{ "what": "The registry's C_16 enclosure (maximusrufus, teorth/optimizationproblems #246, merged 2026-10-26 with both rows asterisked) decided as a second party by two implementations that share no code with each other or with the claimant's checkers (kept as text, never run): every finite claim of the two certificates, stated in corpus/optconst-queue/c16-2026-10-27/claims.json before deciding; the analytic steps (the weighted Cauchy–Schwarz dual bound, the tail bounds, the asymptotic bound g) read and re-derived there, never given a verdict; the convexity step not used (each box decided by subdivision with monotone enclosures)", "generated": "2026-10-27T14:54:37Z", "seconds": 227.7, "node": "v24.14.1", "python": "3.9.6", "deciders": { "js": "a8622101c59098d0ad1131138bc00e708611a15eb51543df2fe25803704630e2", "py": "098fa3d3629e831f506628863a1a8b61b50144f33b4fbe4506f208c8c0d00338", "bgw16a": "b3a7eca375da5830058e88223582a4fe3f75acca9d57efbd4389fac4c432cda3", "ivdec": "f4e514c49e79d8e082d825703d1efaf525eb543299f00fd4713b22b883b2f04f", "eqcertInterval": "9891280d08e59f7f3cf1549257d86af988f93971a4c307abc309cb0550a315cd", "eqcertTranscendental": "7d62c491ab6e79a7873f78111f666eb71924b26cc406a8758b91ebd77ad69b79", "eqcertRational": "59e97fc40328adc7866f148d8ee61156810dbbcd890b4361dca697aa58d71920" }, "pins": "corpus/optconst-queue/c16-2026-10-27/PINS.json", "pinsSha256": "2c7447fec7dcd90aa78391b7aad625d3936ba008a7194184e1bc16771a6013f2", "claims": "corpus/optconst-queue/c16-2026-10-27/claims.json", "claimsSha256": "fd4016c8358117939f3234a465d4b6cdae80f3f8858b4454de6fd130ba56ee67", "registryCommit": "e95af44bc6ae938f545a0fc62cd1b6c1f6088a56", "certificateCommit": "b3728b856bd79a85f8ba94c158e7a51616c0179b", "pr": { "number": 246, "title": "C_16: certified upper bound 2.1562256466 (enclosure with the #226 lower bound)", "author": "maximusrufus", "mergedAt": "2026-10-26T03:54:34Z" }, "printed": { "lower": "2.1562247706", "upper": "2.1562256466", "asteriskedAtPin": true, "readme": { "lower": "2.1562247706", "upper": "2.1562256466" } }, "word": { "lower": "CERTIFIED", "upper": "PARTIAL", "row": "PARTIAL" }, "wordWhy": "lower: every finite claim CERTIFIED by both implementations; the one reading (LR1: Parseval on [0, 2π]², the normalisation) is the definition applied to one explicit u, as in optq-226-16a. upper: the finite core CERTIFIED by both (UF1–UF3: the tiling, all 84 boxes against the printed digits, g(D0); and UF4's U_rational <= the printed digits); the step from it to C_16 <= 2.1562256466 crosses three analytic steps that are READ (UR1 the weighted Cauchy–Schwarz bound, UR2 the tail bounds with Σ|k|⁻⁴ = (2π²/3)G, UR4 F <= g for δ >= e and its supremum on [D0, ∞)), re-derived in claims.json — hence PARTIAL under the house grammar", "verdicts": { "LF1": { "js": "CERTIFIED", "py": "CERTIFIED", "word": "CERTIFIED" }, "LF2": { "js": "REFUSED", "py": "CERTIFIED", "word": "CERTIFIED", "note": "the brackets contain the exact sums and the rational is below the exact value (Python, exact at 2^-200); the JavaScript decider, at double precision, sees the brackets meet its enclosures and cannot separate the rational, which sits ~1.3e-15 below the exact value (REFUSED at its precision)" }, "UF1": { "js": "CERTIFIED", "py": "CERTIFIED", "word": "CERTIFIED" }, "UF2": { "js": "CERTIFIED", "py": "CERTIFIED", "word": "CERTIFIED" }, "UF3": { "js": "CERTIFIED", "py": "CERTIFIED", "word": "CERTIFIED" }, "UF4": { "js": "REFUSED", "py": "REFUSED", "word": "REFUSED", "U_rational": "CERTIFIED (both): 2.1562256465510830 <= 2.1562256466", "boxBounds": { "settledByJs": 82, "settledByPy": 65, "settledByEither": 84, "refutedByEither": 0, "note": "a printed box bound is the claimant's float at R = 2000; the Python decider's looser tail (R = 1000) cannot reach the small-μ boxes and the JavaScript decider's doubles cannot separate the two g-boxes, whose printed bound is g(d1) to the last ulp; no box bound is below its formula in either; UF4 bears on nothing in the enclosure (UF2 decides the boxes against the printed digits directly)" } } }, "readings": { "UR1": { "statement": "for every μ > 0 and δ >= 1: F(δ) <= B_μ(δ) := S1(μ)(1 + μδ)/π − ln δ − ln(1 + ln δ), S1(μ) = Σ_{k≠0} 1/(|k|²(1 + μ|k|²))", "reason": "an inequality between infinite sums; three lines, re-derived here", "used": "yes" }, "UR2": { "statement": "for R >= 2 and T = R − √2: Σ_{|k|>R} 1/(|k|²(1 + μ|k|²)) <= min( (Z2 − P2(R))/μ , π ln(1 + 1/(μT²)) + π√2/T ), where P2(R) = Σ_{0<|k|<=R} |k|⁻⁴ and Z2 = Σ_{k≠0} |k|⁻⁴ = 4ζ(2)β(2) = (2π²/3)G, G Catalan's constant", "reason": "bounds on an infinite sum; (A) rests on a classical identity, (B) on an integral comparison; both re-derived here. The value of G and π are computed, not read.", "used": "yes" }, "UR3": { "statement": "B_μ is convex in δ on [1, ∞), so its maximum over a box is attained at an endpoint (the claimant's step between boxes)", "reason": "correct, and NOT USED: the replay decides every box by subdivision with monotone enclosures (FINITE UF2), so no verdict rests on convexity", "used": "not used: every box is decided by subdivision" }, "UR4": { "statement": "with R0 = 100, T0 = R0 − √2, H = Σ_{0<|k|<=R0} |k|⁻², K = H/π + √2/T0 − 2 ln T0: for every δ >= e, F(δ) <= g(δ) := 1 + K + (K + ln ln δ)/ln δ + 2T0²/(δ ln δ); and sup_{δ >= D0} g(δ) <= g(D0) when ln D0 >= 1 and K + ln ln D0 >= 1", "reason": "the claimant's g is stated in the checker with its derivation in certify_c16_upper.py, which is NOT in the repository; re-derived here from the write-up's 'μ = 1/(δ ln δ)' and found to be the same function", "used": "yes" }, "UR5": { "statement": "a LOWER bound for the same tail, used only for lower ends of enclosures (refutations, red controls), never for a certification: with T' = R + √2, Σ_{|k|>R} 1/(|k|²(1 + μ|k|²)) >= max(0, π ln(1 + 1/(μT'²)) − π√2 · min(1/T', 1/(3μT'³)))", "reason": "an integral comparison, re-derived here", "used": "lower ends only (refutations, red controls)" }, "LR1": { "statement": "For the real zero-mean trigonometric polynomial u = Σ_{0 < |k|² ≤ R²} c_k e^{ik·x}, c_k = 1/(|k|²(1 + μ|k|²)) (c_k = c_{−k}), Parseval on [0, 2π]² gives u(0) = S1, ‖∇u‖² = 4π² S2, ‖Δu‖² = 4π² S3, with (the certificate's names, n = |k|², r₂(n) lattice points of norm n, c_n = q/(n(q + pn)), μ = p/q) S1 = Σ r₂(n) c_n, S2 = Σ r₂(n) n c_n², S3 = Σ r₂(n) n² c_n². Normalising ũ = u/‖∇u‖: Θ(δ_u) ≥ S1²/(4π² S2) at δ_u = S3/S2, so L ≥ F(δ_u) ≥ S1²/(π S2) − ln δ_u − ln(1 + ln δ_u), provided δ_u >= 1.", "reason": "definitional: the sup in Θ and the max in L applied to one explicit u; Parseval on the 2π-torus. The same reading as optq-226-16a (2026-10-16).", "used": "yes" } }, "findings": [ "the enclosure holds as printed: 2.1562247706 <= C_16 <= 2.1562256466 (upper: PARTIAL as above)", "the boxes support 2.1562256458 (the largest endpoint value, box 31, μ = 0.12206997134993264, is 2.1562256407495 … 2.1562256457523); the printed 2.1562256466 and the certificate's U = 2.1562256465510830 are conservative by ~8e-10 (each printed box bound sits ~5–8e-10 above its endpoint maximum: the certifier's float64 error factor)", "the trial polynomial's exact value is 2.15622477064423579034 … 2.15622477064423579035 (Python; JavaScript 2.1562247706441 … 2.1562247706444); the printed 2.1562247706 is its floor at ten places", "the certificate's float fields slip by 1e-12 against their own rationals: L_lower_bound_float_rounded_down = 2.1562247706432345 while float(L_lower_bound_rational) = 2.1562247706442346 (harmless: lower), and U = 2.1562256465520826 while float(U_rational) = 2.1562256465510825 (harmless: higher)", "neither checker's VERIFIED is the printed inequality: the lower checker accepts |L_lo − claimed| < 1e-9, the upper accepts U_v <= U + 1e-9 (at R = 1000); both inequalities are decided here against the printed digits directly", "the certifiers named in the checkers (certify_c16_upper.py, certify_c16_lower.py) are not in the repository; the asymptotic g they use is re-derived in claims.json UR4 and agrees with the checker's formula", "the registry's numerical row ≈ 2.15627 [BDZ2013] lies above the upper end: any value printing as 2.15627 is >= 2.156265 > 2.1562256466" ], "js": { "lower": { "LF1": { "claim": "C_16 >= 2.1562247706", "verdict": "CERTIFIED", "R": 4000, "mu": "1221/10000", "latticePoints": 50265328, "distinctNorms": 3126894, "S1": [ "9.313582031186797", "9.313582031186878" ], "S2": [ "6.294089927426415", "6.294089927426461" ], "S3": [ "24.729665059462636", "24.729665059462811" ], "delta": [ "3.929029509366121", "3.929029509366179" ], "deltaAtLeast1": true, "A": [ "4.386828656220537", "4.386828656220651" ], "enclosure": [ "2.1562247706441", "2.1562247706444" ], "seconds": 6.3 }, "LF2": { "bracketsMeetEnclosures": { "S1": true, "S2": true, "S3": true }, "rationalLeExactRHS": "REFUSED", "floatLeRational": true, "printedLeRational": true, "rationalDecimal": [ "2.1562247706442341", "2.1562247706442346" ], "verdict": "REFUSED" } }, "upper": { "claim": "C_16 <= 2.1562256466", "verdict": "CERTIFIED", "R": 2000, "distinctNorms": 820303, "UF1": { "verdict": "CERTIFIED", "boxes": 84, "from": "1", "to": "490000000" }, "UF2": { "verdict": "CERTIFIED", "boxes": 84, "certified": 84, "subBoxes": 363550, "viaG": 3, "maxDepth": 21, "refusedBoxes": [] }, "UF3": { "verdict": "CERTIFIED", "gD0": [ "2.0586917808338", "2.0586917808340" ], "sideConditions": true, "claimantPrints": 2.058691780833913 }, "UF4": { "U_rationalLePrinted": true, "U_rationalDecimal": [ "2.1562256465510825", "2.1562256465510830" ], "boxBounds": { "CERTIFIED": 82, "REFUTED": 0, "REFUSED": 2 }, "verdict": "REFUSED" }, "K": [ "0.865689609438524", "0.865689609438594" ], "H": [ "31.520223641129469", "31.520223641129587" ], "P2": [ "6.026811254292156", "6.026811254292195" ], "endpointMax": { "box": 31, "enclosure": [ "2.1562256407495", "2.1562256457523" ] }, "supported": "2.1562256458", "supportedVerdict": "CERTIFIED", "printedMinusOneUnit": { "bound": "2.1562256465", "verdict": "CERTIFIED", "how": "implied: 2.1562256458 <= 2.1562256465 and 2.1562256458 is CERTIFIED (the acceptance test is monotone in the bound)" }, "seconds": 56.3, "boxes": [ { "box": 0, "d1": "1", "d2": "11813512380064925/9007199254740992", "mu": "10000.00000000001", "printedBound": 2.0048956558169997, "S1": [ "0.000602634618983", "0.000602634619115" ], "tail": { "used": "A", "A": "0.000000000078540", "B": "0.002223013455565", "lower": "0.000000000078409" }, "B_d1": [ "1.9184373943484", "1.9184373947632" ], "B_d2": [ "2.0048956548149", "2.0048956553588" ], "g_d1": null, "formula": [ "2.0048956548149", "2.0048956553588" ], "UF4": "CERTIFIED", "UF2": "CERTIFIED", "subBoxes": 7, "closedByG": false }, { "box": 1, "d1": "11813512380064925/9007199254740992", "d2": "7309912752694429/4503599627370496", "mu": "2.179217670291749", "printedBound": 2.112942575667142, "S1": [ "2.071900286328824", "2.071900286923527" ], "tail": { "used": "A", "A": "0.000000360404470", "B": "0.002223374290846", "lower": "0.000000359809786" }, "B_d1": [ "2.0332987722615", "2.0332987729919" ], "B_d2": [ "2.1129425742632", "2.1129425751222" ], "g_d1": null, "formula": [ "2.1129425742632", "2.1129425751222" ], "UF4": "CERTIFIED", "UF2": "CERTIFIED", "subBoxes": 11, "closedByG": false }, { "box": 2, "d1": "7309912752694429/4503599627370496", "d2": "17426138630712791/9007199254740992", "mu": "0.7691191302112587", "printedBound": 2.1352907752740595, "S1": [ "4.169446768874040", "4.169446770559156" ], "tail": { "used": "A", "A": "0.000001021167928", "B": "0.002224035988242", "lower": "0.000001019482851" }, "B_d1": [ "2.1046590859416", "2.1046590871477" ], "B_d2": [ "2.1352907733380", "2.1352907746727" ], "g_d1": null, "formula": [ "2.1352907733380", "2.1352907746727" ], "UF4": "CERTIFIED", "UF2": "CERTIFIED", "subBoxes": 21, "closedByG": false }, { "box": 3, "d1": "17426138630712791/9007199254740992", "d2": "5058112939009181/2251799813685248", "mu": "0.4584325861855489", "printedBound": 2.1466053716124147, "S1": [ "5.492704927578160", "5.492704930405469" ], "tail": { "used": "A", "A": "0.000001713228535", "B": "0.002224729026875", "lower": "0.000001710401270" }, "B_d1": [ "2.1323332833191", "2.1323332850174" ], "B_d2": [ "2.1466053691393", "2.1466053709661" ], "g_d1": null, "formula": [ "2.1466053691393", "2.1466053709661" ], "UF4": "CERTIFIED", "UF2": "CERTIFIED", "subBoxes": 37, "closedByG": false }, { "box": 4, "d1": "5058112939009181/2251799813685248", "d2": "23038764881360657/9007199254740992", "mu": "0.3220193801585742", "printedBound": 2.1524732075026782, "S1": [ "6.466729841424558", 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true, "UF3": true, "UF4": true, "boxEndpoints": true, "G": true, "K": true, "reds": true }, "agree": true, "redControls": [ { "id": "upper-below-supported", "what": "C_16 <= 2.1562256457, one unit in the tenth place below the bound the boxes support (2.1562256458): box 31 (its endpoint value 2.1562256407495 … 2.1562256457523) cannot be closed", "expect": "REFUSED or REFUTED", "js": "REFUSED", "fired": true, "box": 31, "py": "REFUSED" }, { "id": "lower-plus-one-unit", "what": "C_16 >= 2.1562247707 from the same trial polynomial (the printed lower bound one unit up): its value 2.1562247706441… is below", "expect": "REFUTED", "js": "REFUTED", "fired": true, "py": "REFUTED" }, { "id": "box-bound-below-its-max", "what": "box 31 (μ = 0.12206997134993264) with its printed bound 2.1562256465510825 replaced by 2.1562256407485436, below the lower end of its endpoint maximum 2.1562256407495", "expect": "REFUTED or REFUSED", "js": "REFUTED", "fired": true }, { "id": "coverage-gap", "what": "box 40 removed from the tiling of [1, D0]", "expect": "REFUTED", "js": "REFUTED", "fired": true, "why": "box 39 ends at 142912841316515349/36028797018963968, box 40 starts at 1146719135389559897/288230376151711744" }, { "id": "tail-dropped-mutant", "what": "a mutant decider with S1(μ) taken as its partial sum over |k| <= 2000 alone (the UR2 tail omitted) certifies C_16 <= 2.1562240, below the certified lower end 2.1562247706 — caught by the crossing; the full decider cannot close box 31 on the same claim (REFUSED)", "expect": "CAUGHT", "js": "CAUGHT", "fired": true, "mutant": "CERTIFIED", "full": "REFUSED" } ], "limits": [ "the upper end's step from the finite core to the constant is analytic (UR1, UR2, UR4) and read, not computed", "the lower end is a value of F at one explicit u; it says nothing about where the maximum is", "the two deciders share the mathematics (the monotone enclosures of B_μ and g; the dual bound) and no code; they differ in the partial-sum radius (2000 here, 1000 in Python), in G (an alternating bracket here, Ramanujan's series there), in the r₂ table (divisor sieve here, octant enumeration there) and in the box method (monotone subdivision here, derivative-sign tests there)" ] }