{ "schema": "cert-machine/delta4-probe/1", "date": "2026-10-05", "functional": { "statement": "F(φ) = lim #mono increasing 4-APs/n² for a ↦ φ(a/n); Δ = {x,t ≥ 0, x+3t ≤ 1}; 1[x₁=x₂=x₃=x₄] = (1 + Σ_{i 0 computed for q = 4, 6)": { "cross": "none", "u": "0", "pdExact": true, "lambdaMinFloat": 1.1666666666666667 } }, "exactMultiClassChecks": [ { "shifts": [ [ 0, 1 ] ], "parityEqual": true, "mod3Equal": true }, { "shifts": [ [ 4, 1 ], [ 7, -2 ], [ 20, 1 ] ], "parityEqual": true, "mod3Equal": true } ], "reading": "φ* is a strict local minimiser to second order in edge directions also against parity and mod-3 splits (exact), and against order-q splits for every q whose local u_q > −7/12 (κ_min = 7/6) (u_4 = 1/12, u_6 = 11/108 exact; other q not computed). Irrational-phase modulation has product pair correlations, so only its local u changes; not computed." } }, "arithmetic": { "literature": { "luPeng": "L. Lu, X. Peng, Monochromatic 4-term arithmetic progressions in 2-colorings of Z_n, arXiv:1107.2888 (JCTA 2012): c₄ ≤ 1/72 ≈ 0.0138889 (their eq. (12)); 7/96 ≤ m₄(Z_p) ≤ 17/150 (Theorem 1, ordered (a,d) ∈ Z_p², d = 0 included); Conjecture 1: inf{m₄(Z_n): 4 ∤ n} = 1/12; Conjecture 2: lim m_k([n]) = lim m_k(Z_n) for k ≥ 4", "butlerGrahamLu": "S. Butler, R. Graham, L. Lu, Unrolling residues to avoid progressions, arXiv:1209.2687: restate 1/72 as the best known for k = 4, \"far superior to the block coloring\"; the rule: colour ℓ by its first nonzero base-11 digit (residue / non-residue)", "erdosproblems1186": "the snapshot of 2026-10-05 lists δ₃ and the F_p bounds 7/192 ≤ δ̃₄ ≤ 17/300 but no [n] bound for k = 4 (neither BCG nor Lu–Peng)" }, "residueZ11": { "F": "1/66", "theorem6": "1/(2·11·3) = 1/66", "equal": true }, "twoLevelUnrolling": { "F": "37/2662", "formula": "(1/121)(10/6 + 1/66) = 37/2662", "equal": true }, "fullUnrolling": { "limit": "1/72", "bruteForce": { "n": 16000, "mono": 3547812, "monoOverN2": 0.013858640625, "nTimesGap": -0.484 } }, "versusBCG": { "ratio": 0.806543195306824, "lowerBy": 0.19345680469317605 }, "lpFirstVariationSamples": { "float": true, "n": 16000, "samples": [ { "x": 0.01, "residue": 1, "firstVariation": 0.031875 }, { "x": 0.01, "residue": 2, "firstVariation": 0.0319375 }, { "x": 0.01, "residue": 0, "firstVariation": 0.0029375 }, { "x": 0.25, "residue": 1, "firstVariation": 0.0643125 }, { "x": 0.25, "residue": 2, "firstVariation": 0.06425 }, { "x": 0.25, "residue": 0, "firstVariation": 0.006125 }, { "x": 0.5, "residue": 1, "firstVariation": 0.0756875 }, { "x": 0.5, "residue": 2, "firstVariation": 0.07575 }, { "x": 0.5, "residue": 0, "firstVariation": 0.0068125 }, { "x": 0.75, "residue": 1, "firstVariation": 0.0644375 }, { "x": 0.75, "residue": 2, "firstVariation": 0.0644375 }, { "x": 0.75, "residue": 0, "firstVariation": 0.005875 }, { "x": 0.99, "residue": 1, "firstVariation": 0.031875 }, { "x": 0.99, "residue": 2, "firstVariation": 0.031875 }, { "x": 0.99, "residue": 0, "firstVariation": 0.0030625 } ], "reading": "every sampled single flip raises the count: first-order stable under local flips" }, "zmAvoidersTable": { "note": "exhaustive backtracking, χ(0) = +1; reproduces W_c(4,2) = 34 (Irawan, arXiv:2509.14595): avoiders exist for m ∈ {5,…,12,14,15,18,21,22,33} and none for 13, 16, 17, 19, 20, 23–32, 34–40", "table": [ { "m": 5, "avoiders": 10, "bestMono": 5, "residueF": "1/30" }, { "m": 6, "avoiders": 19, "bestMono": 6, "residueF": "1/36" }, { "m": 7, "avoiders": 14, "bestMono": 7, "residueF": "1/42" }, { "m": 8, "avoiders": 42, "bestMono": 12, "residueF": "1/32" }, { "m": 9, "avoiders": 21, "bestMono": 21, "residueF": "7/162" }, { "m": 10, "avoiders": 40, "bestMono": 18, "residueF": "3/100" }, { "m": 11, "avoiders": 22, "bestMono": 11, "residueF": "1/66" }, { "m": 12, "avoiders": 137, "bestMono": 24, "residueF": "1/36" }, { "m": 13, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 14, "avoiders": 28, "bestMono": 26, "residueF": "13/588" }, { "m": 15, "avoiders": 100, "bestMono": 39, "residueF": "13/450" }, { "m": 16, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 17, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 18, "avoiders": 3, "bestMono": 54, "residueF": "1/36" }, { "m": 19, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 20, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 21, "avoiders": 56, "bestMono": 57, "residueF": "19/882" }, { "m": 22, "avoiders": 44, "bestMono": 42, "residueF": "7/484" }, { "m": 23, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 24, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 25, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 26, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 27, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 28, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 29, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 30, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 31, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 32, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 33, "avoiders": 88, "bestMono": 93, "residueF": "31/2178" }, { "m": 34, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 35, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 36, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 37, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 38, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 39, "avoiders": 0, "bestMono": null, "residueF": null }, { "m": 40, "avoiders": 0, "bestMono": null, "residueF": null } ] }, "luPengZn": { "B20": { "mono": 36, "over": 400, "printed": "9/100" }, "B22": { "mono": 42, "over": 484, "printed": "42/484 = 21/242" } }, "reading": "the best known δ₄ upper bound is an arithmetic colouring: on each residue class it is solid, and a 4-AP with d ≢ 0 (mod 11) is never monochromatic. A functional of local densities sees it as φ ≡ 0 with every pair statistic 0 — the case where the pointwise relaxation is 0. BCG's 36 blocks are 24.0% above it." }, "fp": { "functional": "exactly, for EVERY colouring of Z_p (p prime): #mono (a,d)/p² = (1/8)(1 + 6μ² + Λ₄(χ)), because (a,d) ↦ (a+id, a+jd) is a bijection of Z_p² — every pair term is μ². The quartic Λ₄ is the whole problem; there is no position variable to centre a block expansion in.", "normalisation": "Lu–Peng m₄(Z_p): ordered (a,d), d = 0 included, random 1/8; erdosproblems δ̃₄ = m₄/2 (random 1/16): 7/192 ≤ δ̃₄ ≤ 17/300", "circleBlocks": { "colouring": "11 equal circle blocks, quadratic-residue pattern", "exact": "15/121", "Lambda4": "-7/121", "bruteForceZ3001": 0.1239803326692946, "reading": "only 0.8% below random: on the circle the QR pattern does nothing; its power on [n] is arithmetic, not positional" }, "luPengB20": { "construction": "B20 repeated, remainder arbitrary (their Theorem 3, odd n)", "p": 4001, "monoOverP2": 0.11332676703356027, "float": true, "printed": "17/150 ≈ 0.113333" }, "sameThreeQuestions": { "functional": "μ² pair terms + the quartic; no reduction (the pointwise relaxation is 0 here too)", "candidate": "Lu–Peng's B20-periodic colouring, verified numerically above; it is periodic, not a circle block colouring", "firstOrder": "not set up: with no position variable the δ₃ expansion has nothing to expand in; the natural analogue (perturbing the periodic word) is a finite search, and the lower-bound side is Fourier/U³ counting (Wolf, Lu–Peng), not a local functional" } }, "feasibility": { "grid": "uniform 1/128 plus the 35 edges", "cells": 163, "cellFourTuplesWithQuarticMass": 34361, "degree2": { "momentMatrix": 164, "note": "δ₃ size; but L(y) cannot be degree 2 here: ∬σσσσ over a cell 4-tuple region is not a function of cell averages" }, "degree4dense": { "momentMatrix": 13530, "moments": 31256555 }, "blockOnlyInner": "the inner zero-gap step needs h₄ (✓ exact) and a PD edge Hessian (✓ exact, λ_min ≈ 0.0822) — but the local C₃ + Q₄ (−2/3 + 1/3 of U = 17/12) enter at Hessian order, so the inner certificate must carry cubic/quartic sliver geometry at edge resolution; and the [−1,1] relaxation of F is not the weak-* closure of ±1 colourings (the quartic is not weakly continuous), so even the block-only theorem needs a different relaxation" }, "search": { "float": true, "method": "search.js: basin hopping (merge / split / jiggle, Newton polish on F); modes: bcg = start at BCG's edges, basin = random 20–49-block start", "noiseFloor": 1e-12, "runs": [ { "mode": "basin", "seed": 11, "hops": 80, "blocks": 18, "F": 0.017940209996684955, "minusBCG": 0.0007199433454917747, "seconds": 303.966 }, { "mode": "basin", "seed": 12, "hops": 80, "blocks": 24, "F": 0.017728060151282705, "minusBCG": 0.0005077935000895248, "seconds": 412.849 }, { "mode": "basin", "seed": 13, "hops": 80, "blocks": 22, "F": 0.017730343473486704, "minusBCG": 0.0005100768222935231, "seconds": 421.424 }, { "mode": "basin", "seed": 14, "hops": 80, "blocks": 22, "F": 0.017987001847663837, "minusBCG": 0.0007667351964706567, "seconds": 377.61 }, { "mode": "basin", "seed": 15, "hops": 80, "blocks": 21, "F": 0.018264632848533753, "minusBCG": 0.0010443661973405724, "seconds": 280.102 }, { "mode": "bcg", "seed": 1, "hops": 80, "blocks": 36, "F": 0.017220266651193167, "minusBCG": -1.3877787807814457e-17, "seconds": 731.777 }, { "mode": "bcg", "seed": 2, "hops": 80, "blocks": 36, "F": 0.017220266651193167, "minusBCG": -1.3877787807814457e-17, "seconds": 710.94 } ], "belowBCG": 0, "reading": "the BCG starts accepted no move in 80 hops (their −1e−17 is float noise); random starts end 3–6% above BCG" }, "verdict": { "target": "delta4-centring-transfer", "verdict": "BLOCKED", "by": [ "the candidate: BCG's 36 blocks (F exactly their printed 0.0172203) are not the δ₄ record — Lu–Peng 2011 give c₄ ≤ 1/72 ≈ 0.0138889 by an arithmetic (residue-unrolling) colouring, re-verified here; a block-density functional cannot represent it", "the reduction: k = 4 has no PRS-type reduction — gap-3 drops (a sum of squares), but the quartic term is not a functional of local densities, and the pointwise relaxation that keeps it evaluates to 0" ], "whatDoesTransfer": "within block colourings BCG's φ* passes every local test exactly: h₄ ≥ 0 with zeros exactly at the 35 edges, edge Hessian PD (λ_min ∈ [0.0822, 0.0823]), PD also against residue-class splits — \"BCG optimal among block colourings\" is a well-posed, open, much weaker question, with degree-4-type certificates" }, "timingsSeconds": { "functional": 0.248, "bruteForce": 10.041, "reduction": 0.001, "candidate": 0.181, "h4table": 0.269, "h4facts": 0.412, "hessian": 6.374, "arithmetic": 2.808, "fp": 0.205, "feasibility": 0.799, "total": 21.349 } }