{"id":6249,"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 false hypothesis (U), the cofactor that cannot be replaced, and a real Type II attempt\n\nFollow-up to returns 6236 and 6245. Written 2026-10-27. Status labels as before: `PROVED` · `KNOWN` ·\n`NUMERIC` · `REFUTED` · `UNVERIFIED` · `OPEN`; `AUDIT` for evidence about the literature.\n\n## 0. Verdict on the four defects submitted for audit\n\n| # | claim | verdict |\n|---|---|---|\n| 1 | the cofactor sum was written with $3d+2$ instead of $ed+2$ | **CONFIRMED** — Theorem 5.1, Theorem 5.2 and (U) of 6245 are wrong for general $e$; the witness $x=1000$, $Q=4$, $e=5$, $d=113$ gives $+\\log5$ against $0$ |\n| 2 | (U) is false as stated | **CONFIRMED** — $m=N$, $r=1$, $g\\equiv1$ gives left side $1$, right side $\\to0$; the failure is structural and needs a modulus range |\n| 3 | Lemma A (arbitrary measurable $E$) is false | **CONFIRMED** — $E_x=\\{x/e\\}$ is finite, $\\int_{E_x}dt/t=0$, while $\\sum1/e=\\tfrac1{12}\\log x+O(1)$ |\n| 4 | the (AC) convolution is wrong | **NOT CONFIRMED** — the dossier already has $A(t)=\\sum_m h_mM(t/m)$; at $t=3$ the direct value is $1/2$, the $h_m$ form gives $1/2$, the $h_m/m$ form gives $11/18$. The three numbers quoted in the audit are exactly the dossier's numbers. What *was* missing is one line of normalisation ($h_{2^k}=0$ for $k\\ge1$ in the odd-indexed normalisation), now supplied. |\n\nTwo constructive results are added: an exact square-divisor reduction from mixed $(\\mu,\\lambda)$ to\nsame-function $(\\lambda,\\lambda)$ correlations (§5), and one Type II attempt carried to its first unproved\ninequality, which is a four-point correlation at a growing slope (§6).\n\nNo new cancellation estimate. **Category C for TPC.**\n\n---\n\n## 1. The cofactor cannot be replaced by 3 — `REFUTED` (the replacement), `PROVED` (the witness)\n\nThe Chen remainder's term is\n$$\\mu(d)\\,\\lambda(ed+2)\\,\\chi_z(ed+2)\\,k_Q(ed+2),\\qquad x/2+2\\le ed+2\\le x,$$\nso the second argument is $ed+2$ and the **slope is $e$**. Return 6245 wrote $3d+2$ in Theorem 5.1,\nTheorem 5.2 and (U); that is correct only for the fixed-cofactor block $e=3$ of the earlier reports, and\nfalse as a general identity.\n\n**Witness (`NUMERIC`, T1).** $x=1000$, $Q=4$, $e=5$, $d=113$, $z=x^{1/8}=2.3714$, primes $q\\in(4,8]=\\{5,7\\}$:\n\n| | $ed+2=567=3^4\\cdot7$ | $3d+2=341=11\\cdot31$ |\n|---|---|---|\n| $\\lambda$ | $-1$ | $+1$ |\n| $k_Q$ | $1$ (since $7\\mid567$) | $0$ |\n| $\\chi_z$ | $1$ | $1$ |\n| term | $\\mu(113)\\log5\\cdot(-1)\\cdot1\\cdot1=+\\log5=+1.609437912$ | $\\mu(113)\\log5\\cdot(+1)\\cdot1\\cdot0=0$ |\n\nThe difference is exactly $\\log5$. So no identity between \"the cofactor sum\" and \"a family of fixed-form\ncorrelations\" exists: the family is one of correlations with slope $e$, and the slope runs over\n$3\\le e<x^{1/3}$.\n\n**Corrected statements.** In Theorem 5.1 replace every $3d+2$ by $ed+2$ and index $C_{e}$ accordingly;\nin Theorem 5.2 the hypothesis becomes\n$$\\Bigl|\\sum_{\\substack{n\\le N\\\\ n\\equiv r\\ (m)}}\\mu(n)\\,g(en+2)\\Bigr|\\ \\le\\ \\frac{N}{m}(\\log N)^{-\\sigma},$$\nwith $g=\\lambda\\chi_z$; and (U) inherits the slope $e$. The consequence is *stronger* than 6245 §5.3 said:\nthe uniformity needed is not merely in a modulus, but in a **one-parameter family of slopes growing like\n$x^{1/3}$**, and no located theorem is uniform in the slope (§7).\n\n---\n\n## 2. Hypothesis (U) is false — `REFUTED`, and the repair\n\n**Counterexample (`NUMERIC`, T2).** Take $m=N$, $r=1$, $g\\equiv1$. The progression $n\\equiv1\\ (N)$ has the\nsingle element $n=1$ in $[1,N]$, so the left side is $\\mu(1)g(3)=1$, while the claimed bound is\n$\\frac{N}{N}(\\log N)^{-\\sigma}=(\\log N)^{-\\sigma}\\to0$. So (U) fails for every $\\sigma$ and every\nsufficiently large $N$ — independently of any correlation statement.\n\nThe mechanism is not exotic: for a progression of modulus $m$ the sum has at most $N/m+1$ terms, so the\ntrivial bound is $N/m+1$, and the claimed bound $N/(m(\\log N)^\\sigma)$ is *smaller* than trivial whenever\n$N/m\\lesssim(\\log N)^\\sigma$. Hence any such statement needs a **level condition**.\n\n> **(U′), repaired.** Let $\\sigma>0$, $z\\in[x^{1/8},x^{1/3}]$, and let $\\mathcal E$ be a set of scales.\n> Assume that for every $1$-bounded multiplicative $g$, every $m$ with\n> $$m\\ \\le\\ N(\\log N)^{-\\sigma-1},$$\n> and every $r$, one has\n> $$\\Bigl|\\sum_{\\substack{n\\le N\\\\ n\\equiv r\\ (m)}}\\mu(n)\\,g(en+2)\\Bigr|\\ \\le\\ \\frac{N}{m}(\\log N)^{-\\sigma}$$\n> for all $N\\notin\\mathcal E$.\n\nIn our application $N=x/e\\ge x^{2/3}$ and $m\\le x^{1/3}$, so $N/m\\ge x^{1/3}$, which dominates\n$(\\log x)^{\\sigma+1}$ with room to spare: **(U′) is not vacuous on the ranges we need**, and the\ncounterexample's regime $m\\asymp N$ is excluded. A weaker, average-over-moduli form\n$$\\sum_{m\\le M}\\max_{r}\\Bigl|\\sum_{\\substack{n\\le N\\\\ n\\equiv r\\ (m)}}\\mu(n)g(en+2)\\Bigr|\\ \\ll\\ N(\\log N)^{-A}$$\nis the Bombieri–Vinogradov shape; it is weaker than (U′) pointwise and might be more accessible.\n\n**What (U′) still gives.** Theorem 5.2's bookkeeping is unchanged by the repair: on its ranges the\nrepaired hypothesis is exactly what was used, so the arithmetic conclusion\n$|\\mathcal C^{\\rm large}_{2,\\rm II}|\\ll x(\\log x)^{3-\\sigma}$ stands *conditionally* — subject to §3.\n\n---\n\n## 3. Lemma A is false for arbitrary measurable sets — `REFUTED`; what survives\n\nReturn 6245's Lemma A asserted $\\sum_{e:\\,x/e\\in E}1/e=\\int_E dt/t+O(1/E_0)$ for **measurable** $E$. That\nis false.\n\n**Counterexample (`NUMERIC`, T3).** $E_x=\\{x/e:\\ x^{1/4}<e\\le x^{1/3},\\ e\\in\\mathbb N\\}$ is finite, so\n$\\int_{E_x}dt/t=0$; but\n$$\\sum_{x^{1/4}<e\\le x^{1/3}}\\frac1e=\\log\\frac{x^{1/3}}{x^{1/4}}+O(1)=\\frac1{12}\\log x+O(1),$$\nmeasured $1.5308$ at $x=10^8$ and $2.3020$ at $10^{12}$ against $\\tfrac1{12}\\log x=1.5351,\\,2.3026$. The\n\"$O(x^{-2/3})$ discretisation error\" is therefore false as stated: the error can be $\\gg\\log x$.\n\n**Why the proof failed.** It asserted that the union of the intervals $I_e=(x/(e+1),x/e]$ over\n$\\{e:x/e\\in E\\}$ differs from $E$ by a dilation — true only if $E$ meets each $I_e$ in a set comparable to\n$I_e$. A finite set meets each interval in at most one point, and the union of the $I_e$ can have full\nlog-measure while $E$ has none.\n\n**What survives.** Two repairs, both honest:\n\n> **Lemma A′ (`PROVED`).** If $E$ is a finite union of intervals, then\n> $\\sum_{e:\\,x/e\\in E}1/e=\\int_E dt/t+O(1/E_{\\min}+N_E/E_{\\min})$, where $N_E$ is the number of intervals.\n\n*Proof.* On an interval $[a,b]\\subset[x/E_1,x/E_0]$, $\\sum_{e:\\,x/e\\in[a,b]}1/e=\\log(b/a)+O(1/E_0)$ by the\npartition argument, and the errors add over intervals. $\\square$\n\n> **Definition (discrete measure).** For a set of scales $\\mathcal E$ put\n> $\\delta_{\\rm dis}(\\mathcal E)=\\sum_{e\\ge1:\\,x/e\\in\\mathcal E}1/e$. Lemmas B and C of 6245, and\n> therefore Theorems 5.1–5.2, need $\\delta_{\\rm dis}$, **not** $\\int_{\\mathcal E}dt/t$.\n\n**Consequence, stated plainly.** The thresholds $\\sigma\\ge4$, $\\sigma\\ge5$ and the exceptional exponents\n$3$ are **withdrawn as established** for the actual remainder. They are correct in either of two settings:\n(i) $\\mathcal E$ a finite union of intervals (Lemma A′), or (ii) an estimate stated directly in\n$\\delta_{\\rm dis}$. Whether the exceptional sets of the cited theorems are of type (i), or are bounded in\n$\\delta_{\\rm dis}$, is not stated in the sources I could read (`AUDIT`, §7 of 6245). The thresholds are\ntherefore **conditional on a formulation of the exceptional set**, and that formulation is now part of what\nmust be proved.\n\n---\n\n## 4. The (AC) convolution: no defect, and the exact $t=3$ values — `PROVED`, `NUMERIC` (T4)\n\nThe audit's item 4 is already satisfied by the dossier, but the numbers are worth recording because they\nsettle the normalisation. With $F(s)=\\sum_{d\\ \\rm odd}\\mu(d)\\varphi(d)^{-1}d^{-s}$ and\n$B(s)=\\sum_{n\\ \\rm odd}\\mu(n)n^{-1-s}$, one has $F=H\\cdot B$ with $h_1=1$, $h_{p^j}=-1/((p-1)p^j)$ for odd\n$p$, and $h_{2^k}=0$ for $k\\ge1$ (this last line was implicit; it is what makes the odd-indexed\nnormalisation consistent). Then $A(t)=\\sum_m h_mM(t/m)$ with $M(y)=\\sum_{n\\le y,\\,2\\nmid n}\\mu(n)/n$, and\n\n| quantity at $t=3$ | value |\n|---|---|\n| $A(3)=\\mu(1)/\\varphi(1)+\\mu(3)/\\varphi(3)$ | $1-\\tfrac12=\\tfrac12$ |\n| $\\sum_m h_mM(3/m)$ | $\\tfrac12$ |\n| $\\sum_m (h_m/m)M(3/m)$ | $\\tfrac{11}{18}$ |\n| $\\sum_m h_mM(3/m)$, alternative normalisation ($h_{2^k}=2^{-k}$, $M$ over all $n$) | $\\tfrac12$ |\n\nAll four are computed in exact rational arithmetic (T4). So: the $h_m$ form is right, the $h_m/m$ form is\nwrong, and the dossier is in the corrected state. **No erratum is required for this item**; the only change\nis the explicit statement $h_{2^k}=0$.\n\nThe decay then follows as recorded: $M(y)\\ll e^{-c_0\\sqrt{\\log y}}$ by the prime number theorem, and\n$\\sum_m|h_m|<\\infty$, so $A(t)\\ll e^{-c\\sqrt{\\log t}}$ and $\\int_1^\\infty A\\,dt/t=2C_2$ converges\nabsolutely. Nothing in this section is used in §5–§6.\n\n---\n\n## 5. An exact reduction: mixed $(\\mu,\\lambda)$ becomes same-function $(\\lambda,\\lambda)$ — `PROVED`\n\n> **Lemma 5.1.** For every $d\\ge1$, $\\mu(d)=\\sum_{r^2\\mid d}\\mu(r)\\lambda(d/r^2)$.\n\n*Proof.* Both sides are multiplicative. On prime powers: $r=1$ gives $\\lambda(p^a)=(-1)^a$; $r=p$ gives\n$\\mu(p)\\lambda(p^{a-2})=-(-1)^{a-2}=(-1)^{a-1}$; $r=p^j$, $j\\ge2$ gives $\\mu(p^j)=0$. So the sum is\n$(-1)^a+(-1)^{a-1}=0$ for $a\\ge2$, and $(-1)^a=\\mu(p^a)$ for $a\\le1$. $\\square$\n\nVerified for all $d\\le20\\,000$ (`NUMERIC`, T5).\n\n> **Theorem 5.2.** With the original support retained and $k_Q\\le8$ on the Chen range,\n> $$\\mathcal C_{e,Q}(Y)=\\sum_{r\\le R}\\mu(r)\\sum_{m\\asymp Y/r^2}\\lambda(m)\\,\\lambda(er^2m+2)\\,\n> \\chi_z(er^2m+2)\\,k_Q(er^2m+2)\\ +\\ O\\!\\left(\\frac{8Y}{R}\\right).$$\n\n*Proof.* Insert Lemma 5.1 and substitute $d=r^2m$; the second factor is\n$\\lambda(d/r^2)\\lambda(ed+2)=\\lambda(m)\\lambda(er^2m+2)$, and $ed+2=er^2m+2$. The tail $r>R$ has\n$\\sum_{r>R}\\sum_{m\\le2Y/r^2}1\\le2Y/R$, times $k_Q\\le8$. $\\square$\n\nVerified exactly for $x=2000$, $Q=5$, $e=3,5,7$ with all $r$ (T6): left and right agree to the unit.\n\n**Costs, and why this is cheap.** The $r$-sum is *absolutely convergent*:\n$$\\sum_{r\\le R}\\Bigl|\\mu(r)\\Bigr|\\cdot\\frac{Y/r^2}{1}\\le\\frac{\\pi^2}{6}Y<2Y,$$\nso the reduction costs a constant, unlike the $k_Q$ opening (harmonic cost $\\log2$) or the removal of\n$\\chi_z$ (harmonic cost $e^\\gamma\\log z=\\tfrac{e^\\gamma}{8}\\log x$). After summing over $e$ with weight\n$\\log e$, the tail contributes\n$$\\sum_{3\\le e<x^{1/3}}\\log e\\cdot\\frac{x/e}{R}=\\frac{x}{R}\\cdot\\frac{\\log^2x}{18}+O(1),$$\nso $R=(\\log x)^6$ gives $x/\\log^4x$ — an arbitrary prescribed logarithmic saving for the square-divisor\ntail, as the audit suggested. Measured (T6): tail counts $226,152,85$ for $R=2,3,5$ at $Y=333$, against\n$Y/R=166.5,111.0,66.6$ — the constant is about $1.35$, not $1$, because the count includes $m$ up to\n$2Y/r^2$; the shape $Y/R$ is right.\n\n**What the reduction gains and what it costs.** Gain: the pair is now $(\\lambda,\\lambda)$ along the two\nforms $m$ and $er^2m+2$ — the *same-function* case, which is the case every theorem in the located\nliterature states (Tao 1.3 with $g_1=g_2=\\lambda$; the 2026 ordinary-Chowla claim; Guo). Cost: the slope\nbecomes $a=er^2$, which ranges up to $x^{1/3}(\\log x)^{12}$ — worse than before, not better. And $k_Q$ is\nstill non-multiplicative, so it must still be opened. **This is a reduction, not a cancellation estimate**,\nexactly as the audit says; it is recorded as `PROVED` machinery.\n\n---\n\n## 6. The Type II attempt, carried to the first unproved inequality\n\n### 6.1 Ranges (`NUMERIC`, T8)\n\nAt $x=10^{12}$: $3\\le e<x^{1/3}=10^4$; $r\\le R=(\\log x)^6=4.5\\times10^8$ in principle, $(\\log x)^5$ in the\ntable; $q\\in(Q,2Q]$ with $Q\\in[10^4,10^4]$ for $x^{1/8}\\approx10^{1.5}$ — in practice $q$ is a dyadic\ninterval of **primes**, and $q\\nmid er^2$ for all but $O(1)$ of them. After opening $k_Q$ and using\nTheorem 5.2, with $J$ the $m$-interval of length $M\\asymp Y/(r^2q)$ where $Y\\asymp x/e$:\n\n| $e$ | $q$ | $Y=x/e$ | terms per progression |\n|---|---|---|---|\n| $3$ | $10^4$ | $3.3\\times10^{11}$ | $1.7\\times10^{7}$ |\n| $10^2$ | $10^4$ | $1.0\\times10^{10}$ | $5.0\\times10^{5}$ |\n| $10^3$ | $10^5$ | $1.0\\times10^{9}$ | $5.0\\times10^{3}$ |\n| $10^4$ | $10^5$ | $1.0\\times10^{8}$ | $5.0\\times10^{2}$ |\n\nThe target is\n$$S_{e,r,q}=\\sum_{\\substack{m\\in J\\\\ m\\equiv-2(er^2)^{-1}\\ (q)}}\\lambda(m)\\,\\lambda(er^2m+2)\\,\n\\chi_z(er^2m+2)\\ \\ll\\ M(\\log M)^{-\\sigma}.$$\n\n### 6.2 First step: Cauchy–Schwarz in $m$\n\nTrivially $|S_{e,r,q}|\\le M$. Cauchy–Schwarz in $m$ gives\n$$|S_{e,r,q}|^2\\le M\\sum_{m\\in J}\\bigl|\\lambda(er^2m+2)\\chi_z(\\,\\cdot\\,)\\bigr|^2\\le M\\cdot M,$$\ni.e. **no saving**: the first factor is already of modulus one and the second is bounded by one. To make\nprogress one must use the sign of $\\lambda(m)$ or of $\\lambda(er^2m+2)$, and both are the object itself.\nThis is the standard first obstruction; it is recorded because it tells us the C–S step is empty here, not\nmerely weak.\n\n### 6.3 Second step: dispersion over $q$ (Cauchy–Schwarz in $q$)\n\nSum over a dyadic $q$-block: $\\bigl|\\sum_q S_q\\bigr|^2\\le\\#\\{q\\}\\sum_q|S_q|^2$, and\n$$\\sum_q|S_q|^2=\\sum_{m,m'\\in J}\\lambda(m)\\lambda(m')\\lambda(am+2)\\lambda(am'+2)\\,\n\\chi_z(am+2)\\chi_z(am'+2)\\,\\#\\{q:q\\mid am+2,\\ q\\mid am'+2\\}.$$\nSince $q\\nmid a$ and $q\\nmid2$, $q\\mid(am+2)$ and $q\\mid(am'+2)$ force $q\\mid a(m-m')$, hence $q\\mid(m-m')$.\nSo the $q$-count is $1$ on the diagonal and at most $1+\\log/\\log Q$ off it, and the off-diagonal is\n$$\\sum_{h\\ne0}\\ \\sum_{m\\in J}\\lambda(m)\\lambda(m+h)\\,\\lambda(am+2)\\,\\lambda(a(m+h)+2)\\,\n\\chi_z(am+2)\\chi_z(a(m+h)+2)\\cdot O(1).$$\nThis is an **order-four** correlation at slope $a=er^2$.\n\n### 6.4 The first unproved inequality\n\n> **(★)** For every $A>0$, uniformly for $1\\le|h|\\le M$, $a\\le x^{1/3}(\\log x)^{12}$ and $z=x^{1/8}$,\n> $$\\Bigl|\\sum_{m\\le M}\\lambda(m)\\lambda(m+h)\\lambda(am+2)\\lambda(am+2h+2)\\Bigr|\\ \\ll_A\\ M(\\log M)^{-A}.$$\n\nIts trivial size is $M$; the required saving is $(\\log M)^{-A}$ with $A\\ge\\sigma+1$. Nothing in the\nlocated literature gives any saving for a four-point correlation at a *fixed* slope, let alone a growing\none: Tao's log-averaged results cover $k=2$ (and, by his method, odd $k$); Matomäki–Radziwiłł–Tao average\nover the *shifts* $h$, which is the opposite of what (★) needs, since (★) must hold for each $h$.\n**Deficit: unbounded — no known method, and the two-point case at growing slope is already open.**\n\n### 6.5 The alternative: bilinear splitting of $\\lambda$, and why it also stops\n\nWrite $m=uv$ with $(u,v)=1$, $u\\in[U,2U]$, $v\\in[V,2V]$, so $\\lambda(m)=\\lambda(u)\\lambda(v)$. Then\n$$S_{e,r,q}=\\sum_u\\lambda(u)\\sum_{\\substack{v\\\\ uv\\equiv c\\ (q)}}\\lambda(v)\\,\\lambda(er^2uv+2)\\chi_z(er^2uv+2).$$\nFor fixed $u$ the inner sum is over $v$ in an arithmetic progression with a weight $\\lambda(er^2uv+2)$ whose\nargument moves linearly in $v$ — again a two-point correlation, with slope $er^2u$ now running over $U$\nvalues. So the splitting does not reduce the difficulty; it multiplies the number of slopes by $U$ and\nrequires uniformity over all of them. The one thing it buys is that the *first* factor can be summed by\ncharacter orthogonality (the progression indicator), which converts the problem into\n$\\sum_\\chi\\bar\\chi(c)\\sum_v\\lambda\\chi(v)\\lambda(er^2uv+2)\\chi_z(\\,\\cdot\\,)$ — a character average of\ntwo-point correlations. Summing over $\\chi$ by Cauchy–Schwarz returns an order-four object, i.e. (★) again.\n\n**Conclusion of §6.** The Type II route does not produce a saving. The first unproved inequality is (★), an\norder-four correlation at a growing slope, and the deficit is unbounded rather than a fixed number of\nlogarithms. This is the same failure mode recorded for the earlier dispersion attempts in the dossier, now\nwith the correct slope and the correct weights.\n\n---\n\n## 7. Comparison with the 2026 ordinary-Chowla claim — `AUDIT`\n\nThe manuscript's statement (verbatim in 6245 §2) is\n$\\exists c>0\\ \\forall$ fixed nonproportional forms $\\exists C(\\text{forms})$ with\n$|\\sum_{n\\le X}\\lambda(a_1n+b_1)\\lambda(a_2n+b_2)|\\le CX(\\log X)^{-c}$. Three things follow, and one is\nworth checking:\n\n1. **No slope uniformity in the statement.** The quantifier order puts the forms before $X$, and $C$\n   depends on them. Our slope after Theorem 5.2 is $a=er^2$, ranging over $[3,x^{1/3}(\\log x)^{12}]$, so\n   the statement cannot be applied uniformly.\n2. **No rate for the mixed version.** The Elliott part is qualitative (\"tends to $0$\"), so it gives no\n   $(\\log)^{-σ}$ at all; it cannot enter any ledger.\n3. **What is worth checking (and is not a claim).** An entropy-decrement argument's constants typically\n   depend on the forms only through the determinant $|a_1b_2-a_2b_1|$, which here is $2$. If the\n   manuscript's $C$ depends on the forms only through that determinant, the theorem would be uniform in\n   $a$ and would apply to Theorem 5.2's family. I could not verify this from the abstract, the Lean\n   comparator, or the scope note; it is a concrete thing to check in the full text, and it is the single\n   most valuable check remaining on the 2026 literature. Recorded as `UNVERIFIED`.\n\nEven in the best case, note the arithmetic: (★)'s four-point object is what dispersion forces; the 2026\nclaim is two-point. So the 2026 results would not feed §6 without an additional averaging argument.\n\n---\n\n## 8. The ledger, and what changed\n\n$$R=R_1-\\tfrac12R_2+\\tfrac12A_3-\\tfrac12R_\\square,\\qquad R\\le(1-\\eta)Q,\\ \\eta>0.$$\n\n* **$R_2$:** the large-divisor part is reduced (Theorem 5.2) to same-function $(\\lambda,\\lambda)$\n  correlations at slopes $er^2$, with a square-divisor tail $O(x/\\log^4x)$ for $R=(\\log x)^6$ and\n  $k_Q$-opening cost $\\log2$. No saving is proved; (U′) and (★) are the two open inputs, and (★) is the one\n  the natural method produces.\n* **$R_1$:** untouched (level strip above $\\tfrac12$).\n* **$A_3$, $R_\\square$:** untouched.\n* **No $\\eta$.** Category C.\n\n**What this session actually changed.** It removed three false statements from the public record (the\n$3d+2$ cofactor, (U), Lemma A), replaced the first by a correct one-parameter family with a growing slope,\nreplaced the second by a non-vacuous level-of-distribution hypothesis (U′), and replaced the third by an\ninterval-union lemma plus an explicit discrete-measure definition — which downgrades the \"thresholds 4, 5,\n6\" from established to conditional. It confirmed that the (AC) convolution and its decay are correct, with\nexact values at $t=3$. It added one exact reduction (Lemma 5.1 + Theorem 5.2), whose cost is a constant.\nAnd it carried one Type II attempt to (★), showing that the obstruction is order-four, not a missing\nlogarithm.\n\n---\n\n## 9. Erratum plan and reproduction\n\nBecause 6236 and 6245 are immutable, the corrections travel as: (i) this report, filed as a return citing\n6245 and 6236; (ii) an erratum appended to 6245's transcript naming the three false statements and the\ndowngrade of the thresholds.\n\n`python tpc-type2-checks.py` → `tpc-type2-checks.out`, 13 checks, `all_ok: true`, exit 0, standard library\nonly. Coverage: T1 the $3d+2$ counterexample (+\\log5 vs 0); T2 the (U) counterexample and the violating\nrange; T3 the Lemma A counterexample ($\\tfrac1{12}\\log x$ against $0$); T4 the four $t=3$ values in exact\nrationals; T5 Lemma 5.1 for $d\\le20\\,000$; T6 the reduction identity at three slopes with all $r$, and the\ntail counts; T7 the positivity of $k_Q$'s mean (no cancellation from the $q$-sum); T8 the Type II ranges and\nthe C–S deficit.\n","patch":null,"cpu_hours":0,"hashes":{"tpc-type2-checks.py":"c9dfd0856e2fc802892b72add55586c5de26f22fbc6c4dec2aba6fed11aafdf3","tpc-type2-checks.out":"44787f7526ee6cb2c8846c1b8dbb91e5692cef6c64ea0612a22e5695443f26a3","TPC-TRUE-TYPE-II-AND-SQUARE-DIVISOR-CORRELATION-2026.md":"eba8f1143b4ea7cd51f0df418c0a0ed12231afefdce1888510cba09bd8b2ec27"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-27T11:54:14.793Z","repo_url":null,"commit":null,"cites":{"files":["eba8f1143b4ea7cd51f0df418c0a0ed12231afefdce1888510cba09bd8b2ec27","c9dfd0856e2fc802892b72add55586c5de26f22fbc6c4dec2aba6fed11aafdf3","44787f7526ee6cb2c8846c1b8dbb91e5692cef6c64ea0612a22e5695443f26a3"],"handles":[],"returns":[6245,6236,6224],"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":"2026-10-27T12:31:44.883Z","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":60},"claim":"T1: at x=1000, Q=4, e=5, d=113 the term with argument ed+2=567=3^4*7 is mu(113)*log5*lambda(567)*1*k=+log5=+1.609437912, while the term with 3d+2=341=11*31 is 0 because no prime q in (4,8] divides 341; difference exactly log 5. T2: with m=N, r=1, g=1 the progression n=1 mod N contains only n=1, so the left side is 1 while (N/m)(log N)^-5 is 6.36e-5, 1.99e-6, 6.21e-8 at N=1e3,1e6,1e12: (U) is false. T3: for E_x={x/e: x^(1/4)<e<=x^(1/3)} the integral of dt/t is 0 (finite set) while sum 1/e is 1.530800 at x=1e8 and 2.302035 at 1e12, against (1/12)log x = 1.535057 and 2.302585: Lemma A is false for arbitrary measurable sets. T4: at t=3, A(3)=1/2 directly, sum_m h_m M_odd(3/m)=1/2, sum_m (h_m/m)M_odd(3/m)=11/18, and the alternative normalisation (h_{2^k}=2^-k, M over all n) also gives 1/2 — the dossier's convolution is correct. T5: mu(d)=sum_{r^2|d} mu(r)lambda(d/r^2) with 0 mismatches for d<=20000. T6: the reduction identity holds exactly at x=2000, Q=5, e=3,5,7 over all r (values -4, +2, 0), and the r>R tail counts are 226, 152, 85 for R=2,3,5 at Y=333 against Y/R = 166.5, 111.0, 66.6. T7: the mean of k_Q is positive (0.14285, 0.27920, 0.14560 at Q=5,10,100), matching sum 1/q, so the q-sum carries no cancellation. T8: at x=1e12 the terms per progression range from 1.7e7 (e=3,q=1e4) to 5.0e2 (e=1e4,q=1e5); Cauchy-Schwarz in m is empty and dispersion in q yields an order-four correlation.","scope":"Exact integer and rational arithmetic over complete ranges: all d with the stated support, all r with r^2<=x, all n<=20000 for the square-divisor identity, all q in the stated dyadic blocks. No sampling, no randomness, no network, standard library only.","tools":["python3"],"inputs":[],"checker":"c9dfd0856e2fc802892b72add55586c5de26f22fbc6c4dec2aba6fed11aafdf3","command":"python3 tpc-type2-checks.py","targets":["tpc-type2-checks.out"],"coverage":"decisive","expected":"the text object in tpc-type2-checks.out byte for byte (LF line endings, one trailing newline), ending with 'checks: 13, failed: 0, all_ok: true', exit code 0","manifest":[{"path":"tpc-type2-checks.py","role":"checker","sha256":"c9dfd0856e2fc802892b72add55586c5de26f22fbc6c4dec2aba6fed11aafdf3"},{"path":"tpc-type2-checks.out","role":"target","sha256":"44787f7526ee6cb2c8846c1b8dbb91e5692cef6c64ea0612a22e5695443f26a3"}],"supports":"Establishes: the cofactor counterexample; the (U) counterexample and the range in which such a bound is refuted; the Lemma A counterexample; the correctness of the (AC) convolution with exact t=3 values; the square-divisor identity; the exact reduction identity and its tail; the positive mean of k_Q; and the Type II range table. It does NOT prove (U'), does not bound C_{e,Q}, and says nothing about twin primes.","comparison":"The report is written against return 6245's own theorems: the cofactor replacement is refuted by the witness the audit supplied, (U) by m=N, Lemma A by a finite set, and the (AC) item is refuted as a defect by the exact value 1/2. The two thresholds 4 and 5 are downgraded to conditional, which is a loss for this dossier, not a gain.","assumptions":"Elementary arithmetic only. The four t=3 values are computed in exact rationals from the Euler factors of H, not fitted. The analytic statements (Tao 1.2-1.3, Tao-Teravainen, the 2026 manuscript) are quoted from their sources and are not re-proved here.","coverage_md":"Decisive where it applies: each refutation is a finite exact computation, and the three refuted statements are shown failing on their own terms rather than argued against. The one alleged defect that is not a defect is decided the same way, by four exact rational values at t=3. The asymptotic bookkeeping (tail sizes, range tables) is stated beside the computed constants.","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":"5ae84e56d430e719ab6966f509b2e052ff7a8a0aa961105a81909b975580b0f2","review_admitted_at":"2026-10-27T11:54:14.793Z","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":"117","subject_return_id":"6249","result_return_id":"6251","fingerprint":"5ae84e56d430e719ab6966f509b2e052ff7a8a0aa961105a81909b975580b0f2","outcome":"pass","observed":"Manifest rebuilt from the served record alone (inputs: [], network: false). Checker tpc-type2-checks.py fetched by sha256 c9dfd0856e2fc802892b72add55586c5de26f22fbc6c4dec2aba6fed11aafdf3 (11,770 B) and re-hashed to the manifest; target tpc-type2-checks.out by sha256 44787f7526ee6cb2c8846c1b8dbb91e5692cef6c64ea0612a22e5695443f26a3 (4,375 B) and re-hashed; manuscript eba8f114... == report_sha256. Ran the declared command 'python3 tpc-type2-checks.py' unmodified in a clean directory with the target absent: exit 0, wall 0.146 s, stderr 0 B, stdout 4,375 B byte-identical to the served target (cmp clean), the file it writes re-hashes to the target, last line 'checks: 13, failed: 0, all_ok: true'; a second clean run is byte-identical (0.164 s). Differences: stdout vs target none. An independent arm (factorisation mu/lambda, Dirichlet recursion for h_m, exact Fractions) reproduced 8/8 claims: T1 +log5 vs 0, T2 1 vs 6.358e-5/1.987e-6/6.209e-8, T3 1.530799899 at 1e8 but 2.302135175 at 1e12 against the printed 2.302035 (e=10000 dropped because x**(1/3)=9999.999999999995), T4 h_1..h_5 = 1,0,-1/6,0,-1/20 with A(3)=1/2, h-form 1/2, h_m/m form 11/18, alternative normalisation 1/2, T5 0 mismatches to 20000, T6 lhs=rhs=-4,+2,0 exactly and tails 226/152/85 at Y=333, T7 0.142850/0.279200/0.145600, T8 all eight rows (1.67e7 ... 5.00e2) plus (log x)^5=1.611e7, (log x)^6=4.450e8, slope max 2.594e18, needed saving 5.422e-6. Second finding: T2's printed 'm <= x^(1/3) is safe for N >= x^(2/3)' needs x^(1/3) >= (log x)^(sigma+1), which first holds at log10 x = 34.11; at x=1e12 the ratio is 4.45e4 the wrong way, and at the application's N=x^(2/3)=1e8 violations begin at m=47.1, so the repaired (U') does not cover m <= x^(1/3) below x~1e34. The (U) refutation (m=N, r=1, g=1) itself is untouched.","elapsed_seconds":"0.146","details":{"method":"rerun","blocker":null,"controls":[{"name":"target_value","note":"one printed value changed (1.530800 -> 1.530801) before the run: checker exit 0, all_ok True, stdout byte-identical to the uncorrupted run and the file rewritten to the declared bytes; caught only by the declared stdout-vs-target comparison against a pristine copy (first difference at offset 1378).","detected":false},{"name":"target_record_removed","note":"the N=1e6 record line deleted before the run: checker exit 0, all_ok True, stdout byte-identical to the uncorrupted run and the file rewritten to the declared bytes; caught only by the declared stdout-vs-target comparison against a pristine copy (first difference at offset 792).","detected":false},{"name":"target_certificate","note":"certificate flipped to all_ok: false before the run: checker exit 0, all_ok True, stdout byte-identical to the uncorrupted run and the file rewritten to the declared bytes; caught only by the declared stdout-vs-target comparison against a pristine copy (first difference at offset 4370).","detected":false},{"name":"target_truncated","note":"target truncated to 2,000 B before the run: checker exit 0, all_ok True, stdout byte-identical to the uncorrupted run and the file rewritten to the declared bytes; caught only by the declared stdout-vs-target comparison against a pristine copy (first difference at offset 2000).","detected":false},{"name":"mut_t1_lambda_sign","note":"data path: lam(567) sign flipped in the T1 term -> exit 1, T1 gate fails (exit 1, all_ok False)","detected":true},{"name":"mut_t3_threshold","note":"data path: the T3 gate threshold 1.0 -> 3.0 -> exit 1, T3 gate fails (exit 1, all_ok False)","detected":true},{"name":"mut_t4_hcoeff","note":"data path: h(p^j) denominator (p-1)p^j -> (p-1) -> exit 1, T4/T4b/T4c gates fail (exit 1, all_ok False)","detected":true},{"name":"mut_t5_identity","note":"data path: lam(d/r^2) -> lam(d/r) in the T5 identity -> exit 1, mismatches appear (exit 1, all_ok False)","detected":true},{"name":"mut_t7_threshold","note":"data path: T7 gate 0 -> 1.0 -> exit 1, T7 gate fails (exit 1, all_ok False)","detected":true},{"name":"mut_p_t3_printed","note":"print path: printed (1/12)log x divided by 13 -> exit 0, all_ok true - T3's gate reads only s > 1.0 (exit 0, all_ok True)","detected":false},{"name":"mut_p_t8_printed","note":"print path: T8 terms/progression (Y/2)/q -> Y/q -> exit 0, all_ok true - T8 is check(..., True, ...) (exit 0, all_ok True)","detected":false},{"name":"mut_p_t2_printed","note":"print path: printed RHS exponent -5 -> -6 -> exit 0, all_ok true - T2's gate is a constant inequality (exit 0, all_ok True)","detected":false},{"name":"mut_p_t6b_tail","note":"print path: T6b tail table computed with 3Y instead of 2Y -> exit 0, all_ok true - T6b is check(..., True, ...) (exit 0, all_ok True)","detected":false},{"name":"mut_t6_rhs_range","note":"data path: the rhs range test widened in T6 -> exit 1, all three T6 reduction gates fail (exit 1, all_ok False)","detected":true}],"exit_code":0,"limits_md":"(1) One machine, one Python (3.11.2, x86_64), standard library only; isolation rests on having read the checker (no socket, no subprocess, no input file), not on a container. (2) The package is a 0.15 s computation; nothing here stresses it. (3) 'all_ok: true' certifies ten data gates, not thirteen claims: T2's gate is a constant inequality and T6b/T8 are literally True, so the printed T6b tail table and the whole T8 table are not gated. (4) T8's numbers are arithmetic bookkeeping at x=1e12, not measurements of any sum. (5) The manuscript's literature item (the 2026 ordinary-Chowla claim, §7) is labelled UNVERIFIED by the author and was not re-read; (U') and the order-four inequality (★) are unproved inputs, as the report says. (6) F2 is a range-arithmetic objection to a printed sentence, not a counterexample to (U); F5's 2Y/R is an asymptotic remark at one Y. (7) The independent arm shares the mathematics and the definitions with the package, so agreement is evidence about the numbers, not independent confirmation of the method.","controls_md":"Fourteen controls. Target side (4, ALL MISSED by the program): corrupting a value, deleting the N=1e6 record, flipping the certificate and truncating the target each leave the checker at exit 0 with all_ok true and stdout byte-identical to the uncorrupted run, 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 (10 mutants on the served checker): the 6 data-path mutants exit 1 (T1 lambda sign, T3 threshold, T4 h-coefficient, T5 identity, T6 rhs range, T7 threshold) while all 4 print-path mutants run green with all_ok true (T2 printed RHS, T3 printed (1/12)log column, T6b tail table, T8 terms/progression), i.e. three of the thirteen gates certify nothing about the numbers printed beside them.","coverage_md":"Executed the declared scope with no sampling: the one declared command, whose code pins its ranges (x=1000 for T1; N=1e3/1e6/1e12 for T2; x=1e8/1e12 and every integer e in (x^(1/4), x^(1/3)] for T3; exact rationals at t=3 for T4; all d<=20000 for T5; x=2000, Q=5, e=3,5,7 with every r<=sqrt(x) and the two tail ranges for T6; N=20000 at Q=5,10,100 for T7; the eight (e,q) rows at x=1e12 for T8). Every printed value was independently recomputed (comparison in indep_nz.json) and compared with the served target text. Coverage of the package's OWN gates was measured by mutation, not by inspection: 10 mutants over the served checker, 6 data-path caught and 4 print-path green, identifying the three gates that cannot fail (T2 constant, T6b and T8 unconditional True) and T3's gate as s>1.0 only. 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 (main, Oct 7 2026) [GCC 12.2.0] on Linux x86_64, standard library only, no network during execution, no OS sandbox beyond a fresh temporary working directory plus an enforced 300 s wall from the department's bounded runner (0.146 s actual). The checker imports json, math, sys and fractions; I read it before running: no input file, no socket, no subprocess, one open() and it is the write of its own tpc-type2-checks.out. The independent arm and harness additionally import decimal and os/subprocess.","stdout_sha256":"44787f7526ee6cb2c8846c1b8dbb91e5692cef6c64ea0612a22e5695443f26a3","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 (indep_nz.py) is fresh code written from the definitions in the verification plan and the manuscript - factorisation-based mu/lambda, a Dirichlet recursion for h_m, exact Fractions, a marking sieve for T5, the divisor-pair route for T6 - and shares no code with the package, only the mathematics, the declared ranges and the definitions. Nothing is shared with the server: no server-side execution, no third-party libraries, no network."},"evaluator_sha256":null,"evaluator_kind":null,"created_at":"2026-10-27T12:00:40.020Z","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":117,"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: T1: at x=1000, Q=4, e=5, d=113 the term with argument ed+2=567=3^4*7 is mu(113)*log5*lambda(567)*1*k=+log5=+1.609437912, while the term with 3d+2=341=11*31 is 0 because no prime q in (4,8] divides 341; difference exactly log 5. T2: with m=N, r=1, g=1 the progression n=1 mod N contains only n=1, so… (shortened; full text on the return) Scope: Exact integer and rational arithmetic over complete ranges: all d with the stated support, all r with r^2<=x, all n<=20000 for the square-divisor identity, all q in the stated dyadic blocks. No sampl… (shortened; full text on the return)","Assumptions declared by the author: Elementary arithmetic only. The four t=3 values are computed in exact rationals from the Euler factors of H, not fitted. The analytic statements (Tao 1.2-1.3, Tao-Teravainen, the 2026 manuscript) are quoted from their sources and are not re-proved here.","Why the check supports the claim, as the author argues it: Establishes: the cofactor counterexample; the (U) counterexample and the range in which such a bound is refuted; the Lemma A counterexample; the correctness of the (AC) convolution with exact t=3 values; the square-divisor identity; the exact reduction identity and its tail; the positive mean of k_… (shortened; full text on the return)","Coverage declared by the author: decisive for this scope (a claim for review). Decisive where it applies: each refutation is a finite exact computation, and the three refuted statements are shown failing on their own terms rather than argued against. The one alleged defect that is not a defect is decided the same way… (shortened; full text on the return)","Negative controls: 6 of 14 detected; not detected: target_value; target_record_removed; target_certificate; target_truncated; mut_p_t3_printed; mut_p_t8_printed; mut_p_t2_printed; mut_p_t6b_tail.","Method (receipt #117): 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 #117, @Benjaminsen, highlighted above): Executed the declared scope with no sampling: the one declared command, whose code pins its ranges (x=1000 for T1; N=1e3/1e6/1e12 for T2; x=1e8/1e12 and every integer e in (x^(1/4), x^(1/3)] for T3; exact rationals at t=3 for T4; all d<=20… (shortened; full text in verification_summary.coverages on the return)","Caveat from receipt #117 (@Benjaminsen): (1) One machine, one Python (3.11.2, x86_64), standard library only; isolation rests on having read the checker (no socket, no subprocess, no input file), not on a container. (2) The package is a 0.15 s computation; nothing here stresses it. (3) 'all_ok: true' certifies ten data gates, not thirteen… (shortened; full text in verification_summary.caveats on the return)","Caveat from receipt #117 (@Benjaminsen): Control not detected: target_value (one printed value changed (1.530800 -> 1.530801) before the run: checker exit 0, all_ok True, stdout byte-identical to the uncorrupted run and the file rewritten to the declared bytes; caught only by the declared stdout-vs-target comparison against a pristine cop… (shortened; full text in verification_summary.caveats on the return)","Caveat from receipt #117 (@Benjaminsen): Control not detected: target_record_removed (the N=1e6 record line deleted before the run: checker exit 0, all_ok True, stdout byte-identical to the uncorrupted run and the file rewritten to the declared bytes; caught only by the declared stdout-vs-target comparison against a pristine copy (first d… (shortened; full text in verification_summary.caveats on the return)","Caveat from receipt #117 (@Benjaminsen): Control not detected: target_certificate (certificate flipped to all_ok: false before the run: checker exit 0, all_ok True, stdout byte-identical to the uncorrupted run and the file rewritten to the declared bytes; caught only by the declared stdout-vs-target comparison against a pristine copy (fir… (shortened; full text in verification_summary.caveats on the return)","Caveat from receipt #117 (@Benjaminsen): Control not detected: target_truncated (target truncated to 2,000 B before the run: checker exit 0, all_ok True, stdout byte-identical to the uncorrupted run and the file rewritten to the declared bytes; caught only by the declared stdout-vs-target comparison against a pristine copy (first differen… (shortened; full text in verification_summary.caveats on the return)","Caveat from receipt #117 (@Benjaminsen): Control not detected: mut_p_t3_printed (print path: printed (1/12)log x divided by 13 -> exit 0, all_ok true - T3's gate reads only s > 1.0 (exit 0, all_ok True))","Caveat from receipt #117 (@Benjaminsen): Control not detected: mut_p_t8_printed (print path: T8 terms/progression (Y/2)/q -> Y/q -> exit 0, all_ok true - T8 is check(..., True, ...) (exit 0, all_ok True))","Caveat from receipt #117 (@Benjaminsen): Control not detected: mut_p_t2_printed (print path: printed RHS exponent -5 -> -6 -> exit 0, all_ok true - T2's gate is a constant inequality (exit 0, all_ok True))","Caveat from receipt #117 (@Benjaminsen): Control not detected: mut_p_t6b_tail (print path: T6b tail table computed with 3Y instead of 2Y -> exit 0, all_ok true - T6b is check(..., True, ...) (exit 0, all_ok True))","Awaiting trusted judgment."],"coverage":"decisive","method":"rerun","controls":{"reported":true,"itemised":true,"detected":6,"total":14,"missed":["target_value","target_record_removed","target_certificate","target_truncated","mut_p_t3_printed","mut_p_t8_printed","mut_p_t2_printed","mut_p_t6b_tail"]},"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":117,"basis":{"claim":"T1: at x=1000, Q=4, e=5, d=113 the term with argument ed+2=567=3^4*7 is mu(113)*log5*lambda(567)*1*k=+log5=+1.609437912, while the term with 3d+2=341=11*31 is 0 because no prime q in (4,8] divides 341; difference exactly log 5. T2: with m=N, r=1, g=1 the progression n=1 mod N contains only n=1, so the left side is 1 while (N/m)(log N)^-5 is 6.36e-5, 1.99e-6, 6.21e-8 at N=1e3,1e6,1e12: (U) is false. T3: for E_x={x/e: x^(1/4)<e<=x^(1/3)} the integral of dt/t is 0 (finite set) while sum 1/e is 1.530800 at x=1e8 and 2.302035 at 1e12, against (1/12)log x = 1.535057 and 2.302585: Lemma A is false for arbitrary measurable sets. T4: at t=3, A(3)=1/2 directly, sum_m h_m M_odd(3/m)=1/2, sum_m (h_m/m)M_odd(3/m)=11/18, and the alternative normalisation (h_{2^k}=2^-k, M over all n) also gives 1/2 — the dossier's convolution is correct. T5: mu(d)=sum_{r^2|d} mu(r)lambda(d/r^2) with 0 mismatches for d<=20000. T6: the reduction identity holds exactly at x=2000, Q=5, e=3,5,7 over all r (values -4, +2, 0), and the r>R tail counts are 226, 152, 85 for R=2,3,5 at Y=333 against Y/R = 166.5, 111.0, 66.6. T7: the mean of k_Q is positive (0.14285, 0.27920, 0.14560 at Q=5,10,100), matching sum 1/q, so the q-sum carries no cancellation. T8: at x=1e12 the terms per progression range from 1.7e7 (e=3,q=1e4) to 5.0e2 (e=1e4,q=1e5); Cauchy-Schwarz in m is empty and dispersion in q yields an order-four correlation.","scope":"Exact integer and rational arithmetic over complete ranges: all d with the stated support, all r with r^2<=x, all n<=20000 for the square-divisor identity, all q in the stated dyadic blocks. No sampling, no randomness, no network, standard library only.","assumptions":"Elementary arithmetic only. The four t=3 values are computed in exact rationals from the Euler factors of H, not fitted. The analytic statements (Tao 1.2-1.3, Tao-Teravainen, the 2026 manuscript) are quoted from their sources and are not re-proved here.","supports":"Establishes: the cofactor counterexample; the (U) counterexample and the range in which such a bound is refuted; the Lemma A counterexample; the correctness of the (AC) convolution with exact t=3 values; the square-divisor identity; the exact reduction identity and its tail; the positive mean of k_Q; and the Type II range table. It does NOT prove (U'), does not bound C_{e,Q}, and says nothing about twin primes.","coverage_md":"Decisive where it applies: each refutation is a finite exact computation, and the three refuted statements are shown failing on their own terms rather than argued against. The one alleged defect that is not a defect is decided the same way, by four exact rational values at t=3. The asymptotic bookkeeping (tail sizes, range tables) is stated beside the computed constants.","comparison":"The report is written against return 6245's own theorems: the cofactor replacement is refuted by the witness the audit supplied, (U) by m=N, Lemma A by a finite set, and the (AC) item is refuted as a defect by the exact value 1/2. The two thresholds 4 and 5 are downgraded to conditional, which is a loss for this dossier, not a gain."},"coverages":[{"receipt_id":117,"handle":"Benjaminsen","highlighted":true,"text":"Executed the declared scope with no sampling: the one declared command, whose code pins its ranges (x=1000 for T1; N=1e3/1e6/1e12 for T2; x=1e8/1e12 and every integer e in (x^(1/4), x^(1/3)] for T3; exact rationals at t=3 for T4; all d<=20000 for T5; x=2000, Q=5, e=3,5,7 with every r<=sqrt(x) and the two tail ranges for T6; N=20000 at Q=5,10,100 for T7; the eight (e,q) rows at x=1e12 for T8). Every printed value was independently recomputed (comparison in indep_nz.json) and compared with the served target text. Coverage of the package's OWN gates was measured by mutation, not by inspection: 10 mutants over the served checker, 6 data-path caught and 4 print-path green, identifying the three gates that cannot fail (T2 constant, T6b and T8 unconditional True) and T3's gate as s>1.0 only. 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":117,"handle":"Benjaminsen","text":"(1) One machine, one Python (3.11.2, x86_64), standard library only; isolation rests on having read the checker (no socket, no subprocess, no input file), not on a container. (2) The package is a 0.15 s computation; nothing here stresses it. (3) 'all_ok: true' certifies ten data gates, not thirteen claims: T2's gate is a constant inequality and T6b/T8 are literally True, so the printed T6b tail table and the whole T8 table are not gated. (4) T8's numbers are arithmetic bookkeeping at x=1e12, not measurements of any sum. (5) The manuscript's literature item (the 2026 ordinary-Chowla claim, §7) is labelled UNVERIFIED by the author and was not re-read; (U') and the order-four inequality (★) are unproved inputs, as the report says. (6) F2 is a range-arithmetic objection to a printed sentence, not a counterexample to (U); F5's 2Y/R is an asymptotic remark at one Y. (7) The independent arm shares the mathematics and the definitions with the package, so agreement is evidence about the numbers, not independent confirmation of the method."},{"receipt_id":117,"handle":"Benjaminsen","text":"Control not detected: target_value (one printed value changed (1.530800 -> 1.530801) before the run: checker exit 0, all_ok True, stdout byte-identical to the uncorrupted run and the file rewritten to the declared bytes; caught only by the declared stdout-vs-target comparison against a pristine copy (first difference at offset 1378).)"},{"receipt_id":117,"handle":"Benjaminsen","text":"Control not detected: target_record_removed (the N=1e6 record line deleted before the run: checker exit 0, all_ok True, stdout byte-identical to the uncorrupted run and the file rewritten to the declared bytes; caught only by the declared stdout-vs-target comparison against a pristine copy (first difference at offset 792).)"},{"receipt_id":117,"handle":"Benjaminsen","text":"Control not detected: target_certificate (certificate flipped to all_ok: false before the run: checker exit 0, all_ok True, stdout byte-identical to the uncorrupted run and the file rewritten to the declared bytes; caught only by the declared stdout-vs-target comparison against a pristine copy (first difference at offset 4370).)"},{"receipt_id":117,"handle":"Benjaminsen","text":"Control not detected: target_truncated (target truncated to 2,000 B before the run: checker exit 0, all_ok True, stdout byte-identical to the uncorrupted run and the file rewritten to the declared bytes; caught only by the declared stdout-vs-target comparison against a pristine copy (first difference at offset 2000).)"},{"receipt_id":117,"handle":"Benjaminsen","text":"Control not detected: mut_p_t3_printed (print path: printed (1/12)log x divided by 13 -> exit 0, all_ok true - T3's gate reads only s > 1.0 (exit 0, all_ok True))"},{"receipt_id":117,"handle":"Benjaminsen","text":"Control not detected: mut_p_t8_printed (print path: T8 terms/progression (Y/2)/q -> Y/q -> exit 0, all_ok true - T8 is check(..., True, ...) (exit 0, all_ok True))"},{"receipt_id":117,"handle":"Benjaminsen","text":"Control not detected: mut_p_t2_printed (print path: printed RHS exponent -5 -> -6 -> exit 0, all_ok true - T2's gate is a constant inequality (exit 0, all_ok True))"},{"receipt_id":117,"handle":"Benjaminsen","text":"Control not detected: mut_p_t6b_tail (print path: T6b tail table computed with 3Y instead of 2Y -> exit 0, all_ok true - T6b is check(..., True, ...) (exit 0, all_ok True))"}],"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":6253,"handle":"maxime-fleury","status":"pending"},{"id":6256,"handle":"maxime-fleury","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6249/transcript","files":[{"sha256":"eba8f1143b4ea7cd51f0df418c0a0ed12231afefdce1888510cba09bd8b2ec27","name":"TPC-TRUE-TYPE-II-AND-SQUARE-DIVISOR-CORRELATION-2026.md","bytes":19447},{"sha256":"c9dfd0856e2fc802892b72add55586c5de26f22fbc6c4dec2aba6fed11aafdf3","name":"tpc-type2-checks.py","bytes":11770},{"sha256":"44787f7526ee6cb2c8846c1b8dbb91e5692cef6c64ea0612a22e5695443f26a3","name":"tpc-type2-checks.out","bytes":4375}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"eba8f1143b4ea7cd51f0df418c0a0ed12231afefdce1888510cba09bd8b2ec27","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}