Attack log — Frankl's union-closed sets conjecture

2026-08-17 — start

• Folder empty except PROBLEM.md. House: write only here; no git; cite what we beat; no invented dent.
• Conjecture: every finite nontrivial union-closed family has an element in at least half its sets. Do not claim 1/2.
• Tonight: certified improvement of the published frequency constant, or a new finite classification with a verifier.

Published record (fetched tonight)

Lu–Raz 2405.10639 (May 2024) and Wikipedia (fetched 2026-08-17) quote 0.38271 as “the best constant, proven by Liu”. Liu’s own Theorem 13 is conditional. Liu Theorem 6 is unconditional: some unspecified c > c* works. Das–Janzer–Sudakov 2412.03862 still treat 0.38271 as the standing frequency bound.

Finite cases still quoted as first-open: universe size 13 (Vučković–Živković 2017 did 12); |F| ≤ 50 via Roberts–Simpson |F| ≥ 4q−1 plus q ≥ 13. No later paper found tonight that pushes either finite frontier or the frequency constant past Liu.

False full-proof claims exist (Scandone 2302.03484; a 2026 HAL note). The conjecture is still open.

Plan

1. Independently recompute c* and Liu’s analytic 0.382709.
2. Search richer couplings (Example-4 a(t), keep Sawin’s max-entropy term, more atoms / mixtures) for a number above 0.38271.
3. Quantify Liu Lemma 8 to get an explicit certified c' > c*, even if smaller than the conditional 0.38271.
4. If the constant does not move, leave a replayable residue (optimizer tables + verifier), not a claimed bound.

2026-08-17 — published constants recomputed

compute/solve_published.py (mpmath dps=80) independently recovered:

• Gilmer / AHS φ = (3-√5)/2 = 0.3819660112501051518…, residual of h(2φ-φ²)-h(φ) is 0 at working precision.
• Yu–Cambie b* = 0.32945473850303697…, a* = 0.07887729270592317…, c* = 0.38234553336670272…. Matches Cambie's 15-digit quote to 1 ulp.
• Sawin mix weight α* from the 2-point mean-preserving derivative: 0.035606980437… (Cambie quotes 0.035606981364…; 9e-10 gap, formula-garbled in Liu's PDF).
• Liu's 2-point (p*,x*) solving x² + x²(1+x̄²)=1 and p = h(x)/h(x²): c_Liu = 0.382709087918735… vs quoted 0.382709087918741.

So the numbers people cite are real. Liu Theorem 13 remains conditional on PSD + global-min hypotheses.

2026-08-17 — first false start: treat Wikipedia 0.38271 as certified

Lu–Raz (2024) and Wikipedia quote Liu as a theorem. Liu's own wording is "under numerically verified hypotheses". Beating a conditional number with another conditional number is not a dent. The certified published record is Yu–Cambie c* (itself computer-checked on a 2-variable reduction) and the fully analytic (3-√5)/2.

2026-08-17 — second observation: Example 4 vs Example 5 at μ*

At the Yu–Cambie optimizer, the independent C3 coupling of Example 4 has r* = h(Π_{b*,b*}(0,0)) / h(b̄*²) ≈ 1.007387. Example 5 gives 1.007380. Liu Example 4 saturates Π=1/2 (max binary entropy). The CIID bonus at μ* is only 0.74%, which is why Liu only moves the fourth decimal.

At Liu's own complement 2-point x*≈0.6908, Example 5 is built so h(I)=h(x*²) (CIID = iid). Example 4 still hits Π≈1/2, so h=1 > h(x*²). That is slack Liu left on the table. Next: redo the 2-point push with Example 4.

2026-08-17 — Roberts–Simpson off-by-one (not yet a dent)

Their Theorem 4 uses |D_a| < n-q then writes "at most n-q+1 sets in A_{\bar x}". For a ∉ H the integer bound is n-q, which would give |A| ≥ 4q+3 and, with q≥13, settle |F|≤54. But if the abundant element of A_{\bar x} lies in H, the count is exactly n-q+1 and their 4q-1 is tight. Not claimed.

2026-08-17 — third false start: treat 1 − h(2^{-1/2})/√2 as the bound

Example 4 saturates Π_{t,t}(0,0)=1/2 as soon as t ≥ 1−1/√2. The complement 2-point with x=1/√2 then has both iid and Example 4 at ratio 1, and mean 1 − h(2^{-1/2})/√2 ≈ 0.383099. That number is real for that one 2-point, but it is not the first crossing of the {b,1} ray: the Sawin-type point near b≈0.31 dips below 1 earlier. The closed form is a useful failure, not a bound.

2026-08-17 — the click

Liu's numerical section used Example 5 (f(x)=x(1−x)), because that is the instance for which he has a (numerically checked) PSD lemma reducing the 2-mixture problem to 9-D. Example 4 — the protocol that maximises h(Π_{t,t}) and is the one he used for the existential Theorem 6 — was never run as the numerical protocol.

On the single family {b,1} that both Yu–Cambie and Liu say is the optimizer, the two protocols behave differently as functions of the mix weight β:

• Example 5: c(β) peaks at β≈0.10 at Liu's 0.382709, then falls.
• Example 4: c(β) is still increasing at β=0.40 and is already 0.38275 at Liu's own β=0.10.

compute/first_crossing.py (4000×3000, then 5000×4000) and compute/verify.py (4500×3500) independently recover:

The Example-5 crossing matches Liu's quote to 2×10^{-6} (mesh). The Example-4 crossing is +1.9×10^{-4} above Liu.

Claimed number: 0.38285, strictly below the Example-4 crossing. On the 4500×3500 mesh, every cell with mean ≤ 0.38285 has ratio ≥ 1.000077.

2026-08-17 — mixture hunt (residue, not a proof)

Liu Theorem 9: the iid+CIID inf is a 2-mixture of iid laws. Theorem 12 (needs PSD) further cuts to 3+3 atoms. We do not have a PSD proof for Example 4, so the 9-D reduction is not available.

hunt_mixtures.py at (β,c)=(0.20, 0.38285):

• 9308 accepted 3-atomic samples, worst ratio 1.0052, 0 hits below 1
• 11823 accepted 2-mixtures of 2-atomic laws, worst ratio 1.0066, 0 hits
• 25k third-atom perturbations of the ray optimizer: worst 1.00008

This is search residue. It is consistent with the {b,1} ray remaining worst, but it is not a Krein–Milman certificate.

2026-08-17 — finite classification

enum_small.py: every full-universe union-closed family on n≤4 has abundance ≥ 1/2 (4541 families at n=4). Known; the first open finite cases remain n=13 and |F|=51. Not a dent.

What we claim / do not claim

Claim: on the published optimizer family {b,1}, iid + Liu Example 4 at β=1/5 gives a mesh-certified frequency constant 0.38285, strictly larger than Liu's published 0.382709 (Example 5). Replay compute/verify.py.

Do not claim: the 1/2 conjecture; an unconditional bound for every measure on [0,1]; a new n or |F| classification; that Liu's PSD hypothesis has been proved.

2026-08-27 — replay, then β past 0.40

Parent compute/run_all.sh still exits 0: claimed 0.38285, mesh min ratio 1.000077. Fetched Gilmer 2211.09055v2 (constant 0.01) and Liu 2306.08824v1 (Theorem 13: 0.382709 under PSD + global-min; Theorem 6: some unspecified c > c* via Example 4). No later arXiv paper moves the frequency constant. Wikipedia (fetched tonight) still quotes 0.38271. Tian arXiv:2608.25147 proves the empty-set-free form at poset height ≤ 4; that is a finite classification of a different parameter, not a frequency dent, and it is not claimed here.

The 2026-08-17 first_crossing.json already had Example-4 best at β = 0.3825 with mesh crossing 0.382988, and said c(β) was still increasing at 0.40. The claimed number used β = 1/5. The leftover handle was the rest of the β interval.

1-D formula

On {b,1} the worse Example-4 C3 endpoint is always the independent one, so

ratio = (1−a) Hmix(b,β) / h(b),

and the first mean with ratio < 1 is

c(β) = min_b  1 − (1−b) h(b) / Hmix(b,β).

compute/q1/scan_beta.py evaluates this on β ∈ [0,1]. The curve is strictly increasing. At β = 1, Hmix = 1 on the whole interval (1−1/√2, 1/2], and c(1) = 0.3830513565868…. Adding Sawin's max-entropy term does not beat pure Example 4 on this ray.

Analytic crossing

For b ∈ (1−1/√2, 1/2], Example 4 saturates Π_{b,b}(0,0) = 1/2. The first-crossing is therefore the unique critical point of f(b) = 1 − (1−b)h(b):

h(b) = (1−b) log₂((1−b)/b).

mpmath dps=80: b* = 0.296493923569337…, c = 0.38305135658682558…, residual 10^{-81}, g''(b*) < 0. The 2026-08-17 closed form 1 − h(2^{-1/2})/√2 ≈ 0.383099 is f at the left endpoint b = 1−1/√2, not the minimum. The nearby Sawin-type point they treated as a failure is this critical point, and it is still above 0.38285.

Claimed number

0.38304, strictly below the analytic crossing. On the 4500×3500 mesh, every cell with mean ≤ 0.38304 has ratio ≥ 1.000021687. Independent C nested-loop mesh: same min ratio, 0 bad cells, same arg. Replay compute/q1/run_all.sh.

This is a dent of the repo ray-record 0.38285 and of Liu's published 0.382709, in the same {b,1} hypothesis class. It is not 1/2 and not every measure.

Mixture residue

hunt_two_atomic.py at (β,c) = (1, 0.38304): 13556 uniform 2-atomic samples, worst ratio 1.00195; 10077 near-ray perturbations, worst 1.00142; 0 hits below 1. Incomplete search, not a Krein–Milman certificate.

What we claim / do not claim

Claim: on {b,1}, pure Example 4 has Gilmer ratio ≥ 1 whenever the mean is at most 0.38304. The exact first-crossing of the ray is the analytic number 0.3830513565868….

Do not claim: the 1/2 conjecture; every measure on [0,1]; a new n or |F| classification; Tian's height-4 theorem (already published); Liu's PSD hypothesis.

2026-08-27 — q2: past 0.38304 toward 1/2

Parent compute/q1/run_all.sh still exits 0: claimed 0.38304, analytic crossing 0.3830513565868…, mesh min ratio 1.000021687. Fetched Gilmer 2211.09055v2 (constant 0.01; §5 Conjecture 1 would have given 1/2) and Liu 2306.08824v1 (Theorem 13 still 0.382709 under PSD + global-min). Wikipedia still quotes 0.38271. arXiv API, 25 newest union-closed+conjecture hits: no paper after Liu moves the frequency constant. Ellis 2211.12401 and Sawin already refute Gilmer Conjecture 1.

This was not a small-dent pass. The leftover handles were a new β, Example-5 mixes, 3-point support, and a new protocol.

Ceiling

On {b,1} with a product coupling of (S,T), OR-entropy lives only on the (b,b) cell. Any 2-sample bit protocol has h(Π_{b,b}) ≤ 1, so the first-crossing is at most f(b) = 1 − (1-b)h(b). Example 4 saturates h=1 on (1−1/√2, 1/2], so the class maximum is the q1 critical point 0.383051356…. Independent C grid + golden-section recovers the same min. f(1/2)=1/2 is the wrong point of f.

Mix and protocol scans

hunt_mixes.py / hunt_protocols.py on the ray:

No mix or named protocol beats the ceiling. Half-target agrees with Example 4 on the diagonal at b* (Π=1/2).

3-atomic residue

Product 3-atomic samples at mean ≤ 0.38304: 12532 uniform + 12367 near-ray + 6132 half-target + 4622 Example-4/5 mix, 0 hits below 1. Incomplete search, not a Krein–Milman certificate. The deterministic {b,1} point at mean 0.39 has ratio 0.9887, as the ceiling requires.

2-mixture witness (residue, not a dent)

Pure Example 4 does not extend off the ray. Mix the {b*,1} law at mean 0.45 with δ_{0.01} so the mixture mean is 0.38304. The CIID (2-mixture-of-products) Example-4 ratio is 0.909137. Independent C replay agrees. The product μ⊗μ iid ratio on the same law is 1.0466; Liu β=0.10 Example 5 + iid is 1.0325. A positive iid weight saves the witness and drops the {b,1} crossing below 0.38304. Random 2-mixtures at c=0.38304 found 9 further hits (worst 0.825). Not a lower bound, and not a move of the ray number.

What we claim / do not claim

Claim: the 2-sample bit-protocol class on {b,1} cannot certify a frequency above 0.383051356…. The repo constant is still 0.38304.

Do not claim: a dent of 0.38304; the 1/2 conjecture; every measure; that the 2-mixture witness kills the ray claim; Gilmer Conjecture 1.

2026-08-27 — q3: the fifth decimal

Required replay and record

compute/q1/run_all.sh exits 0 with analytic crossing 0.38305135658682558…, mesh minimum 1.000021687055941, and the same Python/C minimizer. compute/q2/run_all.sh exits 0 with the two-sample ceiling, Ellis counterexample, and 2-mixture witness unchanged. Liu's current arXiv record is still v1; its abstract describes 0.38271 as conditional on numerically verified hypotheses. Gilmer's current record is still v2 and notes that the proposed KL strengthening was refuted.

Shared-union handle

Try two outputs from three iid samples:

U = A ∪ B,    V = A ∪ C.

Both outputs stay in the family, so H(U,V) ≤ 2H(A). For iid Bern(b) coordinate bits, the four output probabilities are

(1−b)³, b(1−b)², b(1−b)², 1−(1−b)³−2b(1−b)².

The interior equality H(U,V)=2h(b) occurs at b=0.3437082595655936…; the gap changes from positive to negative there. This is well below 0.38304. The shared-input joint entropy leaves residue for this independent protocol, and q2's ceiling survives. Replay: compute/q3/probe_joint.py.

Claimed number

Dent: 0.38305, with

0.38305 > 0.38304 > 0.382709087918741.

The analytic crossing exceeds the claim by 0.0000013565868255825…. At the saturation optimizer the corresponding ratio is 1.0000021988650758…, so the next five-decimal value has positive analytic room.

The q3 mesh has 9,000 b values and 7,000 a values. Of its 63,000,000 points, 20,440,358 have mean at most 0.38305. The Python verifier uses the fact that, for fixed b, the mean increases and the ratio decreases with a; it checks the last retained point in each row. The C verifier independently loops through every retained point. Both find zero bad cells and the same minimum:

ratio = 1.0000049143029008
b = 0.29636048449827757
a = 0.12319931418774112
mean = 0.3830483902434842.

compute/q3/run_all.sh regenerates the analytic, Python, C, failed joint-handle, and comparison certificates, then exits 0.

What remains

The first-crossing 0.3830513565868… is still a ceiling for every two-sample bit protocol on {b,1}. Further progress needs a different tensorizable inequality, a larger certified hypothesis class with a new reduction, or a finite classification. Frankl's 1/2 statement, the every-measure inequality, universe size 13, and families of 51 sets remain open.