# Completed review specification: the arithmetic reduction of the twin sum > Public mirror edition: repository-internal working instructions were removed. See MIRROR.md. **This review round is completed at its recorded scope.** Its checklist is retained so readers can assess what was requested and what was actually checked. Current review assignments belong to [RESEARCH-EXECUTION.md](RESEARCH-EXECUTION.md), lane V. The next open review specification is [review-request-0907.md](review-request-0907.md), covering the 2026-09-07 measurement and source-reading session. [handoff-review-0906.md](handoff-review-0906.md) and reports 20–21 record the bounded review and readiness pass. [round-review-0906.md](round-review-0906.md) records the subsequent scale and coefficient corrections. Read these dispositions before treating any item below as still unchecked. ## 1. What to read, in order 1. [TWIN-REDUCTION.md](TWIN-REDUCTION.md): the whole campaign in one document, with the review state of each claim in its §9. 2. [chain-review-0906.md](chain-review-0906.md): what the three review passes on the reduction chain checked and did not check. 3. The owning notes, in the order the claims depend on each other: [prime-detection-spec.md](prime-detection-spec.md), [polylog-fold-transfer.md](polylog-fold-transfer.md), [shifted-prime-decomposition.md](shifted-prime-decomposition.md), [signed-divisor-grouping.md](signed-divisor-grouping.md), [endpoint-fourier.md](endpoint-fourier.md), [endpoint-pairing.md](endpoint-pairing.md), [prime-power-dispersion.md](prime-power-dispersion.md), [residual-coverage.md](residual-coverage.md), [grouped-divisor-moment.md](grouped-divisor-moment.md), [reachability-coverage.md](reachability-coverage.md), [corner-correlation.md](corner-correlation.md). 4. The review reports themselves, preserved verbatim in [history/reviews-0906/](history/reviews-0906/README.md), only after forming your own view of a step. The outcome register [OUTCOMES.md](OUTCOMES.md) records each result with its limits under its question; [QUESTIONS.md](QUESTIONS.md) is the generated index of every question attacked. ## 2. The claims, ranked by leverage A defect in a higher row invalidates every lower row that depends on it. | rank | claim | calibration | owning record | consequence of a defect | |---|---|---|---|---| | 1 | S(x) = C_2 x + E_dagger(x) + O_H(x/log^H x) for every fixed H, with E_dagger the exact CRT endpoint sum on W_dagger | derived; three passes, no defect | chain-review-0906.md; the notes in §1 item 3 | every downstream statement changes | | 2 | the endpoint remainder has the block shape sum_m A_left(gm) Y(m) with separate left and right coefficients and an h-only harmonic weight | derived; reconstructed exactly in review 06 | residual-coverage.md, endpoint-fourier.md §2, signed-divisor-grouping.md (15)-(18) | every dispersion budget changes | | 3 | the grouped moment bound (2)-(3) for arbitrary bounded coefficients, and the budgets (1+a)/2, a/2+3b/2, a | derived; review 01 | grouped-divisor-moment.md §§1-4 | the controlled region and the ceiling change | | 4 | at a prescribed fixed exponent margin S_0 fails that margin for every nonzero-kernel saving; uniform gamma=2 covers its complement | derived; reviews 06 and 19 | reachability-coverage.md | the direction of the campaign changes | | 5 | on S_0 the remainder equals the full C(n)C'(n-2) residual up to O_H; only s=s'=1 is sum_n mu(n)mu(n-2)L(n)L'(n-2) plus O(x^(19/20+eps)) | derived; review 11 with repairs applied | corner-correlation.md §§1-2 | the parity classification changes | | 6 | the one-sided corner bound implies and is implied by the sufficient margin, given the complement | derived logic; review 11 and consumer-comparison flags | corner-correlation.md §2.2; consumer-comparison.md §4 | the claim that the one-sided form is the conclusion changes | | 7 | the Mobius Bombieri-Vinogradov input follows from Koukoulopoulos Cor. 13.4, Thms 26.2, 26.6, eq. (26.3) with Vaughan's identity for mu | derived from published theorems; review 15 | mobius-bv-derivation.md | one classical contribution loses its citation | | 8 | the priced negatives: Bettin-Chandee Theorem 1 at 129/125, Corollary 1 adds zero area, Type I/II decomposition reaches 41/40, Cauchy-in-h lemma 3/2 at the corner, Heath-Brown edge relocates | derived; reviews 15 and partial | small-divisor-kernel.md, determinant-corollary.md, left-divisor-signs.md, signed-moment.md, heath-brown-edges.md | a route recorded as failed might be open | | 9 | the log-averaged Chowla transfer fails at the named steps | derived, one pass | corner-log-average.md | a route recorded as failed might be open | | 10 | the measurements | measured, pre-registered falsifiers | kernel-sign-control.md, corner-measurement.md | heuristic reading only | ## 3. Dependency checklist at dispatch These were the outstanding checks when this brief was dispatched. Their current dispositions are in the review records linked above and TWIN-REDUCTION section 9. This list is not the current research queue. Each entry names its point of use; numbering is retained for citations. 1. **The right-orientation rewrite of the prime-power and first-branch moments** (residual-coverage §3). Asserted from sign-agnostic ingredients, never written out. It sets the 19/25 product edge through the right zero budget J_R. Reviews 06 and 19 both flag it. 2. **The unweighted-cofactor reading of the moment's phase** (heath-brown-edges.md §3): that e(hz'/(gmu)) is the CRT count with both cofactors summed unweighted, so a = 1 minus the largest free-variable exponent for any identity. One reader. If the shape tolerated a cofactor weight, the Heath-Brown conclusion collapses and the corner under that identity would need only gamma >= 2/3. 3. **Lemmas A and B of signed-moment.md and its heuristic ceiling.** One pass. The ceiling's rigorous half lacks a lower bound on sum |G(u,h)|^2. 4. **The Perron separation with four intersecting cuts** (grouped-divisor-moment §5, residual-coverage §4): the error is asserted from the two-cut remark; the box-by-box order of operations is forced but unwritten. 5. **endpoint-pairing.md §5 (9)**, the divisor-count bound on the full positive Vaaler majorant: hypotheses and side conditions checked, the derivation of (9) itself not read by any reviewer. 6. **prime-dispersion.md (11)**, the completion with periods removed: cited inputs confirmed verbatim (Pascadi Lemmas 3.2-3.3), the full text not read line by line. 7. **signed-divisor-grouping.md §2**, the uniform excluded-prime Mobius mean: checked once (review 02), consumed by every later note. 8. **polylog-fold-transfer.md §§1-3 and §5**: only §4 was reviewed. 9. **corner-log-average.md**: Pilatte's rate constant c <= 0.003832 is derived there from his §2.3 chain, unreviewed; and the eta_0 dependence of the band-swap cost was not measured against varying eta_0. 10. **corner-correlation.md**: status PARTIAL because three literature imports (Pilatte's c, Frantzikinakis-Host, Klurman-Mangerel) were checked at abstract level only; the classification itself is complete. 11. **consumer-comparison.md**: the interval transfer from Murty-Vatwani's conclusion on [1,x] to our dyadic consumer is derived there at medium confidence; the Siegel-zero implication at K=0 at medium-high. The two citing sources for Heath-Brown 1983 disagree on the upper range exponent (q^300 versus q^500); the original was not reachable. 12. **Koukoulopoulos GSM 203 numbering** read from the author's preliminary version by two readers, not from the printed book. Iwaniec-Kowalski §17.2 and Opera de Cribro §§9, 16, 18 were not reachable; either may state the Mobius Bombieri-Vinogradov case directly. 13. **Validator coverage gaps**, not defects: grouped-divisor-validation.js and prime-power-dispersion-validation.js test their harmonic averages on the half-open band [A,2A) while the notes use the closed band; both inequalities were re-run on the closed band and hold. 14. **Baier-Zhao Lemma 2.2** (Vaaler majorant) as a published statement: numbering and page not verified; the inequality was verified numerically and re-derived from the Fejer form. ## 4. Known limits of the internal reviews - Agreement among reviews does not establish independence of their arguments. A new reviewer should reconstruct the decisive steps in rows 1 to 5 of section 2, rather than count prior approvals. - The reviews re-derived decisive inequalities and confirmed theorem statements at primary sources; they did not check the proofs of the imported theorems (Bettin-Chandee Theorem 1 and Corollary 1, the Kloosterman bounds, Koukoulopoulos' theorems, Tao's log-averaged theorems). - Finite validators check identities and rational bookkeeping. None proves an asymptotic rate; every measurement note says so. - The judgement paragraphs (which route is more promising, whether the corner is "the parity wall") are labelled assessment and are not claims for review. ## 5. What the review must return The completed round used the following return format: 1. One-line verdict per reviewed claim: DEFECT FOUND with file, equation and line, or NO DEFECT FOUND IN SCOPE. 2. A table of decisive steps, each CHECKED with the inequality reproduced, DEFECTIVE with the exact failure, or ASSUMED-UNVERIFIED. 3. Imported theorems with primary source, version, theorem number, hypotheses, and whether each hypothesis is met at the point of use. 4. What was not checked. 5. For any defect: which rows of §2 fall with it. Reports from this round are retained in [history/reviews-0906/](history/reviews-0906/README.md). New work uses the question-based paths and integration rules in RESEARCH-EXECUTION. One handler reconciles shared state after review. The integration gates are: ```sh node research/qc.js --index --strict node research/qc/selftest.js node research/audit-numbers.js ``` These verify syntax, custody and finite numbers; they do not verify an analytic argument. ## 6. What is not being asked Not asked: to propose new estimates, to price further interfaces, to run computations beyond what a finite identity check needs, or to judge whether the programme should continue. Those are separate decisions. The publication moratorium is in force; this review is internal.