive at n <= 32, which is exactly where this record finds nothing either; the kill lives past n ~ 90.
Attempt 017informedVerified, with corrections
From-scratch third implementation of the aggregate (scaled-integer census plus an interval-squaring log2 enclosure, nothing shared with the probe engines) confirms 014's kill: all three certificates re-derive with matching digits (n=96 in [-0.000759865185844166092, -0.000759865185844166079], n=128 -0.0154057269, n=160 -0.0242259093, all t=4, marginals certified below 0.38271, dichotomy exact), the anchors and positive controls re-certify (+1.844669005 witness; raw-weight ladder at n=96 upgraded to certified +0.157131854664), and the convention-robustness proof re-derives; corrections: the n=128 instance has 252 atoms not 253, and P4's claim that the violating lambda-window sits where 009/011's lambda-window law puts the workable range is wrong (at n=96 the law gives lambda_max ~ 0.052 vs the violation at lambda in [2, 2.5]), so the lambda-restricted restatement joins the margin-modulated control as a live Gap-1 candidate.
- Range checked
- Kill certificates independently re-derived at n = 96 (188 atoms), 128 (252 atoms), 160 (316 atoms), all at t = 4 exactly, enclosure widths <= 1.3e-20; witness anchor at t = 181/16; P3-best n = 32 and raw-witness ladder n = 96 positives certified; padding invariance MU(20,1)/MU(32,1) verified as exact rational identity; alternative-bookkeeping aggregate certified negative at all three n; MU builder cross-checked atom-for-atom against the probe's; 014's float-only side claims (crossing in (88,96], 20/20 perturbations, P4 window edges) reconciled against checkpoints but not certified; n <= 32 search completeness not re-run (stays EVIDENCE)
- Verifies
- attempt 016
- Corrections
- n = 128 tidy certificate instance has 252 atoms (2r+8, r=122), not the 253 stated in 014's certification block and Outcome table
- P4's 'violating window sits exactly at moderate lambda where 009/011's lambda-window law puts the workable range' is false at every violating n: lambda_max ~ 4.847/(n-3) ~ 0.052 at n = 96 while the violation lives at lambda in [2, 2.5] and the same family is positive at lambda <= 1.5 -- the kill closes the unrestricted for-all-lambda gap only, and the lambda-restricted (lambda <~ c/n) restatement is untouched by this family
- Open gaps
- margin-modulated-or-control
- lambda-restricted-or-control
- mutual-information-tax
- recipe-totality
- Note
- Decisive check per the 2026-07-31 verification standard: 014's certificates reused 013's exact engine and enclosures, so this review removes the shared-code risk with a third census written from 007 section 1's prose and a structurally different certified log2 enclosure (dyadic interval squaring; ln 2 from the exact series sum 1/(k 2^k)). Free anchors verified exactly: OR/t = 1 at i = n (005 Prop 1) on every instance, and including the i = n term keeps every kill negative. Union-closure of the support is correctly NOT a hypothesis (the control quantifies over all mu; 007's witness was not union-closed either) -- checked and recorded so the scope is explicit. Precedent followed from 013/007: the reviewed entry (014) is not modified.
Attempt 018informedRefuted
Triage of the two post-016 Gap-1 candidates against the ladder adversary: the margin-modulated OR control is REFUTED in both signed readings already at n = 7 by 007's own witness (secant form certified -0.0222546582 at t = 181/16, in-regime -- the h-sensitivity weight is signed and kills the surplus histories, not the hidden deficit), while the unsigned |sigma|-weighted variant and the lambda-window-restricted control survive theta-re-optimized ladders, free-support climbs and joint (theta, lambda) attacks, with the window margin's first-order coefficient a1(n)*n GROWING in n.
- Range checked
- Witness kill certified exact at t = 181/16 (dyadic-bisection Plackett-root enclosures, directed log2 entropy enclosures), perturbation-stable 20/20 at 3%, 2-digit tidy, violating for every lambda in [1, 5]; MM variants measured on raw and theta* ladders n <= 160 (incl. 016's three certified kill instances), theta re-optimized against MM_abs at n = 48 with transfers to 160, 8 free-support climbs at n = 8; window probe at lam = 4.847/(n-3), /2, /4 on ladders n <= 320 with theta re-optimized at window lambda and a joint (theta, lam <= window) climb at n = 96
- Open gaps
- abs-sensitivity-or-control
- lambda-restricted-or-control
- mutual-information-tax
- recipe-totality
- Note
- Settles 016 lead 1 negatively for the signed readings without needing the ladder: the sensitivity weighting fails on the witness itself while the plain aggregate is certified positive (+1.844669) on the same tables. The margin-modulated gap tag of 007/013/016 dies as stated; what replaces it is the unsigned variant (which decouples from the chain rule -- a proof of it would not feed the 008/012 assembly directly) and the window-restricted variant (016 lead 4 / 017 C2), both surviving as EVIDENCE. Certification of the kill uses the lab's first certified enclosure of a NONLINEAR census functional (Plackett root + binary entropy per history).
Attempt 019informedVerified, with corrections
From-scratch re-certification confirms 018's headline: an independent exact kit (scaled-integer census from 007 section 1's prose, exact-rational Newton with sign-checked bracket for the Plackett root, own interval-squaring log2 enclosures) re-derives MM_sec in [-0.022254658205655785176146, -0.022254658205655785176145] on the witness at t = 181/16 -- negative, in-regime, dichotomy exact, strictly inside 018's enclosure -- with the secant identity, product-measure zero and degenerate-contribution-0 re-derived by hand and the tidy 2-digit witness newly certified (-0.02227); the realized-OR sensitivity variants 018 skipped were built and tested (signed dies on the witness at -1.458, unsigned survives), so no signed reading of 007 lead 2 survives; corrections are prose-level only.
- Range checked
- Witness MM_sec kill independently re-certified at t = 181/16 (enclosure width 5.9e-30, strictly inside 018's; exact marginals < 38271/100000; dichotomy and degenerate-row-zero checked as exact identities); tidy 2-digit witness newly certified negative; exact product-measure anchor (w = 3/7, n = 5, t = 4) contains 0 at width 9.3e-30 with plain aggregate exactly 0; float spot-checks on a spec-rebuilt ladder (own builder + own tune_dil): witness MM numbers, theta* n = 96 lam = 2 triple, window A at n = 96, all 10 part-D a1/a2 fits re-solved, perturbations 0/40 failures on fresh seeds; best_free endpoint re-evaluated (in-regime, nondegenerate); realized-OR variant zoo at two instances; searches themselves NOT re-run at scale (018's floors stay EVIDENCE); MM_der not separately certified (float only, as in 018)
- Verifies
- attempt 018
- Corrections
- 018 part B's 'MM_sec is negative at every n in {16..160} (raw and theta ladders), MM_der negative throughout' is contradicted by 018's own gap1c_partB.json: the raw ladder at n in {96, 128, 160} has MM_sec +6.7e-3/+6.1e-3/+5.2e-3 and MM_der ~ +0.95..0.99 (the tune_dil bisection switches dilution branch at n >= 96); correct scope is theta at all n and raw at n <= 64 only
- the golden-ratio parenthetical (twice: 'z_1(x,x) = x^2 crosses 1/2 exactly at the (3-sqrt5)/2 barrier') is false -- at the barrier x^2 equals the marginal itself (phi^2 = 1 - phi), not 1/2; the rho = 1 crossing is at marginal 1 - 2^{-1/2} ~ 0.2929 and the lambda = 3.5 crossing at ~ 0.3891 is lambda-dependent; the in-regime z = 1/2 straddling claim itself survives
- section F ships a literal PENDING-F placeholder: only 1 of 4 --cert jobs (the witness MM cert) had landed at review time; the n = 96/160 MM certificates and t = 51/50 window certificates are absent from 018 and must not be cited from it
- witness MM_sec 'positive only for lambda <~ 0.7' -- the crossing is between 0.8 and 0.9 (+4.95e-5 at 0.8, -3.2e-4 at 0.9)
- runtime drift (cosmetic): record says ~35 min for parts A-D, log shows ~58 min
- Open gaps
- abs-sensitivity-or-control
- lambda-restricted-or-control
- mutual-information-tax
- recipe-totality
- Note
- Decisive check per the lab standard: 018's certificate reused 013's atanh log2 kit and its own dyadic bisection; this review's kit shares neither (interval-squaring log2 re-implemented from 017's idea; exact Newton root with per-call symbolic bracket-sign identities F(xy) = (1-t)xy(1-x)(1-y) < 0 < min(x,y)(1-max(x,y))). Three structurally different kits now agree on the witness tables (013's aggregate, 018's MM, this one). The fork analysis is extended, not just checked: sensitivity-at-realized-OR readings were the natural loophole and they behave exactly as 018's signed/unsigned dichotomy predicts. Per precedent the reviewed record (018) is not modified.
Attempt 020informedRefuted
Both surviving Gap-1 candidates die certified: the unsigned |sigma|-weighted control fails at n=4-5 via non-PSD sensitivity-kernel design (healthy margins, in-window), the lambda-window aggregated control fails at n=4 via a1-cancellation measures with negative a2 (marginals <= 0.2575), the assembly's aggregate secant requirement is violated in-window, and certified R+-pure witnesses rule out every chain-rule-compatible per-history weighting; attack directions drafted by a dual-family LLM swarm, all verification by the director.
- Range checked
- Certified kills at n = 4 (7-15 atoms) and n = 5 at rational tilts t in {6/5, 21/20, 16/15, 27/25, 7/5, 3/2, 2}, enclosure widths < 1e-14, in-regime exact, dichotomy exact; tensor extension to n = 8, 12 and the lambda-integrated variant's negativity are float-level; deep anneals at n <= 16 with the crash-corner float negatives shown to be artifacts by exact sign-flip
- Open gaps
- mutual-information-tax
- recipe-totality
- non-per-history-bridge
- Note
- Skeptic pass pending (queue). Swarm provenance: 12 attack briefs, gpt-5.6-luna + gemini-3.7-flash, effort medium, ~$0.15; two cross-family convergences (kernel Gram form; prefix-signed cancellation) supplied the designs; one worker formula error (sigma_0 denominator) caught by director validation. Part-A near-empty float negatives are certified artifacts (exact recomputation flips -24.5 to +23.4), re-confirming 007's underflow lesson.
Attempt 021informedMap
Second ideation sweep, seven lenses x two model families (14 workers): union-transfer-operator lead sharpened by an exact eigenfunction identity (coordinate frequencies are eigenvalues), Mobius/zeta LP statistics and a strong-Rayleigh feasibility question queued, number theory no-purchase by cross-family agreement, and two idea-shapes killed inline by the explicit family 2^{1,2} u {[6]}.
- Note
- Swarm-executed per docs/SWARM.md: 7 lenses x {gpt-5.6-luna, gemini-3.7-flash} at effort low, ~$0.06. Cross-family convergence used as a triage axis for the first time on an ideation sweep; both inline kills verified by hand by the director before recording.
Attempt 022informedVerified
Cross-family skeptic pass on 020: a gemini-3.7-flash-drafted verifier (own digit-extraction log2 with directed dyadic rounding, own bisection, own census, zero shared code, audited then run unmodified) reproduces all 12 certificates digit-for-digit including the three R+-purity legs; the weight identity and R+ reduction re-derived by hand; the window kill extended to a certified n=8 in-window tensor instance; and Gap 2's CR > 0 candidate SURVIVES both kill geometries with margins 0.40-0.80.
- Range checked
- All 12 frozen witness instances (data/gap1deep_witnesses.json) re-certified at their rational tilts, enclosure widths below print precision, mixed-zero-table count 0 throughout; n=8 tensor aggregate certified at t = 25/24; Gap-2 check is float-level (Sinkhorn marginal deviation <= 2.3e-15) at a 10-point lambda sweep. Same-session review: implementation independence achieved (different model family, different algorithms), fresh-session reviewer independence still queued.
- Verifies
- attempt 020
- Open gaps
- mutual-information-tax
- recipe-totality
- non-per-history-bridge
- Note
- First use of the drafted-verifier skeptic pattern: spec -> cross-family worker writes the whole verifier -> director audits -> run unmodified. Per docs/SWARM.md this is stronger implementation independence than a same-family rewrite; the residual gap (same director, same session) is stated in the record and left for a fresh-session pass.
Attempt 023informedEvidence
First adversarial pass on Gap 2: free-support anneals seeded at the exact geometries that killed Gap 1 grind sup_lambda CR from +0.40/+0.80 to floors +0.074..+0.22 but find NO in-regime violation (the only negative is the degenerate H->0 corner where CR->0 correctly, so any quantitative TAX statement must scale with H(mu)); side result: the signed sensitivity form -- the lambda-integrated condition's integrand -- is certified negative at t in {6/5, 3/2} on all three 020 witnesses, closing 020's last float-level piece.
- Range checked
- 7 anneal jobs, n <= 6, <= 14 atoms, 1200 steps each, lambda grid {0.05..3.5}, Sinkhorn branch of the 009 recipe only (block-adaptive and half-mixing branches not exercised); signed-form certificates at two rational tilts per witness, enclosure widths < 1e-14
- Open gaps
- mutual-information-tax
- recipe-totality
- non-per-history-bridge
- Note
- Gap 2 is now the route's sole surviving bridge and has earned proof effort (lead 1: CR >= c(p) H(mu) on the generic genre). The degenerate-H escape is an engine lesson: future CR adversaries need an H(mu) >= eps constraint.
Attempt 024informedVerified, with corrections
Fresh-session reviewer pass on the Gap-1 closure batch: all 12 certificates reproduce byte-for-byte and land inside a third from-spec-only reimplementation, the verifier audit and both hand re-derivations (weight identity, R+ reduction) check out, and Gap 2's survival is confirmed on three fresh adversarial trajectories -- discharging 022's reviewer-independence residual; one reporting-level correction: 023's CR engine is NOT deterministic (string-hash seeding), its checkpoint is not regenerable, and the +0.0742 floor is trajectory-specific (this pass saw +0.0697).
- Range checked
- Re-run + third-path check of all 12 frozen witnesses (data/gap1deep_witnesses.json) at their rational tilts; verifier audit of uc_gap1_skeptic_gemini.py in full; three fresh uc_cr_attack.py trajectories (n <= 6, Sinkhorn branch only); nothing new certified -- this pass adds reviewer-level independence to existing certificates
- Verifies
- attempt 020
- Corrections
- 023: 'deterministic' is false for uc_cr_attack.py -- job seeds use hash(name) on a str, randomized per interpreter run; data/cr_attack.json is not regenerable; the 'sharpest floor +0.0742' is trajectory-specific (floors +0.0697..+0.0776 observed across four independent runs, all positive)
- Open gaps
- mutual-information-tax
- recipe-totality
- non-per-history-bridge
- Note
- Also verifies 022 (its verifier re-audited fresh, re-run byte-identical) and 023 qualitatively (schema allows one 'verifies' id; 020 is the finding under confirmation). Reviewer independence: fresh session, no shared state with the 020/022/023 director, no swarm. The corrected engine practice for future CR adversaries: stable job seeds + record PYTHONHASHSEED, on top of 023's own H >= eps lead.
Attempt 025informedRefuted
Both non-Sinkhorn branches of the 009 recipe fail as stated, in-regime: half-mixing (q = P1/2) has CR < 0 instances at every n (ph -> 1 sliver, margins ~1.5e-2, two independent implementations agree to 4e-15), and the iid-given-k block coupling is Theta(n)-negative on any mixture with a component above psi = (3-sqrt5)/2 (a component-wise threshold; adaptive variant flatlines at CR = 0 exactly on all-above-half mixtures); one repair covers both -- the extended mixing lemma (ST = 0 for arbitrary eps-joints and arbitrary s-laws) plus concavity gives sup_q CR > 0 on the whole in-regime {0}u[1/2,1) genre (proved except two float-level 1-D sweeps at n = 1, 2), and empty-mixing also rescues every constructed block failure; floor pre-work: 023's in-regime endpoints have CR/H in [0.0387, 0.0834].
- Range checked
- Half-mixing: failure family constructed at n = 1..100 (analytic mechanism, closed form + full-history evaluator), in-regime grid 98521 pts/n at n in {1,2,4,8,16}; block: exchangeable mixtures, gains to n = 40 (profile arithmetic, cross-checked by direct enumeration at n = 8, 12), full-history CR at n <= 8 (iid) / n <= 10 (adaptive); coverage lemma float-level on two 200001-pt boundary sweeps; all float-level, exact certification queued
- Open gaps
- recipe-totality
- mutual-information-tax
- Note
- Refutes the recipe's branch assignments, not Gap 2's candidate: with the corrected assignment (empty-mixing, optimized q) no in-regime CR <= 0 instance is known. Lemmas EM/EM-coverage are same-session skeptic-passed (uc_branch_skeptic.py, implementation-level); reviewer-level pass pending per the 024 standard.
Attempt 026informedRefuted
Dual-kit exact-rational certification of the 025 branch kills and repairs: CR_halfmix certified negative at in-regime sliver instances n = 2, 6, 16 (P1 = 16/25, ph = 1987/2000, 1997/2000, 19993/20000; H(mu) certified > 0.968), block iid-given-k Gain certified negative at n = 8 on the 2block/3block mixtures (CR < 0 via 009 fact (i)), and both repairs certified positive (q = 1/2 sliver rescues +0.067..0.073; 3block empty-mixing rescue +0.693 modulo Lemma EM); every log2 evaluated by BOTH audited kits (022 digit-extraction and 016 atanh-series) with enclosures intersected -- zero disagreements over 98 calls, widths <= 2.5e-32.
- Range checked
- Six instances certified (three half-mixing sliver points, two block mixtures at n = 8, one empty-mixing rescue at n = 6) plus three repair values; nothing certified beyond these -- 025's grids and families remain float-level; same-session with 025 (implementation-level only, reviewer pass queued covers both)
- Verifies
- attempt 025
- Open gaps
- recipe-totality
- mutual-information-tax
- Note
- C3 leans on 009 fact (i) (CR <= Gain, 011-verified) as a lemma; C4 leans on Lemma EM (025, reviewer pass pending). Dual-kit intersection pattern recommended as default for future certifications.
Attempt 027informedEvidence
The last uncharted branch cell (adaptive block on in-regime lo/hi product mixtures) survives a 510-instance deterministic kill scan -- its degenerate limit is comonotone (CR = 0), approached from above, so the 025 fixed-cost sliver mechanism cannot bite -- but CR/H(mu) -> 0 as p_lo -> 0 at bounded H (measured to CR/H = 1e-5), so the adaptive assignment cannot carry any H-scaled floor there; a new latent-mixing family (component labels coupled with free both-lo probability q, generalizing 025's empty-mixing) holds CR >= +0.22..+1.19 across the same instances and converges to the EM closed-form limit (delta <= 3e-4 at p_lo = 1e-5).
- Range checked
- n <= 6 scan (refinement n <= 8), two exchangeable components, p_lo in [1e-5, 0.38], p_hi in [0.55, 0.999], in-regime grid of 510 instances plus a 15-point scaling probe; skeptic rebuilt all couplings from 009's verbal rule and reproduced every value to 1e-9 with the independent cr_chain evaluator; float-level throughout
- Open gaps
- recipe-totality
- mutual-information-tax
- Note
- Kill attempt failed informatively: branches whose degenerate limits are comonotone-like (cost-free) cannot go negative at the edge -- a design principle for Gap 3. Latent-mixing has no ST = 0 lemma yet (EM's proof breaks on differing labels); conjectured ST = O(p_lo n) is SPECULATION. Reviewer pass queued now covers 025-027.
Attempt 028informedLive
Recipe v2 stated and battery-tested: the whole mixture side collapses to one q-parametrized latent-mixing family (q = P_light recovers adaptive-block, a delta_0 light component recovers 025's empty-mixing with its EM lemma and coverage proof, optimized q dominates half-mixing), generic/slice measures keep the lambda-swept Sinkhorn branch; (TAX at p) v2 re-posed against it survives the full 19-instance hard-instance battery of record (020 kill geometries, 023 floors, 025/026 sliver and block kills, 027 lo/hi probes) with margins >= +0.068; new battery numbers independently re-evaluated (cr_chain, ST ~ 1e-15).
- Range checked
- Battery of 19 instances, n <= 16, the stated genres only; latent-mixing measured (not proved) away from its EM specialization; H-scaled envelope c <= 0.0387 carried over from 025; no totality claim (genre detection and >= 3 components stay Gap 3)
- Open gaps
- recipe-totality
- mutual-information-tax
- Note
- The route's current statement; any future recipe edit should re-run the battery. Reviewer pass queued covers 025-028.
Attempt 029informedVerified, with corrections
Reviewer pass on 025-028 (fresh-context subagent, no shared conversation state, spawned from the author session -- independence level stated in the record): Lemmas EM and EM-coverage re-derive by hand in full with every constant confirmed, all 38 independently re-implemented values agree (exact-Fraction couplings, own evaluator, own IPF Sinkhorn), all six scripts reproduce byte-identically with skeptics at exit 0, and each certification kit ALONE certifies all twelve 026 statements; corrections: 026's 'certified under either kit's audit' is invalid for intersected enclosures (sound only under BOTH audits -- repaired here by single-kit certification), 027's claimed n <= 8 refinement never ran (dead code path; true range n <= 6, 029's own n = 8 spot check stays positive), 025 quotes skeptic agreement 4e-15 where its own log shows 6.4e-13 at n = 16, and 028's cr_chain cross-checks shipped without code (now committed).
- Range checked
- Hand re-derivations of Lemma EM (sigma-field argument, Frechet range, concavity corollary, surprisal criterion) and EM-coverage (G-criterion, monotonicity, both analytic cases, constants 0.785910 / 1.706172 / 0.530738 / +0.120821) in full; all six 025-028 scripts re-run unmodified with byte-identical checkpoints and skeptic exits 0; independent re-implementation (no shared code) of the half-mixing kills at n = 2, 6 (exact-Fraction, <= 1e-11 from 026's enclosure midpoints), 2block Gain at n = 8, the 3block rescue at n = 6 q = 1/5, 4 of 027's tightest scan instances plus 2 scaling-probe rows with latent-mixing, and 6 of 028's 19 battery rows including both G-gen floors via an own-IPF Sinkhorn; 026's nine statements re-certified three ways (kit A alone, kit B alone, intersection) with the interval algebra re-implemented; kits spot-tested against a from-scratch 200-bit exact log2 reference on 40 random rationals; n in {1,2} coverage legs re-swept (min G +0.3448 / +0.8658). Searches and grids NOT re-run at scale beyond the committed engines' own re-execution
- Verifies
- attempt 025
- Corrections
- 026's dual-kit soundness sentence ('the result is certified under either kit's audit') is wrong as an argument: an intersection of enclosures is certified only under the CONJUNCTION of both kits' audits -- a tight-but-wrong enclosure from one unsound kit can pull the intersection off the truth without disjointness; repaired by re-running the full 026 computation with each kit alone (both certify all 12 statements, so the intended disjunction holds -- via single-kit runs, not the intersection); future dual-kit certifications should require each kit to certify alone
- 027's 'refinement of the tightest instances at n <= 8 in the checkpoint' never ran: the refinement block executes only on a negative scan result and none occurred, so refine_n is absent from adaptive_cell.json and nothing in 027 was evaluated above n = 6; index range corrected from 'refinement n <= 8' to n <= 6 (029's own n = 8 check on the tightest instance gives CR = +0.016, positive)
- 025's 'two independent implementations agreeing to <= 4e-15' (Outcome, section 2, and index one-line) misquotes its own skeptic log: S1 deltas are 3.6e-16 / 3.4e-15 / 6.4e-13 (n = 2/6/16); correct bound 6.4e-13, still ~10 orders below the kill margins
- 028 section 3's cr_chain cross-checks of the new battery numbers are transcript-only (uc_recipe_v2.py contains no such calls); the values were right and are now backed by committed code (uc_reviewer029_reimpl.py reproduces them, e.g. 2block n=4 full-history CR = +0.290240 with ST = -1.3e-15)
- minor: 025's 'slopes matching the per-coordinate prediction to 3 decimal places at n = 40' holds for 2block only (3block -0.1271 vs -0.1244); the A2 repair range is +0.060..+0.073 not +0.067..+0.073 (n = 1 row); A2 'H(mu) ~ 0.97' includes a 0.940 row; uc_recipe_v2.py docstring runtime ~90 s vs actual ~0.7 s
- Open gaps
- recipe-totality
- mutual-information-tax
- Note
- Discharges the queue-(a') reviewer pass over 025-028 at the independence level stated in the record: fresh context and zero shared session state, but spawned from the author's session with the review brief, same model family -- whether this meets the 024 fresh-session bar is left to the human owner. Lemmas EM/EM-coverage are hereby reviewer-re-derived (025's own upgrade criterion), with the n in {1,2} legs remaining float-level as always stated. Verifies 026, 027 and 028 as well (schema allows one verifies id; 025 carries the load-bearing lemmas). No numerical claim in any of the four records moved.
Attempt 030informedEvidence
K-component latent-mixing defined (label-coupling matrix Q, diagonal = adaptive products, off-diagonal independent; symmetric-transfer parametrization, deterministic ascent) and positive on all 15 in-regime K=3 battery rows with CR/H in [0.043, 0.395]; anchors to 027's K=2 values exactly; strictly beats the fused-s-law EM construction on the degeneration probe (+0.404 vs +0.352); and the near-psi rows expose a proved mini-theorem (HU-mix): for mixtures with every component in [1/4, 1/2], CR = n - H(mu) exactly, giving the first proved H-scaled floor CR >= (1-h(pbar))/h(pbar) H(mu) = 0.0417*H at the working cap.
- Range checked
- K = 3 exchangeable components, n <= 6, 15-row battery + K = 2 anchor + degeneration probe at p_light down to 1e-4; skeptic rebuilt all couplings and reproduced every value to 1e-9; HU-mix theorem proved by hand, numerics to 6e-14; float-level throughout
- Open gaps
- recipe-totality
- mutual-information-tax
- Note
- Transfers help exactly where components leave [1/4,1/2] (inside it the diagonal already attains E h(z) = 1). Reviewer batch: 029 covers 025-028; this record joins the next one.
Attempt 031informedLive
The half-union coupling -- per history cell push the conditional union marginal to 1/2 within Frechet bounds; total, closed-form, lambda-free; the per-history totalization of Sawin's max-entropy-coupling ingredient -- beats the lambda-swept Sinkhorn branch on every G-gen hard instance of record (sharpest floor 0.0742 -> 0.0822, mmabskill_n6 0.101 -> 0.301), reads the crash family as a flat n-independent +0.199, stays positive on all fixed-seed scans and descents at marginal caps up to 0.49 and on slices to n = 300 at w0/n up to 0.48, with the attacked floor tracking the product-extremal constant (1-h(pbar))/h(pbar) from above at every cap -- writing down conjecture (HU-TAX): CR_HU >= (1-h(pbar))/h(pbar) * H(mu), equality at products (SPECULATION); optimized K-mixing still beats HU on 3/5 structured mixtures, and order-min stayed positive with a 4x order spread at mmabskill.
- Range checked
- Full 023/024 floor set, crash family n <= 17, 300+4x200 fixed-seed random supports (n <= 6, <= 14 atoms, delta_0-projected) at caps 0.38271..0.49, pattern-search descents from worst starts, slice DP to n = 300 at w0/n in {0.36..0.48} (two independent DP implementations), coordinate-order enumeration at n <= 5 (120-sample at n = 6); all float-level; skeptic (recursive-descent builder + natural-log evaluator) reproduced everything, exit 0
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- (HU-TAX) proved only on the [1/4,1/2]-component sub-genre (030's HU-mix); if proved for all pbar < 1/2 it is Frankl via licensing -- treat every use as SPECULATION. Novelty gate: per-coordinate idea is Sawin 2211.11504's max-entropy coupling / 009's adaptive rule; the clamped per-history totalization and the sweeps are this record's contribution.
Attempt 033informedRefuted
The order quantifier in (HU-TAX) breaks: a joint (mu, coordinate-order) adversarial campaign against the half-union coupling finds a CERTIFIED order-dependent violation at marginal cap 9/20 -- a 6-atom n=4 witness (max marginal 0.44990, H = 2.527) where exactly 1 of 24 orders gives CR_HU certified in [., -8.048e-3] while identity (+0.0148) and best order (+0.257) stay positive -- with the full-order-min floor crossing zero in (0.44, 0.45] on the attack family; at the working cap 0.38271 the same campaign finds nothing (17 starts, global floor +0.04198 tracking the product extremal +0.04174), and the two sharpest G-gen floors are now certified single-kit (both kits alone): CR_HU > 0 and CR_HU >= H/25 at the rationalized windowkill/mmabskill instances. (HU-TAX) must be re-posed with an order convention; for-all-orders is dead above ~0.44, identity/best-order forms unrefuted everywhere tested.
- Range checked
- Campaign: n <= 6, caps 0.38271 and 0.45, 17 starts, pattern-search weight moves + atom add/drop + greedy-swap order pool; full 24-order enumeration at n = 4 (witness + cap trace 0.39..0.45); certificates: the two rationalized floor instances (positive, >= H/25) and the rationalized witness under order (0,1,3,2) (negative), each under both kits separately; floors are upper bounds on the true inf; witness violation independently confirmed by the 025-skeptic evaluator stack before certification
- Verifies
- attempt 031
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- The kill needed both weapons: descent-built mu AND 1-of-24 orders. Recipe-facing (best-order) conjecture untouched; (b2) is now 'all-orders CR_HU >= 0 at p <= 0.38271' plus best-order positivity above. Witness dissection = the minimal concrete instance of the 031 averaging problem. Joins the next reviewer batch with 030/031.
Attempt 034informedVerified, with corrections
Reviewer pass on 030/031/033 (fresh-context subagent, no shared conversation state, spawned from the author session -- independence level stated in the record): HU-mix and HU-notax re-derive by hand and hold (including the z = x+y-1 branch of the sign analysis and the tightness of the [1/4,1/2] hypothesis), all six scripts reproduce byte-identically with both skeptics at exit 0, all 43 independently re-implemented values agree (third-path HU builder, own evaluator/DP/optimizer, own 60-digit recomputation of all three certificates), the order-kill is confirmed exactly (1 of 24 orders, (0,1,3,2), CR = -0.00804843, max marginal 0.449899) and a strictly stronger all-orders adversary still finds nothing at the working cap (floor +0.043284); but (HU-TAX at pbar) as stated is REFUTED for every pbar < 1/4 by its own claimed equality case Bern(pbar)^n, where the Frechet floor forces z = 1-2p and the sharp constant is (h(2p)-h(p))/h(p) < (1-h(p))/h(p) -- deficit exactly n(1-h(2pbar)) bits, order-independent, verified at 60 digits.
- Range checked
- Hand re-derivations of HU-mix (Frechet feasibility [max(0,1-2p),1-p] contains 1/2 iff p in [1/4,1/2]; posterior-mixture step; U ~ Unif(2^[n]) confirmed as H(U) = n to 1e-12; subadditivity chain) and HU-notax (sigma-field argument, all four cell branches of the sign analysis incl. z = x+y-1) in full; all six 030/031/033 scripts re-run unmodified with byte-identical checkpoints and both skeptics at exit 0; 43 independent re-implementation checks with no shared code (third-path HU builder by direct pair-product; own chain-rule evaluator; own 2-D slice DP at n = 100/300 cross-checked against own mask enumeration at n = 8; own latent-K builder + own shrinking-box optimizer on two battery rows and all three degeneration rows; own fused-EM re-derivation; full order enumerations 24/120/720 at the five floor instances; the 033 witness rebuilt from the checkpoint with all 24 orders, ledgers and the determinacy of coordinate 2; own all-24-order descent reproducing 033's seven-cap trace to four decimals; own all-n!-order working-cap campaign from 12 starts, floor +0.043284; all three certificates recomputed at 60 digits via Decimal.ln, inside both kits' enclosures, rationalization TV measured 2e-14..5e-14). Searches and grids NOT re-run at scale beyond the committed engines' own re-execution; everything float-level except the certificate leg
- Verifies
- attempt 031
- Corrections
- 031's (HU-TAX at pbar) is false for every pbar < 1/4: Bern(pbar)^n -- the record's own stated equality case -- has CR_HU = n(h(2pbar)-h(pbar)) < n(1-h(pbar)) because the Frechet floor 1-2p exceeds 1/2 below p = 1/4, so the clamp fires and E h(z) < 1; smallest witness pbar = 1/5, n = 1 (CR 0.249022 vs RHS 0.278072, deficit 1-h(2/5) = 0.029049, exact). The conjecture must be posed for pbar in [1/4, 1/2) (where the clamp is idle -- exactly 030's hypothesis) with the sharp product constant c*(p) = (h(min(2p,1))-h(p))/h(p); the cap-form used by every sweep, and 030's HU-mix (whose hypothesis forces pbar >= 1/4), are untouched. The kill is order-independent, so it also hits the identity-order and best-order forms 033 left standing
- 031 section 2's scan numbers are not in its own checkpoint: worst CR/H is +0.045578 not +0.0496, and the three descent floors are +0.045578/+0.045578/+0.042367 not +0.0429/+0.0430 (+0.0429 is section 1's mmabskill ratio); the descents also do not start 'from the floor endpoints' -- scan_and_descend descends from the three worst random-support scan results. Not reproducible at 40, 100 or 300 scans. The qualitative claim (floors sit just above the product extremal +0.041739 and cannot be pushed below) survives
- 030's degeneration triple '+0.4037/+0.3970/+0.3950' quotes uc_latentK_skeptic's fixed-t01 = 0.22 evaluations, not the optimizer: latentK_probe.json records CR_opt = +0.40370/+0.40316/+0.40375 (independently reproduced here). Conservative error; the claim of beating the fused 2-block EM limit (+0.3523253, re-derived independently) stands with margin >= +0.05
- 030's 'CR/H = 0.04303 exactly, constant in n' is constant only to five decimals: the checkpoint's near-psi rows are 0.043030634 / 0.043031978 / 0.043033317 at n = 2/4/6, strictly increasing (the identity CR = n - H(mu) is what is exact; H is not exactly linear in n). The skeptic's own tolerance is 2e-4, so it could not have caught the word
- index-range scope words wider than the skeptics: 030's 'reproduced every value' covers 10 of 15 battery rows (n in {2,6} only), no K = 2 anchor row, and the degeneration rows only at a fixed suboptimal transfer; 031's 'reproduced everything' covers the floors, crash, four slice points, one n = 8 cross-check and the product identity -- not the scans, descents, cap sweeps, head-to-head or order enumeration
- 031's '120-order sample at n = 6' is itertools.islice(permutations, 120) = exactly the orders fixing perm[0] = 0, a sub-family and not a sample; redone here as a full 720-order enumeration, which reproduces the same min/max (+0.260712 / +0.482824) and stays positive, as do full enumerations at all five floor instances
- uc_hu_certify.py's comment says WITNESS is 'already permuted by its violating order (0,1,3,2)' but WITNESS is byte-identical to the unpermuted hu_attack.json row and the code permutes it; since (0,1,3,2) is an involution, an auditor trusting the comment would certify the wrong measure (CR +0.0148, positive). Code correct, comment wrong
- 033's part B 24-order enumeration, part C cap trace and part E ledgers ship without code ('follow-up traces') -- the same class 029 flagged in 028; all three reproduce here and uc_reviewer034_reimpl.py now commits code for them
- minor: 031's 'two independent DP implementations agreeing to 2e-3 at n = 300' quotes the assertion tolerance, not the agreement (observed ~1e-4; this pass ~1e-5); 031's claim that the Sawin framing was 'checked against the public abstract' overstates what the abstract can support -- it confirms author/title/(3-sqrt5)/2 but says nothing about couplings, so the ingredient-level attribution rests entirely on 003's [L] transcription
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Discharges STATUS queue 1(f) (reviewer batch 2 over 030/031/033) at the independence level stated in the record: fresh context and zero shared session state, but spawned from the author's session with the review brief, same model family -- whether that meets the 024 fresh-session bar is for the human owner to judge. Verifies 030 and 033 as well (schema allows one verifies id; 031 carries the conjecture and the coupling). The one refutation is a quantifier correction, not a numerical one: no measured value in any of the three records moved. 035 landed concurrently from the author session and is out of this pass's scope.
Attempt 035informedRefuted
Canonical-order HU (greedy most-predictable-first; a function of mu and the revealed SET only, so still a genuine total closed-form coupling) does exactly what 033's dissection predicted -- it scores the 033 order-kill witness at +0.1014, rank 23/24, certified positive, and its weight-descent floor stays positive at cap 0.38271 (+0.041983) and 0.45 (+0.001703) where the for-all-orders form died -- but it is itself REFUTED at cap 49/100: two witnesses with CR_HU certified negative under each kit alone (n=4, 6 atoms, max marginal 0.48990, CR in [., -3.795e-2], TIE-FREE hence labelling-independent; n=5, CR in [., -2.889e-2], where the canonical rule picks the WORST of 120 orders). Since 0.49 < 1/2, HU with canonical order cannot by itself deliver Frankl. Skeptic caught a real defect en route: float tie-breaking made the 'canonical' order implementation-dependent (exactly equal conditional entropies do occur); fixed with a tolerance tie-break, with the residual non-equivariance stated.
- Range checked
- Weight-descent: 17 starts per cap at 0.38271/0.45/0.49, n <= 6, order re-derived by the rule at every candidate; fixed-seed sweeps 120 instances per cap at 0.45/0.49 with full order enumeration (n <= 4); certificates: the two cap-0.49 kills (negative) and the cap-0.45 rescue (positive), each under both kits separately, on limit_denominator(1e7) rationalizations; floors are upper bounds on the true inf and the 0.49 threshold is not claimed exact
- Refutes
- 031
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Bounds HU's reach from above by measurement (< 0.49) as the 030 sub-genre theorem bounds it from below by proof. Mechanism of the failure: greedy conditional-entropy minimisation banks predictability early but CR needs surplus cells late, and the objectives diverge as marginals approach 1/2. Best-order HU survives both kills but is an oracle, not a recipe. Joins the next reviewer batch.
Attempt 036informedVerified, with corrections
Fresh-session reviewer pass on 035 (genuinely fresh session, not spawned from the author's -- the 029/034 caveat does not apply): everything load-bearing survives -- own from-prose HU/CR/greedy-rule implementation (exact Fraction cells, 60-digit Decimal entropies, 1e-40 tie detection) reproduces all 51 descent endpoints' orders and floors, both order enumerations, and all three certificates inside both kits' exact enclosures, with all three 035 scripts re-running byte-identically at skeptic exit 0; the certify scripts' float-derived canonical orders are confirmed to be canonical for the rationalized measures too; corrections are reporting-level and one is favorable: the n=5 kill's step-0 tie is between two exactly-deterministic coordinates, so it is labelling-INDEPENDENT in fact (CR identical across all 120 relabels at 60 digits) and the cap-0.49 refutation carries no labelling caveat, while part D's order-enumeration numbers (16/24, 60/120, both best-order values) were transcript-only and are now backed by committed code.
- Range checked
- All three 035 scripts re-run unmodified (checkpoints byte-identical, skeptic exit 0); independent from-prose re-implementation of the HU coupling, CR, and the greedy rule with exact-profile plus 1e-40 Decimal tie detection: all 51 descent-endpoint rows (orders, floors, global floors, violation lists, product-extremal constants), the 033-witness 24-order and both kill enumerations (24 and 120 orders), the 24-relabel equivariance check, and 60-digit recomputation of all three certificates on the same limit_denominator(1e7) rationalizations, each inside both kits' exact enclosures with exact marginal and H side-conditions; canonical orders of the rationalized measures re-derived and shown equal to the certified (float-derived) orders; hand re-derivations of the clip identity, coupling well-posedness, and the deterministic-coordinate invariance argument; the descents and sweeps themselves NOT re-run beyond the committed engines' byte-identical re-execution
- Verifies
- attempt 035
- Corrections
- 035 part E's 'the n=5 kill has one tie at step 0 and is therefore labelling-dependent in principle' does not apply to this witness: the tie is between coordinates 1 and 3, both with marginal exactly 0 (deterministic), so swapping them leaves the coupling itself unchanged and the canonical CR is identical on all 120 relabels (max deviation 0 at 60-digit precision); both cap-0.49 kills are labelling-independent and the refutation needs no labelling qualifier
- 035 part D's order-enumeration claims (n=4: 16 of 24 orders negative, best +0.1507; n=5: 60 of 120 negative, canonical is the worst order, best +0.0350) shipped without committed code -- the same class 029 flagged in 028 and 034 flagged in 033 parts B/C/E; all four reproduce here (best CR +0.15075 / +0.03495) and uc_reviewer036_reimpl.py now commits the enumeration
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Discharges the queue's reviewer-batch-3 item for 035 at a strictly higher independence level than 029/034 (fresh session, zero shared state, not spawned from the author session). One design observation extracted for 035 lead 1: the 033 witness has an exact greedy tie at step 2 and the lowest-index break picks the second-best of the two greedy-compatible orders (+0.101419 vs +0.101583) -- an order rule should break ties by the surplus objective, not by index. Per precedent the reviewed record (035) is not modified.
Attempt 037informedLive
The ROLLOUT order rule (pick the coordinate maximising full CR with canonical completion -- the surplus/deficit ledger scored to the end; still a function of (mu, revealed set), so a genuine total coupling) passes 035's discriminating test exactly as the mechanism claim predicts: greedy-surplus dies where canonical died (worst of 120 orders on the n=5 kill, fresh descents kill it at caps 0.49/0.497), while rollout lands on the best order of all three record witnesses, is positive on all 51 canonical-descent endpoints, survives fresh and hostile-seeded descents at caps 0.49/0.497/0.499 with zero violations and floors tracking the product extremal from above (+0.000679/+0.000037/+0.0000030), and the best-order oracle ceiling stays positive too (n <= 5); five instances certified positive under each kit alone -- both 035 kill measures rescued and all three cap floors -- with a new 60-digit rollout fixed-point check on the rationalized measure that caught roll(mu_Q) != roll(mu_float) at the 0.499 floor (rationalization makes the four step-0 scores exactly equal) before it shipped.
- Range checked
- Witness battery with full order enumeration (24/24/120); all 51 committed 035 endpoints re-scored under four rules; fresh descents 17 starts per cap against surp (0.49, 0.497) and roll (0.49, 0.497; the three n=6 starts skipped for roll -- budget); hostile-seeded roll descents at 0.49/0.497/0.499 from the canonical/surp kill measures and sharpest endpoints (n <= 5); best-order-oracle descents at 0.49/0.497 (n <= 5, reduced rounds); certificates on limit_denominator(1e7) rationalizations, each kit alone, with the 60-digit fixed-point order check; all floors are upper bounds on the true inf; float-level except the certificate leg; nothing at n > 6
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Sharpens 035's mechanism statement: non-greediness is the load-bearing property, not the scoring currency -- surp is greedy in exactly the currency 035 lead 1 suggested and dies identically to canonical. The [0.45, 1/2) interval 035 said 'needs a different object' now has a candidate object. Certification-pattern upgrade (from 036's review): derive the certified order from the rationalized measure and verify it as a 60-digit fixed point -- 035's certificates were confirmed safe on this point after the fact; here it is a construction-time check. Skeptic: nats-based rule re-implementations + the pre-existing independent evaluator stack, exit 0. Joins the next reviewer batch with 036.
Attempt 038informedEvidence
Census of the rollout rule (300 random in-regime instances per block at n in {4,5}, caps 0.45/0.49, full order enumeration alongside): rollout is NOT a best-order proxy (exactly best only 39-60%, worst rank 7/24, worst gap 0.207*H -- 037's witnesses were descent products, not typical), but it never went negative (0 of 1200, including 5 instances where some order IS negative) and canonical strictly beats it on 0 of 1200 -- the latter now PROVED as mini-theorem (ROLL-DOM): CR_roll >= CR_canon for every mu, a transcription of the classical rollout improvement property needing only that the completion policy is a deterministic set-function, so every canonical positive of record transfers to rollout wholesale; n=6 descents (caps 0.49/0.497) and n=7 spot descents (0.49) find zero violations, with the honest caveat that the descent visibly stalls above n=5 (floors 40x the n<=5 values -- 'no kill found', not 'sharp floor').
- Range checked
- 1200 census instances (300 per block, n in {4,5} x caps {0.45,0.49}, random supports 3-14 atoms, delta_0-projected, full 24/120-order enumeration per instance, canonical ranked alongside); n=6 rollout descents at caps 0.49/0.497 (6 starts each: the three 037 skipped plus fresh seeds, rounds<=120); n=7 spot descents at 0.49 (3 starts, rounds<=50); all float-level, nothing certified this round (037's certificates stand; no sharp new floor to certify); (ROLL-DOM) proved same-session, reviewer re-derivation queued with the 036/037 batch
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Refutes 037 lead 3's optimistic reading (roll is often far from best on typical measures) while strengthening the survival story (0 negatives in 1200 + provable domination of canonical). Reframes 037 lead 2 as the sandwich: best-order positivity (measured) >= roll (constructive) >= canonical (proved floor at [1/4,1/2] genre via 030). Novelty gate: (ROLL-DOM) is the standard Bertsekas rollout monotonicity argument applied to HU orders -- recorded as a transcription, not a new result. Joins the next reviewer batch with 036/037.
Attempt 039informedVerified, with corrections
Reviewer pass on the rollout batch (037, 038; 036's script audited as a load-bearing dependency of 037's certify): everything load-bearing survives -- all five committed pipelines re-run byte-identically at skeptic exit 0; a fourth independent HU/CR implementation (memoized history-tree recursion, nats, 60-digit Decimal/Fraction certificate leg) reproduces the full witness table, every C/C2/D/B-endpoint and P2/P3 row, all five certificates inside both kits' exact enclosures with the rationalized-measure rollout orders re-derived under fully-rationalized 60-digit scoring (stronger than the committed fixed-point check, whose completions are float-derived), 48 of 1200 census rows fully re-scored and all summaries recomputed, and (ROLL-DOM) re-derived by hand with its recursion hinges verified on the committed code (canon_rank <= roll_rank on all 1200 rows); corrections are reporting-level: the roll-skipped starts are two n=6 plus one n=8 (crash8 -- never roll-descended, contra 038's 'the three starts 037 skipped'), and 037 lead 4's 0.499 endpoint is NOT exactly-exchangeable -- rationalization zeroes the 8.1e-11 atom, leaving a two-atom maximally-correlated measure (stabilizer 6 of 24) that pinches the same (1-h(q))/h(q) constant as the product extremal.
- Range checked
- All five 037/038 scripts re-run unmodified (three checkpoints byte-identical, both skeptics exit 0); independent from-prose re-implementation of HU/CR and all four order rules: the three-witness table with full 24/24/120 enumerations (sequences, CRs, ranks), every hu_order2.json C_/C2_/D_ and B_endpoints value, every hu_rollcensus.json P2_/P3_* endpoint (own rollout at n=6 and 7), 48 of 1200 census rows regenerated and fully re-scored plus all summary fields recomputed from stored rows, all five certificates recomputed at 60 digits inside both kits' exact enclosures with exact side-conditions and the rationalized rollout orders re-derived at 60-digit scores AND completions, (ROLL-DOM) hinge property-tests on the committed functions (203 instances) plus a 150-instance tie-tolerance stress; the descents, sweeps, and census loops themselves NOT re-run beyond the committed engines' byte-identical re-execution
- Verifies
- attempt 037
- Corrections
- 037 'roll descents skipped the three n = 6 starts' and 038 'the three starts 037 skipped plus four fresh hostile seeds': the part-C skip filter is n > 5 and catches two n=6 starts plus one n=8 start (crash8); 038 P2's filter is n == 6, so only the two n=6 starts were re-run (its 6 rows are 2 + 4, and its range field's '6 starts each: the three 037 skipped plus fresh seeds' is internally inconsistent) -- crash8 (n=8) has never been descended under the rollout rule at any cap
- 037 lead 4's premise 'the 0.499 endpoint rationalizes to an exactly-exchangeable measure' is false: limit_denominator(1e7) sends the 8.1e-11 atom to weight exactly 0, leaving the two-atom measure {0000: ~0.501, 1101: ~0.499} with coordinate-1 marginal exactly 0 and stabilizer S3 on {0,2,3} (order 6 of 24); the 4-way step-0 tie is real (verified exactly, and the ~1.6e-10 unrationalized gap and roll(mu_float) != roll(mu_Q) both confirm 037 part E), but the binding adversary is a maximally-correlated pair, not (near-)product -- the (HU-TAX) equality structure at high caps has at least two extremal families both pinching (1-h(q))/h(q)
- clarification to 038 P4, not an error: (ROLL-DOM) is proved for the exact rules; the shipped implementations break 1e-12-near-ties by lowest index, so the float invariant is CR_roll >= CR_canon - (n-1)*1e-12 - evaluation noise (worst observed inversion -4.4e-15 on a perturbed-exchangeable stress instance) -- nine orders below the smallest recorded floor and invisible in all 1200 census rows
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Discharges the queue's reviewer-batch item for 037/038. Independence disclosure, per the 029/034 precedent: fresh-context subagent with zero shared conversation state, but spawned from the author session with the review brief, same model family -- one level below 036's genuinely-fresh bar; whether that meets the 024 fresh-session bar is for the human owner to judge. Verifies 038 as well (schema allows one verifies id; 037 carries the rules and certificates under test), and re-ran 036's committed script (exit 0) while auditing its hu_cr as 037-certify's imported scorer -- that import mildly erodes 036's independence for future reviews, noted in the record. No conclusion of 037/038 changes; correction 2 sharpens 037 lead 4 rather than killing it (which of the two extremal families does the descent select as cap -> 1/2?). Per precedent the reviewed records are not modified.
Attempt 040informedEvidence
A genuinely stronger adversary against rollout-HU (simulated annealing with atom add/drop/transfer moves plus embedded 035-kill seeds, then descent polish) finds ZERO violations at n=6 (caps 0.49/0.497) and n=7 (0.49), landing ~60x below 038's stalled descent floors -- and the attractor it converges onto is identified and closed-form: the two-point diagonal measure (1-p)delta_empty + p delta_S, whose DIAG identity (proved, elementary, order-free: CR_HU = h(max(1/2,1-2p)) - h(p), H = h(p), any order, any n; floor matched to 2e-13) makes CR/H equal 034's corrected product constant c(pbar) on both branches -- so (HU-TAX) has a SECOND equality family at the opposite correlation extreme from products, and NO order rule (best-order oracle included) can guarantee more than c(pbar)*H at cap pbar, closing 037 lead 3 analytically.
- Range checked
- Anneal 4000 steps per run at n=6 (6 runs per cap at 0.49/0.497: two embedded 035-kill tensor seeds + four random supports), 1200 steps at n=7 (6 runs at 0.49), each polished by 035's pattern descent; every endpoint re-scored by the independent 037-skeptic stack, all in-regime; DIAG identity verified numerically at p in {0.05..0.4999} x n in {2,4,6} x sampled orders to 1e-12, proved same-session (reviewer re-derivation joins the 036-038 batch); the 0.497 floor (+0.000180) is NOT family-saturated (diagonal value at p=0.494 is 0.000104); nothing at n > 7; floors are upper bounds on the true inf
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Upgrades 038's n=6/7 story from 'descent stalled' to 'anneal saturated an identified extremal family'. The two equality families of (HU-TAX) -- products (independent) and diagonals (perfectly correlated) -- sit at opposite correlation extremes and both hit c*(pbar) exactly; lead 1 (the interpolation path between them) is the natural falsification test, and lead 2 records the per-cell proof shape both boundary patterns suggest. Joins reviewer batch 3 with 036/037/038.
Attempt 041informedEvidence
Falsification sweep between the two (HU-TAX) equality families finds NO dip: along both interpolation paths -- the convex mixture product->diagonal and the two-component product-mixture path that stays inside the recipe's latent-mixing genre -- the margin CR_HU/H - c*(p) is zero exactly at the endpoints (the proved equality cases, matched at every n up to 64 via an exact count-pair DP for exchangeable product mixtures) and strictly positive in the interior, GROWING with n (interior min at p=0.49: +2.2e-4 -> +9.1e-3 on the convex path, +1.4e-7 -> +8.7e-6 on the product-mixture path, n=2 -> 64), over 1968 grid points spanning n in {2..64} and caps up to 0.49 with zero negatives; on this evidence products and diagonals are locally rigid, isolated equality cases, not edges of a flat region.
- Range checked
- Two paths x n in {2,4,8,16,32,64} x p in {0.30, 0.38271, 0.45, 0.49} x 41-point grids plus 21-point refinements at each path's global minimum; exact count-pair DP for K <= 3 product mixtures, cross-checked vs the general evaluator at n <= 6 (<= 1e-10, marginals exact) and vs the rollout order for exchangeability (<= 1e-12); skeptic re-scores 50 sampled small-n margins through the independent 037 stack and verifies the closed-form endpoint identities at every n (the large-n DP validation); all float-level; non-exchangeable interpolations and n > 64 untested
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Settles 040 lead 1 positively for the two exchangeable paths at every n tested; the conjectured bound is locally reinforced between its equality cases, which is the shape 040 lead 2's per-cell argument wants. New lead: HU's CR is additive over independent blocks, so tensor products of diagonals and Bern(p) factors are further equality cases -- the equality set is a family of block-tensors, and symmetry-breaking perturbations BETWEEN blocks are the next falsification surface. Joins the next reviewer batch.
Attempt 042informedEvidence
The equality set of (HU-TAX) is a whole combinatorial family, and it resists attack at its own boundary: CR_HU is additive over independent blocks for EVERY revelation order (short proof recorded -- per-coordinate conditionals depend only on own-block prefixes, so the joint cell process is the product of block processes), hence every tensor of diagonal blocks and Bern(p) factors at common p has CR/H = c(p) exactly -- verified to 4.4e-16 over eight partition cases x five orders each including non-contiguous interleavings and the p < 1/4 clamp branch; anneal+polish attacks seeded AT equality points (caps 0.49/0.497, n = 4/6) find zero violations and only slide along the family toward the cap; close-outs: crash8 -- the one start never rollout-descended (039's correction) -- is positive at both caps (+0.000952/+0.000310), and the 0.497 anneal re-seeded at the diagonal saturates that cap's floor at c(0.494) to 1e-12 (040 lead 3), so every attacked cap's floor now sits on the diagonal family.
- Range checked
- Equality: 8 block-tensor cases (n = 4-6, p in {0.20, 0.30, 0.38271, 0.45, 0.49}) x 5 orders each (identity, rollout, 3 random interleavings), all to 4.4e-16, additivity proof same-session (reviewer re-derivation queued); attack: 5 anneal (4000 steps) + polish campaigns seeded at block-tensors, caps 0.49/0.497, n <= 6; close-outs: crash8 rollout descents (n = 8, reduced rounds) at 0.49/0.497, diagonal-seeded 0.497 anneal at n = 6; all float-level; the whole-equality-set characterization is NOT claimed (lead 1)
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Settles 041 lead 1's verify-then-perturb: verified, and the perturbations found nothing. The extremal landscape after 040/041/042: every adversary of record converges onto the block-tensor family from every seeding tried, and the margin grows in every off-family direction measured. The proof-shaped target is now sharp: CR_HU/H >= c*(p) with equality iff block-tensor (lead 1). No cheap close-outs remain on the HU line. Joins the next reviewer batch with 041.
Attempt 043informedVerified, with corrections
Reviewer pass on the interpolation/block-tensor batch (041, 042): everything load-bearing survives -- 041's count-pair-DP collapse and 042's every-order block additivity (plus 040's DIAG identity) re-derived by hand and both correct as recorded, a fifth independent HU/CR implementation (posterior-pair state, own no-pruning count DP) reproduces 132 sampled interp rows to 1e-9 at n up to 64, all of 042 part A through non-contiguous interleavings and the clamp branch to 1e-12, and every B/C endpoint to 1e-8 with all four pipelines re-running byte-identically at skeptic exit 0; corrections land on 042's part-B narrative: the two POLISHED attack endpoints (floors +0.000364 and +0.0000610) are NOT single diagonals and NOT on the equality family -- they are 8-atom equal-marginal measures with pairwise-CORRELATED blocks sitting +1.4e-5 ABOVE c(own marginal), the '+0.000364 = c(0.4887)' match is spurious (c*(0.4887) = 3.686e-4), and 'every endpoint ON the family' holds only for the three unpolished rows (one of which is a diagonal padded by two marginal-0 coordinates, so the equality set also extends beyond the common-p family as stated).
- Range checked
- Hand re-derivations: 041's count-pair collapse for K-product mixtures incl. q in {0,1} components and the H formula; 042's block additivity for every revelation order incl. interleavings, the DIAG identity, and the c(p) tensor corollary. Independent re-implementation (no imports from the reviewed stack): 132 of 1968 hu_interp rows (n in {2,8,64}, p in {0.38271,0.49}, both paths) to 1e-9, all summaries recomputed from stored rows, DP-vs-atom and exchangeability cross-checks at n <= 6, 5 own-built tensor cases x 4 orders to 1e-12, every hu_blocks B/C endpoint re-scored via own rollout to 1e-8 plus own-constant margins, C_sat497 == c(0.494) to 1e-12; all four committed pipelines re-run unmodified with byte-identical checkpoints and both skeptics exit 0; the anneal/polish searches not re-run seed-for-seed beyond that
- Verifies
- attempt 041
- Corrections
- 042 part B / outcome: 'the polished endpoints are single diagonals at p closer to the cap -- e.g. +0.000364 = c(0.4887)' and 'every endpoint ON the family' are false for the two campaigns where the polish improved (cap 0.49 and 0.497, n=6, d2xd2xd2): those endpoints are 8-atom measures on the block support with equal marginals (0.489001 / 0.495947) but pairwise-correlated blocks (P(both on) 0.238143 vs product 0.239122), CR/H - c(own marginal) = +1.47e-5 / +1.36e-5 > 0, and c*(0.4887) = 3.686e-4 != 3.639e-4; the three unpolished endpoints are exact family members
- clarification: the cap-0.49 part-B floor +0.000364 is a descent stall point, not a family value -- it undercuts the anneal's family-saturated floor c*(0.487) = +0.000488 only because the polish is not projection-bounded; 'every attacked cap's floor sits on the diagonal family' survives in the anneal-with-projection sense (040, C_sat497) only
- audit note, not an error in the records: the committed B/C 'violations' flag tests floor < 0, weaker than (HU-TAX)'s CR/H >= c*(own max marginal); re-checked every stored endpoint against its own constant here (all >= 0 to float noise), but intermediate anneal candidates are unchecked and future attacks should instrument the own-constant margin directly
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Discharges the queue's reviewer-batch item for 041/042. Independence disclosure, per the 029/034/039 precedent: fresh-context subagent with zero shared conversation state, but spawned from the author session with the review brief, same model family -- one level below 036's genuinely-fresh bar; whether that meets the 024 fresh-session bar is for the human owner to judge. Verifies 042 as well (schema allows one verifies id; 041 is the batch's lead record and verifies clean, corrections attach to 042's part-B narrative). No conclusion of either record changes: zero violations confirmed and strengthened via own-constant re-checks; the corrected picture leaves the family-convergence story with two counter-instances (lead: the correlated-block stall measures are the closest off-family points of record). Per precedent the reviewed records are not modified.
Attempt 044informedRefuted
Executing the 043 audit caveat window-wide finds the kill the weaker violation flag hid for four records: re-scoring all 212 stored endpoints against c*(own max marginal) under rollout exposes ONE violation -- the D_bestorder_cap_0.497 endpoint bred against the BEST-ORDER oracle and never before scored under rollout -- a 9-atom n=4 measure at max marginal exactly 0.49500 with CR_roll certified negative (-9.597e-4, each kit alone, 60-digit rollout fixed point confirmed, 18 of 24 orders negative, rollout ranking 13/24 on its own kill) while the best order is certified positive (+1.610e-2): rollout-order HU positivity is REFUTED above cap ~0.495, so HU with the rollout order cannot by itself deliver Frankl -- canonical rescued 0.45 and died at 0.49 (035), rollout rescued 0.49 and dies at 0.495, and best-order positivity (which survives this witness and every endpoint of record; census min own-margin +4.5e-2 over 1200 instances) is now the only unrefuted for-all-mu positivity statement on the HU line, exactly as 038's sandwich framing anticipated.
- Range checked
- 212 stored endpoints across hu_order2/hu_rollcensus/hu_roll_anneal/hu_blocks/hu_canon re-scored under rollout vs c*(own fmax) (211 clear, family-saturated anneal endpoints at exactly 0); the 1200-instance 038 census regenerated and re-scored (zero violations, min +4.49e-2); the kill confirmed by both float stacks to 1e-12 before certification, then certified on the limit_denominator(1e7) rationalization under each kit alone with exact marginal < 497/1000 and the positive best-order control; rollout's crossing bracketed only as (0.49, 0.495] by these families; best-order HU not attacked here (lead 1)
- Refutes
- 037
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- The audit lesson lands twice: (i) violation flags must test CR/H < c*(own fmax), not CR/H < 0 -- the weaker flag sat on this kill inside 037's own committed checkpoint; (ii) endpoints bred against one objective must be re-scored under every standing rule before a survival claim is recorded (transfer attacks are free) -- adopted as a standing skeptic step. Recipe v3 carries rollout-HU to a defended cap of 0.49; [0.49, 1/2) currently belongs to no rule; 038 lead 2 (best-order positivity) is the line's main conjecture, with lead 1 here the direct attack on it at caps 0.495-0.499. Joins the next reviewer batch.
Attempt 045informedEvidence
The direct attack on best-order HU -- the line's last unrefuted positivity statement -- at the caps where both fixed rules died: 67 weight-descents on the max-over-all-orders CR/H at caps 0.495/0.497/0.499 (n <= 5, full enumeration inside the objective, hostile-seeded with the 044 kill witness, the sharpest 037 D endpoints, and equality-family members perturbed off-family, every endpoint flagged against c(own fmax) per the 044 lesson) find ZERO kills and ZERO sharp violations; the binding endpoints saturate their own constant to 3e-9 -- the 042 equality family is the oracle adversary's attractor (off-family perturbations descend BACK onto it, own-margin exactly 0), which is precisely the (HU-TAX, best-order) shape; the 044 rollout-kill witness is comfortably positive under best-order (+3.07e-3); 038 lead 2 is promoted to the line's main conjecture (SPECULATION): max-order CR_HU >= c(pbar) H(mu) with equality exactly on the block-tensor family.
- Range checked
- Caps 0.495/0.497/0.499, n <= 5 only (enumeration cost), 21/23/23 descents per cap with 035's move set (the 040 anneal NOT ported to the oracle objective -- queued as lead 1); floors are stall points except the family-saturated ones; no certificates this round (the binding family values are 042's certified ones); skeptic re-derived every endpoint's ratio/fmax/margin by full enumeration through the independent stack, exit 0
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Survival earned against the same families and transfer seeds that killed canonical (035) and rollout (044), with the equality family visibly binding under the oracle -- but the record's own weak points are stated: descent-only adversary and n <= 5. Leads: port the 040 anneal to the oracle objective; sampled-order n = 6 attack; and the first proof probe -- max-order CR_HU >= c*(pbar)H on n = 2 blocks via the 041 count-pair DP, an honest-sized hand calculation. Joins the next reviewer batch with 044.
Attempt 046informedEvidence
The n = 2 case of the promoted best-order conjecture, taken as far as it goes: the closed form reproduces the general evaluator to 2.4e-15, the ORDER QUANTIFIER IS VACUOUS at n = 2 (the identity order alone meets the bound everywhere, worst margin 0 over a 1.38M-point grid, attained at products -- in contrast to n >= 4 where the order choice is decisive), the equality set is exactly products + diagonals + the n = 1 degenerations (each a hand identity, re-checked under both kits alone), and exact-rational branch-and-bound certifies 88.4% of the in-regime volume with ZERO residue (nothing resisted refinement; the rest is unfinished budget). Two lemmas are proved by hand -- N2-ONE-BAD (at most one conditional can be out of regime, so deficit cells contribute exactly 0) and N2-CONC (concavity of psi(t) = h(min(1/2,t)) gives the sharp cell-wise bound, tight iff the two conditionals agree) -- and together they reduce the theorem to a scalar inequality that is FALSE on 21.3% of the region while the original margin is positive there: an exact decomposition identity (verified to 2.6e-15) shows the discarded pair-interaction term t*Delta, t = min(x-1/2, 1-x), is load-bearing, so no argument that averages the cell interaction away can prove even n = 2.
- Range checked
- n = 2 only, arbitrary dependence (three parameters x, u0, u1); closed form checked on 19,886 in-regime samples and a 1.38M-point grid; equality identities at 24 rational points under each kit alone; branch-and-bound at budget 2,000,000 boxes, minimum edge 1/512, largest-box-first, in exact rational interval arithmetic under kit B with volume accounting (certified 0.235977, out-of-regime 0.233095, residue 0, unprocessed 0.030930 of root volume 0.5); the reduction test on 400,000 Case-A points; transversal local probes at eps = 1e-2 and 1e-3; everything else float-level. No theorem is claimed at n = 2 and nothing is claimed at n >= 3
- Corrections
- 034's stated constant c*(pbar) = (h(min(2pbar,1)) - h(pbar))/h(pbar), repeated in 037's Leads section and in STATUS.md, is wrong for every pbar > 1/4 and is NEGATIVE there (-0.181287 at 0.38271, -0.527591 at 0.45, -0.858519 at 0.49), which as written would make (HU-TAX) vacuous on exactly the range 034 re-posed it for. The constant every engine computes, and which reproduces every recorded number (+0.041739 at 0.38271, +0.007278 at 0.45, +0.000289 at 0.49, +0.0000260 at 0.497), is (h(max(1/2, 1-2pbar)) - h(pbar))/h(pbar). Prose-only: no computation in 030-045 evaluated the stated form, so no measured value moves. Root cause: the p < 1/4 branch (h(2p), where the Frechet floor binds) was written as if it were the general formula, with min(.,1) guarding the wrong endpoint
- this record's own first two certification runs used a depth-first queue and reported a residue volume without the unprocessed volume beside it, which read as near-complete coverage; measured, depth-first had certified 0.045% of the box and left 99.8% unprocessed at a 2,000,000-box budget. Fixed here by a largest-box-first queue, and the lesson is recorded: a certified sweep must report certified / out-of-regime / residue / unprocessed volumes together or quote no fraction at all
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Confirms at n = 2 that the difficulty in (HU-TAX) is neither the order quantifier (empty here, essential at n >= 4 per 033/035/044) nor the deficit cells (they contribute exactly 0 here) but the averaging of the pair interaction -- 031's recorded obstruction, now isolated to a two-variable function on a three-parameter box, with the exact target inequality written out in lead 1. Also finds a fourth, LIMITING equality structure at the q -> 1/2 boundary (two disjoint singletons at marginals 1/2, where CR = 0 and c = 0 together): the absolute margin CR - c(q)H degenerates there, so a descent on it drifts to the boundary and reports vanishing margins that are not near-violations -- the mirror of 023's H -> 0 corner lesson, and a guard every future (HU-TAX) adversary needs. Joins the next reviewer batch.
Attempt 047informedEvidence
The n = 2 target of 046, written as A + B + C >= 0 (A = first-coordinate slack, B = diagonal ledger, C = pair interaction; A, C >= 0 by proved sign lemmas so only B can be negative) exposes a DICHOTOMY: over 42,575 deficit cases every single one is covered by A alone or by C alone, none needs both and none fails, so the n = 2 Case-A theorem is exactly 'B < 0 implies max(A, C) >= -B'; the split is by which coordinate carries the maximum marginal -- when q = f0 the slack A vanishes identically and the pair interaction alone covers on all 7,435 cases, giving branch L1 as the smallest unproved statement on the HU line, stated in two variables. A third structural lemma falls by hand (N2-ONE-ABOVE: at most one conditional marginal can exceed q, since their x-weighted mean is f1 <= q), completing the deficit structure begun by 046's N2-ONE-BAD; the decomposition itself is verified to 2.9e-15 against an independent history-recursion evaluator, and a skeptic descent that actively maximises the shortfall bottoms out at +4.3e-07.
- Range checked
- n = 2, Case A (both conditionals in regime), 200,000 in-regime samples per part with fixed seeds; dichotomy census on 42,575 deficit cases; split census by dominant marginal (7,435 at q = f0 / 34,650 at q = f1); tightness minima on both branches plus a skeptic descent on the shortfall; the decomposition identity cross-checked on 30,000 samples against a from-scratch evaluator. All float-level: no certificates this round, no proof of either branch, nothing at n >= 3
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Advances 046 from 'the pair interaction is load-bearing' to 'it is load-bearing exactly when the first coordinate carries the max marginal, and then it suffices alone'. Mechanism reading: at q = f0 the slack A is zero by construction (nothing to donate) so the surplus must come from the coupling's own pair structure -- which is why 031's averaging schemes failed, since averaging destroys exactly that term. Both branches remain unproved; the only tight corner is x -> 1 (the n = 1 degeneration, where t, B and C vanish together), so a proof can be crude elsewhere and should expand in (1-x) there. Joins the next reviewer batch with 046.
Attempt 048informedEvidence
Branch L1 of 047, taken into scale-free form: the ratio C/(-B) has no tight interior point (sampled floor 1.1492 over 600,000 branch samples; a skeptic descent that actively minimises it bottoms at 1.0881), the boundary configuration (p0,p1) = (0,1/2) evaluates to EXACTLY 1/c(q) (verified to 2.2e-16), so the branch infimum is exactly 1 and is approached only as q -> 0 and only LOGARITHMICALLY -- there is no polynomial corner to expand, which corrects 047 lead 1. Two further structure results: at every tight configuration t = 1-x = q (375 versus 0 among ratio < 1.5 cases), so L1 reads qDelta >= -B; and the natural c-free simplification is useless -- tDelta dominates the c-free bound on 0 of 600,000 samples, so c is essential. Along the extremal the whole statement collapses to h(2q) <= 2h(q).
- Range checked
- Branch L1 only (q = f0), 600,000 sampled branch points per part with fixed seeds; the boundary identity at six values of q; the fixed-q minimiser located on a 140x140 mesh at seven values of q (corner (0,1/2) for q <= 0.03, interior above, independently re-classified by the skeptic on a 90x90 mesh); the ratio floor attacked by a minimising descent. All float-level: no certificates, no proof of L1, nothing outside the branch, and the interior minima are mesh minima
- Corrections
- 047 lead 1 advises expanding in (1-x) at x -> 1 as the tight corner of L1. That is the ABSOLUTE margin C + B, which vanishes there only because both terms do -- the same trap 046 section G recorded at the q -> 1/2 boundary. In the scale-free ratio there is no polynomial corner: the approach to the infimum 1 is logarithmic in q (c*(q) = h(2q)/h(q) - 1 with h(2q)/h(q) -> 2 like 1 + 1/log2(1/q)). The simplification actually worth having is P3: at every tight configuration t = 1-x = q
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- The negative result is the useful one: L1 has infimum exactly 1, so no argument with slack to spare can prove it -- every bound must be asymptotically sharp as q -> 0. That kills the crude-bound family outright (P4) and explains why 046's concavity route could not suffice: concavity is tight only at p0 = p1, whereas L1's extremal is (0,1/2), maximally unequal. Along that extremal the statement collapses to h(2q) <= 2h(q), elementary and true; the open part is everything off it. Joins the next reviewer batch with 046/047.
Attempt 050informedEvidence
The size this route had never examined: n = 3, judged against c*(own max marginal) per the 044 standard. Best-order positivity SURVIVES (0 violations across 4,500 census instances with full order enumeration and 16 descents at caps 0.45/0.49, the descents saturating the equality bound at +0.000000 rather than crossing it), and so does rollout (2000/2000) -- so the promoted conjecture now has support at every size the line has examined, n = 2 through 7. But the order quantifier is NOT vacuous here: the identity order fails the own-constant bound at n = 3 (2 of 3000 at cap 0.49, min margin -0.00696), and at the sharpest witness -- a 4-atom measure with max marginal exactly 0.485 -- its CR is certified NEGATIVE (-1.005300143e-2) under each kit alone while the best order is certified positive (+2.775518457e-1). So the vacuity 046 proved is special to n = 2, and n = 3 already behaves like n >= 4.
- Range checked
- n = 3 census 3,000 instances per cap at 0.45/0.49 with full 6-order enumeration, n = 4 contrast 1,500 per cap with 24 orders, 2,000 fresh instances per size for the quantifier measurement, 8 best-order descents per cap at n = 3; all float-level except the certified order-failure witness (both kits alone, exact marginal and H checks in rationals). No certificate for any positive result; nothing above cap 0.49 at n = 3; 16 descents is a thin adversary next to the 17-start campaigns of 033/035
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Bounds 046 from above: the n = 2 vacuity was tempting to read as the start of a pattern and is not one -- one extra coordinate restores order-dependence, which fits 046's own mechanism (N2-ONE-BAD used that both conditionals out of regime forces f1 > 1/2, and that argument has no analogue after a length-2 history). A proof strategy built on the n = 2 lemmas (046/047/048) therefore needs new ingredients at n = 3, not just more algebra. What grows stronger is best-order positivity, now measured at every size 2 through 7, always with the equality family binding and never crossed. Joins the next reviewer batch.
Attempt 051informedEvidence
Same-day self-correction to 050 part D: its best-order descents at n = 3 reached min own-constant margin +0.000000 not by finding an n = 3 extremal but by DEGENERATING -- both sharpest endpoints have an atom driven to weight 0 and at most two coordinates carrying mass, i.e. they are n = 1 instances embedded in n = 3 where equality is automatic. Re-running under an essentiality constraint (every coordinate marginal >= 0.05, at least 4 atoms above 1e-3; not vacuous, it rejects 659 of 3000 random draws) gives the honest n = 3 best-order floor: +0.000018 at cap 0.45 and +0.000249 at cap 0.49 over 20 descents, three orders of magnitude above the degenerate 2.6e-09, still strictly positive, and the sharpest endpoint certified > 0 under each kit alone (CR_HU in [+3.691112270e-05, .]). 050's census, order-quantifier finding and certified negative witness are untouched -- they do not involve the part-D descents.
- Range checked
- The two sharpest 050 part-D endpoints audited from the committed checkpoint; 10 essential-support descents per cap at 0.45 and 0.49 (n = 3, essentiality thresholds 0.05 marginal and 4 atoms above 1e-3, stated as a choice); one certificate on a limit_denominator(1e7) rationalization under each kit alone. Float-level otherwise; 20 descents is not an exhaustive adversary and nothing is claimed above cap 0.49
- Refutes
- 050
- Corrections
- 050 part D reports min own-constant margin +0.000000 with 0 violations and reads the saturation as the equality family binding, consistent with 042. The endpoints are degenerate: cap 0.45 has coordinate marginals [0.4462, 0.0, 0.0] with 2 live atoms and cap 0.49 has [0.4895, 0.4895, 0.0] with 2 live atoms, so both are n = 1 faces where the bound holds with equality automatically. Every number 050 quotes is correct; the reading is not. The corrected sentence: the descent escapes to the n = 1 face rather than finding an n = 3 near-extremal
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Third degeneracy caught in one window and the same shape as the other two: an adversary minimising a margin finds the cheapest way to make it vanish, and 046 section G caught a vanishing constant (q -> 1/2), 046 section G-prime a vanishing coverage (depth-first B&B), and this a vanishing dimension. Standing lesson proposed: descent campaigns on this route carry an explicit non-degeneracy constraint and report the fraction of draws it rejects, alongside 044's own-constant flag and the scale-free margin. Lead: re-audit the 037/038/040/045 descent campaigns for support collapse -- the diagnostic is three lines and would either confirm those floors or correct several at once.
Attempt 052informedEvidence
Effective-dimension audit of every descent endpoint the route has recorded, prompted by 051: of 328 stored endpoints, 96 (29.3%) have a coordinate with marginal exactly 0 and so live at a smaller n than their label, and of the 27 HEADLINE floors that the records actually quote, 11 are quoted at a higher n than the instance uses -- most severely the n = 7 anneal floor of 040, which is a TWO-dimensional instance (and which explains why that run reported exactly the same +0.000488 as its n = 6 counterpart: it was the same instance). Positivity and kill claims are untouched, since every endpoint is still a valid in-regime instance and a certified negative CR at effective n is still certified; what is affected is dimension labelling, so the claims to have probed n = 6 and n = 7 rest on fewer genuinely high-dimensional endpoints than the records imply. Atom count is explicitly NOT used as a criterion -- a diagonal has two atoms and is genuinely n-dimensional -- a mistake this audit made on its first pass and corrected before recording.
- Range checked
- Pure re-analysis of all committed hu_*.json checkpoints: 328 endpoints across 27 blocks from campaigns 033, 035, 037, 038, 040, 042, 045 and 050. Criterion is marginals only (a coordinate with marginal exactly 0 is absent); min marginal below 0.05 but nonzero is reported separately as near-degenerate (33 endpoints) and not folded in. No new search, no certificates
- Corrections
- 11 of 27 recorded headline floors quote a higher n than their endpoint uses: hu_attack cap_0.38271 (4 -> 2), hu_canon B_cap_0.38271 (4 -> 3), hu_order2 C_surp_cap_0.49 and 0.497 (5 -> 3), hu_order2 C_roll_cap_0.497 and C2_roll_cap_0.497 (4 -> 3), hu_roll_anneal A_n6_cap_0.497 (6 -> 4), hu_roll_anneal A_n7_cap_0.49 (7 -> 2), hu_rollcensus P2_cap_0.49 (6 -> 5) and P2_cap_0.497 (6 -> 4), hu_n3_census D_cap_0.45 (3 -> 2). The affected records are 033, 035, 037, 038, 040 and 050; each is left as written per the repo rule. No positivity or kill claim changes -- only the dimension the floor is evidence about
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Fourth cheap-escape caught in this window and the widest: a vanishing constant (046 section G), a vanishing coverage (046 section G-prime), a vanishing dimension in one campaign (051), and a vanishing dimension across most campaigns at once (here). The honest state: the conjecture has never been violated at any effective dimension, but the evidence at n >= 6 is thinner than the record implies. Leads: re-run the 040 anneal and the 038 P2/P3 blocks under the 051 essentiality constraint (expect the floors to rise as they did at n = 3, +2.6e-09 -> +1.8e-05; whether they stay positive is the falsifiable part), and decide whether the adversary prefers degenerate faces because they ARE the extremal structure -- if so, the equality families of 042 are incomplete.
Attempt 053informedVerified, with corrections
Reviewer batch 5 over 044-048, default stance refute: all six committed checkpoints re-run BYTE-IDENTICALLY and all four skeptics exit 0; the 044 kill is recomputed at 50+ digits by an independent exact evaluator (CR_roll = -0.0009596936656816684080411, inside both kits' enclosures), its rollout order re-derived from the 035/037 rules, its witness confirmed already present in 037's committed checkpoint, and all 120 hu_order2 endpoints re-audited to 6.4e-16 finding the same single violation; 045's 67 endpoints, 046's B&B accounting and part-E/G numbers, 047's dichotomy census (needing both = 0) and 048's minimiser table all reproduce through independently written code -- but eight corrections follow, two of them mathematical: 046's Lemma N2-CONC is tight not only at p_a = p_b but on the whole region min(p_a,p_b) >= 1/4 where psi is constant (worked instance (0.30, 0.40)), and 048's boundary identity ratio(q,0,1/2) = 1/c(q) holds only for q <= 1/4, since ratio = min(1/2-q, q)/(q c(q)) (4.94930 vs 7.42395 at q = 0.3), a gap that survived its skeptic because engine and skeptic both sampled only q <= 0.2.
- Range checked
- All five records audited against the committed checkpoints; ten pipelines re-run (uc_hu_ownconst, uc_hu_ownconst_certify, uc_hu_bestorder(+skeptic), uc_hu_n2(+skeptic, 819 s), uc_hu_n2_dichotomy(+skeptic), uc_hu_L1(+skeptic)) with hu_ownconst / hu_ownconst_certify / hu_bestorder / hu_n2 / hu_n2_dichotomy / hu_L1 byte-identical; the five hand statements (N2-ONE-BAD, N2-ONE-ABOVE, N2-CONC and its averaging step, the exact decomposition in BOTH cases, the L1 boundary identity) re-derived from scratch before reading the records' arguments; the 044 certificate recomputed in exact rationals with 80-digit Decimal entropies (neither repo kit) and checked for containment in both kits' enclosures; 045's 67 endpoints re-scored by full 24/120-order enumeration (max deviation 7.2e-16); 046/047/048's censuses re-run on this reviewer's own samplers and grids. Not reviewed: 049-052 (written concurrently), anything at n >= 3, and the transcript-only DFS figures of 046 G' (corroborated only qualitatively by a float-enclosure surrogate).
- Verifies
- attempt 046
- Corrections
- 046's Lemma N2-CONC states equality in s(p_a,p_b) >= (G(p_a)+G(p_b))/2 holds iff p_a = p_b; that clause is false. psi(t) = h(min(1/2,t)) is CONSTANT on [1/2, infinity), so the bound is also tight whenever min(p_a,p_b) >= 1/4 with p_a != p_b -- e.g. p_a = 0.30, p_b = 0.40, where both sides equal 0.073879 exactly. The inequality itself, the averaging step over the cells, and every use made of the lemma survive; the sharpness clause does not, and 048's 'concavity is tight only at p0 = p1' inherits the imprecision. Corrected clause, verified on 200,000 pairs: equality iff p_a = p_b OR min(p_a,p_b) >= 1/4.
- 048's boundary identity ratio(q, p0=0, p1=1/2) = 1/c(q) is stated unqualified but holds only for q <= 1/4. Own derivation: at that configuration Delta = 1 so C = t = min(1/2-q, q) and -B = q c(q), hence ratio = min(1/2-q, q)/(q c(q)), which equals 1/c(q) only while t = q. At q = 0.3 the ratio is 4.94930 against 1/c*(0.3) = 7.42395. The engine tabulates six values q <= 0.1 and the skeptic tests q <= 0.2, so neither probed the failing range. Everything the record concludes from the identity (the corner satisfies L1 strictly; the infimum-1 story at q -> 0) survives.
- 046 F's own correction of 034 over-reaches: the mis-stated constant (h(min(2p,1))-h(p))/h(p) is NEGATIVE only above p = 1/3, not above p = 1/4 (h(2p) = h(p) iff 3p = 1). On (1/4, 1/3) it is positive but too small -- +0.101737 against the correct +0.134707 at p = 0.30 -- so it is wrong there without being vacuous. The correction's substance stands: the stated form differs from the computed one above 1/4, and no computation uses it (17 c/z/sigma definitions in explore/ all compute the max(1/2, 1-2p) form; the sole repo-wide occurrence of the wrong form is a diagnostic print in uc_reviewer034_reimpl.py:733 that feeds no checkpoint).
- 047's 'A >= 0 and C >= 0, so only B can be negative' and its entire census are CASE-A ONLY (the engine draws p0, p1 in (0,1/2)); the index's range field says so but the record's part A and its 'the deficit structure at n = 2 is now completely determined' do not. Outside Case A -- 50.2% of in-regime measures under the uniform (x,u0,u1) parametrization -- the ledger interaction C = t*Delta is negative on 63% of sampled points and 047's closed-form A+B+C is not the margin at all (deviation up to 0.5), because Delta must be read as the ledger form, not the psi closed form. The n = 2 conjecture is not endangered there: the margin stays >= 0 over 200,000 Case-B samples (worst +1.6e-5).
- 048 P3's 'every tight configuration has t = 1-x = q (375 versus 0)' quotes a --fast run: the committed checkpoint says 5,698 versus 0, reproduced exactly by an own full re-run of the same generator. The qualitative claim is unaffected.
- 046's 'worst margin 0 over a 1.38M-point grid' (parts A and B, and this index's one_line and range) describes a superseded skeptic: the committed uc_hu_n2_skeptic.py sweeps N = 60, i.e. 219,539 points, worst -6.939e-16 -- which is exactly what 046 part C quotes as -6.9e-16. 1,379,840 = 110 x 112 x 112 is the N = 111 grid; part A's -6.7e-16 matches neither committed run.
- 044's 'rollout ranks 13/24 on its own kill' is not reproducible under a best-to-worst ranking: all six orders revealing coordinate 1 first give exactly the same CR (-9.596936657e-4), so rollout sits in a six-way tie at descending ranks 7-12. 13 is that block's position counting from the WORST end, under which rollout is above the median -- the opposite of the record's gloss 'worse than the middle'. The kill itself is unaffected (18 of 24 orders negative, confirmed).
- 044's parenthetical that the endpoints at exactly 0 are 'the family-saturated anneal endpoints' misattributes them: the three exactly-zero rows are hu_blocks.json block-tensor rows (B_attack:[(d,2),(d,2)], B_attack:[(d,3),(d,3)] to 5e-20, C_sat497), while the closest hu_roll_anneal endpoint sits at +1.53e-11. The audit's substance (211 of 212 clear) is unaffected.
- Reporting nits: 047's parts B and C are different samples, so 42,575 deficit cases and 7,435 + 34,650 = 42,085 do not reconcile as the prose implies; 046 part E's 'identity verified to 2.6e-15 over 99,970 in-regime samples' corresponds to no committed check (the identity is true -- 6.5e-16 here over both cases -- but the committed cross-check is 047's skeptic S1, 2.89e-15 over 30,000 samples); 045's '+3.07e-3' for the 044 witness is the cap-0.499 row, +5.13e-3 at caps 0.495/0.497.
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Independence caveat, as in 029/034/039/043: fresh-context subagent with zero shared conversation state, but spawned from the author's session with a reviewing brief and the same model family -- whether that meets 024's fresh-session bar is for the human owner to judge. Commissioned as 049 (hence uc_reviewer049_reimpl.py); 050-052 were committed while it ran, so new_attempt.py assigned 053 and 049 is an unused id. The re-implementation imports none of the reviewed engines or either certification kit. Standing proposal from the batch: three of the eight corrections would have been caught by requiring every skeptic to sample at least one region its engine does not, and to say which.
Attempt 054informedEvidence
The falsifiable question 052 posed, answered in the safe direction: re-running the high-n campaigns under 051's essentiality constraint (every coordinate keeps marginal >= 0.03, so the adversary cannot leave the dimension it was launched in) leaves every floor positive and RAISES all of them -- rollout at n = 6 goes +0.030430 -> +0.091924 (cap 0.49) and +0.053740 -> +0.079948 (cap 0.497), rollout at n = 7 goes +0.099692 -> +0.135386, and a best-order spot check at n = 6 gives +0.074824 -- with zero violations and zero endpoints losing a coordinate. So the previously-degenerate high-n claims were not hiding a violation, and the four cheap escapes this window catalogued were all making the conjecture look HARDER to defend than it is, never easier. Also re-checks 053's correction (ii) independently: with t = min(1/2-q, q) the boundary ratio is t/(q c*(q)), which grows above q = 1/4 (4.95 at 0.3, 70.7 at 0.49), so the L1 infimum is still approached only as q -> 0.
- Range checked
- Rollout descents at n = 6 (6 starts per cap at 0.49 and 0.497) and n = 7 (4 starts at 0.49), plus a best-order SPOT CHECK at n = 6 (one start, 12 rounds, all 720 orders enumerated per candidate -- stated as a spot check, not a campaign, because the objective costs 720 CR evaluations per candidate). Essentiality threshold 0.03 is a stated choice. Float-level throughout: no certificates this round. 040's anneal has no constrained counterpart yet
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Closes 052 lead 1 for the 038 campaigns. The direction of the shift is the content: in every case the honest floor is higher than the degenerate one, so the degeneracies manufactured near-misses rather than hiding violations -- the opposite would have been far more serious. Remaining: a real best-order campaign at n = 6 (the spot check needs 8-17 starts, which needs branch-and-bound over orders to afford), an essentiality-constrained rerun of 040's anneal (the source of the worst overstatement, n = 7 -> effective n = 2), and folding the essentiality check into the standing harness so campaigns cannot regress.
Attempt 055informedEvidence
Re-running the 040 anneal with essentiality enforced inside move acceptance (2,933 moves rejected for it, so the constraint was actively binding) gives floors +6.26e-13 at n = 6 cap 0.49, +2.079e-04 at 0.497 and +2.484e-04 at n = 7 -- zero violations, zero endpoints losing a coordinate -- and the sharpest endpoint converges to a DIAGONAL {empty: 0.6174, full: 0.3826} with every marginal 0.3826, i.e. a genuinely 6-dimensional member of the 042 equality family, so margin 0 there is expected rather than a violation. That corrects 054: its constrained figures (+0.091924, +0.079948, +0.135386) are descent STALL POINTS, not floors -- the anneal reaches five orders of magnitude lower under the same constraint. Fifth instance this window of one lesson in a new dress: a weak adversary's stall is not a floor, and where degeneracy made floors look too small, stalling makes them look too large.
- Range checked
- 3 anneals x 900 steps per block at n = 6 (caps 0.49, 0.497) and n = 7 (cap 0.49), essentiality threshold 0.03 enforced in move acceptance; comparison against the 040 checkpoint with each recorded endpoint labelled admissible or excluded (two of three have min marginal exactly 0 and are excluded, so those rows are not like-for-like). Float-level: no certificates this round, nothing above cap 0.497, and no campaign larger than stated
- Refutes
- 054
- Corrections
- 054 reports essentiality-constrained floors of +0.091924 (n = 6 cap 0.49), +0.079948 (n = 6 cap 0.497) and +0.135386 (n = 7 cap 0.49) and reads them as the honest floors. They are stall points of a pattern-search descent: the constrained anneal here reaches +6.26e-13 at n = 6 cap 0.49, five orders of magnitude lower, on a genuinely 6-dimensional instance. 054's comparison against the DEGENERATE recorded floors still holds in direction; what does not hold is treating its numbers as floors
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Also the first demonstration that the equality family is reachable in ESSENTIAL form at n >= 6 -- 042 predicted diagonals are extremal at every n, and this is the first adversary to land on one at that size. Leads: quote every floor with an adversary-strength note (054 and 055 differ by five orders of magnitude on the same quantity, so the search and its budget are part of the claim); point the constrained anneal at the caps above 0.497 that 045 only reached with descents; and certify the n = 6 diagonal endpoint as an exact equality instance via the 042 identity at p = 0.3826.
Attempt 056informedVerified
The n = 6 endpoint reached by the constrained anneal of 055 is certified an EXACT equality instance under each kit alone: rationalized to p = 2047143/5349940, all six coordinate marginals equal p exactly (so it is genuinely 6-dimensional, not a smaller instance embedded in six coordinates -- the failure mode 052 found across the route), CR_HU - c*(p)H is enclosed in [-4.930e-32, +4.930e-32] under kit A and [-5.047e-19, +5.047e-19] under kit B (both containing 0), and CR_HU is certified positive at +0.040109336 with H > 0.9599. So 042's diagonal identity is attained in ESSENTIAL form at n = 6, 055's float +6.26e-13 is exact equality rather than a small gap, and the essential floor at n = 6 is exactly 0 -- meaning any future floor claim above 0 at that size describes the search, not the problem.
- Range checked
- One instance: the 055 A_n6_cap_0.49 sharpest endpoint, rationalized by limit_denominator(1e7), certified under each kit separately with the marginal equality checked exactly in rationals. Nothing certified at n = 7 (055's endpoint there is +2.484e-04 and is not on the family), and no block-tensor equality instance is certified here
- Verifies
- attempt 055
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Closes a gap the window kept re-opening: five times a float near zero was read as meaning something and three of those readings were wrong (a vanishing constant, a vanishing coverage, two vanishing dimensions, a stall). Here the near-zero number means exactly what it appeared to -- and certification is the only way to tell the two cases apart, which is the transferable point. Leads: certify a block-tensor equality instance at n = 6 to confirm the essential equality set is the full family there, and decide whether the n = 7 endpoint misses the family because the anneal did not converge or because the family is harder to reach at that size.
Attempt 057informedEvidence
A seeded control settles what the n = 7 gap of 055 meant: placed ON the equality family, the constrained anneal STAYS there at n = 7 (margins -2.1e-15 and -2.7e-15 after 900 steps, float noise on a quantity certified exactly 0 at n = 6 in 056) exactly as it does at n = 6, so the family is a stable attractor at that size and the essential floor at n = 7 is 0. Unseeded and given 3x the budget (2,700 steps, three runs) the anneal still does not find it, improving only from +2.484e-04 to +1.840e-04. So the n = 7 gap is search convergence, not structure -- reading (i) of the two 056 posed -- and the positive n = 7 numbers on record describe the search rather than the problem. Technique worth keeping: when an adversary fails to reach a known target, seeding it at the target separates cannot-get-there from did-not-get-there, and those have opposite consequences.
- Range checked
- The 042 diagonal identity checked at n = 6 and 7 for p in {0.3826, 0.45, 0.487} (float, margin 0.000e+00); four seeded anneals at 900 steps (n = 6 and 7, two p each); three unseeded anneals at 2,700 steps at n = 7 cap 0.49, all essential. Float-level throughout -- no certificate at n = 7, though the 042 identity argument certified in 056 applies verbatim; nothing above cap 0.49 at n = 7; and only that 3x the budget fails, not that no budget succeeds
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Sixth time this window that a near-zero number, or a failure to reach zero, needed a second measurement to interpret; the through-line is now explicit -- on this route no extremal number means anything until something independent says which of its possible meanings applies. Leads: seed every future high-n campaign at the equality family as a control run alongside the unseeded search (it costs one anneal and says whether a positive floor is real), and test whether the n = 6 to n = 7 difficulty jump is pure volume by adding a collapse-toward-a-diagonal move, which the current move set never proposes.
Attempt 058informedEvidence
One added move settles the question 057 posed and refutes its volume reading: the anneal move set never proposes a diagonal directly (reaching {empty, full} from a ten-atom support needs eight coordinated drops), so adding a single collapse-toward-a-diagonal move and re-running the same campaign with a same-code same-seed control gives -- WITH the move, all 3 runs at n = 7 reach the equality family (floor -1.227e-14, 3 of 3 endpoints are diagonals); WITHOUT it, none do (+2.300e-05, 0 of 3), at identical budget. The move also works at n = 6 without breaking what already worked. So the n = 7 difficulty was the move set, not the doubling volume, and one move closes a gap that tripling the budget did not.
- Range checked
- Three runs x 2,700 steps per block at cap 0.49 only, n = 7 (with move and matched control) and n = 6 (with move), essentiality enforced throughout. Float-level: no certificates, though the 056 identity applies verbatim to the diagonals found. Not tested at other caps, and reaching the conjectured extremal family says nothing about whether a lower point exists
- Refutes
- 057
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Method result worth carrying: an adversary that cannot construct the known extremal family in one move will systematically overstate floors, so the collapse move belongs in the standing anneal and any newly discovered extremal family should be checked against the move set the same way. This is also the cleanest instance of the window pattern and the first where the fix was constructive rather than a caveat -- the matched control both interpreted the number and repaired the instrument that produced it. Leads: re-run the 045 best-order caps 0.495-0.499 with the augmented anneal (that campaign used descents only and those caps are closest to the boundary), and audit the move set against 042's block tensors, which are not reachable in one collapse move either.
Attempt 059informedEvidence
The promoted conjecture put to the strongest adversary this route has built -- the essentiality-constrained anneal with the 058 collapse move, against the best-order objective with full order enumeration per candidate -- at the caps closest to 1/2 that 045 could only reach with descents: ZERO violations in 15 anneals across caps 0.495/0.497/0.499 at n = 4 and 0.495/0.499 at n = 5, and at the hardest cap tested (0.499, within 0.001 of the boundary) the anneal lands exactly on the equality family (-3.220e-15, float noise around the value certified exactly 0 in 056). Across this window the adversary was strengthened four times and each time the floors moved downward onto the family and never through it -- the signature a true conjecture with a known extremal family should produce.
- Range checked
- 15 anneals: 3 x 1500 steps at n = 4 for caps 0.495/0.497/0.499 (24 orders per candidate) and 3 x 750 steps at n = 5 for caps 0.495/0.499 (120 orders per candidate), essentiality enforced with the collapse move enabled. All three of the 045 endpoints at these caps have min marginal exactly 0 and are excluded by essentiality, so the anneal-versus-descent comparison is not like-for-like and is labelled so in the checkpoint. Float-level: no certificate here, nothing at n >= 6 for the best-order objective, nothing above cap 0.499
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- The substance behind 'the promoted conjecture survives' is now stronger than the absence of counterexamples: four successive strengthenings of the adversary (own-constant flagging, essentiality, annealing, the collapse move) each moved the recorded floors down onto the equality family and never through it. What remains open is unchanged -- no proof at any n >= 2, and the best-order objective is untested at n >= 6 beyond a single spot check. Leads: best-order at n = 6 with rollout as a cheap lower bound and order enumeration only near the floor; certify the cap-0.499 equality endpoint by the 056 route; and a cap sweep toward 1/2 where c* -> 0 and the scale-free form of 046 section G becomes mandatory.
Attempt 060informedEvidence
Same-day correction to 059, found while certifying its endpoint: the campaign caps constrain only the instances a run STARTS from -- admissibility is max marginal < 1/2 -- so the adversary is free to wander to small marginals, and the equality endpoint 059 reports at cap 0.499 has its own max marginal at 0.0360, not 0.499. Four of five sharpest endpoints had drifted off their caps, while those that stayed have floors three orders of magnitude larger. Re-running with a marginal FLOOR (max marginal >= cap - 0.005) gives the route's first genuinely near-boundary best-order numbers: zero violations with marginals pinned in [0.492, 0.500], and at cap 0.495 the adversary reaches the equality family INSIDE the band (floor exactly 0, at the diagonal {empty: 0.5082, full: 0.4918}); the positive floors at 0.497/0.499 are budget artifacts since diagonals exist at every p.
- Range checked
- Three anneals per block at n = 4 (caps 0.495/0.497/0.499, 1500 steps, 24 orders per candidate) and n = 5 (cap 0.499, 750 steps, 120 orders), with essentiality AND a marginal floor of cap - 0.005 enforced in admissibility; every endpoint verified to have stayed in the band. Float-level: no certificates, nothing at n >= 6, and 12 anneals do not exhaust the near-boundary region
- Refutes
- 059
- Corrections
- 059 headlines that at cap 0.499, the hardest tested, the anneal lands exactly on the equality family. The endpoint in question has own max marginal 0.0360, so that run did not test the boundary at all -- the cap bounds only the starting instances. 059's zero-violation result stands as a statement about the in-regime space at large; its boundary claim does not. The drift is visible in 059's own checkpoint: sharpest endpoints at own marginals 0.4846 (cap 0.497), 0.0360 (0.499), 0.0930 (n=5 cap 0.495) and 0.0390 (n=5 cap 0.499)
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- The same error 044 named, one level up: 044 established that margins must be judged against the instance own constant rather than the campaign cap, and this is that mistake applied to the search REGION rather than the constant. Three constraints have now leaked in this route -- dimension (052), essentiality (051), and the cap (here) -- each caught only by inspecting endpoints. Standing lead: a single harness check that every recorded endpoint satisfies its campaign stated region would have caught all three. Next: push the band toward 1/2 (caps 0.4995, 0.4999), where c* -> 0 and the scale-free form of 046 section G becomes mandatory.
Attempt 061informedVerified
One committed check for all three constraints that leaked on this route (dimension in 052, essentiality in 051, the cap in 060), run over every stored endpoint: of 393 endpoints, 135 (34.4%) leave the region their own campaign declared -- 96 with an absent coordinate, 31 below the essentiality bar, 8 drifted off their cap. The decisive detail is which rule NEVER fires: R3, max marginal < 1/2, has zero violations, and it is the only one the engines enforce inside their objective functions rather than at the seed. So the principle is evidenced, not just asserted -- a constraint checked when a run starts will leak, a constraint checked inside the objective does not -- and the post-051 checkpoints, which were built that way, are clean while the pre-051 campaigns carry all 135 flags.
- Range checked
- Pure re-analysis of all committed hu_*.json checkpoints: 393 endpoints across every block that stores measures. Four rules (absent coordinate, essentiality 0.03, in-regime, cap drift beyond 0.02 -- the last deliberately loose so it flags only real drift). Blocks predating a rule are reported rather than excluded, since the output is a map of which numbers describe their claimed region, not a verdict on records written before the rule existed. No new search, no certificates
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Seventh and last measurement-discipline finding of the window, and it generalises the other six -- a margin against the wrong constant (044), a vanishing constant (046 G), a vanishing coverage (046 G-prime), a vanishing dimension (051, 052), a stall read as a floor (055), a cap bounding only the start (060). The unifying repair is not more care but putting the constraint where the search cannot avoid it and auditing endpoints rather than intentions. No positivity or kill claim changes, since R3 is clean and every endpoint remains a valid in-regime instance. Leads: gate future campaigns on this check, and re-run the two hu_order2 D_bestorder blocks that feed 037/045 headline claims and carry R1 flags.
Attempt 062informedVerified, with corrections
Discharging 061 lead 2 -- the two flagged blocks that still feed live claims (hu_order2 D_bestorder at caps 0.49 and 0.497, which supply the best-order floors of 037 and 045) recomputed over clean endpoints only: at cap 0.49 the recorded floor +6.942017e-04 already comes from a clean endpoint (product4, all marginals 0.4888) and is sound, while at cap 0.497 the recorded +2.190730e-04 comes from a FLAGGED endpoint whose min marginal is exactly 0, and the honest clean floor is +2.632841e-04. Both are positive and both exceed c*(0.497) = 2.6e-05, so no qualitative claim moves -- the recorded number was 20% sharper than its claimed region could justify. Useful calibration: after seven measurement-discipline findings the live headline numbers are off by 20% in one place and correct in the other, because the conclusions never rested on a single sharpest number.
- Range checked
- Re-minimisation over the clean subsets of two committed blocks (14 endpoints each, 6 and 5 flagged); no new search and no certificates. Says nothing about the other flagged blocks, which belong to superseded campaigns
- Corrections
- the D_bestorder_cap_0.497 headline floor in hu_order2.json, +2.190730e-04, is produced by the seed0 endpoint whose min coordinate marginal is exactly 0 -- an absent coordinate, so the instance is effectively smaller than the n = 4 it is labelled with. Over the 9 endpoints that satisfy the region 061 checks, the floor is +2.632841e-04 (product4, marginals 0.4969). The cap-0.49 block needs no restatement: its recorded floor already comes from a clean endpoint
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Closes the endpoint-hygiene thread opened by 051: 061 supplies the gate, this supplies the one live restatement it implied. The calibration is worth keeping -- the route conclusions were robust to a third of its endpoints being mislabelled precisely because no conclusion rested on a single sharpest number. Open mathematics unchanged: no proof at any n >= 2, the n = 2 obstruction mapped in 046-048, and the best-order objective untested at n >= 6 beyond a spot check.
Attempt 063informedEvidence
Best-order at n = 6 -- the size that previously had a single spot check -- made affordable by one observation: rollout LOWER-BOUNDS best-order, so whenever the rollout margin is already non-negative the best-order bound holds with no enumeration at all. Across 6,000 essential in-regime instances at caps 0.49 and 0.497 the lower bound sufficed on 100% of them, the 720-order enumeration was never needed, there are zero violations, and the min own-constant margins are +0.014946 and +0.037327; an n = 5 control that enumerates all 120 orders regardless confirms the shortcut never disagrees (1,000 of 1,000). So the expensive objective was never the obstacle, and the route now has census evidence with no violation at every size from n = 2 to 7.
- Range checked
- 3,000 essential in-regime instances per cap at n = 6 (0.49 and 0.497) plus a 1,000-instance n = 5 control with full 120-order enumeration; essentiality enforced and margins judged against each instance own constant. This is a random CENSUS, not an adversarial campaign -- no descent or anneal floor at n = 6 best-order is claimed. Float-level, no certificates, nothing above cap 0.497
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- The sufficiency fraction is itself the interesting statistic: at exactly 100% over 7,000 instances, rollout on random essential measures is not merely close to best-order but already clears the conjectured bound, consistent with the 038 census finding that rollout is often though not always the best order. Leads: run an adversarial campaign (not a census) at n = 6 best-order by annealing on the ROLLOUT margin and enumerating orders only at the endpoint, which buys an adversarial n = 6 floor at roughly rollout cost; and watch the sufficiency fraction, since any instance where rollout is negative but best-order is not would be the first separation of the two statements at n >= 6.
Attempt 064informedEvidence
The first ADVERSARIAL best-order result at n = 6, made affordable by annealing on the rollout margin (one order per candidate rather than 720) and enumerating all 720 orders only at the endpoint -- valid because best-order dominates rollout pointwise, so a non-negative rollout floor already proves the best-order floor non-negative. Four anneals x 1,200 steps per cap: at 0.49 the campaign REACHES the equality family (floor -3.031e-14 at a diagonal with marginal 0.0403, float noise around the value certified exactly 0 in 056) and at 0.497 it stalls at +3.203e-03, with zero violations at both. The endpoint gaps between best-order and rollout are exactly 0 at the sharpest endpoints, so the cheap surrogate loses nothing where it binds -- which is what makes n >= 6 tractable at all.
- Range checked
- Four anneals x 1,200 steps per cap at n = 6, caps 0.49 and 0.497, essentiality enforced with the collapse move enabled, objective = rollout margin with a single 720-order enumeration at each endpoint. Float-level: no certificates (the diagonal endpoint is float, though the 056 identity applies verbatim), nothing above cap 0.497, and 8 anneals do not exhaust the space
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Sharper than the 063 census at both caps, as a campaign should be (-3.0e-14 vs +0.014946 at 0.49; +3.2e-03 vs +0.037327 at 0.497). The new technical fact is the vanishing gap: the expensive objective and its cheap lower bound agree exactly at the binding configurations, so annealing on rollout and enumerating once at the end gives the same floor at 1/720 the cost per candidate. Leads: push the construction to n = 7 and 8 (rollout cost is now the only cost, and a single 40,320-order endpoint enumeration is still affordable), and collect any instance where the gap is positive AND the rollout margin negative, which would be the first separation of the two statements in the region that matters.
Attempt 065informedEvidence
Best-order campaigns at n = 7 and n = 8, where domination removes the enumeration entirely: since the best-order margin dominates the rollout margin pointwise, a non-negative ROLLOUT floor is already a proof that the best-order floor is non-negative, so no order enumeration is needed at any point and it is performed only to measure the surrogate slack. At cap 0.49 the n = 7 campaign REACHES the equality family (rollout floor -4.485301e-14, with the 5,040-order endpoint enumeration confirming the gap is exactly 0 at all three endpoints, as at n = 6) and n = 8 stalls at +5.996704e-02 on half the budget with no enumeration performed. Zero violations at both. The cost of testing the promoted conjecture is now independent of the order count, so the barrier that confined this statement to n <= 5 is gone.
- Range checked
- Three anneals per size at cap 0.49: n = 7 at 900 steps with a 5,040-order endpoint enumeration, n = 8 at 450 steps with none. Essentiality enforced, collapse move enabled, margins against each instance own constant. Float-level, no certificates, nothing above cap 0.49 at these sizes, and the n = 8 figure is a half-budget stall rather than a floor (the 055 lesson applies)
- Open gaps
- mutual-information-tax
- recipe-totality
- Note
- Closes the window with the best-order statement -- the route last unrefuted for-all-mu positivity claim -- tested adversarially at every size from 2 to 8 with no violation anywhere, and with the cost barrier removed by an observation rather than by compute. The gap between best-order and rollout vanishes at the binding endpoints at n = 7 as at n = 6, now measured at two sizes. Leads: n = 9 and 10 are reachable by the same route since only the rollout evaluation costs anything, and giving n = 8 a full budget would test whether every size converges to the equality family -- if so, that becomes a statement worth trying to prove rather than to test. Open mathematics unchanged from 046-048.
Go deeper
Problem statement and published background · Full prior-art notes · Machine-readable index
Found something, or think we got it wrong?
Both are welcome, and the second especially. Nothing here has been peer-reviewed by a human mathematician, so errors are expected rather than embarrassing — one record in this library was already corrected by a later review.
Challenge something on this page · Report an attempt you ran · Browse open issues
scripts/blind.sh union-closed ../workMore on blind and informed mode.