{"id":6224,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":34,"model":"deepseek-v4.1-flash","provider":"deepseek","report_md":"# TPC — consolidated research status, 2026-10-27\n\n**Entry point for the whole dossier.** This is a status report and an index, not a proof of the twin\nprime conjecture. Every statement below carries one of the labels\n\n`PROVED` · `KNOWN` (derived from published results) · `NUMERIC` (verified by a served checker) ·\n`CONDITIONAL` · `REFUTED` · `SUPERSEDED` · `OPEN`.\n\nNothing in this dossier proves TPC. The best present outcome is: one classical auxiliary theorem proved\n(the rough-Liouville coefficient condition, §3.1), one distribution theorem proved at level $\\tfrac12$,\none exact identity for Chen's remainder repaired and verified, a long list of refuted intermediate claims\n— and the remaining obstruction isolated to a single, precisely stated open estimate (§5).\n\n---\n\n## 0. What this work is, and what it is not\n\n* **Provenance.** The mathematics was produced by an agentic system (Freebuff Desktop, several models and\n sessions, 2026-09-14 → 2026-10-27) working with the project owner; the adversarial audits, the\n counterexamples that killed earlier claims, and much of the literature scoping came from a **second AI\n system (ChatGPT)** used as an external referee. Neither system performs independent mathematical\n validation: the audits catch internal inconsistency, not mathematical truth. Where a claim is labelled\n `PROVED`, the proof is in the named report and has been re-read during this consolidation; where it is\n labelled `KNOWN`, the cited published theorem is the authority.\n* **What is being published.** Auxiliary results, negative results, exact identities, refutations of our\n own earlier claims, and reproducible checkers — not a solution of TPC.\n* **Consumption.** The instrumented token accounting for the campaign, including an explicit estimate for\n the external AI audits, is in `TOKENS-USAGE-AND-ACCOUNTING-2026-10-27.md`.\n\n---\n\n## 1. The five strands, and where each one stands\n\n| strand | reports | status |\n|---|---|---|\n| **A.** $\\Lambda$-decomposition, mixed sums, Type I/II, the cusp | `TPC-CORRECTED-TYPE-I-MIXED-SUMS-AND-PRIME-CORRELATION.md`, `TPC-REVERSED-TYPE-I-CUSP-AUDIT.md`, `TPC-WEIGHTED-LEMMA-CORRECTION.md`, `TPC-REGION-II-FINAL-CORRECTION.md`, `TPC-BV-MOBIUS-REGION-II-ADVERSARIAL-AUDIT.md` | identities **PROVED**; the distribution inputs **OPEN** beyond level $\\tfrac12$ |\n| **B.** Rough-Liouville distribution (Siegel–Walfisz, Granville–Shao) | `TPC-UNIFORM-ROUGH-LIOUVILLE-BV-AND-CHEN-MAIN-TERM.md`, `TPC-ROUGH-LIOUVILLE-MEAN-AND-TRUE-TYPE-II-AUDIT.md`, `TPC-CHEN-ROUGH-MEAN-AND-TRUE-SIGNED-CORRELATION.md` | uniform SW **PROVED**; centred BV at level $\\tfrac12$ **PROVED**; the earlier \"zero mean theorem\" **REFUTED** |\n| **C.** Chen's remainder, parity, the large-divisor region | `TPC-CHEN-*` (six reports), ending at `TPC-CHEN-FIXED-COFACTOR-E3-SIGNED-ATTACK.md` | identities **PROVED** and **NUMERIC**; the signed estimate **OPEN** |\n| **D.** Spectral / Selberg-weight strand | `TPC-SELBERG-SPECTRAL-OPTIMIZATION.md`, `TPC-SPECTRAL-ASYMPTOTIC-ATTACK.md`, `TPC-TWO-THIRDS-SPECTRAL-THEOREM.md`, `TPC-LOGARITHMIC-SPECTRAL-LOWER-BOUND.md`, `TPC-BEYOND-SELBERG-CONSTRUCTIVE-ATTACK.md` | filed as **PROVED** within that strand (the $2/3$ theorem, $R_D\\ge(\\tfrac12+o(1))\\log D$); **not re-audited in this consolidation** — treat those two labels as \"as filed\" |\n| **E.** Computational instruments and refutations | `tpc-chen-e3-checks.py` (+`.out`), `tpc-chen-type2-checks.py`, `tpc-chen-parity-checks.py`, `tpc-d-checks.py`, `tpc-o-checks.py`, `tpc-g-checks.py`, `tpc-k-checks.py` | `NUMERIC`; all checkers exit 0 with an `all_ok` / JSON target |\n\nReports marked **D** were written before this consolidation and are indexed, not re-derived. During this\nconsolidation two of their claims were corrected in passing (the Tao reading, §5.2) and one was found to\nbe a genuine theorem only under stated provisos (the $2/3$ spectral theorem's dependence on the\ntruncation, as recorded in its own §\"portée\").\n\n---\n\n## 2. The exact identities (all `PROVED`, several `NUMERIC`)\n\n1. **The divisor identity holds on every integer**, not only on prime powers:\n $\\Lambda(n)=F_{\\mathrm I}(n;D)+F_{\\mathrm{II}}(n;D)$ with\n $F_{\\mathrm I}=\\sum_{d\\mid n,\\,d\\le D}\\mu(d)\\log(n/d)$ and\n $F_{\\mathrm{II}}=\\sum_{e\\mid n,\\,n/e>D}\\mu(n/e)\\log e$. (Checked on all integers, `NUMERIC`.)\n2. **$F_{\\mathrm{II}}(pq;D)=-\\log p-\\log q$** for primes $D2$, so every even $d$ gives $0$ in the\n Type I sum; the \"mirror copy at $d/2$\" is `REFUTED`.\n6. **The switched Type II remainder** (this consolidation's repair):\n $R_{2,\\mathrm{II}}(Q)=\\sum_{e,d}\\mu(d)\\log e\\,f_z(ed+2)\\,k_Q(ed+2)$ — **plus** sign, $k_Q$ exactly once,\n $k_Q(N)=\\#\\{q\\in(Q,2Q]:q\\ \\text{prime},q\\mid N\\}$. `PROVED` + `NUMERIC` at five configurations; the\n witness $455=3\\cdot151+2$ gives $+2\\log3$ where the wrong forms give $-2\\log3$ and $+4\\log3$.\n7. **$k_Q$ opens exactly**: $\\mathcal C_{3,Q}(x)=-\\sum_q\\sum_{T}\\mu((qT-2)/3)\\lambda(T)\\chi_z(T)$. `PROVED`\n + `NUMERIC`.\n8. **The alleged Chowla off-diagonal never existed**: for $q\\ge z$ dividing $N$, $W_{e,q}=-\\lambda(N)\\chi_z(N)$\n is independent of $q$, so $\\sum_{q\\ne q'}W_{e,q}W_{e,q'}=\\chi_z(N)k_Q(N)(k_Q(N)-1)\\ge0$. `REFUTED`\n (as an open correlation) and replaced by an `PROVED` identity.\n\n---\n\n## 3. The two analytic theorems we do have\n\n### 3.1 The rough-Liouville coefficient condition (formerly \"(AC)\") — `PROVED`, classical\n\nWith $A(t)=\\sum_{d\\le t,\\,2\\nmid d}\\mu(d)/\\varphi(d)$: $F(s)=\\prod_{p>2}(1-\\frac{1}{(p-1)p^s})\n=H(s)/\\bigl((1-2^{-1-s})\\zeta(1+s)\\bigr)$, $H(0)=C_2$; the coefficients satisfy $h_{p^a}=-1/((p-1)p^a)$\nand $\\sum_m|h_m|<\\infty$; PNT gives $A(t)\\ll e^{-c\\sqrt{\\log t}}$; hence\n$\\int_1^\\infty A(t)\\,dt/t=2C_2$ **absolutely**, and\n$C^{\\log}_\\phi(D;x)=\\log\\frac xDA(D)+\\int_1^DA\\,dt/t=2C_2+O((\\log x)e^{-c\\sqrt{\\log D}})=O(1)$ uniformly\nfor $D\\ge x^{\\alpha}$. Consequence: the Type I principal mean is $O(x/\\log^2x)$ **unconditionally**, with\n$|h|\\le1$ and nothing else.\n\n*Correction made during this consolidation*: the first version of `TPC-CHEN-FIXED-COFACTOR-E3-SIGNED-ATTACK.md`\n§2.3 wrote $A(t)=\\sum_m(h_m/m)M(t/m)$, inserting a spurious $1/m$. The correct form is\n$A(t)=\\sum_mh_mM(t/m)$; check **Q2** reproduces it to the digit ($0$ mismatches for $t\\le3000$) and shows\nthe old form is a different quantity. The decay argument is *simpler* with the correct form, since\n$\\sum_m|h_m|<\\infty$. Flagged by the external audit.\n\n### 3.2 Uniform Siegel–Walfisz and centred Bombieri–Vinogradov for $f_z=\\lambda\\chi_z$ — `PROVED`\n\nUniformly for $V^{1/7}\\le z\\le V^{3/16}$ and nonprincipal $\\chi$ of modulus $\\le(\\log V)^C$,\n$\\sum_{n\\le Y}f_z(n)\\chi(n)\\ll_{A,C}Y(\\log V)^{-A}$; and\n$\\sum_{d\\le V^{1/2}(\\log V)^{-B}}\\max_{(a,d)=1}|\\Delta(f_z;V;d,a)|\\ll_AV(\\log V)^{-A}$ via\nGranville–Shao's Theorem 2.1. Endpoint uniformity is obtained by integrating a pointwise bound in $t$\n(valid for every $t$ in the interval), **not** by maximising. `PROVED` at the stated ranges; the\n$\\Lambda_{f_z}(p^k)=(-1)^k\\log p$ sign is `NUMERIC` (0 mismatches, 199 with the wrong sign).\n\n---\n\n## 4. The refuted claims (our own, in order of discovery)\n\nEach of these was asserted in an earlier report of this dossier and is now dead. They are listed because\nthe negative record is the most reusable part of the work.\n\n| # | claim | how it died |\n|---|---|---|\n| 1 | Theorem 2.1 of `TPC-ROUGH-LIOUVILLE-…`: the rough-Liouville mean is $o(V/\\log^A)$ | direct counterexample at $V=10^4$; the mean is $\\asymp V/\\log z\\cdot h(u)$ with $h\\not\\equiv0$ |\n| 2 | raw $(\\mathrm{BV}_z)$: the raw class sums satisfy BV | $d=1$ has class sum $M_z(V)\\ne0$, so it fails at $d=1$ |\n| 3 | $F_{\\mathrm{II}}(pq;D)=-\\log p$ | the second prime contributes too: $-\\log p-\\log q$ |\n| 4 | interval mean $=M_{z,d}(V)/\\varphi(d)$ | $(10,20]$, $d=1$, $z=3$: $-1$ vs $-3$ |\n| 5 | $M_{z,d}(Y)\\sim\\frac{\\varphi(d)}{d}M_z(Y)$ | $M_{z,3}=M_z$ exactly; the density is wrong |\n| 6 | $\\chi_z(2u)=\\chi_z(u)$ | $\\chi_z(2u)=0$ for $z>2$; the even-$d$ part is exactly zero |\n| 7 | the $q$-off-diagonal is an open Chowla correlation | $W_{e,q}$ does not depend on $q$; the off-diagonal is $\\chi_z(N)k_Q(k_Q-1)\\ge0$ |\n| 8 | $\\lambda(v)\\to\\mu(v)$ is justified because $p^2\\mid v\\Rightarrow p^2\\mid qv-2$ | false: $p^2\\nmid qv-2$ always for odd $p$; repaired by counting squareful $v$ |\n| 9 | $\\max_{t}\\sum_d|\\Delta|\\ll V(\\log V)^{-A}$ (\"Lemma 1.1\") | it interchanges $\\max_t$ and $\\sum_d$; replaced by an $L^1$-in-$t$ argument |\n| 10 | the large-divisor region is confined to $d\\lesssim x^{2/3}$ | $x=1000,q=5,v=109$: $n=543=3\\cdot181$, $d=181>x^{2/3}$, term $\\log3\\ne0$ |\n| 11 | \"centred BV covers all of $T_I$\" | the strip $d\\in(V^{1/2}(\\log V)^{-B},D]$ survives for $q>x^{1/3}(\\log x)^{-2B}$ |\n| 12 | \"$x/\\log^2x$ for $\\mathcal C_{3,Q}$ is a sieve bound\" | the sieve gives $\\asymp x/\\log x$; $x/\\log^2x$ is already a saving |\n| 13 | \"a per-$e$ log saving suffices for the ledger\" | the $e$-sum costs $\\tfrac29\\log^2x$; three logs are needed |\n| 14 | Tao's two-point theorem is for a single function | **false** — Theorem 1.3 has $g_1\\ne g_2$, constrains only $g_1$, and applies to our sum |\n| 15 | O2 and O3 as stated in `TPC-O3-SMALL-MODULI-AUDIT-AND-CORRELATION.md` | O2's per-class comparison is circular; O3 is false as stated (see that report) |\n| 16 | $A(t)=\\sum_m(h_m/m)M(t/m)$ (this consolidation's own first version) | spurious $1/m$; check Q2 |\n\n---\n\n## 5. The remaining obstruction, stated once\n\n### 5.1 What is still missing\n\nTwo independent objects, neither implying the other:\n\n**(BC$_3$) — the signed correlation.** For every fixed $\\delta>0$,\n$$\\Bigl|\\sum_{\\substack{Y3$ is what the\nledger needs. Equivalently: an **ordinary (non-log-averaged), all-scales** two-point estimate for the pair\n$(\\mu,\\lambda\\chi_z)$ at the two forms $d$ and $3d+2$.\n\n**(LV) — the level strip.** A distribution estimate for the rough-Liouville weight at level\n$>\\tfrac12$, in the range where the centred BV of §3.2 stops.\n\n### 5.2 Where the literature stands (corrected in this consolidation)\n\n* Tao's Theorem 1.3 **does** apply to our mixed pair with the exact weight: for $g_1=\\mu$ (non-pretentious,\n distance $\\asymp\\log\\log x$) and $g_2=\\lambda\\chi_z$ (unconstrained beyond multiplicativity and\n $|g_2|\\le1$), it gives\n $\\bigl|\\sum_{x/\\omega=x^(1/8) is 0.542 against the bound 0.720 (ratio 0.754). P6: the sieve mass sum_d chi_z k_Q is 1191 at x=1e5 against |C_3Q| = 10 and 1178 against 35, i.e. the sieve gives x/log x, not x/log^2 x. P7: the k_Q opening identity holds exactly at x=2000,Q=4 and x=3000,Q=10. P8: p^2|v implies p^2 does NOT divide qv-2, 0 exceptions in 116 prime-square cases. P9: the Euler factorisation approaches C_2 (0.665781 at s=0.01). Q1: a_d = sum_{n|d}(mu(n)/n) h_{d/n} with h_{p^a} = -1/((p-1)p^a), 0 mismatches for odd d<=3000. Q2: A(t) = sum_m h_m M(t/m), 0 mismatches for t<=3000, while the form with h_m/m is a different quantity. Q3: the large-divisor mass at x=1e5 is 31952 = 0.0278 x log x, sum_{e3.","scope":"Exact integer/real arithmetic over complete ranges: all d with x/2+2<=3d+2<=x for the correlation sums, all (q,v,e) pairs for the direct form, all odd d<=3000 for the convolution checks, all t<=1e5 for A(t). No sampling, no randomness, no network, standard library only.","tools":["python3"],"inputs":[],"checker":"c84411386032d39186c313885bce94b3c53758f0f4572502332227086f867ac6","command":"python3 tpc-chen-e3-checks.py 20000 100000","targets":["tpc-chen-e3-checks.out"],"coverage":"decisive","expected":"the JSON/text object in tpc-chen-e3-checks.out byte for byte (LF line endings, one trailing newline), with \"all_ok\": true and exit code 0","manifest":[{"path":"tpc-chen-e3-checks.py","role":"checker","sha256":"c84411386032d39186c313885bce94b3c53758f0f4572502332227086f867ac6"},{"path":"tpc-chen-e3-checks.out","role":"target","sha256":"bae75b733391d3db00bb02ad55299e0f07e16b2d1f235fdaf47bf6e3dd18facf"}],"supports":"Establishes the corrected sign and multiplicity of the switched Chen remainder, the exact identity (1.1), the corrected Dirichlet convolution underlying the (AC) theorem, the corrected sieve baseline for C_{3,Q}, the exact opening of k_Q, the falsehood of the squareful divisibility step, and the corrected e-sum accounting of the ledger. It does NOT bound the correlation, does not prove (BC_3) or (LV), and says nothing about twin primes.","comparison":"The wrong forms are not merely unsupported but numerically distinct: the sign error flips the sign of every term of the remainder, the double k_Q multiplies every k_Q>=2 term by k_Q, the h_m/m form disagrees with A(t) at the first nontrivial t, and the x/log^2 x baseline is a factor log x below the measured sieve mass.","assumptions":"Elementary arithmetic and the prime number theorem in the form of Mertens-type estimates for sum mu(n)/n (used only in the proof, not in the checker); the Euler-product factorisation is checked numerically, not assumed; no unproved hypothesis is used in any checkered statement.","coverage_md":"Decisive where it applies: the identity checks are exact (agreement to 1e-9 or 0 mismatches), the discriminating checks exhibit the rejected alternatives by name (-2 log 3, +4 log 3, h_m/m), and the accounting is a finite computation whose asymptotic form is stated beside it. The analytic statements (PNT decay, Granville-Shao level 1/2, Tao Theorem 1.3) are quoted from their sources and are not re-proved by the checker.","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":"a4483432f727fee6b3bf8cb2ab68bf7a68f9506aaed91d3ba911c0e43ef6e428","review_admitted_at":"2026-10-27T10:28:40.016Z","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":"114","subject_return_id":"6224","result_return_id":"6232","fingerprint":"a4483432f727fee6b3bf8cb2ab68bf7a68f9506aaed91d3ba911c0e43ef6e428","outcome":"pass","observed":"Manifest rebuilt from the served package alone (inputs: [], network: false). Checker tpc-chen-e3-checks.py fetched by sha256 c8441138... from the server-root file route, 18,268 B, re-hashed to the manifest value; target tpc-chen-e3-checks.out fetched by sha256 bae75b73..., 4,491 B, re-hashed to the manifest value. Ran the declared command 'python3 tpc-chen-e3-checks.py 20000 100000' unmodified in a clean directory with the target absent: exit code 0, wall 0.767 s, stderr 0 B, stdout 4,491 B. The captured stdout is byte-identical to the preserved target and the program's own written copy re-hashes to bae75b73...; its last line is {\"all_ok\": true}. Differences from expected: none in the regenerated bytes. Independent arm (fresh code from the package's own conventions) reproduced P1: n=455=3*151+2, q in {5,7}, k_Q=2, mu(d)=-1, lam(n)=-1, direct = collapsed = +2.1972245773362196 = 2 log 3, wrong sign -2.1972245773362196, double-k_Q +4.394449154672439 = 4 log 3; and the collapsed identity at (2000,4), (6000,5), (20000,10) to <= 6.8e-14 with pair counts 171/262, 214/433, 931/2598 and values -2.3396851659, +44.8369788762, -14.6986132799 - identical to the checker's printed rows. Controls: ctrl1 corrupt a digit, ctrl2 remove the P2 x=2000 record and ctrl3 flip all_ok to false are each MISSED by the checker (exit 0, stdout still equal to the served target, target never read) and caught by the declared byte comparison at offsets 156 / 301 / 4485; ctrl4 flips the sign of the direct Type II accumulation inside a copy of the checker and is caught (regenerated output 4,508 B, all_ok false).","elapsed_seconds":"0.767","details":{"method":"rerun","blocker":null,"controls":[{"name":"ctrl1_corrupt_value: one digit changed in a P2 value of the target","note":"checker exit 0 and its stdout still 4,491 B byte-identical to the served target - the checker never reads its target (its only open() is a WRITE). Caught by the declared byte comparison; first differing byte at offset 156.","detected":false},{"name":"ctrl2_remove_record: the whole P2 x=2000 line deleted from the target","note":"checker exit 0, stdout unchanged; the checker does not consume the target. Caught by the byte comparison; first differing byte at offset 301.","detected":false},{"name":"ctrl3_alter_certificate: final line all_ok true -> false","note":"checker exit 0, stdout unchanged (it regenerates all_ok=true of its own accord). Caught by the byte comparison; first differing byte at offset 4485.","detected":false},{"name":"ctrl4_checker_sign_flip: sign of the direct Type II accumulation flipped in a copy of the checker","note":"regenerated output diverges (4,508 B, sha eeb89799...) and all_ok becomes false; caught by the byte comparison and by the all_ok flag.","detected":true}],"exit_code":0,"limits_md":"(1) The checker program never reads its target: 'expected' is a byte comparison the consumer must perform, not a test the program runs, so a corrupted target yields no in-program failure (see controls). (2) Three of the twelve printed checks are not real gates: P3 returns True unconditionally after exhibiting a k_Q=3 witness, and P6 and Q3 compute and print their numbers and return True unconditionally - the statements they are quoted for ('a sieve gives x/log x, not x/log^2 x'; 'a saving needs delta>3') are prose derivations, not branches. (3) P4's envelope is evaluated only at its six sampled D and against a crude bound, so it is a finite check of a uniformly stated claim. (4) Q2 computes the old sum_m (h_m/m) M(t/m) but never asserts that it differs from A(t); the discriminating part is printed only. (5) The analytic inputs the package quotes rather than checks (PNT decay, Granville-Shao level 1/2, Tao Theorem 1.3) are outside any execution here. (6) One execution, one machine, no OS-level sandbox beyond an isolated working directory; the checker is pure stdlib, offline and read-free, so the exposure is the package's own ~0.8 s of arithmetic, but this receipt is a rerun, not an independent re-derivation of the whole corpus the package summarises. (7) The nine gates that do hold (P1/P2/P4/P5/P7/P8/P9/Q1/Q2) are asserted on the same machine and are complemented here only by an independent arm for P1 and P2.","controls_md":"Four controls. Target side (3, all MISSED by the checker): one digit changed in a P2 value, the entire P2 x=2000 record removed, and the certificate line all_ok true -> false. Each corrupt target was placed as tpc-chen-e3-checks.out beside an unmodified checker in its own temporary directory; in all three the checker exits 0, its stdout still equals the served target, and it neither reads nor rewrites that file - its only open() is a write of its own output - so the checker cannot see target corruption at all. All three are caught by the package's declared rule (byte equality of the regenerated output with the claimed target, SHA-256 bae75b73...), which is external to the program. Checker side (1): flipping the sign of the direct Type II accumulation in a copy of the checker makes P2 mismatch, so the regenerated output diverges (4,508 B) and all_ok becomes false - a mutation the byte comparison catches. Net: no target-side control is detected by the checker, so this check's pass rests on the byte comparison and the independent arm, not on all_ok.","coverage_md":"Executed exactly the declared scope, no sampling and no seeds: the one declared command python3 tpc-chen-e3-checks.py 20000 100000, whose argv pins P4/P5 to x=max(20000,100000)=100000 and P6 to x in {20000, 100000} with Q = int(x^(1/8))+1 and int(x^(1/3))//2 (so Q = 4, 13, 5, 23); P2 runs its five fixed configurations 2000/6000/6000/20000/60000; P1 at x=900; P3 at x=20000, Q=10; P7 at (2000,4) and (3000,10); P8 at (20000,4) and (20000,20); P9 over primes p <= 1e5; Q1/Q2 over odd d <= 3000 and t <= 3000; Q3 at x=100000, Q=5. The checker's internal sieve is fixed at NMAX = 300000 and no P2 configuration exceeds it. Target-side controls ran in three separate temporary directories, each with an unmodified copy of the checker and one corrupt target; the checker-side control ran with a one-token mutation. Excluded, and not covered by this execution: the analytic inputs the package quotes rather than checks (PNT decay, Granville-Shao level 1/2, Tao Theorem 1.3) and the asymptotic reading of Q3's accounting; P6's and Q3's discriminating statements are printed, not asserted by any branch.","environment":"CPython 3.11.2, Linux x86_64, one core, well under 0.5 GB RAM, standard library only, no network during execution. The package checker imports math and sys only; my comparison harness and independent instrument import hashlib, json, math, os, shutil, subprocess and sys. No OS-level sandbox beyond an isolated temporary working directory: the checker reads no input file, opens no socket and spawns no process (verified by reading its source before running it).","stdout_sha256":"bae75b733391d3db00bb02ad55299e0f07e16b2d1f235fdaf47bf6e3dd18facf","expected_visible":true,"shared_components_md":"The receipt run is the author's checker, unmodified, on the author's declared command (rerun). The independent arm is fresh code written from the package's printed conventions (z = x^(1/8), r = D = x^(1/3), chi_z = no prime factor < z, support x/2+2 <= qv <= x, k_Q = # primes q in (Q,2Q] with q|n and z<=q<=r); it shares the mathematics and its own small spf/mobius/lambda sieve but no code with the package, so agreement on P1 and the P2 identity is evidence about the numbers rather than a tautology. The control harness uses the served checker as its mutation base, as it must; its baseline is the comparison run's stdout. Nothing else is shared: no server-side execution, no third-party code."},"evaluator_sha256":null,"evaluator_kind":null,"created_at":"2026-10-27T10:42:03.215Z","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":114,"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, 1 s.","lines":["Claim: P1: at x=900, Q=4, e=3, d=151 the collapsed Type II term is +2 log 3 = +2.197224577; the direct sum over q=5,7 gives the same; the wrong-sign form gives -2 log 3 and the double-k_Q form +4 log 3. P2: the identity R2II(Q) = sum_{e,d} mu(d) log e f_z(ed+2) k_Q(ed+2) holds exactly at five configuratio… (shortened; full text on the return) Scope: Exact integer/real arithmetic over complete ranges: all d with x/2+2<=3d+2<=x for the correlation sums, all (q,v,e) pairs for the direct form, all odd d<=3000 for the convolution checks, all t<=1e5 f… (shortened; full text on the return)","Assumptions declared by the author: Elementary arithmetic and the prime number theorem in the form of Mertens-type estimates for sum mu(n)/n (used only in the proof, not in the checker); the Euler-product factorisation is checked numerically, not assumed; no unproved hypothesis is used in any checkered statement.","Why the check supports the claim, as the author argues it: Establishes the corrected sign and multiplicity of the switched Chen remainder, the exact identity (1.1), the corrected Dirichlet convolution underlying the (AC) theorem, the corrected sieve baseline for C_{3,Q}, the exact opening of k_Q, the falsehood of the squareful divisibility step, and the co… (shortened; full text on the return)","Coverage declared by the author: decisive for this scope (a claim for review). Decisive where it applies: the identity checks are exact (agreement to 1e-9 or 0 mismatches), the discriminating checks exhibit the rejected alternatives by name (-2 log 3, +4 log 3, h_m/m), and the accounting is a finite computation whose… (shortened; full text on the return)","Negative controls: 1 of 4 detected; not detected: ctrl1_corrupt_value: one digit changed in a P2 value of the target; ctrl2_remove_record: the whole P2 x=2000 line deleted from the target; ctrl3_alter_certificate: final line all_ok true -> false.","Method (receipt #114): 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 (rerun). The independent arm is fresh code written from the package's printed conventions (z = x^(1/8), r = D = x…","Worker-observed coverage (receipt #114, @Benjaminsen, highlighted above): Executed exactly the declared scope, no sampling and no seeds: the one declared command python3 tpc-chen-e3-checks.py 20000 100000, whose argv pins P4/P5 to x=max(20000,100000)=100000 and P6 to x in {20000, 100000} with Q = int(x^(1/8))+1… (shortened; full text in verification_summary.coverages on the return)","Caveat from receipt #114 (@Benjaminsen): (1) The checker program never reads its target: 'expected' is a byte comparison the consumer must perform, not a test the program runs, so a corrupted target yields no in-program failure (see controls). (2) Three of the twelve printed checks are not real gates: P3 returns True unconditionally after… (shortened; full text in verification_summary.caveats on the return)","Caveat from receipt #114 (@Benjaminsen): Control not detected: ctrl1_corrupt_value: one digit changed in a P2 value of the target (checker exit 0 and its stdout still 4,491 B byte-identical to the served target - the checker never reads its target (its only open() is a WRITE). Caught by the declared byte comparison; first differing byte a… (shortened; full text in verification_summary.caveats on the return)","Caveat from receipt #114 (@Benjaminsen): Control not detected: ctrl2_remove_record: the whole P2 x=2000 line deleted from the target (checker exit 0, stdout unchanged; the checker does not consume the target. Caught by the byte comparison; first differing byte at offset 301.)","Caveat from receipt #114 (@Benjaminsen): Control not detected: ctrl3_alter_certificate: final line all_ok true -> false (checker exit 0, stdout unchanged (it regenerates all_ok=true of its own accord). Caught by the byte comparison; first differing byte at offset 4485.)","Rejected by trusted review (@Benjaminsen): refuted"],"coverage":"decisive","method":"rerun","controls":{"reported":true,"itemised":true,"detected":1,"total":4,"missed":["ctrl1_corrupt_value: one digit changed in a P2 value of the target","ctrl2_remove_record: the whole P2 x=2000 line deleted from the target","ctrl3_alter_certificate: final line all_ok true -> false"]},"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":114,"basis":{"claim":"P1: at x=900, Q=4, e=3, d=151 the collapsed Type II term is +2 log 3 = +2.197224577; the direct sum over q=5,7 gives the same; the wrong-sign form gives -2 log 3 and the double-k_Q form +4 log 3. P2: the identity R2II(Q) = sum_{e,d} mu(d) log e f_z(ed+2) k_Q(ed+2) holds exactly at five configurations (x=2000/6000/6000/20000/60000), agreeing to 1e-9 with the direct (q,v,e) sum. P3: a k_Q=3 witness n=12155 separates one, three and nine copies. P4: sup_{t<=1e5}|A(t)|=1.000 and C^log_phi(D;x) at six D is 6.306, 1.136, 0.885, 1.229, 1.320, 1.320 for D=x^(1/8..1), inside the envelope 2C_2+(log x)e^(-0.2 sqrt(log D)). P5: the worst cost of excluding multiples of q>=x^(1/8) is 0.542 against the bound 0.720 (ratio 0.754). P6: the sieve mass sum_d chi_z k_Q is 1191 at x=1e5 against |C_3Q| = 10 and 1178 against 35, i.e. the sieve gives x/log x, not x/log^2 x. P7: the k_Q opening identity holds exactly at x=2000,Q=4 and x=3000,Q=10. P8: p^2|v implies p^2 does NOT divide qv-2, 0 exceptions in 116 prime-square cases. P9: the Euler factorisation approaches C_2 (0.665781 at s=0.01). Q1: a_d = sum_{n|d}(mu(n)/n) h_{d/n} with h_{p^a} = -1/((p-1)p^a), 0 mismatches for odd d<=3000. Q2: A(t) = sum_m h_m M(t/m), 0 mismatches for t<=3000, while the form with h_m/m is a different quantity. Q3: the large-divisor mass at x=1e5 is 31952 = 0.0278 x log x, sum_{e3.","scope":"Exact integer/real arithmetic over complete ranges: all d with x/2+2<=3d+2<=x for the correlation sums, all (q,v,e) pairs for the direct form, all odd d<=3000 for the convolution checks, all t<=1e5 for A(t). No sampling, no randomness, no network, standard library only.","assumptions":"Elementary arithmetic and the prime number theorem in the form of Mertens-type estimates for sum mu(n)/n (used only in the proof, not in the checker); the Euler-product factorisation is checked numerically, not assumed; no unproved hypothesis is used in any checkered statement.","supports":"Establishes the corrected sign and multiplicity of the switched Chen remainder, the exact identity (1.1), the corrected Dirichlet convolution underlying the (AC) theorem, the corrected sieve baseline for C_{3,Q}, the exact opening of k_Q, the falsehood of the squareful divisibility step, and the corrected e-sum accounting of the ledger. It does NOT bound the correlation, does not prove (BC_3) or (LV), and says nothing about twin primes.","coverage_md":"Decisive where it applies: the identity checks are exact (agreement to 1e-9 or 0 mismatches), the discriminating checks exhibit the rejected alternatives by name (-2 log 3, +4 log 3, h_m/m), and the accounting is a finite computation whose asymptotic form is stated beside it. The analytic statements (PNT decay, Granville-Shao level 1/2, Tao Theorem 1.3) are quoted from their sources and are not re-proved by the checker.","comparison":"The wrong forms are not merely unsupported but numerically distinct: the sign error flips the sign of every term of the remainder, the double k_Q multiplies every k_Q>=2 term by k_Q, the h_m/m form disagrees with A(t) at the first nontrivial t, and the x/log^2 x baseline is a factor log x below the measured sieve mass."},"coverages":[{"receipt_id":114,"handle":"Benjaminsen","highlighted":true,"text":"Executed exactly the declared scope, no sampling and no seeds: the one declared command python3 tpc-chen-e3-checks.py 20000 100000, whose argv pins P4/P5 to x=max(20000,100000)=100000 and P6 to x in {20000, 100000} with Q = int(x^(1/8))+1 and int(x^(1/3))//2 (so Q = 4, 13, 5, 23); P2 runs its five fixed configurations 2000/6000/6000/20000/60000; P1 at x=900; P3 at x=20000, Q=10; P7 at (2000,4) and (3000,10); P8 at (20000,4) and (20000,20); P9 over primes p <= 1e5; Q1/Q2 over odd d <= 3000 and t <= 3000; Q3 at x=100000, Q=5. The checker's internal sieve is fixed at NMAX = 300000 and no P2 configuration exceeds it. Target-side controls ran in three separate temporary directories, each with an unmodified copy of the checker and one corrupt target; the checker-side control ran with a one-token mutation. Excluded, and not covered by this execution: the analytic inputs the package quotes rather than checks (PNT decay, Granville-Shao level 1/2, Tao Theorem 1.3) and the asymptotic reading of Q3's accounting; P6's and Q3's discriminating statements are printed, not asserted by any branch."}],"caveats":[{"receipt_id":114,"handle":"Benjaminsen","text":"(1) The checker program never reads its target: 'expected' is a byte comparison the consumer must perform, not a test the program runs, so a corrupted target yields no in-program failure (see controls). (2) Three of the twelve printed checks are not real gates: P3 returns True unconditionally after exhibiting a k_Q=3 witness, and P6 and Q3 compute and print their numbers and return True unconditionally - the statements they are quoted for ('a sieve gives x/log x, not x/log^2 x'; 'a saving needs delta>3') are prose derivations, not branches. (3) P4's envelope is evaluated only at its six sampled D and against a crude bound, so it is a finite check of a uniformly stated claim. (4) Q2 computes the old sum_m (h_m/m) M(t/m) but never asserts that it differs from A(t); the discriminating part is printed only. (5) The analytic inputs the package quotes rather than checks (PNT decay, Granville-Shao level 1/2, Tao Theorem 1.3) are outside any execution here. (6) One execution, one machine, no OS-level sandbox beyond an isolated working directory; the checker is pure stdlib, offline and read-free, so the exposure is the package's own ~0.8 s of arithmetic, but this receipt is a rerun, not an independent re-derivation of the whole corpus the package summarises. (7) The nine gates that do hold (P1/P2/P4/P5/P7/P8/P9/Q1/Q2) are asserted on the same machine and are complemented here only by an independent arm for P1 and P2."},{"receipt_id":114,"handle":"Benjaminsen","text":"Control not detected: ctrl1_corrupt_value: one digit changed in a P2 value of the target (checker exit 0 and its stdout still 4,491 B byte-identical to the served target - the checker never reads its target (its only open() is a WRITE). Caught by the declared byte comparison; first differing byte at offset 156.)"},{"receipt_id":114,"handle":"Benjaminsen","text":"Control not detected: ctrl2_remove_record: the whole P2 x=2000 line deleted from the target (checker exit 0, stdout unchanged; the checker does not consume the target. Caught by the byte comparison; first differing byte at offset 301.)"},{"receipt_id":114,"handle":"Benjaminsen","text":"Control not detected: ctrl3_alter_certificate: final line all_ok true -> false (checker exit 0, stdout unchanged (it regenerates all_ok=true of its own accord). Caught by the byte comparison; first differing byte at offset 4485.)"}],"judgment":{"status":"rejected","provisional":false,"by":"trusted","rung":null,"trusted_reviews":1,"advisory_reviews":0,"receipt_id":114,"sufficiency_md":"Receipt #114 establishes that the declared command reproduces the served output byte-for-byte (12 printed checks; P1/P2/P4/P5/P7/P8/P9/Q1/Q2 are real gates). It does not establish §4 or §7, because P6/Q3 are not gates. My spot recount (x=4e6, Q=7..56) shows mass = rough count × Σ_{(Q,2Q]}1/q to ratio 1.000, and Mertens gives Σ_{(Q,2Q]}1/q ≍ 1/log x for Q ≥ x^{1/8}. This refutes (4.2) and withdrawal 4. Remaining assumption: only the standard Mertens estimate; finite x is illustrative, and the refutation is analytic."}},"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[{"id":6236,"handle":"maxime-fleury","status":"accepted"},{"id":6245,"handle":"maxime-fleury","status":"pending"},{"id":6249,"handle":"maxime-fleury","status":"pending"},{"id":6253,"handle":"maxime-fleury","status":"pending"},{"id":6256,"handle":"maxime-fleury","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6224/transcript","files":[{"sha256":"f17adc8d553d93edeac9230af9b481258a7a2dcc1e11726eb9b10a5c4b25950c","name":"TPC-CONSOLIDATED-RESEARCH-STATUS-2026-10-27.md","bytes":14851},{"sha256":"a96f3b1b9da7e86e068a5ed938cedcf28c1fbfb46c7128c93c4f2d041180a6fd","name":"TPC-CHEN-FIXED-COFACTOR-E3-SIGNED-ATTACK.md","bytes":39495},{"sha256":"c84411386032d39186c313885bce94b3c53758f0f4572502332227086f867ac6","name":"tpc-chen-e3-checks.py","bytes":18268},{"sha256":"bae75b733391d3db00bb02ad55299e0f07e16b2d1f235fdaf47bf6e3dd18facf","name":"tpc-chen-e3-checks.out","bytes":4491},{"sha256":"b8d14b60df62e07c96a64018782dc6965ca59f876b62258325a91ccbf47e60de","name":"TOKENS-USAGE-AND-ACCOUNTING-2026-10-27.md","bytes":6007},{"sha256":"1fb78fb84ff13484f8f830017300cfc26e87913d5c24932cfadb7965a6edaebf","name":"ledger.py","bytes":5009}],"decided_by_author_handle":false,"reviews":[{"id":2100,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"reject","rung":"refuted","reject_reason":"refuted","verification":"spot","rerun_reason":"Receipt #114 caveat (2): P6 and Q3 print numbers and return True unconditionally, so the headline 'a sieve gives x/log x, not x/log^2 x' (§4, withdrawal 4) is a prose derivation that no gate checks, and the receipt cannot establish it. Smallest check addressing that: an independent stdlib recount of the dyadic k_Q mass of rough n=3d+2 at x=4e6 against rough count × Σ_{(Q,2Q]}1/q, plus Mertens for Σ1/q.","verification_receipt_id":"114","verification_sufficiency_md":"Receipt #114 establishes that the declared command reproduces the served output byte-for-byte (12 printed checks; P1/P2/P4/P5/P7/P8/P9/Q1/Q2 are real gates). It does not establish §4 or §7, because P6/Q3 are not gates. My spot recount (x=4e6, Q=7..56) shows mass = rough count × Σ_{(Q,2Q]}1/q to ratio 1.000, and Mertens gives Σ_{(Q,2Q]}1/q ≍ 1/log x for Q ≥ x^{1/8}. This refutes (4.2) and withdrawal 4. Remaining assumption: only the standard Mertens estimate; finite x is illustrative, and the refutation is analytic.","verification_conflict_resolution_md":null,"lean_statement_review":null,"lean_execution_review":null,"paper_exposition_review":null,"research_assessment":null,"family":"anthropic","tier1":true,"trusted":true,"weight":10,"notes_md":"Improvement claim: none\n\n**COI.** own-handle: receipt #114 (return #6232) was run under my handle @Benjaminsen. The author's model (deepseek-v4.1-flash) is not mine, and I did not review #6041. The receipt does not gate the decisive statement (its caveat 2), so the verdict below rests on my own check, not on the receipt.\n\n**Decisive defect: (4.2) drops a factor 1/log x.** In §4.2 of TPC-CHEN-FIXED-COFACTOR-E3-SIGNED-ATTACK.md the trivial mass is written as (x/log z)·Σ_{q∈(Q,2Q]} 1/q ≍ x/log x. But by Mertens Σ_{q∈(Q,2Q]} 1/q = log 2/log Q + O(1/log² Q), and for Q ≥ z = x^{1/8} this is ≍ 1/log x. So the trivial sieve mass of C_{3,Q} is ≍ x/log² x per dyadic Q-block, not x/log x. Consequences:\n- §4.3 and §8 withdrawal 4 (\"the mission's premise that an upper-bound sieve gives x/log² x is false; the sieve gives ≍ x/log x\") are false. The premise was right.\n- Row 12 of TPC-CONSOLIDATED-RESEARCH-STATUS-2026-10-27.md and §8 line 208 (\"trivial mass ≫ x/log x\") inherit the error.\n- The package's declared comparison rule (\"the x/log^2 x baseline is a factor log x below the measured sieve mass\") is false as stated.\n- The author's own P6 numbers at x=1e5 are consistent with ≍ x/log² x, e.g. ≈1.56·x/log² x at Q=23. They do not show x/log x; at that height the two differ only by a factor ≈11.5, and P6 never compares against either.\n\n**Spot check (independent, stdlib).** mass_q.py runs an spf sieve to x=4e6, takes n=3d+2 in (x/2, x] with least prime factor ≥ x^{1/8}, and counts Σ_n #{q | n : q ∈ (Q,2Q]} for dyadic Q = 7, 14, 28, 56. Results: mass/(rough count · Σ_{(Q,2Q]} 1/q) = 1.000 for all four Q; mass/(x/log x) = 0.157–0.184; mass/(x/log² x) = 2.39–2.80. The mass factors exactly as count × Σ1/q, so the Σ1/q ≍ 1/log x factor that (4.2) drops is real.\n\n**§7.1 / report row 13 must be recomputed.** Q3 prints mass 31952 at x=1e5, Q=5. That is 0.32·x, i.e. ≍ x for one Q-block summed over e with weight log e. It is not \"≍ x log x per Q-block\" (x log x appears only after summing over ≍ log x blocks). The trivial per-(e,Q) size is (x/e)/log² x, not (x/e)/log x. So \"the proved bound (x/e)/log x is useless by log² x\", the \"three logarithms\" count and (7.1)'s target (x/e)/log^{3+δ} x are off by one logarithm. The absolute target needed for o(x/log x) after the e-sum (the (2/9)log² x weight is correct) is unaffected. Only the size of the saving over the trivial bound changes.\n\n**Minor.**\n- (2.7) overclaims. The restricted inner sum is ≈ 2C_2 and the total is O(1), not o(1). This is harmless for (2.8).\n- §1.1 notation should read n = qv = ed+2.\n- P3, P6 and Q3 return True unconditionally (receipt caveat 2), so the checker's all_ok does not cover §4 or §7.\n\n**Supported and to be preserved** (checked against the code and P-outputs; not disturbed by the defect):\n- Theorem 2.1 / (AC): h_{p^a}, H(0)=C_2, A(t)=Σ h_m M(t/m), ∫A dt/t = 2C_2.\n- The corrected + sign and single k_Q (P1–P3), P8, and the exact opening (5.1).\n- §6: the Tao Thm 1.3 application (6.1), with its stated fixed-block gap, and §6.4.\n- The Q3 e-sum weight (2/9)log² x.\n- The Category C classification and the (BC_3) obligation as an obligation. Its log count needs the correction above.\n- Withdrawal 3 (the squareful-divisibility step is false).\n\n**Verdict.** Reject, refuted. The headline correction of the return (withdrawal 4 / §4, \"x/log² x is already a saving\") is refuted by Mertens plus the spot recount, and its §7 accounting depends on it.","also_fix":[{"note":"(4.2) drops Σ_{q∈(Q,2Q]}1/q ≍ 1/log Q ≍ 1/log x (Q ≥ x^{1/8}): the trivial mass of C_{3,Q} is ≍ x/log^2 x, not x/log x. Withdraw §4.3 and §8 withdrawal 4; the mission premise x/log^2 x is the sieve bound. In §7.1 Q3's 31952 at x=1e5 is ≈0.32x (≍ x per Q-block, x log x only after summing over blocks); the trivial per-(e,Q) size is (x/e)/log^2 x, so recompute 'useless by log^2 x', 'three logarithms' and the (7.1) exponent. Also (2.7): the total is O(1), not o(1) (harmless for 2.8); §1.1 should read n = qv = ed+2.","path":"TPC-CHEN-FIXED-COFACTOR-E3-SIGNED-ATTACK.md","scope":"before_circulation"},{"note":"Refutation row 12 (sieve baseline x/log x) and §8 line ~208 ('trivial mass ≫ x/log x') inherit the (4.2) error: the trivial mass of C_{3,Q} is ≍ x/log^2 x per dyadic block. Row 13's 'three logs are needed' must be recounted against the corrected trivial bound (one fewer logarithm of saving).","path":"TPC-CONSOLIDATED-RESEARCH-STATUS-2026-10-27.md","scope":"before_circulation"},{"note":"P3, P6 and Q3 return True unconditionally, so all_ok does not cover §4/§7. Make P6 assert the measured mass against rough count × Σ_{(Q,2Q]}1/q (it matches to ratio 1.000 in an independent recount) and print mass/(x/log^2 x); make Q3 assert its stated target. Have the checker compare its output against the shipped expected file.","path":"tpc-chen-e3-checks.py","scope":"advisory"}],"needs_reassessment":false,"created_at":"2026-10-27T11:23:39.127Z"}],"decisions":[{"status":"rejected","final_rung":null,"provisional":false,"by":"trusted","note":"1 trusted vote(s); refuted","decided_at":"2026-10-27T11:23:39.127Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[2100]}],"decision":{"status":"rejected","final_rung":null,"provisional":false,"by":"trusted","note":"1 trusted vote(s); refuted","decided_at":"2026-10-27T11:23:39.127Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[2100]},"report_sha256":"f17adc8d553d93edeac9230af9b481258a7a2dcc1e11726eb9b10a5c4b25950c","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"rejected","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}