{
 "what": "The optimization-constants registry's C_31 rows of 2026-10-18 (PR #231, [J2026]: γ₂ ≤ 0.825101 at h = 9 and ≤ 0.824120 at h = 10, both asterisked), replayed: Lueker's automaton M_h rebuilt from his paper alone in two implementations that share no code with each other or with the claimant (instruments/lueker/mh.py, the reference; mh.c, held to it), calibrated on Lueker's own table for h = 2..8, run at h = 9 and h = 10 with our own witness vectors, the certificate pass under directed rounding, the bound evaluated outward, the printed rows decided.",
 "generated": "2026-10-22T10:42:47Z",
 "seconds": 176.1,
 "machine": {
  "platform": "macOS-27.0-arm64-arm-64bit",
  "cpus": 8,
  "python": "3.9.6",
  "model": "Apple M2",
  "memoryBytes": 17179869184
 },
 "compiler": {
  "command": "/usr/bin/clang -O2 -ffp-model=strict -o /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data/mh-runner instruments/lueker/mh.c -lpthread -lm",
  "version": "Apple clang version 21.0.0 (clang-2100.0.123.102)"
 },
 "code": {
  "mh.c": "d1aa7051b26ce93afa56954b10f8ffac006d688e1e33c32f22841e69122b03eb",
  "mh.py": "08faf521d55ac83c1590686c3b2c5d9ae514424e8ac88ad35726f5434f753259",
  "runner": "adf344639ec3f87003296792a996d6456ab50a6b963f29ab3789a8dbe9cb8efc"
 },
 "paper": {
  "title": "Improved bounds on the average length of longest common subsequences",
  "author": "George S. Lueker",
  "journal": "J. ACM 56(3), Article 17 (2009)",
  "doi": "10.1145/1516512.1516519",
  "readAs": "the author's extended version dated 2008-12-18 (41 pages), Wayback Machine capture of 2009-02-05; its Theorem 2, Figure 8 and Figure 9 are the journal's Theorem 3.3, Figure 7 and Table II",
  "url": "https://web.archive.org/web/20090205160100id_/http://www.ics.uci.edu/~lueker/papers/lcs/lcs.pdf",
  "sha256": "c662fc3b247307437876ec075da9ca376e4ca90de314a3a2b867d757c1692a7e"
 },
 "sources": {
  "dir": "corpus/optconst-queue/c31-2026-10-21",
  "claimantCommit": "cd99c689",
  "registryCommit": "46dbf03669219cb2bcf33fd5bbf65209d137f994",
  "pr": {
   "number": 231,
   "title": "C_31: improved upper bound 0.824120 (Lueker's method at h = 9 and 10)",
   "mergedAt": "2026-10-18T05:48:50Z",
   "mergeCommit": "87ce7072a011871a9edbc7af85b5422253f521f8",
   "author": "leandrejack-afk"
  },
  "files": {
   "upstream/31a.md": "87b45b99d718d1fbf78050d79b20012e263680e78cc7d1c8edee90d9a41eae89",
   "upstream/README.md": "2364c27121c08724869e42d785b113cba915c747d2592b052c9c1e785e9d3eea",
   "claimant/README.md": "37dcddfbba47dca7559b7c4c1ed0153dd0021ebc5c3e9b264e9e8834be338c07",
   "claimant/Makefile": "a818a689a3f3b73e53767bd892c79c00b8fcbfc6e6236f01bec6e5700200557b",
   "claimant/certificates/SHA256SUMS.txt": "c313cc64ba1bcb9f4449a5084318ea2959adb2cf9c0b7ca500f598846827c8fb",
   "claimant/certificates/certify_h9.out": "80f151044cad2926ff808795f97bcf5b58e5bbc47f62d5f65011764596894746",
   "claimant/certificates/certify_h10.out": "05570f674225808f80c20ad59eec28cf218194871c8546e88b2db42a706e4f34",
   "claimant/certificates/weights_up_h9_z0.3298.txt": "4527978829778b09a63b7b85e5eb2937e92e043b1f085ce800d77712d0a99b4e",
   "claimant/certificates/weights_up_h10_z0.3376.txt": "1fa072bf1fe9a02e46443760ec1417426a521accad9e721be5ea2527fc51adfd",
   "claimant/README-472c18de.md": "8590bc10bbe0304b635b65a19769c28c69e45edb6dbea077a2c38103d35d82d3"
  }
 },
 "method": {
  "theorem": "Lueker's Theorem 2 (journal Theorem 3.3), after Dančík–Paterson: M covers Σ*, z ∈ (0, 1/2), λ ∈ (0, 1), v > 0 with A_z v ≤ λ v component-wise ⇒ γ₂ ≤ 2 log(2z)/log λ; any positive v is a witness (eq. 41)",
  "automaton": "Figure 8 (journal Figure 7), transition by transition; states (x, y, dx, dy); one representative per complement orbit (x̄, ȳ, dx, dy), the symmetry Lueker names in §3.2 and checks here on every state at h = 4",
  "weights": "z^{a+2} for a < h−1, z^{h+1}/(1−z) at a = h−1 (eq. 36), rounded up from exact for the certificate",
  "certificate": "λ_cert = max_s (A_z v)_s / v_s with FE_UPWARD in every thread (self-tested); λ_free the same in round-to-nearest inflated by (1+2^-52)^(4h−1); ρ_lower = min_s with FE_DOWNWARD, a lower bound on the spectral radius; the bound 2 ln(2z)/ln λ in 60-digit decimal, widened outward",
  "words": "CERTIFIED: our certificate reaches the printed bound; REFUSED: it does not, or an input is malformed; REFUTED: the claimant's λ below our certified lower bound on ρ(A_z)"
 },
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  "REFUSED": 0,
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   "h": 9,
   "constant": "C_31",
   "ref": "J2026",
   "claim": "γ₂ ≤ 0.825101 (Lueker's method at h = 9)",
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   "claimantLambdaMinusOurs": "1.540e-11",
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    "0x1.060d3c48dfff4p-11",
    "0x1.61dfe6e994111p-13",
    "0x1.dddf4df149621p-15",
    "0x1.42a8cc2dc56ap-16",
    "0x1.48e4f2c096861p-17"
   ],
   "claimant": {
    "lambda": "0x1.8ac915e584bc4p-2",
    "lambdaDecimal": "0.38553270543451057",
    "states": 400253317
   },
   "bound": [
    "0.8241195406941601630209750195431848205262",
    "0.8241195406941601630209750195431848205262"
   ],
   "boundUp": "0.8241195407",
   "lambdaCertUp": "0.385532597279844",
   "rhoLowerDown": "0.385532597279807",
   "printed": "0.824120",
   "verdict": "CERTIFIED",
   "why": "γ₂ ≤ 0.824119540695 by our certificate, at or below the printed 0.824120",
   "claimantLambdaMinusOurs": "1.082e-07",
   "claimantLambdaAboveOurLowerBound": true,
   "statesAgree": true,
   "sample": {
    "h": 10,
    "states": 1000,
    "transitions": 38000,
    "seed": 20261021,
    "mismatches": 0
   },
   "seconds": {
    "build": 13613.2,
    "eigen": 12206.9,
    "certify": 46.4
   }
  }
 ],
 "redControls": [
  {
   "id": "weights-off-by-one-power",
   "what": "the weights read as z^{a+1} instead of z^{a+2} at h = 4",
   "expect": "DIFFERS from Lueker",
   "luekerLambda": "0.217017510",
   "got": "0.821413740",
   "fired": true
  },
  {
   "id": "truncation-weight-not-summed",
   "what": "the truncated input weighted z^{h+1} instead of z^{h+1}/(1−z) at h = 4",
   "expect": "DIFFERS from Lueker",
   "luekerLambda": "0.217017510",
   "got": "0.215991524",
   "fired": true
  },
  {
   "id": "claim-below-the-certificate",
   "what": "Lueker's h = 8 certificate against a printed bound 10^-6 below his",
   "expect": "REFUSED",
   "got": "REFUSED",
   "why": "our certificate gives γ₂ ≤ 0.826279178561, above the printed 0.826279: not reached by this certificate",
   "fired": true
  },
  {
   "id": "lambda-below-the-spectral-radius",
   "what": "a claimant λ one part in 10^9 below our certified lower bound on ρ(A_z) at h = 8",
   "expect": "REFUTED",
   "got": "REFUTED",
   "why": "the claimant's certified λ = 0.34183225837681519 is below our certified lower bound 0.34183225871864747 on the spectral radius of A_z: no positive vector satisfies A_z v ≤ λ v",
   "fired": true
  },
  {
   "id": "z-at-one-half",
   "what": "z = 1/2, outside Theorem 2's (0, 1/k)",
   "expect": "REFUSED",
   "got": "REFUSED",
   "why": "z outside (0, 1/2): Theorem 2 needs 0 < z < 1/k with k = 2",
   "fired": true
  },
  {
   "id": "lambda-at-one",
   "what": "λ = 1, outside (0, 1)",
   "expect": "REFUSED",
   "got": "REFUSED",
   "why": "λ outside (0, 1): Theorem 2 needs λ(z) in (0, 1)",
   "fired": true
  },
  {
   "id": "vector-with-a-zero",
   "what": "a witness vector with one zero component (h = 5)",
   "expect": "REFUSED",
   "got": "REFUSED",
   "why": "mh: v not positive and finite",
   "fired": true
  },
  {
   "id": "vector-wrong-length",
   "what": "a witness vector one component short (h = 5)",
   "expect": "REFUSED",
   "got": "REFUSED",
   "why": "mh: v read: size differs from the state count",
   "fired": true
  },
  {
   "id": "all-ones-vector",
   "what": "the all-ones vector as the witness at h = 8: a valid certificate (λ = the largest row sum), too weak for the printed bound",
   "expect": "REFUSED",
   "got": "REFUSED",
   "lambda": "0x1.9a359a8137a0dp-2",
   "bound": [
    "0.9695596334708344583242636497500245546543",
    "0.9695596334708344583242636497500245546543"
   ],
   "why": "our certificate gives γ₂ ≤ 0.969559633471, above the printed 0.826280: not reached by this certificate",
   "fired": true
  }
 ],
 "redControlsFired": "9/9",
 "log": [
  {
   "cmd": "build 2 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 8",
   "stderr_tail": ""
  },
  {
   "cmd": "eigen 2 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 0.2118 600 1e-13 8",
   "stderr_tail": "  iter 32 ratio [0.13120421149656478, 0.13120421149657246] spread 5.860e-14 0 s"
  },
  {
   "cmd": "certify 2 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 0.2118 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data/v_h2.bin 8",
   "stderr_tail": ""
  },
  {
   "cmd": "build 3 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 8",
   "stderr_tail": ""
  },
  {
   "cmd": "eigen 3 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 0.2422 600 1e-13 8",
   "stderr_tail": "  iter 71 ratio [0.17775261044260832, 0.17775261044262408] spread 8.869e-14 0 s"
  },
  {
   "cmd": "certify 3 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 0.2422 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data/v_h3.bin 8",
   "stderr_tail": ""
  },
  {
   "cmd": "build 4 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 8",
   "stderr_tail": ""
  },
  {
   "cmd": "eigen 4 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 0.2642 600 1e-13 8",
   "stderr_tail": "  iter 101 ratio [0.2170175100984329, 0.21701751009845147] spread 8.556e-14 0 s"
  },
  {
   "cmd": "certify 4 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 0.2642 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data/v_h4.bin 8",
   "stderr_tail": ""
  },
  {
   "cmd": "build 5 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 8",
   "stderr_tail": ""
  },
  {
   "cmd": "eigen 5 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 0.2828 600 1e-13 8",
   "stderr_tail": "  iter 136 ratio [0.25410060606001322, 0.25410060606003682] spread 9.285e-14 0 s"
  },
  {
   "cmd": "certify 5 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 0.2828 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data/v_h5.bin 8",
   "stderr_tail": ""
  },
  {
   "cmd": "build 6 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 8",
   "stderr_tail": ""
  },
  {
   "cmd": "eigen 6 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 0.2980 600 1e-13 8",
   "stderr_tail": "  iter 172 ratio [0.28715560866856055, 0.28715560866858897] spread 9.898e-14 0 s"
  },
  {
   "cmd": "certify 6 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 0.2980 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data/v_h6.bin 8",
   "stderr_tail": ""
  },
  {
   "cmd": "build 7 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 8",
   "stderr_tail": ""
  },
  {
   "cmd": "eigen 7 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 0.3104 600 1e-13 8",
   "stderr_tail": "  iter 211 ratio [0.31602274410587972, 0.31602274410590908] spread 9.292e-14 5 s"
  },
  {
   "cmd": "certify 7 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 0.3104 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data/v_h7.bin 8",
   "stderr_tail": ""
  },
  {
   "cmd": "build 8 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 8",
   "stderr_tail": "6419098 states, 2 to go, 8 s"
  },
  {
   "cmd": "eigen 8 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 0.3209 600 1e-13 8",
   "stderr_tail": "  iter 251 ratio [0.3418322587186462, 0.34183225871868] spread 9.890e-14 56 s"
  },
  {
   "cmd": "certify 8 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 0.3209 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data/v_h8.bin 8",
   "stderr_tail": ""
  },
  {
   "cmd": "certify 9 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 0.3298 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data/v_h9.bin 8",
   "stderr_tail": ""
  },
  {
   "cmd": "certify 10 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data 0.3376 /private/tmp/claude-501/-Users-carlostoledo-Projects-cert-machine/8e6103f2-2be3-4982-be32-838cc5c0061f/scratchpad/c31/data/v_h10.bin 8",
   "stderr_tail": ""
  }
 ]
}