# Adversarial verification of the multiplicative band transfer **Twin-prime infinitude and the sufficient signed margin remain OPEN. This review adds no estimate, no controlled region and no margin.** It checks one newly written derivation against its own stated scope and against the printed hypotheses of the two theorems it imports. **Current qualification:** [round-review-0906.md](round-review-0906.md) retains the checked derivation (1)--(6) but corrects the overly broad dyadic exclusion in check l below. Bounded-interval stability gives a dyadic scale-average saving with another exponent loss. No every-dyadic bound, o(N) or twin margin follows. Proper-prime-power branches also already have a separate negligible fixed-corner bound. Baseline: commit `10f64919495ee7e5300120ecb6aead1ca7e91c94`, `git status --short` empty at dispatch. Reviewed note: [prime-band-transfer.md](prime-band-transfer.md), status PARTIAL. ## 1. Verdicts | # | step | verdict | one-line reason | |---|---|---|---| | a | multiplicativity and 1-boundedness of g_{B,t} | PASS | L_B is additive on coprime arguments and mu is multiplicative, so g(mn)=g(m)g(n) for (m,n)=1; \|g\|<=1 since \|mu\|<=1 and the exponential is unimodular | | b | Fourier identity (1) | PASS | psi=u*chi with chi smooth and 1 on [0,2] exists, hat psi is Schwartz hence L^1, and L_B(n)<=log n<=2 log X for n<=2X+2 puts ell_B in [0,2] where psi(u)=u | | c | pretentious-distance comparison (2) | PASS | D uses only prime values, g_{B,t}(p) and mu(p) differ only for p in B and by at most 2 in modulus, and \|inf f - inf g\| <= sup\|f-g\|; Mertens makes 2*sum_{p in B}1/p = 2 log(beta/alpha)+o(1) | | d | MRT (1.12) supplies exp M >> (log X)^{7/24} but not log X | PASS | (1.12) read at source gives M >= (1/3-eps) loglog X + O(1) for conductors q <= (log X)^{1/125}; at Y=X^2 and eps=1/24 this is 7/24 > 1/4, and 1/3-eps never reaches exponent 1 | | e | source theorem structure, shift 2, modulus 1, functions fixed per X | PASS | Theorem 3.1(ii) read at source: L^{-c} bound off a set of logarithmic measure << log X * L^{-c}, W in [L^c], integers b,h1,h2 = O(L^c) with h1 != h2, natural average on (N,2N], g1,g2 fixed given X | | f | integral (3) | PASS | \|A\|<=2 on the exceptional set of measure << log X (log X)^{-c}, and the theorem's (log X)^{-c} off it against ((1/2) log X) of measure; both terms are (log X)^{1-c}, uniformly in t,s | | g | Tonelli order for (4) | PASS | the integrand \|hat psi(t)\|\|hat psi(s)\|\|A_{t,s}(N)\| is nonnegative and jointly measurable, so the t,s integral passes outside the N integral without any common exceptional set | | h | window-to-band replacement and CRT error versus N >= sqrt(X) | PASS | the symmetric difference is the upper strip of multiplicative width at most 2 plus at most two integers above V; the main term is O_eta(N log X) and the CRT error is O(X^{29/100+4eta+o(1)} log^2 X) <= N X^{-1/5} log^2 X at every N >= sqrt(X) | | i | mesh count, symmetric-difference mass, disjointness, error (5) | PASS | O(1/delta) intervals cover theta in [1/2,1]; endpoint log-shift O(delta log X + 1) gives symmetric-difference mass O(delta log X + 1) against band mass O_eta(log X); left and right bands stay separated because 1/20+2eta < 3/25 | | j | summing over frozen pairs, delta=(log X)^{-c/2}, boxed (6) | PASS | each mesh interval's integral is bounded by the full nonnegative integral, so the cost is exactly 1/delta; at delta=(log X)^{-c/2} the two competing terms are equal and 1/log X is dominated because c<1 | | k | proper-square error and endpoint rounding absorbed | PASS | corner-correlation (5) carries O(x^{19/20+eps}) at x=2N, which after the N log^2 X normalization is X^{-1/40+eps'}, below every log power | | l | continuous versus dyadic in section 4 | CORRECTED SCOPE | a bare integral does not give every-dyadic control, but interval stability additionally gives a dyadic scale average; round-review-0906 §2 derives it. The no-o(N) and no-margin conclusions stand. | **First actual gap: there is no load-bearing gap.** The chain (1) to (6) follows at the stated scope from the two imported theorems plus elementary steps. Four steps are true but unstated in the note, and one sentence is imprecise; all five are recorded in section 3 and none of them changes a bound. ## 2. What was read at source Fetched with `curl` and hashed locally on 2026-09-06. Both PDF digests reproduce the custody hashes recorded in the reviewed note. | source | version and URL | what was read | SHA-256 of the PDF | |---|---|---|---| | Tao and Teravainen, *Quantitative correlations and some problems on prime factors of consecutive integers* | v2, 2026-04-25, `https://arxiv.org/html/2512.01739v2` and `https://arxiv.org/pdf/2512.01739v2` | Theorem 3.1 in full including axioms (i) and (ii), (3.1)-(3.4), Remarks 3.2 (all three bullets), the definitions of the pretentious distance and of M(g;X) and M(g;X,Q) | `ce10e83b10c6544e1dbff037a5e4efa0e387892e0fc596ae09dce76025d7b41e` | | Matomaki, Radziwill and Tao, *An averaged form of Chowla's conjecture* | `https://arxiv.org/pdf/1503.05121` (latest version; digest matches the note's recorded v3) | equation (1.12) with its displayed justification, the definitions of D(f,g;X) and M(g;X,Q), and the statement of Theorem 1.6 around it | `8a2633b1594615fe0c340bbca01ad059be5bd66d3495bd028e7e9d2264f1e688` | No fetch failed. Nothing below rests on an abstract, a search summary or a secondary description. Printed statements used, quoted in substance: - Theorem 3.1(ii): for 1-bounded multiplicative g1,g2 and 1 <= L <= log X, if delta_N = 0 for all X^{0.4} <= N <= X and exp(M(g1;X^2,log^{1/125}X)) >> L, then for a sufficiently small absolute c > 0 there is E contained in [sqrt X, X] with (1/log X) integral_E dt/t << L^{-c} such that for any W in [L^c] and integers b,h1,h2 = O(L^c) with h1 != h2, (W/N) sum_{N chi(n)n^{it}; X)^2, with D(f,g;X)^2 = sum_{p<=X} (1 - Re(f(p) conj(g(p))))/p. The definitions in the two papers agree verbatim. - MRT (1.12): for g = lambda and Q, M as in their Theorem 1.6, for every eps > 0, M >= inf over |t| <= X, q <= Q, chi of sum_{exp((log X)^{2/3+eps}) <= p <= X} (1 + Re chi(p)p^{it})/p, which is at least (1/3 - eps) loglog X + O(1), read from the arXiv PDF of 1503.05121v3, section 1.1, display (1.12), PDF page 6, in this note's notation; the source justifies it by the Vinogradov-Korobov zero-free region, and it applies because chi has conductor q <= (log X)^{1/125}. ## 3. Checked hazards, and the five things the note leaves unsaid **H1. Condition (3.2) is scoped to axiom (i) only. Checked, and it matters.** Theorem 3.1's technical condition g1(p) = 1 for exp(log^{1/11} X) <= p <= exp(log^{1/10} X) is printed after (3.1) and before the label (ii). In the LaTeXML markup of the v2 HTML both (3.1) and (3.2) sit inside the list item `S3.I1.ix1`, which carries the tag `(i)`; axiom (ii) is the separate item `S3.I1.ix2`. The paper's own third bullet in Remarks 3.2 derives a Chowla-type bound for Liouville from Theorem 3.1(ii), and lambda(p) = -1 violates (3.2), so (3.2) cannot bind case (ii). This is load-bearing for the transfer: g_{B,t}(p) = -1 for every p below the band, including that whole range, so a global (3.2) would have refuted the application outright. The reviewed note does not mention (3.2) at all; recording the check here closes that hole. Calibration: verified against the source markup and against the paper's own use. **H2. The shift assignment is not written down, and either choice is harmless.** The note's A_{t,s}(N) = N^{-1} sum_{N V at the top of the n-range, and contributes a second discrepancy set. That set is contained in (V, n/D_0] with n/D_0 <= V + 1 + o(1), so at most two integers wide, which is what "vanishing floor adjustments" has to mean. The finite validator confirms that at two scales every disagreement between the exact window and the pure band lies in the upper strip or in the two integers above V, and that a strip factor of 1.001 in place of 2 fails, so the width 2 is sharp rather than generous. Derived plus verified on finite instances; not a defect. **H5. One imprecise sentence in section 4.** "the bound on the normalized absolute average |A_w|/N is only O_eta(log^{2-c_*}X)" reads as a pointwise bound on |A_w(N)|/N. What (6) gives is a bound on the continuous logarithmic scale average of |A_w(N)|/(N log^2 X). The next two sentences of the same paragraph state the pointwise limitation correctly, so no downstream claim depends on the loose reading, but the sentence should say "scale-averaged". Wording only. **H6. An observation about the MRT source, not load-bearing here.** MRT's parenthetical in section 1.1 after (1.11) states that M(g;X,Q) is non-decreasing in Q. From their own printed definition M is an infimum over a set of characters that grows with Q, hence non-increasing in Q. The transfer does not use that parenthetical: (1.12) is a direct lower bound whose stated justification is the zero-free region for conductors q <= (log X)^{1/125}, which is exactly the conductor range Theorem 3.1(ii) requires, and Tao and Teravainen cite (1.12) for precisely M(lambda; X^2, log^{1/125} X). Recorded so that a later reuse of the source does not lean on the parenthetical. Two further points, both consistent with the note and with review 21 F7. Theorem 3.1(ii) puts the non-pretentiousness hypothesis on g1 alone; g2 needs only 1-boundedness and multiplicativity, so the note establishes more than it needs when it covers both. And Remarks 3.2 records that Pilatte obtained L^{-c} errors with L = log X in the model case of Liouville. That is a different theorem about a different function, reached without (3.3), and it does not license L = log X for g_{B,t} through Theorem 3.1(ii). The note's refusal to take L = log X is correct on the input it names. ## 4. The arithmetic behind checks h, i and j, written out At x = 2N, with w = 6/25 on the left and w = 1/20 on the right and 0 < eta < 1/400 fixed. *Window against band.* The exact window of corner-correlation (5) is {r prime : r > V, D_1 < n/r <= D_0}. Its lower edge coincides with the band's lower edge, since r > floor(x^w) and r > x^w agree on integers. Its upper edge is r < n/D_1, which for n in (N, 2N] sweeps [(1/2) x^{w+2eta}, x^{w+2eta}(1+o(1))], a multiplicative interval of ratio 2. The one remaining piece is r < n/D_0 with n/D_0 <= V + 1 + o(1), the sliver of H4. *Cost of the swap.* Bounding |L_B L'_{B'} - L_w L'_w| <= Delta * L'_{B'} + L_w * Delta' and swapping the order of summation, each piece is sum_{r,r'} (log r)(log r') (N/(rr') + O(1)) with r != r'. Mertens gives sum over a multiplicative-width-2 strip of (log r)/r = log 2 + O(1) and over a full band = 2eta log x + O(1), so the main term is O_eta(N log X). The error term is O(#pairs * log^2 X) with #pairs <= X^{29/100+4eta+o(1)}. At eta = 1/400 the exponent is exactly 3/10, so at the worst scale N = sqrt X the ratio #pairs/N is at most X^{3/10-1/2} = X^{-1/5}, and the note's quoted X^{-1/10} in (5) is the weaker and therefore safe statement. This is the check the brief names "CRT error versus N >= sqrt(X)": the comparison does go against N and not against X, and it survives at the bottom of the range with a fixed power to spare. *Disjointness at the bottom of the range.* The left band sits above X^{w/2} = X^{3/25} = X^{0.12} when N >= sqrt X; the right band sits below X^{1/20+2eta} <= X^{11/200} = X^{0.055}. So r != r' throughout and CRT applies with error O(1) per pair. *Mesh.* theta = log N/log X ranges over [1/2,1], so O(1/delta) intervals of length delta cover it. Freezing moves each log-endpoint by O(delta log X + 1), giving symmetric-difference reciprocal-log mass O(delta log X + 1) against a band mass O_eta(log X); after the N log^2 X normalization this is O_eta(delta + 1/log X), and the CRT error contributes O(X^{-1/5}). Summing (4) over the O(1/delta) frozen pairs costs the factor 1/delta because each mesh interval's nonnegative integral is at most the whole-range integral. Balancing delta^{-1}(log X)^{-c} against delta gives delta = (log X)^{-c/2} and c_* = c/2, and 1/log X is dominated because c < 1. ## 5. What (6) gives, exactly, and what it does not Derived, conditional on the two imports, at fixed eta and for all large X: the normalized continuous logarithmic scale average over N in [sqrt X, X] of |A_w(N)|/(N log^2 X) is O_eta((log X)^{-c_*}) for some fixed c_* > 0 whose value is not extracted. Against the unsigned envelope of the same subfamily, of order eta^2 N log^2 N by corner-correlation section 1.5 and review 21 F6, that is a saving of a factor (log X)^{-c_*} in the scale average. It does not give, and the note does not claim: - a bound at every prescribed N. The later sampling argument gives a dyadic scale average and unbounded good dyadic scales, at a weak relative rate; membership in an exceptional set alone does not exclude this; - o(N) at any scale, since the scale-averaged bound on |A_w|/N is only O_eta((log X)^{2-c_*}); - an estimate for prime-r branches with s > 1 or s' > 1 and their non-squarefree inputs. Proper prime powers are separately negligible by corner-correlation §1.1; formula (1) is not their representation; - a new controlled divisor region, a change to any cut, or a signed margin. ## 6. Still open after this review - The sufficient signed twin margin C_2 x + E_dagger(x) >= c_0 x/log^K x. - Remaining prime-r branches with s>1 or s'>1, and the complement of S_0 at the global precision. The proper-prime-power bound is completed. - A sufficiently strong rate and full-coefficient bound on the scales needed by the consumer. The later dyadic scale-average consequence is too weak for this and is not an every-dyadic pointwise estimate. - The numerical value of c_*, which is inherited from an absolute constant in Theorem 3.1 that the source does not make explicit. Falsifiers for this review, and their status: a global reading of the source's condition (3.2) would refute the application, and the source markup and the paper's own Liouville corollary rule that reading out (checked, section 3 H1); a failure of uniformity in (2) over the frozen bands would break the mesh, and Mertens supplies it (checked, H3); a CRT error of size N in (5) would break the transfer, and the pair count is a fixed power below N at N = sqrt X (checked, section 4); a non-integrable hat psi would break (1), and psi in C_c^infinity rules it out (checked, b). The imported theorems are not reproved and their proofs were not read. ## 7. Verification The finite ingredients are checked in [next-transfer-review-validation.js](next-transfer-review-validation.js), with its output embedded by `node research/qc/embed.js`. It checks multiplicativity of g_{B,t} on coprime pairs and 1-boundedness across four bands and five values of t; fires two controls, that g_{B,t} is not completely multiplicative and that mu*L_B is not multiplicative; verifies the window-to-band geometry of H4 exactly at two scales with an active tightened-strip control; and checks the exponent arithmetic of section 4 by cross-multiplied integer rationals. It is finite algebra. It does not test the Fourier integral, the pretentious distance comparison, either imported theorem or any asymptotic claim, and it does not prove (6).