{ "what": "the open pull-request queue of teorth/optimizationproblems on 2026-10-15 \u2014 #210, #217, #221, #225, #226 \u2014 decided where a finite exact object exists, read where the claim is a short proof, refused where the evidence is not in the submission; #226's 16a and 70a, refused as not attempted on 10-15, decided on 2026-10-16", "preregistration": "corpus/optconst-queue/preregistration.json", "pins": "corpus/optconst-queue/PINS.json", "registryCommit": "2c1968cd", "rows": [ { "id": "optq-225-1a", "pr": 225, "constant": "C_1a", "claim": "C_1a <= 1.5027435 (an upper bound exhibited by an explicit 65,536-step function with exact rational heights; Atris records @ 1fa857a0)", "definition": "the autocorrelation constant (registry 1a: the largest C with max (f*f) >= C (\u222b_{-1/4}^{1/4} f)^2 for every f >= 0)", "verdict": "CERTIFIED", "value": "645423412969467179759478160871915520/429496729600000035389440000000000729", "valueDecimal": "1.502743486709583379743219592078", "slack": "11416380946843545535536171151000023/858993459200000070778880000000001458000000", "n": 65536, "lag": 25072, "how": "the 65,536 heights as integers over one denominator; the autoconvolution computed as ONE exact integer product by Kronecker substitution and the maximum taken over every grid shift (piecewise linearity puts the supremum on the grid); value 1.502743486709583379743219592078, claimed 1.5027435", "registryBefore": "1.502862 (upper)", "claimantPrinted": "1.502743486709583379743219592079" }, { "id": "optq-225-1b", "pr": 225, "constant": "C_1b", "claim": "C_1b <= 0.38085856 (an upper bound exhibited by an explicit 114,688-step function with exact rational heights; Atris records @ 1fa857a0)", "definition": "Erd\u0151s's minimum overlap constant (registry 1b: the largest C with sup_x \u222b f g(x + \u00b7) >= C for f + g = 1 on an interval of length 2, \u222bf = 1)", "verdict": "CERTIFIED", "value": "10919976354672639675467789613614419/28672000000000000000000000000000000", "valueDecimal": "0.380858550316428560109786189090", "slack": "277647360324532210386385581/28672000000000000000000000000000000", "n": 114688, "lag": 36048, "how": "the 114,688 heights as integers over one denominator; the correlation with 1 - h computed as ONE exact integer product by Kronecker substitution and the maximum taken over every grid shift (piecewise linearity puts the supremum on the grid); value 0.380858550316428560109786189090, claimed 0.38085856", "registryBefore": "0.380868 (upper)", "claimantPrinted": "0.380858550316428560109786189091" }, { "id": "optq-221-3c", "pr": 221, "constant": "C_3c", "claim": "C_3c >= 1.674894211307116 from two laws on 51 and 47 points of Z^2 (weights multiples of 10^-50) mixed in counts 39041372760532870645 : 60958627239467129355 of 10^20 base-55 digits (Jaswal, Zenodo 23289305)", "definition": "the entropy formulation of registry 3c: C_3c >= H(X-Y) / max(H(X), H(Y), H(X+Y), H(X+2Y)) for any finitely supported law; for independent digits the entropies add when the digit map is injective, which max |L| = 27 <= (55-1)/2 guarantees", "verdict": "CERTIFIED", "rho": [ [ "1.674894211307116228764343737060", "1.674894211307116228764343737061" ] ], "precisionBits": 128, "maxAbsForm": "27", "injective": true, "laws": [ 51, 47 ], "how": "per-law pushforwards exact; entropies as directed-rounding intervals at 128 bits (instruments/sumdiff + bigfloat, no code of the claimant's); the mixture by rational weights; \u03c1 enclosed in [1.674894211307116228764343737060, 1.674894211307116228764343737061] against 1.674894211307116", "registryBefore": "1.6747338950414058* (upper 1.75)", "claimantPrinted": "1.674894211307116228764343737060957..." }, { "id": "optq-226-20a", "pr": 226, "constant": "C_20a", "claim": "C_20a >= 8: X = E - 1 (E standard exponential) is isotropic log-concave with Var(X^2) = 8, and products of independent copies give Var(|X|^2) = 8n (Patankar)", "definition": "registry 20a: the smallest C with Var(|X|^2) <= C n for every isotropic log-concave X in every dimension", "verdict": "CERTIFIED", "values": { "claim": "Var(X^2) = 8 for X = E - 1, E standard exponential; hence C_20a >= 8", "mean": "0", "variance": "1", "E[(E-1)^4]": "9", "Var(X^2)": "8", "logConcave": "the density e^{-t} 1_{t>=0}: log-linear on its support, hence log-concave (by definition)", "verdict": "CERTIFIED" }, "how": "E[E^k] = k! exactly; E[(E-1)^4] = 9, E[(E-1)^2] = 1, mean 0: Var(X^2) = 8; independence adds variances. Log-concavity of e^{-t} on [0, \u221e) is the definition (log-linear)", "registryBefore": "2 (lower)" }, { "id": "optq-226-20c", "pr": 226, "constant": "C_20c", "claim": "C_20c >= sqrt(3): mu uniform on [-sqrt3, sqrt3] is isotropic log-concave and A = (-inf, 0] has mu(A) = 1/2, mu+(A) = 1/(2 sqrt3) (Patankar)", "definition": "registry 20c: the KLS constant, sup over n of the worst inverse Cheeger constant mu+(A)/min(mu(A), 1 - mu(A)) over isotropic log-concave mu and Borel A", "verdict": "CERTIFIED", "values": { "claim": "psi_mu >= sqrt(3) for mu uniform on [-sqrt3, sqrt3] and A = (-inf, 0]; hence C_20c >= sqrt(3)", "variance": "1", "mu(A)": "1/2", "cheegerRatioSquared": "1/3", "psiSquared": "3", "logConcave": "a uniform density on an interval is log-concave (constant on its support)", "verdict": "CERTIFIED" }, "how": "in Q(sqrt3) through squares: variance 3/3 = 1, (mu+(A)/min(mu(A), 1-mu(A)))^2 = 1/3, so the inverse Cheeger constant of this A is sqrt3 and psi >= sqrt3", "registryBefore": "sqrt(pi/2) \u2248 1.2533 (lower)" }, { "id": "optq-226-23b", "pr": 226, "constant": "C_23b", "claim": "C_23b <= 1/2: for odd n every off-diagonal entry of G = A^T A is odd, \u03a3(\u03bb_k \u2212 n) = 0 and \u03a3(\u03bb_k \u2212 n)^2 >= n(n\u22121); Bhatia--Davis gives (\u03bb_max \u2212 n)(n \u2212 \u03bb_min) >= n \u2212 1, whence \u03ba(A)^2 >= 1 + 2 sqrt(n\u22121)/n and \u03ba(n) >= 1 + (2 \u2212 sqrt2)/sqrt n for odd n >= 3 (Patankar)", "definition": "registry 23b: with \u03ba(n) the least condition number of an n \u00d7 n \u00b11 matrix and \u03ba(n) = 1 + f(n)/n^\u03b1, the exponent constant", "verdict": "CERTIFIED", "scope": "BY READING, not computed: each step is a standard inequality applied correctly \u2014 the parity of a sum of n terms \u00b11 for odd n; trace of G \u2212 nI zero since diagonal entries equal n; the Frobenius norm bound from n(n\u22121) odd entries of absolute value >= 1; Bhatia--Davis (the variance of numbers in [\u03bb_min, \u03bb_max] with mean n is at most (\u03bb_max \u2212 n)(n \u2212 \u03bb_min)); the last two lines are algebra. A reading, held as such", "how": "the five sentences of the PR text read against the definition; no number is computed", "registryBefore": "1 (upper)" }, { "id": "optq-226-16a", "pr": 226, "constant": "C_16a", "claim": "C_16a >= 2.156224 from a truncated BDZ2013 conditional extremal on [0, 2\u03c0]\u00b2: u = \u03a3_{0 < |k|\u00b2 \u2264 4000\u00b2} c_k e^{ik\u00b7x}, c_k = 1/(|k|\u00b2(|k|\u00b2 + 41/5)); L \u2265 S_0\u00b2/(\u03c0 S_1) \u2212 ln \u03b4 \u2212 ln(1 + ln \u03b4), \u03b4 = S_2/S_1 (Patankar)", "definition": "registry 16a: L = max_{\u03b4 \u2265 1} [4\u03c0\u0398(\u03b4) \u2212 ln \u03b4 \u2212 ln(1 + ln \u03b4)], \u0398(\u03b4) = sup |u(0)|\u00b2 over zero-mean u on T\u00b2 with \u2016\u2207u\u2016\u00b2 = 1, \u2016\u0394u\u2016\u00b2 = \u03b4; T\u00b2 = [0, 2\u03c0]\u00b2 as in BDZ2013 (2.4), (3.14), whose 2.15627 is the registry's numerical row", "verdict": "CERTIFIED", "values": { "constant": "C_16a (Brezis\u2013Gallou\u00ebt\u2013Wainger remainder on the 2\u03c0-torus)", "claim": "C_16a >= 2.156224", "verdict": "CERTIFIED", "K": 4000, "mu": "41/5", "latticePoints": 50265328, "latticePointsByGauss": 50265328, "distinctNorms": 3126894, "scaleBits": 200, "decimalPrecision": 60, "S0": [ "1.136251665485918381930719", "1.136251665485918381930720" ], "S1": [ "0.093658875460082085641875", "0.093658875460082085641876" ], "S2": [ "0.368248886713245279667341", "0.368248886713245279667342" ], "delta": [ "3.931809824795461349799786", "3.931809824795461349799787" ], "deltaAtLeast1": true, "A=S0^2/(pi S1)": [ "4.387834655467140676376667", "4.387834655467140676376668" ], "enclosure": [ "2.156224753793913363046129", "2.156224753793913363046130" ], "width": "1.4325148E-52", "seconds": 7.0, "torus": "[0, 2\u03c0]\u00b2 (BDZ2013 (2.4), (3.14); the registry's 2.15627 row is BDZ's)", "tail": "none: u is a trigonometric polynomial, every sum finite" }, "decidedOn": "2026-10-16", "claimantSource": { "derivations": "vidigoat/optimization-records @ 39fb848 tao/PROOFS.md, pinned at corpus/optconst-queue/sources/vidigoat-39fb848/ with the two scripts as .py.txt (never run)", "pointer": "teorth/optimizationproblems pull/226#issuecomment-6207378455 (2026-10-16T03:25:05Z)" }, "scope": "COMPUTED: the three lattice sums as fixed-point enclosures (floor and ceiling of every term at 2^200; 50,265,328 lattice points in 3,126,894 norm classes from a sieve, the count checked against Gauss's circle formula), \u03c0 by Machin in integers, ln correctly rounded \u00b11 ulp; READ: Parseval on the 2\u03c0-torus, the normalisation \u0169 = u/\u2016\u2207u\u2016 (so \u0398(\u03b4) \u2265 S_0\u00b2/(4\u03c0\u00b2 S_1) and \u03b4 = S_2/S_1), \u03b4 \u2265 1 established from its enclosure", "how": "standard library only (instruments/optq/bgw16a.py, no code of the claimant's); the right-hand side enclosed in [2.156224753793913363046129, 2.156224753793913363046130] against 2.156224; \u03b4 in [3.931809824795461349799786, 3.931809824795461349799787]", "registryBefore": "(\u03b2+\u03c0)/\u03c0 \u2248 1.82283 (lower)", "claimantPrinted": "2.1562247\u2026 (\u03b4 \u2248 3.93181)" }, { "id": "optq-226-70a", "pr": 226, "constant": "C_70a", "claim": "C_70a >= 1.201927 from the cube of side a and the ball of radius r in n = 300,000 with a/r = 13/1000: for every |det u| = 1, |u(B_1) + rB| \u2265 \u03a3_j C(n, j) a^j \u03ba_{n\u2212j} r^{n\u2212j} (Steiner, zonotope faces, Sz\u00e1sz, AM\u2013GM), so C_70 \u2265 (\u03a3_j \u2026)^{1/n}/(a + \u03ba_n^{1/n} r) (Patankar)", "definition": "registry 70a: the infimum of C such that every pair of centrally symmetric convex bodies in every dimension n has a linear u with |det u| = 1 and |u(B_1) + B_2|^{1/n} \u2264 C(|B_1|^{1/n} + |B_2|^{1/n}); a pair for which no u does better than c is a lower bound", "verdict": "CERTIFIED", "values": { "constant": "C_70a (reverse Brunn\u2013Minkowski)", "claim": "C_70a >= 1.201927", "verdict": "CERTIFIED", "n": 300000, "a/r": "13/1000", "decimalPrecision": 50, "enclosure": [ "1.201927142572502844960989", "1.201927142572502844960990" ], "width": "1.7E-48", "log10 of the Steiner sum": [ "-482224", "-482224" ], "log10 of kappa_n": [ "-636701", "-636701" ], "seconds": 1.0, "read": "Steiner; V_j of a zonotope as the sum of its j-face parallelotope volumes; Sz\u00e1sz on the Gram matrix; AM\u2013GM \u2014 each applied as stated, so the cube is in its worst position for every |det u| = 1" }, "decidedOn": "2026-10-16", "claimantSource": { "derivations": "vidigoat/optimization-records @ 39fb848 tao/PROOFS.md, pinned at corpus/optconst-queue/sources/vidigoat-39fb848/ with the two scripts as .py.txt (never run)", "pointer": "teorth/optimizationproblems pull/226#issuecomment-6207378455 (2026-10-16T03:25:05Z)" }, "scope": "COMPUTED: the Steiner sum at n = 300,000 by outward-rounded decimal recurrences (\u03ba_m = \u03ba_{m\u22122}\u00b72\u03c0/m, C(n, j+1) = C(n, j)(n\u2212j)/(j+1), t^j), \u03c0 by Machin, the n-th roots through correctly rounded ln and exp; READ: the four-step chain \u2014 Steiner's formula, V_j of a zonotope as the sum of its j-face parallelotope volumes, Sz\u00e1sz's inequality on the Gram matrix, AM\u2013GM \u2014 each applied as stated; n = 2 agrees with the closed form \u221a(t\u00b2 + 4t + \u03c0)/(t + \u221a\u03c0)", "how": "standard library only (instruments/optq/rbm70a.py, no code of the claimant's); the bound enclosed in [1.201927142572502844960989, 1.201927142572502844960990] against 1.201927", "registryBefore": "1 (lower)", "claimantPrinted": "1.201927" }, { "id": "optq-217-28a", "pr": 217, "constant": "C_28a", "claim": "C_28a <= 9: Borsuk's covering assertion fails in dimension 9 \u2014 OpenAI's BorsukNine (rank-one projectors onto lines of R^4 under the Frobenius metric; Lean OAI.BorsukNine.main_theorem), recorded by kleinwaks", "definition": "registry 28a: the smallest dimension in which Borsuk's conjecture fails; a counterexample in dimension d gives C_28a <= d", "verdict": "CERTIFIED", "scope": "by this repository's own earlier record: certs/openai-math-kernel.json (the openai/math audit, lane K, 2026-10-10: the Lean project rebuilt here and checked by two kernels, the statement read MATCHES against the paper's claim with no definition holes); the PR text re-read today against that row", "kernelRow": { "challenge": "BorsukNine", "word": "CERTIFIED", "why": "Comparator exit 0; nanoda and Lean's kernel both accept", "definitionHoles": 0, "statementLines": 35 }, "how": "nothing new computed: the certificate is the Lean development already decided; the PR's wording (\"counterexample to Borsuk's covering assertion\", dimension 9) agrees with the theorem as read then", "registryBefore": "4 <= C_28a <= 63" }, { "id": "optq-210-lambda", "pr": 210, "constant": "\u039b (de Bruijn--Newman, registry 21a)", "claim": "\u039b <= 319/2000 = 0.1595 by a \"certified record package\" (Mosaic Intelligence, 2026-10-03; the registry records 0.2 (PT2021) with Mosaic's own 0.1875* row; the route: collision exclusion with an interior zero-free check at X = 6000000185827 and an exterior polydisc floor for a 690,988-term Dirichlet polynomial, carried by a \"circular heat\" lemma and a \"real projection\" lemma)", "definition": "registry 21a: the de Bruijn--Newman constant; \u039b <= c is a theorem about the zeros of H_t for t >= c", "verdict": "REFUSED", "scope": "the evidence is not in the submission", "inventory": { "files": 104, "frozenFiles": 63, "scripts": [ "verify/confine.py.txt", "verify/record_score.py.txt", "assembly_ce_319_2000/verify.py.txt", "assembly_ce_319_2000/math/verify.py.txt", "assembly_ce_319_2000/math/second_line.py.txt" ], "imports": { "verify/confine.py.txt": [ "__future__", "os", "re", "runpy", "shutil", "sys", "sysconfig", "threading" ], "verify/confine.py.txt:defs": 4, "verify/record_score.py.txt": [ "__future__", "argparse", "fractions", "hashlib", "json", "os", "pathlib", "re", "shutil", "subprocess", "sys", "time" ], "verify/record_score.py.txt:defs": 20, "assembly_ce_319_2000/verify.py.txt": [ "copy", "fractions", "pathlib" ], "assembly_ce_319_2000/verify.py.txt:defs": 17, "assembly_ce_319_2000/math/verify.py.txt": [ "flint", "fractions", "json", "pathlib", "sympy" ], "assembly_ce_319_2000/math/verify.py.txt:defs": 26, "assembly_ce_319_2000/math/second_line.py.txt": [ "fractions", "json", "mpmath", "pathlib" ], "assembly_ce_319_2000/math/second_line.py.txt:defs": 24 }, "bankCitations": [ "713", "729", "744", "745", "755", "767", "774", "786", "791" ], "mentions": { "zeta": 16, "Dirichlet": 7, "H_t": 171, "Polymath": 24, "PT2021": 222, "bank": 1370 }, "scorerGates": [ "_report_ok", "anchor_ok", "cover_ok", "coverage_ok", "criterion_ok", "frozen_ok", "line_ok", "lint_ok", "record_lines_ok", "rh_ok", "script_ok", "torus_ok" ], "totalChecksRunPrinted": "168846" }, "how": "the package read file by file (nothing run): the scorer (verify/record_score.py, 20 defs) and the two math checkers (math/verify.py with python-flint arb, math/second_line.py with mpmath.iv) evaluate auxiliary inequalities on FROZEN coefficient tables (walls, clocks, norms, debits, Taylor leaves) and compare rationals, hashes and box covers; no function of the three evaluates \u03b6, H_t or the Dirichlet polynomial itself \u2014 the torus floor, the separator at X and the Polymath15 error constants enter as 63 frozen outputs whose derivations cite a private \"bank\" (713, 729, 744, 745, 755, 767, 774, 786, 791) that is not in the bundle. The anchor \u039b <= 3/16 is the claimant's own earlier row, merged unreviewed. What an outside computation could decide first: the torus-floor inequality of bank767 (|F_1| \u2212 \u03a3_{r>1} |F_r| > 1/10 on T^4) by an interval cover; the separator at x = X for t in [0, 3/16] including y = 0; the error constants p, q from Polymath15 Theorem 1.3 \u2014 each a campaign of its own", "registryBefore": "0.2 (PT2021); 0.1875* (Mosaic, merged 2026-07-17)" } ], "counts": { "CERTIFIED": 9, "REFUTED": 0, "REFUSED": 1 }, "decider": { "stepfun.py": "01be053114a8899abf909e06584e23ac0f659a89471329d36420c59b30cb6c57", "twolaw.js": "ad15c46f2469dc9ca944ae0450d40d7045871a52c047abdab2d9d954f86391f5", "exact226.py": "47d486f23ef0edb1cf5ac3410ff5f5daf1829209ce2cc728c919f5b5b6bd12f6", "ivdec.py": "f4e514c49e79d8e082d825703d1efaf525eb543299f00fd4713b22b883b2f04f", "bgw16a.py": "b3a7eca375da5830058e88223582a4fe3f75acca9d57efbd4389fac4c432cda3", "rbm70a.py": "2a773e559b0e1d6aac58d717a05ff8c235a4066b03cea1130b0408c4a52c63b5" }, "python": "3.9.6", "seconds": 156.5, "generated": "2026-10-16T04:35:21Z" }