# The import map: which foreign mathematics is worth aiming at this object **Scope.** This file is the standing, graded map of candidate mathematics imports, and that is all it is. It holds no mathematics of its own beyond the pricing of each candidate. One row per candidate field, one short paragraph per row. The five imports that have already run are carried as the calibration set, with their landed verdicts, so a new row can be graded against something rather than against nothing. What to work on next is [../TODO.md](../TODO.md); what is closed is [OUTCOMES.md](OUTCOMES.md); how to search is [SEARCH-CONVENTIONS.md](SEARCH-CONVENTIONS.md); which candidates became paper proposals is [../paper/proposals/PROPOSALS.md](../paper/proposals/PROPOSALS.md). **Why this file exists.** Five foreign imports ran on 2026-08-19 (chaining, Suen, thinning, max-plus, B-free), and twelve more rows ran across the two nights following. None moved the exponent. Every one banked something: a proven identity, a derived constant, a published anchor, or a wall address. That hit rate is the reason to do this structurally instead of by inspiration, and the reason to grade expected payoff by TYPE rather than by probability of success. --- ## 0. How to read a row **Structural fit** is the first gate and the one that kills most candidates. - **EXACT-IDENTITY** — the moiré object *is* the foreign object, up to notation. The copy theorem is a max-plus semiring identity; the fold is 2-of-`p` thinning; `maxsum_m` is a scan statistic. A vocabulary match is not this. - **STRONG-ANALOGY** — the two objects share a mechanism and a named theorem transfers with one stated modification, which the row names. - **VOCABULARY-ONLY** — the words coincide and the mathematics does not. These are rejected and do not appear in this table; they are listed with their reasons in `history/staging/import-map-construction.md`, so the same candidate is not proposed twice. **Circularity pre-check** is the second gate, and it is the day's hardest-won lesson: four candidate hypotheses were unmasked in one wave as the Zone Postulate wearing different clothes (the `L = 1` residue count, `H″` at `m = 1` and at `m = 2`, and any constant bound on `δ`). Every row carries a verdict on the question *does the import's needed hypothesis, written honestly, imply the postulate?* - **CLEAN** — the payoff does not need a hypothesis of that strength. A finite computation, a derived constant, a published anchor, a wall address. - **TPC-STRENGTH** — the hypothesis implies the postulate and is not known to be weaker. Legal to pursue only if there is an independent reason to think it softly provable, and the row must say what that reason is. - **CIRCULAR** — the hypothesis is the postulate restated. Reject or re-scope. **Expected payoff, by type.** Grade the type, not the probability. - **THEOREM** — a proved statement the corpus did not have. - **DERIVED-CONSTANT** — a number that was fitted becomes a number that is derived. - **WALL-ADDRESS** — the obstruction restated in a convention that owns it, so the next attempt starts at the wall instead of walking to it. - **PUBLISHED-ANCHOR** — the corpus's object located in print, or an owning-convention row for [SEARCH-CONVENTIONS.md](SEARCH-CONVENTIONS.md). - **CLOSURE** — a family of routes closed at once, with a mechanism. **Cost** is the size of a first experiment shaped to four hours, not the size of the mathematics. **Verification legend on the theorem column.** **[SOURCED]** the statement was read at a publisher page, an arXiv abstract page, or an authors' hosted full text during map construction. **[SOURCED-BIB]** the bibliographic data was verified but the statement itself was not opened. **[MEMORY]** written from memory and not reached at any source. **[CONJECTURE-ONLY]** (added 2026-08-21 with row 14, the first wholly-conjectural source): every statement in the source is conditional or heuristic — the row is priceable only as PUBLISHED-ANCHOR and/or WALL-ADDRESS, and THEOREM / DERIVED-CONSTANT are structurally unavailable to it. The per-row provenance is in `history/staging/import-map-construction.md`. ## 0a. The regrade rule, and the prune rule **Every landed import regrades its own row.** A row leaves UNTRIED the moment an experiment runs against it, and its verdict is written in as LANDED with the payoff that was actually banked, whether or not the route survived. The five calibration rows are what that looks like. **A row graded VOCABULARY-ONLY needs new structural evidence to re-enter.** Not a new argument for why it would be nice, and not a different theorem from the same field. A statement of the form "the moiré object is the foreign object, here is the identification" is what re-entry costs. **All thirteen original rows have run (the last landed 2026-08-20), so the map is SPENT as a queue and stands as the calibration record.** The standing rule for anything new: a future import gets a graded row here, with its circularity pre-check, BEFORE it runs — never a run out of inspiration. Four rows have been added under that rule and all four are resolved: row 14 landed 2026-08-21, rows 15 and 17 landed 2026-08-28, and row 16 is STALE — it was priced as an experiment two in-house notes had already executed, which is the failure mode the rule is meant to catch on the other side. **The map is pruned on every visit, under the corpus's forward-only charter.** A row whose route is closed leaves this file and becomes one line in [OUTCOMES.md](OUTCOMES.md), pointing at its record. This file holds candidates and the calibration set, never a history of candidates. *Amended 2026-08-20: a closed row is retained here with its closing mechanism when the mechanism is what stops the route being re-proposed; it still gets its OUTCOMES.md line. Rows leave only when the line alone suffices.* --- ## 1. The calibration set: what a landed import looks like Five ran, none moved the exponent, all five banked payoff (the twelve landings since have not overturned any of this table's lessons; the EXACT-IDENTITY reading in particular held at every one). Read this table before grading a new row, because it is the only calibration the grades have. | field | the theorem imported | moiré object | target hole | fit | circularity | payoff banked | status | |---|---|---|---|---|---|---|---| | probability in metric spaces | Dudley's entropy bound; Talagrand's generic chaining | the position supremum in the maximal law | the `ℓ¹ → ℓ²√log` wall | EXACT-IDENTITY | CLEAN | WALL-ADDRESS + DERIVED-CONSTANT + CLOSURE | **LANDED, route closed** | | correlation inequalities | Suen's inequality; Janson RSA 13 (1998); the lopsided local lemma | the anchored `δ`, the kill events | Assumption A | STRONG-ANALOGY | the `δ` target came back **TPC-STRENGTH** | WALL-ADDRESS + PUBLISHED-ANCHOR + CLOSURE | **LANDED, target refuted** | | point-process thinning | renewal thinning, `φ → qφ/(1−rφ)` | the fold as 2-of-`p` deletion | `H″` | EXACT-IDENTITY | `H″(m=2)` in moment form came back **CIRCULAR** | THEOREM (Fold Moment Identity) + DERIVED-CONSTANT | **LANDED, coupling closed** | | max-plus algebra | tropical Perron–Frobenius; Fekete; Kingman | the copy theorem, the tile transfer matrix | TODO 1d, limit existence | EXACT-IDENTITY | constant-free CLEAN; with any constant **TPC-STRENGTH** | THEOREM (A5 Theorem A sharpened) + WALL-ADDRESS | **LANDED, 1d lead live** | | B-free dynamics | Mirsky measures, tautness, heredity, complexity | the twin comb as a two-class sieve | Assumption A, `H″` | EXACT-IDENTITY at period scale | CLEAN | PUBLISHED-ANCHOR (an exclusion in print) + CLOSURE | **LANDED, route closed** | Records: `history/staging/import-chaining.md`, `import-suen.md`, `import-thinning.md`, `import-maxplus.md`, `import-bfree.md`. **What the calibration set teaches, in three lines.** An EXACT-IDENTITY fit predicts a banked payoff and predicts nothing about the route surviving: four of five were exact fits and four of five routes closed. The circularity check fired on three of five rows and would have saved most of the wasted motion had it run first. And the payoff that recurred most was WALL-ADDRESS, which is the cheapest thing an import produces and the one the corpus keeps using. --- ## 2. The live map Ordered by expected payoff. All seventeen original rows have now run or been overtaken: rows 1 to 13 landed by 2026-08-20, row 14 on 2026-08-21, rows 15 and 17 on 2026-08-28, and row 16 is stale, its experiment executed elsewhere before the row could be scheduled. Rows 22 and 23 are the 2026-09-06 full-coefficient follow-ups and are outside the original-row counts. Rows 20 and 21 were added 2026-08-29 under §0a's standing rule; both were priced by a recon pass and are resolved at recon grade, not by an experiment, so neither is counted as a landing. Their numbers are the ones their drafts carry, which leaves 18 and 19 reserved for `history/staging/recon-0828-farfields.md` §5's two drafts, still unwritten. The cost column is the price set BEFORE each run, kept for calibration; what a resolved row actually took is in its status cell. | # | field | the importable theorem, and where its statement lives | moiré object | target hole | fit | circularity | payoff | cost | status | |---|---|---|---|---|---|---|---|---|---| | 1 | extreme value theory of scan statistics | Leadbetter's `D(u_n)`/`D′(u_n)` conditions and the extremal index; Berman's condition `r_h = o(1/log h)`; Berman, *Ann. Math. Statist.* 35 (1964) 502–516 **[SOURCED]**; Leadbetter–Lindgren–Rootzén, Springer 1983, chs. 3–4 **[SOURCED-BIB]** | `maxsum_m(T_x)`, the maximum of a moving sum over the cyclic gap word | the disputed `√m` factor (TODO 0c) | EXACT-IDENTITY | CLEAN | THEOREM + DERIVED-CONSTANT + PUBLISHED-ANCHOR | 4 h | **LANDED 2026-08-19** — kill criterion passed at T₂₃ and blind T₂₉ (H = 0.3216/0.3367 vs √m's 0.5); banked the complementary-window duality `maxsum_m + minsum_{D−m} = W`; the rule `H = 0.2205 + 0.0061 ln D` was pre-registered, validated one level out (T₂₉, 0.21 s.e.), and then KILLED TWICE blind — H(T₃₁) = 0.3460 misses its sealed band at 4.96 band-s.e. and H(T₃₇) = 0.3565 misses its own at 3.63 s.e., two engines, one verdict (`history/staging/scanstat2.md`, `scanstat-t37.md`); the DERIVED-CONSTANT claim is withdrawn — no single exponent should be quoted at all, since H is a grid-dependent summary of a curve (H(m≤16) − H(m≤64) > 0 at all six levels) and the ladder bends below every line (post-hoc six-level refit accommodates T₃₇ only at 2.20 s.e.); the tail factor √(2 ln D) and its one named repair √(2 ln θD) are REFUTED by magnitude and by sign (the measurement sits ABOVE the θ ≤ 1 ceiling at small m and widens with D); what stands: √m refuted at seven exact levels (0.2661 → 0.3565), the duality verified at all 1484 m, the crossover measured as a plateau with mirrored ramps, and the streaming extremal-index method; the row's own offered identity Σγ(k)=0 is TRUE and its conclusion REFUTED (permutation-invariance counterexample); `history/staging/import-scanstat.md`; adversarial pass: `history/staging/adversary-wave2.md`. PRIOR-ART (2026-08-19, `history/staging/identifications-prior-art.md`): the duality is OWNED — it is the complement identity of the circular scan statistic (Cressie, J. Appl. Probab. 14 (1977); Naus; Wallenstein–Naus; GNW 2001 chs. 8–10, 17) — so no live sentence may present it as new; the exponent law H = a + b·ln D is NOVEL-SO-FAR, absent in the conditional-scan convention BY ARITHMETIC (exchangeability forces (D−m)/(D−1)), adjacent owner hyperuniformity / local number variance, where no sieved-set instance exists | | 2 | constrained coding and symbolic dynamics | capacity `= log` of the Perron root of the constraint graph: Shannon, BSTJ 27 (1948) **[MEMORY]**, Lind–Marcus, CUP 1995, ch. 4 **[SOURCED-BIB]**; the longest-run law, Flajolet–Sedgewick Prop. V.2 **[SOURCED, verbatim]** | the alternation-legal window of the old gap word, which IS the per-fold `L` | `H″`, the window statistic's growth (TODO 0b) | EXACT-IDENTITY | TPC-STRENGTH at the target; CLEAN for the shape | DERIVED-CONSTANT + WALL-ADDRESS | 4 h | **LANDED 2026-08-19** — graph EXACT (strictly sofic, capacity ln 2 p-free, word count 2^{n+1}−1, PAIRS census reproduced 7/7; PRIOR-ART 2026-08-19: the strict-soficity and the capacity are REPRODUCTIONS of printed results — the language is the B = 1 charge constraint / alternate-mark-inversion, Marcus–Roth–Siegel §2.3 p. 47 and §3.2 p. 75 — what stays banked is the 3/p → 2/p rate correction, the weight-(1,1,2) multiplicity, and the wall address); the map's own rate formula KILLED by its own criterion (t = 2.69 on ln p) and the corrected first-moment law predicts exact L to a flat 1.50 ONLY with the measured letter weights — the shape claim dies with the f = 3/p input (f falls 170× where 3/p falls 16× — the corrected census makes the fall LARGER, `fdecay-deep.md` — so L ≍ polylog); `history/staging/import-sofic.md`; adversarial pass: `history/staging/adversary-wave2.md` | | 3 | the repulsive lattice gas | Shearer's exact criterion, *Combinatorica* 5 (1985) 241–245 **[SOURCED-BIB]**, with its tightness; Scott–Sokal, *J. Stat. Phys.* 118 (2005) 1151–1261 **[SOURCED]**; Regts, *PTRF* 186 (2023) 621–641 **[SOURCED]** | the sieve's own inclusion–exclusion as a hard-core partition function; Bonferroni depth as cluster-expansion truncation | the local-lemma wall, the `4.2665 → 2` gap | EXACT-IDENTITY | CLEAN | CLOSURE (a family) + WALL-ADDRESS | 4 h | **LANDED 2026-08-19, route closed** — on the complete dependency graph Shearer's exact criterion IS the union bound (Scott–Sokal Ex. 3.1), so the wall is x = 13 as an identity, not x = 7; the family closes by three mechanisms — tightness for resampling oracles, chordality + mutual exclusivity for Moser–Tardos, and for Achlioptas–Iliopoulos the sequel's unconditional γ_i ≥ μ(f_i) (the record's atomicity lemma has an unproven hypothesis here; the closure survives without it, per the adversarial pass); Regts's zero-free polydisc reaches exactly to Mertens; banked θ_Shearer ≈ 1.41 (proven floor 2/√e = 1.21306), NOVEL-SO-FAR: the H* ladder is absent from OEIS at six indexings and no exactly-computed Shearer region on an arithmetic family exists in zbMATH/arXiv/OEIS full text, searched as "Shearer's region" (`history/staging/identifications-prior-art.md`; MathSciNet unreached); `history/staging/import-shearer.md`; adversarial pass: `history/staging/adversary-wave2.md` | | 4 | Stein's method for Poisson approximation | Arratia–Goldstein–Gordon, *Ann. Probab.* 17 (1989) 9–25 **[SOURCED-BIB]**; Barbour–Holst–Janson, OUP 1992 **[SOURCED-BIB]** | multi-kill extinction in a fixed window; the mixed super-`W` joint deficit | the two-parameter extinction form; the `~3.8` joint-deficit constant | STRONG-ANALOGY | CLEAN for both constants; TPC-STRENGTH if pushed to `H″` | DERIVED-CONSTANT, twice | 4 h | **LANDED 2026-08-19, kill FIRED** — b₃ sits at 0.85–0.9998 of its own ceiling at every window (both objects share ONE uniform residue draw, so the non-neighbourhood reconstructs the indicator and Chen–Stein's error term is void; the same product-space hypothesis that closed rows 3/10); AGG formulas now [SOURCED, verbatim, via the authors' 1990 Statistical Science restatement, constant 1.4 real]; what the first-moment arithmetic banked anyway, relabelled: extinction A DERIVED at 2.2091e-2 vs the record's fitted 2.4312e-2 (c at 0.37σ, beats the geometric null both ways), and a zero-parameter candidate for the ~3.8 — 4·Σ_{x k+1` (p. 24); `C_{2,H} = 2∏p(p−2)/(p−1)² ≈ 1.32032363` (p. 27, = 2C₂, ceiling coefficient `1/C_{2,H} = 0.7574` at `q = 2, ϕ = 1`). Kourbatov–Wolf, *Mathematics* 7 (2019) 400 = arXiv:1901.03785 **[SOURCED, verbatim, at page image; CONJECTURE-ONLY — the paper contains zero theorems]** | `Z₂(p)`'s envelope process — CUSTODY 3's identity `env(p)` = largest record wholly below `p′²`, EXACT at all 27,292 zones (`zonegap-01.js`), makes the OBJECT identification exact; the zone indexing is OURS (§2 below) | the staircase-correction model `zonegap-01.md` §7 names as not-reached; prereg band-setting for the next sweep; guard duty on measurements | fit: **EXACT-IDENTITY at the object, CONJECTURE-ONLY at every statement** | circularity: **SPLIT — CIRCULAR in any proof chain, CLEAN as instrument** (the binding cell, §3 below) | payoff: PUBLISHED-ANCHOR + WALL-ADDRESS (the almost-all quantifier wall) | 4 h | **LANDED 2026-08-21** — parameter-free calibration ran (`research/import-kw-01-calibrate.js`): `G/T̄ ∈ [0.78, 1.05]` from record 10 to record 82 (p ~ 6·10³ to 7·10¹⁶), records split 33/37/12 below-`T_c`/inside/above-`T̄`, 15 sign changes of `G − T̄`; 0 of 82 records breach eq (20), ceiling load creeps 0.544 → 0.685 → 0.759 by ladder thirds (the Shanks shape, → 1); **the recon's "their 0.7574 vs our 0.7504" is a RANGE ACCIDENT** (coefficient vs load; the same load statistic reads 0.8463 at record 75 and tends to 1 under their own eq (21)); band check `c3` vs `T̄(p²)/ln³p`: measured/trend 0.943, 0.904, 0.921, 0.867 (over-read 6–13%, `T_c` bracketing below); Gumbel sign-level consistent only (skew +0.588 on a correlated running-max path, not their ensemble); `history/staging/import-kw-zonegap.md` | | 15 | the distribution of the fractional parts of `x/n` and `x/p` | Saffari–Vaughan II, *Ann. Inst. Fourier* **27** (1977) 1–30, DOI 10.5802/aif.649, **Theorem 10 (p. 7)** and **(1.27)**; `c_α` defined in part I (1.9), DOI 10.5802/aif.634 **[SOURCED, both re-fetched and hashed 2026-08-28]** | the branch phase `⌊W/q⌋ mod M_T`, which cap-36 reduces to `⌊M·{W'/q}⌋`, `W' = W/M_T` | the Skeleton Equidistribution Conjecture; TODO item 4 | **EXACT-IDENTITY at the marginal face**, confirmed at the page: `x_SV = W/M_T`, `y = √W`, `α = j/M_T`. The joint face has no theorem in the source | **SPLIT.** CLEAN where the theorem reaches, and measured to be worth `≤ 5.68%` of one branch there; the deep half stays UNRESOLVED, read at the lower rung as **suspected TPC-STRENGTH** | PUBLISHED-ANCHOR + WALL-ADDRESS; **no THEOREM**, the column struck because Theorem 10's lower range condition reads `M > W^{1/12}` and fails at every fixed `M` | 4 h, spent | **LANDED 2026-08-28, no route; prediction HIT** — the prereg's kill (any single covered branch above 10% of `Σ_T Σ_q Snum_T`) did not fire: the largest covered branch is 0.20% / 0.70% / 5.68% / 1.10% at @13, @17, @19, @23, covered aggregate `+0.2% / −0.9% / −8.1% / −0.3%`, and the containment `{M_T ≤ √W} ⊆ {M_T ≤ lB}` is strict, machine-checked on 1,133,872 pairs. The THEOREM column is struck rather than repriced: Theorem 10's range condition `x^{6/11+ε} < y ⩽ x` reads `W^{1/12} < M_T ≤ √W` under the dictionary, so no fixed-`M` asymptotic follows, no line of `natal-cap-36-skeleton-door.md` Measurement D is upgraded, and the `exp(−C(log x/log log x)^{1/3})` saving certifies no finite level; the surviving statement is also one marginal while the door is joint. The wall is sharper than the priced range-condition failure: `M_T > lB` is exactly `q > W/M_T`, so on the open side the phase never wraps and there is no fractional part left to equidistribute. `history/staging/import-fracparts.md`, `import-map-rows-15-17.md` | | 16 | the theory of records | Rényi's record theorem: for an i.i.d. sequence with continuous marginal the record indicators are independent with `P(record at n) = 1/n`, so every record-TIME statistic is distribution-free (Rényi, *Théorie des éléments saillants d'une suite d'observations*, Aarhus colloquium 1962, 104–117 **[MEMORY]**; the same statement as developed in Arnold–Balakrishnan–Nagaraja, *Records*, Wiley 1998, DOI 10.1002/9781118150412 **[SOURCED-BIB]**, and restated in the survey Godrèche–Majumdar–Schehr, *J. Phys. A* **50** (2017) 333001, DOI 10.1088/1751-8121/aa71c1 **[SOURCED-BIB]**). The trended replacements: the `F^α` scheme, Deheuvels–Nevzorov, *J. Math. Sci.* **81** (1996) 2368–2378 and **88** (1998) 29–35, DOIs 10.1007/bf02362342 and 10.1007/bf02363259 **[SOURCED-BIB]**; Ballerini–Resnick, *Records from improving populations*, *J. Appl. Probab.* **22** (1985) 487–502, DOI 10.1017/s0021900200029272, and *Records in the presence of a linear trend*, *Adv. Appl. Probab.* **19** (1987) 801–828 **[SOURCED-BIB]** | the record ladder of `Z₂`, which `zonegap-03-model.md` §1 makes the whole of the object (`D ≡ 0`, so `Z₂ = env` exactly), read as a record process rather than as an envelope | **TODO Z5**, the 6.0% record-location deficit: `z` mean `−1.298` against a matched-null `−0.212 ± 0.244` and trend load `A = 0.9295` against `0.9895 ± 0.0182`, measured against one specific null (exponential gaps at `ābar = ln²x/(2C₂)`, Gumbel block maxima, 200 reps) | **STRONG-ANALOGY.** The theorems are for independent draws and the twin-gap sequence is neither independent nor stationary; the modification the row names is that the import is used to price the NULL, never the object | **CLEAN.** A record-time law is a statement about the observed prime record sequence and carries no bound on `G₂`; it cannot imply an every-window statement in either direction. The row-14 caution applies in its instrument form and is adopted verbatim: legitimate as an instrument, never as a link in a proof chain | WALL-ADDRESS only; the DERIVED-CONSTANT half is spent elsewhere and is not banked here | 4 h | **STALE.** The row proposes as an experiment a move that had already run: `record-location-null.md` executed the null side on 2026-08-28 and the 6.0% survives every correction available from the ensemble at `z` between −3.3 and −4.3, and `lit-kourbatov-shortfall.md` closed the prior-art half the same day by identifying the deficit with Kourbatov's published `b` (JIS 16 (2013) 13.5.2 §§5.1–5.2, `b ≈ 1.2597`, `μ* = −1.659` below `10^15`). Z5's null branch is closed; its mechanism branch is open. Both notes are outside output custody — no OUTPUT banner, figures hand-pasted — so their numbers stay in their files. **Updated 2026-08-29:** the null branch is not closed as flatly as this cell says: `measure-record-null2-0829.md` (inside output custody, unlike the two notes above) makes the lattice share a law-dependent bracket, 16.1% to 25.1% of b, and leaves the shape assumption behind every b reading open at 1.6 sigma | | 17 | repulsive point processes, and hyperuniformity as the surviving half | the two-point inequality for Hermitian determinantal measures, `P(i,j∈S) = K_ii K_jj − \|K_ij\|² ≤ P(i)P(j)`, proven in house from the `2×2` determinant and anchored in Soshnikov, *Russian Math. Surveys* **55** (2000) 923–975 and Lyons, *Publ. Math. IHÉS* **98** (2003) 167–212 **[SOURCED at abstract, theorem numbers MEMORY]**; the positive half is the Torquato–Stillinger classification by the growth of the local number variance, Torquato, *Physics Reports* **745** (2018) 1–95, read at the author's arXiv full text and cited by ITS pages, not the journal's: arXiv:1801.06924 eq. (14) p. 10, §5.3 p. 23, §5.3.1 eq. (88) p. 25, §5.5 p. 28, eq. (252) p. 78 **[SOURCED at the page image; the published article is 95 pages on different pagination and these are not its page numbers]** | the tile `T_x ⊂ ℤ/W` as a stationary point process under a uniform translate, with pair correlation `g = W(d)` (`variance-note.md` Thm 1) and structure factor `S(ν) = δ∏_p f̂_p(ν_p)` (`varE-spectral.md` §1) both exact | the family of routes modelling the survivor process as repulsive, and the owning-convention row for the local number variance | **EXACT-IDENTITY at the statistic; NEGATIVE at the model**, and two-sided rather than one-sided | **CLEAN.** A pair-correlation evaluation at a fixed level is a finite computation | CLOSURE (a family, both signs, with a mechanism) + PUBLISHED-ANCHOR | 2 h | **LANDED 2026-08-28** — the prereg `g(6) > 2` at every level passed and the kill criterion (`g(d) > 1` at any `d`, any level) fired on the first line: `g` takes the value 0 or a value at least `2.3812`, nothing in `(0,1)`, at every level, with `d = 6` the exact minimiser and `g(6) = 6∏_{5≤p≤x}(1−4/(p−2)²) = 2.661728` at `x = 11`. The closure is two-sided: Hermitian determinantal, negatively-associated and strongly Rayleigh models die on `g(6) ≥ 2.38 > 1`, permanental and positively-associated models die on `g(2) = 0 < 1`. Two of the row's own claims were corrected by its run — the extension to one-dimensional Gibbs and Coulomb gases is WITHDRAWN (a repulsive pair potential does not force `g ≤ 1` at finite density), and the reading of the sub-Poisson `Var/E` as hyperuniformity is DEMOTED: every fixed level is class I only because it is periodic, exact and empty, while along the diagonal family `L = y^u` the variance is linear in `L` with ratio 0.152 to 0.396 and no member approaches a hyperuniform limit, so the word for the family is sub-Poisson. Also banked: the distance correction to `import-map-construction.md` §1 and the `Var/E = Σ_{ν≠0} S(ν)K_L(ν/M)` identity. Nothing about the conjecture. `history/staging/import-repulsive.md`; producer `history/staging/import-repulsive.js` | | 20 | extremal majorants and minorants: Beurling, Selberg, Vaaler, Graham-Vaaler, Carneiro-Littmann, and the Hilbert-space reformulations | Selberg's majorant `C_E(z) = ½{B(β − z) + B(z − α)}`, entire of exponential type `2π`, with `χ_E ≤ C_E` everywhere, transform continuous and supported on `[−1, 1]`, and the identity `∫(C_E − χ_E) = 1`; and the large sieve constant it establishes, `A(N, δ) = N − 1 + δ^{-1}`, sharp. Vaaler, *Some extremal functions in Fourier analysis*, *Bull. Amer. Math. Soc.* **12** (1985) 183-216, §1 pp. 183-185 **[SOURCED, verbatim at the AMS free full text, sha256 in the record's §6]** | the Lemma V sawtooth `r_{d₁,d₂}(x)` and its frequency set `{a/e : e \| P(z), e ≤ D₁D₂}` | the position quantifier on `R_H` | **STRONG-ANALOGY at the object**, and blocked arithmetically at the working point before any of the mathematics is used | **SPLIT.** CLEAN at the mean-square form, which is the form Lemma V is written in and which is already PROVEN and already known useless for `G₂` (`sift-limit-attack.md` §7e, the almost-all exponent is 0). **TPC-STRENGTH** at the sup form, which is what the corpus's working point actually assumes and which inherits `attack-wrongdirection-audit.md` §2 item 6, `need_sharp/z²` measured 0.4913 to 0.6144 at `z = 13..43` | CLOSURE with mechanism, for the family; plus a PUBLISHED-ANCHOR line owed to `SEARCH-CONVENTIONS.md`, the owning convention for the corpus's L4 Fejér-mass identity being extremal majorants and minorants. No THEOREM, no DERIVED-CONSTANT | 0.75 h | **PRICED 2026-08-29, resolved at recon grade, no experiment run.** The deciding arithmetic: Selberg's construction pays exactly `1/δ` at type `2πδ`, removing one modulus from the Lemma V frequency set needs `δ < 1/(D₁D₂)`, and at the operative `θ_total = 2s/u ≈ 2.4` to `2.8` that puts the error above `D₁D₂ = z^{2s} > H`, i.e. above the window itself. That step is **[DERIVED HERE]** hand arithmetic on a SOURCED identity and a CITED working point, it has had no second reader, and its falsification check (a producer printing `D₁D₂/H` and `1/δ` beside `H/m`) has **NOT** run; if it failed the row would reopen at the mean-square form only, never at the sup form. Two further blocks stand behind it and do not depend on it: the family's payoff is the `ℓ² → ℓ²` large sieve constant while the wall's address is `ℓ¹ → ℓ²`, which is the ground `import-map-construction.md` line 72 already rejected the large sieve on; and `sift-limit-attack.md` §7e holds the Fejér mass exactly, `Σ_{a ≢ 0 mod e} F_H(a/e) = h(e − h)`, so the import offers an inequality where an equality stands, and it does not touch the absolute-value step that kills the `u_sup` route at `C^{π(z)}`, `C ≈ 2.05`. **The CLOSURE is graded no higher than a family already partly resident**, per the red team: the Lemma V remainder is itself Vaaler-completed (`lemmaV-neighbours.md`, and `smoothness-front.md` §3.3 reading Vaaler 1985 through Graham-Kolesnik Theorem A.6), and `smoothness-front.md`'s own verdict already prices that member, "the Vaaler coefficients clear Pascadi's condition maximally but buy nothing". What is new is the Selberg majorant of an interval indicator and the `1/δ` arithmetic, not the field. `history/staging/recon-0829-farfields2.md` §3c, §5, §7; red team `history/staging/redteam-0829-measure-c.md` §4c, §5g | | 21 | the eigenstructure of the gap transfer matrix, in the cycles-of-gaps convention | `M_J = R · Λ · L` with `LR = I`, upper-triangular entries of `R` and `L` binomial and independent of the prime, and eigenvalues `a_{kj} = ∏_{q=17}^{p_k} (q − j − 1)/(q − 2)` with `a_{kj} > a_{k,j+1}` and `a_{kj} → 0`; Table 1 prints eight values at `p_k = 999,999,999,989` from `p₀ = 13`, `a_{k2} = 0.10206751799779` first among them, and the authors state that convergence of the gap ratios is governed by `a_{k2}` and is slow. Holt and Rudd, *Eratosthenes sieve and the gaps between primes*, arXiv:1408.6002 §5.1, Table 1 and the Figure 3 caption **[SOURCED, verbatim at the arXiv full text, sha256 in the record's §6]** | the fold's histogram operator, `a3-09-histogram-operator.md`, whose own ledger already records it as a rediscovery of Holt-Rudd §5 | none reached; the row is an anchor and a wall address and was never priced as a target | **EXACT-IDENTITY at the operator**, and wrong precision and wrong functional at the conclusion | **CLEAN.** An eigenvalue of a finite matrix with an explicit product formula carries no hypothesis of postulate strength, and it carries no conclusion about the anchor either | PUBLISHED-ANCHOR, which is the strongest available here, plus a WALL-ADDRESS; and a `SEARCH-CONVENTIONS.md` row owed, "cycles of gaps" and "eigenstructure of `M_J`". No THEOREM, no DERIVED-CONSTANT | 1 h | **PRICED 2026-08-29, resolved at recon grade; the novelty-check producer has NOT been written.** Three deciding facts, none of which a sharper analysis narrows. The spectral gap is `1 − a₂ = 1 − 2.82/ln z + o(1/ln z)`, a polylogarithmic rate against killer 2's threshold `ε < 1/W = e^{−(1+o(1))x}` (`attack-wrongdirection-audit.md` §1 Axis C). The functional `a₂` governs is the population ratio `w_{g,1}(∞) = N_g/N₂`, a first-moment count of gaps of each bounded size, and `G₂` is a maximum. And `M_J` is restricted to spans `\|s\| < 2p` throughout, so `G₂` is outside its state space, while `J → ∞` is not a limit of this eigenstructure because `(q − j − 1)/(q − 2)` turns negative for `j ≥ q − 1`. **The anchor is graded below the recon's first draft, and the correction is the red team's:** `history/staging/lit-pdf-holt-rudd.md` line 259 already carries, verbatim and page-numbered, the per-prime eigenvalue `a_j = (p − j − 1)/(p − 2)` for the bidiagonal `M_J`, so what this row adds to the corpus is the product-over-primes closed form, the printed numerical values and the rate, beside the binomial eigenvectors `PRIOR-ART.md` had already attributed (`redteam-0829-measure-c.md` §1h row 1). The rate constant is **[SCRATCHPAD-GRADE]**: `a₂ · ln z` reads 2.816534, 2.819348, 2.820118, 2.820203 at `z = 10⁴` to `10⁷` from a scratchpad producer, reproduced digit for digit on an independent sieve by the red team (`redteam-0829-measure-c.md` §1h row 5), with the extrapolation to `p_k` matching the printed `a_{k2}` at a relative `1.8 × 10⁻⁵`; two measurements, no embedded producer, and nothing in the row's verdict rests on the constant rather than on the `1/ln z` shape, which is Mertens. `history/staging/recon-0829-farfields2.md` §3e, §5, §6, §7 | | 22 | two-parameter quadratic sieve / Barban–Vehov mean square | Graham estimate, integer case: Chen An [2206.10104v1](https://arxiv.org/html/2206.10104v1), (1.1) and Theorem 1.1, primary statement read | logarithmically averaged full corner coefficient after Mobius inversion | unsigned full-coefficient energy before signed correlation | EXACT-IDENTITY for the smoothed cofactor weight | CLEAN for the auxiliary norm | DERIVED bound, not a twin margin | analytic derivation and bounded algebra check; not a timed census | **LANDED 2026-09-06**: [corner-coefficient-energy.md](corner-coefficient-energy.md) gives O_eta(x log x) squared norms and product bound, retaining all branches; short-endpoint range repaired explicitly. Signed saving and complement OPEN; row 23 prices the sharp transition norm. The easy quadratic-main-term O(D2^2) remainder fails at these long cutoffs. Imported proof not independently re-proved. | | 23 | sharp truncated Mobius divisor sums | de la Breteche–Dress–Tenenbaum [author PDF](https://tenenb.perso.math.cnrs.fr/PPP/Sxz.pdf), (1.5), Theorem 1.1; full source read | full sharp corner after inversion | long-cutoff finite mean square | EXACT-IDENTITY for the coefficient | CLEAN for one-point norms | DERIVED bounds, not a signed margin | analytic derivation and exact finite identities | **LANDED 2026-09-06**: [sharp-corner-transition.md](sharp-corner-transition.md) prices sharp norms and refutes negligible L2 smoothing transfer for sufficiently small fixed eta; signed correlation OPEN. | | 24 | classical prime distribution: divisor-weighted maximal Bombieri–Vinogradov | Koukoulopoulos, *The Distribution of Prime Numbers*, AMS GSM 203 (2019), Exercise 26.2, p. 286, [author preliminary PDF](https://dms.umontreal.ca/~koukoulo/documents/publications/primes.pdf); Tao, [Notes 3](https://terrytao.wordpress.com/2015/01/10/254a-notes-3-the-large-sieve-and-the-bombieri-vinogradov-theorem/), Exercises 20 and 23 **[SOURCED: exercise statements]** | ordinary-prime progression-error input in centered-discrepancy-estimate section 3a, with q=m[b^2,g] | source locator for the prefix maximum and divisor multiplicity | EXACT-IDENTITY for this input after elementary weight/normalization transfer | CLEAN for the auxiliary estimate | PUBLISHED-ANCHOR only | 15–20 min source and algebra check; no computation | **SOURCE MATCH 2026-09-17; review pending.** The multiplicity c(q)<=tau(q)^3<=tau_8(q); fixed epsilon>0 places Q_0=2x^(1/2-epsilon)log^(3L)x below every fixed BV logarithmic cutoff. Abel summation, proper-prime-power and centering costs are paid by logarithmic precision and the fixed power gap. This covers the BV error step, not the separate tails, signed main term or Lambda(n-2)mu(n) remainder. No new theorem or twin margin; the earlier landings count is unchanged. | **Counts (recounted 2026-08-28; the seventeen original rows only, since rows 20 and 21 were added 2026-08-29 at recon grade and are not counted as landings; their own cells are STRONG-ANALOGY/SPLIT and EXACT-IDENTITY/CLEAN).** Seventeen rows, none UNTRIED: fourteen resolved by 2026-08-21, rows 15 and 17 landed 2026-08-28, and row 16 is STALE, priced as an experiment that had already run in house. Fits: seven EXACT-IDENTITY (row 14 at the object only, row 15 at the marginal face only, row 17 at the statistic only), ten STRONG-ANALOGY, zero VOCABULARY-ONLY (twelve rejected at that grade and listed with reasons in `history/staging/import-map-construction.md`, one of which — determinantal point processes — is not rejected but promoted to row 17). Circularity as now graded, one grade per row by the grade its cell leads with: twelve CLEAN, three TPC-STRENGTH (rows 2, 5, 7), two SPLIT (rows 14 and 15), which is the seventeen; no row is graded CIRCULAR outright, though row 14's split half is CIRCULAR in any proof chain. Four cells are split rather than single, and rows 2 and 4 are counted under the grade they lead with: rows 2, 4, 14 and 15. Payoff, counted as the payoff column now reads: three THEOREM (rows 1, 9, 13), four DERIVED-CONSTANT (rows 1, 2, 4, 7), ten WALL-ADDRESS, six PUBLISHED-ANCHOR, four CLOSURE. Row 15 was priced with a THEOREM before the range exponent was read at the page image, so the as-first-priced THEOREM count was four; row 16's payoff cell names DERIVED-CONSTANT only to say it is not banked here, and it is not counted. Payoff AS BANKED differs again, and the difference is the point: rows 1 and 7 withdrew DERIVED-CONSTANT, row 12 banked WALL-ADDRESS instead of the priced PUBLISHED-ANCHOR, row 7 banked a PUBLISHED-ANCHOR it was not priced for, row 15 lost its THEOREM to a misread range exponent, and row 16 banked only the wall address its two executing notes left standing. --- ## 3. The rows, one paragraph each ### 1. Extreme value theory of scan statistics `maxsum_m(T_x) = max_i (g_i + … + g_{i+m−1})` is the maximum of a moving sum, and the owning convention for that object is **scan statistic** in probability and **MOSUM** in change-point analysis, neither of which appears anywhere in this corpus. Leadbetter's `D(u_n)`/`D′(u_n)` machinery applies directly, and it says something specific about the corpus's own dispute: `D′` fails at lags below `m`, because consecutive windows share `m−1` summands, so the extremal index is below 1, but the extremal index enters only as `+ln θ/√(2 ln D)`, which is lower order. **Both halves of the form break ON THE CYCLIC TILE FAMILY — `sd_m = σ√m` at seven levels, and the tail factor `√(2 ln D)` by magnitude and by sign (`history/staging/scanstat2.md`; the head/anchored window setting is a different object with its own constants, per U-FRAME's no-transfer rule).md` §5 measured it (`0.2661, 0.2804, 0.3001, 0.3216, 0.3367, 0.3460, 0.3565` against `0.5` over seven exact levels, each more than eight standard errors low; `history/staging/scanstat2.md`, `scanstat-t37.md`). There is a one-line identity offered during map construction, now ADJUDICATED (2026-08-19, `history/staging/import-scanstat.md` §1): the identity `Σ_{k=0}^{D−1} γ(k) = 0` is TRUE and trivial (it is Parseval at frequency zero), and its offered conclusion `Var(S_m)/m → 0` is REFUTED — the hypothesis is invariant under every permutation of the gap word while `sd_m` is not, and a reshuffled real tile measures exponent 0.499 against the real word's 0.300. What the identity actually proves is the complementary-window duality `maxsum_m + minsum_{D−m} = W`, `sd_m = sd_{D−m}`, which forces any growth law to turn over at `m = D/2`; the sub-0.5 exponent is real, level-dependent, and its explanation still needs a decay scale the identity does not supply. Two independent recon lines during construction disagreed about where the exponent is heading — one read the rise `0.266 → 0.322` with tile size as a finite-size crossover toward `0.5`, the other argued periodicity forces it to `0` once `m` approaches `D` — and the identity above is what reconciles them, so the honest instruction is to fit `H` as a function of `D` and `m` rather than quote one number. Circularity is CLEAN because `a3-05-bound-L.md` §7's floor already says a maxsum law does not deliver the u-frame; the payoff is the object. ### 2. Constrained coding and symbolic dynamics The Alternation Lemma of [kappa-not-L.md](kappa-not-L.md) is literally a right-resolving two-state labelled graph: from residue `a` only gaps `≡ 0` or `−2` are legal, from `a−2` only `0` or `+2`, so the class word is a walk on two states and the legal language is a sofic shift. That is an exact identification, not a resemblance, and it gives the per-fold `L` a name in a mature field. The honest caution is what capacity theory does *not* give: it bounds the count of legal words of length `n` by `λⁿ`, never the length of the longest legal window, and no theorem was found bounding the longest word of a sofic shift that is also a factor of a given periodic word. What transfers instead is the first-moment composition with the longest-run law, and the shape it predicts is the row's whole value. The qualifying-gap density is exactly `3` residues in `p` ([kappa-not-L.md](kappa-not-L.md), three progressions of modulus `6p` with weights 1, 1, 2), and `ln D = Σ_{q ≤ p} ln(q−2) ≍ p` by Chebyshev, so the first-moment longest legal run WOULD sit at `≍ p/ln p` — IF the qualifying fraction were `3/p`. **The experiment killed exactly that input (LANDED 2026-08-19, `history/staging/import-sofic.md`): the map's rate formula fails its own pre-registered flatness test (t = 2.69), the corrected law `ln(pD)/ln(p/2)` predicts exact L to a flat factor 1.50 only when fed the MEASURED letter weights, and the measured qualifying fraction falls 170× (more, after the census's shift-alias correction, `fdecay-deep.md`) where 3/p falls 16× — so the first-moment prediction is `L ≍ polylog`, a DOWNGRADE of U-FRAME §12's reading, not an advance on §5a's requirement.** The map's constant arithmetic was also wrong (it substituted `p` for `ln D(T₉₇) = 79.77`; the corrected factor was 3.2 to 5.2, before the shape died). What the row banked instead: the graph is exact and strictly sofic, capacity `ln 2` independent of `p`, legal-word count `2^{n+1}−1`, the weight-(1,1,2) multiplicity derived, and the wall restated — `L` is governed entirely by the letter measure, which is `f-decays`' problem, not symbolic dynamics'. Circularity: proving `L ≤ c p/ln p` unconditionally closes the u-frame chain and therefore implies the postulate, so the target is TPC-STRENGTH and legal only as the corpus's declared live route; the CLEAN payoff is the derivation of the shape and of A5's constant, which needs no such hypothesis. Price the row against [gate-multiplies.md](gate-multiplies.md) §8 first: even a perfect `L` does not save the tile chain below `p ≈ 800`. ### 3. The repulsive lattice gas, and Shearer's tightness `history/staging/import-suen.md` §8 located the local-lemma wall at the **admissible conditioning-set modulus**: the lopsided hypothesis is quantified over all subsets `S` of non-neighbours, which needs the window to equidistribute modulo `∏_{q∈S} q = x#`, while the pairwise-drawn graph is empty already at `H = x²`. Scott–Sokal is the theorem that makes that address canonical — the local lemma's conclusion holds for a dependency graph and probabilities **if and only if** the independent-set polynomial is non-vanishing in the corresponding polydisc — and Shearer's criterion is the exact threshold, with a tightness clause that decides a whole family at once: outside Shearer's region there is a probability space realising the same graph and the same probabilities in which the good event has probability zero. **Given only the dependency graph and the marginals, the set-wise quantifier is not removable.** That closes entropy compression, Moser–Tardos and the resampling-oracle programme as a class, which is row 10. What stays live is the other direction: Regts's theorem that absence of complex zeros implies strong spatial mixing, which would substitute quantitative decoupling between distant prime constraints for the joint independence the product-space local lemma gets free. The first thing to compute is whether the zero-free radius for the corpus's own structure reaches past the Mertens threshold `Σ 2/q < 1`, which is where six independent routes already land ([sift-limit-attack.md](sift-limit-attack.md) §7). ### 4. Stein's method for Poisson approximation Two named constants in [../TODO.md](../TODO.md) are fitted and want deriving: the extinction law's two-parameter form, and the `~3.8` multiple of the forced mutual-exclusion scale that the mixed super-`W` joint deficit sits at. Both are first-moment objects with a pair correction, which is exactly the shape Arratia–Goldstein–Gordon's `b₁`, `b₂`, `b₃` decomposition prices, and the pair sum the method needs is already proven in this corpus as the Fold Moment Identity's `Ψ`. The fit is STRONG-ANALOGY rather than exact because the kill events are not indicators of a single independent field, and `b₃` is only zero when the neighbourhood is a genuine independence neighbourhood, which for kills mod `q` it is not below the CRT scale. Circularity needs care and the row states where the line is: a Poisson approximation for kills in a window at `λm̄ ≥ 1` would be `H″`, and `H″` in moment form is already recorded as implying the postulate; the two constants are finite-window objects and are CLEAN. Take the constants and stop there. ### 5. Spectral theory of automorphic forms `history/staging/import-chaining.md` §4.3 relocated the maximal law's loss off the position supremum and onto the `(e,a)` sum over moduli and frequencies, and named the missing statement: an `ℓ¹ → ℓ²√log` bound on the arithmetic of `Θ_e(a)`. Map construction searched for a theorem of that shape in four conventions and found one family only. The large sieve is `ℓ² → ℓ²` and does not have the shape; Gallagher's larger sieve is a different object; `ℓ²`-decoupling requires a positive-definite second fundamental form and carries an `N^ε` loss that would eat the saving even if the geometry were there; Weil and Deligne give pointwise square-root bounds whose sum is still `ℓ¹`. **The one published machine that extracts cancellation from an `ℓ¹` sum over moduli is spectral: Deshouillers–Iwaniec via the Kuznetsov formula, as used in Bombieri–Friedlander–Iwaniec.** Its saving is a power of the modulus in a restricted range rather than a clean `√log`, so this row is an address and an anchor rather than a route, and the sharp maximal law it would serve is TPC-implying (`history/staging/phase1-T4-maximal-law.md` §2). The cheap and decisive first move does not need any of that machinery: measure whether `Θ_e(a)` exhibits square-root cancellation across the family at all. ### 6. Discrepancy theory Spencer's theorem is the canonical statement that removes exactly a `√log` from a union bound over a family, in the author's own words improving "the basic probabilistic method with which `K = c(ln n)^{1/2}`", and that is the wall's exact shape. The row is nonetheless graded STRONG-ANALOGY and priced as a closure, because the failing step is named and is not repairable inside the field: **every theorem in this family asserts the existence of a good sign vector, and the corpus does not get to choose its signs — arithmetic hands them over.** Banaszczyk, Bansal and Lovett–Meka all inherit that hypothesis. Two hours spent writing this down properly is two hours that stops the family being re-proposed, and the same paragraph disposes of decoupling and of the large sieve for the same wall. ### 7. Concentration on product spaces McDiarmid's bounded-differences inequality lost to Chebyshev on this object once already (`paper/anchored-note.md`), and the reason is the variance proxy: `n` coordinates each with a bounded effect. Talagrand's convex distance inequality has a different exposure, and the certifiable-function corollary is the one to reach for, because the survivor deficit over the rotation ensemble is exactly a certifiable statistic — a witness that the count is at least `k` is a set of primes, of size proportional to `k`, not a reading of all `π(x)` coordinates. The variance proxy becomes `c²ℓ·E[X]` rather than `n`. The caveat belongs in the row rather than in a footnote: `c` is the worst-case effect of one prime's class choice, and if one prime can swing the statistic hard then `c` blows up and this loses to Chebyshev for the same reason McDiarmid did. The published repair for exactly that failure is the exceptional-set variant in Bruhn–Joos, and the experiment should price `c` before anything else. *Outcome (2026-08-20, `history/staging/row7-recon.md`): it did — d(q₁) misses the theorem's own slack by a factor growing like √N, the published repair's exponent is ≍ ln²x/x and falling, and the family closed.* ### 8. Erdős covering systems, the distortion method The corpus has already dived this field ([covering-dive.md](covering-dive.md) §3) and recorded the negative: no covering-systems result bounds the length of a finite interval coverable by two classes per prime at polynomial scale, searched in the covering-systems convention and tabled in [SEARCH-CONVENTIONS.md](SEARCH-CONVENTIONS.md) §3. *Outcome (2026-08-19, `history/staging/import-distortion.md`): the technique was tried — it speaks at two classes and dies on the CRT ambient; the whole yield is G₂ ≤ 6·2^{2(π(x)−2)}+6, beating only the period bound.* BBMST's distortion method is the only machinery found anywhere in this map that already handles **an adversary choosing the shifts**, without paying a union bound, by tracking the uncovered set's measure through the CRT tower prime by prime. Two hypotheses block a direct application and the row states both: the theorem needs distinct moduli while two classes mod `p` is multiplicity 2, and the corpus's `C = Σ µ(p)·2/p` diverges like `2 lnln x`, so `e^{−4C}/2` degrades to about `(log x)^{−8}` — a density, and density says nothing about the longest run. Klein–Koukoulopoulos–Lemieux handle multiplicity `s` by a modification of the same method, which is the half of the repair that exists. ### 9. The interpolation method TODO 1d wants limit existence for the exponent without a numerical value, because a value would be too strong. Bayati–Gamarnik–Tetali is the canonical machine for exactly that: it establishes scaling limits for combinatorial optima without computing them. Its hypotheses are the row's whole content, and the third one is fatal in the obvious form — it needs a **random ensemble with exchangeable hyperedges**, which is what manufactures the composition law that `history/staging/import-maxplus.md` §3 showed the fold does not have. So the method cannot touch `G₂(x#)` directly. What it can touch is the rotation ensemble, which is a genuine random model this corpus already uses, and a proof that the ensemble's exponent has a limit is a theorem about an object the programme quotes. Circularity is CLEAN because limit existence carries no number. ### 10. Algorithmic local lemma Kept as its own row because it was an explicit seed and because the pricing is worth recording rather than re-deriving. Moser–Tardos does not weaken the set-wise quantifier; it buys it with product structure, requiring every event to be determined by a subset of mutually independent variables with dependency equal to shared variables. Applied here that asks the window to be a product space over the residues mod every `q ≤ x`, which is `H ≥ x#` and not `H = x²` — the same address in different coordinates. Harvey–Vondrák's resampling oracles need lopsided association, which quantifies over all monotone events on the non-neighbours and is strictly stronger than all subsets. Kolipaka–Szegedy show Moser–Tardos is efficient exactly up to Shearer's bound, which by row 3 is where the quantifier becomes unremovable. One sub-item was not verified during construction and is the only reason this row is not simply closed: Achlioptas–Iliopoulos claim a framework that "works when the underlying state space is entirely unstructured", and their main theorem's hypotheses were not opened. *Outcome (2026-08-19/20): it IS now closed — A–I's hypotheses were opened, the causality digraph is complete here, and the binding inequality is the sequel's unconditional γ_i ≥ μ(f_i); `history/staging/import-shearer.md` §4.* ### 11. Combinatorial optimization The identification is exact and pleasant: coverage of a fixed interval `[1, L]` is a monotone submodular function, choosing one pair `{a_p, a_p−2}` per prime is a partition matroid, and `[1, L]` is coverable if and only if the optimum equals `L`. A `(1 − 1/e)` guarantee would then certify non-coverability whenever an algorithm's coverage falls below `0.632 L`, turning the greedy oracle — which [OUTCOMES.md](OUTCOMES.md) records as exact to `x ≈ 53` and unreliable beyond — into a certified upper bound at levels past A144311's `x = 79`. **The row is dead on arithmetic and is carried so that it is not proposed again.** The achievable coverage of a long interval is `1 − ∏_{3≤p≤x}(1−2/p)`, which reads `0.800` at `x = 5` and rises to `0.959` at `x = 79`. It never falls below `0.632` at any level, so the certificate cannot fire anywhere, and the reason is structural: the problem asks to distinguish 100% coverage from 98% coverage, and a constant-factor approximation guarantee cannot see that difference. One hour writes it up as a wall address. ### 12. Diophantine approximation The kinship with the lonely runner problem is real rather than verbal: both ask that a line in a torus miss a union of axis-parallel obstacles, and both sit in the regime where the total obstacle measure exceeds 1 so that union bounds fail. Three mismatches stop it being an instance, and they are worth having written down: the lonely runner has no shifts, since all runners start together, while the adversary here picks `a_p` freely; the lonely runner asks for one good time while the postulate asks that every window of `H` contain a survivor; and the whole toolkit is calibrated to small `n`, with thirteen cases settled in sixty years and the general lower bound improved only by a `1/(n² (lnln n)²)` nudge. **The value is calibration, and it is worth two hours: the one-class-per-modulus version of this corpus's covering question is a famous open problem with a sixty-year record of case-by-case progress, which is independent evidence about what `4. *Outcome (2026-08-20, `history/staging/row12-recon.md`): spent — WALL-ADDRESS realised (the field's own stuck factor of 2 is caused by prime velocities; the multi-obstacle version is answered with the union bound exactly tight), plus the Lonely Rabbit published anchor.*2665 → 2` will not come from.** ### 13. Hypergraph covering The corpus already cites the Pippenger–Spencer covering theorem once, inside [covering-dive.md](covering-dive.md) §4.1 as the ingredient behind FGKMT's lower bound for one class per prime. What is untried is the two-class version as a construction: FGKMT's stated main new ingredient is a generalisation of Pippenger–Spencer used to cover an interval by one residue class per prime, and the corpus's own live lower bound is a transfer of Kalmynin–Konyagin that `covering-dive.md` §Q4 flags as a reading of a published proof rather than a consequence of a published theorem. A two-class Pippenger–Spencer would replace that reading with a theorem. Circularity is CLEAN in the strongest sense: no lower bound on `G₂` can imply the postulate, which is an upper-bound statement. *Outcome (2026-08-20, `history/staging/import-hypergraph.md`, adversarial pass `history/staging/redteam-0820-math.md` §3): the priced THEOREM against the K–K residual was mispriced — the covering engine and the K–K smooth-band reading live on disjoint mechanisms at disjoint scales — but the first-moment half of the dictionary landed the row's real prize: the theorem-provenance chain `G₂(x#) ≫ x ln x` from published ingredients only, adversary-confirmed, now registered at `paper/proposals/prop-xlnx-lower-bound.md`. Status cell above carries the corrected yield-cap floor.* --- ## 4. The top five: what was pre-registered and what happened Rows 1, 2, 3, 4 and 5. Each has a three-sentence experiment below: the mapping, what is pre-registered, and what kills it. Pre-registration is not optional here and the reason is on the record: `G₂(41#) = 546` meant something only because its window was written down first. **Row 1, scan statistics.** *Mapping:* identify `maxsum_m` with the scan statistic of the cyclic gap word, check `D(u_n)` directly on `T_13..T_23`, and prove or refute the identity `Σ_k γ(k) = 0` and its consequence `Var(S_m)/m → 0`. *Pre-registration:* fit `sd_m = c·m^{H}` with `H` allowed to depend on `D` on `T_13..T_19`, and write down the predicted `maxsum_m` at `T_23` for `m = 1, 2, 4, 8, 16, 32, 64` before computing it. *Kill:* if the two-parameter fit does not beat the one-parameter `σ√m` law out of sample at `T_23` by more than its own error bar, the correction is measured but not established, and the row closes with the identity as its only payoff. *Outcome (2026-08-19/20, `history/staging/import-scanstat.md`, `scanstat2.md`, `scanstat-t37.md`): the kill criterion passed at T₂₃ and blind T₂₉; the offered identity was TRUE and its conclusion REFUTED; and the rule the run derived was itself killed twice blind at T₃₁ and T₃₇ — the banked payoff is the duality, the seven-level refutation of √m, and the method.* **Row 2, constrained coding.** *Mapping:* build the two-state labelled graph from the qualifying-gap law, compute its Perron root and the first-moment longest legal run `ln D / ln(1/f)` at each of the nine computable folds. *Pre-registration:* predict the ratio of that first moment to the exact per-fold `L` at all nine folds before looking, and predict that the ratio is flat in `p` rather than growing like `ln p`. *Kill:* if the ratio grows with `p` at all, the `p/ln p` shape claim is wrong and the row drops to a wall address; if it is flat but above 5, the constant is out of reach and the row closes. *Outcome (2026-08-19, `history/staging/import-sofic.md`): the graph is exact and strictly sofic, the map's own rate formula died by its own flatness criterion, and the shape claim died with the f = 3/p input; the language family is printed (MRS B = 1 charge constraint).* **Row 3, the repulsive lattice gas.** *Mapping:* write the corpus's kill events as a hard-core lattice gas on the dependency graph of `history/staging/import-suen.md` §8 and compute Shearer's region `R(G)` exactly at `x = 5, 7, 11, 13, 17, 19, 23, 29`. *Outcome (2026-08-19, `history/staging/import-shearer.md`):* the pre-registered coincidence with the Mertens threshold `Σ_{5≤p≤x} 2/p < 1` CONFIRMED and upgraded to an identity — on the complete dependency graph Shearer's exact region IS the union bound (Scott–Sokal Ex. 3.1). The rider was wrong twice: the last feasible level is x = 11 and the wall is x = 13 (the map's x = 7 was the SUFFICIENT asymmetric LLL, which stays at x = 7 under free weight optimisation — two levels earlier than the exact criterion, and a property of the sufficient condition only). The kill clause cannot fire on the complete graph, and its matching-graph firing is §8(i)'s wrong TPC proof, not evidence for Regts, whose zero-free polydisc reaches exactly to Mertens and not past it. **Row 4, Stein's method.** *Mapping:* write the multi-kill count in a fixed window as `W = Σ X_a` with the neighbourhood of dependence taken from the Fold Moment Identity's pair structure, and compute `b₁`, `b₂`, `b₃` at the four windows where the extinction law was validated. *Pre-registration:* predict the two-parameter extinction form's parameters from `b₁` and `b₂` alone, and predict the joint deficit's multiple of the forced scale, before comparing either to the measured `~3.8` and to the fitted extinction parameters. *Kill:* if `b₃` is not small at the windows where the law was validated, the Poisson approximation has no error term there and the derivation is not available. *Outcome (2026-08-19, `history/staging/import-stein.md`): the kill clause FIRED — b₃ sits at 0.85–0.9998 of its own ceiling at every window, so no Stein-derived constant exists; the first-moment arithmetic banked the derived (A, c) and the 3.8 candidate anyway, relabelled.* **Row 5, the `Θ_e(a)` measurement.** *Mapping:* compute `Θ_e(a)` over the full `(e,a)` family at `z = 13, 17, 19, 23` and form the ratio `‖Θ‖₁ / (‖Θ‖₂ · √(#(e,a)))`, which is 1 under perfect alignment and `O(1/√#)` under square-root cancellation. *Pre-registration:* write down which of the two regimes is expected at each level, and the threshold ratio below which an `ℓ¹ → ℓ²` theorem could deliver the needed `√log`, before computing. *Kill:* a ratio at or above 0.5 at every level kills the whole `ℓ¹ → ℓ²` programme for this wall, which is worth more than a partial success and retires rows 5 and 6 together. *Outcome (2026-08-19, `history/staging/import-l1l2.md`): the kill line (ρ ≥ 0.5) did NOT fire — ρ falls 0.435 → 0.065 — and the prereg's own sealed note records that its inference was INVERTED (Cauchy–Schwarz makes the flat family the favourable case); the row closed anyway on the identity ‖Θ·S_H‖₂ = rms(R_H).* --- *This document states current understanding. Superseded rows, retired grades and the reasons they changed are in [history/CHANGELOG.md](history/CHANGELOG.md), indexed by document; the construction record, including the twelve candidates rejected as VOCABULARY-ONLY, is `history/staging/import-map-construction.md`.*