union-closed 005: refute the pointwise Plackett odds-ratio control (gap a) - #4
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…ap a) Attempt 005 attacks queue item 1(a). The claimed pointwise control OR <= 2^lambda for the tilt family fails in both directions: diagonal histories force OR >= 2^lambda (Cauchy-Schwarz in the tilt kernel's inner product), and an explicit 4-atom family drives OR to 2^(lambda(3-n)) on a positive-mass history. Sharp universal range recorded; near-product first-order cancellation keeps the perturbative lead alive. Gap restated as averaged odds-ratio control. Awaiting independent skeptic pass (006). Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01LD7YKChwfSKyjraRaBHoHR
Independent re-derivation of every proposition plus a from-scratch engine (different Sinkhorn scheme, off-grid lambdas, sparse supports). All seven load-bearing claims CONFIRMED; two minor corrections (equality-iff in Prop 2 needs lambda > 0; part-B prose figure mislabeled). The Karlin-Rinott citation checked against the actual paper; binary-cube case re-proved via Ahlswede-Daykin independently. Bonus: the restated averaged-control gap survives its own first falsifiable test on the crash family. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01LD7YKChwfSKyjraRaBHoHR
First attempts on the problem, both run blind (source ac7a822). 001: canonical word census to length 26 (17,527 words), two-layer float-screen + exact-certificate coverage map of the obtuse half-disk, 376 certified boxes. Coverage staircase stalls at 135 deg; the length-14 family dies at 112.4989 deg, a blind rediscovery of the published constructive frontier. Kill-condition verdict: needed word length grows hyperbolically in the angle (not super-exponentially), but certified box width collapses exponentially in length - length buys angle reach, not area. 002: adversarial review, every layer re-implemented from scratch; all claims confirmed, death law survives two out-of-sample prediction rows, certified-area figure corrected (~4.3e-6, not 2.2e-6). New mechanism tags: unfolding-word-census, certified-box-coverage. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01LD7YKChwfSKyjraRaBHoHR
First attempts on the problem, both run blind (source ac7a822). 001: exhaustive census of centrally symmetric conv(+-S) over all S of <= 11 antipodal pairs of nonzero {0,+-1}^4 points, enumerated up to the hyperoctahedral group B4 (18.6M orbits, 1.77M distinct bodies, Burnside-verified level counts, zero exact-arithmetic punts). Minimum volume product is exactly 32/3, attained by 1113 orbits, every one GL-certified Hanner; nearest non-attainer sits 4535/31104 above the bound. Kill condition met: nothing near the bound that is not Hanner, so the census stops rather than widening by reflex. 002: adversarial review - own Burnside code, from-scratch exact volume products for all 1176 near-bound bodies, all 1113 Hanner certificates re-verified, 240-body float audit (max dev 8.9e-15); every headline number confirmed, two wording corrections. Raw eval slices (1.05 GB) and orbit mask lists (155 MB) are regenerable and not committed; see the filing note in 001. New mechanism tags: volume-product-census, group-orbit-enumeration. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01LD7YKChwfSKyjraRaBHoHR
Three lines run, all skeptic-verified: union-closed gap (a) refuted as stated and restated as averaged control (005/006); mahler-4d and billiards-triangles first attempts, both blind, both confirmed (001/002 each). Queue reshaped: blind census items replaced by the leads they generated; union-closed item 1 now points at the corrected gap set. New insights: first blind-mode data points, and the silent-watcher ops lesson. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01LD7YKChwfSKyjraRaBHoHR
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Attempt 005 attacks queue item 1(a). The claimed pointwise control
OR <= 2^lambda for the tilt family fails in both directions: diagonal
histories force OR >= 2^lambda (Cauchy-Schwarz in the tilt kernel's
inner product), and an explicit 4-atom family drives OR to 2^(lambda(3-n))
on a positive-mass history. Sharp universal range recorded; near-product
first-order cancellation keeps the perturbative lead alive. Gap restated
as averaged odds-ratio control. Awaiting independent skeptic pass (006).
Co-Authored-By: Claude Fable 5 noreply@anthropic.com
Claude-Session: https://claude.ai/code/session_01LD7YKChwfSKyjraRaBHoHR