2026-07-31 cycle: union-closed gap (a′) refuted float-free; billiards death law proven; two-track plan queued - #5
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First run of the IDEATE.md procedure anywhere in the lab, on the union-closed problem right after gap (a) died as stated (005/006). Six lens agents (algebra, analysis, dynamics, geometry-topology, graph-theory, number-theory) briefed per the procedure; operator-theory deferred with the reason recorded. Three queue-worthy routes survived the filter: pairwise-closure LP/SOS certification, the union-transfer-operator eigenvalue field (coatom reformulation hand-checked), and bipartite-MIS decomposition. Six secondary leads and five no-purchase verdicts recorded, including a proven product-weight no-go (power set gives x/(1+x) < 1/2 for any product weight) that kills the multiplicative/Dirichlet toolbox here; the lens agent's claimed numeric witness for that no-go did NOT reconcile and was replaced by the trivial proof - the filter step is where that was caught. New mechanism tag: field-lens-sweep. Gap-attack attempts 007-009 and billiards 003 are in flight in this same cycle; their index entries land with their records. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_014b21rgRWhdp58knNjSnuq7
The smallest-budget line of the cycle, run as a probe. Gap 2 gets its first precise candidate statement (TAX at p) via a chain-rule assembly value CR = per-coordinate gains minus the tax, plus a previously unnamed second tax ST = Gain - CR >= 0. Findings, all finite-n EVIDENCE pending skeptic review: the candidate is smoothing-sensitive on 002's own certificate gadgets (so the 002 no-go does not kill it); it survives all four adversary genres tested; the pure slice tilt is provably tax-free (T_A = T_B = 0, exchangeable-future argument) and the half-mixing coupling provably has ST = 0; the binding loss channel on O(1)-entropy families is the second tax, measured ~0.40 log2 n on slices. Two mini-lemmas (SL/HM) proved in-record, not yet reviewed; the crash family narrows the positive-CR lambda-window toward 0 as n grows, entangling recipe-totality with the tax. Evaluator cross-validated 4 ways incl. reproducing 005's crash-OR identity to 1.8e-15. New tags: chain-rule-assembly, mutual-information-tax. Skeptic pass queued as 011. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_014b21rgRWhdp58knNjSnuq7
Full adversarial pass on the Gap-2 probe: both mini-lemmas (SL, HM) independently re-proved by hand (SL proves more than claimed - the conditional A-law is fully uniform given (a,b)); every headline number re-derived by a from-scratch engine (worst diff 1.4e-10); the cell 009 left untested (tilt-recipe CR on Sawin gadgets) run here and closes POSITIVE to n=300; the lambda-window narrowing is sharpened to a law, lambda_max ~ 4.847/(n-3), sup CR at lambda=0 for n>=14; the 0.40 log2 n second-tax scaling survives a 3x range extension to n=240 (slope is lambda-dependent: 0.380 at lambda=1.5). Corrections are reporting-level: grid-edge values quoted as suprema (errs conservative), the slice p-set misreported, an lgamma floor at n=60000 behind 'computed exactly', and one unstated legality hypothesis (m_k=1/2 needs p_k >= 1-1/sqrt(2)) that held on every instance used but would fail silently on mixtures with a component marginal < 0.293 - flagged as a reuse trap. 009 stays as written per the hard rules; status VERIFIED_WITH_CORRECTIONS, verifies 009. Gap 2 findings are now ledger-eligible. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_014b21rgRWhdp58knNjSnuq7
…ts mapped
The perturbative-assembly line. 005 Prop 6 was proved modulo one
standard smoothness step; that step is identified (existence/
uniqueness/differentiability of the symmetric Sinkhorn potential in mu)
and proved from scratch via a quantitative IFT with explicit resolvent
bound, giving Theorem P6-prime: |log2 OR - lambda| <= 924*
min(p,1-p)^{-3n}*delta^2 uniform over directions at fixed n. 005 leads
3 and 4 both run (they are distinct; the queue line conflated them):
the end-to-end assembly check passes on every marginal-capped instance
(78 runs, n<=8, delta<=0.2), and the centered kernel suppresses the
crash OR to 2^{lambda(3-n)/n}. Correction to the queue framing:
rho*(0.383)=1.0422, not 1.03. The naive n-uniform pointwise delta^2
budget is killed (deviation grows ~0.3 bits/coordinate), but the
coordinate-averaged downward budget flattens - the perturbative form
of gap a-prime - and a conditional assembly theorem assembles budgets
B1-B4 into (S-coup at p) on an n-independent delta-ball, every
dependency labelled. Status LIVE; proofs pending skeptic pass (012).
New tags: perturbative-expansion, implicit-function-theorem. New gap:
sinkhorn-tv-stability.
Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_014b21rgRWhdp58knNjSnuq7
…0-atom witness
The restated Gap 1 (M_i >= lambda for all mu, i) is first made
well-posed (the zero-mass bookkeeping at i >= 3 is settled by a proved
degeneracy dichotomy: excluded histories are unidentifiable, not
extreme, and the normalization is forced by the i=n and product-mu
anchors), then proved for four structured subclasses (supports <= 4
atoms - explaining 006's S8 - potential-MTP2 mu, i in {1,n}, and first
order in lambda with a perfect-square coefficient), and then REFUTED
in the record-relevant regime: an explicit 10-atom mu on 2^[7] with
all marginals <= 0.318 < 0.38271 has M_5 = lambda - 0.122033 at
lambda = 3.5, violating for every lambda >= 0.03; stable under 3%
perturbation. The 444-instance standard kill battery never crossed the
boundary - the witness came from reducing the two-prefix case to free
slices. The i-AGGREGATED control survives (+1.84 on the witness
itself) and becomes the proposed replacement gap, consistent with
008's independent finding that the coordinate-averaged downward budget
is the object that flattens.
Methodological lesson recorded: cross-engine float agreement is not
independence - the first witnesses were shared-IEEE underflow
artifacts; caught by a physical parameter sweep, guards now in the
engine. Skeptic pass (013) must re-verify the witness in exact
arithmetic. Route stays LIVE: a bridge died, not the interface.
New tag: gram-mass-pairing. New gaps: aggregated-or-control,
margin-modulated-or-control.
Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_014b21rgRWhdp58knNjSnuq7
…free The decisive check closes 007's confessed vulnerability (both of its engines shared IEEE floats; its first witnesses were shared-underflow artifacts): the 10-atom witness is re-verified end-to-end in exact rational arithmetic with directed-rounding log2 enclosures (widths < 1e-40). At rational tilts t = 4 and t = 16 the violation of M_i >= lambda is a fully rational statement with lambda = 2, 4 exactly - no irrational number anywhere. Certified at 12 exact tilts; positive at small t exactly where 007's first-order theorem forces it. New strengthening: the refutation survives the conjecture-friendliest alternative bookkeeping (degenerate histories scored at lambda still give -0.067, certified) - it does not hinge on the conditioning-out convention. Both invariances hold as exact rational identities; all four partial theorems re-derived by hand; the i-aggregated control is certified positive (+1.844669) on the witness and survives seeded attacks; 006's S8 upgraded from evidence to corollary of Theorem B. Scope nuance recorded: 008's P6-prime does not literally discharge Theorem C's smoothness step (d/d-mu vs d/d-lambda). No corrections. Status VERIFIED, verifies 007. Gap (a-prime) is now a skeptic-confirmed dead end; the aggregated control is the live replacement. New tag: exact-rational-arithmetic. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_014b21rgRWhdp58knNjSnuq7
… proven as necessity Queue item 9. Both pre-registered out-of-sample predictions HELD: W(10,9) dies at 162.90 and W(12,12) at 2160/13 = 166.1538..., each within ~1.5e-12 deg of the law (adaptive alive-testing was needed - the fixed grids of 001/002 under-measure deaths by ~0.003 deg because the alive windows pinch). Mechanism found in closed form: a division-free Laurent-ring unfolding (circumdiameter normalization) collapses the corridor criterion to a glide-axis offset sitting inside every gate projection, with the three binding functions factoring exactly - e.g. p(A1) - m = cos(a*alpha) sin((b+1)*beta) sin(alpha+beta). This yields a GENERAL law gamma_d(a,b) = 180 - 90(a+b)/(a(b+1)) unifying 001's two slice formulas, confirmed out-of-sample on W(5,3) -> 144 and W(4,2) -> 135 (members on the mirror half never scanned by 001/002), and a necessity theorem - the corridor is EMPTY at every gamma >= gamma_d - machine-certified for 15 family members via exact-rational interval arithmetic. The hyperbolic length-vs-angle growth law along W(a,a) is established on the upper side. Honest gaps labelled: exact death = gamma_d needs alive points approaching gamma_d (certified alive stops 1e-6 short); identities I1-I3 and Lemma C are per-member certified, SPECULATION for general (a,b); a > 2b+3 uncovered. Skeptic pass queued as 004. New tags: symbolic-trig-closed-form, certified-interval-inequality, out-of-sample-prediction. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_014b21rgRWhdp58knNjSnuq7
…nstants fall The heaviest proof content survives every attack: Lemmas A-C and Theorem P6-prime re-derived line-by-line by hand, the bound stress-tested on 120 instances to the theorem boundary with zero violations - the Prop-6 closure should now be treated as verified at fixed n. The 78-run assembly table, calibration block (the queue's rho ~ 1.03 was indeed wrong), and centered-kernel closed form all confirm, and a 104-run hunt found zero marginal-capped negatives. The kills land on the forward-looking half. An orbit-symmetrized engine extends the budget census from n = 9 to n = 32 and REVERSES 008's flattening reading: the averaged downward OR budget's increments stop shrinking and grow again past n ~ 22, killing budget (B1) as stated at evidence level by 008's own falsification criterion - the pre-asymptotic worry 008 itself flagged was real. The tax budget (B2) is first-order in delta, not quadratic (halving ratios -> 2), and must be restated linear. The conditional theorem's box-uniform tau_half step is analytically false at its own parameters (corner counterexample at s0 = 0.10; ceiling 0.0843; the claimed numerical check does not exist in the code), and delta_0 = 0.022 drops to ~0.004 under 008's own formula with the dropped terms restored. Two further reporting corrections (out-of-class worst-net row; 831-vs-924 constant chase). Status VERIFIED_WITH_CORRECTIONS, verifies 008; six corrections. 008 stays as written per the hard rules. Net ledger effect: P6-prime is a real theorem; the perturbative assembly route needs a new n-uniform budget object - note the tension with 007/013's certified i-aggregated positivity, which lives at fixed small n. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_014b21rgRWhdp58knNjSnuq7
…rvives Every kill attempt failed. The glide-to-corridor reduction re-derived by hand from 001's corridor definition (strictness airtight: an endpoint offset gives at most point-overlap, never positive width); the mod-360 case tree hand re-enumerated with preconditions used exactly where claimed, and a 1.94M-point adversarial sign search with boundary/corner targeting found zero dropped-branch hits (positive control fires). Identities I1-I3 re-proven in a structurally independent composed-map ring for all 15 members, plus a proof-grade off-torus grid-interpolation certification for W(2,1) sharing no representation with either ring. Lemma C re-certified for 4 members with a fully independent interval stack (own Machin pi, own Taylor enclosures); the H2-prime end zone is comfortably negative. The death-definition identity with 001's census criterion confirmed; all four out-of-sample deaths re-measured to ~1e-12; the mirror-half (x > 1/2) windows for W(5,3)/W(4,2) confirmed as real and previously unscanned. Two wording-level corrections, nothing load-bearing: one sentence cites 1e-8 where the certificate is 1e-6, and 'corridor is empty at gamma >= gamma_d' overstates - the theorem proves no POSITIVE width; a zero-width touching corridor at gamma_d is not excluded (a float probe at the (18,18) corner shows exactly that). Status VERIFIED_WITH_CORRECTIONS, verifies 003. The death-angle necessity law is now a skeptic-confirmed result of the lab. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_014b21rgRWhdp58knNjSnuq7
Five lines run, four skeptic-verified, one MAP. Union-closed: gap (a') refuted float-free (007/013), Theorem P6-prime proved and the naive n-uniform budgets killed (008/012), gap (b) formalized and surviving (009/011); the route stays LIVE with the i-aggregated control as the restated gap. Billiards: the death-angle law is now a proven, skeptic-confirmed necessity theorem with a general formula (003/004). First ideation sweep ran (010). Queue reshaped around the refutation, the corrected budgets, the sweep leads, and the death-law follow-ups. New insights: exact-rational-arithmetic as the verification standard, the small-n-flattening trap, the site-test escaping quirk. Three new dead ends recorded. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_014b21rgRWhdp58knNjSnuq7
Item 11: complete the parametric death theorem (sufficiency at gamma_d via the corner argument; general-(a,b) proofs of I1-I3 and Lemma C; the a > 2b+3 branch). Item 12, the exploratory track: invert the death-law machinery - use the Laurent-ring binding factorization as a design tool to hunt for word families whose factors vanish only past 135 degrees; one certified alive point at gamma > 135 would push the constructive frontier past the census stall, and a proven cap would sharpen the accumulation-point question in item 14. Later items renumbered; billiards table row points at the plan. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_014b21rgRWhdp58knNjSnuq7
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What this adds
One full research cycle: eight attempt records across two problems (union-closed 007–013, billiards-triangles 003–004), the lab's first cross-field ideation sweep, a rewritten ledger, and a documented two-track plan for the billiards follow-up (queue items 11–12) ready for the next instance to execute.
Problem: union-closed (Frankl) and triangular billiards
Mode: informed (all lines; blind mode not used this cycle)
Status claimed: REFUTED (007, skeptic-verified by 013) · LIVE (008, verified with corrections by 012) · EVIDENCE (009, verified with corrections by 011) · MAP (010) · VERIFIED (billiards 003, verified with corrections by 004)
Headlines:
The bar
SPECULATIONis labelled inline (e.g. general-(a,b) identities in billiards 003; conditional-theorem dependencies in 008).VERIFIEDdescribes a range, not the conjecture. Ranges stated in every index entry (e.g. necessity certified for 15 members; witness certified at 12 rational tilts).modeis honest. All informed; recorded per attempt.verifiesset in the index.prior-art.jsonupdated for both problems: mechanism tags (7 new, registered inmechanisms.json), statuses, gaps, leak_terms.python -m pytest tests/ -qpasses — 158 passed, 7 skipped.What is not claimed
No bound on Frankl and no breakthrough: the route's ceiling (0.4315) remains a model ceiling, not a theorem, and this cycle removed a candidate bridge (a′) rather than adding one. The billiards theorem is necessity only — a zero-width touching corridor exactly at γ_d is not excluded (004's correction), the general-(a,b) statement rests on per-member-certified identities, and nothing is claimed about unstable orbits or the a > 2b+3 branch. Sweep routes are candidates, not results. 008's conditional theorem depends on budgets that 012 showed need restating; its constants as printed in 008 are superseded by 012's corrections.
Obstruction / open gaps
Union-closed: the live gaps are now the i-aggregated OR control (certified positive only at n ≤ 7 — probe n ≳ 20 before proof effort, per the small-n-flattening lesson), the δ-linear tax budget, recipe totality, and Sinkhorn TV-stability. Billiards: sufficiency at γ_d, general-(a,b) proofs of I1–I3/Lemma C, and the a > 2b+3 branch — queued as item 11, with the exploratory "design a family past 135°" track as item 12. The next instance can start from STATUS.md queue items 11–12 directly.
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https://claude.ai/code/session_014b21rgRWhdp58knNjSnuq7
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