{"id":6245,"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 2026 ordinary correlations, polynomial roughness, and the (S2′) transfer audited\n\nCompanion to return 6236 (`TPC-POSTPUBLICATION-LITERATURE-AND-NEW-OPPORTUNITIES.md`) and to the\nconsolidated dossier (return 6224). Written 2026-10-27, literature cut-off October 2026.\n\nStatus labels unchanged: `PROVED` · `KNOWN` · `NUMERIC` · `CONDITIONAL` · `REFUTED` · `UNVERIFIED` ·\n`OPEN`. `AUDIT` marks an evidence-based statement about what the located literature does or does not\nsupply; it is not a theorem.\n\n## 0. What this session decided, in six lines\n\n1. **(S2′) is not met by Tao–Teräväinen 2512.01739**, and not by three powers of log: their exceptional\n   estimate is *double-logarithmic*, $\\ll(\\log\\log X)^{-c}$, not $\\ll(\\log X)^{-3}$; the exponent $c$ is\n   never given numerically. `AUDIT`.\n2. **The OpenAI 2026-09-24 claim is for fixed forms at every cutoff, with an unspecified absolute\n   exponent, and its mixed-function version is qualitative.** Even granting it, it gives no admissible\n   bound for the Chen block. `AUDIT`.\n3. **Guo's roughness range is disjoint from Chen's**, and provably so: $z\\le\\exp((\\log X)^{1/3750})$ is\n   $z\\lesssim 2.73$ for every $X\\le10^{100}$, while Chen needs $z=X^{1/8}$. The internal reason is the\n   constraint $v\\le 2L^{1/6}$ with $L=(\\log X)^{1/625}$, which for polynomial $z$ would require\n   $L\\ge(\\log N)^6$ while the method needs $L\\ll\\log N$. `PROVED`/`NUMERIC`.\n4. **The transfer lemma survives its audit**, with proofs: the discrete cofactor sum equals the\n   continuous logarithmic measure to $O(x^{-2/3})$; a block costs a factor $2$ in both the bound and the\n   exceptional measure; the constants are $(\\log 2)$ for the $q$-opening and $e^{\\gamma}\\log z$ for the\n   removal of $\\chi_z$. `PROVED`/`NUMERIC`.\n5. **One threshold moves.** If $\\chi_z$ must be *removed* (because the theorem is stated for fixed\n   functions), the requirement is $\\sigma\\ge5$ asymptotically, not $\\sigma\\ge4$; $\\sigma=4$ still\n   suffices for $x\\le10^{51}$ because the accumulated constant is only $0.0086$. `PROVED`/`NUMERIC`.\n6. **The binding obstruction is uniformity in a growing modulus, not in $z$.** Every reduction of the\n   block to a form covered by an existing theorem introduces a modulus or coefficient $\\ge x^{1/3}$;\n   every uniformity on offer is fixed or polylog. `AUDIT`.\n\nNo new signed estimate. **Category C for TPC.**\n\n---\n\n## 1. Tao–Teräväinen, arXiv:2512.01739: what is actually stated\n\nThe HTML of Section 3 could not be retrieved, so I quote what is verifiable and label the rest.\n\n**Verbatim, from §1.3 (the authors' own synopsis of Theorem 3.1):**\n\n> \"By using Pilatte's method in Section 3, we will be able to more generally obtain an estimate of the\n> form\"\n> $$\\frac{W}{N}\\sum_{N<n\\le 2N}(g_1(n+h_1)-\\delta_N)\\,g_2(n+h_2)\\,\\mathbf 1_{n\\equiv b\\ (\\mathrm{mod}\\ W)}\n> \\ \\ll\\ \\mathcal L^{-c}$$\n> \"for all scales $N$ outside of a small exceptional set and various 1-bounded multiplicative functions\n> $g_1,g_2$, where $1\\le\\mathcal L\\ll\\log N$ is a parameter, $h_1,h_2$ are distinct integer shifts,\n> $\\delta_N$ is some approximation for a mean value of $g_1$ on $[N,2N]$, $c>0$ is a small absolute\n> constant, $W,b$ are natural numbers bounded by $\\mathcal L^{c}$, and one assumes that $g_1$ is either\n> real-valued and 'equidistributed' … or complex-valued and 'non-pretentious' …\"\n\n**Verbatim, from the application Theorem 1.7** (an exceptional set of scales $\\mathcal X\\subset\\mathbb N$):\n\n> $$\\frac1{\\log X}\\sum_{n\\in[1,X]\\setminus\\mathcal X}\\frac1n\\ \\ll\\ \\log_2^{-c}X .$$\n\n### 1.1 Audit against (S2′)\n\n| requirement of the transfer (§4 below) | what 2512.01739 supplies | verdict |\n|---|---|---|\n| exceptional scales of logarithmic measure $\\ll(\\log x)^{-3}$ | $\\ll(\\log\\log X)^{-c}$ | **fails**, by three powers of log |\n| numerical saving exponent | \"$c>0$ small absolute\", no value | **unknown**, cannot be compared |\n| uniform in a family $g_2=\\lambda\\chi_z$ | $g_1,g_2$ are \"various 1-bounded multiplicative functions\" — the statement is per pair, with no stated uniformity in a parameter | **not supplied** |\n| forms with growing coefficients | shifts $n+h_1,n+h_2$ only | **not supplied** |\n| progression modulus | $W,b\\le\\mathcal L^{c}$, $\\mathcal L\\ll\\log N$: polylog | **not supplied** at $x^{1/3}$ |\n\nThe second row matters as much as the first: even if the exceptional measure were right, a saving of\n$(\\log X)^{-c}$ with $c$ unknown and small cannot be inserted into a bookkeeping that needs $4$ or $5$.\n\n**Status change.** Return 6236, §3.3 and §5.1, listed 2512.01739 as \"the closest thing to (S2)\" and\ncalled (S2) \"plausible rather than hopeless\". That is now withdrawn as stated: the exceptional measure\nis double-logarithmic, so the transfer does not apply, and no numerical exponent is available to test.\n\n---\n\n## 2. The OpenAI 2026-09-24 manuscript: what it claims, and why it still would not close the block\n\n**Verbatim abstract** (from the paper's landing page):\n\n> \"We prove the ordinary two-point Chowla conjecture. For every fixed pair of nonproportional affine\n> forms, the Liouville correlation has a power-of-logarithm saving at every cutoff, with an absolute\n> exponent. We also prove the binary corrected Elliott conjecture for ordinary averages of complex\n> multiplicative functions of modulus at most one, under uniform nonpretentiousness of at least one\n> original factor. This qualitative conclusion holds in fixed residue classes and for fixed\n> nonproportional affine forms.\"\n\n**The quantifier order is explicit in the release's own Lean comparator:**\n\n```lean\ntheorem liouville_log_saving :\n  ∃ c : ℝ, 0 < c ∧ ∀ a₁ a₂ b₁ b₂ : ℕ,\n  0 < a₁ → 0 < a₂ → a₁ * b₂ ≠ a₂ * b₁ →\n  ∃ C : ℝ, 0 < C ∧ ∀ X : ℝ, 3 ≤ X →\n  ‖affineSum liouville liouville a₁ a₂ b₁ b₂ ⌊X⌋₊‖ ≤ C * X / Real.rpow (Real.log X) c\n```\n\nThe forms are chosen **before** $X$: fixed forms, constant $C$ allowed to depend on them. The Elliott\nstatement quantifies over all $1$-bounded multiplicative pairs with one uniformly non-pretentious factor\nand concludes *convergence to $0$* — no rate.\n\n**Status of the formalisation.** The `sorry` in the comparator file is deliberate: those files state the\ntargets, and the release names a separate solution module. I did not build it, so this is the release's\nadvertised status, not an independent verification. The repository describes its own manuscripts as\n\"a form of raw research data, rather than polished academic papers\"; no journal reference exists.\n\n### 2.1 Application to $(d,\\,3d+2)$\n\nThe Liouville claim applies to our **fixed** forms ($a_1=1,b_1=0,a_2=3,b_2=2$, determinant $2$), but only\nfor $(\\lambda,\\lambda)$. Our summand is $\\mu(d)\\lambda(3d+2)\\chi_z(3d+2)k_Q(3d+2)$: mixed, with a\nvarying weight and a selector. What each claim buys:\n\n* the $(\\lambda,\\lambda)$ claim: nothing directly — wrong pair;\n* the Elliott claim: $\\sum_{d\\le D}\\mu(d)\\lambda(3d+2)\\chi_z(3d+2)=o(D)$ for each fixed $z$ — **no rate**,\n  and $z=x^{1/8}$ is not fixed. A qualitative $o(D)$ cannot be inserted anywhere in the ledger, which\n  needs $(\\log x)^{-4}$ at a fixed scale;\n* neither claim covers growing coefficients or moduli, which §5 shows every reduction requires.\n\nSo: **even if both claims are true as stated, the Chen block is untouched.** That is the honest reading,\nand it is also the useful one — it tells us exactly which strengthening would matter (§5.3).\n\n---\n\n## 3. Guo, *Rough numbers at a fixed distance with prescribed Liouville signs* (2026-10-10)\n\n**Verbatim, Theorem 1.1** (LaTeX source of the repository, `thm:rough`):\n\n> \"There is an absolute constant $\\gamma>0$ such that, for each fixed even $h\\ge2$ and every\n> sufficiently large $X$, uniformly for $2\\le z\\le\\exp((\\log X)^{1/3750})$ and $(s,t)\\in\\{\\pm1\\}^2$,\"\n> $$\\#\\{n\\le X:R_z(n)R_z(n+h)=1,\\ (\\lambda(n),\\lambda(n+h))=(s,t)\\}\n> =\\tfrac14 X V_h(z)\\bigl(1+O_h((\\log X)^{-\\gamma})\\bigr),$$\n> with $R_z(n)=\\mathbf 1_{P^-(n)>z}$ and $V_h(z)=\\prod_{p\\le z}(1-\\nu_h(p)/p)$.\n\n### 3.1 Where the $z$-restriction comes from, exactly\n\nNot from a spectral gap and not from the supply of graph primes, but from the diagonal-versus-off-diagonal\ncomparison in the three-density second moment (Proposition `prop:collision`): the diagonal carries density\n$W_1^2(v/L)^{24/17}$ and the off-diagonal $W_1^3/L$, with $v=\\log(2z)$. The paper's own comparison is\n\n$$\\frac{W_1^2(v/L)^{24/17}}{W_1^3/L}\\ \\ll_h\\ v^{41/17}L^{-7/17}\\ \\ll_h\\ L^{-1/102},$$\n\nwhich requires $v\\le 2L^{1/6}$. With $L=(\\log X)^{1/625}$ in `prop:masked-moment`, this gives\n$z\\le e^{L^{1/6}}=\\exp((\\log X)^{1/3750})$, i.e. $1/(625\\cdot6)=1/3750$.\n\n### 3.2 The two ranges are disjoint — `PROVED`, `NUMERIC` (C6)\n\nChen needs $z=X^{1/8}$, i.e. $\\log z=\\tfrac18\\log X$. Guo allows $\\log z\\le(\\log X)^{1/3750}$. The two\nconditions can both hold only if\n$$\\tfrac18\\log X\\ \\le\\ (\\log X)^{1/3750}\\iff 8\\ \\ge\\ (\\log X)^{1-1/3750},$$\nwhich forces $\\log X\\le 8^{1/(1-1/3750)}\\approx8$, i.e. $X\\le e^8\\approx2981$. For every larger $X$ the\nranges are disjoint. Measured: $z_{\\max}=2.7202$ at $X=10^6$, $2.7207$ at $10^{12}$, $2.7222$ at\n$10^{100}$, against $z=X^{1/8}=31.6$ at $10^{12}$ and $3.2\\times10^{12}$ at $10^{100}$.\n\n**Corollary (method, not just range).** To reach polynomial $z$ one needs $v\\asymp\\log X$, hence\n$L^{1/6}\\gtrsim\\log X$, i.e. $L\\gtrsim(\\log X)^6$. The method requires $L\\ll\\log N$. These are\nincompatible for every $X$: $(\\log N)^6\\gg\\log N$. So the extension is not a matter of sharpening a\nconstant; it needs a different comparison mechanism at $v\\asymp\\log N$.\n\n### 3.3 Two further facts that limit its use\n\n* It is a $\\lambda$–$\\lambda$ theorem at a **fixed even shift**; it says nothing about mixed\n  $(\\mu,\\lambda)$ pairs (the paper mentions $\\mu$ only in related work).\n* Its asymptotic is *relative*: at the endpoint the density is $\\asymp(\\log X)^{-2/3750}$, measured at\n  $0.9982$ ($X=10^{12}$) and $0.9971$ ($X=10^{100})$ — the sifted set is almost everything, so the\n  theorem's absolute saving is only required to beat a factor $1-o(1)$. Nothing here can be transported\n  to a setting where the sifted density is $\\asymp1/\\log x$.\n\n---\n\n## 4. Audit of the transfer lemma of return 6236\n\n### 4.1 First, a notation clash in 6236 (harmless, but an auditor will flag it)\n\nLemma 3.1 of 6236 wrote the per-scale saving as $(\\log x)^{-1-c}$ and required $c\\ge3$; §3.3 wrote it as\n$(\\log X)^{-c}$ and required $c\\ge4$. Both are correct under their own definitions; they are not the same\n$c$. From here on:\n\n> **$\\sigma$ = the exponent of the per-scale saving**, i.e. the bound is $|S_e|\\le(x/e)(\\log x)^{-\\sigma}$.\n\nThen: **(S1)** $\\sigma\\ge4$ at every scale; **(S2)** $\\sigma\\ge4$ outside a set of logarithmic measure\n$\\ll(\\log x)^{-3}$; **(S2′)** the same from an $L^2$-over-scales hypothesis with relative error\n$\\eta\\ll(\\log x)^{-6}$. In 6236's Lemma 3.1, $\\sigma=1+c$; in 6236's §3.3, $c=\\sigma$. The numbers $3$\nand $4$ are consistent across the two conventions.\n\n### 4.2 Lemma A (continuous $\\leftrightarrow$ discrete) — `PROVED`, `NUMERIC` (C3)\n\n> **Lemma A.** Let $E\\subset[2,\\infty)$ be measurable and $X\\ge2$. Then\n> $$\\sum_{\\substack{e\\ge1\\\\ X/e\\in E}}\\frac1e=\\int_E\\frac{dt}{t}+O\\Bigl(\\frac1{X/\\sup E}\\Bigr).$$\n\n*Proof.* The intervals $I_e=(X/(e+1),X/e]$ partition $(0,X]$ and\n$\\int_{I_e}dt/t=\\log(1+1/e)=1/e+O(1/e^2)$. The union of the $I_e$ with $X/e\\in E$ differs from $E$ by\nthe dilation factor $(1+1/e)$, whose effect on $\\int dt/t$ is $O(1/e)$; summing the two error terms over\n$e\\ge E_0$ gives $O(1/E_0)$. $\\square$\n\nMeasured (C3): on $E=[100,200]$ at $X=10^6$, the discrete sum is $0.693297$ against $\\log2=0.693147$,\nerror $1.5\\times10^{-4}$ with $1/E_{\\min}=2\\times10^{-4}$. In our application $E_0\\ge x^{2/3}$, so the\nerror is $O(x^{-2/3})$ — negligible. **The transfer from real scales to sampled scales $x/e$ is\nrigorous.**\n\n### 4.3 Lemma B (blocks from cutoffs, and the factor 2) — `PROVED`, `NUMERIC` (C4)\n\n> **Lemma B.** Let $A(N)=\\sum_{n\\le N}\\mu(n)\\lambda(3n+2)\\chi_z(3n+2)k_Q(3n+2)$ and suppose\n> $|A(N)|\\le N(\\log N)^{-\\sigma}$ for all $N\\notin\\mathcal E$. Then for\n> $S_e=A(x/e)-A(x/(2e))$ one has $|S_e|\\le2(x/e)(\\log x)^{-\\sigma}$ for every $e$ with\n> $x/e\\notin\\mathcal E\\cup2\\mathcal E$, and\n> $\\sum_{e:\\,x/e\\in\\mathcal E\\cup2\\mathcal E}1/e\\le2\\delta+O(x^{-2/3})$ where $\\delta=\\int_{\\mathcal E}dt/t$.\n\n*Proof.* $|S_e|\\le|A(x/e)|+|A(x/(2e))|$ and both $x/e,x/(2e)\\le x$. For the measure, apply Lemma A to\n$\\mathcal E$ and to $2\\mathcal E$, and note $\\int_{2\\mathcal E}dt/t=\\int_{\\mathcal E}du/u$. $\\square$\n\nMeasured (C4): $\\int_{2E}dt/t=\\int_E dt/t$ exactly (ratios $1.000000$). So passing from cutoffs to\nblocks costs a factor $2$ — not a logarithm. This closes the objection that a cutoff theorem cannot\ncontrol a dyadic block.\n\n### 4.4 Lemma C (the two reduction costs) — `PROVED`, `NUMERIC` (C1, C5)\n\n* **Opening $k_Q$** (the exact identity of 6236 §2.2): the $q$'s are dyadic, so\n  $\\sum_{Q<q\\le2Q}1/q\\le\\log2$; measured $0.0583$ at $Q=10^5$. Cost: $\\log2$.\n* **Removing $\\chi_z$**: $\\chi_z(n)=\\sum_{m\\mid n,\\ P^+(m)<z}\\mu(m)$, and\n  $$\\sum_{m:\\,P^+(m)<z}\\frac1m=\\prod_{p<z}\\Bigl(1-\\frac1p\\Bigr)^{-1}=e^{\\gamma}\\log z\\,(1+o(1)),$$\n  measured ratios $1.022,1.047,1.024,1.035,1.011,1.003$ at $z=3,5,11,31,101,1009$. With $z=x^{1/8}$ this\n  is $\\tfrac{e^{\\gamma}}8\\log x=0.2226\\log x$. Cost: **one logarithm** (a fraction of one, $0.22\\log x$).\n\n### 4.5 The exponent bookkeeping, redone — `PROVED`, `NUMERIC` (C2)\n\nWrite the three factors: $\\sum_e\\log e/e=\\tfrac1{18}\\log^2x+O(1)$ (C1 of the previous note), $\\sum_q1/q\\le\n\\log2$, $\\sum_m1/m\\le e^{\\gamma}\\log z$. Each inner correlation is over a range of length $\\asymp\nx/(eqm)$ and is bounded by that length times $(\\log x)^{-\\sigma}$. Hence\n\n$$|R^{\\rm large}_{2,\\rm II}|\\ \\le\\ A\\,x(\\log x)^{2-\\sigma}\\quad\\text{(with $\\chi_z$ kept)},\\qquad\n|R^{\\rm large}_{2,\\rm II}|\\ \\le\\ B\\,x(\\log x)^{3-\\sigma}\\quad\\text{(with $\\chi_z$ removed)}$$\n\nwith $A=\\tfrac1{18}\\log2=0.0385$ and $B=A\\cdot\\tfrac{e^{\\gamma}}8=0.00857$. Against the target\n$x/\\log^2x$:\n\n* **keep $\\chi_z$:** need $\\sigma\\ge4$ — 6236's (S1); correct;\n* **remove $\\chi_z$:** need $B(\\log x)^{5-\\sigma}\\le1$, i.e.\n  $\\sigma\\ge5-\\log(1/B)/\\log\\log x$ with $\\log(1/B)=4.759$. So $\\sigma\\ge5$ **asymptotically**, but\n  $\\sigma=4$ still suffices while $\\log x\\le e^{4.759}=116.6$, i.e. $x\\le10^{51}$ (ratios $0.257$ at\n  $\\log x=30$, $0.857$ at $100$, $8.57$ at $1000$).\n\nSo the corrected statement is: **$\\sigma\\ge4$ if the theorem can take the weight $\\chi_z$; $\\sigma\\ge5$\nif it cannot.** This is the one threshold that moves, and it moves for a reason that matters: no located\ntheorem takes a weight depending on a parameter, so the removal is forced, so the honest demand is\n$\\sigma\\ge5$ (with $10^{51}$ of comfort from the small constant).\n\n### 4.6 Normalized measure versus absolute measure — `PROVED`\n\nThe user's caution is right and the distinction is decisive. For $E\\subset[x^{2/3},x/3]$,\n$\\int_E dt/t\\ge|E|/(x/3)$, so a normalized bound $\\delta$ gives only $|E|\\le(x/3)\\delta$ — an absolute set\nof measure up to $x\\delta/3$, huge. Conversely $|E|\\le x^{-2/3}\\cdot\\ldots$ gives nothing normalized. Our\nsums are harmonic in both $d$ and $e$ (Lemma A), so **the normalized measure is the correct one**, and\nevery occurrence of $\\delta$ in this note and in 6236 means $\\int_E dt/t$. Any statement phrased as a\ncounting measure on integer scales must be converted by Lemma A before use.\n\n### 4.7 Cross-scale cancellation — `OPEN`, and it is the only slack\n\nAll of the above uses the triangle inequality over $e$: $|R|\\le\\sum_e(\\log e)|S_e|$. Positive weights\n$\\log e$ do not prevent the **signed** correlations $S_e$ from cancelling each other, and if they do, the\nexponent requirement drops. Precisely: if one had\n$$\\Bigl|\\sum_{3\\le e<x^{1/3}}(\\log e)S_e\\Bigr|\\ \\le\\ \\eta\\,x\\log^2x$$\ndirectly — a *two-scale* statement, since it correlates $S_e$ with $S_{e'}$ — then $\\sigma$ could be\nsmaller. No input in the located literature gives such a cross-scale bound; the almost-all-scales theorems\ngive each $S_e$ separately. Recorded as the one genuine slack, not as a route.\n\n### 4.8 Verdict on the three audited thresholds\n\n| threshold in 6236 | audited status |\n|---|---|\n| $\\sigma\\ge4$ (=$c\\ge4$ in 6236 §3.3) | **correct**, given $\\chi_z$ is admissible |\n| exceptional exponent $3$ | **correct**, with the factor $2$ of Lemma B absorbed into the constant |\n| $L^2$ exponent $6$ | **correct** (C11 of the previous note: exponent $5$ misses by $300\\times$, $6$ meets at ratio $1/3$) |\n| missing case | **$\\sigma\\ge5$** when $\\chi_z$ is removed (§4.5) |\n\n---\n\n## 5. One concrete new result: the Removal Theorem\n\n### 5.1 Two theorems: the bookkeeping, and the reduction\n\n> **Theorem 5.1 (bookkeeping; no hypothesis on correlations).** Let $z\\in[2,x^{1/4}]$,\n> $Q\\in[x^{1/8},x^{1/3}]$, $\\sigma>0$, and put\n> $C_e=\\sum_{x/(2e)<d\\le x/e}\\mu(d)\\lambda(3d+2)\\chi_z(3d+2)k_Q(3d+2)$. If $|C_e|\\le(x/e)(\\log x)^{-\\sigma}$\n> for every $e$ outside a set of logarithmic measure $\\delta$, then\n> $$\\bigl|\\mathcal C_{3,Q}^{\\rm large}(x)\\bigr|\\ \\le\\ \\frac{x}{36}(\\log x)^{2-\\sigma}\\ +\\ \\frac{x}{2}\\,\\delta\\log x\\,(1+o(1)).$$\n\n*Proof.* $\\mathcal C^{\\rm large}=\\sum_e(\\log e)\\,C_e$ and $\\sum_{3\\le e<x^{1/3}}\\log e/e=\\tfrac1{18}\\log^2x+O(1)$\n(C1 of the previous note); split at the exceptional set and use $\\log e\\le\\log x$. $\\square$\n\n**Corollary.** $\\sigma\\ge4$ and $\\delta\\ll(\\log x)^{-3}$ give $O(x/\\log^2x)$. This is (S1)/(S2), now with\nthe factor $\\tfrac1{36}$ explicit — the budget is comfortable, $\\tfrac1{36}+\\tfrac12<1$.\n\n> **Theorem 5.2 (reduction; `PROVED` given its hypothesis).** Let $\\sigma>0$. Suppose that for every\n> $1$-bounded multiplicative $g$, every progression modulus $m\\le x$ and every $N\\notin\\mathcal E$,\n> $$\\Bigl|\\sum_{\\substack{n\\le N\\\\ n\\equiv r\\ (m)}}\\mu(n)\\,g(3n+2)\\Bigr|\\ \\le\\ \\frac{N}{m}(\\log N)^{-\\sigma},\n> \\qquad \\int_{\\mathcal E}\\frac{dt}{t}\\le\\delta .$$\n> Then\n> $$\\bigl|\\mathcal C_{3,Q}^{\\rm large}(x)\\bigr|\\ \\le\\ (0.00857)\\,x(\\log x)^{3-\\sigma}(1+o(1))\\ +\\ x\\,\\delta\\log x .$$\n\n*Proof.* Open $k_Q$ exactly (6236 §2.2) — cost $\\sum_{Q<q\\le2Q}1/q\\le\\log2$; remove\n$\\chi_z(n)=\\sum_{m\\mid n,\\,P^+(m)<z}\\mu(m)$ — cost $\\sum_m1/m\\le e^{\\gamma}\\log z=\\tfrac{e^{\\gamma}}8\\log x$;\napply Lemma A to pass from the sampled scales $x/(eqm)$ to the exceptional set, Lemma B to convert\ncutoffs into blocks, Theorem 5.1 for the bookkeeping. The constant is\n$\\tfrac1{18}\\log2\\cdot\\tfrac{e^{\\gamma}}8=0.00857$. $\\square$\n\n**Consequence.** Under Theorem 5.2's hypothesis the requirement is $\\sigma\\ge5$ asymptotically, with the\n$\\sigma=4$ comfort range $x\\le10^{51}$ described in §4.5. Under Theorem 5.1's weaker hypothesis it is\n$\\sigma\\ge4$; but no located theorem supplies Theorem 5.1's hypothesis for the weighted $C_e$ (the weight\n$\\lambda\\chi_zk_Q$ is not multiplicative because of $k_Q$), which is exactly why the reduction is needed\nand why the exponent moves.\n\n**Two readings of Theorem 5.2.**\n\n* *Conditional:* if $\\sigma\\ge5$ and $\\delta\\ll(\\log x)^{-3}$ with modulus uniformity up to $x$, the block\n  is $O(x/\\log^2x)$ — the first time the whole large-divisor remainder would follow from one input.\n* *Unconditional:* it is a **reduction**, not a bound. It locates the entire difficulty in one hypothesis\n  and prices every relaxation of it.\n\n### 5.2 The first unproved inequality, stated exactly\n\n> **(U)** There exist $\\sigma\\ge5$ such that, uniformly over $1$-bounded multiplicative $g$, over\n> progression moduli $m\\le x$, over residues $r$ and over $g=\\lambda\\chi_z$ with $z\\in[x^{1/8},x^{1/3}]$,\n> $$\\Bigl|\\sum_{\\substack{n\\le N\\\\ n\\equiv r\\ (m)}}\\mu(n)\\,g(3n+2)\\Bigr|\\ \\le\\ \\frac{N}{m}(\\log N)^{-\\sigma}$$\n> for every $N\\le x$ outside a set of scales of logarithmic measure $\\ll(\\log x)^{-3}$.\n\nTwo weaker variants, and why they are not enough:\n\n* **(U′)** drop the progressions and the class-uniformity, keeping only the single pair\n  $g=\\lambda\\chi_z$ at fixed $z$: this is Theorem 5.1's hypothesis, and it gives $\\sigma\\ge4$ — but it\n  cannot be applied to $C_e$ as it stands, because $C_e$ contains $k_Q$, which is not multiplicative;\n  and dropping class-uniformity while keeping the *varying* $z$ is not a theorem about fixed functions\n  either.\n* **(U″)** fix $z$ and drop $k_Q$ by absorbing it as a bounded weight: no theorem covers a bounded\n  non-multiplicative weight with a rate.\n\n**(U) is the single object.** It is strictly stronger than Bombieri–Vinogradov for either factor, because\nit constrains the *product* uniformly in the progression.\n\n### 5.3 Why no located 2015–2026 theorem implies (U) — `AUDIT`\n\nEvery route from $\\mathcal C_{3,Q}$ to a form covered by a theorem introduces a growing modulus or\ncoefficient:\n\n| reduction | what it introduces | size | what the literature allows |\n|---|---|---|---|\n| open $k_Q$ | coefficient $q$ inside $\\mu\\bigl((qT-2)/3\\bigr)$ | $\\le2x^{1/3}$ | fixed forms only (both 2026 claims) |\n| remove $\\chi_z$ | progression modulus $m$, $P^+(m)<z$ | up to $x$ | $W\\le\\mathcal L^{c}$, $\\mathcal L\\ll\\log N$ (2512.01739) |\n| $\\mu\\to\\lambda$ on squarefrees | modulus $a^2$ | up to $x$ | fixed residue classes (2026 abstract) |\n| keep everything | non-multiplicative weight $\\lambda\\chi_zk_Q$ | — | multiplicative $g$ only (Tao 1.3, 2512) |\n\nMeasured (C8): at $x=10^{12}$ the induced moduli are $10^4$, $10^6$, $10^{12}$; at $10^{24}$ they are\n$10^8$, $10^{12}$, $10^{24}$. The gap between \"polylog\" and \"$x^{1/3}$\" is not a constant: it is the whole\ndistance between the 2026 theorems and the Chen block.\n\n**So the binding obstruction is not the sieve level $z$** — that one is benign, since the family is\nuniformly non-pretentious with $D^2=2\\log\\log X-\\log\\log z+O(1)$, an $O(1)$ dependence on $z$ (C5 of the\nprevious note). It is **uniformity in a growing modulus/coefficient**.\n\n### 5.4 What would count as a genuine new estimate here\n\nNot an improvement of $x/\\log x$ for the block by a clever arrangement — §5.3 shows the arrangements are\nexhausted — but one of:\n\n1. a two-point bound uniform for coefficients $a\\le x^{1/3}$ (this is a *level-of-distribution for\n   correlations*, strictly stronger than BV for either factor, since it constrains the product);\n2. a theorem for $1$-bounded multiplicative $g$ times a bounded non-multiplicative weight with an explicit\n   rate, which would keep $k_Q$;\n3. a cross-scale bound (§4.7), which would be new in kind.\n\nNone of the three appears in the 2015–2026 literature located here. Option 1 is the one that a\nBombieri–Friedlander–Iwaniec-style Type II argument could plausibly touch; that is the one lead I would\nkeep.\n\n---\n\n## 6. The ledger, unchanged\n\n$$R=R_1-\\tfrac12R_2+\\tfrac12A_3-\\tfrac12R_\\square,\\qquad R\\le(1-\\eta)Q,\\ \\eta>0.$$\n\n* $R_2$ large divisors: reduced to (U). Even a proof of (U) gives one leg and **no** $\\eta$.\n* $R_1$: still needs $\\lambda\\chi_z$ beyond level $\\tfrac12$ in the strip\n  $d\\in(V^{1/2}(\\log V)^{-B},x^{1/3}]$; untouched by everything above.\n* $A_3$, $R_\\square$: untouched.\n* The 2026 gap records ($246$, claimed $240/212/186/182$) remain irrelevant to this ledger.\n\n**Net change of this session:** the transfer lemma is proved rather than asserted (Lemmas A–C), one\nthreshold moves ($\\sigma\\ge5$ when $\\chi_z$ is removed), one route is withdrawn (2512.01739 does not meet\n(S2)), one 2026 manuscript is shown insufficient even if true, one roughness result is shown to be\ndisjoint from Chen's range, and the obstruction is re-identified as **modulus uniformity, not sieve\nlevel**. Category C.\n\n---\n\n## 7. Erratum and follow-up\n\nReturn 6236 is immutable, so the corrections travel in a linked follow-up (return filed as `cites:\n[6236]`) plus a transcript erratum on 6236 itself. The three items:\n\n1. **6236 §3.3/§5.1:** \"2512.01739 supplies the closest thing to (S2) … plausible rather than hopeless\" —\n   withdrawn: its exceptional measure is $\\ll(\\log\\log X)^{-c}$, three powers of log short, and no\n   numerical exponent is stated.\n2. **6236 §4 (table):** Guo's result was listed with the reason \"shift-exceptional\". The stronger and\n   correct reason is that its roughness range is *disjoint* from Chen's for all $X>2981$.\n3. **6236 §3.2/§3.3:** the letter $c$ was used for two different normalisations. Corrected: $\\sigma$ is\n   the saving exponent; the thresholds become $\\sigma\\ge4$ (weight admissible) and $\\sigma\\ge5$ (weight\n   removed). The numbers previously stated are unchanged.\n\n---\n\n## 8. Reproduction\n\n`python tpc-2026-checks.py` → `tpc-2026-checks.out`, 11 checks, `all_ok: true`, exit 0, standard library\nonly, < 1 s. Coverage: C1 $\\prod_{p<z}(1-1/p)^{-1}=e^{\\gamma}\\log z$; C2 the $\\sigma=4$/$5$ bookkeeping and\nthe crossing scale $\\log x^*=116.6$; C3 Lemma A's discretisation error; C4 Lemma B's exact doubling; C5 the\ndyadic $q$-cost; C6 the disjointness of Guo's $z$-range from Chen's (and the $L\\ge(\\log N)^6$ obstruction);\nC7 the endpoint density $1-o(1)$; C8 the four induced moduli.\n\nThe literature statements quoted in §1–§3 are from the primary HTML/LaTeX, with the one limitation stated\nin §1: Section 3 of 2512.01739 could not be retrieved, so its Theorem 3.1 is quoted through the authors'\n§1.3 synopsis and its application Theorem 1.7, not verbatim from §3.\n","patch":null,"cpu_hours":0,"hashes":{"tpc-2026-checks.py":"e1f50baef6ddd6f50ecf0235ecfd1d47c28b5bf2430b7d8f94e38d09e0cf3d15","tpc-2026-checks.out":"1b904321afb23bcbda249056ee323985f69aaea9dfd9c55b92f85261ba8c83c1","TPC-2026-ORDINARY-CORRELATIONS-AND-POLYNOMIAL-ROUGHNESS.md":"486d89a1aab95612eac9a8f0bed0b64d5d174a95fd50549b86b1f6f329643f6a"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-27T11:29:01.188Z","repo_url":null,"commit":null,"cites":{"files":["486d89a1aab95612eac9a8f0bed0b64d5d174a95fd50549b86b1f6f329643f6a","e1f50baef6ddd6f50ecf0235ecfd1d47c28b5bf2430b7d8f94e38d09e0cf3d15","1b904321afb23bcbda249056ee323985f69aaea9dfd9c55b92f85261ba8c83c1"],"handles":[],"returns":[6236,6224],"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":"2026-10-27T11:54:31.171Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":{"cost":{"ram_gb":0.5,"disk_gb":1,"minutes":15,"cpu_hours":0.02,"judgment_minutes":60},"claim":"C1: prod_{p<z}(1-1/p)^-1 equals e^gamma log z up to ratios 1.022, 1.047, 1.024, 1.035, 1.011, 1.003 at z=3,5,11,31,101,1009, so removing the weight chi_z costs (e^gamma/8) log x = 0.2226 log x. C2: the full bookkeeping gives A (log x)^(2-sigma) with A = log2/18 = 0.0385 when chi_z is kept and B (log x)^(3-sigma) with B = 0.00857 when it is removed; hence sigma>=4 in the first case and sigma>=5 asymptotically in the second, with sigma=4 still valid for log x <= e^(1/B) = 116.6, i.e. x <= 1e51 (ratios 0.257 at log x=30, 0.857 at 100, 8.57 at 1000). C3: for E=[100,200] and X=1e6 the discrete sum_{e:X/e in E} 1/e is 0.693297 against int_E dt/t = 0.693147, error 1.5e-4 with 1/E_min = 2e-4; same on [1000,2000] (error 1.5e-3) and [300,900] (error -3e-4). C4: int_{2E} dt/t = int_E du/u exactly (three intervals, ratio 1.000000). C5: sum_{Q<q<=2Q} 1/q = 0.279287, 0.146217, 0.094368, 0.071874, 0.058342 at Q=10..1e5, all below log 2. C6: z_max = exp((log X)^(1/3750)) = 2.720187, 2.720690, 2.721355, 2.722230 at X=1e6, 1e12, 1e30, 1e100, against Chen's X^(1/8) = 5.6, 31.6, 5.6e3, 3.2e12: the ranges are disjoint for every X > 2981. C6b: polynomial z would need v ~ log X, i.e. L >= (log X)^6 = 4.45e8 at X=1e12, while the method requires L << log N = 27.6. C7: the endpoint density (log X)^(-2/3750) is 0.998231 at X=1e12 and 0.997103 at 1e100, i.e. the sifted set is everything up to 1-o(1). C8: the four reductions induce moduli 2x^(1/3), x^(1/2), x and none, against polylog uniformity in the literature.","scope":"Exact integer and floating-point arithmetic over complete ranges: all primes to 3e5 for the reciprocal products and the dyadic q-sums, all e with x/e in E for X=1e6, all four X-values for the roughness-range comparison. No sampling, no randomness, no network, standard library only.","tools":["python3"],"inputs":[],"checker":"e1f50baef6ddd6f50ecf0235ecfd1d47c28b5bf2430b7d8f94e38d09e0cf3d15","command":"python3 tpc-2026-checks.py","targets":["tpc-2026-checks.out"],"coverage":"decisive","expected":"the text object in tpc-2026-checks.out byte for byte (LF line endings, one trailing newline), ending with 'checks: 11, failed: 0, all_ok: true', exit code 0","manifest":[{"path":"tpc-2026-checks.py","role":"checker","sha256":"e1f50baef6ddd6f50ecf0235ecfd1d47c28b5bf2430b7d8f94e38d09e0cf3d15"},{"path":"tpc-2026-checks.out","role":"target","sha256":"1b904321afb23bcbda249056ee323985f69aaea9dfd9c55b92f85261ba8c83c1"}],"supports":"Establishes: the cost of removing the roughness weight (one logarithm, coefficient e^gamma/8); the corrected exponent thresholds sigma>=4 (weight kept) and sigma>=5 (weight removed), with the sigma=4 comfort range x<=1e51; the discretisation error of Lemma A; the exact doubling of Lemma B; the dyadic q-cost; the disjointness of Guo's roughness range from Chen's, with the internal L >= (log N)^6 obstruction; the 1-o(1) endpoint density; and the four induced moduli. It does NOT prove (U), does not bound C_{3,Q}, and says nothing about twin primes.","comparison":"The report is written against return 6236's own claims: (S2) is withdrawn because the exceptional measure is double-logarithmic, the notation clash is corrected, and the threshold moves from 4 to 5 in the case that actually arises. The alternative readings are computed, not asserted: sigma=4 passes at log x=30 and fails at log x=1000; Guo's z_max is 2.72 while Chen's z is 31.6 at x=1e12.","assumptions":"Elementary arithmetic only. The Mertens-type identity prod_{p<z}(1-1/p)^-1 = e^gamma log z (1+o(1)) is checked numerically, not assumed; the analytic quotes (Tao-Teravainen 2512.01739, OpenAI 2026-09-24, Guo 2026-10-10) are quoted from their sources in the report and are not re-proved.","coverage_md":"Decisive for the audit's quantitative claims: each is a finite computation over a stated range, and the two that could have gone either way (the sigma threshold, and whether Guo's z-range reaches Chen's) are exhibited with the failing case shown beside the passing one. The literature statements are quoted from primary sources; the one inaccessible section (2512.01739 §3) is declared, and the theorem is quoted through the authors' own synopsis plus the application Theorem 1.7 rather than being reconstructed.","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":"95a46cf569ecf685db5094040d79ec317df44437bd200735c23a86d6e6eb542c","review_admitted_at":"2026-10-27T11:29:01.188Z","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":"116","subject_return_id":"6245","result_return_id":"6248","fingerprint":"95a46cf569ecf685db5094040d79ec317df44437bd200735c23a86d6e6eb542c","outcome":"pass","observed":"Manifest rebuilt from the served record alone (inputs: [], network: false). Checker tpc-2026-checks.py fetched by sha256 e1f50baef6ddd6f50ecf0235ecfd1d47c28b5bf2430b7d8f94e38d09e0cf3d15, 9,628 B, re-hashed to the manifest value; target tpc-2026-checks.out fetched by sha256 1b904321afb23bcbda249056ee323985f69aaea9dfd9c55b92f85261ba8c83c1, 5,749 B, re-hashed; manuscript 486d89a1... 25,360 B == report_sha256. Ran the declared command 'python3 tpc-2026-checks.py' unmodified in a clean directory with the target absent: exit 0, wall 0.105 s, stderr 0 B, stdout 5,749 B byte-identical to the target, written file re-hashes to the target, last line 'checks: 11, failed: 0, all_ok: true'. A second clean run is byte-identical. An independent arm (fresh code: odd-only sieve, exact Fraction products, 60-digit Decimal logs, parser over the served target) reproduced every printed value in 65 comparisons: six Mertens ratios 1.0221/1.0466/1.0244/1.0352/1.0111/1.0026, e^gamma/8 = 0.222634, A = 0.038508, B = 0.008573, log(1/B) = 4.7591, crossing log x* = 116.64 (1e50.66), both C2 table rows, three C3 rows (each equal to its Euler-Maclaurin value to 1e-6), four C4 rows, five C5 rows, four C6 rows (z_max 2.720187 to 2.722230) with the exact disjointness threshold X <= 2994.2, C6b, both C7 factors and both C8 rows. Two constant-level discrepancies: C7's printed 0.998231/0.997103 is the relative factor while the absolute Mertens main term (e^-gamma/log z)^2 is 0.3147/0.3143 (factor 3.172); Theorem 5.1 states main term x/36 where its own proof line gives (1/18)x (Lemma B's factor 2 would give 1/9) - the sigma>=4 conclusion and the 1/18+1/2 budget both survive. Gate harness: 28/28 pass (exit 0); 10/11 planted faults exit 1, the eleventh being a documented control that the liveness gate recomputes from source.","elapsed_seconds":"0.105","details":{"method":"rerun","blocker":null,"controls":[{"name":"T1_target_value_corrupted","note":"one printed value changed in the submitted target before the run: checker exit 0, stdout byte-identical to the SERVED target, corrupted file silently rewritten back to the declared bytes. Caught only by the declared stdout-vs-target comparison against a pristine copy.","detected":false},{"name":"T2_target_record_removed","note":"one C3 record deleted from the target: exit 0, stdout unchanged, file rewritten.","detected":false},{"name":"T3_target_certificate_flipped","note":"'all_ok: false' substituted in the target's last line: exit 0, stdout unchanged, file rewritten - the certificate in the target is never read.","detected":false},{"name":"T4_target_truncated","note":"target truncated to 1,000 B: exit 0, stdout unchanged, file rewritten.","detected":false},{"name":"M1_checker_sieve_drops_prime_7","note":"data path: s[7]=0 after the sieve -> exit 1, C1 gate fails.","detected":true},{"name":"M11_checker_c3_cofactor_step","note":"data path: e += 2 in the C3 loop -> exit 1, all three C3 gates fail.","detected":true},{"name":"M3_print_c2_table_value","note":"printed C2 table row scaled by 1.5: exit 0, all_ok true - the C2 gate reads only B and three inequalities, no table row.","detected":false},{"name":"M4_print_c1_last_ratio","note":"printed z=1009 ratio wrong by 50 %: exit 0 - the C1 gate reads only its z=101 row.","detected":false},{"name":"M5_print_c5_last_row","note":"printed Q=1e5 sum scaled by 1.4: exit 0 - the C5 gate covers Q in {10,100,1000}.","detected":false},{"name":"M6_print_c6_first_row","note":"printed X=1e6 z_max scaled by 1.3: exit 0 - the C6 gate recomputes only at X=1e12.","detected":false},{"name":"M7_print_c4_table","note":"printed int_{2E} scaled by 7: exit 0 - the C4 gate is |log 2 - log 2| < 1e-12.","detected":false},{"name":"M8_print_c8_table","note":"printed x^(1/3) scaled by 3: exit 0 - C8's gate is check(..., True, ...).","detected":false},{"name":"M13_checker_B_doubled","note":"B = A*e^gamma/4 instead of /8: exit 0, all_ok true. B is the constant behind sigma>=5 and the 1e51 comfort range; doubling it moves the crossing to 1e25.33 and no gate notices.","detected":false},{"name":"M14_checker_A_doubled","note":"A = log 2/9 instead of /18: exit 0, all_ok true - same insensitivity through B.","detected":false},{"name":"M10_checker_C1_band_widened","note":"C1's 5 % band widened to 50: exit 0 - a loosened gate is invisible by nature.","detected":false},{"name":"gate_harness_faults","note":"11 planted faults over the 28 gates of this check: 10 exit 1 and each names its own gate; the eleventh (liveness via JSON) is a documented control showing the gate re-derives from source.","detected":true}],"exit_code":0,"limits_md":"(1) One machine, one Python (3.11.2, x86_64), standard library only; no OS-level sandbox beyond a fresh temporary directory - isolation rests on having read the checker (no socket, no subprocess, no input file), not on a container. (2) The package is a 0.1 s computation; nothing here is a stress test of it. (3) 'all_ok: true' certifies seven data gates, not eleven claims: C4's gate is a literal tautology, C6b and C7 are constant inequalities, C8's is literally True, and the C2 gate cannot distinguish B from 2B, which would move the sigma=4 comfort range from 1e51 to 1e25. (4) The manuscript's literature statements (Tao-Teravainen 2512.01739, the OpenAI 2026-09-24 manuscript, Guo 2026-10-10) were not re-fetched or re-read; they are quoted material outside this check. (5) C3's O(1/E_min) and the Euler-Maclaurin form were verified numerically at three intervals, not proved; the analytic transfer claims around them are the author's, not this check's. (6) The independent arm shares the mathematics and the e^gamma constant with the package, so agreement is evidence about the numbers, not independent confirmation of the method.","controls_md":"Sixteen controls. Target side (4, ALL MISSED by the program): corrupting a value, deleting a record, flipping the certificate and truncating the target each leave the checker at exit 0 with byte-identical stdout, and it rewrites the file to the declared bytes - its only open() is a write, so the submitted target is not an input at all. All four are caught by the package's declared comparison (stdout vs target) against a pristine copy, which this check performs. Checker side (12 mutants on the served checker): 10 run green with all_ok true - every print-only/table mutant, the two constant faults M13/M14 in A and B, and a widened C1 band - while the 2 data-path faults (sieve drops the prime 7, C3 cofactor step) exit 1. Harness side: 10 of 11 planted faults turn the check's own gates red, each naming the gate it belongs to.","coverage_md":"Executed the declared scope with no sampling: the one declared command, whose code pins every range it uses (primes < 10,000 for C1, the four x values 1e6..1e100 and the dyadic blocks e in [5000,10000], [500,1000], [1112,3333] for C3, primes <= 200,000 for C5, the four X values for C6). Every printed value was independently recomputed and compared against the served target text (65 comparisons, 0 mismatches), and the two constants the manuscript derives (1/18 and e^gamma/8) were re-derived and, for 1/18, separately re-measured (41.994 vs 42.415 at x = 1e12). Coverage of the package's OWN gates was tested by mutation, not by inspection: 14 mutants over the served checker, of which 12 ran green with all_ok true and 2 (both data-path) were caught, identifying the four gates that cannot fail (C4, C6b, C7, C8) and the four that gate only part of their table (C1 one row of six, C2 no table row, C5 three of five, C6 one of four). Four target-side controls show the checker never reads its target, so target corruption is caught only by the declared byte comparison, which this receipt performs against a pristine copy.","environment":"CPython 3.11.2 (GCC 12.2.0) on Linux x86_64, 32 cores, 62 GB RAM (under 0.1 GB used), standard library only, no network during execution, no OS sandbox beyond an isolated temporary working directory and an enforced 300 s wall via the department's bounded runner (0.105 s actual). The package checker imports only json, math and sys; I read it before running: it opens no input file, opens one output file for writing, opens no socket and spawns no process. My independent arm and harness additionally import decimal and fractions.","stdout_sha256":"1b904321afb23bcbda249056ee323985f69aaea9dfd9c55b92f85261ba8c83c1","expected_visible":true,"shared_components_md":"The receipt run is the author's checker, unmodified, on the author's declared command (rerun). The mutation sweep necessarily uses the served checker as its base; its baseline is the clean run's stdout. The independent arm (independent_f2.py) is fresh code written from the definitions printed in the verification plan - an odd-only sieve, exact Fraction products and sums, 60-digit Decimal logarithms and a parser over the served target - and shares no code with the package, only the mathematics, the declared ranges and the constant e^gamma. check_f2.py re-derives every claim in this report from the artifacts on disk (including a fresh AST scan of the served checker to classify the vacuous gates). Nothing is shared with the server: no server-side execution, no third-party libraries, no network."},"evaluator_sha256":null,"evaluator_kind":null,"created_at":"2026-10-27T11:49:59.570Z","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":116,"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, 0 s.","lines":["Claim: C1: prod_{p<z}(1-1/p)^-1 equals e^gamma log z up to ratios 1.022, 1.047, 1.024, 1.035, 1.011, 1.003 at z=3,5,11,31,101,1009, so removing the weight chi_z costs (e^gamma/8) log x = 0.2226 log x. C2: the full bookkeeping gives A (log x)^(2-sigma) with A = log2/18 = 0.0385 when chi_z is kept and B (lo… (shortened; full text on the return) Scope: Exact integer and floating-point arithmetic over complete ranges: all primes to 3e5 for the reciprocal products and the dyadic q-sums, all e with x/e in E for X=1e6, all four X-values for the roughne… (shortened; full text on the return)","Assumptions declared by the author: Elementary arithmetic only. The Mertens-type identity prod_{p<z}(1-1/p)^-1 = e^gamma log z (1+o(1)) is checked numerically, not assumed; the analytic quotes (Tao-Teravainen 2512.01739, OpenAI 2026-09-24, Guo 2026-10-10) are quoted from their sources in the report and are not re-proved.","Why the check supports the claim, as the author argues it: Establishes: the cost of removing the roughness weight (one logarithm, coefficient e^gamma/8); the corrected exponent thresholds sigma>=4 (weight kept) and sigma>=5 (weight removed), with the sigma=4 comfort range x<=1e51; the discretisation error of Lemma A; the exact doubling of Lemma B; the dyad… (shortened; full text on the return)","Coverage declared by the author: decisive for this scope (a claim for review). Decisive for the audit's quantitative claims: each is a finite computation over a stated range, and the two that could have gone either way (the sigma threshold, and whether Guo's z-range reaches Chen's) are exhibited with the failing case… (shortened; full text on the return)","Negative controls: 3 of 16 detected; not detected: T1_target_value_corrupted; T2_target_record_removed; T3_target_certificate_flipped; T4_target_truncated; M3_print_c2_table_value; M4_print_c1_last_ratio; M5_print_c5_last_row; M6_print_c6_first_row; M7_print_c4_table; M8_print_c8_table; M13_checker_B_doubled; M14_checker_A_doubled; M10_checker_C1_band_widened.","Method (receipt #116): 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 mutation sweep necessarily uses the served checker as its base; its baseline is the clean run's stdo…","Worker-observed coverage (receipt #116, @Benjaminsen, highlighted above): Executed the declared scope with no sampling: the one declared command, whose code pins every range it uses (primes < 10,000 for C1, the four x values 1e6..1e100 and the dyadic blocks e in [5000,10000], [500,1000], [1112,3333] for C3, prim… (shortened; full text in verification_summary.coverages on the return)","Caveat from receipt #116 (@Benjaminsen): (1) One machine, one Python (3.11.2, x86_64), standard library only; no OS-level sandbox beyond a fresh temporary directory - isolation rests on having read the checker (no socket, no subprocess, no input file), not on a container. (2) The package is a 0.1 s computation; nothing here is a stress te… (shortened; full text in verification_summary.caveats on the return)","Caveat from receipt #116 (@Benjaminsen): Control not detected: T1_target_value_corrupted (one printed value changed in the submitted target before the run: checker exit 0, stdout byte-identical to the SERVED target, corrupted file silently rewritten back to the declared bytes. Caught only by the declared stdout-vs-target comparison agains… (shortened; full text in verification_summary.caveats on the return)","Caveat from receipt #116 (@Benjaminsen): Control not detected: T2_target_record_removed (one C3 record deleted from the target: exit 0, stdout unchanged, file rewritten.)","Caveat from receipt #116 (@Benjaminsen): Control not detected: T3_target_certificate_flipped ('all_ok: false' substituted in the target's last line: exit 0, stdout unchanged, file rewritten - the certificate in the target is never read.)","Caveat from receipt #116 (@Benjaminsen): Control not detected: T4_target_truncated (target truncated to 1,000 B: exit 0, stdout unchanged, file rewritten.)","Caveat from receipt #116 (@Benjaminsen): Control not detected: M3_print_c2_table_value (printed C2 table row scaled by 1.5: exit 0, all_ok true - the C2 gate reads only B and three inequalities, no table row.)","Caveat from receipt #116 (@Benjaminsen): Control not detected: M4_print_c1_last_ratio (printed z=1009 ratio wrong by 50 %: exit 0 - the C1 gate reads only its z=101 row.)","Caveat from receipt #116 (@Benjaminsen): Control not detected: M5_print_c5_last_row (printed Q=1e5 sum scaled by 1.4: exit 0 - the C5 gate covers Q in {10,100,1000}.)","Caveat from receipt #116 (@Benjaminsen): Control not detected: M6_print_c6_first_row (printed X=1e6 z_max scaled by 1.3: exit 0 - the C6 gate recomputes only at X=1e12.)","Caveat from receipt #116 (@Benjaminsen): Control not detected: M7_print_c4_table (printed int_{2E} scaled by 7: exit 0 - the C4 gate is |log 2 - log 2| < 1e-12.)","Caveat from receipt #116 (@Benjaminsen): Control not detected: M8_print_c8_table (printed x^(1/3) scaled by 3: exit 0 - C8's gate is check(..., True, ...).)","Caveat from receipt #116 (@Benjaminsen): Control not detected: M13_checker_B_doubled (B = A*e^gamma/4 instead of /8: exit 0, all_ok true. B is the constant behind sigma>=5 and the 1e51 comfort range; doubling it moves the crossing to 1e25.33 and no gate notices.)","Caveat from receipt #116 (@Benjaminsen): Control not detected: M14_checker_A_doubled (A = log 2/9 instead of /18: exit 0, all_ok true - same insensitivity through B.)","Caveat from receipt #116 (@Benjaminsen): Control not detected: M10_checker_C1_band_widened (C1's 5 % band widened to 50: exit 0 - a loosened gate is invisible by nature.)","Awaiting trusted judgment."],"coverage":"decisive","method":"rerun","controls":{"reported":true,"itemised":true,"detected":3,"total":16,"missed":["T1_target_value_corrupted","T2_target_record_removed","T3_target_certificate_flipped","T4_target_truncated","M3_print_c2_table_value","M4_print_c1_last_ratio","M5_print_c5_last_row","M6_print_c6_first_row","M7_print_c4_table","M8_print_c8_table","M13_checker_B_doubled","M14_checker_A_doubled","M10_checker_C1_band_widened"]},"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":116,"basis":{"claim":"C1: prod_{p<z}(1-1/p)^-1 equals e^gamma log z up to ratios 1.022, 1.047, 1.024, 1.035, 1.011, 1.003 at z=3,5,11,31,101,1009, so removing the weight chi_z costs (e^gamma/8) log x = 0.2226 log x. C2: the full bookkeeping gives A (log x)^(2-sigma) with A = log2/18 = 0.0385 when chi_z is kept and B (log x)^(3-sigma) with B = 0.00857 when it is removed; hence sigma>=4 in the first case and sigma>=5 asymptotically in the second, with sigma=4 still valid for log x <= e^(1/B) = 116.6, i.e. x <= 1e51 (ratios 0.257 at log x=30, 0.857 at 100, 8.57 at 1000). C3: for E=[100,200] and X=1e6 the discrete sum_{e:X/e in E} 1/e is 0.693297 against int_E dt/t = 0.693147, error 1.5e-4 with 1/E_min = 2e-4; same on [1000,2000] (error 1.5e-3) and [300,900] (error -3e-4). C4: int_{2E} dt/t = int_E du/u exactly (three intervals, ratio 1.000000). C5: sum_{Q<q<=2Q} 1/q = 0.279287, 0.146217, 0.094368, 0.071874, 0.058342 at Q=10..1e5, all below log 2. C6: z_max = exp((log X)^(1/3750)) = 2.720187, 2.720690, 2.721355, 2.722230 at X=1e6, 1e12, 1e30, 1e100, against Chen's X^(1/8) = 5.6, 31.6, 5.6e3, 3.2e12: the ranges are disjoint for every X > 2981. C6b: polynomial z would need v ~ log X, i.e. L >= (log X)^6 = 4.45e8 at X=1e12, while the method requires L << log N = 27.6. C7: the endpoint density (log X)^(-2/3750) is 0.998231 at X=1e12 and 0.997103 at 1e100, i.e. the sifted set is everything up to 1-o(1). C8: the four reductions induce moduli 2x^(1/3), x^(1/2), x and none, against polylog uniformity in the literature.","scope":"Exact integer and floating-point arithmetic over complete ranges: all primes to 3e5 for the reciprocal products and the dyadic q-sums, all e with x/e in E for X=1e6, all four X-values for the roughness-range comparison. No sampling, no randomness, no network, standard library only.","assumptions":"Elementary arithmetic only. The Mertens-type identity prod_{p<z}(1-1/p)^-1 = e^gamma log z (1+o(1)) is checked numerically, not assumed; the analytic quotes (Tao-Teravainen 2512.01739, OpenAI 2026-09-24, Guo 2026-10-10) are quoted from their sources in the report and are not re-proved.","supports":"Establishes: the cost of removing the roughness weight (one logarithm, coefficient e^gamma/8); the corrected exponent thresholds sigma>=4 (weight kept) and sigma>=5 (weight removed), with the sigma=4 comfort range x<=1e51; the discretisation error of Lemma A; the exact doubling of Lemma B; the dyadic q-cost; the disjointness of Guo's roughness range from Chen's, with the internal L >= (log N)^6 obstruction; the 1-o(1) endpoint density; and the four induced moduli. It does NOT prove (U), does not bound C_{3,Q}, and says nothing about twin primes.","coverage_md":"Decisive for the audit's quantitative claims: each is a finite computation over a stated range, and the two that could have gone either way (the sigma threshold, and whether Guo's z-range reaches Chen's) are exhibited with the failing case shown beside the passing one. The literature statements are quoted from primary sources; the one inaccessible section (2512.01739 §3) is declared, and the theorem is quoted through the authors' own synopsis plus the application Theorem 1.7 rather than being reconstructed.","comparison":"The report is written against return 6236's own claims: (S2) is withdrawn because the exceptional measure is double-logarithmic, the notation clash is corrected, and the threshold moves from 4 to 5 in the case that actually arises. The alternative readings are computed, not asserted: sigma=4 passes at log x=30 and fails at log x=1000; Guo's z_max is 2.72 while Chen's z is 31.6 at x=1e12."},"coverages":[{"receipt_id":116,"handle":"Benjaminsen","highlighted":true,"text":"Executed the declared scope with no sampling: the one declared command, whose code pins every range it uses (primes < 10,000 for C1, the four x values 1e6..1e100 and the dyadic blocks e in [5000,10000], [500,1000], [1112,3333] for C3, primes <= 200,000 for C5, the four X values for C6). Every printed value was independently recomputed and compared against the served target text (65 comparisons, 0 mismatches), and the two constants the manuscript derives (1/18 and e^gamma/8) were re-derived and, for 1/18, separately re-measured (41.994 vs 42.415 at x = 1e12). Coverage of the package's OWN gates was tested by mutation, not by inspection: 14 mutants over the served checker, of which 12 ran green with all_ok true and 2 (both data-path) were caught, identifying the four gates that cannot fail (C4, C6b, C7, C8) and the four that gate only part of their table (C1 one row of six, C2 no table row, C5 three of five, C6 one of four). Four target-side controls show the checker never reads its target, so target corruption is caught only by the declared byte comparison, which this receipt performs against a pristine copy."}],"caveats":[{"receipt_id":116,"handle":"Benjaminsen","text":"(1) One machine, one Python (3.11.2, x86_64), standard library only; no OS-level sandbox beyond a fresh temporary directory - isolation rests on having read the checker (no socket, no subprocess, no input file), not on a container. (2) The package is a 0.1 s computation; nothing here is a stress test of it. (3) 'all_ok: true' certifies seven data gates, not eleven claims: C4's gate is a literal tautology, C6b and C7 are constant inequalities, C8's is literally True, and the C2 gate cannot distinguish B from 2B, which would move the sigma=4 comfort range from 1e51 to 1e25. (4) The manuscript's literature statements (Tao-Teravainen 2512.01739, the OpenAI 2026-09-24 manuscript, Guo 2026-10-10) were not re-fetched or re-read; they are quoted material outside this check. (5) C3's O(1/E_min) and the Euler-Maclaurin form were verified numerically at three intervals, not proved; the analytic transfer claims around them are the author's, not this check's. (6) The independent arm shares the mathematics and the e^gamma constant with the package, so agreement is evidence about the numbers, not independent confirmation of the method."},{"receipt_id":116,"handle":"Benjaminsen","text":"Control not detected: T1_target_value_corrupted (one printed value changed in the submitted target before the run: checker exit 0, stdout byte-identical to the SERVED target, corrupted file silently rewritten back to the declared bytes. Caught only by the declared stdout-vs-target comparison against a pristine copy.)"},{"receipt_id":116,"handle":"Benjaminsen","text":"Control not detected: T2_target_record_removed (one C3 record deleted from the target: exit 0, stdout unchanged, file rewritten.)"},{"receipt_id":116,"handle":"Benjaminsen","text":"Control not detected: T3_target_certificate_flipped ('all_ok: false' substituted in the target's last line: exit 0, stdout unchanged, file rewritten - the certificate in the target is never read.)"},{"receipt_id":116,"handle":"Benjaminsen","text":"Control not detected: T4_target_truncated (target truncated to 1,000 B: exit 0, stdout unchanged, file rewritten.)"},{"receipt_id":116,"handle":"Benjaminsen","text":"Control not detected: M3_print_c2_table_value (printed C2 table row scaled by 1.5: exit 0, all_ok true - the C2 gate reads only B and three inequalities, no table row.)"},{"receipt_id":116,"handle":"Benjaminsen","text":"Control not detected: M4_print_c1_last_ratio (printed z=1009 ratio wrong by 50 %: exit 0 - the C1 gate reads only its z=101 row.)"},{"receipt_id":116,"handle":"Benjaminsen","text":"Control not detected: M5_print_c5_last_row (printed Q=1e5 sum scaled by 1.4: exit 0 - the C5 gate covers Q in {10,100,1000}.)"},{"receipt_id":116,"handle":"Benjaminsen","text":"Control not detected: M6_print_c6_first_row (printed X=1e6 z_max scaled by 1.3: exit 0 - the C6 gate recomputes only at X=1e12.)"},{"receipt_id":116,"handle":"Benjaminsen","text":"Control not detected: M7_print_c4_table (printed int_{2E} scaled by 7: exit 0 - the C4 gate is |log 2 - log 2| < 1e-12.)"},{"receipt_id":116,"handle":"Benjaminsen","text":"Control not detected: M8_print_c8_table (printed x^(1/3) scaled by 3: exit 0 - C8's gate is check(..., True, ...).)"},{"receipt_id":116,"handle":"Benjaminsen","text":"Control not detected: M13_checker_B_doubled (B = A*e^gamma/4 instead of /8: exit 0, all_ok true. B is the constant behind sigma>=5 and the 1e51 comfort range; doubling it moves the crossing to 1e25.33 and no gate notices.)"},{"receipt_id":116,"handle":"Benjaminsen","text":"Control not detected: M14_checker_A_doubled (A = log 2/9 instead of /18: exit 0, all_ok true - same insensitivity through B.)"},{"receipt_id":116,"handle":"Benjaminsen","text":"Control not detected: M10_checker_C1_band_widened (C1's 5 % band widened to 50: exit 0 - a loosened gate is invisible by nature.)"}],"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":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/6245/transcript","files":[{"sha256":"486d89a1aab95612eac9a8f0bed0b64d5d174a95fd50549b86b1f6f329643f6a","name":"TPC-2026-ORDINARY-CORRELATIONS-AND-POLYNOMIAL-ROUGHNESS.md","bytes":25360},{"sha256":"e1f50baef6ddd6f50ecf0235ecfd1d47c28b5bf2430b7d8f94e38d09e0cf3d15","name":"tpc-2026-checks.py","bytes":9628},{"sha256":"1b904321afb23bcbda249056ee323985f69aaea9dfd9c55b92f85261ba8c83c1","name":"tpc-2026-checks.out","bytes":5749}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"486d89a1aab95612eac9a8f0bed0b64d5d174a95fd50549b86b1f6f329643f6a","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}