# G2: the state of the object *(Consolidation note. This file assembles what is known about G2, the two-class maximum twin-slot gap, from the eleven notes of 2026-08-17 plus the older load-bearing ones. It computes nothing new. Every claim carries a calibration marker (PROVEN, VERIFIED by exact computation, CERTIFIED, MEASURED, INFERRED, CONJECTURED, REFUTED) and a pointer to the file it came from.)* ## 0. What is hard here, first The achieved upper exponent is 4.266450284… from DHR. No argument here has improved it. A fixed exponent below 2 would prove twin primes, but the present sieve bound supplies no such result. This is a limitation of that argument, not an impossibility theorem for every use of the exact tile. The target distinctions are owned by [../README.md](../README.md) Status: unbounded occupied intervals suffice for infinitude, whereas uniform gap control asks for more. A constant at exponent 2 cannot be silently turned into a fixed-power or little-o saving. **Current arithmetic assessment (2026-09-09).** The complete reduction S=C2*x+R+O_H(x/log^H x) has an OPEN signed lower consumer. The [grouped-divisor moment](grouped-divisor-moment.md) controls a region with explicit fixed-margin cuts; it does not bound the full remainder. [Global cutoff averaging](global-cutoff-averaging.md) includes all outside arithmetic. The [C3 majorant](global-smooth-majorant.md) gives a full absolute O(x) bound for its profile, with no sufficient signed constant. The [cofactor transfer](cofactor-progression-transfer.md) has a growing polylogarithmic range but a rate weaker than o(x); its mixed tails and outside contribution remain open. The [two-family](paired-factor-budget.md) and [Dickman](supported-coefficient-dickman.md) calculations do not pay the required full signed balance. Their exact limitations, including the difference between an insufficient lower bound and a negative upper bound, are retained in the owning notes. The [moving-cutoff repair](moving-cutoff-parity.md) specifies the centered prime–Mobius discrepancy D_y. D_y>=-4x/25+o(x) on unbounded dyadic scales would suffice, but is OPEN. Its checked formula and tolerance are not an arithmetic estimate. The full residual and centered formulations are linked in [RESEARCH-HANDOFF.md](RESEARCH-HANDOFF.md) section 3; their gains cannot be counted independently. The [2026-09-09 integration](research-round-validation.md) accepts the centered truncation and repaired low Type I estimate, leaving the exact S=C2*x+B+O_A(x/log^A x) with B unestimated. The structured moment adds a fourth residual cut; the target box exponent remains 407/400 and the corrected remaining moment saving is greater than 7/200. This is regional control, not a sufficient global margin. C's norm obstruction and E's two all-depth test failures retain their corrected scopes. [RESEARCH-EXECUTION.md](RESEARCH-EXECUTION.md) records the completed dispatch and candidate next obligations. The [router](README.md), [question ledger](QUESTIONS.md) and [outcomes](OUTCOMES.md) locate the owning arguments. No G2 upper exponent or twin-prime proof status changed. ## Reference inventory — read for a specific claim The current assessment is section 0 above. The inventory below retains bounds, measurements and earlier routes with their original evidence. Use [OUTCOMES.md](OUTCOMES.md) and [../TODO.md](../TODO.md) for current dispositions before treating an older route description as an assignment. **PROVEN.** - `G₂(x#) ≪_ε x^{4.26645+ε}` — no earlier published upper bound at any exponent was found for the two-class problem, searched in the owning convention **bounded number of residue classes per prime** per `SEARCH-CONVENTIONS.md` §1/§3, never in ours alone. `paper/beta2-note.md`, `dhr-verification.md`. - `G₂ ≥ g` pointwise, so the one-class constructions transfer free: FGKMT's `x log x logloglog x/loglog x`, and the `x log x/logloglog x` that the Erdős Problems page for #687 reports (edited 2026-08-31). Read at source 2026-09-07: that bound is not a paper. It is an anonymous AI-authored 48-page PDF ("A Tilted Residue-Class Construction for Long Prime-Free Intervals", GitHub DottedCalculator/ai-math, posted 2026-08-26, not on arXiv, not refereed), whose covering theorem `Y(X) ≥ c₀ X log X/log₃X` is exactly the one-class primorial statement and has been formalised in Lean by a third party (plby/lean-proofs, `Erdos4Tilted.lean`; build not reproduced here). It transfers verbatim through `G₂ ≥ g` at that calibration: formalised, unrefereed. Above both, `G₂(x#) ≫ x ln x` from published ingredients only (K–K Corollary 1 + Mertens + PNT + the CRT identity — composition ours, adversary-confirmed at every ingredient's page image, **NOT refereed**; `history/staging/import-hypergraph.md` §4, `redteam-0820-math.md` §3.3, `paper/proposals/prop-xlnx-lower-bound.md`); and above both, `G₂(P(y)) ≫ y (ln y)³ (lnlnln y)²/(lnln y)⁴` for `y ≥ 10^{134.1}` by substituting `Ω_p = {a_p, a_p−2}` into Kalmynin–Konyagin. Two logs above the free bound, adversarially checked, **NOT refereed**, and re-deriving the inside of a published proof where the x ln x chain consumes statements only. `two-class-lower-bounds.md` §§3, 4c. - The mean-square Lemma V, with `B ≤ 9A(z)²(E(z)−1) = O((log z)⁸)`, unconditional, explicit and uniform in the level `s`, re-derived adversarially on 2026-08-29 and holding for any weights supported on divisors of `P(z)` with `|λ_d| ≤ 1`. The companion finding that `B` was never the binding term is VERIFIED at z = 13..47, not proven. `history/staging/attack-AB-bounded.md` §1.1; `history/staging/redteam-0829-theorem1.md` §§2, 9 (the L1-L5 chain re-derived too: PROVEN with no hypothesis; a period mean, not a supremum, so Lemma V proper is untouched). - The complementary-window duality `maxsum_m + minsum_{D−m} = W`, VERIFIED at all 1484 `m` on T₁₃ — and **not ours**: it is the complement identity of the circular scan statistic (Cressie 1977). `IMPORT-MAP.md` row 1. - Shearer's exact criterion on a complete dependency graph **is** the union bound, which is what closes the whole local-lemma family. `history/staging/import-shearer.md` §4. - The Fold Moment Identity. `paper/proposals/prop-thinning-null.md`. - The Tail-Count Transport, `N_new(θ) ≤ (q−2)·N(θ) + 2·Σ_L Q_L(θ)`, the sharp per-level evaluator, exact at eight consecutive folds. `U-FRAME.md` §11. - The Exact Invariance Lemma: bands scale by exactly `(p−2)` and fossil depth is frozen at birth. `GLOSSARY.md` §stratum, `ATTACKS2.md` attack 2. - (PROPOSAL grade, not proven here, moved out of this list 2026-08-30 per `coherence-0830.md` D1:) per-fold `L` as the longest alternation-legal window with the 3/2 constant a Perron root, `paper/proposals/prop-exact-fold-L.md` §2, its equality lemma proven in a staging record and verified at 36 cells, no second reader. - `G₂(37#) = 528` carries an exhaustive maximality certificate: `maxsum₁` over all 217,929,355,875 gaps, from an engine sharing no code with the exact-ladder producers. `history/staging/scanstat-t37.md`. **MEASURED, and blind-validated against sealed bands.** - The extinction rate law holds at **five windows over four decades of Y**; at `W = 2·10¹¹` the last multi-kill fold measured 631 against the sealed [571, 877]. It carries its own systematic in writing: a **~20% count overprediction at five of five windows** — and the per-fold piece is now found and blind-validated (2026-08-20): the law's per-fold error is a **deterministic fold-factor field** M_p (= λ_derived/λ_model), with Poisson dispersion around the corrected mean — at a fresh pre-registered anchor the sealed NB fold-factor bands scored **33 of 37 folds inside the 90% bands and 0 of 37 outside the 99.73%**, where the old `±3√λ` clause read 29/37 (it had failed at 43.2% at `2·10¹¹` and is retired for future larger-Y preregs; it was never separately refuted at `Y ≤ 2·10⁸`). The field is constant in window length to within ~2–3% relative sd over `2·10⁷…2·10¹¹`, and constant across the three anchors tested to within the **~6–10% resolution one replicate pair affords** — "deterministic" is the surviving model class, not yet a measured identity in the anchor direction. The ~20% aggregate correction is the exposure-weighted mean of M over the scored folds. **The field is now PARTIALLY DERIVED (2026-08-20 night, `history/staging/mp-derivation.md`; adversarial verdict `history/staging/redteam-0820-night-empirical.md` §T1):** M_p ≈ k·W1(θ_p)·exp(−δ·θ_p/m̄_p), where W1 is the **exact endpoint comb** of the two kill classes (zero parameters, PROVEN residue counting; the free exponent on ln W1 reads 1.031 where the derivation says exactly 1) and (k, δ = 0.277) are the law's own two constants re-estimated — one depth exponent beside the scale. The field's roughness drops sd(ln M) = 0.560 → 0.177 on the train split, and the formula then HIT a second pre-registered blind test at a fresh anchor 1.32·10¹¹ (prereg committed ALONE at `303711b`): **34 of 37 inside the 90% bands** (needed ≥ 28; incumbent-truth expectation 34.6), 0 of 37 outside 99.73%, margin +25.4 nats over the constant-bias rival. Custody residual, on record: producer-02 first exists in git 5m21s after the seal, so "the producer did not exist at the seal" is unverifiable from git — mitigated, not proven, by the sealed bands being a deterministic function of artifacts committed half a day earlier and by the outcome sitting AT the incumbent expectation rather than above it. The field is a function of θ, not of p, by the MATCHED CONTROL — the raw twin contrast 0.155-vs-0.764 is confounded (θ-sharing ⟺ Δp = 2): non-twin pairs matched on predicted M agree like twins (rms 0.181 vs 0.155), and control-pair variance decomposes onto the θ-arithmetic at ratio 1.09. The residual sub-field stays measured-only: sd ≈ 0.18, a far-tail flattening, fold 631; under the formula's own PLN variance only two of the four Poisson-flagged folds (409, 631) stay above 3. `history/staging/foldL-window5.md`, `history/staging/perfold-error-model.md`, corrections `history/staging/redteam-0820-empirical.md` §T1; `paper/proposals/prop-thinning-null.md` §5. - The joint deficit's closed form `1 − J = 4·Σ_{x 32` and every published `f-decays` point from `x = 37` up was off by 0.62× to 1.05×. Recomputed alias-free at all 42 levels. The standing lesson is verify at a level where the feared mechanism can fire. `history/staging/fdecay-deep.md`. **HEURISTIC, held behind one open step** (the identification IS the θ = 2 replacement error, `attack-0830-varE-identification.md`, HELD)**.** The variance-to-mean ratio of twin-slot counts on the comb-restricted diagonal window has a zero-parameter closed form, lim Var/E = Pr[GD(2) > 2] = 1 − e^{−2γ}(9/2 − 4 ln 2) = 0.45546, confirmed to 2.9e−11 by three routes, residuals ≤ 0.002 at nine exact points from x = 13 against the Monte-Carlo model (+0.0029 at x = 13 against the exact model), exact at one excluded class where it reproduces Gorodetsky's λ(u); the model's own limit theorem is PROVEN (`history/staging/varE-limit-theorem.md`: ln n/ln y → GD(2) with rate O(lnln y/ln y), E[g] → λ₂(u) with no correction, the n < L band exact at nine levels at 0.063/ln y), and the θ = 2 mean-coefficient replacement stays open (`varE-theta2-proof.md`: two of its three groups closed unconditionally, the CRT-mixed lags a class discrepancy whose dominant part (82% of the measured remainder at x = 19) sits below 2L in the unbalanced range and is a friable-integers-in-progressions question (`attack-0830-varE-identification.md`, HELD); measured error O(1/ln y), falling through x = 31). The earlier fitted reading 0.611 is refuted as an inference: the fit protocol run on the model returns 0.615 and wins the frozen forecast too. Not in print at θ = 2, searched in the owning convention (variance of y-rough integers in short intervals, Gorodetsky's; `SEARCH-CONVENTIONS.md` §1). `history/staging/varE-spectral.md`, `varE-theta2-step.md`, `lit-dickman-variance.md`, `redteam-0828-varE.md`. **PROVEN, conditional, and a barrier rather than a bound.** On the hypothesis Granville uses (infinitely many real characters with a Siegel zero β ≥ 1 − κ/log q, κ arbitrarily small), interval sifting problems with two classes per prime attain the dimension-2 sieve bounds up to u = 2, so β_interval(2) ≥ 2: the first barrier statement on Face 4, sitting below the open band (2, 4.26645], where the transfer dies on a Chowla-strength correlation. `history/staging/attack-barrier-kappa2.md`. **DEAD.** 72 routes (recounted 2026-08-30 evening: 67 on 08-28, two added 08-29, four 08-30), one line each with its mechanism and record, in `OUTCOMES.md`. Added 2026-08-28: the QR refinement of the anchored caps (Z3), closed by a pre-registered test with a matched control. By family: **chaining** (the entropy integral of the true metric already exceeds the union bound, so chaining's ceiling sits below the union bound's floor); **the local-lemma family** (Shearer, Moser–Tardos, resampling oracles, entropy compression — all die on the complete dependency graph); **the concentration family** (McDiarmid, Azuma, Talagrand, Warnke, Kutin, Kim–Vu — one shared uniform residue draw defeats every local neighbourhood, and Banks–Ford–Tao arXiv:1908.08613 §5 had already run the programme on this ensemble); **spectral ℓ¹→ℓ²** (the conversion *is* the sharp maximal law and its true constant sits below the TPC line); **coverings** (the economy dies at `x = 13` where `Σ 2/p` crosses 1; the adversary side is SAFE by `x^{1−o(1)}` and widening); **thinning as a route** (the merge event is a function of the gap value and is mutually singular with every solvable coupling). Four candidate hypotheses were unmasked in one wave as the Zone Postulate wearing other clothes, which is why every `IMPORT-MAP.md` row now carries a circularity pre-check. **Earlier exponent route (parked under TODO 0).** Lemma V's missing piece is the quantifier — almost-all against worst-position — and the instrument family that would supply it is now named and priced. Deshouillers–Iwaniec 1982 Theorem 12, Maynard's Lemma 6.12 and Pascadi's 2026 Corollary 18 all require the **modulus and the inverted variable each to carry a fixed smooth profile**; Rosser weights are arbitrary functions of the whole modulus and supply none. If both were smooth, the published bound would still price at exponent 5.0907 > β₂ and the consumer cannot take a smooth profile (`attack-0829n-rml-proof.md` §4.5, HELD). The open arrow is now written with its quantifiers as REC(s, u₀) (same note §3); an adverse derived floor is on file (`attack-0830-rec-cheapest.md` §4) and has now been red-teamed on both halves (`redteam-0830-floor-sign.md`: the exact half PROVEN; `redteam-0830-floor-growth.md` and, on 2026-09-04, `redteam-0904-floor-growth-2.md` graded by `redteam-0904-item0.md`: the growth half survived two adversarial passes at every admissible level split and stays DERIVED, with the hypothesis s ≤ 3 added; the proof-gap reading is refuted, since REC quantifies over the remainder of the certificate the floor bounds). At that rung REC, F1, F2 and RML below β₂ are false rather than unproven, the route is a truth gap (`OUTCOMES.md`, 2026-08-30, reworded 2026-09-04), and its falsity lives beyond a level that depends on (s, u₀), 10^15.3 at (3.0, β₂) and 5.6e31 at (2.698721, 4.2165), where no computation reaches. This front is therefore CLOSED at rung derived-and-red-teamed-twice; the second pass that its earlier wording asked for has run and found no break, so only a non-two-class input reopens it. The affordable rough mass is 12.49% of ours for DI and — corrected 2026-08-20 — 32.50% for Pascadi, leaving a shortfall of `H^{0.818235}`: Theorem A's own admissible range carries a third term, `X ≪ max(1, q/N, q²/N³)`, that supplies `Y_N = max(1, q/N²)` for arbitrary sequences free, moving the frontier 0.354437 → 0.393922. And that axis is EXHAUSTED: `Y_N` enters Corollary 18's `ℐ²` only through the exceptional factor, and past 0.393922 the `Y_N`-free regular-spectrum main term `CS(C+DR)(RS+N)` binds whatever `Y_N` does — exact arithmetic (`1.212157 + 2σ ≤ 2`), so no coefficient information moves it further. `history/staging/lemmaV-neighbours.md`, `history/staging/smoothness-front.md`; the exponent chain and the source readings independently confirmed, `history/staging/redteam-0820-math.md` §2. Beside it, open and now carrying the named legal route at the exponent: the **1d bounded-defect Fekete target**. The submultiplicativity defect is an identity on the Overshoot slack, `D(s,t) = S(s) + S(t) − S(st)`, so the whole of TODO 1d is a statement about the sublinear part of `ln G₂`; the bounded-defect Fekete lemma is now stated and PROVEN with (H-mono) discharged, and the remaining hypothesis was WEAKENED to power pairs (2026-08-20 night) — **(H-sub-pow)**, `f(b^{k+1}) ≤ f(b^k) + f(b) + K` for all `b ≥ 2, k ≥ 1`, conclusions unchanged, since the lemma's proof consumes pairs only at `(b^k, b)` and the liminf half uses none — so conditional on (H-sub-pow), the route's single remaining gap, the exponent limit exists, and de Bruijn–Erdős is not needed (the reachable range supports a bounded defect). Its `t = 2` slice is **the doubling window** (§3a): proving `Ĝ(2s) ≤ C₂·Ĝ(s)` on the base-2 power chain for any `C₂ < 2^{β₂} = 19.2455` is trap-free in its all-s form only (the eventual form is TPC-implying below C₂ = 4, TODO D; custody data force `C₂ ≥ 348/66 = 5.2727`, so `log₂C₂ ≥ 2.3985 > 2`) and every provable point beats β₂; the two bridges tried, the K*-product and the per-fold composition, are closed (`OUTCOMES.md`, 2026-08-29/30). `history/staging/import-interp.md` §0, `history/staging/fekete-1d.md` §5, `history/staging/attack-hsub-01.md`, `history/staging/redteam-0820-night-proofs.md` §2. **The compute lever, and it is looser than the ladders suggest.** The exact ladder runs to 43# (`G₂(43#) = 618`); 41# and 43# are recomputations against A144311's 22 terms rather than new values, which makes them a check on both sides. The `162 GB` price recorded for the @29 census was the in-memory instrument's and not the object's — the segmented census runs @29 and @31 at 62 MB. Standing lever: any question phrased about a bounded window `[0, x^k)` rather than about the whole tile runs on a segmented sieve and reaches about three decades further, and the segmented engine walks the exact 29# period in O(1) memory. Check any stalled ladder against that before calling it compute-bound. `TODO.md` standing compute note, `LOCALIZED-GAP.md` §10. **And the second difficulty, which is about our measurements rather than about G₂.** Three quantities this programme quotes are not constants: the extreme-value `c` is a surface `c(x, lnD)` reading 0.45 to 1.09 across the reachable range (§3c); the exponent reads 1.50 on the 22 trusted terms and about 1.2 on certified ones, a third of that gap priced (§6); and the all-positions exponent θ is above 2 and rising with no visible asymptote (§7). Every number below carries the coordinates it was measured at, because two apparent contradictions in this programme were nothing but missing coordinates. --- ## 1. The object, and why it matters ### 1a. Definition Fix a prime x and write W = x# for the primorial. The **tile** T_x is the residue pattern of width W (classically: the primorial wheel mod x#). A **twin slot** is a residue r with gcd(r(r+2), W) = 1, that is, r and r+2 are both coprime to W. The census of twin slots in one period is D_x = ∏_{3 ≤ q ≤ x}(q − 2), OEIS A059861, the Schemmel totient (1869). > **G2(x#)** is the largest cyclic gap between consecutive twin slots of T_x. The omitted classes are fixed by the arithmetic and not chosen: for p = 2 one class is forbidden, and for each odd prime q ≤ x two classes are forbidden, r ≡ 0 and r ≡ −2 mod q. So G2 is the maximum-gap function of a sieve of dimension κ = 2 whose two omitted classes sit at distance exactly 2. **Three gap objects are in play, and they are not interchangeable.** **G2(x#), the whole-tile gap** — the object above: one number per tile, the maximum taken over the full period x#. Every exponent statement, ladder term and bracket in this file is about G2 unless it says otherwise. Exact in custody to 43#; 22 trusted terms to x = 79 via A144311. **h2, the free-class adversary** — Ziller and Morack's OEIS A288815: the same maximum-gap question with each prime free to CHOOSE its two omitted classes, where G2's classes are forced to {0, −2} by the arithmetic. It dominates G2 at every level and serves as the control ladder (§6: its exponent figure is 1.57 against the trusted ladder's 1.50). Any upper-bound machinery that never uses the class positions is really bounding h2, and pays h2's price. **Z2(x), the zone gap** — new, 2026-08-21 (Chris): the same maximum-gap question restricted to the zone (x, x′²) rather than the whole tile. One number per zone, not per tile. Because the zone lies below the frontier, its twin slots are genuine twin prime pairs, so Z2 is measured from the primes themselves with no tile built — thousands of levels where the tile ladder has fourteen — and an interior gap of the zone is a gap of the tile's slot sequence, so the interior part of Z2 never exceeds G2(x#). Definition and the not-G2 distinction: `GLOSSARY.md` §"Core objects", the "Zone gap" entry. The measurement artifact is `zonegap-01.js`; its boundary convention, data and law belong to that artifact and are not restated here. ### 1b. The Gap Reformulation **PROVEN reduction** (`research/ZONE-POSTULATE.md` §3). Both x# − 1 and x# + 1 are coprime to x#, so the tile carries a twin slot at its edge, and reading forward from the origin the first twin slot lies within G2 of it. Every hole of T_x in the zone (x, x′²) is a genuine prime, and every twin slot lying wholly inside the zone is a genuine twin prime pair, because a composite x-rough number is at least x′². Hence > **G2(x#) < x′² − 2 ⟹ Zone Postulate at x ⟹ (for infinitely many x) TPC.** Since ln(x#) = θ(x) ~ x, the window x² is (ln W)², so the sufficient condition is a two-class Jacobsthal exponent below 2. ### 1c. The exact logical status of the Zone Postulate **PROVEN, both directions, elementary** (`research/ZONE-POSTULATE.md` §2). - **Weak form** (the zone is occupied for infinitely many x) is **equivalent to the Twin Prime Conjecture**. Forward, occupied zones sit above x and x → ∞, so the twins found are unbounded. Backward, given a twin pair (r, r+2) with r > 3, take x the largest prime below r, so x′ = r and the pair lies in the zone of x. - **Strong form** (every zone occupied) implies TPC and imposes an additional occupancy requirement. A converse or logical independence is not proved here. A G2 bound holding at every prime gives the strong form; a bound holding only infinitely often gives the weak form. Uniformity in tile position does not itself imply uniformity in the prime level. `research/THE-DIALS.md` §1 dial 4 prices the difference as **free slack with no known mechanism**: an argument allowed to fail on a sparse set of x would be strictly weaker and still sufficient, and the two mechanisms looked for both fail. An almost-all-positions bound cannot be steered to the origin, because certification pins the anchor at 0 and there is exactly one usable window per tile. Large prime gaps widen the window by 2(x′−x)/x → 0, where a factor of three is needed. ### 1d. Why the object is the right one to hold `research/THE-DIALS.md` §5 measured three candidate windows to 10⁸. The square-to-square family (n², (n+1)²) has twelve genuine failures, last at n = 122, and reproduces OEIS A091592 term for term. The pane (n², (n+2)²) has one failure, at n = 26. The zone (x, x′²) has none, and its worst case across 1,228 windows sits at 1.13 times the Hardy-Littlewood count, never below its own expectation. A uniform gap bound cannot prove a statement that has exceptions, so the zone is the widest window whose statement has the same shape as the only tool we have. The comfort is margin and margin is not a dial (`THE-DIALS.md` §2). --- ## 2. The ladder All fourteen exact terms — the custody ladder; the trusted extension to x = 79 (A144311's 22 terms, adopted 2026-08-20) lives in `research/a144311-full-ladder.js`. The last four — 31#, 37#, 41#, 43# — were computed here; 41# and 43# on 2026-08-18, each twice on disjoint natal masks. G2 from `research/05-twin-jacobsthal.js` and `05b-twin-jacobsthal-segmented.js`, provenance table in `research/oeis-G2-submission.md`. The one-class column h(x#) is Jacobsthal at primorials, OEIS A048670, also tabulated by Holt (arXiv:1402.1970 §4). The ceiling column h2(x#) is Ziller and Morack's adversarial paired Jacobsthal, OEIS A288815 = 6·A072753 + 6. | n | x | G2(x#) | h(x#) one class | h2(x#) ceiling | x′² | x′²/G2 | G2/h | |---|---|---|---|---|---|---|---| | 1 | 2 | 2 | 2 | 2 | 9 | 4.50 | 1.00 | | 2 | 3 | 6 | 4 | 6 | 25 | 4.17 | 1.50 | | 3 | 5 | 12 | 6 | 18 | 49 | 4.08 | 2.00 | | 4 | 7 | 30 | 10 | 30 | 121 | 4.03 | 3.00 | | 5 | 11 | 42 | 14 | 66 | 169 | 4.02 | 3.00 | | 6 | 13 | 66 | 22 | 150 | 289 | 4.38 | 3.00 | | 7 | 17 | 108 | 26 | 192 | 361 | 3.34 | 4.15 | | 8 | 19 | 150 | 34 | 258 | 529 | 3.53 | 4.41 | | 9 | 23 | 204 | 40 | 366 | 841 | 4.12 | 5.10 | | 10 | 29 | 258 | 46 | 450 | 961 | 3.72 | 5.61 | | 11 | 31 | 348 | 58 | 570 | 1369 | 3.93 | 6.00 | | 12 | 37 | 528 | 66 | 708 | 1681 | 3.18 | 8.00 | | 13 | 41 | 546 | 74 | 894 | 1849 | 3.39 | 7.38 | | 14 | 43 | 618 | 90 | 1044 | 2209 | 3.57 | 6.87 | Three pointwise relations hold at every shared term and all three are elementary. - **G2 ≥ g(x#) = h(x#) (PROVEN, VERIFIED at all 22 shared terms).** Twin slots are a subset of holes, so an interval with no hole has no twin slot, and the twin-slot gap covering the largest hole gap is at least as long. `two-class-lower-bounds.md` §1. **The ratio's trend is settled by the full trusted ladders (2026-08-20, `research/external-ladders-01.js`):** the old 14-term reading "peaks at x = 37 and then falls twice, no trend established" was an artifact of the data ending at 43 — seven of the eight trusted terms x ≥ 53 sit ABOVE the old 8.00 peak, the maximum moves to the last term (8.55 at x = 79), and the tail reads as a slow upward drift: the one-class and two-class maximal gaps are separating, slowly. - **G2 ≤ h2(x#) (VERIFIED at all 21 shared terms, to x = 73; `research/external-ladders-01.js`).** Offset 2 is one of the offsets h2 maximises over. `exponent-control.md` §3. h2/G2 sits in [1.63, 1.81] at the seven trusted levels x = 47..73; the low outlier of the whole column is x = 37 at 1.341 — the same level that spikes G2/h and c2′. The three are not independent: all three are ratios carrying G2(37#) = 528, and of nine instruments free of that value none reads high at 37, the largest being h(37#) itself at z = −2.20 in the low direction and −1.08 on the range-matched ladder (MEASURED, `history/staging/measure-g2-provenance-0829.md` §4). What survives is one object: G2(37#) overshoots a blind seven-term extreme-value forecast by z = +6.58, the largest such residual on any of the three ladders, and that overshoot is unexplained. - **h2 ≥ h (VERIFIED at all 21 terms, elementary).** The adversary choosing two classes may take the first to be the one-class optimum. **G2(37#) = 528 additionally carries an INDEPENDENT EXHAUSTIVE maximality certificate** (2026-08-19): `maxsum₁ = 528` taken over all 217,929,355,875 gaps of T₃₇ by the scan-statistic engine, which shares no code with either exact-ladder producer. It certifies the maximum over the whole tile rather than agreeing with a published term, which makes it the strongest custody fact in this table (`history/staging/scanstat-t37.md`, `U-FRAME.md` §8). A third engine, a wheel-210 segmented striking sieve sharing no code with either, re-derived the whole ladder exhaustively at x = 11..37 on 2026-09-04 with every self-test exact (slot count, Σg = W, gap count, maximum) and measured the argmax sets: mirror-invariant, no fixed-point hit, and from x = 23 zero congruence pairs among the maximal windows, those zeros forced by both kill classes being present; a fourth engine (fold replication over the gap word) reproduced every position at x = 11..31 (`history/staging/measure-0904-argmax.md`, `redteam-0904-argmax.md`; MEASURED, understanding only). The last two terms are ours. G2(31#) came from the segmented method over 2.0·10¹¹ positions in about 80 minutes, G2(37#) from a mod-30 lattice walk over 7.42·10¹² positions in about 54 minutes, both with the slot count matched against A059861 exactly (`research/oeis-G2-submission.md`). The sequence **is** in OEIS: it is **A144311 + 1**, Carter 2008, 22 terms. The 2026-08-16 negative "at any of three offsets" was run in our own convention and none of the three was the −1 shift that returns A144311 at once, so `research/oeis-G2-submission.md` is a duplicate and must not be sent (`research/U-FRAME.md` §6a, `research/SEARCH-CONVENTIONS.md` §2). The margin column is flat. Its minimum over the computed ladder is 3.18 at x = 37, and on Ziller and Morack's dominating h2 the same margin against p_{n+1}² is flat at about 2.2 over nineteen terms, slope +0.018 ± 0.045 on [23, 73], with minimum 1.880 at x = 17 and never revisited (`exponent-control.md` §6). --- ## 3. Growth ### 3a. Everything known, in one table | statement | value | calibration | source | |---|---|---|---| | upper bound | G2(x#) ≪_ε x^{β₂+ε}, β₂ = 4.26645028414864191641 | **PROVEN** (DHR dimension-2 lower-bound sieve, input verified line by line against the primary source) | `paper/beta2-note.md` | | upper bound, conjectural | h2(n) < p_n² − p_n, hence G2 below it | CONJ, verified to x = 73 | Ziller and Morack 2017, Conjecture 6 | | **the target** | G2(x#) < x′² − 2, i.e. exponent 2 | what route A needs | `ZONE-POSTULATE.md` §3 | | **the named legal route at the exponent (2026-08-20 night)** | the doubling window: Ĝ(2s) ≤ C₂·Ĝ(s) on the base-2 power chain for any C₂ ∈ [5.2727, 2^{β₂} = 19.2455) gives limsup ln G2(x#)/ln x ≤ log₂C₂ ∈ [2.3985, 4.2665) | **OPEN, trap-free** — TPC through the slice needs C₂ < 4, excluded by custody data (Ĝ(32)/Ĝ(16) = 348/66 = 5.2727, red-team verified as the sup of Ĝ(2s)/Ĝ(s) over ALL data, sitting on the (2^k, 2) chain); every provable point improves β₂ | `history/staging/attack-hsub-01.md` §3; `history/staging/redteam-0820-night-proofs.md` §2b–2c | | lower bound, best proven | G2(x#) ≫ x·log x·logloglog x / loglog x | **PROVEN**, free by monotonicity from G2 ≥ g plus Ford-Green-Konyagin-Maynard-Tao (JAMS 31, 2018) via Rankin 1938 and Pintz 1997 | `two-class-lower-bounds.md` §3 Full write-up `paper/kk-lower-bound.md` (2026-08-28). | | lower bound, published-ingredient chain, above the free transfer | G2(x#) ≫ x·ln x, effective constant | **PROVEN from published ingredients consumed as theorems (K–K Cor 1 + Mertens + PNT + the §1 CRT identity), composition ours, adversary-confirmed end to end, NOT refereed** — held one adversarial pass at the door; finite cover independently rebuilt and replayed clean | `history/staging/import-hypergraph.md` §4; `history/staging/redteam-0820-math.md` §3.3–3.4; `paper/proposals/prop-xlnx-lower-bound.md` | | lower bound, derived here, two logs above the free one | G2(P(y)) ≫ y(ln y)³(lnlnln y)²/(lnln y)⁴ for y ≥ 10^{134.1} | **DERIVED HERE, NOT REFEREED** — K–K's §2 trichotomy re-derived for Ω_p = {a_p, a_p−2}, Case 2 empty and \|Ω^III_p\| = 2 verified to 10⁶, adversarially checked twice; sits ≈ ln²x above the x·ln x chain, on weaker (proof-reading) provenance | `two-class-lower-bounds.md` §4c | | lower bound, best constructed | 356,712 at x = 4001 | **CERTIFIED**, explicit (a_p) replayed by an independent routine, zero uncovered, at all sixteen levels | `two-class-lower-bounds.md` §5 | | lower bound, honest Rankin analogue | x(log x)² logloglog x / loglog x | INFERRED, contingent on the Rankin/FGKMT refinements transferring | `two-class-lower-bounds.md` §4b | | lower bound, conjectural truth | x(log x)^{3+o(1)} | CONJ, Maier-Pomerance accounting run in dimension 2 | `two-class-lower-bounds.md` §4b | | raw exponent fit, G2, 22 trusted terms, x-frame | 1.777 ± 0.029 | MEASURED, and the standard error is worthless | `exponent-control.md` §5 | | raw exponent fit, h2, 19 terms, x-frame | 1.847 ± 0.035 | MEASURED, same caveat | `exponent-control.md` §5 | | control-corrected exponent, h2 | **1.57 ± 0.06** stat, systematic unquantified | MEASURED | `exponent-control.md` §5 | | control-corrected exponent, G2, 22 terms | **1.50 ± 0.05** stat | MEASURED | `exponent-control.md` §5 | | control-corrected exponent, certificate ladder to x = 4001 | 1.11 to 1.25, falling with range | MEASURED, on the greedy construction rather than on G2 | `two-class-lower-bounds.md` §5 | | hard floor on the exponent | ≥ 1 | **PROVEN** via h2 ≥ h and exponent(h) = 1 + o(1) | `exponent-control.md` §3 | | proportional-bias correction, 0.67 to 0.83 | **REFUTED** by that floor | REFUTED | `exponent-control.md` §3 | | extreme-value law, on the diagonal | max ≈ c·m·(θ(x) − ln m), c₂′ = 0.4814 ± 0.049 for G2 | MEASURED, 8 exact terms, cv 10.2%, whole-period windows only | `two-class-lower-bounds.md` §6, `maxgap-law.md` §4 | | c is a surface c(x, lnD), not a constant | 0.446 to 1.083 inside the single tile x = 29; 0.577 (x = 23) to 0.989 (x = 6421) at fixed lnD | MEASURED, REFUTES any universal c | `maxgap-law.md` §4 | | mean twin-slot gap | m₂(x) = x#/∏(q−2) ~ e^{2γ} ln²x/(2C₂) ≈ 2.40 ln²x | **VERIFIED** | `ZONE-POSTULATE.md` §6, `localized-04-maxsum.md` §3 | | conjectural shape | G2(x#) = x(log x)^{3+o(1)}, with the o(1) strongly negative on everything computable | CONJ | `two-class-lower-bounds.md` §4b, §6 | ### 3b. The control, which is the reason not to trust any of the fits The estimator does not work at these sizes, and that is provable, because the same estimator run on 58 terms of an object whose answer is known reports 1.282 when the truth is 1 (`exponent-control.md` §1). The bias grows and then sticks: | window width | windows | mean | sd | bias | |---|---|---|---|---| | 10 | 49 | 1.262 | 0.094 | +0.262 | | 19 | 40 | 1.280 | 0.058 | +0.280 | | 21 | 38 | 1.282 | 0.053 | +0.282 | | 30 | 29 | 1.283 | 0.036 | +0.283 | The scatter collapses by a factor of three across those widths and the centre does not move. On the same data the pure power law beats the family containing the truth by 47 AIC units with white residuals (27 sign runs against 28.5 expected, lag-1 autocorrelation 0.118) and its exponent is wrong by 0.28. So a clean power-law fit with white residuals carries no evidence about an asymptotic exponent at these sizes. **House practice.** Any exponent read off a ladder in this repo is quoted alongside the control line: *58 terms, true exponent 1, measured 1.282 ± 0.008, no drift.* Two further readings from the same note. On nineteen terms of h2 the models c·p^1.847 and c·p·log^2.448 p are separated by **0.1 AIC units** while differing by a factor of x^0.85 in the limit, so the data does not determine the exponent. And at matched n and range the frozen quadratic loses 74.7 AIC units on the one-class control against only 12.2 on h2, so two classes do look far more quadratic than one, without that being a measurement of the exponent. Exponent 2 is disfavoured by one-sidedness rather than by a standard error. The control's bias is positive in all 40 windows, minimum 1.197, because a positive power of log inside the truth biases a finite-range power fit upward. Nothing about the two-class object suggests a negative log power, so the bias transfers in sign and the true exponent lies **below** the raw 1.847. ### 3c. The extreme-value law, and the fact that c is a surface Reading Ford's random-dart prediction J(T) ~ e^γ T log T as extreme-value statistics generalises it: for a sifted set of mean gap m inside a period W, the number of gaps is W/m and the maximum of that many exponential gaps is about m·ln(W/m). Tested against every exact value that exists (`two-class-lower-bounds.md` §6), every row a **whole period**, that is lnD/θ(x) → 1: | c | object | terms | range | mean | cv | range of c | |---|---|---|---|---|---|---| | c₁ | one class g(x#), A048670 | 46 | x ∈ [11, 229] | 0.3718 | 7.2% | [0.3359, 0.4873] | | c₂ | free 2-class h2(x#), A288815 | 17 | x ∈ [11, 73] | 0.8511 | 7.3% | [0.7784, 1.0157] | | c₂′ | our G2(x#) | 8 | x ∈ [11, 37] | 0.4814 | 10.2% | [0.4463, 0.5939] | **c is a surface c(x, lnD) and the law is a diagonal law (MEASURED, `maxgap-law.md` §4).** Two coordinates move it in opposite directions. At fixed x, cutting one exact period into blocks and growing lnD makes c fall monotonically from 1.083 to 0.446 inside the single tile x = 29, terminating on the exact G2(29#) = 258. At fixed lnD ≈ 11, growing x makes c rise 0.577, 0.587, 0.745, 0.803, 0.857, 0.951, 0.989 at x = 23, 29, 97, 199, 401, 1601, 6421. Along the **diagonal**, where the window is the whole period, lnD = θ(x) − ln m grows because more primes enter, and the two effects cancel. That cancellation is the whole content of the law's apparent constancy. It is not an extreme-value statement, and the off-diagonal exponents (b ≈ −0.4 in lnD, a ≈ +0.4 in ln ln x) predict a 32% fall along the diagonal leg x = 101 → 271 where the exact terms deliver a 2.5% rise. > **Report c with its coordinates (x, lnD, lnD/θ(x)) or do not report it.** The > localized files read 0.74 to 1.17 and the exact tiles read 0.45 to 0.59 for the > same twin-slot object. Those are the same surface at different positions, and > on matched statistics both spreads are ±22.5%, not one tight and one loose. **What the constant is (MEASURED, `maxgap-law.md` §5).** c is the reciprocal of the gap tail's exponential rate at the extreme. c₁ = 0.372 says the one-class gap tail decays like exp(−2.7 g/m), not exp(−g/m), and the extreme-value solve D·S(t) = 1 returns the exact whole-period maximum at both x = 23 objects, ratio 1.000. **The caveat that stops this being an asymptotic.** For one class the conjectural truth is x ln²x, so c₁ should eventually grow like ln x. It does drift upward, at a tenth of the required rate: c₁ ~ (log p)^{0.12 ± 0.02} on the top 47 exact terms, against the exponent 1 Maier-Pomerance needs. A synthetic ladder built to have exponent exactly 1 returns 0.981 ± 0.006 from the identical estimator, so the shortfall is a measurement rather than a resolution limit. The law describes the accessible range, not the limit, which agrees with `exponent-control.md` §2 finding the frozen Maier-Pomerance shape refuted on the same data (rms 0.281, maximum log residual 1.239, a factor of 3.5). **Where the missing log lives (INFERRED from a PROVEN source, `maxgap-law.md` §7).** In Ford, Green, Konyagin and Tao's own accounting the truth is y ~ e^γ·x·T1·r, so in the law's units c = T1·r/log x. T1 is Rankin's alignment gain, PROVEN and capped by the smooth-number step at log x·log₃x/(log₂x)², which is less than one log. r is the number of survivors one large prime can kill, proven only at r = 2 and conjectured by Maier and Pomerance at (log x)^{1+o(1)}. Both are maxima over indices a random-dart model does not carry. Evaluated with r frozen at 2, the ledger returns c = 0.3674 at x = 229 against the measured 0.3722, and it predicts flatness across exactly the range where flatness is observed because log₃x/(log₂x)² is stationary at x = 181. It doubles c only past x ≈ 10¹⁷. ### 3d. What the second residue class costs Three independent routes, one answer, and the answer is **one logarithm, not a power of x**. - **Rankin accounting run in dimension 2** (`two-class-lower-bounds.md` §4b, INFERRED). The second class multiplies the survivor density after the small primes by 1/log z, so it multiplies the coverable length by log x. The Maier-Pomerance ledger, whose one-class exponent decomposes as 1 + 1, gains one row in dimension 2 and closes at 3. - **The certificate ladder** (MEASURED). (Y2/Y1)/ln x plateaus at 2.18, flat from x = 113 onward. Fitting the ratio as a power of ln x gives 1.39 on the whole ladder, 1.17 from x = 229, 1.09 from x = 773, and 0.88 on the top five points. - **The Poisson law** (MEASURED). m₂/m₁ ~ e^γ ln x/(2C₂) ~ 1.35 ln x, and both objects sit at a flat multiple of their own prediction. In the **discrepancy** channel the price is entirely different, and the two must not be quoted as one phenomenon. There sup₂/sup₁ crosses over at x = 11 and then grows by ×1.177 per fold, asymptotically √(3/2) = 1.2247 per fold, that is (3/2)^{π(x)/2}, exponential in π(x) (`discrepancy-two-class.md` §7). --- ## 4. Structure ### 4a. The two merge facts **Fact A (PROVEN).** For x ≥ 3, no two twin slots of T_x are 2 apart. If s and s+2 were both twin slots then s, s+2, s+4 would all be x-rough and they cover every residue class mod 3, so one is divisible by 3. **Fact B (PROVEN).** Folding T_x by p deletes exactly the slots in classes {0, −2} mod p, and those classes are 2 apart. In an interval of length below p − 2, two kills would be either p apart (impossible in the interval) or exactly 2 apart, which Fact A rules out. **An interval of length below p − 2 contains at most one kill.** (`LOCALIZED-GAP.md` §2.) Fact A is what makes the two-class version of Holt and Rudd's fusion mechanism *cleaner* than the one-class original: their minimum span between fusions is 2p_{k+1} because the minimum gap between generators is 2, ours is p − 2 because the two kill classes sit only 2 apart. **The Mirror-Sweep Lemma (PROVEN 2026-08-20; re-derived from scratch and verified exhaustively by an independent engine).** The tile mirror σ(s) = W−2−s conjugates the fold's deletion classes {a, a−2} mod p into {w−a, w−a−2}, w = W mod p, so the residue-deleted maxsum family satisfies Δ_m(x,p,a) = Δ_m(x,p,(w−a) mod p) for all m, for every alignment outside the specials {1, w−1, w+1, p−1} — **at most four** (three whenever w ∈ {2, p−2}; the producer's own 5→7 row prints {1,3,6}). In copy order the per-copy degradation curve is a **palindrome**, C(k) = C(p−1−k), broken only at the seam-striking copies, where the identity degrades to a one-sided inequality with an exact correction term: the seam side never loses. The single symmetry-breaking point is the edge slot W−1, the natal cohort's birth canal. Every argmax set from 7→11 up is closed under a ↦ w−a; the 5→7 exception is the licensed one (its unique argmax IS the seam-striking alignment). The fold-ladder verification's single-engine caveat at 31→37 is resolved: the structural red team's from-scratch direct-definition tracker reproduces the full 29→31 curve (31 of 31 alignments) and six 31→37 values spanning the whole shape — max, min, both tiers, bulk — all agreeing, the rest halved by the proven mirror. Same pass, kill-decoupling: the per-copy kill count stops predicting degradation at depth — exactly flat at 17→19 for every tie-break permutation, REVERSED at 23→29 for every tie-break (19→23's flatness is tie-break-dependent, range straddling zero). `history/staging/attack-0c-holesweep.md` §4; `history/staging/redteam-0820-structural.md` §1. ### 4b. The Localized Merge Lemma, and its true turn-on point Write M(T_x, Y) for the largest twin-slot gap of T_x among gaps starting below Y, and maxsum_m for the largest sum of m consecutive such gaps. > **Lemma (PROVEN, `LOCALIZED-GAP.md` §3).** If M(T_x, Y) ≤ (p − 2)/4 then > M(T_p, Y) ≤ maxsum₂(T_x, Y). **The lemma has two conditions and they turn on a decade apart** (`localized-04-maxsum.md` §6). The **hypothesis-side** condition M(T_x, Y) ≤ (p−2)/4 is read at the old level and is the only one a chain can use in advance; it first holds at **x = 13933** at Y = 10⁹ (k = 2.17), VERIFIED, against a prediction of 13,630 from solving x* = 9.6·ln²x*·ln(Y/m̄), 2% out. The **conclusion-side** condition M(T_p, Y) < p−2 is read at the new level, is four times weaker, and first holds at x = 1453 at k = 2; `localized-03-merge-lemma.js` line 74 tests that one. Verifying it verifies the lemma and does not give a gate, because it refers to the level not yet reached. At k = 3 the same equation gives x* ≈ 2.4·10⁴ with Y ≈ 1.4·10¹³, out of computational reach by four decades. The lemma is true and the regime where it applies is not one we can survey. ### 4c. Why the chain dies, and it is cheaper than the sieve wall **Deficit Lemma (PROVEN, `localized-04-maxsum.md` §7).** For 1 ≤ m ≤ D/2, maxsum_m ≥ m·m̄·(1 − O(m/D)), by averaging and nothing else. VERIFIED: the minimum of R(m) = maxsum_m/(m·m̄) over m ≤ 1024 measures 1.187, 1.280 and 1.296 at x = 16001, 997 and 3499, never below 1. **Traverse Bound (PROVEN).** Telescoping the merge lemma forces j·m̄ ≤ α·q_{j+1} with α = 1/4, so **j ≤ x₀/(9.6 ln²x₀) folds whatever C is, even C = 1**, against the π(x) ≈ x/ln x folds the route needs. Short by a factor 9.6 ln x, which is 93 at x = 16001. Measured chain survival is **0 folds for every x ≤ 12143 at Y = 10⁹, and exactly 1 fold at x = 13933 and x = 16001, against the 1863 that π(16001) requires**. **The obvious repair is closed (PROVEN).** Weakening the gate to M ≤ α·p does not help at any α, because the gate feeds back. If the chain proves M ≤ B(x), Fact B permits B/x kills per gap, the telescope index is B/ln x, and the bound returned is about 2.4·R·B·ln x. The map B ↦ 2.4·R·B·ln x is expanding for every x ≥ 2, so there is no fixed point. The obstruction is not the size of the gate, it is that **the gate multiplies wherever the index accumulates against a fixed base.** **How far that principle reaches, and it is narrower than it sounds (`gate-multiplies.md` §2, §6).** Stated as four hypotheses on a proof strategy (accumulating index, a deficit floor, linear cost feedback, and a traverse count exceeding the budget), the no-fixed-point argument closes this chain and A4's tile analogue, and it does **not** reach a recursion that re-bases at every fold, where the running bound cancels out of the cost instead of compounding. Three escapes exist and all three are occupied by something in the repo: an exact identity (the copy theorem for the whole maxsum family, and A9's transfer operator), a B-free per-fold cost, and re-basing. The u-frame recursion escapes it. **What closes the tile analogue instead is simpler and harder to escape.** > **Overshoot Budget (PROVEN given the measured G2 law, `gate-multiplies.md` §5).** > Any valid chain of upper bounds ending in G2(x#) < x² passes through quantities > all at least G2(x#), so its total multiplicative overshoot over the truth is > capped at x²/G2(x#). MEASURED, the slack ln(x²/G2(x#)) reads 1.058, 0.940, > 0.984, 0.878, 0.953, 1.182, 1.016, 0.953 across x = 11 to 37. Flat over the > whole reachable ladder, with a predicted asymptote of ln(1/0.55) = 0.598 nats. So an accumulating-index chain must pin its cumulative index to within a factor 1.82 of the truth over π(x) folds, and the budget is a fixed number of nats spread over an infinite ladder however the chain is decomposed. The only unconditional tool available, the spacing bound of Fact B, is loose by 3.5× to 6.7× at a single fold (MEASURED at four folds, deepest T_23 by 29). **There is no growing margin anywhere in this route, at any level, in any formulation.** ### 4d. The growth law of maxsum, which survives the chain **MEASURED, and not a fit** (`localized-04-maxsum.md` §3): > maxsum_m = m·m̄ + σ·√(2 m ln D), so R(m) = 1 + (σ/m̄)·√(2 ln D / m), > with σ/m̄ = 0.892, 0.920, 0.949 at x = 997, 3499, 16001. Over m ≥ 2 ln D it holds to R/R_EV ∈ [0.981, 1.110], [1.038, 1.119] and [0.990, 1.064] at those three levels. R(m) falls from about 15 at m = 1 to about 1.2 at m = 1024 and never plateaus, so sup_m R(m) = R(1), and R(1) is a property of the **window** and not of the level: R(1)/ln D ∈ [0.736, 1.170] across Y = 10⁷, 10⁸, 10⁹ and x from 89 to 16001, a factor 180 in x. Those are off-diagonal coordinates in the sense of §3c, so that 0.74 to 1.17 is not the same number as the tile's 0.46 and neither travels to the other's window. Below m ≈ 2 ln D the maximum is a single tail event, one sparse patch owning the whole small-m curve. Above it the argmax migrates and the excess is √m Gaussian fluctuation. This retires the standing question `U-FRAME.md` §9 used to carry, the repo's named single hole, one rung with no ladder. §9 now records the hole as closed and cites this result as what closed it. The ladder exists. It was never the obstruction. ### 4e. The Origin Excess Lemma, its ceiling, and the reversal at the zone width > **PROVEN, VERIFIED 14/14** (`maier-matrix.md` §4). Let y < x, let y′ be the > least prime above y, and let **S ≤ y′²**. Then #(A ∩ [0,S)) = D_y(S) − L, where > L counts twin slots t of T_y with t < S such that t or t+2 is struck, and a > y-rough integer m ≤ y′² is struck by a prime in (y, x] only if m is itself a > prime in (y, x] or m = y′² with y′ ≤ x. > > *Non-vacuity, a remark and not a hypothesis:* the conclusion has content only > when **y′² > x**. An x-rough integer below x is 1, which is not a slot, so > D_x(S) = 0 for every S ≤ x, and at S = y′² ≤ x the lemma reads 0 = 0. All > fourteen verification cells satisfy y′² > x. So L ≤ 2(π(x) − π(y) + 1), a count of primes, against D_y(S) which grows with S. Measured L is 0, 1 or 2 across all fourteen cells. The m = y′² boundary case is load-bearing: dropping it fails 4 of the 14 cells. **The lemma has exactly one hypothesis, S ≤ y′², and it travels with three companion statements of three different logical types** — the non-vacuity remark above, the corollary below, and the Scale Collision Proposition. The canonical enumeration is `maier-matrix.md` §4a and the derivations are `origin-excess.md` §§6a-6c; do not keep a private count of them here. **The advantage is capped by an absolute constant, not by (ln x/ln y)² (MEASURED, `origin-excess.md` §0).** The binding constraint is the corollary's positivity threshold: the guaranteed lower bound D_y(y′²) − 2(π(x) − π(y) + 1) is positive only for x below a threshold x*, and ln x*/ln y ≈ 1.44 at every level computed. So the advertised (ln x/ln y)² is **capped at about 2.2**, and the largest ratio in the fourteen-cell table is **1.372**. No choice of (y, x) reaches past 2.2. **Why the collision is structural rather than an artifact of this lemma.** > **Survival Quotient Identity (PROVEN, VERIFIED 39/39, `origin-excess.md` §2).** > With ρ_z(S) := (D_z(S)/S)/δ_z the local-to-global density ratio of level z, > > **origin(S)/mean(S) = ρ_x(S)/ρ_y(S)** for every y < x and every S ≤ y#. The origin's excess is not a property of the origin. It is one survival curve read at two levels, and by the repo's Unification Law ρ_z(S) depends only on u = ln S/ln z. The Zone Postulate asks for S = x′², that is u_x = 2, which is where ρ takes its minimum. **So at the zone's width the origin does not merely lose its advantage, it reverses (MEASURED, decisive).** Against the full ensemble of all x# translates, origin/mean = ρ(2), and after stripping the finite-size Hardy-Littlewood factor that reads 0.79303, 0.79475, 0.79668, 0.79863, 0.79922 at x = 1487 to 6037 against > **ρ(2) → e^{2γ}/4 = 0.79305: the origin carries 21% LESS than the mean > density at exactly the width the programme needs.** The trough is at S ≈ 3.5 y′², not a decay to 1, and the band of y for which the origin still beats the mean at S = x′² shrinks monotonically toward y = x across nine levels (max ln x/ln y = 1.219 down to 1.137). At S = x′² the class a_q = 0 that the origin occupies is the worst of the q available classes at the smallest prime q = y′, rank 17 of 17 at y = 13, x = 23. The rank measurements agree. At y = 13, x = 23 the origin is the largest of 7429 rows at S = 289 and the 5752nd, below the mean, at S = 1000; at S = x′² it ranks 2254 of 7429; and at y = 13, x = 19 it sits below the mean outright. The Origin Excess Lemma needs S ≤ y′² and the Zone Postulate needs S = x′², so both together force y ≥ x, at which point the matrix is a tautology. That is the **Scale Collision Proposition** (PROVEN, `origin-excess.md` §6c): y < x with x prime gives y′ ≤ x < x′, hence y′² < x′² strictly, which is a fact about the two parameters and not about this construction, so no better lemma repairs it. **The informative regime and the origin-distinguished regime are disjoint, and pushing to the zone width lands on the one point where the origin is worst.** ### 4f. The Maier matrix, classified completely **PROVEN + VERIFIED across eleven divisors of 13#** (`maier-matrix.md` §6). The matrix depends on the column modulus q only through d = gcd(q, x#), and returns exactly one number, the mean of the window count over the x#/d windows at multiples of d, which is the Level Ledger histogram h_d. | d = gcd(q, x#) | ensemble | output | |---|---|---| | d = 1 | all W translates | the global mean exactly, ρ = 1, zero information | | d = y#, proper divisor | the W/d windows at multiples of d | ρ_y(S), the coarser tile's local density | | d = W | one window | the target itself, the tautology 0 = 0 | The transfer identity at d = y# is Σ_r #(A ∩ [r·y#, r·y# + S)) = D_y(S)·∏_{y **VERDICT: SAFE against the construction side.** The gap between the best > available lower bound and the threshold is x^{1−o(1)} and widening. To threaten > the Zone Postulate a two-class construction would have to beat the one-class > construction by a **power** of x, and every route checked says the second class > is worth exactly one logarithm. **The adversarial minimum at the origin, exact (2026-08-20).** The same free-class adversary, turned against the anchored Natal@5 comb: minimising survivors over ALL scour-class choices gives **advmin@11 = 16, exact, with two disjoint proof stacks** — the producer's branch-and-bound with proven pruning plus a replay-verified witness, and the structural red team's independent union-only branch-and-bound, completed over the full 10-prime space with zero leaves below 16 (`history/staging/attack-advmin-1113.md`; `history/staging/redteam-0820-structural.md` §2a). The staircase's certified anchored floor at @11 is 34 (`paper/staircase-note.md` Thm 8), so the floor sits **18 ABOVE the class-blind ceiling**, and the corrected quantifier (red-team wording): *any bound whose certified floor is a single number valid simultaneously for every choice of scour classes (a class-uniform conclusion) is instantiated by the adversarial witness, so its ceiling is advmin@11 = 16 < 34: class-uniform caps cannot re-certify the staircase floor, let alone close 34 → 45. A cap whose conclusion depends on the chosen classes is not touched — but reading the classes is exactly what "anchored-aware" means, so only anchored-aware caps remain.* At @13 the certified bracket is [21, 152] against floor 110 and truth 307: at least 155 of the 197 floor-to-truth points are anchored-only. Class-uniform-conclusion joint caps are DEAD as a route (`OUTCOMES.md`); the live successor is TODO's anchored-arguments attack. **The anchored-point floor family, and the classic ladder's plateau (2026-08-20 night).** The composition the advmin verdict demanded exists, and it is one move: the **unified-cap lemma** (PROVEN, red-team re-derived — `history/staging/attack-anchored-01.md`; `history/staging/redteam-0820-night-proofs.md` §1) extends the staircase's cofactor injection to m = 1, so the self-strike allowance s(q) disappears into a per-prime cap family with `fresh ≤ capU_K ≤ cap_K` at every depth and every prime, @11 and @13. Its floors are **ANCHORED-POINT floors per Theorem 8's own quantifier** — separated from, not contradicted by, the 16-witness, which does not instantiate them: 36 @11 and 115 @13 at K = 0 (above the staircase's 34/110), exact truth 45/307 reached first at K = 8/28, and every @11 floor exceeds advmin's 16 — a floor family class-uniform caps provably cannot reach. The classic ladder (s(q) kept as an allowance) provably plateaus at 41 @11 and 296 @13 at every depth — pool saturation plus exhaustion — and the shortfall is the named blocking structure with digit-exact accounting: **wheel-excluded self slots** plus **twin-collision shadows**, 6 − 2 = 4 = 45 − 41 @11 and 17 − 6 = 11 = 307 − 296 @13. The **forcing ladder** — the exact adversarial minimum with the first j scour primes anchored — runs 16→20→20→22→23→25→30→35→38→42→45 (j = 0..10), fully reproduced by an independent branch-and-bound; anchoring q = 13 alone costs the adversary +4, and every single anchored prime costs at least +1. Mirror symmetrization gains exactly 0 at the anchored cap and is REFUTED as an improvement channel (`OUTCOMES.md`): the covariance σ(K_q(a)) = K_q(w−a) is a one-line theorem, so the zero gain is structural, not empirical. ### 5b. What is closed The closed routes are indexed in [OUTCOMES.md](OUTCOMES.md) — one line per route, verdict, mechanism, and the record file that carries the full argument. That index is the authority; this section exists so citations of "§5b" keep resolving. ## 6. The exponent, and the one gap left in reading it ### 6.1 The trusted ladders read 1.50 and the certificate ladder reads 1.2 Both readings are of **one object**, not two. By the CRT identity of `two-class-lower-bounds.md` §1, G2(x#) − 1 is exactly the longest interval coverable by choosing a residue pair per prime, so the certified greedy ladder is a proxy for the same exponent that the exact terms measure, not a different quantity. What differs is the proxy and the range. | reading | object | range | control-corrected | |---|---|---|---| | exact terms, h2 | free two-class ceiling | x ≤ 73, 19 terms | **1.57 ± 0.06** stat | | trusted terms, G2 | the object itself | x ≤ 79, 22 terms (A144311) | **1.50 ± 0.05** stat | | certified greedy ladder | a lower certificate for G2 | x ≤ 4001, 16 levels | **1.11 to 1.25**, falling with range | Two measured effects account for about a third of the difference and they point opposite ways. A certified lower-bound ladder biases the exponent **down** by about 0.09 at a 28% terminal shortfall, priced on the one-class control (`h2-scoping.md` §5b). The trusted ladders are still short, and the control's own reading rises with prefix length, 1.245 at 22 terms to 1.282 at 56, so the short ladders carry up to +0.04 of the other sign. Adding the first back gives about 1.29 from the certificates against 1.50 from the trusted G2 terms. > **Quote 1.57 for h2, 1.50 for G2, and note that the long certificate ladder > reads 1.2 with about a third of the gap priced as the greedy's own downward > bias.** The residual 0.2 is unexplained and is not worth chasing: every > reading, corrected or raw, sits below 2 and above 1, and the gap that matters is > 4.2665 against 2. The headline does not rest on the eight trusted terms this repository has not reproduced. Refitting by provenance block with the same estimator and the same matched-width control gives 1.533 on custody terms alone, 1.525 on custody plus Alekseyev, 1.465 on the eight external terms alone and 1.500 dropping the x = 61 term (Ĝ(64) = 1080 in item D's coordinate), against 1.498 on all 22 (MEASURED, `history/staging/measure-g2-provenance-0829.md` §3). The same index sets run on the one-class control spread 0.932 to 1.233, so the split is a null against a floor set by the estimator's own range sensitivity, not a demonstration of independence. ### 6.2 The law's two constants were a coordinate error, and it is settled `c` in max ≈ c·m·(θ(x) − ln m) is a surface c(x, lnD), not a constant, and the two values quoted in the repo were the same formula read on two different curves through it (§3c, `maxgap-law.md` §4). On matched statistics the two spreads are identical at ±22.5%; there was never a tightness disagreement, only a range compared against a coefficient of variation. **Quote the Poisson form with its (θ(x) − ln m) factor and the coordinates (x, lnD, lnD/θ(x)), never a bare constant.** On the diagonal, where the window is the whole period, c₂′ = 0.4983 over all 20 exact diagonal terms to x = 79 (cv 7.4%, trend (ln x)^{+0.227}). **The bare-constant reading that used to sit here, G2 ≈ 1.2 x ln²x, is superseded as of 2026-08-18**, and the strength of that supersession was itself corrected on 2026-08-19. Quotable: **G2(x#) = x ln^{2+o(1)}x with the o(1) positive**, best single description **≈ 0.76 x ln²x lnln x** over x = 11..79, two significant figures on the constant and not three. **The claim that two INDEPENDENT instruments agreed is REFUTED**: the exact frame cancels out of every reported offset, so the diagonal is the same comparison in a sharper error bar rather than a second witness, and it returns attack E's own fitted parameter. The AICc figures 31.8 and 10.6 must not be quoted as exclusions either — the one-class control, whose conjectured truth IS x ln²x, is thrown out by 32.8, and the power law's margin falls to 3.3 on deleting x = 11 while the whole ranking reverses at x = 5..79. What survives is the frame-free ranking, a factor of eight, and the x-exponent tending to 1 at every window tested. Nor is any of it an asymptotic: the FGKMT ledger overtakes such a reading, and attack E's §7 shows x ln²x lnln x and x ln³x are not separable in principle here (`two-class-lower-bounds.md` §6, `history/staging/redteam-2026-08-18.md` §§1–2). ## 7. The theta ladder: it does not turn over On `ZONE-POSTULATE.md` §5's question, whether the all-positions exponent θ turns over below 2: θ is the window exponent z^θ at which the vector-sieve certificate is positive at every position, and the Gap Reformulation needs θ < 2. **The unconditional column, measured self-consistently, sits below 2 at every exactly-measured level.** The instrument that reported a crossing took its window from no caller (`theta-ladder-sup.js` fixed `u = 3.2`), so its suprema were measured at windows 61×–123× too long. Self-consistent exhaustive full-period walks give need/z² = 0.3550, 0.4360, 0.5485, 0.4877, 0.4637, 0.6119 at z = 13..31 — all exact, all inside the budget, worst margin 39% — and the certificate is positive at every one of the 223,092,870 positions at H = 0.46 z². The crossing is real and sits in (31, 47]: at z = 47 a prefix walk finds min T = 0 at H = 1.041 z², and a prefix minimum upper-bounds the period minimum (`history/staging/theta-selfconsistent.md`). The old ladder's prefix rows are lower bounds on an overstated quantity and bound nothing (`theta-ladder.md` §5b). **What retires the route is the maximal law's price, not a crossing: the sharp Gaussian maximal law is TPC-implying** (`history/staging/phase1-T4-maximal-law.md`). **The conditional column is where the route loses ground.** need/z² runs 1.596, 1.773, 1.707, 2.379, 3.477, 4.188, 5.130, 6.396 at z = 19 to 47, and θ rises monotonically from z = 29 to 2.482. **Where the growth lives.** Decomposing need = 2√(2 lnW)·√⟨ρ²⟩/M, two of the three factors are tame; the whole problem is ⟨ρ²⟩, which tracks the divisor-pair count N growing like z^{4.6} with a cancellation factor flat near 75 to 125 rather than improving, against a budget that allows z³. **What is not established.** "θ settles" is not observed. The fitted growth exponent of c(z) is not stationary, running 0.94 ± 0.38 on the first four points and 3.07 ± 0.28 on the last four, so the four fits that reject θ → 2 at 95% reject a model rather than a limit. Extending the ladder cannot repair that: z = 71 costs 36 hours and moves ln z only from 3.85 to 4.26. **θ is above 2 over the whole measured range, it is rising, and it has no visible asymptote.** --- ## 8. What is genuinely ours Two prior-art audits ran over this material, the 2026-08-13 three-sweep audit and the 2026-08-17 Holt corpus sweep plus reference sweep (`research/PRIOR-ART.md`). Stated conservatively, and with the owners named first. **Not ours.** Holt owns the cycle of gaps, the fold recursion, the fusion mechanism (Holt and Rudd 2014, Lemma 3.1 of arXiv:1408.6002), the closure theorem, the transfer operator, the population models, the interval of survival, and the one-class discrepancy front ΔΦ. Maier owns the matrix, and Granville and Soundararajan own the layout we used. Buchstab owns the survival curve, computed by Cheer and Goldston and used by Maier. Ziller and Morack own h2, A288815, and the reduction "h2(n) < p_n² − p_n for n ≥ 3 implies Goldbach and the twin prime conjecture". Schemmel owns the census ∏(q−2). Rankin, Erdős, Pintz and Ford-Green-Konyagin-Maynard-Tao own the one-class lower bounds. Iwaniec owns g(q) ≪ ln²q. Diamond, Halberstam and Richert own the dimension-2 sieve, with the exact β₂ due to Booker and Browning. The CRT collapse that makes G2 an adversarial covering problem was already in this repo under the name PAIRED (`research/attack2-rankin2d.js`). **Ours, at the calibration marked.** The table discipline, because `README.md` sends readers here on the promise of a calibration marker on every line: **one object per row, one calibration per cell.** A row whose subject contains "and" is a defect, and a calibration cell containing "+" or "given" names a second object that needs its own row. | item | calibration | note | |---|---|---| | G2 as a studied object | **NOT OURS — OEIS A144311, Andrew Carter, September 2008** | the object is published and has been for eighteen years, under the `G2 − 1` convention: "the longest sequence of consecutive integers, each equal to 1 or −1 modulo at least one of the first n primes". Their m = our r+1; same fixed classes {0,−2}. Five of our waves searched `G2` and never `G2 − 1` | | the four exact terms computed here | VERIFIED | 31#, 37#, 41#, 43#. **A144311 already carries 22 terms**, so 41# and 43# were recomputations, not new values — they agree, which is a check on both. Terms 15–22 extend the ladder to x = 79 and are ours to verify, not to claim | | G2(x#) ≪_ε x^{4.2665+ε} | PROVEN | the first upper bound at any exponent for the two-class problem. **Re-tested 2026-08-18 in the owning vocabulary** (A144311's, not ours) and it survives: no published upper bound at any exponent, and the one bound-shaped heuristic (Paseman, MO 88323) is asymptotically WEAKER — his exponent is unbounded, ours fixed. **But the TECHNIQUE is not ours**: MO 37679 answer 52890 (zeb, 2011) already reads `j(x#) ≪ x^{4.032}` off a sieve's error exponent at dimension ONE. **Claim the dimension-2 instantiation, not the method**, and never table 4.032 beside 4.2665 — the closeness is coincidence | | the mean-square Lemma V, `⟨R²⟩_H ≤ B(z,s)·H` for every H, z, s | **PROVEN, 2026-08-18** | five identities and two triangle inequalities; `B(z,3.0)` flat at 1.3833–1.4883 over z = 13..37. Its L4 also proves (V1) and (V2) with the equality case, which `sift-limit-lemmaV.js` only asserted. **Prior art NOT checked — do not claim novelty until it is** | | `u_sup`, an unconditional worst-position remainder bound with no maximal law | **MEASURED, and CLOSED as a route** | rises at all eight steps over z = 13..43 to 3.2026; constant model's RSS 49× the best fit's; the bounded family's best asymptote is 5.46, **above** β₂. Real as a bound, useless as a road. The Fourier gain over the divisor-pair bound is 23–44× and **falling**, not a flat 80× | | G2(x#) ≥ g(x#) pointwise, and with it the first lower bound of any kind for a two-class Jacobsthal function | PROVEN, trivial | absent from the literature and from both OEIS entries (A288815, A072753) | | the "difference 2 only, infinitely often only" weakening of Ziller and Morack's reduction | PROVEN, logically lightweight | strictly weaker hypothesis than theirs, and not found elsewhere | | the two-class Rankin accounting, giving log x and not a power | INFERRED | Maier-Pomerance's own ledger run in dimension 2 for the first time. **"Their ledger" is load-bearing:** Kalmynin-Konyagin run a dimension-2 Mertens ledger of their own (arXiv:2302.00459), so the unqualified form of this claim is false | | the two-class driving-term lemma | PROVEN | cleaner than Holt's one-class version because Fact A rules out the second kill | | the driving-term route to a bound on G2 | **REFUTED** | it certifies gaps of size O(x) and cannot reach x² (`two-class-lower-bounds.md` §7) | | the certified G2 lower-bound ladder to x = 4001, sixteen levels, independently replayed | CERTIFIED | extends the two-class data 55-fold beyond Ziller and Morack's 21 exact terms | | the measurement of c in max ≈ c·m·ln(W/m), and the finding that it is a **diagonal** constant on a surface c(x, lnD) | MEASURED | the law itself is Ford's random-dart prediction. Ours is the measurement across 46 exact one-class and 17 exact two-class terms, the block statistic that separates the two coordinates, and the identification of c as the reciprocal of the gap tail's rate at the extreme | | the Overshoot Budget | PROVEN as an implication, its input MEASURED | a lifetime slack measured at 0.88 to 1.19 nats caps every accumulating-index chain, and it survives any decomposition | | the no-fixed-point argument, four-hypothesis form | PROVEN | closes accumulating-index chains and A4's tile analogue; it does **not** reach a recursion that re-bases at every fold (§4c) | | the Survival Quotient Identity origin/mean = ρ_x(S)/ρ_y(S) | PROVEN, VERIFIED 39/39 | one line of rewriting, and it turns the origin's excess into a statement about one survival curve read at two levels | | the exact unconditional θ suprema over complete periods to z = 31, with prefix bounds to z = 71 | VERIFIED | removes the Gaussian maximal law from the sighting entirely, which no amount of laddering the conditional column could do | | route A's shadow implication into the explicit one-class Jacobsthal bound | PROVEN as an implication | §5 route A; the explicit-constant question under it is closed and the floor stands | | Facts A and B in their two-class form | PROVEN | mechanism is Holt and Rudd's; the two-class form is not in their corpus. Fact B is loose by 3.5× to 6.7× at a single fold | | the two-class Localized Merge Lemma | PROVEN | the localized-window application is on Holt's own open-problem list — **arXiv:2603.25915 §4.1–4.2 (2026, Holt alone)**, where he defines `g_max(I,p)` and asks its trend, NOT 1408.6002, whose §1.1 list has six items and none localized (PDF read 2026-08-18) | | the Deficit Lemma and the Traverse Bound | PROVEN | one-line averaging, and it kills the chain unconditionally | | the growth law maxsum_m = m·m̄ + σ√(2m ln D) | MEASURED, unfitted | retires U-FRAME §9's named single hole | | the Origin Excess Lemma | PROVEN, VERIFIED 14/14 | one hypothesis, S ≤ y′², non-vacuous when y′² > x; enumerated by logical type at `maier-matrix.md` §4a | | the Origin Excess Corollary's ceiling at about 2.2 | PROVEN, on one MEASURED input | the input is the flatness of ln x\*/ln y ≈ 1.44 over nine levels; measured maximum 1.372 over fourteen cells | | the reversal to ρ(2) = 0.79305 at the zone width | MEASURED, decisive | the origin carries 21% less than the mean density at exactly the width the programme needs | | the complete classification of the Maier matrix on a periodic set by d = gcd(q, x#) | PROVEN, VERIFIED | sharper than "it does not work" | | the two-class discrepancy ΔΦ₂, with its ×3 ceiling and the identity R_k(p) → k+1 | PROVEN | the two-class object was not found in the convention that owns its one-class neighbour, Holt's `ΔΦ(x,p)` at arXiv:2308.07570, and `SEARCH-CONVENTIONS.md` §1 carries no row for the two-class form, so read that as our reach; the ceiling is the Möbius term count | | the per-fold sup and sd growth of ΔΦ₂ | MEASURED | ×1.8356 and ×1.6612 two-class, against the √R_k limit 1.7321 (§4g) | | the control-estimator method as house practice | method | quote the 58-term control line beside any ladder exponent | | the Inertness Lemma and the u-frame presentation | PROVEN | `U-FRAME.md` §2, §3 | | the complementary-window duality `maxsum_m + minsum_{D−m} = W` | **NOT OURS** | the complement identity of the circular scan statistic: Cressie, *J. Appl. Probab.* 14 (1977); Naus; Wallenstein–Naus; Glaz–Naus–Wallenstein 2001 chs. 8–10, 17. Verified here EXACT at all 1484 `m` on T₁₃, which is a reproduction. No live sentence may present it as new (`IMPORT-MAP.md` row 1) | | the wheel as the extremal object of a closed-form Diophantine problem | **NOT OURS — the Lonely Rabbit problem** | Rab(n) = 1/w(n), conjectured Cusick (Acta Arith. 22 (1972) 1–9), PROVED Schark (Monatsh. Math. 78 (1974) 131–146), asymptotic Rab(n) ~ e^{−2γ}/(n log log n). The extremal modulus is a primorial wheel and the constant is Mertens, so the wheel-is-extremal intuition has a PUBLISHED-ANCHOR — and it is an anchor, not a bound on G2 (`PRIOR-ART.md`, `history/staging/row12-recon.md` §5) | | the exhaustive maximality certificate for G2(37#) = 528 | **CERTIFIED, ours** | `maxsum₁` over all 217,929,355,875 gaps of T₃₇, from an engine sharing no code with either exact-ladder producer. A144311 owns the TERM; what is ours is the independent proof that 528 is maximal over the whole tile rather than a value that matched (`history/staging/scanstat-t37.md`) | | the two-class substitution into Kalmynin–Konyagin, giving G2(P(y)) ≫ y(ln y)³(lnlnln y)²/(lnln y)⁴ | **DERIVED HERE, checked twice, NOT REFEREED** | K–K own the §2 trichotomy and the machine; ours is the re-derivation at Ω_p = {a_p, a_p−2}, where Case 2 is empty and the third class has size exactly 2. Two logs above the free FGKMT bound. The referee gap is named: Halberstam–Richert Thm 2.2 and the Selberg remainder's ξ-versus-z condition at κ = 4 (`two-class-lower-bounds.md` §4c) | | the Zone Postulate weak-form equivalence, stated as a biconditional | PROVEN, lightweight | not found as a biconditional, searched under twin-Legendre; `SEARCH-CONVENTIONS.md` §1 carries no owning convention for the biconditional form, so that negative is in our framing. The framing is a device rather than a result | The honest summary of the ownership question: the mechanisms are almost all borrowed and correctly attributed, and what is ours is the specific object G2, its data, the two-sided bounds on it, and four small lemmas that each close a door. --- ## 9. Open questions, ranked Ranked by what each would buy, not by how hard it is. An item that closes keeps its slot and is marked CLOSED, so a reader who remembers the ranking finds the answer where the question was. This list is the state side of the programme and `TODO.md` is the plan side; they overlap, and neither is a subset of the other. **1. Any unconditional two-class exponent below 4.2665.** Buys the only thing that matters. The band (2, 4.2665] is a proof gap rather than a truth gap. `research/sift-limit-attack.md` §2 finds nothing published that blocks it and no κ = 2 extremal example known (Halberstam 2003 p. 117); the published lower bounds on β₂ (Selberg's reciprocal convention, Brady 2017: 1.8196; β(2) ≥ 2 from the one-sided dimension axiom, 2026-09-04) sit at or below the band's lower endpoint, searched as tabled in `SEARCH-CONVENTIONS.md` §1 and §3. β₂ itself is unimproved since Diamond-Halberstam 2008, with every post-2008 source worse at κ = 2 (`SEARCH-CONVENTIONS.md` §4); nobody has shown it cannot move for this problem either. The visible road is full Brüdern-Fouvry decoupling at 1 + √e ≈ 2.649, which removes about 71% of the open band without finishing. Cost: unknown, and the whole programme. **2. Whether an explicit elementary bound already gives g(x#) < x′² — CLOSED, and the answer is no.** The item would have retired route A's second difficulty floor if any explicit elementary bound delivered the constant-1 form at primorials. None does. Kanold (2^{√k}), Stevens (k^{Θ(log k)}) and Paseman (arXiv:1311.5944, k^{O(log log k)}) are all far weaker than exponent 2; the only exponent-2 statements are Vaughan 1977 for general n and Iwaniec 1971 Theorem 2 / 1978 at primorials, both with inexplicit constants, and the constant is what the question turns on. The floor stands and route A's price is unchanged (§5 route A, `TODO.md` 000b). **3. Close the residual 0.2 between the trusted and certificate exponents (§6.1).** 1.50 on trusted G2 terms against about 1.29 on certificates once the greedy's own downward bias is added back. Buys a number the repo can quote without a footnote, and settles whether the greedy's shortfall really is constant-factor: the free-choice greedy is flat at 0.73 against the ILP optima over 19 terms while the branch-and-bound lower ladder's shortfall grows 0% to 28% over n = 9 to 21, and those two readings are not obviously compatible. Cost: extending the greedy against the exact ILP optima, which already exists as a routine. **4. Prove that window/G2 is unbounded.** `ZONE-POSTULATE.md` §8 item 3 (folded there from item 4 on 2026-08-29). Strictly weaker than route A and still TPC-implying: unbounded means G₂(x#) < x′² − 2 at infinitely many x, which is weak ZP (labelled 2026-08-29, `history/staging/object-g2-read-0829.md` §5). It would show the Gap Reformulation is not asymptotically self-defeating. Today's evidence for it is one-directional and was flagged as such: the certificate is a lower bound on G2, so x²/certificate is an *upper* bound on x²/G2, and its growth (3.8, 7.8, 18.1, 44.9 at x = 37, 229, 1009, 4001) does not by itself prove x²/G2 grows. **5. Find a mechanism that spends the infinitely-often slack.** An argument allowed to fail on a sparse set of x is strictly weaker than any gap bound and still sufficient, because the weak form is equivalent to TPC. The slack is free and there is no known mechanism: both candidates have been checked and both fail (§1c, `THE-DIALS.md` dial 4). This item is the search for a third. **6. Compute G2(41#), the thirteenth term — CLOSED 2026-08-18, and the Poisson law survived its own test.** G2(41#) = 546 and G2(43#) = 618 are exact (§2), each computed twice on disjoint natal masks, and **both landed inside their pre-registered Poisson windows** — 546 against the 476–633 band below, central 513. A144311 already carried both terms, so they are recomputations that agree rather than new values, which is a check on both sides. The cost estimate below missed by two orders of magnitude in the cheap direction: the run took 2m31s against the six hours priced here, which is the cost-estimates-err-cheap rule's best exhibit. The band's own discriminator resolved toward the outlier reading rather than the level shift. Everything from here to the end of the item is the pre-computation pricing, kept because §9's rule is that a closed item keeps its slot. Worth less than it looks for the exponent: measured on 48 control cases, adding an eleventh term to a ten-term fit moves the exponent by 0.022 on average and 0.078 at worst, against a bias of +0.262. Its real value is as a falsifiable test of the Poisson law, which predicts **476 to 633, central 513**, below both the free-choice ceiling of 894 and the older multiplier extrapolation of about 740. The band splits the test: about **488** if x = 37's c₂′ = 0.594 is an outlier and about **633** if it is a genuine level shift, and one term discriminates them. The ledger drift over the single step 37 → 41 is 4%, so the surface structure of §3c does not blunt it. **Cost: about 6 hours by the streaming leg**, which is the instrument to price against, because it scales in slots rather than in positions. T₃₇ holds D = ∏_{3≤p≤37}(p−2) = 217,929,355,875 slots against T₃₁'s 6,226,553,025, a factor of exactly 35, so the measured 519 s that recovered G₂(37#) from T₃₁ (`U-FRAME.md`, streaming-leg custody) scales to 5.05 h, and 5.6 h carrying the 41/37 deletion-state factor. The 37-hour figure also in circulation is the lattice walk over positions, which is linear in tile width. The residual risk is memory rather than time at 2.2·10¹¹ slots, and the segmented engine walks a full period in O(1) memory. At six hours this is a same-day falsifiable test of the Poisson law rather than an overnight commitment. **7. Whether sup/sd stays bounded in the discrepancy channel.** Nothing proven there beyond the ×2 and ×3 ceilings, and the right-base doubled-exponent gap has an exact cause. It is the same shape as the sifting gap in a place simple enough to attack. `discrepancy-two-class.md` §10. **8. M(x, x^k) for fixed k ≥ 3 under the single-alignment recursion — MEASURED 2026-08-17, and the route to a growth law through it is CLOSED.** The head sits in copy 0, which receives only the alignment a = 0, so its recursion is single-alignment rather than a maximum over p alignments. Measured to x = 1613 (Y = 4.2e9, 253 folds, `research/localized-single-alignment.md`): the multiplier IS different in shape — exactly 1 at 94% of folds, total spend 107% of the 3 ln ln budget against the tile shape's two-orders overshoot, and the kill census is countable at 0.81 p²/ln²p dead flat — but 55% of the growth and the birth of every deep record sit in the boundary term the window's own growth creates, which is not a fold statement; it is the localized gap problem re-posed at the new scale. The recursion reduces M to itself plus a countable correction. Standing caveat unchanged: at k = 2 the head is crystallised and TPC-hard by construction. **9. The compute observation, which outlives the lemmas it came from.** Every localized object needs a segmented sieve of [0, x^k) by primes up to x and never needs the tile of width x#. Any question in this repo phrased about a bounded window rather than about the whole tile can be asked three decades further out than it currently is (`LOCALIZED-GAP.md` §10). The same observation on the other side: `maxgap-law.md`'s segmented engine walks the exact period 29# = 6,469,693,230 in O(1) memory, so whole-tile statistics are reachable two levels past where the repo was building tiles. --- ## 10. Sources and reproduction All of them read in full for this note. | file | what it holds | |---|---| | `research/exponent-control.md` / `.js` | the exponent, calibrated against the one-class control, and the frame correction | | `research/two-class-lower-bounds.md` / `.js` | G2 ≥ g, the imported lower bound, the certificate ladder, the Poisson law, route A's second floor | | `research/localized-04-maxsum.md` / `.js` | the maxsum growth law, the Deficit Lemma, the Traverse Bound | | `research/LOCALIZED-GAP.md` | Facts A and B, the Localized Merge Lemma, the refutation of the chain | | `research/maier-matrix.md` / `.js` | the Origin Excess Lemma, the scale collision, the complete classification | | `research/origin-excess.md` / `.js` | the Survival Quotient Identity, the ceiling at 2.2, and ρ(2) = 0.79305 | | `research/discrepancy-two-class.md` / `.js` | the two-class discrepancy channel and the Level Ledger's true looseness | | `research/gate-multiplies.md` / `-01..03.js` | the no-fixed-point argument in four hypotheses, the Overshoot Budget, and the surviving form of TODO 0b's per-fold rate | | `research/maxgap-law.md` / `.js` | c as a surface, the block statistic that separates its coordinates, and the FGKMT ledger | | `research/theta-ladder.md` | the conditional θ ladder to z = 47 and the exact unconditional suprema to z = 71 | | `research/h2-scoping.md` | why extending A288815 is both infeasible and non-diagnostic | *Files added since 2026-08-18 are indexed in [../README.md](../README.md)'s file table, not here; §0 above is the current map.* Older and load-bearing: `research/ZONE-POSTULATE.md`, `research/THE-DIALS.md`, `research/U-FRAME.md`, `research/PRIOR-ART.md`, `paper/beta2-note.md`, `research/oeis-G2-submission.md`. ``` node research/exponent-control.js # ~1 s node research/two-class-lower-bounds.js # 3 s, ladder to x = 571 node research/two-class-lower-bounds.js --full # 6 min, ladder to x = 4001 node research/localized-04-maxsum.js 1e9 scan 8 16000 # 18 s node research/localized-04-maxsum.js 1e9 997,16001 1024 16000 # 100 s node --max-old-space-size=6144 research/maier-matrix.js # 6 s node --max-old-space-size=6144 research/origin-excess.js # 13 s node --max-old-space-size=6144 research/discrepancy-two-class.js # 17.9 s node research/gate-multiplies-02.js 29 # 17 s node research/gate-multiplies-03.js # 4.3 s, streams T_29 node --max-old-space-size=8192 research/maxgap-law.js --big # 147 s, exact periods to 29# node research/window-check.js 1e11 # ~10 min, the Zone Postulate check ``` --- *This document states current understanding. Superseded claims, retired numbers and the reasons they changed are in [history/CHANGELOG.md](history/CHANGELOG.md), indexed by document.*