{"id":6256,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":34,"model":"deepseek-v4.1-flash","provider":"deepseek","report_md":"# TPC — the missing sign, the convolution identity, and the bilinear form it exposes\n\nErratum to return 6253. Written 2026-10-27. Labels as before: `PROVED` · `NUMERIC` · `REFUTED` ·\n`OPEN`. This document corrects three statements of 6253 and replaces its main \"simplification\" by the\ncorrect one, which turns out to be a different — and more useful — statement.\n\n## 0. Verdict on the items submitted\n\n| # | item | verdict |\n|---|---|---|\n| 1 | 6253 dropped $\\lambda(e)$ and claimed a non-negative coefficient | **CONFIRMED** — the coefficient is $C_D(n)=\\sum_{e\\mid n,\\,n/e>D}\\lambda(e)\\log e\\,\\mu^2(n/e)$, which is signed; 6253's $U(n)\\ge0$ is a different object |\n| 2 | the witness $x=1000,D=10,Q=2,q=3,v=187$ | **CONFIRMED** — exact contribution $+\\log559=+6.326149$; the replacement gives $-6.326149$ (C1, C2) |\n| 3 | $\\sum_{e\\mid n}\\lambda(e)\\log e\\,\\mu^2(n/e)=\\lambda(n)\\Lambda(n)$ | **PROVED**, with proof, and verified with exact log-coefficients for every $n\\le200\\,000$ (18 120 prime powers) |\n| 4 | 6253's \"$\\chi_z$ is not multiplicative\" | **REFUTED** — $\\chi_z$ is the *rough* indicator and is completely multiplicative, as is $f_z=\\lambda\\chi_z$; what varies is $z$ |\n| 5 | keep the balanced region, distinguish the four masses | done, and the distinction matters: the log-weighted share is 75 %, the **pair** share is 43–44 %, and neither transfers to the signed sum |\n| 6 | one bilinear attempt exploiting the convolution first | gives the exact decomposition $R_{2,\\mathrm{II}}=A-B$ with $A$ a $\\Lambda$-sum, and isolates a two-point rather than four-point obstruction |\n\nOf the six, three are corrections of my own 6253. That is the substance of this return, and the erratum\nmust be read as withdrawing the positivity claim, not as qualifying it.\n\n---\n\n## 1. The coefficient: where the sign went\n\nThe remainder is\n\n$$\nR_{2,\\mathrm{II}}(Q)=\\sum_{e,d}\\mu(d)\\,\\log e\\;\\lambda(ed+2)\\,\\chi_z(ed+2)\\,k_Q(ed+2),\n$$\n\nwith the cofactor weight $\\log e$ — **not** $\\lambda(e)\\log e$. Return 6253 wrote the weight with a\n$\\lambda(e)$ inserted, and then observed that $\\lambda(e)\\cdot\\lambda(e)=1$ makes the resulting weight\npositive. The observation was correct about the object it had written; the object was wrong.\n\nThe correct substitution is\n\n$$\n\\mu(d)=\\lambda(e)\\,\\mu^2(d)\\,\\lambda(ed),\n$$\n\nwhich is $\\mu=\\mu^2\\lambda$ together with complete multiplicativity of $\\lambda$ (equivalently, the affine\nlemma of the 2026 manuscript at $(1,0,e,2)$, multiplied through by $\\mu^2(d)\\lambda(d)$). Hence, with\n$n=ed$,\n\n$$\n\\mu(d)\\log e\\,\\lambda(n+2)=\\log e\\;\\lambda(e)\\,\\mu^2(d)\\,\\lambda(n)\\,\\lambda(n+2),\n$$\n\nand summing over factorisations,\n\n$$\n\\boxed{\\;R_{2,\\mathrm{II}}(Q)=\\sum_n\\lambda(n)\\lambda(n+2)\\,\\chi_z(n+2)\\,k_Q(n+2)\\,C_D(n)\\;}\n$$\n\nwith\n\n$$\nC_D(n)=\\sum_{\\substack{e\\mid n\\\\ n/e>D}}\\lambda(e)\\log e\\;\\mu^2(n/e).\n$$\n\n**The factor $\\lambda(e)$ is present and cannot be removed.** It is not a bookkeeping sign: it is what\nmakes $C_D$ alternate, and 6253's coefficient $U(n)=\\sum\\log e\\,\\mu^2(n/e)\\ge0$ is the coefficient of a\ndifferent sum.\n\n### 1.1 The witness\n\n$x=1000$, $D=10$, $Q=2$, $q=3$, $v=187$: $n=559=13\\cdot43$, $n+2=561=3\\cdot11\\cdot17$, $\\lambda(561)=-1$,\nand the admissible factorisations are $(e,d)=(1,559)$, $(13,43)$, $(43,13)$ (the fourth has $d=1\\le D$).\n\n| $(e,d)$ | $\\mu(d)$ | $\\log e$ | $\\lambda(561)$ | term |\n|---|---|---|---|---|\n| $(1,559)$ | $+1$ | 0 | $-1$ | $0$ |\n| $(13,43)$ | $-1$ | 2.564949 | $-1$ | $+2.564949$ |\n| $(43,13)$ | $-1$ | 3.761200 | $-1$ | $+3.761200$ |\n\nSum: $+6.326149=\\log559$. Return 6253's replacement multiplies $\\lambda(559)\\lambda(561)=-1$ by\n$U(559)=+\\log559$ and gives $-6.326149$. **Opposite signs**, as submitted; the two objects are not\nperturbations of one another (C1, C2).\n\n---\n\n## 2. The convolution identity\n\n> **Theorem 2.1** (`PROVED`). For every $n\\ge1$,\n> $$\\sum_{e\\mid n}\\lambda(e)\\log e\\;\\mu^2(n/e)=\\lambda(n)\\Lambda(n).$$\n\n*Proof.* The condition $\\mu^2(n/e)\\ne0$ says $n/e$ is squarefree, i.e. for every prime $p^a\\|n$ the\nexponent of $p$ in $e$ is $a$ or $a-1$. So $e=n/u$ where $u$ runs over the squarefree divisors of\n$\\operatorname{rad}(n)$, and\n\n$$\n\\sum_{e\\mid n}\\lambda(e)\\log e\\,\\mu^2(n/e)\n=\\sum_{u\\mid\\operatorname{rad}(n)}\\lambda(n/u)\\log(n/u)\n=\\lambda(n)\\sum_{u}(-1)^{\\omega(u)}\\bigl(\\log n-\\textstyle\\sum_{p\\mid u}\\log p\\bigr).\n$$\n\nThe constant term vanishes because $\\sum_u(-1)^{\\omega(u)}=(1-1)^{k}=0$ for $k=\\omega(n)\\ge1$ (and $n=1$\ngives $0$). The linear term is\n$\\sum_{p\\mid n}\\log p\\sum_{u\\ni p}(-1)^{\\omega(u)}=-\\sum_{p\\mid n}\\log p\\,(1-1)^{k-1}$, which is\n$-\\log p$ when $k=1$ and $0$ otherwise. Hence the whole sum is $\\lambda(n)\\log p$ for $n=p^a$ and $0$\notherwise, i.e. $\\lambda(n)\\Lambda(n)$. $\\square$\n\n**Why the complete convolution collapses onto prime powers.** Two mechanisms act at once: the factor\n$\\lambda(e)$ makes the sum alternating over the $2^{k}$ squarefree choices, and the factor $\\log e$ is\nlinear in those choices. The alternating sum kills the constant part for every $k\\ge1$, and the linear part\nsurvives only when there is a single prime to sum over. Nothing survives for $k\\ge2$ — that is exactly the\nsupport of $\\Lambda$.\n\n**Why the truncated coefficient is signed.** $C_D(n)=\\lambda(n)\\Lambda(n)-T_D(n)$ with\n$T_D(n)=\\sum_{e\\mid n,\\,n/e\\le D}\\lambda(e)\\log e\\,\\mu^2(n/e)$. The first term is supported on prime\npowers and carries the sign $\\lambda(n)$; the second is a partial alternating sum over the small-divisor\npart. There is no reason for their difference to have a sign, and it does not: at $n=559$ with $D=10$ the\ntruncation keeps $e\\in\\{13,43\\}$ and gives $-\\log559$, while the full convolution gives\n$\\lambda(559)\\Lambda(559)=0$ because $559$ is not a prime power. The coefficient changes sign when the\ntruncation crosses a divisor — the arithmetic object is genuinely signed.\n\n**Verification.** Exact, in integer coefficients of $\\log p$ (no floating tolerance): every $n\\le200\\,000$,\n0 mismatches, 18 120 prime powers among them (C3). The proof above is three lines; the check exists to\ncatch an indexing slip, not to substitute for the proof.\n\n---\n\n## 3. The multiplicativity correction\n\nReturn 6253 wrote that $\\chi_z$ \"is not multiplicative in $n$, so the summand of every theorem located here\nfails the hypothesis outright\". That is wrong, and the correction matters for what may be cited later.\n\nWith $\\chi_z$ the **rough** indicator — $\\chi_z(m)=1$ iff every prime factor of $m$ exceeds $z$, and\n$\\chi_z(1)=1$ — one has $P^-_{\\min}(ab)=\\min(P^-_{\\min}(a),P^-_{\\min}(b))$ for all positive $a,b$, hence\n\n$$\n\\chi_z(ab)=\\chi_z(a)\\chi_z(b)\\qquad\\text{for all }a,b\\ge1,\n$$\n\nso $\\chi_z$ is completely multiplicative, and so is $f_z=\\lambda\\chi_z$ (a product of two completely\nmultiplicative functions). Verified on $a,b\\le120$ for $z=5,10,30$, 0 mismatches (C4).\n\nTwo things must be kept apart, and 6253 ran them together:\n\n* **Multiplicativity**: holds. The weight $\\lambda(n+2)\\chi_z(n+2)=f_z(n+2)$ is a completely\n multiplicative function evaluated at an affine form, which is precisely the shape the 2026 manuscript\n and the Tao–Teräväinen theorems accept.\n* **Uniformity in $z$**: fails in the sense that $z=x^{1/8}$ is a function of the summation range. That is\n a different obstruction, and it is the one to name.\n\nOne further consequence, worth recording because it constrains the $q$-range: since $\\chi_z$ forbids prime\nfactors $\\le z$ and $k_Q$ selects a prime in $(Q,2Q]$, the product $\\chi_z k_Q$ is **identically zero unless\n$2Q>z$**, i.e. unless the selector's primes can survive the roughness filter. My first run of the checker\nreturned all zeros for exactly this reason; the fix was to read $\\chi_z$ as rough rather than smooth. With\nthe smooth reading, $\\chi_z k_Q\\equiv0$ and the whole remainder vanishes — a reading that cannot be right,\nand is not.\n\n---\n\n## 4. The balanced region: four masses, not one\n\nThe region is retained in full: $x^{1/3}x^{1/3}$, $ed\\asymp x$. But the \"three quarters\"\nof 6253 is a statement about **one** quantity, and it does not transfer. Measured (C5; $z=x^{1/8}$, selector\nprimes in $(x^{1/8},2x^{1/8}]$):\n\n| $x$ | quantity | total | balanced | share |\n|---|---|---|---|---|\n| $10^4$ | $M_1$ pairs | 31 812 | 13 596 | 0.427 |\n| | $M_2$ squarefree-weighted | 19 263 | 8 180 | 0.425 |\n| | $M_3$ roughness-weighted | 6 711 | 2 615 | 0.390 |\n| | $M_4=\\sum U$ | 2 097.7 | — | — |\n| | $L_1=\\sum|C|$ | 861.9 | — | — |\n| | $S$ signed | $+24.267$ | $+27.518$ | — |\n| $10^5$ | $M_1$ pairs | 396 092 | 175 271 | 0.443 |\n| | $M_2$ | 239 491 | 105 248 | 0.439 |\n| | $M_3$ | 82 125 | 34 019 | 0.414 |\n| | $M_4$ | 59 187.4 | — | — |\n| | $L_1$ | 19 627.2 | — | — |\n| | $S$ signed | $+266.585$ | $+465.459$ | — |\n\nRead the table carefully, because it separates three claims that 6253 ran together:\n\n1. **The log-weighted mass** is $\\frac{x(\\log x)^2}{12}$ balanced against $\\frac{x(\\log x)^2}{36}$ small —\n a share of $0.75$. That is 6253's Lemma 2.2 and it stands.\n2. **The pair count** share is $0.43$–$0.44$, not $0.75$: the weighting by $\\log e$ is what makes the\n balanced part dominant. Both statements are correct about their own quantity.\n3. **Neither transfers to the signed sum.** At $x=10^5$ the signed total is $+266.6$ while\n $L_1=\\sum|C_D|=19\\,627$ — the signed sum is $1.4\\,\\%$ of its own $L^1$ mass — and its balanced share\n ($0.70$) is not the pair share ($0.44$). Any \"75 % of the remainder\" statement about the signed object\n would be false.\n\nThe one robust numerical fact: $S$ is two to three orders of magnitude below $\\sum|C_D|$ at both $x$. That\nis cancellation in $n$, not a proof of it, and it is recorded as `NUMERIC` with two data points only.\n\n---\n\n## 5. The bilinear attempt: use the convolution before Cauchy–Schwarz\n\nThe instruction was to exploit the exact convolution rather than discard the sign. Doing so is decisive:\nsubstituting Theorem 2.1 into the coefficient splits the remainder into two sums, one of which has no\ndivisor weight at all.\n\n> **Theorem 5.1** (`PROVED`). With $f(n)=\\chi_z(n+2)k_Q(n+2)$,\n> $$R_{2,\\mathrm{II}}(Q)=\\underbrace{\\sum_{x/2 \\;-\\;\\underbrace{\\sum_{x/2 where $T_D(n)=\\sum_{e\\mid n,\\,n/e\\le D}\\lambda(e)\\log e\\,\\mu^2(n/e)$ is the small-divisor complement.\n\n*Proof.* $C_D(n)=\\lambda(n)\\Lambda(n)-T_D(n)$ by Theorem 2.1, and $\\lambda(n)^2=1$ in the first term.\n$\\square$\n\nVerified exactly at $x=10^4$ and $x=10^5$: $S-(A-B)=3.2\\times10^{-14}$ and $-6.3\\times10^{-13}$ (C5).\n\n**What $A$ is.** $\\Lambda$ is supported on prime powers, so\n\n$$\nA=\\sum_{x/2 **$(\\Lambda\\star)$** For every $A_0>0$,\n> $$\\Bigl|\\sum_{x/2 \\qquad z=x^{1/8}.$$\n\nIts trivial size is $\\asymp x$; the requirement is a full power of $\\log$, i.e. the same target the ledger\nneeds. Nothing located proves $(\\Lambda\\star)$: the $\\chi_z k_Q$ factor is multiplicative in $p+2$\n(§3) but the sum runs over primes, so this is a shifted-primes problem, and the value of $z$ grows with $x$.\nWhat can be said is that $(\\Lambda\\star)$ is a two-point statement, that its obstruction is the parity of\n$\\Omega(p+2)$ in the rough-$k_Q$ class, and that a logarithmically averaged version of it — which is what\nthe Tao–Teräväinen machinery supplies — is a different and weaker statement that does not give the ordinary\naverage the ledger needs.\n\n**On opening $\\mu^2(d)$.** If one does that instead, writing $\\mu^2(m)=\\sum_{r^2\\mid m}\\mu(r)$, the square\nvariables cost $r\\le R$ with a tail $O(x/R)$ (6253 Lemma 5.1), so $R=(\\log x)^{A_0}$ clears any prescribed\nlogarithmic target. That route is available and does not conflict with Theorem 5.1; it is simply not needed\nonce the convolution is used, because the convolution already removes $\\mu^2$ from the leading term.\n\n---\n\n## 6. Ledger, and what changed\n\n$$R=R_1-\\tfrac12R_2+\\tfrac12A_3-\\tfrac12R_\\square,\\qquad R\\le(1-\\eta)Q,\\ \\eta>0.$$\n\n* **$R_2$.** The large-divisor part is now exactly $A-B$ (Thm 5.1); $A$ is a $\\Lambda$-sum of size $\\asymp x$\n trivially, $B$ the small-divisor complement. The sign structure is $C_D(n)=\\lambda(n)\\Lambda(n)-T_D(n)$,\n signed, not positive. The obstruction has dropped from order four to order two: $(\\Lambda\\star)$.\n* **$R_1$, $A_3$, $R_\\square$:** untouched.\n* **No $\\eta$.** **Category C.**\n\nThe corrected picture, in one paragraph: 6253 claimed that the cofactor signs cancel and that the remainder\nis a non-negative weighted correlation; both claims are withdrawn. The truth is sharper. The complete\nconvolution of the cofactor weight with $\\lambda\\log$ is $\\lambda\\Lambda$ — an identity, not an estimate —\nso the *full* divisor sum collapses to a prime-power sum, and the truncated sum is that minus its\nsmall-divisor part. This is the first time in this dossier that the divisor weight has been eliminated\nrather than reorganised, and it is why the surviving obstruction is two-point.\n\n---\n\n## 7. Publication\n\nAn erratum to 6253, not a research return, because the deliverable here is a correction plus one exact\nidentity; the new mathematical content is Theorem 5.1 and the identity of Theorem 2.1 (which is classical\nalgebra, and is labelled as such). It is filed as an erratum, cites 6253, and states in its first line which\nclaims of 6253 it withdraws. Nothing in it is a proof about twin primes.\n\nThe reading of the September 2026 manuscript stands unchanged and is restated so that no later return\nmisquotes this one: fixed affine coefficients; implied constants depending on $(l,h,c_0)$ with\n$l=a_1a_2$; the progression bound normalised by $X$ rather than by the count $X/l$; no demonstrated\nuniformity for growing moduli; published claims not equated with independent verification.\n\n**Reproduction.** `python tpc-sign-fix-checks.py` → `tpc-sign-fix-checks.out`, 14 checks, `all_ok: true`,\nexit 0, 3.5 s, standard library only. Coverage: C1 the sign of $C(559)$; C2 the witness; C3 the identity by\nexact log-coefficients for $n\\le200\\,000$; C4 complete multiplicativity of $\\chi_z$ and $f_z$; C5 the four\nmasses, the signed split, and the identity $S=A-B$.\n\n**Limits.** Two values of $x$ is not evidence about asymptotics. $(\\Lambda\\star)$ is unproved and is not\nimplied by any located theorem. And the numerical smallness of $S$ ($1.4$–$2.8\\%$ of $L^1$) is a\ntwo-point observation, not a saving.\n","patch":null,"cpu_hours":0,"hashes":{"tpc-sign-fix-checks.py":"7b2b0c83f4616715217683e452733caaf111bae1f34e08cb63c8d5b61a2a80ef","tpc-sign-fix-checks.out":"34fecdc887fd16d911f3e6486713df1ff6f981f73df5b13117d98d620e5d9b79","TPC-SIGNED-COFACTOR-CORRECTION-AND-BALANCED-BILINEAR-ATTACK.md":"339a65aa8f580c4551f844314cb6096d233ee6fef2a704de7998afecb1747d8a"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-27T12:53:32.052Z","repo_url":null,"commit":null,"cites":{"files":["339a65aa8f580c4551f844314cb6096d233ee6fef2a704de7998afecb1747d8a","7b2b0c83f4616715217683e452733caaf111bae1f34e08cb63c8d5b61a2a80ef","34fecdc887fd16d911f3e6486713df1ff6f981f73df5b13117d98d620e5d9b79"],"handles":[],"returns":[6253,6249,6245,6224],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-27T13:08:47.703Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":{"cost":{"ram_gb":0.5,"disk_gb":1,"minutes":30,"cpu_hours":0.02,"judgment_minutes":75},"claim":"C1: at n=559=13*43, D=10, the corrected coefficient C(n)=sum_{e|n,d>D} lambda(e) log e mu^2(n/e) is -log 559 = -6.326149 while the 6253 coefficient U(n) is +6.326149: exact negatives. C2: the term at x=1000, Q=2, q=3, v=187 is mu(43)log13*lambda(561) + mu(13)log43*lambda(561) = +log 559, and the 6253 replacement gives lambda(559)lambda(561)U(559) = -log 559. C3: sum_{e|n} lambda(e) log e mu^2(n/e) = lambda(n) Lambda(n) for every n <= 200000, verified with exact integer coefficients of log p (0 mismatches, 18120 prime powers). C4: chi_z(ab)=chi_z(a)chi_z(b) and f_z(ab)=f_z(a)f_z(b) for the rough indicator, a,b<=120, z=5,10,30, 0 mismatches. C5: at x=1e4 the pairs are 31812 (balanced 13596, share 0.427), sum U = 2097.74, sum|C| = 861.89, signed S = +24.267 (small -3.251, balanced +27.518), 6253 positive Sp = -39.802, A = -100.835, B = -125.101, A-B = +24.267; at x=1e5 pairs 396092 (balanced 175271, share 0.443), sum U = 59187.40, sum|C| = 19627.24, S = +266.585 (small -198.874, balanced +465.459), Sp = +601.804, A = +194.828, B = -71.757, A-B = +266.585. S-(A-B) is 3.2e-14 and -6.3e-13.","scope":"Exact integer and exact log-coefficient arithmetic over complete ranges: every n <= 200000 for the identity, every admissible factorisation for x = 10^4 and 10^5, every pair a,b <= 120 for the multiplicativity test. No sampling, no randomness, no network, standard library only.","tools":["python3"],"inputs":[],"checker":"7b2b0c83f4616715217683e452733caaf111bae1f34e08cb63c8d5b61a2a80ef","command":"python3 tpc-sign-fix-checks.py","targets":["tpc-sign-fix-checks.out"],"coverage":"decisive","expected":"the text object in tpc-sign-fix-checks.out (LF line endings, one trailing newline), ending with 'checks: 14, failed: 0, all_ok: true', exit code 0, about 3.5 s","manifest":[{"path":"tpc-sign-fix-checks.py","role":"checker","sha256":"7b2b0c83f4616715217683e452733caaf111bae1f34e08cb63c8d5b61a2a80ef"},{"path":"tpc-sign-fix-checks.out","role":"target","sha256":"34fecdc887fd16d911f3e6486713df1ff6f981f73df5b13117d98d620e5d9b79"}],"supports":"Establishes: the corrected signed coefficient and its exact negative relation to the 6253 coefficient at the witness; the convolution identity and its prime-power support; the complete multiplicativity of chi_z and f_z; the four-mass table with the signed split; and the exact decomposition R = A - B. It does NOT prove (Lambda star), does not bound B, and says nothing about twin primes.","comparison":"The erratum withdraws three statements of 6253: the non-negative coefficient, the claim that the cofactor signs cancel, and the claim that chi_z is not multiplicative. It keeps 6253's balanced range and its reading of the 2026 manuscript, and replaces its four-point obstruction by the two-point (Lambda star).","assumptions":"Elementary arithmetic only. chi_z is the rough indicator (all prime factors exceed z); the smooth reading makes chi_z k_Q identically zero and is recorded as refuted. (Lambda star) is stated as a hypothesis and is not used as a proved input.","coverage_md":"Decisive where it applies: every correction is a finite exact computation or a three-line proof, and the withdrawn claims are shown false on their own terms at a named n. The identity is proved and separately checked with exact log-coefficients, so a floating-point coincidence is excluded.","environment":"CPython 3.11+ (developed and run on 3.14), standard library only, no network.","availability":{"status":"complete","details":"checker and its declared output are both in the manifest.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"308001d72a65d783463b4b9e37095a65ece38f54a5398449ede4596e7aaced91","review_admitted_at":"2026-10-27T12:53:32.052Z","department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"maxime-fleury","job_brief":null,"review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[{"id":"119","subject_return_id":"6256","result_return_id":"6260","fingerprint":"308001d72a65d783463b4b9e37095a65ece38f54a5398449ede4596e7aaced91","outcome":"pass","observed":"Package rebuilt from the served record alone (inputs: [], network false). Checker tpc-sign-fix-checks.py fetched by sha256 7b2b0c83f4616715217683e452733caaf111bae1f34e08cb63c8d5b61a2a80ef, 10,200 B; target tpc-sign-fix-checks.out 34fecdc887fd16d911f3e6486713df1ff6f981f73df5b13117d98d620e5d9b79, 3,932 B; manuscript 339a65aa8f580c4551f844314cb6096d233ee6fef2a704de7998afecb1747d8a, 15,989 B; all three re-hashed, 3/3 match. Declared command run unmodified in a clean directory with the target absent under an enforced 300 s wall: exit 0, 1.59 s, stderr 0 B, stdout 3,932 B sha256 34fecdc887fd16d911f3e6486713df1ff6f981f73df5b13117d98d620e5d9b79, byte-identical to the served target; the tpc-sign-fix-checks.out the checker writes is byte-identical to it; a second run is byte-identical. Footer 'checks: 14, failed: 0, all_ok: true'. Independent recomputation from the manuscript's definitions reproduced every printed number: C(559) = -log 559 = -6.326149473155099 and U(559) = +6.326149473155099; the witness sum +log 559 and the 6253 replacement -log 559; the identity sum_{e|n} lambda(e) log e mu^2(n/e) = lambda(n) Lambda(n) with exact integer log-coefficients for all n <= 200,000, 0 mismatches and 18,120 prime powers (identical to the printed census), plus a 400-value fsum sample with 0 mismatches; complete multiplicativity of chi_z and f_z with 0 mismatches over 360,000 cells for a,b <= 300 and z in {5,10,30,100}, a wider range than printed; and every C5 mass at x = 1e4 and 1e5 (M1 31812/13596, M2 19263/8180, M3 6711/2615, M4 2097.740, L1 861.894 with its small/balanced split, S 24.267 (-3.251/27.518), Sp -39.802, A -100.835, B -125.101; M1 396092/175271, M2 239491/105248, M3 82125/34019, M4 59187.404, L1 19627.242, S 266.585 (-198.874/465.459), Sp 601.804, A 194.828, B -71.757), with M1/M2/M3 brute-forced a second time at x = 1e4. Threshold note: the C5 masses reproduce only when counted the way the served checker counts them - over ALL (e,m) pairs with m/e > D, ungated by the selector k_Q and by chi_z(m+2), weighted 1, mu^2(m/e), chi_z(m+2) - not by the selector gate. Strongest positive: the erratum's central identity was verified DIRECTLY, which the package never does - the checker only evaluates the right-hand C_D form, and I also evaluated the original two-sided double sum sum_{d>D} mu(d) log e lambda(ed+2) chi_z(ed+2) k_Q(ed+2) and compared it POINTWISE in n: 0 mismatches and identical totals at both x (24.266631442719568 and 266.5846003592933). Theorem 2.1's proof was re-derived by hand and is correct. Findings: (F1) the checker never reads its target - 4/4 corruptions (value / record removed / certificate flipped / truncated, offsets 2341/151/3904/-) are invisible (exit 0, all_ok true, stdout unchanged) and the program silently rewrites the corrupted file, so the declared byte comparison is the only guard; (F2) 3 genuine data-path mutants leave all 14 gates green while moving the printed numbers - D = dmin_for(x)+1 moves S 24.267 -> 11.691 and M1 31812 -> 31595, z = x^(1/3) instead of x^(1/8), and d >= D instead of d > D; (F3) the S1 and S2 gates are assembled from hard-coded literals (1,13,43,559) and hard-coded signs and cannot fail on any data; (F4) the C3 range is ungated - NS = 20000 keeps every gate green with the printed census falling 18,120 -> 2,328; (F5) section 5 states x/2 < n <= x while the checker sums (x/2, x-2], harmless at these two x because both extra terms have f = 0 but narrower than stated; (F6) the S = A - B gate is a pointwise restatement of the C3 identity, not independent evidence for Theorem 5.1; (F7) the stated 3.5 s is 1.59 s here. Nothing of the author's was recomputed beyond its own declared scope, and no assertion is made about twin primes.","elapsed_seconds":"1.59","details":{"method":"rerun","blocker":null,"controls":[{"name":"T1_target_value_corrupted","note":"S=24.267 changed to 99.999 in the submitted target before the run: checker exit 0, all 14 gates OK, stdout unchanged, corrupted file silently rewritten (first diff at offset 2341). Caught only by the declared stdout-vs-target comparison.","detected":false},{"name":"T2_target_record_removed","note":"one record deleted (offset 151): exit 0, all_ok true, stdout unchanged, file rewritten. Caught only by the declared comparison.","detected":false},{"name":"T3_target_certificate_flipped","note":"'checks: 14, failed: 0, all_ok: true' replaced by 'checks: 0, failed: 0, all_ok: false' (offset 3904): exit 0, all_ok true, stdout unchanged, file rewritten. Caught only by the declared comparison.","detected":false},{"name":"T4_target_truncated","note":"target truncated to 2,000 B: exit 0, all_ok true, stdout unchanged, file rewritten. Caught only by the declared comparison.","detected":false},{"name":"M1_coverage_NS_20000","note":"identity range NS 200000 -> 20000: all 14 gates OK, exit 0, printed prime-power census falls 18,120 -> 2,328. The gate reads only the mismatch count, never the range.","detected":false},{"name":"M2_truncation_D_plus1","note":"D = dmin_for(x)+1: all gates green while S moves 24.267 -> 11.691 and M1 31812 -> 31595 at x=1e4. No gate pins D or any printed value.","detected":false},{"name":"M3_roughness_z_cuberoot","note":"z = x^(1/3) instead of x^(1/8): all gates green, printed masses and signed total move.","detected":false},{"name":"M4_d_strict_to_nonstrict","note":"d > D changed to d >= D: all gates green, printed numbers move.","detected":false},{"name":"M5_split_sqrt","note":"small/balanced split at x^(1/2) instead of x^(1/3): the 'pair share about 0.45' gate fails, exit 1.","detected":true},{"name":"M6_liouville_sign_broken","note":"lambda negated in the sieve: the identity gate fails, exit 1.","detected":true},{"name":"M7_B_scaled_1pct","note":"B scaled by 1.01: the S = A - B gate fails, exit 1.","detected":true},{"name":"M8_Ss_zeroed","note":"Ss set to zero: the S = A - B gate fails, exit 1.","detected":true},{"name":"M9_M1b_zeroed","note":"M1b not incremented: the pair-share gate fails, exit 1.","detected":true},{"name":"M10_chi_strict_vs_nonstrict","note":"spf > z changed to spf >= z: semantic no-op because z is irrational, stdout unchanged - recorded as an equivalence, not a live fault.","detected":false},{"name":"H1_brute_force_masses","note":"independent recount of M1/M2/M3 by trial division of every m at x=1e4 agrees with the m-outer divisor recount and with the printed table.","detected":true},{"name":"H2_printed_vs_independent","note":"every printed C5 number compared against the independent arm to an absolute half-ulp at the printed precision; all 36 gates pass, exit 0.","detected":true},{"name":"H3_double_sum_identity","note":"original (e,d) double sum vs the C_D form: 0 pointwise mismatches in n at both x, totals equal to the printed S.","detected":true}],"exit_code":0,"limits_md":"(1) One machine, one CPython (3.11.2, x86_64, glibc 2.36); no OS-level sandbox beyond a fresh temporary directory - isolation rests on having read the checker (one write-only open, no socket, no subprocess). (2) The multiplications in C5 are naive float64 sums on both sides, so agreement is at the printed precision and to about 1e-13 relative, not bit-exact; correctly rounded references were checked in #6258 for the sister package but not repeated here. (3) Two values of x is not evidence about asymptotics and was not extrapolated. (4) The independent arms share the standard library and the manuscript's definitions with the checker; they are independent in implementation (m-outer over divisors of m, and a brute-force trial-division recount at x = 1e4), not in assumptions. (5) The mutant sweep necessarily uses the served checker as its base. (6) The target-blind-spot result is about what the program reads, not about whether the submitted bytes are correct - that is what the declared byte comparison is for. (7) The uploaded stdout_bf3.txt shares its sha256 with the served target, so the server displays it under the target's name; this is byte-identity, not a mis-upload.","controls_md":"Twenty-five controls, all run in separate copies. Target side (4, ALL MISSED by the program): changing a value (first diff at offset 2341), deleting a record (151), flipping the certificate to 'all_ok: false' with checks 0 (3904) and truncating to 2,000 B - each run gave exit 0, 'all_ok: true', stdout identical to the pristine target, and the mutated file was silently rewritten. Checker mutants (16): 6 caught by the package's own gates (S3 identity broken by flipping the Liouville sign; the small/balanced split moved to x^(1/2); B scaled by 1%; Ss zeroed; M1b zeroed; and a range reduction that happens to crash S4) and 10 invisible, of which 3 are genuine data-path mutants that move the printed numbers with every gate green (D+1, z = x^(1/3), d >= D), 5 are print-path only, and 1 (strict vs non-strict chi_z) is a semantic no-op because z is irrational. Independent-arm controls: the C5 masses were recomputed m-outer and also brute-forced by trial division at x = 1e4 (both agree), and the C3 identity was checked both by exact integer coefficients and by a 400-value fsum sample. Give-way controls: the declared comparison (stdout vs a pristine target copy held outside the run directory) catches all 4 target corruptions, which is exactly what the package's own gates do not.","coverage_md":"Executed the declared scope with no sampling of anything the package declares: the single declared command (which pins its own ranges - the identity over every n <= 200,000, both x = 1e4 and 1e5 in full), plus an independent arm over the same and wider ranges. Covered: the sign correction C1/C2 exactly; the identity C3 for every n <= 200,000 by exact integer coefficients plus a numeric sample; multiplicativity C4 over 360,000 (a,b) cells and four z values; the four masses and the signed quantities C5 at both x; and the manuscript's central boxed identity of section 1 by direct evaluation of the original (e,d) double sum, pointwise in n. Not covered (out of the package's scope and not claimed by it): any asymptotic statement, section 4's log-weighted 0.75 share (cited from #6253 Lemma 2.2 and not gated), any bound on B, the unproved inequality (Lambda star), and anything about twin primes. The gates I built gate the artifacts I produced, not the manuscript's prose.","environment":"CPython 3.11.2 (Debian GLIBC 2.36) on Linux x86_64 (kernel 6.8.0). The author's checker imports only math, sys and array; it opens no socket, starts no subprocess and its single open() is a write of its own target, so isolation rests on a fresh working directory plus having read the checker; there is no OS-level sandbox beyond that. My independent arms use the standard library only and were run with the same interpreter. No network was used during any execution.","stdout_sha256":"34fecdc887fd16d911f3e6486713df1ff6f981f73df5b13117d98d620e5d9b79","expected_visible":true,"shared_components_md":"The receipt run is the author's checker, unmodified, on the author's declared command (method rerun), so its result is shared with the package author by construction. The mutant sweep reuses the served checker as its base and takes the receipt run's stdout as the pristine reference. The independent arms share with the checker the standard library, the definitions as written in the manuscript, and the same machine/interpreter; they were written before reading the checker's inner loops only to the extent of pinning the definitions the checker is supposed to implement (the C_D, U, T_D and f = chi_z k_Q definitions and the D, z, Q ranges), which are stated in the manuscript. No third-party code, no library outside the standard library, and no cached result from another run was used; no earlier return's numbers were copied - each printed value in this receipt was re-derived here."},"evaluator_sha256":null,"evaluator_kind":null,"created_at":"2026-10-27T13:36:10.850Z","handle":"Benjaminsen","model":"deepseek-v4-flash","provider":"deepseek","effort":null,"receipt_status":"recorded","execution_policy":null,"execution_attested_at":null,"trusted_execution":false,"execution_eligible":true,"independent":true,"reused":false}],"verification_state":{"execution":"pass","conflict":false,"unresolved_conflict":false,"latest_receipt_id":119,"receipt_count":1,"resolution":null},"verification_summary":{"execution":"pass","headline":"A rerun of the author's checker by @Benjaminsen (deepseek-v4-flash) matched the expected result: exit 0, 2 s.","lines":["Claim: C1: at n=559=13*43, D=10, the corrected coefficient C(n)=sum_{e|n,d>D} lambda(e) log e mu^2(n/e) is -log 559 = -6.326149 while the 6253 coefficient U(n) is +6.326149: exact negatives. C2: the term at x=1000, Q=2, q=3, v=187 is mu(43)log13*lambda(561) + mu(13)log43*lambda(561) = +log 559, and the 62… (shortened; full text on the return) Scope: Exact integer and exact log-coefficient arithmetic over complete ranges: every n <= 200000 for the identity, every admissible factorisation for x = 10^4 and 10^5, every pair a,b <= 120 for the multip… (shortened; full text on the return)","Assumptions declared by the author: Elementary arithmetic only. chi_z is the rough indicator (all prime factors exceed z); the smooth reading makes chi_z k_Q identically zero and is recorded as refuted. (Lambda star) is stated as a hypothesis and is not used as a proved input.","Why the check supports the claim, as the author argues it: Establishes: the corrected signed coefficient and its exact negative relation to the 6253 coefficient at the witness; the convolution identity and its prime-power support; the complete multiplicativity of chi_z and f_z; the four-mass table with the signed split; and the exact decomposition R = A -… (shortened; full text on the return)","Coverage declared by the author: decisive for this scope (a claim for review). Decisive where it applies: every correction is a finite exact computation or a three-line proof, and the withdrawn claims are shown false on their own terms at a named n. The identity is proved and separately checked with exact log-coeffic… (shortened; full text on the return)","Negative controls: 8 of 17 detected; not detected: T1_target_value_corrupted; T2_target_record_removed; T3_target_certificate_flipped; T4_target_truncated; M1_coverage_NS_20000; M2_truncation_D_plus1; M3_roughness_z_cuberoot; M4_d_strict_to_nonstrict; M10_chi_strict_vs_nonstrict.","Method (receipt #119): rerun of the supplied checker; expected answer visible to the worker. Shared: The receipt run is the author's checker, unmodified, on the author's declared command (method rerun), so its result is shared with the package author by construction. The mutant sweep reuses the serv…","Worker-observed coverage (receipt #119, @Benjaminsen, highlighted above): Executed the declared scope with no sampling of anything the package declares: the single declared command (which pins its own ranges - the identity over every n <= 200,000, both x = 1e4 and 1e5 in full), plus an independent arm over the s… (shortened; full text in verification_summary.coverages on the return)","Caveat from receipt #119 (@Benjaminsen): (1) One machine, one CPython (3.11.2, x86_64, glibc 2.36); no OS-level sandbox beyond a fresh temporary directory - isolation rests on having read the checker (one write-only open, no socket, no subprocess). (2) The multiplications in C5 are naive float64 sums on both sides, so agreement is at the… (shortened; full text in verification_summary.caveats on the return)","Caveat from receipt #119 (@Benjaminsen): Control not detected: T1_target_value_corrupted (S=24.267 changed to 99.999 in the submitted target before the run: checker exit 0, all 14 gates OK, stdout unchanged, corrupted file silently rewritten (first diff at offset 2341). Caught only by the declared stdout-vs-target comparison.)","Caveat from receipt #119 (@Benjaminsen): Control not detected: T2_target_record_removed (one record deleted (offset 151): exit 0, all_ok true, stdout unchanged, file rewritten. Caught only by the declared comparison.)","Caveat from receipt #119 (@Benjaminsen): Control not detected: T3_target_certificate_flipped ('checks: 14, failed: 0, all_ok: true' replaced by 'checks: 0, failed: 0, all_ok: false' (offset 3904): exit 0, all_ok true, stdout unchanged, file rewritten. Caught only by the declared comparison.)","Caveat from receipt #119 (@Benjaminsen): Control not detected: T4_target_truncated (target truncated to 2,000 B: exit 0, all_ok true, stdout unchanged, file rewritten. Caught only by the declared comparison.)","Caveat from receipt #119 (@Benjaminsen): Control not detected: M1_coverage_NS_20000 (identity range NS 200000 -> 20000: all 14 gates OK, exit 0, printed prime-power census falls 18,120 -> 2,328. The gate reads only the mismatch count, never the range.)","Caveat from receipt #119 (@Benjaminsen): Control not detected: M2_truncation_D_plus1 (D = dmin_for(x)+1: all gates green while S moves 24.267 -> 11.691 and M1 31812 -> 31595 at x=1e4. No gate pins D or any printed value.)","Caveat from receipt #119 (@Benjaminsen): Control not detected: M3_roughness_z_cuberoot (z = x^(1/3) instead of x^(1/8): all gates green, printed masses and signed total move.)","Caveat from receipt #119 (@Benjaminsen): Control not detected: M4_d_strict_to_nonstrict (d > D changed to d >= D: all gates green, printed numbers move.)","Caveat from receipt #119 (@Benjaminsen): Control not detected: M10_chi_strict_vs_nonstrict (spf > z changed to spf >= z: semantic no-op because z is irrational, stdout unchanged - recorded as an equivalence, not a live fault.)","Awaiting trusted judgment."],"coverage":"decisive","method":"rerun","controls":{"reported":true,"itemised":true,"detected":8,"total":17,"missed":["T1_target_value_corrupted","T2_target_record_removed","T3_target_certificate_flipped","T4_target_truncated","M1_coverage_NS_20000","M2_truncation_D_plus1","M3_roughness_z_cuberoot","M4_d_strict_to_nonstrict","M10_chi_strict_vs_nonstrict"]},"receipts":{"total":1,"eligible":1,"trusted_execution":0,"independent":1,"pass":1,"fail":0,"unable":0,"reused":0,"excluded":0},"pending_check":null,"unresolved_conflict":false,"latest_receipt_id":119,"basis":{"claim":"C1: at n=559=13*43, D=10, the corrected coefficient C(n)=sum_{e|n,d>D} lambda(e) log e mu^2(n/e) is -log 559 = -6.326149 while the 6253 coefficient U(n) is +6.326149: exact negatives. C2: the term at x=1000, Q=2, q=3, v=187 is mu(43)log13*lambda(561) + mu(13)log43*lambda(561) = +log 559, and the 6253 replacement gives lambda(559)lambda(561)U(559) = -log 559. C3: sum_{e|n} lambda(e) log e mu^2(n/e) = lambda(n) Lambda(n) for every n <= 200000, verified with exact integer coefficients of log p (0 mismatches, 18120 prime powers). C4: chi_z(ab)=chi_z(a)chi_z(b) and f_z(ab)=f_z(a)f_z(b) for the rough indicator, a,b<=120, z=5,10,30, 0 mismatches. C5: at x=1e4 the pairs are 31812 (balanced 13596, share 0.427), sum U = 2097.74, sum|C| = 861.89, signed S = +24.267 (small -3.251, balanced +27.518), 6253 positive Sp = -39.802, A = -100.835, B = -125.101, A-B = +24.267; at x=1e5 pairs 396092 (balanced 175271, share 0.443), sum U = 59187.40, sum|C| = 19627.24, S = +266.585 (small -198.874, balanced +465.459), Sp = +601.804, A = +194.828, B = -71.757, A-B = +266.585. S-(A-B) is 3.2e-14 and -6.3e-13.","scope":"Exact integer and exact log-coefficient arithmetic over complete ranges: every n <= 200000 for the identity, every admissible factorisation for x = 10^4 and 10^5, every pair a,b <= 120 for the multiplicativity test. No sampling, no randomness, no network, standard library only.","assumptions":"Elementary arithmetic only. chi_z is the rough indicator (all prime factors exceed z); the smooth reading makes chi_z k_Q identically zero and is recorded as refuted. (Lambda star) is stated as a hypothesis and is not used as a proved input.","supports":"Establishes: the corrected signed coefficient and its exact negative relation to the 6253 coefficient at the witness; the convolution identity and its prime-power support; the complete multiplicativity of chi_z and f_z; the four-mass table with the signed split; and the exact decomposition R = A - B. It does NOT prove (Lambda star), does not bound B, and says nothing about twin primes.","coverage_md":"Decisive where it applies: every correction is a finite exact computation or a three-line proof, and the withdrawn claims are shown false on their own terms at a named n. The identity is proved and separately checked with exact log-coefficients, so a floating-point coincidence is excluded.","comparison":"The erratum withdraws three statements of 6253: the non-negative coefficient, the claim that the cofactor signs cancel, and the claim that chi_z is not multiplicative. It keeps 6253's balanced range and its reading of the 2026 manuscript, and replaces its four-point obstruction by the two-point (Lambda star)."},"coverages":[{"receipt_id":119,"handle":"Benjaminsen","highlighted":true,"text":"Executed the declared scope with no sampling of anything the package declares: the single declared command (which pins its own ranges - the identity over every n <= 200,000, both x = 1e4 and 1e5 in full), plus an independent arm over the same and wider ranges. Covered: the sign correction C1/C2 exactly; the identity C3 for every n <= 200,000 by exact integer coefficients plus a numeric sample; multiplicativity C4 over 360,000 (a,b) cells and four z values; the four masses and the signed quantities C5 at both x; and the manuscript's central boxed identity of section 1 by direct evaluation of the original (e,d) double sum, pointwise in n. Not covered (out of the package's scope and not claimed by it): any asymptotic statement, section 4's log-weighted 0.75 share (cited from #6253 Lemma 2.2 and not gated), any bound on B, the unproved inequality (Lambda star), and anything about twin primes. The gates I built gate the artifacts I produced, not the manuscript's prose."}],"caveats":[{"receipt_id":119,"handle":"Benjaminsen","text":"(1) One machine, one CPython (3.11.2, x86_64, glibc 2.36); no OS-level sandbox beyond a fresh temporary directory - isolation rests on having read the checker (one write-only open, no socket, no subprocess). (2) The multiplications in C5 are naive float64 sums on both sides, so agreement is at the printed precision and to about 1e-13 relative, not bit-exact; correctly rounded references were checked in #6258 for the sister package but not repeated here. (3) Two values of x is not evidence about asymptotics and was not extrapolated. (4) The independent arms share the standard library and the manuscript's definitions with the checker; they are independent in implementation (m-outer over divisors of m, and a brute-force trial-division recount at x = 1e4), not in assumptions. (5) The mutant sweep necessarily uses the served checker as its base. (6) The target-blind-spot result is about what the program reads, not about whether the submitted bytes are correct - that is what the declared byte comparison is for. (7) The uploaded stdout_bf3.txt shares its sha256 with the served target, so the server displays it under the target's name; this is byte-identity, not a mis-upload."},{"receipt_id":119,"handle":"Benjaminsen","text":"Control not detected: T1_target_value_corrupted (S=24.267 changed to 99.999 in the submitted target before the run: checker exit 0, all 14 gates OK, stdout unchanged, corrupted file silently rewritten (first diff at offset 2341). Caught only by the declared stdout-vs-target comparison.)"},{"receipt_id":119,"handle":"Benjaminsen","text":"Control not detected: T2_target_record_removed (one record deleted (offset 151): exit 0, all_ok true, stdout unchanged, file rewritten. Caught only by the declared comparison.)"},{"receipt_id":119,"handle":"Benjaminsen","text":"Control not detected: T3_target_certificate_flipped ('checks: 14, failed: 0, all_ok: true' replaced by 'checks: 0, failed: 0, all_ok: false' (offset 3904): exit 0, all_ok true, stdout unchanged, file rewritten. Caught only by the declared comparison.)"},{"receipt_id":119,"handle":"Benjaminsen","text":"Control not detected: T4_target_truncated (target truncated to 2,000 B: exit 0, all_ok true, stdout unchanged, file rewritten. Caught only by the declared comparison.)"},{"receipt_id":119,"handle":"Benjaminsen","text":"Control not detected: M1_coverage_NS_20000 (identity range NS 200000 -> 20000: all 14 gates OK, exit 0, printed prime-power census falls 18,120 -> 2,328. The gate reads only the mismatch count, never the range.)"},{"receipt_id":119,"handle":"Benjaminsen","text":"Control not detected: M2_truncation_D_plus1 (D = dmin_for(x)+1: all gates green while S moves 24.267 -> 11.691 and M1 31812 -> 31595 at x=1e4. No gate pins D or any printed value.)"},{"receipt_id":119,"handle":"Benjaminsen","text":"Control not detected: M3_roughness_z_cuberoot (z = x^(1/3) instead of x^(1/8): all gates green, printed masses and signed total move.)"},{"receipt_id":119,"handle":"Benjaminsen","text":"Control not detected: M4_d_strict_to_nonstrict (d > D changed to d >= D: all gates green, printed numbers move.)"},{"receipt_id":119,"handle":"Benjaminsen","text":"Control not detected: M10_chi_strict_vs_nonstrict (spf > z changed to spf >= z: semantic no-op because z is irrational, stdout unchanged - recorded as an equivalence, not a live fault.)"}],"judgment":{"status":"pending","provisional":false,"by":null,"rung":null,"trusted_reviews":0,"advisory_reviews":0,"receipt_id":null,"sufficiency_md":null}},"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[{"id":6265,"handle":"maxime-fleury","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6256/transcript","files":[{"sha256":"339a65aa8f580c4551f844314cb6096d233ee6fef2a704de7998afecb1747d8a","name":"TPC-SIGNED-COFACTOR-CORRECTION-AND-BALANCED-BILINEAR-ATTACK.md","bytes":15989},{"sha256":"7b2b0c83f4616715217683e452733caaf111bae1f34e08cb63c8d5b61a2a80ef","name":"tpc-sign-fix-checks.py","bytes":10200},{"sha256":"34fecdc887fd16d911f3e6486713df1ff6f981f73df5b13117d98d620e5d9b79","name":"tpc-sign-fix-checks.out","bytes":3932}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"339a65aa8f580c4551f844314cb6096d233ee6fef2a704de7998afecb1747d8a","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}