# The 2026-08-29 registry debts, landed: two import rows, one prior-art sentence, sixteen escape labels, and three of seven REFUTED wording fixes *(2026-08-29. Staging note, HELD, no adversarial pass. This is an APPLY pass against four records written earlier the same day. It ran no producer, computed nothing, and opened nothing. Four files were edited: `research/IMPORT-MAP.md`, `research/PRIOR-ART.md`, `research/REFUTED.md` and `research/history/staging/recon-0829-escapes.md`. No git command was run. `research/history/CHANGELOG.md` was not edited; the paragraph for it is section 6 below. Every number carried into an edited file is `[CITED]` from the record that measured it and none was recomputed, per the standing compute rule. The one place this pass produced new content rather than carrying it is section 3, the escape labels, and nine of those sixteen are marked `[FRESH]` in the table itself.)* --- ## 0. Counts | debt | record it comes from | landed | left | |---|---|---|---| | `IMPORT-MAP.md` rows 20 and 21 | `recon-0829-farfields2.md` section 5 drafts, corrected by `redteam-0829-measure-c.md` | 2 rows written, plus the section 2 preamble amended for the count and the reserved numbers | nothing | | `PRIOR-ART.md`, the Holt-Rudd spectrum | `recon-0829-farfields2.md` section 3e, corrected at `redteam-0829-measure-c.md` section 1h | 1 sentence added to the A9 demotion paragraph, plus three words in the correspondence table's cell | nothing | | `recon-0829-escapes.md` section 6, sixteen labels | owed by `applied-0829-measure-c.md` E22 | 16 labels with a one-clause reason each, plus a caption | nothing | | `REFUTED.md`, seven HOLD proposals | `applied-0829-refuted.md` section 2, from `refuted-audit-0829-1/2/3/4.md` section 3 | 3 applied as the audit's exact old to new | 4 for Chris, verbatim in section 4 | **Row count in `REFUTED.md`: 67 data rows before and 67 after**, counted as the table lines after the header and separator, one line each. No row was added, removed, split or merged, no verdict word changed, and no route moved between CLOSED, REFUTED and RETIRED. **What did not move.** Nothing here bears on the exponent band `(2, 4.26645]`, on `Z2`, or on any open question. Three of the four debts are registry hygiene and the fourth, section 3, is a labelling pass whose only forward value is that a reader moving between the two 2026-08-29 recon notes now finds the same frame in both. Two of the three labels that came out `(ii)` say that a literature escape, if its conclusion held in this object, would settle the conjecture, which is a statement about how expensive the escape is and not a route. --- ## 1. `research/IMPORT-MAP.md`: rows 20 and 21, as written Both rows were priced by `recon-0829-farfields2.md` section 5 and both are written at the grade the red-teamed note supports and no higher. Neither is recorded as a LANDED import, because neither ran an experiment: the status cell of each says `PRICED 2026-08-29, resolved at recon grade`, so the map's own count of nineteen landings is unchanged. **Row 20, extremal majorants and minorants.** Theorem column: Selberg's majorant `C_E` at type `2π` with `∫(C_E − χ_E) = 1` and the large sieve constant `A(N, δ) = N − 1 + δ^{-1}`, Vaaler *Bull. AMS* 12 (1985) section 1 **[SOURCED, verbatim, hashed in the record]**. Object: the Lemma V sawtooth and its frequency set. Fit: STRONG-ANALOGY at the object, blocked arithmetically at the working point. Circularity: **SPLIT**, CLEAN at the mean-square form and **TPC-STRENGTH** at the sup form, the second inheriting `attack-wrongdirection-audit.md` section 2 item 6. Payoff: **CLOSURE with mechanism plus PUBLISHED-ANCHOR**, no THEOREM and no DERIVED-CONSTANT. Three things are carried into the status cell that keep the row from reading higher than the record supports: the deciding arithmetic is `[DERIVED HERE]` hand arithmetic with no second reader and its falsification check has **NOT** run; the two blocks behind it (`ℓ² → ℓ²` against the wall's `ℓ¹ → ℓ²`, and an inequality offered where the Fejér-mass equality stands) do not depend on it; and the CLOSURE is explicitly graded as the closure of a family already partly resident, per `redteam-0829-measure-c.md` section 5g. **Row 21, the eigenstructure of the gap transfer matrix.** Theorem column: `M_J = R · Λ · L` with binomial eigenvectors and eigenvalues `a_{kj} = ∏_{q=17}^{p_k}(q − j − 1)/(q − 2)`, Holt and Rudd arXiv:1408.6002 section 5.1 and Table 1 **[SOURCED, verbatim, hashed]**. Object: the fold's histogram operator. Target hole: none reached, and the row says so. Fit: EXACT-IDENTITY at the operator, wrong precision and wrong functional at the conclusion. Circularity: CLEAN. Payoff: **PUBLISHED-ANCHOR plus WALL-ADDRESS**. The status cell carries the red team's correction rather than the recon's first reading: `lit-pdf-holt-rudd.md` line 259 already held the per-prime eigenvalue `(p − j − 1)/(p − 2)`, so what the row adds is the product over primes, the printed values and the rate. The rate constant is marked **[SCRATCHPAD-GRADE]** in the cell, with the note that it is two measurements and no embedded producer, and with the observation that nothing in the row's verdict rests on the constant rather than on the `1/ln z` shape. **One preamble amendment.** Section 2's opening paragraph said "All seventeen rows have now run or been overtaken". It now says seventeen **original** rows, records that rows 20 and 21 were added under section 0a's standing rule and are resolved at recon grade rather than by an experiment, and states that 18 and 19 stay reserved for `recon-0828-farfields.md` section 5's two drafts, which have not been written. Without that sentence the numbering reads as a gap. --- ## 2. `research/PRIOR-ART.md`: the sentence added Placed at the end of the **A9 is demoted** paragraph, which is the prose the correspondence table's line 194 attribution points into. Verbatim as written: > The eigenvalues are theirs as well, not only the eigenvectors: section 5.1 and > Table 1 of 1408.6002 print the whole spectrum in closed form, > `a_j = prod_{17<=q<=p} (q-j-1)/(q-2)`, with `a_2 = 0.10206751799779` at > `p = 999,999,999,989`, so the operator's mixing rate `a_2 ~ 2.82/ln p` > **[SCRATCHPAD-GRADE**, the constant flat to four decimal places across four > decades `z = 10^4` to `10^7`, and extrapolating to the printed value at a > relative `1.8e-5`; measured twice, never inside an embedded producer**]** is a > consequence of a published formula and priority for it is Holt and Rudd's, as > it is for the eigenvectors, while what this corpus held beforehand was the > per-prime `(p-j-1)/(p-2)` of the bidiagonal form and not the product over > primes. The pointers appended to it are `history/staging/lit-pdf-holt-rudd.md` line 259, `history/staging/recon-0829-farfields2.md` section 3e, and `history/staging/redteam-0829-measure-c.md` section 1h for the correction. **Also changed, three words in the table.** The correspondence row for A9 read `transfer matrix M_J, binomial eigenvectors, "driving terms" | 1408.6002 §5, 2014`. It now reads `transfer matrix M_J, binomial eigenvectors, closed-form eigenvalues a_j, "driving terms" | 1408.6002 §5 and §5.1 Table 1, 2014`. The attribution is widened; nothing in it is claimed for this corpus. **Calibration of the added sentence.** The closed form and the printed value are **[SOURCED, verbatim]** at the record's hashed PDF. The constant 2.82 and the `1.8e-5` agreement are **[SCRATCHPAD-GRADE]** and stay so: the recon's producer lives in a session scratchpad and the red team's reproduction was an independent sieve, which is two measurements and not an embedded, `qc`-gated producer. The `1/ln p` shape itself is Mertens and is not at issue. Priority is stated as Holt and Rudd's in the sentence itself. --- ## 3. `research/history/staging/recon-0829-escapes.md` section 6: the sixteen labels The `[OPEN, the full column is owed]` paragraph left by `applied-0829-measure-c.md` E22 is replaced by a three-column table, sixteen rows, with the grading rule restated from `attack-wrongdirection-audit.md` section 0's last bullet: **(i)** the escape's conclusion, if it held in this object, would still not imply TPC through any bridge this corpus holds; **(ii)** it would; **(iii)** the note cannot tell. | row | label | the reason, compressed | |---|---|---| | 3a | **(i)** | its only named consumer is 3d's vector sieve, whose payout item 10 places inside the legal band; no landing below exponent 2 is claimed `[FRESH]` | | 3b | **(ii)** | an asymptotic count of primes in the twin sequence is the Hardy-Littlewood asymptotic, which gives TPC with no bridge `[FRESH]` | | 3c | **(i)** | the conclusion is a `P_2` and not a pair, and Chen's theorem is PROVEN while TPC is open `[FRESH]` | | 3f | **(i)** | a `kappa = 2` construction below `beta_2` lands in the band item 10 makes legal `[CITED, item 10]` | | 4a | **(i)** | the window arrives at `X^theta`, and Axis A's line is `H <= w^2 - w` `[CITED, Axis A]` | | 4b | **(i)** | granted at two classes the threshold is 4.26645, above Axis A's line at 2 `[CITED, Axis A]` | | 4d | **(i)** | Linnik is PROVEN while TPC is open and sits downstream of the one-class target `[FRESH]` | | 4e | **(ii)** | `eps W < 1` certifies the anchored window, which item 8 grades TPC-strength and stronger `[CITED, item 8]` | | 4f | **(i)** | existence of a good configuration somewhere, where Axis D admits only `min_x N(x) >= 1` `[CITED, Axis D]` | | 4g | **(i)** | an `L^1` average over starting points, Axis C's almost-all side `[CITED, Axis C]` | | 5a | **(ii)** | the major-arc output on the twin system is the Hardy-Littlewood asymptotic `[FRESH]` | | 5b | **(i)** as the theorem reads, **(ii)** at the modulus the escape would need | at `q <= (log x)^A` it already holds here and buys the priced tail refinements; at modulus `x#` over a window of about `x'^2` it is the Zone Postulate `[FRESH]` | | 5c | **(i)** | existence of favourable rows, and the anchor is fixed by arithmetic rather than chosen `[CITED, Axis D and Axis B]` | | 5d | **(i)** | a two-sided existence-of-irregularity statement, PROVEN in print while TPC is open `[FRESH]` | | 5e | **(i)** | the conclusion is empty here, and an empty conclusion implies nothing `[FRESH]` | | 5f | **(i)** | the anchor's orbit is a single point, so the conclusion is degenerate here `[FRESH]` | **What the caption in the note says, and why it says it.** Twelve rows are (i), three are (ii), one is split, none is (iii). Seven reasons are citations to `attack-wrongdirection-audit.md` naming the axis or item; **the other nine are fresh judgements made this pass against that audit's axes, marked `[FRESH]` in the table, and they have had no second reader.** That is stated in the caption rather than left for a reader to infer, because the audit's section 2 grades this programme's own targets and not the literature's escapes, which is exactly why E22 landed only three rows by citation and left sixteen. **The one thing a reader should not over-read.** A **(ii)** label opens nothing. All three of the (ii) rows fail their own hypothesis check in sections 3 to 5 of that note, and the label is a reason an escape is expensive rather than a reason to spend a session on it. No verdict, killer, landing killer or number in the note was changed, and the note's ledger block was not touched. **Where the grading is thinnest.** Row 3a, because section 3a never states what exponent a `kappa = 2` bilinear construction would deliver; the label rests on the consumer the row names and would move to (ii) if such a construction delivered an asymptotic rather than a lower bound, which section 3a does not claim. Row 5b is carried as a split rather than forced to one label, following `recon-0829-farfields2.md` section 3c's own precedent for a target whose legal form and paying form differ. --- ## 4. `research/REFUTED.md`: three applied, four left for Chris ### 4a. Applied, as the audit's exact old to new All three change wording or a number's calibration only. No verdict word moved, no route changed register, nothing reopened, and each row is still one line. **Row 23, `:49`, improving `β₂`** (audit 2). Applied verbatim. - old: `unimproved since Diamond–Halberstam 2008; everything after is worse at κ = 2 (Blight 4.45, Franze 4.516, Opera de Cribro 4.83)` - new: `closed for us, not proven unimprovable: unimproved since Diamond–Halberstam 2008 and everything after is worse at κ = 2 (Blight 4.45, Franze 4.516, Opera de Cribro 4.83)` The row's verdict word stays CLOSED. What the qualifier changes is the reader's reading of why: the row previously read as though the cap were known unimprovable, and the audit's finding is that the record says only that nothing better is in print. **Row 17, `:43`, `u_sup` as an unconditional worst-position bound** (audit 2). Applied to the audit's `new` string. The audit's `old` string no longer matched the file, because the counting-slip half was already corrected by audit 1's APPLY, which `applied-0829-refuted.md` section 2 states; what was still open and is now applied is the MEASURED marker and the basis-independence rewording. - in the file before this pass: `it rises at all eight steps over nine levels on a flat ~2.05 per-prime factor, and the divergence is basis-independent, the (h,m) basis worse by a flat 7.26` - now: `measured rising at all eight steps over nine levels on a flat ~2.05 per-prime factor, and not a basis artifact, the (h,m) basis being strictly more expensive by a flat 7.26` `history/staging/attack-hm-basis.md` line 429 still carries its own sentence "the divergence is basis-independent **across this pair**", which is the record's own scoped wording and is not the unqualified claim the row carried; that file is outside this pass's fence and needed no change. **Row 20, `:46`, the loss-budget LP at x = 43** (audit 2). Applied verbatim. The 33 readings, the 0.62 to 0.98 error bar and the non-convergence are all at the record (`history/staging/lp-push-x43.md` sections 6 and its measured table; `research/lp-push-x43.js`), and were carried, not recomputed. - old: `floor 3.3152, with DP1 carrying 71% of the loss against a measured losing consumer; θ is the only place left to push` - new: `calibrated floor 3.3152 over 33 readings, error bar 0.62 to 0.98 and not converged, with DP1 carrying 71% of the loss; inside the LP's information class θ is the only place left to push` ### 4b. Left for Chris, verbatim, with the audit's reason Four, none applied, none partially applied. Each is reproduced with the audit's own flag text so a ruling needs no reopening of the audits. **1. Audit 1, HOLD, outside the row set.** Reason not to apply here: the target is a canonical live file outside this pass's fence, and the audit itself assigns the call to Chris. > `G2-STATE.md`:35-38 lists the Lemma V item in its PROVEN block as one claim. > Three rungs are bundled there: the mean-square form (PROVED, > `sift-limit-attack.md` §7e), the bound on B (INFERRED chain, VERIFIED steps, > `attack-AB-bounded.md` §1.1) and the never-binding finding (VERIFIED at > z = 13..47). Splitting the entry is a live-layer edit to a canonical file and > is Chris's call. **2. Audit 2, row 25, `research/REFUTED.md:51`, HOLD, and it is a producer fix first.** Reason not to apply here: the row change is downstream of a producer edit that must be re-stamped under `research/qc/embed.js`, and the producer is outside this pass's fence; applying the row alone is exactly what the audit says would put the file and the index out of step. > The constant 3.594 in `research/attack-beta2-05-covering-prune.js`:370 and :608 > is not the root of the equation the same lines declare. Correcting the row > without re-stamping the producer would put the file and the index out of step, > so the producer's header text should be corrected under `research/qc/embed.js` > and the row follow. If both are done: `β_pure diverges like 7.19 lnln x` > becomes `β_pure diverges like 7.18 lnln x`. Nothing else in the corpus consumes > the constant, checked by grep on 7.19 and 3.594. **3. Audit 3, line 77, HOLD.** Reason not to apply here: the audit offers no old-to-new pair, and what it proposes is that the index start carrying per-member reading levels at all, which is a format decision for `REFUTED.md` rather than a wording fix to one row. The audit assigns that decision to Chris. > The honest scope is that three of the six named members (Warnke, Kutin, Kim-Vu) > are closed by the divergent-shape argument plus Bruhn-Joos's own p. 15 > assessment, not by having been read at source, and the record's > `[SOURCED-BIB]` tags already say so. The closure is not in doubt. Held rather > than applied because a REFUTED row is a one-line index and per-member sourcing > grades belong in `IMPORT-MAP.md` row 7 and in `row7-recon.md` §0, where they > already are. Chris's call if he wants the index to carry reading levels; if he > does, the clause to add is *"three of the six named members are SOURCED-BIB, > closed by the shape argument rather than at source"*. **4. Audit 4, row 60, line 86, HOLD, for Chris.** Reason not to apply here: the replacement removes a number two other documents quote, and the audit ties the removal to a decision about a third file that is outside this pass's fence. > - old: *Möbius-signed weights are Fourier-flat — every mode's share of the factor's mass is ≤ 2.8e−3 against the ≳ 0.4 a usable smooth component needs, and the exact blocking term is the high-frequency remainder of the completed expansion, which keeps 1 − o(1) of the ℓ² mass and is exactly as inadmissible as the original* > - new: `Möbius-signed weights are Fourier-flat: the aggregated low-frequency mass share sits at or below the flat null at every occupied block and at 31 of 31 fixed-s slices (0.57 to 1.15 times the null, median 0.82) where a usable smooth component needs it at scale C̃/K ≫ 100, and the largest single mode, k = 0 included, carries 2.8e−3; the exact blocking term is the high-frequency remainder of the completed expansion, which keeps 1 − o(1) of the ℓ² mass and is exactly as inadmissible as the original` > > HOLD, not APPLY, because the fix removes a number (`≳ 0.4`) that two documents > now quote, and whoever removes it should decide at the same time whether > `attack-bilinear-transplant.md:419` is corrected or left as a HELD note with a > rider. The `≳ 0.4` has no derivation at any record; if it is kept it needs one. ### 4c. The deferred item, untouched `applied-0829-refuted.md` section 2's last entry, audit 1 row 16 at `research/REFUTED.md:42`, is not a HOLD: it was flagged APPLY and deferred by instruction pending `redteam-0829-theorem1.md`. That red team is being written by another agent as this pass runs, so the row is left exactly as it stands and is not counted among the seven. --- ## 5. Gate `node research/qc.js` after all four file edits, 13 checks: ``` refs 0 finding(s) quotes 0 finding(s) crosslinks 0 finding(s) scripts 0 finding(s) transfers 0 finding(s) calibration 0 finding(s) absence 0 finding(s) sourcing 0 finding(s) embeds 1 finding(s) widths 0 finding(s) ledger 0 finding(s) provenance 0 finding(s) search-convention 0 finding(s) TOTAL 1 ``` **refs 0, crosslinks 0, calibration 0 and search-convention 0, as required.** Every pointer added by this pass resolves: the record pointers in `IMPORT-MAP.md` rows 20 and 21, those in the `PRIOR-ART.md` sentence, and the audit references inside the section 6 label table. **The one finding is an `embeds` finding in another agent's producer and is not this pass's**: `research/zonegap-04-sweep-1e12.js`:352, reading "code changed since the tail was embedded". This pass edited no `.js` file. That counter moved between 1 and 5 across the five gate runs taken here while those producers were being rewritten and re-stamped elsewhere, and `refs` likewise stood at 1 before the first edit (`ZONE-POSTULATE.md`:49) and was cleared by another pass, not by this one. **Row count in `REFUTED.md`: 67 data rows, unchanged**, counted as the table lines after the header and separator. --- ## 6. CHANGELOG paragraph, for the orchestrator to append *(Not appended by this pass. `research/history/CHANGELOG.md` was not edited.)* **2026-08-29, four registries.** The registry debts left by the day's recon and audit notes were landed. `research/IMPORT-MAP.md` gained rows 20 and 21, extremal majorants and minorants at CLOSURE plus PUBLISHED-ANCHOR, and the eigenstructure of Holt and Rudd's gap transfer matrix at PUBLISHED-ANCHOR plus WALL-ADDRESS, both priced by `history/staging/recon-0829-farfields2.md` and written at the grade `history/staging/redteam-0829-measure-c.md` left them, neither counted as a landing and neither opening a route; the section 2 preamble was amended for the count and for the two numbers that stay reserved. `research/PRIOR-ART.md` widened its Holt and Rudd attribution to name the printed spectrum `a_j = prod (q-j-1)/(q-2)`, the printed `a_2 = 0.10206751799779`, and the mixing rate 2.82/ln p at SCRATCHPAD-GRADE, with priority stated as Holt and Rudd's and with the corpus's prior holding narrowed to the per-prime eigenvalue already recorded at `lit-pdf-holt-rudd.md` line 259. `history/staging/recon-0829-escapes.md` section 6 received the sixteen `(i)/(ii)/(iii)` labels its E22 carry had left open, twelve (i), three (ii) and one split, of which seven reasons cite `attack-wrongdirection-audit.md` and nine are fresh judgements marked as such; no verdict, killer or number in that note changed. `research/REFUTED.md` took three of the seven HOLD proposals from the four refuted audits, all wording or calibration and none touching a verdict word: row 23 gained "closed for us, not proven unimprovable", row 17 gained the MEASURED marker and dropped an over-read of basis-independence, and row 20 gained its 33 readings, its 0.62 to 0.98 error bar and the LP information-class scope on the theta clause. The other four were left for Chris because they touch a canonical file, a producer needing a re-stamp, the index's format, or a number two other documents quote. The table stands at 67 rows. `node research/qc.js` reads 0 on refs, quotes, crosslinks, scripts, transfers, calibration, absence, sourcing, widths, ledger, provenance and search-convention, with one embeds finding in a producer this pass did not touch. Record: `history/staging/applied-0829-registries.md`.