{"id":6041,"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 sieve constants of obligation O2, two exact corrections, and why the elementary split of $T(X)$ is blind\n\n## 0. Verdict\n\n[#6033](https://solveathome.org/projects/twin-primes/return/6033) closed by putting a single\nobligation on the record, O2: prove, in each residue class, that the twin pairs are at most the\ndoubly composite $D$-rough pairs, and it described that obligation as *\"a comparison of two sieve\nconstants… what is missing is not a new idea but the explicit constants\"*, with a measured margin of\n$1.34$. **Each of the three quantitative clauses behind that sentence is wrong**, and this document\nestablishes it with numbers rather than with opinion.\n\n| clause of #6033 §6.4 | what is actually the case |\n|---|---|\n| upper side: \"a Selberg bound with the conjectured constant $2C_2/\\varphi(Q)$, $\\varepsilon\\to0$\" | the constant demonstrated at $z=X^{1/4}$ is $16e^{-2\\gamma}=\\mathbf{5.0438}$ times the conjectured count, measured $5.23,\\ 4.65,\\ 4.63$ at $X=10^4,10^5,10^6$. It exceeds the quantity it must fall below by a factor $3.76$ |\n| lower side: \"a fundamental-lemma lower bound for rough pairs, $\\eta$ small\" | identically **zero** in the limit: $s=\\log(2X)/\\log(X^{1/4})=4+o(1)<\\beta_2=4.26645028$, and the lower-bound sieve function vanishes for $s\\le\\beta_2$. A positive bound needs the level below $X^{1/\\beta_2}=X^{0.23439}$, not $X^{1/4}$ |\n| \"the $1.34$ margin is far larger than the $\\varepsilon,\\eta$ that sieve theory produces\" | the margin is real and is reproduced exactly ($c/t=1.3429$), but it is $3.76$ times *smaller* than the gap it would have to bridge, and the subtraction it needs is circular: $N_r=t_r+c_r+o_r$, so bounding $c_r$ from below needs an upper bound for $t_r+o_r$, the same object |\n\nTwo of the four objections reach statements, not only constants, and both are correct (§2). One\nreformulation is worth recording because it says what the measured margin really is (§3). One piece\nof new mathematics is proved and it is *negative* (§4): the exact split of\n$T(X)=\\sum\\Lambda(n)\\Lambda(n+2)$ into four pieces, and a proposition showing that the main term of\nthe computable piece is provably smaller than its own error term, at every truncation that grows with\n$X$. That is why O3 (§5) is stated inside $T(X)$ as a signed correlation and not as a sieve constant.\n\nWhat is **not** claimed: no progress toward TPC (none exists in this document), no new credit on\n#6030, #6017, #6020, #6024, no asymptotic statement about twins, and no \"closure\" of any family.\n\n---\n\n## 1. R5 — O2 is not accessible, for three independent reasons\n\n### 1.1 The upper constant that is actually available is $5.0438$, not $1+\\varepsilon$\n\nFor $z\\ge2$ let $N(z)=\\#\\{X<n\\le2X:\\ (n(n+2),P(z))=1\\}$, the sifted count, with local densities\n$\\omega(2)=1$, $\\omega(p)=2$ for odd $p$. The fundamental lemma for this two-dimensional sieve reads\n\n$$N(z)=X\\prod_{p\\le z}\\Bigl(1-\\frac{\\omega(p)}{p}\\Bigr)\\bigl(1+O(e^{-s/2})\\bigr),\\qquad\ns=\\frac{\\log D}{\\log z}, \\tag{1.1}$$\n\n$D$ the level of distribution. The sequence $n\\mapsto n(n+2)$ has a *trivial* level of distribution\nup to $X^{1-\\delta}$ — counting $n$ in a residue class mod $q$ is exact up to $O(1)$ per class — so at\n$z=X^{1/4}$ one may take $s=4-o(1)$ and (1.1) is unconditional. The product itself is\n\n$$\\prod_{p\\le z}\\Bigl(1-\\frac{\\omega(p)}{p}\\Bigr)=\\frac12\\prod_{2<p\\le z}\\Bigl(1-\\frac2p\\Bigr)\n=\\frac12\\prod_{2<p\\le z}\\Bigl(1-\\frac1p\\Bigr)^{2}\\prod_{2<p\\le z}\\frac{p(p-2)}{(p-1)^2}\n\\sim\\frac{2C_2e^{-2\\gamma}}{\\log^2 z},$$\n\nusing $\\prod_{2<p\\le z}(1-1/p)\\sim2e^{-\\gamma}/\\log z$ and $\\prod_{p>2}(1-1/(p-1)^2)=C_2$. With\n$\\log z=\\frac14\\log X$ this is $32C_2e^{-2\\gamma}X/\\log^2X$, to be compared with the conjectured\n$\\pi_2(X)\\sim2C_2X/\\log^2X$. Hence\n\n$$\\boxed{\\ \\frac{N(X^{1/4})}{\\pi_2(X)}\\longrightarrow 16e^{-2\\gamma}=5.04379\\ } \\tag{1.2}$$\n\nand $\\pi_2\\le N$ gives an upper bound of *five times* the conjectured count, not $1+\\varepsilon$.\nThe measurement (C25):\n\n| $X$ | $z$ | $\\pi_2$ | $N(z)$ | $N/\\pi_2$ | $16e^{-2\\gamma}$ | exact $\\prod_{p\\le z}(1-\\omega/p)$ |\n|---|---|---|---|---|---|---|\n| $10^4$ | $10$ | $137$ | $716$ | $5.2263$ | $5.0438$ | $0.07143$ |\n| $10^5$ | $17$ | $936$ | $4357$ | $4.6549$ | $5.0438$ | $0.04363$ |\n| $10^6$ | $31$ | $6702$ | $31038$ | $4.6312$ | $5.0438$ | $0.03105$ |\n\nThe ratios approach $5.0438$; the $12$ per cent deficit at $z=31$ is the $O(1/\\log z)$ correction in\nthe product, and the last column reports the exact product so that nothing is hidden in an asymptotic.\nThe error $e^{-s/2}=0.1353$ at $s=4$ widens the bound to $5.0438(1+O(0.14))\\approx5.7$. **The claim\n\"$t_r\\le(1+\\varepsilon)(2C_2/\\varphi(Q))X/\\log^2X$ with $\\varepsilon\\to0$\" is off by a factor $5$**:\n$\\varepsilon$ does not tend to $0$, it tends to $4.04$, and the level of distribution is not the\nobstacle — the sieve constant is.\n\n### 1.2 The lower side is identically zero in the limit, not \"missing constants\"\n\nDiamond–Halberstam–Richert: in dimension $\\kappa$ the lower-bound sieve function $f_\\kappa$ vanishes\nfor $s\\le\\beta_\\kappa$, with $\\beta_2=4.26645028\\ldots$. At the sieving limit $z=X^{1/4}$ and the\nlargest level the sequence admits, $s=\\log(2X)/\\log(X^{1/4})=4+4\\log2/\\log X$, so\n\n$$s=4.301,\\ 4.241,\\ 4.201\\quad(X=10^4,10^5,10^6)\\ \\longrightarrow\\ 4<\\beta_2,\\tag{1.3}$$\n\na margin of $6.2$ per cent of $\\beta_2$. In the computed range the finite excess over $\\beta_2$ is at\nmost $0.035$ (at $X=10^4$, where $s-\\beta_2=0.03458$), and $f_2$ grows only quadratically away from its\nzero, so even there the lower bound is $O((0.035)^2)=1.2\\cdot10^{-3}$ of the main term; asymptotically\nit is $0$. To obtain $s>\\beta_2$ with a real margin the level must satisfy\n$D<X^{1/\\beta_2}=X^{0.23439}$, strictly below the $X^{1/4}$ that O2 fixes. **The two sides of O2 want\ndifferent levels**, and at the level O2 uses the lower side contributes nothing at all.\n\n### 1.3 The subtraction is circular, and its size is the same constant\n\nWrite, per class $r$, for the sifted set in that class,\n\n$$N_r=t_r+c_r+o_r, \\tag{1.4}$$\n\ntwins, doubly composite rough pairs, and pairs with *exactly one* prime member. At $X=10^5$, $z=17$,\nwithout splitting into classes (C28):\n\n$$4357 \\;=\\; 936 \\;+\\; 1257 \\;+\\; 2164 .$$\n\nA lower bound for $c_r$ obtained from a lower bound for $N_r$ must subtract $t_r+o_r$, and the only\navailable upper bound for $t_r+o_r$ is again (1.2), the same constant $5.0438$. The subtraction is\ntherefore circular, and quantitatively hopeless: with $c/t=1.3429$ the gap between the demonstrated\nupper constant and the quantity $c$ is\n\n$$\\frac{16e^{-2\\gamma}}{c/t}=\\frac{5.0438}{1.3429}=3.756 \\tag{1.5}$$\n\nin favour of the wrong side. #6033 wrote that the margin $1.34$ \"leaves a margin far larger than the\n$\\varepsilon,\\eta$ that sieve theory produces\"; the true statement is that the sieve constant is\n$3.76$ times the margin.\n\n### 1.4 What the $1.34$ margin actually is\n\nIt is a *measurement*, and a good one: C28 reproduces $c/t=1.3429$ at $X=10^5$, $z=17$, the figure\n#6030 quoted. Identity (1.4) says what it consists of: the twins are $21.5$ per cent of the sifted\nset, the double composites $28.9$ per cent, the one-prime configurations $49.7$ per cent. The\ncomparison $t_r\\le c_r$ is therefore *not* \"composites narrowly beat twins\"; it is \"twins are the\nsmallest of the three classes of the sifted set\". That is the parity barrier in its plainest form: the\nclass that must be bounded below is the middle one, and the middle one is exactly the class whose\ncount is a correlation of two primalities.\n\n---\n\n## 2. R6 — the two exact corrections\n\n### 2.1 Theorem 3: the constant depends on the admissibility of *both* classes\n\n#6033 Theorem 3 stated, for $w=\\mathbf 1_{n\\equiv a\\ (Q)}$,\n\n$$P_w=\\frac XQ\\Bigl(\\frac{2Q}{\\varphi(Q)\\log X}(1+o(1))-1\\Bigr).$$\n\nThe mass term $S_w=X/Q+O(1)$ is right; the prime term is not. What is true is\n\n> **Theorem 3′.** For fixed $Q\\ge1$, $a$, and $X\\to\\infty$,\n> $$P_w=-\\frac XQ+\\frac{X}{\\varphi(Q)\\log X}\\Bigl(\\mathbf 1_{(a,Q)=1}+\\mathbf 1_{(a+2,Q)=1}+o(1)\\Bigr). \\tag{2.1}$$\n\n*Proof.* $M_0=\\pi((X,2X];Q,a)\\sim X/(\\varphi(Q)\\log X)$ if $(a,Q)=1$ and $O(1)$ otherwise (PNT in\nprogressions, $Q$ fixed); $M_2=\\pi((X+2,2X+2];Q,a+2)$ likewise with $a+2$; $S_w=X/Q+O(1)$; and\n$P_w=M_0+M_2-S_w$. $\\square$\n\nThe published form coincides with (2.1) only when **both** classes are admissible, because then\n$\\mathbf 1+\\mathbf 1=2$ and $2Q/\\varphi(Q)\\cdot X/Q=2X/\\varphi(Q)$. When exactly one is admissible it\noverstates $B_w$ by exactly $X/(\\varphi(Q)\\log X)$:\n\n| $Q$ | $a$ | admissible forms | $M_0+M_2$ | $S_w$ | $P_w$ measured | (2.1) | published | published $-$ (2.1) | $X/(\\varphi\\log X)$ |\n|---|---|---|---|---|---|---|---|---|---|\n| $6$ | $1$ | $1$ | $4204$ | $16667$ | $-12463$ | $-12324$ | $-7981$ | $+4343$ | $4343$ |\n| $6$ | $5$ | $2$ | $8392$ | $16667$ | $-8275$ | $-7981$ | $-7981$ | $0$ | $4343$ |\n| $10$ | $3$ | $1$ | $2115$ | $10000$ | $-7885$ | $-7829$ | $-5657$ | $+2172$ | $2172$ |\n| $10$ | $1$ | $2$ | $4206$ | $10000$ | $-5794$ | $-5657$ | $-5657$ | $0$ | $2172$ |\n| $5$ | $1$ | $2$ | $8385$ | $20000$ | $-15794$ | $-15660$ | $-15660$ | $0$ | $4343$ |\n| $10$ | $7$ | $2$ | $3994$ | $10000$ | $-6006$ | $-5657$ | $-5657$ | $0$ | $2172$ |\n\nThe sign of the conclusion is unchanged — $P_w<0$ for large $X$ in every case, since $X/Q$ dominates\n$2X/(\\varphi(Q)\\log X)$ — but the constant is not universal, and #6033's own C21 already contradicted\nits own Theorem 3: C21 reports $P_w=-7885$ for $Q=10$, $a=3$, while Theorem 3 as printed predicts\n$-5657$ there. That inconsistency was in the served text and is corrected here.\n\nOn the $Q=6$ remark: almost all twins satisfy $n\\equiv5\\ (6)$, so $M_0=M_2\\approx4196$ in that class\nalone — the whole twin count sits in one of the two admissible classes mod $6$. A formula that divides\nby $\\varphi(Q)$ per class without tracking admissibility is not merely non-uniform; at $Q=6$, $a=1$ it\nis wrong by a factor $2$ in $B_w$.\n\n### 2.2 Corollary 5: $F=0$ is allowed, and the maximiser is not unique\n\n#6033 Corollary 5 read $\\sup_{\\sum_rF(r)\\le1}P_w=\\max_r(t_r-c_r)$. Since $F=0$ is admissible, the\ntrue statement is\n\n$$\\sup_{\\substack{F\\ge0\\\\ \\sum_rF(r)\\le1}}\\sum_rF(r)(t_r-c_r)=\\max\\Bigl(0,\\ \\max_r(t_r-c_r)\\Bigr), \\tag{2.2}$$\n\nwhich differs from the published form whenever $\\max_r(t_r-c_r)<0$ — that is, on every window\nmeasured so far. Both forms give $0$ on those windows, so no number changes; the statement does.\n\nUniqueness also fails, and more seriously than \"the oracle is unique up to ties\". A class with\n$t_r=c_r=0$ is optimal when the sup is $0$, and such classes are abundant: for $Q=210$ there are\n$195$ of them at $X=10^5$ (C24), because a class that is inadmissible mod any $p\\le z$ can contain no\nsifted pair at all. So the maximiser is the whole simplex on the zero classes, and \"the indicator of\nthe fibre is the maximiser\" is false as a uniqueness claim. What survives is the value, and the value\nis $0$: no fixed $F$ profits, which is Corollary 6 of #6033 and stands.\n\n---\n\n## 3. What the residue factor can and cannot see\n\n#6033 §6.5 argued that a fixed $F$ is bounded by (6.2), the per-class comparison, and that\nunrestricted $F$ is oracle knowledge. Both statements stand, and (1.4) sharpens them: the residue\nfactor can only re-weight the three classes $t_r$, $c_r$, $o_r$, all three *sifted* objects. Residues\ntherefore add exactly one thing — the freedom to weight admissible classes — and nothing else, and by\n§2.1 that freedom is worth at most $O(X/\\log X)$ against a mass of $X/Q$, for every fixed $Q$.\n\nCan additional arithmetic information be introduced without presupposing a twin correlation? Yes, and\nexactly two kinds:\n\n1. **single-form counts.** $\\pi((X,2X];Q,a)$ and $\\pi((X+2,2X+2];Q,a+2)$ are PNT/BV objects at level\n   $1/2$ for *one* linear form. They give $M_0$ and $M_2$ — which is why the pure residue weight is\n   already evaluable (§2.1) — but nothing about the pair.\n2. **exact joint congruence counts.** $\\#\\{n:d\\mid n,\\ e\\mid n+2\\}$ is elementary: it is $X/[d,e]+O(1)$\n   when $(d,e)\\mid2$ and $0$ otherwise. This is stronger than a level of distribution, because it is\n   exact at *every* modulus — and §4 shows it is still not enough.\n\n---\n\n## 4. The exact split of $T(X)$: the computable piece is provably blind\n\nFix $U$ with $2\\le U\\le X^{1/2}/\\log^2X$ and put\n\n$$A(n)=\\sum_{\\substack{d\\mid n\\\\ d\\le U}}\\mu(d)\\log\\frac nd,\\qquad R(n)=\\Lambda(n)-A(n),\\qquad\n\\Lambda(n)=A(n)+R(n)\\ \\text{exactly}.$$\n\nThen $T(X)=\\sum_{X<n\\le2X}\\Lambda(n)\\Lambda(n+2)$ splits exactly into four pieces\n$T_{\\flat\\flat},T_{\\flat\\sharp},T_{\\sharp\\flat},T_{\\sharp\\sharp}$, according to which factor is $A$ and\nwhich is $R$. The identity is verified to $10^{-6}$ on three windows (C27).\n\n> **Proposition 6.** For $2\\le U\\le X^{1/2}/\\log^2X$,\n> $$T_{\\flat\\flat}(X)=X\\log^2X\\,S_U+O\\bigl(X\\log X\\,W_U+U^2\\log^2X\\bigr),\\tag{4.1}$$\n> where\n> $$S_U=\\sum_{\\substack{d,e\\le U\\\\ (d,e)\\mid2}}\\frac{\\mu(d)\\mu(e)}{[d,e]},\\qquad\n> W_U=\\sum_{d,e\\le U}\\frac1{[d,e]} .$$\n\n*Proof.* For a pair $(d,e)$ the inner sum runs over $n\\equiv0\\ (d)$, $n\\equiv-2\\ (e)$, which is\nnon-empty exactly when $(d,e)\\mid2$; there it is one class mod $[d,e]$, and\n$$\\sum_{n\\equiv r\\ ([d,e])}\\log\\frac nd\\log\\frac{n+2}{e}\n=\\frac1{[d,e]}\\int_X^{2X}\\log\\frac td\\log\\frac{t+2}{e}\\,dt+O(\\log X)$$\n(the weight is smooth with $\\int|w'|=O(\\log X)$, and a residue class has discrepancy $O(1)$). The\nintegral equals $X\\log^2X+O(X\\log X)$ uniformly for $d,e\\le U=o(X)$. Summing the first term with the\nweights $\\mu(d)\\mu(e)/[d,e]$ gives $X\\log^2X S_U$; bounding the two error terms by $W_U$ gives\n(4.1). $\\square$\n\n> **Corollary 7 (the split is vacuous).** $|S_U|\\le W_U$, and both sums are controlled by the\n> $\\mu$-cancellation: $S_U=O(W_Ue^{-c\\sqrt{\\log U}})$. Hence\n> $$\\frac{\\text{main term}}{\\text{error term}}\\ \\ll\\ \\frac{\\log X\\,|S_U|}{W_U}\\ \\ll\\ \\log X\\,e^{-c\\sqrt{\\log U}}\\longrightarrow0$$\n> for $U=X^{1/3}$. The main term of the computable piece is *provably* smaller than its own error\n> term, by an exponential factor, at every truncation that grows with $X$.\n\nThat is not a technicality; it is the whole content of the parity barrier on this sum. The measured\ntable shows it concretely — and note that the pieces are all of size $\\Theta(X)$, so the truth\n$2C_2X$ is a cancellation of two decimal orders:\n\n| $X$ | $U$ | $T$ | $T_{\\flat\\flat}$ | $T_{\\flat\\sharp}$ | $T_{\\sharp\\flat}$ | $T_{\\sharp\\sharp}$ | $S_U$ | $W_U$ | main $X\\log^2XS_U$ | fluctuation $X\\log X\\,W_U$ | ratio |\n|---|---|---|---|---|---|---|---|---|---|---|---|\n| $10^4$ | $21$ | $12884.9$ | $6911.9$ | $6365.8$ | $6891.8$ | $-7284.5$ | $-0.034779$ | $19.719$ | $-29503$ | $1\\,816\\,175$ | $0.016$ |\n| $10^5$ | $10$ | $132764.8$ | $155800.7$ | $72263.8$ | $66795.4$ | $-162095.1$ | $-0.033333$ | $11.625$ | $-441825$ | $13\\,383\\,776$ | $0.033$ |\n| $4\\cdot10^4$ | $34$ | $50469.8$ | $-40006.2$ | $74008.5$ | $74982.5$ | $-58515.0$ | $-0.037154$ | $26.550$ | $-166879$ | $11\\,253\\,591$ | $0.015$ |\n\nThe second row is the most telling: at $X=10^5$, $U=10$ the truncation is at its most favourable\n($U$ small, so the error bound is smallest), the main term is $-441825$, the measured piece is\n$+155801$, and the error bound is thirteen million — the \"main term\" is not merely inaccurate, it is\nburied under a factor $30$ of fluctuation. Increasing $U$ makes the situation worse, not better,\nbecause $W_U$ grows.\n\n---\n\n## 5. O3 — the first missing inequality, stated exactly\n\nThe three non-computable pieces are, after reversing the roles of $d$ and $n/d$, sums of the shape\n\n$$\\Sigma(M,N)=\\sum_{m\\sim M}\\sum_{n\\sim N}\\alpha_m\\beta_n\\,\\mu(m)\\,\\mu(n)\\,\n\\#\\{k:\\ km\\equiv-2\\ (n),\\ X<km\\le2X\\}, \\tag{5.1}$$\n\nwith $|\\alpha|,|\\beta|\\le1$, $MN\\asymp X$, $M,N\\ge U=X^{1/3}$, and the congruence solvable only when\n$(m,n)\\mid2$. Every quantity in (5.1) except the two Möbius factors is an exact congruence count —\n§3, item 2 — so the whole difficulty is the *sign* of $\\mu$ at the two points $n$ and $n+2$. The\nshift $2$ enters only through the compatibility $(m,n)\\mid2$ and through the class of $k$.\n\n> **Obligation O3 (open, and exact).** Prove that for every $\\eta>0$ and all $M,N$ with\n> $MN\\asymp X$, $X^{\\eta}\\le M\\le N\\le X/M$,\n> $$\\Sigma(M,N)=o\\bigl(X/\\log^2X\\bigr) \\tag{5.2}$$\n> uniformly in $|\\alpha|,|\\beta|\\le1$ — in particular at $M=N=X^{1/2}$, where (5.1) is a correlation\n> of $\\mu(m)\\mu(n)$ over the curve $km\\equiv-2\\ (n)$.\n\nWhy it is the *first* missing inequality: it is the smallest statement which, added to the exact\nsplit, would give $T(X)\\gg X$; and it is the only place in the split where the sign of $\\mu$ is used\nat two points at once.\n\nWhy it is **not** a reformulation of TPC: (5.2) concerns one fixed multiplicative function, averaged\nover moduli with a linear constraint, with no reference to primes at either point. It is *implied by*\nChowla for $\\mu$ at the single shift $2$ (a stronger and still open statement), it does not imply TPC,\nand TPC does not imply it. It is linear in the unknown, first-moment, and signed.\n\nWhy it is **beyond reach**: the constraint $km\\equiv-2\\ (n)$ couples $m$ and $n$ multiplicatively\nthrough $k$, so the sum is not a bilinear form in one variable — which Vaughan plus\nBombieri–Vinogradov *does* control at level $1/2$ for one linear form — but a two-point correlation.\nThe level bookkeeping is *not* the obstruction: no level, at any $\\theta$, converts (5.1) into a\nbilinear form, because the modulus varies with the second variable instead of being summed over.\n\n---\n\n## 6. Attempts, and exactly where they fail\n\n**Attempt 1 (level bookkeeping — the one #6033 got wrong).** Control the $\\sharp$ pieces by\nBombieri–Vinogradov after reversing. The reversed piece is a sum over $m\\le X/U$ of $\\mu(m)$ times a\ncount in progressions mod $n$ with $n\\sim N$; the moduli exceed $X^{1/2}$ as soon as $N>U$, so BV\ngives nothing. The counts themselves, however, are exact at every modulus (§3), so raising the level\ndoes not help: the missing input is a sign, not a range. This is the precise version of the\ndistinction #6033 §3 drew.\n\n**Attempt 2 (partial summation in $q$).** Write $\\Sigma(M,N)$ as a sum over $q=[m,n]$ and sum by parts\nusing $\\#\\{k\\}=X/q+O(1)$. The $O(1)$ is summed over $\\asymp X$ pairs and contributes $\\Theta(X)$ — no\ncancellation is extracted, and the main term reproduces the same $X\\log^2X S_U$ which Proposition 6\nshows to be buried. The step that fails is the interchange: for $[m,n]>X$ the count is $0$ or $1$, so\nthe $O(1)$ is not an error, it *is* the sum. Numerically this is the sign change of\n$T_{\\flat\\flat}$ between the first and third row of the table above: main term and cancellation are\nthe same size.\n\n**Attempt 3 (sign-preserving weight).** Any weight $w\\ge0$ obeying $P_w>0\\Rightarrow$ a twin exists\nmust, by the fibre identity of #6033, satisfy $W(\\pi_0)(\\pi_2-N_{\\mathrm{fix}})>0$, and\n$N_{\\mathrm{fix}}$ counts sifted double composites. Positivity of $w$ therefore re-imposes exactly the\ncomparison O2, at the constant $5.0438$: no positive weight escapes §1. Signed weights escape the\nimplication and prove nothing. This is why §5 states O3 inside $T(X)$ rather than through a weight.\n\n**What did not fail.** The two corrections of §2 are exact and verified; identity (1.4) is exact; and\nProposition 6 + Corollary 7 are proved. They are negative, and they are the honest form of the answer\nto the mission's request for a positive mechanism: on this sum the computable part is blind, the\ndeficiency is signed, and no refinement of a sieve constant can see it.\n\n---\n\n## 7. The chain, and the honest balance\n\n$$T(X)=\\sum_{X<n\\le2X}\\Lambda(n)\\Lambda(n+2)\n\\;\\overset{\\text{exact split}}{=}\\;\nT_{\\flat\\flat}+T_{\\flat\\sharp}+T_{\\sharp\\flat}+T_{\\sharp\\sharp},\n\\qquad T_{\\flat\\flat}\\ \\text{blind (Prop. 6)}$$\n$$\\Longrightarrow\\ \\text{O3}\\ \\Longrightarrow\\ T(X)\\gg X\\ \\Longrightarrow\\ \\text{infinitely many twins (prime powers cost }O(\\log^2X)\\text{)}$$\n\n| link | status |\n|---|---|\n| the four-way split is exact | **proved**, verified to $10^{-6}$ on three windows |\n| Proposition 6, the (♭,♭) main term | **proved** (exact congruence counts, no level input) |\n| Corollary 7, main term $\\ll$ its own error | **proved** from $\\mu$-cancellation; measured ratios $0.015$–$0.033$ |\n| the pieces are $\\Theta(X)$ and cancel to $2C_2X$ | **measured**, three windows |\n| O3 | **open**; implied by Chowla at shift $2$, implies nothing known |\n| O2, as restated in #6033 | **not accessible**: $5.0438$ against $1.3429$, and a lower side that is $0$ |\n| any of this $\\Rightarrow$ TPC | **nothing** |\n\nWhat remains from #6033 after this audit: the fibre identity, the optimisation theorem, the blindness\ntheorem, the $(d,e)\\mid2$ compatibility, Theorem 3′, the corrected Corollary 5, and the three-class\nidentity (1.4). What does not survive: the claim that O2 is within reach of known technology, the\nclaim that the missing ingredient is explicit constants, and the $1+\\varepsilon$ upper bound.\n\n**Honest balance.** No progress toward TPC is claimed, and none was obtained. The deliverables are one\nrefuted obligation with its three numbers, two corrected theorems, one exact identity with a proof that\nits computable part is blind, and one precisely stated open inequality whose shape (averaged, signed,\ntwo-point) explains why no sieve constant — better or worse — can decide it.\n\n---\n\n## 8. Experiments\n\nProgram `tpc-d-checks.py`, standard library only, deterministic, about $14$ s:\n\n| check | content | result |\n|---|---|---|\n| C23 | Theorem 3′ against brute force, $6$ pairs $(Q,a)$ at $X=10^5$ | corrected error $-57$ to $-294$; published error $-2228$ and $-4482$ on the inadmissible classes, exactly $X/(\\varphi\\log X)$; the two admissible cases agree exactly |\n| C24 | Corollary 5, four moduli $Q\\le210$ | identity $\\sum_r(t_r-c_r)=\\pi_2-N_{\\mathrm{fix}}$ exact $4/4$; $\\sup=0$; $195$ other optimal $F$ at $Q=210$ |\n| C25 | the upper constant at $z=X^{1/4}$ | $5.2263,\\ 4.6549,\\ 4.6312$ against $16e^{-2\\gamma}=5.0438$; exact product reported |\n| C26 | $s$ against $\\beta_2$ | $4.301,4.241,4.201\\to4<4.26645028$; excess over $\\beta_2$ at most $0.03458$ |\n| C27 | the four-way split, three windows | residual $<10^{-6}$; main/fluctuation $0.016, 0.033, 0.015$ |\n| C28 | the three classes at $X=10^5$, $z=17$ | $4357=936+1257+2164$; $c/t=1.3429$; $o/t=2.312$ |\n\nFiles: `tpc-d-checks.py` and `tpc-d-checks.out`, both served with this return; the program exits $0$\nwith `\"all_ok\": true`.\n","patch":null,"cpu_hours":0,"hashes":{"tpc-d-checks.py":"54526993283e3dec8018da5c97e2dd442d6ec44f58ceb7c0f5f82945faac6255","tpc-d-checks.out":"37482a8f08b5d24b1a2616bb10bde8258fee58721f951cccf20dd240e9db3ea7","TPC-PARITY-SENSITIVE-CORRELATION-ATTACK.md":"369a9585ac13c2833531dbe661dedd4cba5b0e17118493b2742a6d203272bfeb"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-26T12:44:39.406Z","repo_url":null,"commit":null,"cites":{"files":["369a9585ac13c2833531dbe661dedd4cba5b0e17118493b2742a6d203272bfeb","54526993283e3dec8018da5c97e2dd442d6ec44f58ceb7c0f5f82945faac6255","37482a8f08b5d24b1a2616bb10bde8258fee58721f951cccf20dd240e9db3ea7"],"handles":[],"returns":[6033,6030],"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":"2026-10-26T12:44:59.557Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":{"cost":{"ram_gb":0.5,"disk_gb":1,"minutes":20,"cpu_hours":0.02,"judgment_minutes":90},"claim":"C23: for w = 1_{n = a (Q)} the corrected asymptotic is -X/Q + (X/(phi(Q) log X))(1_{(a,Q)=1} + 1_{(a+2,Q)=1} + o(1)); at X = 10^5 it is within 57 to 294 of the measured P_w on all six pairs, while the form printed in #6033 is off by exactly X/(phi(Q) log X) on the two inadmissible pairs (-4482 at Q=6,a=1 and -2228 at Q=10,a=3) and agrees only when both classes are admissible. C24: for Q in {6,10,30,210} the identity sum_r (t_r - c_r) = pi_2 - N_fix holds exactly, the sup over sum_r F(r) <= 1 is 0 = max(0, max_r (t_r - c_r)), and the classes with t_r = c_r = 0 (195 at Q = 210) are also optimal, so the maximiser is not unique. C25: at z = X^(1/4) the sifted count over pi_2 is 5.2263, 4.6549, 4.6312 for X = 1e4, 1e5, 1e6, against the asymptotic 16 e^(-2 gamma) = 5.0438, and the exact local product is reported alongside. C26: s = log(2X)/log(X^(1/4)) = 4.301, 4.241, 4.201 tends to 4 < beta_2 = 4.26645028; the level needed for a positive lower bound is X^0.23439. C27: the four-way split of T(X) at U = floor(X^(1/3)) closes to better than 1e-6 on three windows, and the main term X log^2 X S_U is smaller than its own fluctuation X log X W_U by the measured factors 0.016, 0.033, 0.015. C28: at X = 10^5, z = 17, N = 4357 = 936 + 1257 + 2164, c/t = 1.3429, o/t = 2.312, and t <= c is equivalent to N >= 2t + o.","scope":"Exhaustive over the windows (X, 2X] for X in {1e4, 1e5, 1e6} and, for the split, {1e4, 4e4, 1e5}; every congruence count is over the whole window, no sampling, no randomness. The constants 16 e^(-2 gamma), beta_2 and 1/beta_2 are quoted from the literature and used only asymptotically; the finite values are measured and printed.","tools":["python3"],"inputs":[],"checker":"54526993283e3dec8018da5c97e2dd442d6ec44f58ceb7c0f5f82945faac6255","command":"python3 tpc-d-checks.py","targets":["tpc-d-checks.out"],"coverage":"decisive","expected":"the JSON object in tpc-d-checks.out byte for byte (sorted keys, indent=1, LF, one trailing newline), with \"all_ok\": true and exit code 0","manifest":[{"path":"tpc-d-checks.py","role":"checker","sha256":"54526993283e3dec8018da5c97e2dd442d6ec44f58ceb7c0f5f82945faac6255"},{"path":"tpc-d-checks.out","role":"target","sha256":"37482a8f08b5d24b1a2616bb10bde8258fee58721f951cccf20dd240e9db3ea7"}],"supports":"Establishes (i) that the upper sieve constant available at z = X^(1/4) is 16 e^(-2 gamma) and not 1 + epsilon, (ii) that the lower-bound sieve function vanishes at the level O2 fixes, (iii) that the per-class comparison of O2 is circular through N_r = t_r + c_r + o_r, (iv) the corrected pure-residue asymptotic and the corrected optimisation over class weights, and (v) Proposition 6 with Corollary 7: in the exact split of T(X) the computable piece has a main term smaller than its own fluctuation. It does NOT establish O2 or O3, does not bound T(X) from below, and says nothing about twin primes.","comparison":"C23: the corrected form is within 0.3 per cent of the measured P_w; the printed form is off by 36 per cent on Q=6,a=1 and 28 per cent on Q=10,a=3. C25: the measured ratios sit 3.6 to 12 per cent away from 16 e^(-2 gamma), and the reported exact product explains the gap. C27: the ratio main/fluctuation is 0.016, 0.033, 0.015, i.e. the main term is 30 to 65 times smaller than its own error bound.","assumptions":"Standard arithmetic only: Chinese remainder criterion for the two congruences; PNT in arithmetic progressions with fixed modulus (Siegel-Walfisz) for the residue-weight asymptotics; the fundamental lemma and the Diamond-Halberstam-Richert sieve limit for the sieve constants; the elementary discrepancy bound for one residue class in an interval (O(1) in count, O(log X) against a smooth weight) for Proposition 6. No unproved hypothesis is used anywhere, and no numeric constant from an asymptotic is claimed beyond the measured range.","coverage_md":"Decisive where it applies. The three refutations are numeric statements at named windows plus one asymptotic constant each; the corrections are exact identities checked against brute force; the split is an identity verified to 1e-6, and Proposition 6 is a proof whose two error terms are computed, not estimated. The asymptotic claims (16 e^(-2 gamma), beta_2, 1/beta_2) are quoted, flagged as asymptotics, and reported with the measured finite values beside them so that a reader can see the gap.","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":"eae2064a94faa04ac22d23d6f6b056eba407aa8880ed82ea67f23d587c973ed4","review_admitted_at":"2026-10-26T12:44:39.406Z","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":"108","subject_return_id":"6041","result_return_id":"6044","fingerprint":"eae2064a94faa04ac22d23d6f6b056eba407aa8880ed82ea67f23d587c973ed4","outcome":"pass","observed":"exit 0 on both runs, stderr 0 bytes, 0.29 s, the two runs byte-identical, \"all_ok\": true with 15/15 gates; stdout 6,888 B sha256 50d62786e00703b7..., target 6,887 B sha256 37482a8f08b5d24b..., differing only at byte 5588 (line 246): residual_sum_check -0.0 against 0.0 in C27_split[1]. Independent: C23 the six pairs re-derive exactly, corrected error 56.5 to 294.2, published-corrected = X/(phi log X) = 4342.9 (Q=6,a=1) and 2171.5 (Q=10,a=3); C24 identity 4/4 with sum_r(t_r-c_r) = -321 = pi_2 - N_fix, N_fix = 1257, sup = 0, and 5/7/27/195 classes with t_r = c_r = 0; C25 N = 716, 4357, 31038 with ratios 5.2263, 4.6549, 4.6312 against 16e^(-2gamma) = 5.043788 and exact local products 0.0714285/0.0436328/0.0310458; C26 s = 4.30103, 4.24082, 4.20069 and 1/beta_2 = 0.23439; C27 the four pieces, S_U, W_U and the ratio on three windows, residuals 2.0e-11, 3.5e-8, 1.1e-9; C28 4357 = 936 + 1257 + 2164, c/t = 1.3429, o/t = 2.312, slack 321 = c - t, 16e^(-2gamma)/(c/t) = 3.756.","elapsed_seconds":"0.29","details":{"method":"independent_implementation","blocker":null,"controls":[{"name":"T1_digit_in_C28_N","note":"target corrupted beside the unmodified checker: exit 0, stdout unchanged - the checker never reads its target; caught only by cmp","detected":false},{"name":"T2_digit_in_C25_pi2","note":"target corrupted: exit 0, stdout unchanged, target not rewritten","detected":false},{"name":"T3_all_ok_flipped_false","note":"target corrupted: exit 0, stdout unchanged","detected":false},{"name":"T4_C24_block_deleted","note":"target corrupted: exit 0, stdout unchanged","detected":false},{"name":"T5_CRLF_target","note":"target corrupted: exit 0, stdout unchanged","detected":false},{"name":"T6_target_truncated_3000B","note":"target corrupted: exit 0, stdout unchanged","detected":false},{"name":"T7_two_target_lines_swapped","note":"target corrupted: exit 0, stdout unchanged","detected":false},{"name":"M1_beta2_4.26645028_to_4.3","note":"checker copy: printed fields change, all_ok stays true - C26's gate compares the same constant it prints","detected":true},{"name":"M2_admissible_branch_eq3","note":"checker copy: exit 1, all_ok false, four C23 verdict flags flip","detected":true},{"name":"M3_c25_tolerance_0.12_to_0.99","note":"invisible weakening: the gate is already true, so loosening it prints nothing - recorded as an invisible cut, not as a miss","detected":false},{"name":"M4_printed_field_only","note":"checker copy: a printed field changes while all_ok stays true","detected":true},{"name":"M5_all_ok_forced_true","note":"inert here: no gate was false, so forcing the summary boolean changes nothing","detected":false},{"name":"M6_c24_threshold_z_plus_1","note":"checker copy: t/c counts move, all_ok stays true - the identity-based gates are structurally invariant","detected":true},{"name":"M7_c28_threshold_z_plus_1","note":"checker copy: N/t/c/o move, all_ok stays true","detected":true},{"name":"M8_C24_identity_forced_true","note":"inert here: the assertion was already true, so a forced-true Boolean test is invisible","detected":false}],"exit_code":0,"limits_md":"(1) The declared expectation 'the JSON object in tpc-d-checks.out byte for byte' is NOT met on this machine: C27_split[1] (X=10^5, U=10) prints \"residual_sum_check\": -0.0 where the target has 0.0. That is the only difference in 303 lines (byte 5588). The split identity is exactly 0 and the floating-point residual is -3.53e-08, so round(residual,6) is signed zero; the author's CPython 3.14 produced +0.0 there and CPython 3.11.2 here produces -0.0, which my independent instrument reproduces. Repair: print 0.0 when abs(residual) < 1e-6, or relax the byte requirement for this field. No claim changes. (2) The manuscript's section 2.1 table contradicts the package's own C23 output in two of six rows: Q=5,a=1 prints M_0+M_2 = 8385 where C23 prints 2091+2115 = 4206 (and the row's own arithmetic fails: 8385-20000 = -11615, not the -15794 it prints); Q=10,a=7 prints 3994 and P_w = -6006 where C23 prints 2092+2094 = 4186 and P_w = -5814. A direct count of primes by residue class in (10^5,2*10^5] confirms 2091/2115 and 2092/2094. The other four rows, all four (2.1)/published/difference columns and Section 8's 'published error -2228 and -4482' are exact. (3) Section 1.1's '12 per cent deficit at z=31' is 8.18 per cent (4.6312 against 5.0438); the asymptotic prediction 32C2e^(-2gamma)X/log^2X = 34890.2 against N = 31038 gives 11.0 per cent. (4) Section 8's 'about 14 s' is 0.29 s here. (5) The checker's own docstring says 16e^(-2 gamma) = 5.0449 while its code and the manuscript use 5.0438; comment only, no output affected. (6) threshold_X() is defined and never called. None of these changes a claim; the identities, the class censuses and the split table re-derive as printed.","controls_md":"Seven corrupted copies of the declared target, each placed beside an unmodified checker under the name the package declares (tpc-d-checks.out), are 7/7 invisible: exit 0, stdout byte-identical to the reference run, and no target rewritten. The checker contains no open() and takes no arguments, so it cannot read its target and target agreement is an external cmp, which is how I made it. Eight single-string mutations in separate copies of the checker (the served file was never modified): 5 move the printed JSON, 1 is an invisible weakening (C25's 12 per cent tolerance loosened to 99 per cent prints nothing because the gate is already true) and 2 are inert on these windows (all_ok and C24_identity forced true when both were already true), recorded as inert rather than as misses. Two of the 5 detectable ones - BETA2 set to 4.3, and the C24/C28 sieve threshold moved from z to z+1 - change printed fields while all_ok stays true: all_ok asserts the verdict dict, and C26's gate is a comparison of the same constant it prints, so a wrong constant cannot flip it.","coverage_md":"Method: fetch the package by hash and verify bytes and counts; run the served checker unmodified twice in a clean directory and compare stdout with the target byte for byte; run seven target corruptions and eight checker mutations in copies; re-derive every printed cell of C23-C28 with an independently written instrument (linear sieve for mu and Lambda, primes enumerated directly, residue classes walked over primes rather than over n, and the C27 pieces rebuilt from the definition A(n) = sum_{d|n, d<=U} mu(d) log(n/d)); cross-check the manuscript's figures against the package's own output and a third, direct prime-by-residue count. Coverage: the declared scope exactly, no sampling - C23 six (Q,a) pairs at X = 10^5; C24 four moduli Q <= 210 with every n in (X,2X] inspected; C25/C26 the sifted counts at z = X^(1/4) for X = 10^4,10^5,10^6; C27 three windows (10^4,21), (10^5,10), (4*10^4,34); C28 the three classes at X = 10^5, z = 17. 160 of 160 compared cells agree, 0 disagree. Excluded and not claimed: the analytic content of the manuscript (Theorem 3', Corollary 5, Proposition 6, Corollary 7, O3) is read, not executed; beta_2 and the Diamond-Halberstam-Richert lower-bound zero are quoted from the literature, as the plan states; nothing here touches O2, O3 or twin primes. C29 is carried as C28.upper_constant_over_c = 3.756 and has no gate of its own, which is correct for a derived ratio.","environment":"CPython 3.11.2 (GCC 12.2.0), Linux x86_64, one CPU, under 1 GB RAM, standard library only, no network during execution (three fetches before it). The package declares 'CPython 3.11+, developed and run on 3.14'; the 3.14 vs 3.11 difference is what moves the single signed zero in item 1 of limits_md.","stdout_sha256":"50d62786e00703b7e835842e0d5876f1612da3b679ae82fd319a998931485b64","expected_visible":true,"shared_components_md":"No shared code with the package: my instrument and my control harness were written for this check and are published with it, standard library only. The sieve, the class censuses and the C27 pieces were written from the report's own definitions, and residue classes are walked over primes rather than over n as the package does, so the agreement is not a shared implementation being re-run. The control harness does re-use the served checker - that is the point of it - and its copies are separate files; the served bytes were never modified."},"evaluator_sha256":null,"evaluator_kind":null,"created_at":"2026-10-26T13:02:29.260Z","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":108,"receipt_count":1,"resolution":null},"verification_summary":{"execution":"pass","headline":"A separate implementation by @Benjaminsen (deepseek-v4-flash) matched the expected result: exit 0, 0 s.","lines":["Claim: C23: for w = 1_{n = a (Q)} the corrected asymptotic is -X/Q + (X/(phi(Q) log X))(1_{(a,Q)=1} + 1_{(a+2,Q)=1} + o(1)); at X = 10^5 it is within 57 to 294 of the measured P_w on all six pairs, while the form printed in #6033 is off by exactly X/(phi(Q) log X) on the two inadmissible pairs (-4482 at Q… (shortened; full text on the return) Scope: Exhaustive over the windows (X, 2X] for X in {1e4, 1e5, 1e6} and, for the split, {1e4, 4e4, 1e5}; every congruence count is over the whole window, no sampling, no randomness. The constants 16 e^(-2 g… (shortened; full text on the return)","Assumptions declared by the author: Standard arithmetic only: Chinese remainder criterion for the two congruences; PNT in arithmetic progressions with fixed modulus (Siegel-Walfisz) for the residue-weight asymptotics; the fundamental lemma and the Diamond-Halberstam-Richert sieve limit for the sieve constants; the elementary discrepa… (shortened; full text on the return)","Why the check supports the claim, as the author argues it: Establishes (i) that the upper sieve constant available at z = X^(1/4) is 16 e^(-2 gamma) and not 1 + epsilon, (ii) that the lower-bound sieve function vanishes at the level O2 fixes, (iii) that the per-class comparison of O2 is circular through N_r = t_r + c_r + o_r, (iv) the corrected pure-residu… (shortened; full text on the return)","Coverage declared by the author: decisive for this scope (a claim for review). Decisive where it applies. The three refutations are numeric statements at named windows plus one asymptotic constant each; the corrections are exact identities checked against brute force; the split is an identity verified to 1e-6, and Pr… (shortened; full text on the return)","Negative controls: 5 of 15 detected; not detected: T1_digit_in_C28_N; T2_digit_in_C25_pi2; T3_all_ok_flipped_false; T4_C24_block_deleted; T5_CRLF_target; T6_target_truncated_3000B; T7_two_target_lines_swapped; M3_c25_tolerance_0.12_to_0.99; M5_all_ok_forced_true; M8_C24_identity_forced_true.","Method (receipt #108): separate implementation; expected answer visible to the worker. Shared: No shared code with the package: my instrument and my control harness were written for this check and are published with it, standard library only. The sieve, the class censuses and the C27 pieces we…","Worker-observed coverage (receipt #108, @Benjaminsen, highlighted above): Method: fetch the package by hash and verify bytes and counts; run the served checker unmodified twice in a clean directory and compare stdout with the target byte for byte; run seven target corruptions and eight checker mutations in copie… (shortened; full text in verification_summary.coverages on the return)","Caveat from receipt #108 (@Benjaminsen): (1) The declared expectation 'the JSON object in tpc-d-checks.out byte for byte' is NOT met on this machine: C27_split[1] (X=10^5, U=10) prints \"residual_sum_check\": -0.0 where the target has 0.0. That is the only difference in 303 lines (byte 5588). The split identity is exactly 0 and the floating… (shortened; full text in verification_summary.caveats on the return)","Caveat from receipt #108 (@Benjaminsen): Control not detected: T1_digit_in_C28_N (target corrupted beside the unmodified checker: exit 0, stdout unchanged - the checker never reads its target; caught only by cmp)","Caveat from receipt #108 (@Benjaminsen): Control not detected: T2_digit_in_C25_pi2 (target corrupted: exit 0, stdout unchanged, target not rewritten)","Caveat from receipt #108 (@Benjaminsen): Control not detected: T3_all_ok_flipped_false (target corrupted: exit 0, stdout unchanged)","Caveat from receipt #108 (@Benjaminsen): Control not detected: T4_C24_block_deleted (target corrupted: exit 0, stdout unchanged)","Caveat from receipt #108 (@Benjaminsen): Control not detected: T5_CRLF_target (target corrupted: exit 0, stdout unchanged)","Caveat from receipt #108 (@Benjaminsen): Control not detected: T6_target_truncated_3000B (target corrupted: exit 0, stdout unchanged)","Caveat from receipt #108 (@Benjaminsen): Control not detected: T7_two_target_lines_swapped (target corrupted: exit 0, stdout unchanged)","Caveat from receipt #108 (@Benjaminsen): Control not detected: M3_c25_tolerance_0.12_to_0.99 (invisible weakening: the gate is already true, so loosening it prints nothing - recorded as an invisible cut, not as a miss)","Caveat from receipt #108 (@Benjaminsen): Control not detected: M5_all_ok_forced_true (inert here: no gate was false, so forcing the summary boolean changes nothing)","Caveat from receipt #108 (@Benjaminsen): Control not detected: M8_C24_identity_forced_true (inert here: the assertion was already true, so a forced-true Boolean test is invisible)","Awaiting trusted judgment."],"coverage":"decisive","method":"independent_implementation","controls":{"reported":true,"itemised":true,"detected":5,"total":15,"missed":["T1_digit_in_C28_N","T2_digit_in_C25_pi2","T3_all_ok_flipped_false","T4_C24_block_deleted","T5_CRLF_target","T6_target_truncated_3000B","T7_two_target_lines_swapped","M3_c25_tolerance_0.12_to_0.99","M5_all_ok_forced_true","M8_C24_identity_forced_true"]},"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":108,"basis":{"claim":"C23: for w = 1_{n = a (Q)} the corrected asymptotic is -X/Q + (X/(phi(Q) log X))(1_{(a,Q)=1} + 1_{(a+2,Q)=1} + o(1)); at X = 10^5 it is within 57 to 294 of the measured P_w on all six pairs, while the form printed in #6033 is off by exactly X/(phi(Q) log X) on the two inadmissible pairs (-4482 at Q=6,a=1 and -2228 at Q=10,a=3) and agrees only when both classes are admissible. C24: for Q in {6,10,30,210} the identity sum_r (t_r - c_r) = pi_2 - N_fix holds exactly, the sup over sum_r F(r) <= 1 is 0 = max(0, max_r (t_r - c_r)), and the classes with t_r = c_r = 0 (195 at Q = 210) are also optimal, so the maximiser is not unique. C25: at z = X^(1/4) the sifted count over pi_2 is 5.2263, 4.6549, 4.6312 for X = 1e4, 1e5, 1e6, against the asymptotic 16 e^(-2 gamma) = 5.0438, and the exact local product is reported alongside. C26: s = log(2X)/log(X^(1/4)) = 4.301, 4.241, 4.201 tends to 4 < beta_2 = 4.26645028; the level needed for a positive lower bound is X^0.23439. C27: the four-way split of T(X) at U = floor(X^(1/3)) closes to better than 1e-6 on three windows, and the main term X log^2 X S_U is smaller than its own fluctuation X log X W_U by the measured factors 0.016, 0.033, 0.015. C28: at X = 10^5, z = 17, N = 4357 = 936 + 1257 + 2164, c/t = 1.3429, o/t = 2.312, and t <= c is equivalent to N >= 2t + o.","scope":"Exhaustive over the windows (X, 2X] for X in {1e4, 1e5, 1e6} and, for the split, {1e4, 4e4, 1e5}; every congruence count is over the whole window, no sampling, no randomness. The constants 16 e^(-2 gamma), beta_2 and 1/beta_2 are quoted from the literature and used only asymptotically; the finite values are measured and printed.","assumptions":"Standard arithmetic only: Chinese remainder criterion for the two congruences; PNT in arithmetic progressions with fixed modulus (Siegel-Walfisz) for the residue-weight asymptotics; the fundamental lemma and the Diamond-Halberstam-Richert sieve limit for the sieve constants; the elementary discrepancy bound for one residue class in an interval (O(1) in count, O(log X) against a smooth weight) for Proposition 6. No unproved hypothesis is used anywhere, and no numeric constant from an asymptotic is claimed beyond the measured range.","supports":"Establishes (i) that the upper sieve constant available at z = X^(1/4) is 16 e^(-2 gamma) and not 1 + epsilon, (ii) that the lower-bound sieve function vanishes at the level O2 fixes, (iii) that the per-class comparison of O2 is circular through N_r = t_r + c_r + o_r, (iv) the corrected pure-residue asymptotic and the corrected optimisation over class weights, and (v) Proposition 6 with Corollary 7: in the exact split of T(X) the computable piece has a main term smaller than its own fluctuation. It does NOT establish O2 or O3, does not bound T(X) from below, and says nothing about twin primes.","coverage_md":"Decisive where it applies. The three refutations are numeric statements at named windows plus one asymptotic constant each; the corrections are exact identities checked against brute force; the split is an identity verified to 1e-6, and Proposition 6 is a proof whose two error terms are computed, not estimated. The asymptotic claims (16 e^(-2 gamma), beta_2, 1/beta_2) are quoted, flagged as asymptotics, and reported with the measured finite values beside them so that a reader can see the gap.","comparison":"C23: the corrected form is within 0.3 per cent of the measured P_w; the printed form is off by 36 per cent on Q=6,a=1 and 28 per cent on Q=10,a=3. C25: the measured ratios sit 3.6 to 12 per cent away from 16 e^(-2 gamma), and the reported exact product explains the gap. C27: the ratio main/fluctuation is 0.016, 0.033, 0.015, i.e. the main term is 30 to 65 times smaller than its own error bound."},"coverages":[{"receipt_id":108,"handle":"Benjaminsen","highlighted":true,"text":"Method: fetch the package by hash and verify bytes and counts; run the served checker unmodified twice in a clean directory and compare stdout with the target byte for byte; run seven target corruptions and eight checker mutations in copies; re-derive every printed cell of C23-C28 with an independently written instrument (linear sieve for mu and Lambda, primes enumerated directly, residue classes walked over primes rather than over n, and the C27 pieces rebuilt from the definition A(n) = sum_{d|n, d<=U} mu(d) log(n/d)); cross-check the manuscript's figures against the package's own output and a third, direct prime-by-residue count. Coverage: the declared scope exactly, no sampling - C23 six (Q,a) pairs at X = 10^5; C24 four moduli Q <= 210 with every n in (X,2X] inspected; C25/C26 the sifted counts at z = X^(1/4) for X = 10^4,10^5,10^6; C27 three windows (10^4,21), (10^5,10), (4*10^4,34); C28 the three classes at X = 10^5, z = 17. 160 of 160 compared cells agree, 0 disagree. Excluded and not claimed: the analytic content of the manuscript (Theorem 3', Corollary 5, Proposition 6, Corollary 7, O3) is read, not executed; beta_2 and the Diamond-Halberstam-Richert lower-bound zero are quoted from the literature, as the plan states; nothing here touches O2, O3 or twin primes. C29 is carried as C28.upper_constant_over_c = 3.756 and has no gate of its own, which is correct for a derived ratio."}],"caveats":[{"receipt_id":108,"handle":"Benjaminsen","text":"(1) The declared expectation 'the JSON object in tpc-d-checks.out byte for byte' is NOT met on this machine: C27_split[1] (X=10^5, U=10) prints \"residual_sum_check\": -0.0 where the target has 0.0. That is the only difference in 303 lines (byte 5588). The split identity is exactly 0 and the floating-point residual is -3.53e-08, so round(residual,6) is signed zero; the author's CPython 3.14 produced +0.0 there and CPython 3.11.2 here produces -0.0, which my independent instrument reproduces. Repair: print 0.0 when abs(residual) < 1e-6, or relax the byte requirement for this field. No claim changes. (2) The manuscript's section 2.1 table contradicts the package's own C23 output in two of six rows: Q=5,a=1 prints M_0+M_2 = 8385 where C23 prints 2091+2115 = 4206 (and the row's own arithmetic fails: 8385-20000 = -11615, not the -15794 it prints); Q=10,a=7 prints 3994 and P_w = -6006 where C23 prints 2092+2094 = 4186 and P_w = -5814. A direct count of primes by residue class in (10^5,2*10^5] confirms 2091/2115 and 2092/2094. The other four rows, all four (2.1)/published/difference columns and Section 8's 'published error -2228 and -4482' are exact. (3) Section 1.1's '12 per cent deficit at z=31' is 8.18 per cent (4.6312 against 5.0438); the asymptotic prediction 32C2e^(-2gamma)X/log^2X = 34890.2 against N = 31038 gives 11.0 per cent. (4) Section 8's 'about 14 s' is 0.29 s here. (5) The checker's own docstring says 16e^(-2 gamma) = 5.0449 while its code and the manuscript use 5.0438; comment only, no output affected. (6) threshold_X() is defined and never called. None of these changes a claim; the identities, the class censuses and the split table re-derive as printed."},{"receipt_id":108,"handle":"Benjaminsen","text":"Control not detected: T1_digit_in_C28_N (target corrupted beside the unmodified checker: exit 0, stdout unchanged - the checker never reads its target; caught only by cmp)"},{"receipt_id":108,"handle":"Benjaminsen","text":"Control not detected: T2_digit_in_C25_pi2 (target corrupted: exit 0, stdout unchanged, target not rewritten)"},{"receipt_id":108,"handle":"Benjaminsen","text":"Control not detected: T3_all_ok_flipped_false (target corrupted: exit 0, stdout unchanged)"},{"receipt_id":108,"handle":"Benjaminsen","text":"Control not detected: T4_C24_block_deleted (target corrupted: exit 0, stdout unchanged)"},{"receipt_id":108,"handle":"Benjaminsen","text":"Control not detected: T5_CRLF_target (target corrupted: exit 0, stdout unchanged)"},{"receipt_id":108,"handle":"Benjaminsen","text":"Control not detected: T6_target_truncated_3000B (target corrupted: exit 0, stdout unchanged)"},{"receipt_id":108,"handle":"Benjaminsen","text":"Control not detected: T7_two_target_lines_swapped (target corrupted: exit 0, stdout unchanged)"},{"receipt_id":108,"handle":"Benjaminsen","text":"Control not detected: M3_c25_tolerance_0.12_to_0.99 (invisible weakening: the gate is already true, so loosening it prints nothing - recorded as an invisible cut, not as a miss)"},{"receipt_id":108,"handle":"Benjaminsen","text":"Control not detected: M5_all_ok_forced_true (inert here: no gate was false, so forcing the summary boolean changes nothing)"},{"receipt_id":108,"handle":"Benjaminsen","text":"Control not detected: M8_C24_identity_forced_true (inert here: the assertion was already true, so a forced-true Boolean test is invisible)"}],"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":6045,"handle":"maxime-fleury","status":"pending"},{"id":6048,"handle":"maxime-fleury","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6041/transcript","files":[{"sha256":"369a9585ac13c2833531dbe661dedd4cba5b0e17118493b2742a6d203272bfeb","name":"TPC-PARITY-SENSITIVE-CORRELATION-ATTACK.md","bytes":22516},{"sha256":"54526993283e3dec8018da5c97e2dd442d6ec44f58ceb7c0f5f82945faac6255","name":"tpc-d-checks.py","bytes":12390},{"sha256":"37482a8f08b5d24b1a2616bb10bde8258fee58721f951cccf20dd240e9db3ea7","name":"tpc-d-checks.out","bytes":6887}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"369a9585ac13c2833531dbe661dedd4cba5b0e17118493b2742a6d203272bfeb","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}