{"id":1828,"job_id":3289,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #3289 (pursue route 128): #83 and #153 restored at source, registry regenerated with the real generator; both layouts then match all four accepted dispositions, xchan and shadow unchanged\n\n**Caveat first.** The served edition is **not** yet repaired. This return supplies the repair as a patch (`restore-83-153.patch`) plus the full-corpus regeneration evidence. Until an integrator applies it, the served notes and registry still read PARTIAL for two of the four IDs. Everything below is a snapshot of the served tree at 2026-09-26 ~13:10Z. No mathematical status is adjudicated here. The accepted decisions on #83 and #153 are read as the record.\n\n## 1. Sources and decisions re-read (current hashes)\n\n| path | served | history | accepted revision | open findings |\n|---|---|---|---|---|\n| research/centered-discrepancy-estimate.md | v3 = v1 `0e472838` PARTIAL | 3 versions, none after the 09-16 cut | #83 v2 `dd6e1a4c` ANSWERED (accepted, verified, `integration: null`, `superseded_by: null`) | #196 |\n| research/global-factor-signs.md | v3 = v1 `0509638b` PARTIAL | 3 versions | #153 v2 `0a4f04bf` ANSWERED (same) | #1, #197 |\n| research/history/staging/xchan-at29-prereg.md | v5 `d671b418` ANSWERED (#1668, after #190) | 5 | preserved | - |\n| research/history/staging/shadow-prereg.md | v2 `ccdb59c0` ANSWERED SHAPE-ONLY (#225) | 2 | preserved | - |\n| research/QUESTIONS.md | v8 `ccf2cf0f` (#1764, full regeneration 09-26) | 8 | - | - |\n| research/qc/questions.js | v4 `eeaf2882` (#1734; #1630 used v3 `1d2d785e`) | 4 | - | - |\n\nNo reviewed supersession exists for #83 or #153. The evidence: no later version on either path, `superseded_by` is null, and the three open findings each ask for \"restore or record a supersession\". The diffs from served to accepted touch **only ledger lines**, and the note bodies are byte-identical. Centered changes 3 lines: `status`, `parity` (modulus bound `x^(1/2-eps)` -> `2x^(1/2-eps)`) and `verdict` (\"Yes, at the stated scope. ...\"). Global changes 2 lines: `status` and `verdict` (\"All three parts are answered at their stated scope. ...\"). Restoring the exact v2 bytes therefore loses nothing newer.\n\n## 2. Full-corpus regeneration (the actual generator, unmodified)\n\n`regen3289.py` is a stdlib adaptation of #1764's `regen-now.py` that uses public endpoints and needs no credential. It refetched all 680 paths of #1764's manifest from `/docs` and verified each body against `x-content-sha256` (0 mismatches). Coverage: the served listings of `research/` (141 .md) and `research/history/staging/` (442) are all in the mirror, apart from the output `QUESTIONS.md`. It then ran `node research/gen-questions-index.js` on three copies. Each run printed \"554 questions from 581 indexed notes, 0 unindexed\".\n\n- **A, corpus as served:** differs from served `ccf2cf0f` in 3 lines (lines 73, 436, 577): `Q-mobius-bv-derivation` in both layouts and one `Q-dhr-verification` row. Their notes changed after #1764's regeneration. That is ordinary registry lag outside this scope; it is recorded, not adjudicated.\n- **B, only #83/#153 restored:** differs from A in **exactly 4 lines**: 2 IDs x 2 layouts, PARTIAL -> ANSWERED. Each verdict cell carries the accepted verdict text (4/4). The xchan and shadow rows in B are **byte-identical** to the served rows. B = `cda1e669`.\n- **C, negative control:** B with the shadow source set to OPEN. The shadow rows go OPEN in both layouts, so the generator follows the source and an output-only edit cannot survive regeneration.\n\nFour IDs x two layouts in B: ANSWERED/ANSWERED for centered, global, xchan and shadow. That equals every accepted source disposition, with current source hashes for xchan/shadow and the accepted hashes for #83/#153.\n\n`restore-83-153.patch` (unified diff against the served bytes) passes `git apply --check`. Applied, it yields exactly `dd6e1a4c`, `0a4f04bf` and `cda1e669`. It answers findings #1, #196 and #197. The QUESTIONS hunk also carries A's 3 catch-up lines.\n\n## 3. #1630's fixtures as regression\n\n`job-3269-ledger-join.js` (#1630) was run on the current generator v4 and registry v8. It fails on three **frozen counts**, all moved by the xchan repair: stale-vs-current-source 1 -> 0, stale-vs-accepted-source 3 -> 2, and served-vs-#80 status diffs 0 -> 1 (xchan: #80 OPEN, now ANSWERED). `job-3289-ledger-join.js` changes only those three constants (`job-3289-ledger-join.patch`), and every other assertion passes: parse, both layouts, generated == source, 2 restorations -> all ANSWERED, and the shadow mutation control.\n\n## 4. Counts, kept separate\n\n- **Historical mirror reversions** (#1567's 9 still serving at the time): 3 have since been repaired by later versions (QUESTIONS.md, xchan-at29-prereg, beta2-note). 6 still serve v3 = v1: centered (#83), global (#153), corner-correlation (#92), fold-arithmetic-bridge (#101), derive-0904-L7-transfer (#152), prop-staircase-note (#13). This patch covers 2.\n- **Selected-row status discrepancies** against accepted dispositions: served now, 2 IDs (4 copies); in B, 0.\n- **Other three ledger notes** (`others-ledger-3289.json`): accepted and served `status` are equal (PARTIAL) for #92/#101/#152. derive-0904 differs in one ledger line. corner-correlation and fold-arithmetic-bridge have body changes. Restoring them changes no registry status, but may change verdict text. Not measured here.\n\n**Rung:** verified (deterministic reads plus the unmodified generator, hash-bound). **Not claimed:** that the restored ANSWERED statuses are mathematically right (that was #83/#153's reviewers' call); anything about the 3 catch-up lines; that the served edition is repaired.\n\n**Proposer's named requirements:** `node` (v25.2.0, used). `docs-endpoint`, `history-endpoint` and `files-endpoint` are the public `/docs/<path>?raw=1`, `/history/<path>` and `/files/<sha>`. I adapted #1764's regen script rather than rebuilding it, and reused #1630's checker.\n\n35 returns wait for a verdict.\n\nTranscript: scrubbed by sah-py-1.0.5 (credentials, session/account identifiers and local paths outside the working folder removed).","patch":"diff -ru a/research/QUESTIONS.md b/research/QUESTIONS.md\n--- a/research/QUESTIONS.md\n+++ b/research/QUESTIONS.md\n@@ -36,7 +36,7 @@\n |---|---|---|---|---|\n | C | `Q-agent-start` What must an incoming agent read and preserve before continuing the reviewed arithmetic campaign? | ANSWERED | Onboarding and document ownership only. The second dispatch is completed and independently reviewed on 2026-09-09. RESEARCH-EXECUTION.md owns current dispositions and candidate next obligations; RESEARCH-HANDOFF.md owns the updated exact contract. No assignment is started by that review. Twin-prime infinitude and required signed estimates remain OPEN. | [AGENT-START.md](AGENT-START.md) |\n | C | `Q-bilinear-fold-first-attack` Does assigning each Mobius factor to its first fold, or applying a direct second-moment bound, establish the missing one-sided estimate? | ANSWERED | The first-fold identity yields no saving; a direct second moment has a negligible diagonal and an open two-prime off-diagonal. Fixed-fold covariance is an exact residue-imbalance square and the beta weights balance at each fixed modulus. Growing-depth joint control remains OPEN; B>=-C2*x+o(x) is not improved. | [bilinear-fold-attack.md](bilinear-fold-attack.md) |\n-| C | `Q-centered-discrepancy-estimate` Can the top of D_y be removed using ordinary prime distribution after an exact divisor flip, and what estimate remains? | PARTIAL | Equations (3)-(4), the exact split and flip, survive review and finite controls. A repair of the top-range bound T^top=O_(A,eps)(x/log^A x) is DERIVED in section 3a (2026-09-08) with the endpoint atom retained, odd square divisors, the corrected reciprocal-totient local factors with (m',g)=1, explicit truncation powers, a Cauchy multiplicity device, and a uniform Mobius mean proved from the q=1 Siegel-Walfisz statements; its inputs are the terminal-point BV of Tao Notes 3 Theorem 17 with the prefix form derived by rounding. The handler (lane V) read the derivation on 2026-09-08, reconstructed (BV*), (3a.9), (3a.14) and (3a.16), checked the limiting constant of (3a.9) numerically at eight (b,g) pairs and found no defect; it is accepted at its stated scope, with ineffective constants. The payoff is D_y=D^(e_1)+O_(A,eps)(x/log^A x) only, so the handoff consumer is equivalent to D^(e_1)>=-4x/25+o(x). The fixed-endpoint signed discrepancy, D_y>=-4x/25+o(x) and twin-prime infinitude remain OPEN. | [centered-discrepancy-estimate.md](centered-discrepancy-estimate.md) |\n+| C | `Q-centered-discrepancy-estimate` Can the top of D_y be removed using ordinary prime distribution after an exact divisor flip, and what estimate remains? | ANSWERED | Yes, at the stated scope. Equations (3)-(4), the exact split and flip, survive review and finite controls. The top-range bound T^top=O_(A,eps)(x/log^A x) is DERIVED in section 3a (2026-09-08) with the endpoint atom retained, odd square divisors, the corrected reciprocal-totient local factors with (m',g)=1, explicit truncation powers, a Cauchy multiplicity device, and a uniform Mobius mean proved from the q=1 Siegel-Walfisz statements; its inputs are the terminal-point BV of Tao Notes 3 Theorem 17 with the prefix form derived by rounding. It was read twice independently on 2026-09-08, by the handler (lane V), who reconstructed (BV*), (3a.9), (3a.14) and (3a.16), and by reader V2 (history/reviews-0908/01), who reconstructed every step, and is accepted at its stated scope with ineffective constants; the 2026-09-09 integration review (research-round-validation.md section 7) repaired M<=2x^(1/2-eps) and the log^(L+3) count, both absorbed by fixed power slack. The retained finite check of the limiting constant -H_(b,g)(0) of (3a.9) is at four (b,g) pairs at u=1e6. What remains is D_y=D^(e_1)+O_(A,eps)(x/log^A x): the handoff consumer is equivalent to D^(e_1)>=-4x/25+o(x), the fixed-endpoint signed discrepancy carried by Q-fixed-endpoint-discrepancy. That estimate, D_y>=-4x/25+o(x) and twin-prime infinitude remain OPEN. | [centered-discrepancy-estimate.md](centered-discrepancy-estimate.md) |\n | C | `Q-centered-discrepancy-measurement` At reachable x, is the OPEN sufficient input D_y>=-4x/25 numerically violated, is the absolute form (13) numerically plausible, and does D_y carry structure beyond a random-sign model? | ANSWERED | MEASURED to x=2^38. Neither pre-registered falsifier fires. D_y/x stays within 0.0043 of zero for the measured j>=26. Over 30<=j<=38, D_y/x differs from -(T1/x-C2) by at most 2e-4, with maximum 1.84401e-4 at j=30; the classical term dominates this finite comparison. 2026-09-12: that dominant part is the Mobius truncation term of (8) at u=y, computable without primes, to within 1.1e-4 x for j>=31 (\\|B\\|/x = 1.02e-4 at j=31); with it subtracted, the remainder is a mix of S-C2 x and B at random-sign size | [centered-discrepancy-measurement.md](centered-discrepancy-measurement.md) |\n | C | `Q-chain-review-0906` Do the written derivations from S(x) down to E_dagger hold as stated, with every imported theorem used inside its hypotheses and every discarded term at the claimed O_H(x/log^H x) rate, and where does the chain remain unreviewed? | ANSWERED | Three independent adversarial passes found no defect in the six classical contributions from S(x) to E_>, none in the grouped moment (1)-(20) including the E_* to E_dagger step, and none in the residual-coverage link E_> to E_*, whose block-shape derivation was reconstructed and verified exactly. Every imported theorem was confirmed in its primary source except the Mobius Bombieri-Vinogradov citation, which points at an exercise sheet and needs a published replacement. Stale o(x) rates in three note bodies were replaced. A checked argument is not a refereed theorem; the OPEN signed margin is untouched. | [chain-review-0906.md](chain-review-0906.md) |\n | C | `Q-chen-fold-benchmark` Can the fold framework reproduce a classical Chen lower bound on long intervals with every imported hypothesis and error budget stated? | ANSWERED | The classical implication is derived below using three explicitly imported theorem statements, with the combinatorial minorant proved and its integral margin certified by exact rational arithmetic. This is a benchmark, not a new prime theorem, and it does not prove twins or a short-zone result. | [chen-fold-benchmark.md](chen-fold-benchmark.md) |\n@@ -61,7 +61,7 @@\n | C | `Q-fold-arithmetic-bridge` Does one parity-table bridge from the anchored fold ledger yield a sufficient twin lower bound with named arithmetic inputs? | PARTIAL | Exact identities retained. The one unread sieve input of the pricing, Bombieri--Vinogradov for k-fold X^(1/u)-rough products, is derived from Wu's Lemma 2.3 (section 3a); the same input, sieved in the composite variable, gives the contamination aggregate constant 4 for each fixed k,u, replacing the displayed 12.86 to 19.72. With these inputs, elementary bounds give Q_cov(u)<1 and c*_real(u)<4 for every u>4 (section 4a, independently reviewed 2026-09-09 with rational certificates), so neither sufficient ratio test succeeds at any depth. This closes the two tests, not the decorrelation hypotheses, and supplies no twin estimate. | [fold-arithmetic-bridge.md](fold-arithmetic-bridge.md) |\n | C | `Q-full-coefficient-average` Can aggregating the complete coefficients before a correlation theorem remove the explicit cofactor count, and what additional estimate is needed? | PARTIAL | Exact factor and rounded-endpoint Fourier identities retained, with c_(i,0)=3/5 and an explicit composite-filtered weighted sum. The full family is not 1-bounded, but the sufficient phase condition admits at least k=0,+/-1 on the left and l=0,+/-1,...,+/-6 on the right for every Mellin twist; the earlier zero-only claim is corrected. Composite filtering and the required correlation rate remain unmatched. Proposition 6.5, independently reviewed including on 2026-09-09, proves coefficient norm at least (log x)^(2/5) for representations by 1-bounded functions on all smooth inputs. This does not exclude density-one representations, paid growing components or a jointly treated Fourier sum. No sufficient signed twin margin follows. | [full-coefficient-average.md](full-coefficient-average.md) |\n | C | `Q-global-cutoff-averaging` Does the wider corpus suggest changing the global decomposition before extending the small-cofactor estimate, and can that change be made without an unpaid transition or outside term? | PARTIAL | Derived: independently averaging the two initial Mobius cutoffs over fixed exponent intervals gives S=C2*x+sum G_L(n)G_R(n-2)+O_A(x/log^A x), uniformly on dyadic scales, with all cofactor and prime-power branches included. Each full G has squared norm O(x log x). The existing sharp-corner lower norm then forces cancellation between the corner coefficient and the rest at the same input. This does not estimate their shifted product. The global signed residual was already O(x) by the sieve upper bound and positivity; the new norm representation is not an improved signed bound. Prefer a bounded attempt on this global coefficient pair and a one-sided consumer; the twin margin remains OPEN. | [global-cutoff-averaging.md](global-cutoff-averaging.md) |\n-| C | `Q-global-factor-signs` Which full factor configurations cause negative global products, can prime-power exceptions be paid, and does a pair-trigger upper bound control the negative part? | PARTIAL | Derived: G_i(n)=Lambda_(>W_i)(n)-F_i(s_i(n))*log t_i(n)-E_i(n), with all small-prime factors in s_i and all primes of t_i exceeding W_i. The correction is supported on a small-prime power exceeding W_i; its full shifted effect is O_epsilon(x^(39/40+epsilon)). After also paying proper prime powers, the residual is a composite-filtered signed smooth/rough cofactor sum. A ten-small-prime cell has F=-84 although every pair lies below a, refuting the proposed pair-trigger majorant. Prime filters contribute explicit terms when bounding the negative part; dropping them is an upper bound, not an identity. No scale-x signed improvement or twin margin follows. | [global-factor-signs.md](global-factor-signs.md) |\n+| C | `Q-global-factor-signs` Which full factor configurations cause negative global products, can prime-power exceptions be paid, and does a pair-trigger upper bound control the negative part? | ANSWERED | All three parts are answered at their stated scope. Derived: G_i(n)=Lambda_(>W_i)(n)-F_i(s_i(n))*log t_i(n)-E_i(n), with all small-prime factors in s_i and all primes of t_i exceeding W_i; on regular composite inputs a global product is negative exactly when the two F values have opposite signs. The correction is supported on a small-prime power exceeding W_i; its full shifted effect is O_epsilon(x^(39/40+epsilon)), so the prime-power exceptions are paid. After also paying proper prime powers, the residual is a composite-filtered signed smooth/rough cofactor sum. A ten-small-prime cell has F=-84 although every pair lies below a, refuting the proposed pair-trigger majorant. Prime filters contribute explicit terms when bounding the negative part; dropping them is an upper bound, not an identity. What remains is carried elsewhere: switching-negative-mass.md proves the negative-only target false at these cutoffs, and global-smooth-majorant.md gives an absolute O(x) budget for the C3 profile with a non-vanishing majorant, leaving its signed constant OPEN. No scale-x signed improvement or twin margin follows. | [global-factor-signs.md](global-factor-signs.md) |\n | C | `Q-global-smooth-majorant` Can a higher-order majorant control the complete global coefficient pair at scale x while retaining every factor configuration and the prime filters? | PARTIAL | Derived using the corrected Henriot upper theorem: a C3 probability average of the same admissible initial cutoffs gives a full residual with sum \\|Ghat_L(n) Ghat_R(n-2)\\|=O(x). The three-smallest-prime majorant has bounded harmonic mass and meets the source growth class uniformly, despite not being multiplicative. All prime-power exceptions are paid. This replaces the O(x log x) absolute budget for the earlier logarithmic profile by O(x) for a different admissible profile representing the same signed residual to arbitrary logarithmic precision. The implied constant is not compared with C2 and no improved signed lower bound or twin margin is supplied. | [global-smooth-majorant.md](global-smooth-majorant.md) |\n | C | `Q-grouped-divisor-moment` Does the full gcd-normalized moment proposed by the literature audit hold, and what exact portion of the twin-prime remainder does it control? | ANSWERED | The proposed moment is derived from classical completion with all coefficient sectors and uniform twists included. Full rectangles are controlled when delta<19/25 and delta+3nu<161/100, in addition to the preceding region. A concrete extra cut d<=floor(x^(151/200)), de^3<=floor(x^(321/200)) controls the entire d~e~x^(2/5) benchmark and leaves an explicit smaller-domain endpoint remainder. The uniform product threshold stays below 19/25. At delta=8/25,nu=9/20, the next deficit is confined to small-common-divisor nonzero kernels; their required saving and the global twin margin remain OPEN. Finite validation does not prove asymptotic rates. | [grouped-divisor-moment.md](grouped-divisor-moment.md) |\n | C | `Q-handoff-review-0906` Does the handed-back arithmetic campaign survive an independent check of its main regional estimate and the conclusions used to choose the next research direction? | ANSWERED | The bounded handoff audit is completed in reports 20 and 21: the checked local reduction and regional mechanisms survive, named source statements were verified, and the consumer, corner support, rate, shrinking-margin and identity-piece overclaims were corrected. Joint Cauchy is now priced and adds no region. The multiplicative band transfer has a separate PARTIAL owner with a weaker continuous-scale payoff. Imported deep theorems remain imports; this is not corpus-wide certification or a twin margin. | [handoff-review-0906.md](handoff-review-0906.md) |\n@@ -70,7 +70,7 @@\n | C | `Q-joint-factor-estimate` Can an additional estimate for actual factor configurations control the full signed balance of the C3 residual? | ANSWERED | DERIVED: for the single-large-prime delta-rough family F, the residual splits exactly as R_F = C2*x*H_delta - Z_F^p + o(x), where the Type I part C2*x*H_delta is evaluated by ordinary prime BV on each cofactor interval and the twisted Mobius constant 2C2 (corrected 2026-09-08 after V2: Wu Lemma 2.3 is squarefree-only and did not cover the non-squarefree b_R moduli; prime BV covers all moduli), and Z_F^p is the signed prime-partner sum. The complement carries the opposite Type I part -C2*x*H_delta, so grouping by factor configuration transfers Type I mass and leaves the twisted prime correlation B_L unchanged. With the matched upper sieve the family bound is C2*x(1+H_delta-(40/9)H^+), certified below -2/5 C2*x: INSUFFICIENT BOUND. Even a sieve constant 2 in place of 40/9 needs H^abs<1 and leaves the complement unpaid. Independently reviewed 2026-09-09 with the uniform delta-range and complement corrections below. Twin infinitude and every sufficient margin remain OPEN. | [joint-factor-estimate.md](joint-factor-estimate.md) |\n | C | `Q-kernel-sign-control` At delta=8/25, nu=9/20, is the moment Mfrak and its small-common-divisor kernel X_small measurably smaller with the actual coefficient b_u=A_right(gu) than with \\|b_u\\| or with \\|b_u\\| times an independent seeded random sign, and does the actual/random ratio decrease with x over the feasible dyadic range? | ANSWERED | MEASURED, negative. Over seven dyadic scales j=18..30 and four boxes, the pre-registered slope of log2(actual/random) for X_small has mean + sd >= 0 in all four families, so the run gives no heuristic support for extra Mobius-sign cancellation in the small-j kernel. The actual coefficient's \\|X_small\\| ranks like one more random draw. The measurement is weak where it matters: the finite moment is 97.66-102.58 percent equal-frequency class, X_small carries 0.02-2.88 percent, J0=floor(x^(1/20)) is 1 at j=18 and 2 at j=20..30, and Z=2 collapses the prime-power sector of (13) to r=2. Equation (21) and the global twin margin remain OPEN and untouched. | [kernel-sign-control.md](kernel-sign-control.md) |\n | C | `Q-left-divisor-signs` Can the Möbius structure of the left coefficient A_left(gm) replace the first Cauchy inequality, so that its sqrt(M) loss is not paid, and what does that buy at the target box (delta,nu)=(8/25,9/20) and at the corner a=b=1? | PARTIAL | Partly, and not where it is needed. Two lemmas are derived: a Type I bound with no zero-frequency budget, usable at the target box whenever the arbitrary block is shorter than x^(1/4), and a Type II bound that is uniform in the Perron twist heights. Applied to the actual coefficient they confine the target-box deficit to sectors with left prime power r>=x^(43/200) and right prime power q>x^(3/100), and lower the worst block exponent there from 103/100 to 41/40. The corner a=b=1 is untouched: grouped, Type I, Type II, Bettin-Chandee/Wright and even a hypothetical per-block square-root bound all exceed 1 there. No region is added, E_dagger and the consumer (20) are unchanged, and the global twin margin remains OPEN. | [left-divisor-signs.md](left-divisor-signs.md) |\n-| C | `Q-mobius-bv-derivation` Is there a published theorem statement of Bombieri-Vinogradov for the Mobius function at level T^(1/2)/log^B T with a maximum over coprime classes, and if not, does that estimate follow from published theorems by a derivation whose every hypothesis is checked? | ANSWERED | No published theorem statement was located in five channels searched in the owning convention named in research/SEARCH-CONVENTIONS.md; the closest is Granville-Shao's published assertion that the result is known, which carries no proof locator for the Mobius case. The estimate is derived here from four numbered results of Koukoulopoulos, GSM 203 (Corollary 13.4, Theorem 26.2, Theorem 26.6, equation (26.3)) plus Vaughan's identity for mu, with B(A)=A+6 and an ineffective constant. This is a derivation from published theorems, not a refereed theorem, and it is read from the author's preliminary version of that book. | [mobius-bv-derivation.md](mobius-bv-derivation.md) |\n+| C | `Q-mobius-bv-derivation` Is there a published theorem statement of Bombieri-Vinogradov for the Mobius function at level T^(1/2)/log^B T with a maximum over coprime classes, and if not, does that estimate follow from published theorems by a derivation whose every hypothesis is checked? | ANSWERED | A published theorem statement of the max form exists: Granville-Shao, *When does the Bombieri-Vinogradov Theorem hold for a given multiplicative function?*, Forum Math. Sigma (2018), arXiv:1706.05710v1, Theorem 1.1(b) - with f = mu and f*1_P = -1_P, together with Siegel-Walfisz for mu, it gives max-form Mobius Bombieri-Vinogradov at Q = x^(1/2)/(log x)^B (credit return #1183, review 420, not the channel sweep below). Eleven channels were searched in the owning convention named in research/SEARCH-CONVENTIONS.md for the exact class-sup statement (M); that negative is recorded in §2. Shao, arXiv:1607.01814 (2016) §1 asserts BV for mu at level X^(1/2)(log X)^(-B) with class sup and the locator Iwaniec-Kowalski ch. 17; fixed endpoint, exceptional-set form, not proved there (return #157). The estimate is derived here from four numbered results of Koukoulopoulos, GSM 203 (Corollary 13.4, Theorem 26.2, Theorem 26.6, equation (26.3)) plus Vaughan's identity for mu, with B(A)=A+6 and an ineffective constant; the derivation is retained as the carrier of the max-over-y clause only. This is a derivation from published theorems, not a refereed theorem, and it is read from the author's preliminary version of that book. | [mobius-bv-derivation.md](mobius-bv-derivation.md) |\n | C | `Q-moving-cutoff-parity` Does a proof-level reading of the shifted-Mobius sieve give a valid, explicit error-tolerant arithmetic target, and does its divisor switching preserve the moving boundary? | ANSWERED | CHECKED prior art: Murty--Vatwani already supplies the conditional squarefree/parity mechanism and fixed-residue reduction. Its printed p. 654 swap omits n+h>ey; an exact finite counterexample verifies failure of that equality. DERIVED dyadic repair keeps the boundary and gives S=C2*x-2*C2*M+D_y+O_A(x/log^A x). A one-sided D_y>=-4*x/25+o(x) on unbounded dyadic scales is sufficient, using the certified margin C2*(1-A2)>33/200. That discrepancy estimate remains OPEN. Vatwani's recovered author preprint clarifies the one-odd-exponent cases and unfilled near-level-one hypotheses. No new signed bound is obtained; literature novelty is unestablished. | [moving-cutoff-parity.md](moving-cutoff-parity.md) |\n | C | `Q-next-correlation-source-map` Is there a primary-source statement with stronger scale or coefficient quantifiers than the ones the prime-band transfer actually consumed? | ANSWERED | Scoped negative for the inspected statements: function-class and rate mismatches remain. The coordinating review withdraws the all-scales/dyadic-omitting requirement as a necessary gate, since bounded-interval stability yields a weak dyadic scale average from the existing input. No sufficient rate or full-coefficient theorem is matched. Proper-prime-power terms are already negligible on the fixed corner; other prime-r cofactor branches remain open. | [next-correlation-source-map.md](next-correlation-source-map.md) |\n | C | `Q-next-transfer-review` Does prime-band-transfer.md (1)-(6) follow at exactly its stated scope, and does any step fail against an adversarial reading of its hypotheses? | ANSWERED | The derivation (1)-(6) survives at its stated continuous-average scope. The coordinating round review corrects check l: interval stability additionally gives a dyadic scale average, with a further exponent loss. There is no every-dyadic estimate, o(N), full-corner control or twin margin. Proper-prime-power branches have a separate negligible fixed-corner bound. | [next-transfer-review.md](next-transfer-review.md) |\n@@ -382,7 +382,7 @@\n | `Q-calm-vs-kill` | PARTIAL | What does the anchored calm buy against annihilation? | Two theorems, the @11 closure and the X-Limitation Theorem at @11, @13, @17 and @19, and two refutations, calm implying concentration and minimizer loudness; the overlap-floor candidate that would prove non-annihilation dies on the anchor's own drift, with the anchored rotation t = 0 its worst case at @17, leaving the X-floor open. | none | [natal-cap-31-calm-vs-kill.md](natal-cap-31-calm-vs-kill.md) |\n | `Q-capK-bv` | ANSWERED | Is the prime-regime cap_K over the full wheel an unconditional asymptotic via the fundamental lemma over Bombieri-Vinogradov? | PROVEN at short-note grade for q_K = W^{o(1)}, and EMPTY in every computable range (needs s >= 10.82, first cleared at x = 263; the lemma's floor s >= 10 is first reached at x = 239); the deep ladder is untouched. | 11 (retired) | [thm-capK-bv.md](history/staging/thm-capK-bv.md) |\n | `Q-capture-identity` | ANSWERED | Does the transplanted cap family collapse onto a classical object, and what does the collapse say about the depth law and the wall? | PROVEN, elementary: floor_K = T − X(K) on the half-open window with both members in [Q², Q′²), a convention that is load-bearing; the wall is renamed, not weakened; Σ capU counts each member against its own threshold, not against a uniform p_K, the two differing 11% to 37% at K*. | Z2 | [quadpoint-identity-01.md](history/staging/quadpoint-identity-01.md) |\n-| `Q-centered-discrepancy-estimate` | PARTIAL | Can the top of D_y be removed using ordinary prime distribution after an exact divisor flip, and what estimate remains? | Equations (3)-(4), the exact split and flip, survive review and finite controls. A repair of the top-range bound T^top=O_(A,eps)(x/log^A x) is DERIVED in section 3a (2026-09-08) with the endpoint atom retained, odd square divisors, the corrected reciprocal-totient local factors with (m',g)=1, explicit truncation powers, a Cauchy multiplicity device, and a uniform Mobius mean proved from the q=1 Siegel-Walfisz statements; its inputs are the terminal-point BV of Tao Notes 3 Theorem 17 with the prefix form derived by rounding. The handler (lane V) read the derivation on 2026-09-08, reconstructed (BV*), (3a.9), (3a.14) and (3a.16), checked the limiting constant of (3a.9) numerically at eight (b,g) pairs and found no defect; it is accepted at its stated scope, with ineffective constants. The payoff is D_y=D^(e_1)+O_(A,eps)(x/log^A x) only, so the handoff consumer is equivalent to D^(e_1)>=-4x/25+o(x). The fixed-endpoint signed discrepancy, D_y>=-4x/25+o(x) and twin-prime infinitude remain OPEN. | C | [centered-discrepancy-estimate.md](centered-discrepancy-estimate.md) |\n+| `Q-centered-discrepancy-estimate` | ANSWERED | Can the top of D_y be removed using ordinary prime distribution after an exact divisor flip, and what estimate remains? | Yes, at the stated scope. Equations (3)-(4), the exact split and flip, survive review and finite controls. The top-range bound T^top=O_(A,eps)(x/log^A x) is DERIVED in section 3a (2026-09-08) with the endpoint atom retained, odd square divisors, the corrected reciprocal-totient local factors with (m',g)=1, explicit truncation powers, a Cauchy multiplicity device, and a uniform Mobius mean proved from the q=1 Siegel-Walfisz statements; its inputs are the terminal-point BV of Tao Notes 3 Theorem 17 with the prefix form derived by rounding. It was read twice independently on 2026-09-08, by the handler (lane V), who reconstructed (BV*), (3a.9), (3a.14) and (3a.16), and by reader V2 (history/reviews-0908/01), who reconstructed every step, and is accepted at its stated scope with ineffective constants; the 2026-09-09 integration review (research-round-validation.md section 7) repaired M<=2x^(1/2-eps) and the log^(L+3) count, both absorbed by fixed power slack. The retained finite check of the limiting constant -H_(b,g)(0) of (3a.9) is at four (b,g) pairs at u=1e6. What remains is D_y=D^(e_1)+O_(A,eps)(x/log^A x): the handoff consumer is equivalent to D^(e_1)>=-4x/25+o(x), the fixed-endpoint signed discrepancy carried by Q-fixed-endpoint-discrepancy. That estimate, D_y>=-4x/25+o(x) and twin-prime infinitude remain OPEN. | C | [centered-discrepancy-estimate.md](centered-discrepancy-estimate.md) |\n | `Q-centered-discrepancy-measurement` | ANSWERED | At reachable x, is the OPEN sufficient input D_y>=-4x/25 numerically violated, is the absolute form (13) numerically plausible, and does D_y carry structure beyond a random-sign model? | MEASURED to x=2^38. Neither pre-registered falsifier fires. D_y/x stays within 0.0043 of zero for the measured j>=26. Over 30<=j<=38, D_y/x differs from -(T1/x-C2) by at most 2e-4, with maximum 1.84401e-4 at j=30; the classical term dominates this finite comparison. 2026-09-12: that dominant part is the Mobius truncation term of (8) at u=y, computable without primes, to within 1.1e-4 x for j>=31 (\\|B\\|/x = 1.02e-4 at j=31); with it subtracted, the remainder is a mix of S-C2 x and B at random-sign size | C | [centered-discrepancy-measurement.md](centered-discrepancy-measurement.md) |\n | `Q-certificate-engine` | PARTIAL | What did natal-cap-28 assemble, and at what calibration does the certificate engine stand? | There is no fully analytic certificate law and nothing here is closed: what reads as one object is nine at five calibrations with one of the nine refuted, the Certified-Head Theorem is PROVEN on 1/3/6 certified primes at @17/@19/@23, and every engine output carries the status of its weakest input, the HEURISTIC Buchstab Transfer Hypothesis. | none | [certificate-engine.md](certificate-engine.md) |\n | `Q-chain-review-0906` | ANSWERED | Do the written derivations from S(x) down to E_dagger hold as stated, with every imported theorem used inside its hypotheses and every discarded term at the claimed O_H(x/log^H x) rate, and where does the chain remain unreviewed? | Three independent adversarial passes found no defect in the six classical contributions from S(x) to E_>, none in the grouped moment (1)-(20) including the E_* to E_dagger step, and none in the residual-coverage link E_> to E_*, whose block-shape derivation was reconstructed and verified exactly. Every imported theorem was confirmed in its primary source except the Mobius Bombieri-Vinogradov citation, which points at an exercise sheet and needs a published replacement. Stale o(x) rates in three note bodies were replaced. A checked argument is not a refereed theorem; the OPEN signed margin is untouched. | C | [chain-review-0906.md](chain-review-0906.md) |\n@@ -433,7 +433,7 @@\n | `Q-derive-0904-r0-extension` | ANSWERED | Which pieces of the R0 decomposition (the tile form, the zone form, head, tail, Z2, and the composites R0/R1/R2/stretch) carry the parity obstruction's forbidden extension, and is any single piece both legal on the TPC-strength axis and exempt from it? | No piece of R0 is both legal and exempt with content: the criterion reduces to a coordinate count (Proposition A) under which every statement forcing a twin pair into a range where roughness implies primality is covered, and the exempt residue is exactly the statements that assert no such forcing (R0's identity, lower bounds, boundary artefacts) plus the tile form, whose exemption is a property of the statement and provably does not close under deduction (Proposition B), so it survives only for routes with no sieve weight anywhere. | Z2 | [derive-0904-r0-extension.md](history/staging/derive-0904-r0-extension.md) |\n | `Q-destroyer-census` | ANSWERED | Who destroys each channel pair, what does the zone kill budget certify, and where does the counting certificate die? | The census reproduces digit-exactly on an independent engine; the destroyer rule is positional, not temporal, the certificate holds at 15 zones and dies at p = 67 where B/C first crosses 1, and no first-mover excess survives the derived null. | Z4 | [destroyer-census-01.md](history/staging/destroyer-census-01.md) |\n | `Q-determinant-corollary` | ANSWERED | Does Bettin-Chandee Corollary 1, applied to d*k-e*t=2 after moving the non-smooth part of beta_V and beta_Z onto the coefficient side, control any part of the endpoint remainder that the current estimates do not? | The transfer is legal and exact: the smoothness hypothesis the literature scout recorded as the blocker is removable, at the cost eta=x^kappa and a boundary error O(x^(1-kappa)log^C x). Priced, it adds nothing. Requiring all four pieces and all bands gives 22max(a,b)+17min(a,b)<20, whose supremum of delta+nu is 2869/3900 < 19/25, so the region is strictly inside the already controlled delta+nu<19/25; the added area is exactly 0. A split by the size of the moved prime power also adds nothing, because every per-band budget is nondecreasing in the band exponents and the top band reproduces the full box. The log-log piece alone is controlled on 12.97 percent of the domain outside the current region, which is one of four pieces and changes nothing for R. W_dagger does not shrink, E_dagger is unchanged, and the sufficient twin margin remains OPEN. | C | [determinant-corollary.md](determinant-corollary.md) |\n-| `Q-dhr-verification` | ANSWERED | Does the beta2 note's use of the Diamond-Halberstam-Richert dimension-2 sieve survive verification against the primary sources? | The note's Theorem survives: every mismatch found is a remainder-weight convention absorbed by the z^{eps/2} slack, beta2 = 4.26645028414864191641 on Booker-Browning's rigorous table, and the fundamental-lemma fallback exponent corrects to about 19+eps rather than 18+eps. | none | [dhr-verification.md](dhr-verification.md) |\n+| `Q-dhr-verification` | ANSWERED | Does the beta2 note's use of the Diamond-Halberstam-Richert dimension-2 sieve survive verification against the primary sources? | The note's Theorem survives: every mismatch found is a remainder-weight convention absorbed by the z^{eps/2} slack, beta2 = 4.26645028414864191641 on Booker-Browning's rigorous table, and the fundamental-lemma fallback exponent corrects to 18 + 10 ln K + eps (at least 28.98 + eps, since the block {3} forces K >= 3) rather than 18+eps, and to 19 + eps once the residue class is fixed mod prod_{p<23} p (K(23) = 1.1039848905 < e^{0.1}; return #166, certified). | none | [dhr-verification.md](dhr-verification.md) |\n | `Q-dim2-standalone` | ANSWERED | Is the dimension-2 instantiation of the DHR sieve a real step, or an exercise? | An exercise: Diamond-Halberstam's Example 1.2 sets up the same sieve problem and Theorem 9.1 is stated for every half-integer kappa >= 1 with the density condition as its only hypothesis, so what beta2-note adds is two parameter choices and one dimension-free observation that is zeb's at kappa = 1; the honest response is to state the k-class family, whose k = 1 row is a known theorem. | none | [attack-dim2-standalone.md](history/staging/attack-dim2-standalone.md) |\n | `Q-discrepancy-two-class` | ANSWERED | What is the two-class analogue of Holt's signed discrepancy DeltaPhi, and do the three candidate growth laws on the table agree? | One object in three norms rather than three agreeing measurements: the proven Level Ledger bound 2*3^(pi(x)-1) and natal-cap-29's spectral level law share the integer 3 provably, since R_k(p) tends to k+1, so the apparent agreement is an identity; the custody gate reproducing Holt's Table 2 passed. | none | [discrepancy-two-class.md](discrepancy-two-class.md) |\n | `Q-dispersion-range` | ANSWERED | How far do the dispersion and coefficient-sector estimates extend across divisor sizes, and does a better orientation or gcd budget enlarge their overlap? | Averaging the nonzero gcd losses over the original divisor gives a different dispersion bound with cross-prime exponent (5/4)a+(1/2)b+3/50, where a=delta+6/25 and b=nu+1/20. Explicit diagonal, prime-cut and sparse-sector inequalities control a region of full rectangles, including d~x^0.282, e~x^0.522 at product scale x^0.804. Its cross-prime and left-sparse budgets are 0.9985 and 0.9988. The supremum of product exponents in this sufficient region is 5397/6700; it is not a uniform product cutoff. Balanced supports separately allow 12/25<s<94/175. The global remainder and twin margin remain OPEN. | C | [dispersion-range.md](dispersion-range.md) |\n@@ -478,7 +478,7 @@\n | `Q-gate-multiplies` | ANSWERED | Does \"the gate multiplies\" close the entire u-frame recursion branch, or only the merge chain? | Only the merge chain and one relative of it: the no-fixed-point argument does not reach TODO 0b at all, though 0b is wrong as stated for an unrelated reason and the error is a factor of ln u; what survives is the copy theorem for maxsum (VERIFIED 40 of 40), which needs a residue-deleted maxsum bound that does not pass through a kill count. | 0b | [gate-multiplies.md](gate-multiplies.md) |\n | `Q-gate-repair` | ANSWERED | Were the verify-the-verifier repairs landed with the gate ending fully green? | COMPLETE: node research/qc.js --full reads FULL GATE PASSED at the close, TOTAL 0 across the 11 checks, selftest 39 known positives firing and 32 controls silent, audit-numbers 246/246, and no bound tail left amber. | none | [gate-repair-finale.md](history/staging/gate-repair-finale.md) |\n | `Q-global-cutoff-averaging` | PARTIAL | Does the wider corpus suggest changing the global decomposition before extending the small-cofactor estimate, and can that change be made without an unpaid transition or outside term? | Derived: independently averaging the two initial Mobius cutoffs over fixed exponent intervals gives S=C2*x+sum G_L(n)G_R(n-2)+O_A(x/log^A x), uniformly on dyadic scales, with all cofactor and prime-power branches included. Each full G has squared norm O(x log x). The existing sharp-corner lower norm then forces cancellation between the corner coefficient and the rest at the same input. This does not estimate their shifted product. The global signed residual was already O(x) by the sieve upper bound and positivity; the new norm representation is not an improved signed bound. Prefer a bounded attempt on this global coefficient pair and a one-sided consumer; the twin margin remains OPEN. | C | [global-cutoff-averaging.md](global-cutoff-averaging.md) |\n-| `Q-global-factor-signs` | PARTIAL | Which full factor configurations cause negative global products, can prime-power exceptions be paid, and does a pair-trigger upper bound control the negative part? | Derived: G_i(n)=Lambda_(>W_i)(n)-F_i(s_i(n))*log t_i(n)-E_i(n), with all small-prime factors in s_i and all primes of t_i exceeding W_i. The correction is supported on a small-prime power exceeding W_i; its full shifted effect is O_epsilon(x^(39/40+epsilon)). After also paying proper prime powers, the residual is a composite-filtered signed smooth/rough cofactor sum. A ten-small-prime cell has F=-84 although every pair lies below a, refuting the proposed pair-trigger majorant. Prime filters contribute explicit terms when bounding the negative part; dropping them is an upper bound, not an identity. No scale-x signed improvement or twin margin follows. | C | [global-factor-signs.md](global-factor-signs.md) |\n+| `Q-global-factor-signs` | ANSWERED | Which full factor configurations cause negative global products, can prime-power exceptions be paid, and does a pair-trigger upper bound control the negative part? | All three parts are answered at their stated scope. Derived: G_i(n)=Lambda_(>W_i)(n)-F_i(s_i(n))*log t_i(n)-E_i(n), with all small-prime factors in s_i and all primes of t_i exceeding W_i; on regular composite inputs a global product is negative exactly when the two F values have opposite signs. The correction is supported on a small-prime power exceeding W_i; its full shifted effect is O_epsilon(x^(39/40+epsilon)), so the prime-power exceptions are paid. After also paying proper prime powers, the residual is a composite-filtered signed smooth/rough cofactor sum. A ten-small-prime cell has F=-84 although every pair lies below a, refuting the proposed pair-trigger majorant. Prime filters contribute explicit terms when bounding the negative part; dropping them is an upper bound, not an identity. What remains is carried elsewhere: switching-negative-mass.md proves the negative-only target false at these cutoffs, and global-smooth-majorant.md gives an absolute O(x) budget for the C3 profile with a non-vanishing majorant, leaving its signed constant OPEN. No scale-x signed improvement or twin margin follows. | C | [global-factor-signs.md](global-factor-signs.md) |\n | `Q-global-smooth-majorant` | PARTIAL | Can a higher-order majorant control the complete global coefficient pair at scale x while retaining every factor configuration and the prime filters? | Derived using the corrected Henriot upper theorem: a C3 probability average of the same admissible initial cutoffs gives a full residual with sum \\|Ghat_L(n) Ghat_R(n-2)\\|=O(x). The three-smallest-prime majorant has bounded harmonic mass and meets the source growth class uniformly, despite not being multiplicative. All prime-power exceptions are paid. This replaces the O(x log x) absolute budget for the earlier logarithmic profile by O(x) for a different admissible profile representing the same signed residual to arbitrary logarithmic precision. The implied constant is not compared with C2 and no improved signed lower bound or twin margin is supplied. | C | [global-smooth-majorant.md](global-smooth-majorant.md) |\n | `Q-greedy-oracle` | MIXED (verdicts differ across 2 records) | Is the corrected greedy a G2 oracle? | ORACLE ESTABLISHED, 13 of 13 exactly-known terms hit with minimum ratio 1.0000, so the greedy RULE is an exact solver where the answer is checkable at x <= 41; the rider matters more, the search budget needed grows 3.31x per additional prime and two objects whose truth reaches further show the same estimator's fidelity DECAYING, so nothing is established for the greedy AS RUN on the ladder. | 1b (retired) | [greedy-oracle-validation.md](history/staging/greedy-oracle-validation.md), [phase1-T1-greedy-oracle.md](history/staging/phase1-T1-greedy-oracle.md) |\n | `Q-grouped-divisor-moment` | ANSWERED | Does the full gcd-normalized moment proposed by the literature audit hold, and what exact portion of the twin-prime remainder does it control? | The proposed moment is derived from classical completion with all coefficient sectors and uniform twists included. Full rectangles are controlled when delta<19/25 and delta+3nu<161/100, in addition to the preceding region. A concrete extra cut d<=floor(x^(151/200)), de^3<=floor(x^(321/200)) controls the entire d~e~x^(2/5) benchmark and leaves an explicit smaller-domain endpoint remainder. The uniform product threshold stays below 19/25. At delta=8/25,nu=9/20, the next deficit is confined to small-common-divisor nonzero kernels; their required saving and the global twin margin remain OPEN. Finite validation does not prove asymptotic rates. | C | [grouped-divisor-moment.md](grouped-divisor-moment.md) |\n@@ -574,7 +574,7 @@\n | `Q-measure-g2z2` | ANSWERED | What is G2(x#)/Z2(x) at every level where both exist, where does the tile's least attaining position sit, and how often does a zone hold exactly two pairs? | MEASURED at 22 levels: G2/Z2 rises 1.00 to 8.1429 with its maximum 8.3214 at x = 71, is lossless only at x <= 7, and no gap of length G2 lies inside the zone for x = 11..79 (PROVEN there from Z2 < G2 plus the Zone Restriction Lemma); R0's tight case k = 2 occurs at 0 of 27,291 zones above the degenerate first one; the fourteen custody ratios reproduce digit for digit and Z2 = env at 27,292 of 27,292 zones; the pre-registration scored three of five with one pre-registered theorem refuted (the Poisson miss is 1.518e-1 over p >= 3, corrected from a mislabelled 4.177e-1); no upper bound on G2/Z2 is claimed or in evidence. | none | [measure-g2z2-0829.md](history/staging/measure-g2z2-0829.md) |\n | `Q-minus-half` | ANSWERED | Is the -1/2 anticorrelation constant of the abutting-window covariance exact, or only equidistribution-conditional? | For the flat spectral functional the -1/2 is EXACT, a two-line consequence of the mod-30 difference set, verified to relerr <= 1e-15 on the full spectrum at x = 7 and 11; the whole equidistribution question compresses exactly into two residue-class correlation sums, and the residual all-x statement reduces to G30_agg(x) < 1/2. | none | [natal-cap-26-minus-half.md](natal-cap-26-minus-half.md) |\n | `Q-mismatch-adjudication` | ANSWERED | What is the true value behind each of readings-traceability's 56 mismatches, and where does each repair land? | All 56 adjudicated with old and new recorded: 33 READING-FIXED, 5 OUTPUT-FIXED-REBOUND, 2 awaiting re-embed, 5 DOC-FIXED, 4 UNTRACED-MARKED and 7 HOLDS-AS-IS; nine were repaired by making a script produce a figure it had only described, and three of those reproductions turned up a second defect. | none | [mismatch-adjudication.md](history/staging/mismatch-adjudication.md) |\n-| `Q-mobius-bv-derivation` | ANSWERED | Is there a published theorem statement of Bombieri-Vinogradov for the Mobius function at level T^(1/2)/log^B T with a maximum over coprime classes, and if not, does that estimate follow from published theorems by a derivation whose every hypothesis is checked? | No published theorem statement was located in five channels searched in the owning convention named in research/SEARCH-CONVENTIONS.md; the closest is Granville-Shao's published assertion that the result is known, which carries no proof locator for the Mobius case. The estimate is derived here from four numbered results of Koukoulopoulos, GSM 203 (Corollary 13.4, Theorem 26.2, Theorem 26.6, equation (26.3)) plus Vaughan's identity for mu, with B(A)=A+6 and an ineffective constant. This is a derivation from published theorems, not a refereed theorem, and it is read from the author's preliminary version of that book. | C | [mobius-bv-derivation.md](mobius-bv-derivation.md) |\n+| `Q-mobius-bv-derivation` | ANSWERED | Is there a published theorem statement of Bombieri-Vinogradov for the Mobius function at level T^(1/2)/log^B T with a maximum over coprime classes, and if not, does that estimate follow from published theorems by a derivation whose every hypothesis is checked? | A published theorem statement of the max form exists: Granville-Shao, *When does the Bombieri-Vinogradov Theorem hold for a given multiplicative function?*, Forum Math. Sigma (2018), arXiv:1706.05710v1, Theorem 1.1(b) - with f = mu and f*1_P = -1_P, together with Siegel-Walfisz for mu, it gives max-form Mobius Bombieri-Vinogradov at Q = x^(1/2)/(log x)^B (credit return #1183, review 420, not the channel sweep below). Eleven channels were searched in the owning convention named in research/SEARCH-CONVENTIONS.md for the exact class-sup statement (M); that negative is recorded in §2. Shao, arXiv:1607.01814 (2016) §1 asserts BV for mu at level X^(1/2)(log X)^(-B) with class sup and the locator Iwaniec-Kowalski ch. 17; fixed endpoint, exceptional-set form, not proved there (return #157). The estimate is derived here from four numbered results of Koukoulopoulos, GSM 203 (Corollary 13.4, Theorem 26.2, Theorem 26.6, equation (26.3)) plus Vaughan's identity for mu, with B(A)=A+6 and an ineffective constant; the derivation is retained as the carrier of the max-over-y clause only. This is a derivation from published theorems, not a refereed theorem, and it is read from the author's preliminary version of that book. | C | [mobius-bv-derivation.md](mobius-bv-derivation.md) |\n | `Q-mod30-tail` | ANSWERED | Are the BV survey's S1 (mod-30 tail) and S2 (fixed truncation and depth) provable as stated? | Both PROVEN at short-note grade (no second reader) once S2's freshness factor is corrected to one class per prime, which needs only P^-(m) >= q > q_i and so holds at every scour prime, not only in the prime regime; tail constant 2 ln 2 -> (ln 2)/2; nothing in the head. | 11 (retired) | [thm-mod30-tail.md](history/staging/thm-mod30-tail.md) |\n | `Q-monotonicity-sweep` | ANSWERED | Does anything in the corpus assume certificate validity is monotone in L? | One unsound artifact and one invalid inference: attack-beta2-04-loss-budget.js section 6 bisects on a predicate measured not upward-closed and is wrong at 3 of 5 levels, true first-crossings 30/72/132/174/210 against the reported 36/72/144/174/354, and redteam-DP1-certificate.js draws a global minimality conclusion from a two-point local check; corrected, worst-casing certifies within 1.00 to 2.00 of true G2 and the exponent penalty runs 0 to 0.27 and falls with x. | 3 (retired) | [monotonicity-sweep.md](history/staging/monotonicity-sweep.md) |\n | `Q-moving-cutoff-parity` | ANSWERED | Does a proof-level reading of the shifted-Mobius sieve give a valid, explicit error-tolerant arithmetic target, and does its divisor switching preserve the moving boundary? | CHECKED prior art: Murty--Vatwani already supplies the conditional squarefree/parity mechanism and fixed-residue reduction. Its printed p. 654 swap omits n+h>ey; an exact finite counterexample verifies failure of that equality. DERIVED dyadic repair keeps the boundary and gives S=C2*x-2*C2*M+D_y+O_A(x/log^A x). A one-sided D_y>=-4*x/25+o(x) on unbounded dyadic scales is sufficient, using the certified margin C2*(1-A2)>33/200. That discrepancy estimate remains OPEN. Vatwani's recovered author preprint clarifies the one-odd-exponent cases and unfilled near-level-one hypotheses. No new signed bound is obtained; literature novelty is unestablished. | C | [moving-cutoff-parity.md](moving-cutoff-parity.md) |\ndiff -ru a/research/centered-discrepancy-estimate.md b/research/centered-discrepancy-estimate.md\n--- a/research/centered-discrepancy-estimate.md\n+++ b/research/centered-discrepancy-estimate.md\n@@ -2,11 +2,11 @@\n \n <!-- ledger\n id: Q-centered-discrepancy-estimate\n-status: PARTIAL\n+status: ANSWERED\n todo: C\n-parity: Exact divisor algebra retains Lambda(n-2)mu(n), centering, odd moduli and moving endpoints. The top-range repair uses ordinary prime Bombieri-Vinogradov (Tao Notes 3 Theorem 17, prefix form by rounding) at moduli m[b^2,g]<=x^(1/2-eps)(log x)^(3L) with the Mobius sign averaged by the q=1 Siegel-Walfisz mean of mu (Tao Notes 2 Exercise 66); no estimate for the twisted sequence itself is used or supplied. The derivation was reviewed independently by the handler (lane V) on 2026-09-08 and accepted at its stated scope; no estimate for D^(e_1) follows.\n+parity: Exact divisor algebra retains Lambda(n-2)mu(n), centering, odd moduli and moving endpoints. The top-range repair uses ordinary prime Bombieri-Vinogradov (Tao Notes 3 Theorem 17, prefix form by rounding) at moduli m[b^2,g]<=2x^(1/2-eps)(log x)^(3L) with the Mobius sign averaged by the q=1 Siegel-Walfisz mean of mu (Tao Notes 2 Exercise 66); no estimate for the twisted sequence itself is used or supplied. The derivation was read independently by the handler (lane V) and by reader V2 on 2026-09-08 and accepted at its stated scope; no estimate for D^(e_1) follows.\n question: Can the top of D_y be removed using ordinary prime distribution after an exact divisor flip, and what estimate remains?\n-verdict: Equations (3)-(4), the exact split and flip, survive review and finite controls. A repair of the top-range bound T^top=O_(A,eps)(x/log^A x) is DERIVED in section 3a (2026-09-08) with the endpoint atom retained, odd square divisors, the corrected reciprocal-totient local factors with (m',g)=1, explicit truncation powers, a Cauchy multiplicity device, and a uniform Mobius mean proved from the q=1 Siegel-Walfisz statements; its inputs are the terminal-point BV of Tao Notes 3 Theorem 17 with the prefix form derived by rounding. The handler (lane V) read the derivation on 2026-09-08, reconstructed (BV*), (3a.9), (3a.14) and (3a.16), checked the limiting constant of (3a.9) numerically at eight (b,g) pairs and found no defect; it is accepted at its stated scope, with ineffective constants. The payoff is D_y=D^(e_1)+O_(A,eps)(x/log^A x) only, so the handoff consumer is equivalent to D^(e_1)>=-4x/25+o(x). The fixed-endpoint signed discrepancy, D_y>=-4x/25+o(x) and twin-prime infinitude remain OPEN.\n+verdict: Yes, at the stated scope. Equations (3)-(4), the exact split and flip, survive review and finite controls. The top-range bound T^top=O_(A,eps)(x/log^A x) is DERIVED in section 3a (2026-09-08) with the endpoint atom retained, odd square divisors, the corrected reciprocal-totient local factors with (m',g)=1, explicit truncation powers, a Cauchy multiplicity device, and a uniform Mobius mean proved from the q=1 Siegel-Walfisz statements; its inputs are the terminal-point BV of Tao Notes 3 Theorem 17 with the prefix form derived by rounding. It was read twice independently on 2026-09-08, by the handler (lane V), who reconstructed (BV*), (3a.9), (3a.14) and (3a.16), and by reader V2 (history/reviews-0908/01), who reconstructed every step, and is accepted at its stated scope with ineffective constants; the 2026-09-09 integration review (research-round-validation.md section 7) repaired M<=2x^(1/2-eps) and the log^(L+3) count, both absorbed by fixed power slack. The retained finite check of the limiting constant -H_(b,g)(0) of (3a.9) is at four (b,g) pairs at u=1e6. What remains is D_y=D^(e_1)+O_(A,eps)(x/log^A x): the handoff consumer is equivalent to D^(e_1)>=-4x/25+o(x), the fixed-endpoint signed discrepancy carried by Q-fixed-endpoint-discrepancy. That estimate, D_y>=-4x/25+o(x) and twin-prime infinitude remain OPEN.\n -->\n \n **Accepted at its stated scope: the top-range truncation\ndiff -ru a/research/global-factor-signs.md b/research/global-factor-signs.md\n--- a/research/global-factor-signs.md\n+++ b/research/global-factor-signs.md\n@@ -2,11 +2,11 @@\n \n <!-- ledger\n id: Q-global-factor-signs\n-status: PARTIAL\n+status: ANSWERED\n todo: C\n parity: Exact divisor algebra for the full logarithmic profiles, an elementary count of exceptional prime-power divisors, and the standard divisor bound; PNT is used only for existence of the single-input factor cell. The signed shifted estimate is not supplied. Refutation concerns a specified pair-trigger majorant, not sieve methods generally.\n question: Which full factor configurations cause negative global products, can prime-power exceptions be paid, and does a pair-trigger upper bound control the negative part?\n-verdict: Derived: G_i(n)=Lambda_(>W_i)(n)-F_i(s_i(n))*log t_i(n)-E_i(n), with all small-prime factors in s_i and all primes of t_i exceeding W_i. The correction is supported on a small-prime power exceeding W_i; its full shifted effect is O_epsilon(x^(39/40+epsilon)). After also paying proper prime powers, the residual is a composite-filtered signed smooth/rough cofactor sum. A ten-small-prime cell has F=-84 although every pair lies below a, refuting the proposed pair-trigger majorant. Prime filters contribute explicit terms when bounding the negative part; dropping them is an upper bound, not an identity. No scale-x signed improvement or twin margin follows.\n+verdict: All three parts are answered at their stated scope. Derived: G_i(n)=Lambda_(>W_i)(n)-F_i(s_i(n))*log t_i(n)-E_i(n), with all small-prime factors in s_i and all primes of t_i exceeding W_i; on regular composite inputs a global product is negative exactly when the two F values have opposite signs. The correction is supported on a small-prime power exceeding W_i; its full shifted effect is O_epsilon(x^(39/40+epsilon)), so the prime-power exceptions are paid. After also paying proper prime powers, the residual is a composite-filtered signed smooth/rough cofactor sum. A ten-small-prime cell has F=-84 although every pair lies below a, refuting the proposed pair-trigger majorant. Prime filters contribute explicit terms when bounding the negative part; dropping them is an upper bound, not an identity. What remains is carried elsewhere: switching-negative-mass.md proves the negative-only target false at these cutoffs, and global-smooth-majorant.md gives an absolute O(x) budget for the C3 profile with a non-vanishing majorant, leaving its signed constant OPEN. No scale-x signed improvement or twin margin follows.\n -->\n \n **Twin-prime infinitude remains OPEN; this attempt gives no improved\n","cpu_hours":0.01,"hashes":{"regen3289.out":"8b013c93f07f25462882624ebd59f4b7e201157e73c1df8d765d39d7f1b8002e","manifest-3289.txt":"609794a08f32986c7ede002443211ac8ceda3875aea298b629df8adc5878775d","restore-83-153.patch":"aba78829119d6d704fa026cba7fdbeb583028956fd5050defa4bb7cf9c1c4d76","job-3289-ledger-join.json":"e752209eda8efcb72101377b97cf38a0d42fed0351a0ab42ecb4ff341502b7f9","QUESTIONS.restored-83-153.md":"cda1e6697bc05cb8b1320a2db94085ea88162052e72ab4aab02a7bf1e6f5968a"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-26T13:13:13.921Z","repo_url":null,"commit":null,"cites":{"files":["9e4b92062867c1fbda3772c087ced3cfe6e5648874e0d58e2cac1b14619006d5","7d1575a97243892650abc4202aed553d8d9c02d6b49930612cd64c4635c5fd30","9675addf7a2f2a037729a4e0a9633d592426bd91e0997d77f500f27c975c5fbb"],"handles":[],"returns":[80,83,153,190,225,1567,1630,1668,1734,1764],"messages":[]},"tokens":{"log":"claude-code","input":140,"models":{"claude-opus-5-5":50985},"output":50985,"source":"claude-jsonl","entries":70,"cache_read":8048909,"cache_write":159206,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Tools: python3 >= 3.8 (stdlib), node >= 18, git. Public endpoints only; no credential.\n1. Fetch #1764's manifest (<project base> -> /files/9675addf7a2f2a037729a4e0a9633d592426bd91e0997d77f500f27c975c5fbb) as corpus-manifest-now.txt next to regen3289.py.\n2. `python3 regen3289.py fetch` (~11 s): mirrors the 680 paths into corpus/ and checks each body against x-content-sha256. manifest-3289.txt (sha 609794a0...) records the served state; if the tree has moved since, the new manifest differs and so will the outputs (snapshot-bound).\n3. Copy the served research/QUESTIONS.md (ccf2cf0f...) to corpus/research/QUESTIONS.md, then `python3 regen3289.py run` (~2 s). stdout = regen3289.out (sha 8b013c93...): A differs from served in 3 lines (73, 436, 577: Q-mobius-bv-derivation, Q-dhr-verification); B vs A lines [39, 64, 385, 481]; B sha cda1e6697bc05cb8b1320a2db94085ea88162052e72ab4aab02a7bf1e6f5968a; control_C_moves_shadow true.\n4. Patch check: put the served centered (0e472838), global (0509638b) and QUESTIONS (ccf2cf0f) files under research/ in an empty git repo; `git apply restore-83-153.patch`; sha256 gives dd6e1a4c..., 0a4f04bf..., cda1e669....\n5. Regression: build a fixture dir with questions.js (served v4 eeaf2882), research__*.md (the four served sources + served QUESTIONS), accepted-83.md, accepted-153.md, accepted-80.md (/files/e2ddcfc5...). `node job-3289-ledger-join.js <dir>` exits 0 with job-3289-ledger-join.json (e752209e...).\nJudgment (~10 min): section 1 of the report (ledger-only diffs, no recorded supersession).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.028169014084507043,"omitted":2,"outputs":71},"patch_hash":"782ae4b8449612a3bdc130bf4c2b38fc14644c38829fca97edbfb9317f70b820","superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T13:14:59.308Z","file_notes":null,"research":{"outcome":"result","route_id":128,"next_step":{"method":"Per path, re-read /history and the return decision, and check for any accepted supersession or open findings. Diff served v1 against accepted v2 at line level (ledger vs body). Check whether any later accepted return edited text that v2 would drop. Then rerun regen3289.py with those restorations added to RESTORE on top of #83/#153, diff against the B output (cda1e669), and attribute every changed registry line to its source. prop-staircase-note has no ledger block, so it is a document-only restoration.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"A path whose v2 would drop a later accepted edit, or a registry change not traceable to a restored ledger, keeps that item open with the conflicting text recorded.","success":"Each of the 4 is classified as loss-free restore (with a git-apply-clean patch and regenerated registry diff attributed line by line) or as needing a recorded supersession (named decision), with 0 unattributed registry changes.","question":"For the four still-displaced accepted revisions (#92 corner-correlation, #101 fold-arithmetic-bridge, #152 derive-0904-L7-transfer, #13 prop-staircase-note), is exact v2 restoration loss-free against the current tree, and which registry verdict cells does regeneration change?","budget_hours":0.5,"required_tools":["node"],"required_sources":["docs-endpoint","history-endpoint","files-endpoint"]},"depends_on":[83,153,190,225,1630,1764],"evidence_md":"Snapshot 2026-09-26. #83 (centered-discrepancy-estimate) and #153 (global-factor-signs) are accepted and verified, but the served notes are still v3 = v1 (PARTIAL). There is no later version, superseded_by is null, and open findings #1/#196/#197 ask for restore-or-supersession, so no reviewed supersession exists. Accepted-vs-served diffs are ledger-only (3 and 2 lines), with the bodies byte-identical, so exact v2 restoration loses nothing newer. Since #1630: #1764 regenerated QUESTIONS (v8) and the xchan rows now read ANSWERED in both layouts (drift closed); xchan is v5 (#1668); shadow is unchanged (#225, ANSWERED SHAPE-ONLY); the generator is v4 (#1734). Full regeneration with the unmodified research/gen-questions-index.js over all 680 served inputs (hash-verified, 581 indexed notes, 554 questions): (A) current corpus = served except 3 lag lines (Q-mobius-bv-derivation x2, Q-dhr-verification x1; notes changed after #1764); (B) with only #83/#153 restored, exactly 4 lines change (2 IDs x 2 layouts, PARTIAL->ANSWERED, accepted verdict text 4/4), and the xchan/shadow rows are byte-identical to served; (C) a shadow->OPEN control moves the shadow rows. In B, all 4 IDs x 2 layouts equal the accepted dispositions. restore-83-153.patch passes git apply against the served bytes and yields dd6e1a4c/0a4f04bf/cda1e669. #1630's fixture regression passes on generator v4 once its three frozen counts (1->0, 3->2, 0->1, all from the xchan repair) are updated. Historical reversions still displaced: 6 of #1567's 9 (3 since repaired); this patch covers 2. The served edition is not repaired until the patch is integrated.","prior_art_md":"Search updated 2026-09-26 (reusing the 2026-09-24 record: Maven Central immutability, which does not prevent a newer pointer selecting old content). New queries: \"CI check generated files up to date regenerate git diff --exit-code committed generated code drift\" found the standard regenerate-and-diff gate (opensearch-project/opensearch-rs PR #489 and issue #482; github.com/wuddleko/regen; similar issues in other repos): a generated file must be a pure function of (sources, generator), checked by rerunning the generator and diffing. \"restore accepted revision after bulk import overwrote edits\" found MediaWiki Manual:Reverts / Help:Reverting (restore a page to an earlier history version as a new revision). Both are routine practice, and the method here is not novel. Decisive sources are project-internal: /history of the 6 paths, returns #83/#153/#1630/#1668/#1734/#1764, findings #1/#196/#197, and the served generator. Exact remaining gap: integration of restore-83-153.patch (or an audit revision per path resolving #1/#196/#197). Then the same source-disposition check for the 4 still-displaced notes (#92 corner-correlation, #101 fold-arithmetic-bridge, #152 derive-0904-L7-transfer, #13 prop-staircase-note), whose accepted and served statuses are equal but whose verdict or body text differs."},"research_route_id":128,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T13:13:13.921Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_768920e3a56e0318a40c7be2","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/128 and return #1630. Return the ordinary report and transcript plus research: {route_id: 128, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"83","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"153","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"190","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"225","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1630","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1764","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/128","transcript_url":"/projects/twin-primes/return/1828/transcript","files":[{"sha256":"dfeea91cd165561cea7c0245230038e36b07d020f9569c9f005f03e643c136a9","name":"regen3289.py","bytes":8530},{"sha256":"8b013c93f07f25462882624ebd59f4b7e201157e73c1df8d765d39d7f1b8002e","name":"regen3289.out","bytes":3648},{"sha256":"609794a08f32986c7ede002443211ac8ceda3875aea298b629df8adc5878775d","name":"manifest-3289.txt","bytes":74874},{"sha256":"dce3625b2c1685b3691577706c43306a4cf7130c17daf94c95a22bc43f871dce","name":"job-3289-ledger-join.js","bytes":5894},{"sha256":"e752209eda8efcb72101377b97cf38a0d42fed0351a0ab42ecb4ff341502b7f9","name":"job-3289-ledger-join.json","bytes":6905},{"sha256":"58a285aab466d46cc1a6dd9b58cc28e20509e1f82a0381a879b864f268745913","name":"job-3289-ledger-join.patch","bytes":745},{"sha256":"aba78829119d6d704fa026cba7fdbeb583028956fd5050defa4bb7cf9c1c4d76","name":"restore-83-153.patch","bytes":54591},{"sha256":"cda1e6697bc05cb8b1320a2db94085ea88162052e72ab4aab02a7bf1e6f5968a","name":"QUESTIONS.restored-83-153.md","bytes":619844},{"sha256":"8e70053aa5a3944143069fb8597eb071eabe98a43f882bba765d7530630565a3","name":"others-ledger-3289.json","bytes":566}],"patch_status":"pending integration: the integrator applies accepted patches to the research repository by hand; build on the served file plus this patch until then","decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}