# G₂ read, 2026-08-29: the object, its variants, its wall, its legal open set > **RIDER 2026-09-04 (orchestrator, from `recon-0904-sifting-limit-floor.md` > and `redteam-0904-sifting-limit.md`).** §4a's last paragraph (lines ~532-534) > reproduces Face 4's sentence and inherits its defect: "no κ = 2 extremal > example is in print" stands (Halberstam 2003 p. 117), but lower bounds on > β(κ) at κ > 1 ARE in print in Selberg's reciprocal convention, Brady 2017 > gives β₂ ≥ 1.8196, and β(2) ≥ 2 is immediate from the one-sided dimension > axiom; so the sentence reads "no lower bound on β(2) above 2 is known". Also > §8 Q2 (line ~1000): the multiplicity of the maximum is NOT unknown, it is on > the ladder to x = 43 in `exact-g2-ladder.js` (`nmax`), `measure-g2z2-0829.js` > and `ATTACKS3.md` §A1; only the positional half of Q2 was open, and > `measure-0904-argmax.md` measures it. *(Reading note, HELD, internal. Wave of 2026-08-29, one of four parallel object reads. It computes nothing new: every number carries a pointer to the file that produced it. Calibration markers per `CLAUDE.md`: PROVEN, VERIFIED (exact computation), CERTIFIED, MEASURED, HEURISTIC, INFERRED, CONJECTURED, REFUTED, OPEN. Where this note and `README.md` §Status or `research/G2-STATE.md` §0 disagree, those two win and this note is the thing to fix.)* ## 0. What is open **The band is the whole content, and nothing in this reading moved it.** `G₂(x#) ≪_ε x^{4.26645+ε}` is PROVEN (`paper/beta2-note.md` §1, §3), the Gap Reformulation needs exponent 2 with a constant strictly below 1 (`research/G2-STATE.md` §1b), and the open band is `(2, 4.26645]`. The corpus has run roughly fifty recorded passes at it and the band has not narrowed once (`research/G2-STATE.md` §0; `README.md` §Status). Below, the best PROVEN lower bound is `G₂(x#) ≫ x log x logloglog x/loglog x`, free by `G₂ ≥ g` plus FGKMT; the two rungs above it are DERIVED HERE and NOT REFEREED (`research/two-class-lower-bounds.md` §§3, 4c). Measured, the exponent reads 1.50 ± 0.05 stat on 22 trusted terms with the systematic unquantified, against 1.57 ± 0.06 on the h₂ control (`research/exponent-control.md` §5). So the distance between what is measured and what is proven is about 2.8 in the exponent, and the distance between what is measured and what is needed is about 0.5 in the other direction. **The three killers, as they bite this object specifically.** 1. **The class-blind cap (killer 1, the operative one).** Every upper-bound argument the corpus has produced consumes exactly one property of the tile: that each odd prime removes two residue classes. That is a dimension-2 sieve hypothesis, it caps at β₂ = 4.26645 by construction, and it is blind to where the two classes sit. Coordinates and the exact hypothesis list are §4 below. [PROVEN that the current bound uses only this; `paper/beta2-note.md` §2, §6 item 1.] 2. **The wrong quantifier (killer 2).** G₂ is a maximum over positions in one period. Every "almost all intervals" theorem, every ensemble statement and every measure-theoretic bound admitting an exceptional fraction ε decides the worst position only if `εW < 1`, and the corpus's own second-moment bound misses that by e^3025 at x = 19 read self-consistently in the window ensemble (`paper/wall-note.md` §2 Face 1, corrected 2026-08-27). Lemma V's mean-square form is PROVEN and the missing piece is named as the quantifier, not a factor (`research/G2-STATE.md` §5 route A). 3. **The parity floor at the finish (killer 3).** Exponent 2 with constant below 1 is TPC-strength by the Gap Reformulation (`history/staging/attack-wrongdirection-audit.md` §1 Axis A, verbatim: `G₂(w#) = o((log w#)²)` or `≤ (1−ε)(log w#)² i.o.` ⟹ TPC). Tao's general parity obstruction does not name the bare tile statement, because a property defined by congruences alone forbids no Liouville sign pattern; it does name the zone form in full, because there the extension is the set of twin prime pairs (`paper/wall-note.md` §2 head, read at source 2026-08-27). The exemption is one step narrower than it looks and does not cover any route that bounds sums against a non-negative sieve weight. **What this note does not do.** - It computes nothing new. Every number carries a pointer to the file that produced it, and §6 lists the cheap unknowns without running them. No producer script was written and none was run. - It does not close, open or re-price any route. `research/REFUTED.md` is the authority on what is closed and `TODO.md` on what is queued; this note edits neither. - It is not a prior-art claim. Everything in §7 is checked against `research/PRIOR-ART.md` and `research/SEARCH-CONVENTIONS.md` first, and where the reading is this corpus's framing rather than an owning convention's, it says so. - Its §8 questions are **understanding** questions. None of them is a proposed attack, and each was checked against `research/REFUTED.md` and against the full `Q-` id list of `research/QUESTIONS.md`. - One class of check it did not run: it did not re-execute any producer to confirm that a quoted number matches its embedded OUTPUT block. The inconsistencies in §5 are all document-against-document, not document-against-artifact. ## 1. Definitions: every named variant of the object **What is wrong here first: the corpus carries at least three naming defects on this object, and one of them is a genuine definitional collision rather than a typographical one.** They are listed under the table and repeated in §5. | symbol | exact definition | where defined | relation to `G₂(x#)` | |---|---|---|---| | `G₂(x#)` | largest cyclic gap between consecutive twin slots of the tile `T_x`, a twin slot being `r` with `gcd(r(r+2), x#) = 1` | `G2-STATE.md` §1a; `paper/beta2-note.md` §1 | the object | | `A144311(n)` | "longest sequence of consecutive integers, each ≡ 1 or −1 mod at least one of the first n primes", Carter 2008, 22 terms to x = 79 | `G2-STATE.md` §8; `U-FRAME.md` §6a | **equality**, `G₂(x#) = A144311(n) + 1`. The object is published and has been since 2008 | | `Ĝ(s)`, `s` a real ≥ 2 | `Ĝ(s) = G₂(P(s)#)` with `P(s)` the largest prime ≤ s: `G₂` re-indexed on the integers, a step function | `attack-wrongdirection-audit.md` §3.3; `fekete-1d.md` §5 | **equality at prime arguments**, a right-continuous step interpolation elsewhere. Custody chain: Ĝ(2,4,8,16,32,64) = 2, 6, 30, 66, 348, 1080 | | `S(s)` | `S(s) = ln(s²/Ĝ(s))`, the Overshoot slack | `fekete-1d.md` §§3, 5 | an exact reparametrisation: `S > 0` at s ⟺ `Ĝ(s) < s²` at s | | `g(x#)`, written `h(x#)` in most tables | one-class Jacobsthal at primorials, largest gap between consecutive x-rough residues, OEIS A048670 | `G2-STATE.md` §2; `covering-dive.md` §1.1 | **`G₂ ≥ g` pointwise, PROVEN**, twin slots ⊂ holes. VERIFIED at all 22 shared terms | | `H(r)` (literature's `h(k)`) | `max_{ω(n)=r} j(n)`, Jacobsthal maximised over all n with r prime factors; carries Iwaniec's `≪ (k log k)²` | `covering-dive.md` §1.1 (Hajdu–Saradha at source) | **no direct relation**. `H(r) = g(p_r#)` is a theorem for r ≤ 23 and FALSE at r = 24 (Hajdu–Saradha 2012), so the two must not be written as one symbol | | `h₂(x#)` | Ziller–Morack paired Jacobsthal at primorials, OEIS A288815 = 6·A072753 + 6, 21 terms to x = 73 | `covering-dive.md` §2.1 (their Def. 2.2, read from the PDF); `GLOSSARY.md` §G₂ | **`G₂ ≤ h₂` pointwise, VERIFIED at all 21 shared terms.** The *reason* is stated two incompatible ways in the corpus: see defect (ii) below | | free two-class | any pair `{a_p, b_p}` per prime, Resta's ILP, A072753 | `two-class-lower-bounds.md` §1 table | `≥ h₂` "up to the 6a+6 bridge", per that table; OEIS states the bridge as an identity. Both readings give `≥ G₂` | | `Z₂(p)` | the same maximum-gap question restricted to the zone `(p, p′²)` rather than the whole tile; one number per zone | `GLOSSARY.md` §"Zone gap"; `G2-STATE.md` §1a | **not `G₂`.** Its *interior* gaps are gaps of the tile's slot sequence, so the interior part satisfies `Z₂ ≤ G₂(p#)`; the boundary convention belongs to `zonegap-01.js` | | `M(T_x, Y)` | largest twin-slot gap of `T_x` among gaps **starting below** Y: the localized gap | `LOCALIZED-GAP.md` §3; `G2-STATE.md` §4b | `M(T_x, Y) ≤ G₂(x#)`, with equality once `Y` reaches the whole period. `M(x, x^k)` is the same object at `Y = x^k` | | `maxsum_m` | largest sum of `m` consecutive twin-slot gaps (of `T_x`, or of the gaps starting below Y) | `localized-04-maxsum.md` §3; `G2-STATE.md` §4b | **`maxsum₁ = G₂` on the whole period, by definition.** `maxsum_m ≥ G₂` for every m ≥ 1 | | the scan statistic | the circular scan: the maximum count of slots in a window, dual to `maxsum` by `maxsum_m + minsum_{D−m} = W` | `IMPORT-MAP.md` row 1 (Cressie 1977); `history/staging/scanstat-t37.md` | the duality is **PROVEN and NOT OURS**; `maxsum₁ = 528` over all 217,929,355,875 gaps of `T₃₇` is the exhaustive maximality certificate for `G₂(37#)` | | `Y2(x)` | the certified greedy lower construction: the largest `[1, m]` covered by an explicit `(a_p)` with classes `{a_p, a_p − 2}` | `two-class-lower-bounds.md` §5 | **`Y2 ≤ G₂(x#) − 1` always** (each run is a certified covering), with equality PROVEN at thirteen levels x ≤ 43. The certificate ladder runs to x = 5003 | | `θ` (the ladder's) | the window exponent `z^θ` at which the vector-sieve certificate is positive at **every** position of the period | `theta-ladder.md`; `G2-STATE.md` §7 | an upper-bound *instrument* for `G₂`: `θ < 2` would give the Gap Reformulation. Measured above 2 and rising, no visible asymptote | | `need/z²` | the certificate's own required window as a fraction of `z²`, measured self-consistently on exhaustive full-period walks | `history/staging/theta-selfconsistent.md`; `G2-STATE.md` §7 | 0.3550 to 0.6119 at z = 13..31, all exact, all inside budget; the crossing sits in (31, 47] | | `ρ̃_z(y)` | the H-free sawtooth potential of the Brüdern–Fouvry certificate at s = 3.0 | `history/staging/rho-maximal-law.md` §1 | RML(α): `sup_y \|ρ̃_z(y)\| ≤ C·z^α`. The pricing lemma turns any α < β₂ into `G₂(z#) ≪ z^α ln²z`, every step PROVEN except the law | | `R_H(x)` | the H-dependent remainder of the same certificate, at the operative window | `history/staging/phase1-T4-maximal-law.md` §2 | its **sharp** Gaussian maximal law is TPC-implying and already false as literally stated (violated at z = 19 by 0.6%). `ρ̃` and `R_H` are two objects; the corpus's phrase "the sieve Gaussian maximal law" names both | | `m₂(x)` | the mean twin-slot gap, `x#/∏(q−2) ~ e^{2γ}ln²x/(2C₂) ≈ 2.40 ln²x` | `ZONE-POSTULATE.md` §6 | VERIFIED. The normaliser in every extreme-value reading of `G₂` | | `c₂′` | the diagonal extreme-value coefficient in `max ≈ c·m·(θ(x) − ln m)` for `G₂` | `maxgap-law.md` §4; `G2-STATE.md` §3c | **a surface `c(x, lnD)`, not a constant.** Quote it with coordinates or not at all | **Defect (i): one object, two letters, and one of the letters is the literature's name for a different object.** The one-class Jacobsthal at primorials appears as `g(x#)` in `two-class-lower-bounds.md` §1 and as `h(x#)` in `G2-STATE.md` §2's table header, and `G2-STATE.md` §2 writes "`G2 ≥ g(x#) = h(x#)`" to reconcile them in place. Meanwhile the literature's `h(k)`, the one carrying Iwaniec's `≪ (k log k)²`, is `H(r) = max_{ω(n)=r} j(n)` and is a **different function**: `covering-dive.md` §1.1 records, from Hajdu–Saradha 2012 read at source, that the identification holds for r ≤ 23 and fails at r = 24. Nothing measured moves, since every computed level is below r = 24, but the identity is a theorem with a known counterexample and the corpus writes it as a definition. [PROVEN false as a definition; `covering-dive.md` §1.1.] **Defect (ii): `h₂` is defined two incompatible ways in the live layer, and the corpus records both without noticing.** `covering-dive.md` §2.1, verified from the arXiv PDF, gives Ziller–Morack's Def. 2.2: `j₂(n)` is the least m such that every paired progression `⟨a,b⟩_m` with `2 | (b−a)` contains a coprime pair, so the offset `d = b − a` is **one global even number** and the kill pair mod p is `{−a, −a−d}` at that same d for every p. Under that reading `h₂` is a maximum over even offsets, `G₂` is its `d = 2` slice, and `G₂ ≤ h₂` because 2 is one of the offsets, which is exactly what `G2-STATE.md` §2 and `two-class-lower-bounds.md` §1 say. But `GLOSSARY.md` §G₂, `U-FRAME.md` §6a and `G2-STATE.md` §1a all define `h₂` as **free choice of two residues per prime**, which is a strictly larger adversary (it permits different distances at different primes) and is the object `two-class-lower-bounds.md` §1 lists on a *separate row* as A072753, above `h₂`. Since OEIS gives `A288815 = 6·A072753 + 6` as an exact renormalisation, the corpus is simultaneously asserting that the two are the same sequence and that one dominates the other through a bridge. **No downstream number moves**: both readings deliver `G₂ ≤ h₂`, which is all any consumer uses. What is at stake is whether the h₂ control measures a d = 2-free adversary or a fully free one, and therefore what the 1.57 control exponent is an exponent *of*. [OPEN, definitional. Settled by reading A072753's OEIS definition line and ZM Def. 2.2 side by side; not attempted here.] **Defect (iii): "the sieve Gaussian maximal law" names two objects with opposite verdicts.** On `R_H` at the operative window it is TPC-implying; on the H-free `ρ̃` it is not, its win condition being α < β₂ rather than α < 2. That was adjudicated at `attack-wrongdirection-audit.md` §3.6 and the resolution is per-object. Any sentence about "the maximal law" that does not name `R_H` or `ρ̃` is under-specified. [ANSWERED there; carried here because the ambiguous phrase still appears in prose.] **One symbol the corpus uses cleanly, recorded because it is easy to confuse.** `maxsum₁` and `G₂` are the same number on the whole period and different numbers on a bounded window `[0, Y)`, where `maxsum₁ = M(T_x, Y) ≤ G₂(x#)`. Every localized reading in `localized-04-maxsum.md` is the windowed one, and `G2-STATE.md` §3c flags that the localized `c` of 0.74 to 1.17 and the tile `c` of 0.45 to 0.59 are the same surface at different coordinates, not two measurements of one constant. ## 2. Proven statements, with direction **What is wrong here first: exactly one of the statements below is refereed, and it is not one of ours.** `β₂ = 4.26645028414864191641` is Diamond–Halberstam's, with the digits due to Booker–Browning; the DHR Theorem 9.1 that carries it is published. Everything the corpus derived on top of it, in both directions, is either PROVEN-here-and-unreviewed-outside or explicitly NOT REFEREED, and two of the three lower bounds say so on their own face (`research/G2-STATE.md` §3a rows 6 and 7). ### 2a. The upper bound, and exactly which hypothesis it uses > **`G₂(x#) ≪_ε x^{β₂+ε}` for every ε > 0.** [PROVEN, sieve input verified line > by line against the primary source; `paper/beta2-note.md` §§2–3, honesty > section §6.] Checked against `paper/beta2-note.md` §2, the hypothesis list is exactly this and nothing more: 1. `A = {r(r+2) : x < r ≤ x+H}`, `X = H`, `z = p_n + 1`. 2. `ω(2) = 1`, `ω(p) = 2` for odd p, `ω(d) = ∏_{p|d} ω(p)`; the two classes mod an odd p are distinct because `0 ≢ −2`. 3. `|A_d| = (ω(d)/d)·H + r_d` with `|r_d| ≤ ω(d) ≤ 2^{ν(d)}`. 4. Condition Ω(κ) at κ = 2 in the product form (DH Definition 1.3, eq. (1.5), p. 8), quantified over **all** pairs `2 ≤ w₁ < w`, which follows from Mertens applied to `∏(1 − 2/p)`. 5. `0 ≤ ω(p) < p`. **The precise statement of "two classes per prime and nothing else" is: the distance 2 between the two classes enters at exactly one place, item 2, and only to certify that there are two of them rather than one.** Nowhere in §2, §3 or the assembly does the value 2 appear again; the density product `V(z) = (1/2)∏_{2 1/2 except κ = 1 (Ford 2023, quoted at source), and a κ = 2 extremal example is recorded ABSENT in the owning convention. Corrected 2026-08-27; earlier drafts read it as a floor. [`paper/wall-note.md` §2 Face 4; `research/SEARCH-CONVENTIONS.md` §3.] - **The constants are inexplicit**, as in Iwaniec's own theorem, and the dependence of the `O((log log y)²/(log y)^{1/6})` implied constant on the Ω-condition constants is not explicit in anything the corpus has read. [`paper/beta2-note.md` §6 items 3, 4.] - **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, never the method, and never table 4.032 beside 4.26645. [`research/G2-STATE.md` §8.] ### 2b. Finite-level upper bounds > `G₂(19#) ≤ 210` and `G₂(23#) ≤ 420`. [PROVEN, exact-strata re-insertion, which > survives as a finite theorem after the route around it was CLOSED for infinite > regress; `research/REFUTED.md`, `history/staging/attack-beta2-03-exact-strata.md`.] Recorded because it is the only unconditional upper bound on `G₂` at a named level that is not the asymptotic one, and because it is loose by a factor of 1.40 and 2.06 against the truths 150 and 204. ### 2c. Lower bounds, each with its status | statement | direction | refereed? | |---|---|---| | `G₂(x#) ≥ g(x#)` pointwise | floor | **PROVEN, trivial, ours**; the ingredient (twin slots ⊂ holes) is one line. Absent from the literature and from both OEIS entries. VERIFIED at all 22 shared terms per `G2-STATE.md` §2 | | `G₂(x#) ≫ x log x logloglog x/loglog x` | floor | **PROVEN**, free by monotonicity from the row above plus FGKMT (JAMS 31, 2018) via Rankin 1938 and Pintz 1997. The ingredients are refereed; the transfer is one line and is ours. Per the audit, the first lower bound of any kind for a two-class Jacobsthal function | | `G₂(x#) ≫ x·ln x`, effective constant | floor | **PROVEN from published ingredients consumed as theorems** (K–K Corollary 1, Mertens, PNT, the CRT identity). Composition ours, adversary-confirmed at every ingredient's page image, **NOT REFEREED** | | `G₂(P(y)) ≫ y(ln y)³(lnlnln y)²/(lnln y)⁴` for `y ≥ 10^{134.1}` | floor | **DERIVED HERE, NOT REFEREED**, and on weaker provenance than the row above: it re-derives the inside of Kalmynin–Konyagin's §2 trichotomy at `Ω_p = {a_p, a_p − 2}`, where Case 2 is empty and `\|Ω^III_p\| = 2` (verified to 10⁶). The referee gap is named: Halberstam–Richert Thm 2.2 and the Selberg remainder's ξ-versus-z condition at κ = 4 | | `G₂(5003#) ≥ 479,340` | floor, one level | **CERTIFIED**, explicit `(a_p)` replayed by an independent routine, zero uncovered. Seventeen levels; `two-class-lower-bounds.md` §5 | The three asymptotic floors are nested and the corpus states the nesting correctly: FGKMT's is free, `x ln x` sits above it, and the K–K substitution sits about `ln²x` above that. All three are **below** the measured ladder by a wide margin and none of them constrains the upper side. ### 2d. The elementary reformulations, each PROVEN - **The CRT collapse.** `G₂(x#) − 1` equals the maximum length of an interval `[1, m]` coverable by choosing, for each prime `p ≤ x`, the residue pair `{a_p, a_p − 2}` mod p. [PROVEN, elementary, `two-class-lower-bounds.md` §1.] **Not new to the repo**: `research/attack2-rankin2d.js` already records it under the name PAIRED. - **The Gap Reformulation.** `G₂(x#) < x′² − 2 ⟹` Zone Postulate at x `⟹` (i.o.) TPC. [PROVEN, `ZONE-POSTULATE.md` §3; `G2-STATE.md` §1b.] The Zone Postulate's weak form is **equivalent** to TPC and its strong form is strictly stronger; every `G₂` bound proves the strong form, because a gap bound is uniform by construction. - **Fact A.** For x ≥ 3, no two twin slots of `T_x` are 2 apart (else s, s+2, s+4 cover every class mod 3). [PROVEN.] - **Fact B.** Folding by p deletes exactly the classes `{0, −2}` mod p, and an interval of length below `p − 2` contains at most one kill. [PROVEN, `LOCALIZED-GAP.md` §2.] MEASURED loose by 3.5× to 6.7× at a single fold. - **The Mirror-Sweep Lemma.** `σ(s) = W − 2 − s` conjugates the fold's deletion classes, so the residue-deleted maxsum family satisfies `Δ_m(x,p,a) = Δ_m(x,p,(w−a) mod p)` outside at most four specials, and the per-copy degradation curve is a palindrome. [PROVEN 2026-08-20, re-derived from scratch and verified exhaustively by an independent engine; `history/staging/attack-0c-holesweep.md` §4.] - **The Deficit Lemma and the Traverse Bound.** `maxsum_m ≥ m·m̄·(1 − O(m/D))` by averaging, and telescoping the merge lemma forces `j ≤ x₀/(9.6 ln²x₀)` folds against the `π(x)` the route needs. [PROVEN, `localized-04-maxsum.md` §7.] - **The Overshoot Budget.** Any valid chain of upper bounds ending in `G₂(x#) < x²` has total multiplicative overshoot capped at `x²/G₂(x#)`. [PROVEN as an implication, on a MEASURED input: the slack `ln(x²/G₂(x#))` reads 0.878 to 1.182 nats across x = 11..37, flat, predicted asymptote 0.598 nats; `gate-multiplies.md` §5.] - **The pricing lemma for RML.** Any `RML(α)` with `α < β₂` gives `G₂(z#) ≪ z^α ln²z`. [PROVEN except the law itself; `history/staging/rho-maximal-law.md` §2.] ### 2e. The ladder and its certificates Fourteen exact terms in custody to `x = 43`; 22 trusted terms to `x = 79` by adopting A144311 (`research/a144311-full-ladder.js`, 2026-08-20). Grades, which are not uniform: | levels | grade | |---|---| | `x = 37` | **EXHAUSTIVE maximality certificate here**: `maxsum₁ = 528` over all 217,929,355,875 gaps of `T₃₇`, from a scan-statistic engine sharing no code with either exact-ladder producer. The strongest custody fact on the ladder, because it certifies the maximum over the whole tile rather than agreeing with a published term. [`history/staging/scanstat-t37.md`] | | `x ≤ 43` | computed here, `31#` by the segmented method, `41#` and `43#` each twice on disjoint natal masks; and independently in A144311 by a branch-and-bound sharing no method. Agreement is a check on both sides, so these are recomputations, not new values | | `x ≤ 43` | additionally, the repaired greedy attains `G₂ − 1` exactly at thirteen levels, and at `x = 43` reaches the optimum 617 at the uniform R = 2048 budget (14 of 14 sealed-scope exact) | | `x = 47..53` | A144311, Alekseyev 2009. Literature grade, single effort | | `x = 59..79` | A144311 a(17)–a(22), Wang 2024. Literature grade, single effort, and the doubling chain's `Ĝ(64) = 1080` rests on a(18) alone | **No level above 43 has been reproduced here**, and the corpus says so: the greedy oracle, exact at 14 of 14 sealed-scope terms, matches the published optima at only 1 of 8 over `x = 47..79`, with slope −0.0235 ± 0.007 per level, which CLOSED it as an exact solver past `x ≈ 53` [`research/REFUTED.md`; `history/staging/greedy-oracle-validation.md`]. The corpus reads that as the greedy failing, which is the right reading given the budget model, but it leaves the eight literature terms uncorroborated inside this repository. ## 3. Measured laws **What is wrong here first: not one of the numbers below is a proof, and one of them is provably an artefact of its own estimator.** The exponent's estimator returns 1.282 when run on 58 terms of an object whose true exponent is 1, with white residuals and no drift, so a clean power-law fit at these sizes carries no evidence about an asymptotic exponent (`research/exponent-control.md` §§1–2). Every reading here is quoted with its n and its artifact, and the house rule is that any ladder exponent travels beside the control line: *58 terms, true exponent 1, measured 1.282 ± 0.008, no drift.* ### 3a. The exponent | reading | n, range | value | artifact | |---|---|---|---| | raw log-log fit, `G₂`, x-frame | 20 fitted points, x ∈ [5, 79] | 1.777 ± 0.029 nominal, **and the standard error is worthless** | `exponent-control.md` §5 | | control-corrected, `G₂` | same, bias read at matched width 20 | **1.50 ± 0.05 stat, systematic unquantified** | `exponent-control.md` §5 | | control-corrected, `h₂` | 19 terms, x ∈ [5, 73] | **1.57 ± 0.06 stat** | `exponent-control.md` §5 | | θ-frame corrected, `G₂` / `h₂` | same | 1.43 / 1.49 | `exponent-control.md` §5 | | certificate ladder | 16 levels to x = 4001 | 1.11 to 1.25, **falling with range** | `two-class-lower-bounds.md` §5 | | hard floor | — | **≥ 1, PROVEN** via `h₂ ≥ h` and exponent(h) = 1 + o(1) | `exponent-control.md` §3 | **Why the raw fit cannot be trusted, stated as the corpus states it.** The control's bias grows and then sticks: +0.262, +0.267, +0.280, +0.282, +0.283 at window widths 10, 12, 19, 21, 30, while the scatter collapses by a factor of three [MEASURED, 58 terms of A048670]. The estimator converges, and it converges to 1.28. 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. 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 `x^{0.85}` in the limit. [All `research/exponent-control.md` §§1–2.] **Exponent 2 is disfavoured by one-sidedness, not 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 true exponent lies **below** the raw 1.777. [MEASURED + INFERRED, `exponent-control.md` §3.] **The residual 0.2 between the trusted ladders (1.50) and the certificates (≈ 1.29 once the greedy's own downward bias is added back) is unexplained**, and `G2-STATE.md` §6.1 rules it not worth chasing: every reading, corrected or raw, sits below 2 and above 1. A third of the gap is priced as a certified lower-bound ladder's downward bias of about 0.09 at a 28% terminal shortfall, against a +0.04 upward from the short ladders' own prefix effect [`h2-scoping.md` §5b]. ### 3b. Per-fold growth The right budget for a per-fold multiplier `c(p) = G₂(p#)/G₂(p_prev#)` is `ln c(p) ≤ 2 ln p/p`, obtained by differentiating the partial-sum condition rather than dividing it by π(u) [PROVEN, `gate-multiplies.md` §7]. Against it: - this corpus's ladder, folds p = 7..37: `ln c/sharp` = 1.65, 0.77, 1.15, 1.48, 1.06, 1.13, 1.01, 1.35, 2.14 [MEASURED, 9 folds, `gate-multiplies.md` §7]; - Ziller–Morack's 21 exact terms, folds p = 41..73: 1.39, 0.97, 1.36, 0.75, 1.15, 1.10, 1.18, 1.00, 0.61, **mean 1.06** [MEASURED, 9 folds, same section]. > **The multiplier spends 77% to 214% of its sharp budget and is not trending > down**, on both ladders, with no exponent fit entering the reading. Read > instead against the softer `2 ln²p/p` the same folds spend 30% to 85%, which is > where the impression of a falling trend came from, and that softer budget is > REFUTED as sufficient (TODO 0b as stated is false by a factor `ln u`; > `research/REFUTED.md`). The asymptotic headroom is `α/2` and nothing else: 100% of budget at α = 2, 75% at the calibrated 1.50, 65% at the bottom of the bracket. So whether headroom exists asymptotically **is** the exponent question. [INFERRED from a PROVEN identity, `gate-multiplies.md` §7.] ### 3c. The extreme-value law, and `c` as a surface `max ≈ c·m·(θ(x) − ln m)`, tested against every exact value that exists: | c | object | terms | range | mean | cv | range | |---|---|---|---|---|---|---| | `c₁` | one class, A048670 | 46 | x ∈ [11, 229] | 0.3718 | 7.2% | [0.3359, 0.4873] | | `c₂` | free two-class, A288815 | 17 | x ∈ [11, 73] | 0.8511 | 7.3% | [0.7784, 1.0157] | | `c₂′` | `G₂` | 8 | x ∈ [11, 37] | 0.4814 | 10.2% | [0.4463, 0.5939] | [MEASURED, `two-class-lower-bounds.md` §6.] On the diagonal over all 20 exact terms to x = 79, `c₂′ = 0.4983`, cv 7.4%, trend `(ln x)^{+0.227}` [`G2-STATE.md` §6.2]. > **`c` is a surface `c(x, lnD)`, and the constancy on the diagonal is a > cancellation rather than a law** [MEASURED, and it REFUTES any universal `c`; > `maxgap-law.md` §4]. At fixed x = 29, cutting one exact period into blocks and > growing `lnD` makes `c` fall monotonically 1.083 → 0.446. At fixed `lnD ≈ 11`, > growing x makes it rise 0.577 → 0.989 over x = 23..6421. The off-diagonal > exponents 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 object; on matched statistics both spreads are ±22.5%. **The best single description of the ladder** is `G₂(x#) ≈ 0.76 x ln²x lnln x` over x = 11..79, two significant figures and not three, with `G₂(x#) = x ln^{2+o(1)}x` and the o(1) positive. The claim that two independent instruments agreed on this 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. The AICc exclusions 31.8 and 10.6 must not be quoted either. [`G2-STATE.md` §6.2; `history/staging/redteam-2026-08-18.md` §§1–2.] ### 3d. The doubling data `Ĝ` on the base-2 chain, custody plus one literature term: | s | 2 | 4 | 8 | 16 | 32 | 64 | |---|---|---|---|---|---|---| | `Ĝ(s)` | 2 | 6 | 30 | 66 | 348 | 1080 | | `Ĝ(2s)/Ĝ(s)` | 3.0000 | 5.0000 | 2.2000 | **5.2727** | 3.1034 | — | [VERIFIED at `history/staging/attack-doubling-01.md` §1, independently re-derived at `redteam-0820-night-proofs.md` §2b and again in `attack-wrongdirection-audit.md` §3.3.] The sup 5.2727 = 348/66 sits at s = 16, on the chain, so `log₂ C₂ ≥ 2.3985 > 2`: no provable doubling constant is TPC-implying through the all-s slice. `Ĝ(64) = 1080` rests on A144311 a(18), literature grade, and the next chain datum is `G₂(127#)`, out of reach. A second measured reading beside it, and it is a hardness signal rather than a result: `C₂` shows no drift while the a-priori per-step bound `K*` drifts up, `ln(K*+1)` slope 0.6881 ± 0.1328, and an a-priori `K*` bound is itself "the entering primes' two-class covering-run problem on the level-s slot sequence, the same two-class Jacobsthal problem one level up" [MEASURED + self-similarity observation, `attack-doubling-01.md` §3, quoted at `attack-wrongdirection-audit.md` §3.3]. ### 3e. Two ratio readings that are often quoted and should not travel alone - **`G₂/g` drifts slowly upward.** The 14-term reading "peaks at x = 37 and then falls twice, no trend established" was an artefact of the data ending at 43; on the full trusted ladders seven of the eight terms x ≥ 53 sit above the old 8.00 peak and the maximum moves to the last term, 8.55 at x = 79. [MEASURED, `research/external-ladders-01.js`, reported `G2-STATE.md` §2.] - **`G₂/x²` is flat on the exact terms**, slope +0.05 ± 0.11 over eleven exact terms with the instrument sharp at 11.6σ on its control [MEASURED, `TODO.md` 1d]. On the certificate ladder `Y2/x²` falls by a factor 20.1 monotonically while `Y2/(x ln²x)` rises by only 1.21, so the construction is `x·polylog` [MEASURED, `G2-STATE.md` §5a]. Those two are not in tension: the first is the object, the second is a lower certificate whose fidelity degrades with level. - **The zone margin `x′²/G₂` is flat over the whole computed ladder**, minimum 3.18 at x = 37; on `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] [MEASURED, `exponent-control.md` §6]. Flat margin is budget, never evidence (`TODO.md` §THE TARGET). ## 4. The wall on G₂, and the legal open set **What is wrong here first: the corpus's own ranked list of "what would count as progress" contains an entry that is TPC-strength and is not labelled as such.** `research/ZONE-POSTULATE.md` §8 item 4 and `research/G2-STATE.md` §9 item 4 both read "a proof that the window/`G₂` ratio is unbounded, which is weaker than route A". It is weaker than route A, and it is **not** weaker than TPC: `x′²/G₂` unbounded means `G₂(x#) ≤ ε·x′²` for infinitely many x at every ε, hence `G₂(x#) < x′² − 2` at infinitely many x, hence the Zone Postulate at infinitely many x, hence TPC by `ZONE-POSTULATE.md` §2's forward direction. That is verdict **(ii)** on the audit's scheme, and it matches Axis A verbatim (`≤ (1−ε)(log w#)² i.o. ⟹ TPC`). The item was not among the ten the wrong-direction audit tested, so nothing has flagged it. [PROVEN as an implication, one line, from statements the corpus already holds. §5 carries the file:line pair.] ### 4a. Killer 1: what every argument so far has used **The property consumed, stated exactly.** Reading `paper/beta2-note.md` §2 and §6 against the rest of the upper-bound corpus, every upper-bound argument the programme has produced consumes one or more of the following, and nothing else: | P1 | **the class count.** Each odd prime removes exactly two residue classes; `ω(p) = 2`, `ω(2) = 1`. This is the whole input to β₂: the distance 2 appears once, to certify `0 ≢ −2` so that there are two classes and not one | | P2 | **the density product** `V(z) = (1/2)∏(1 − 2/p) ≍ 1/log²z`, which is a function of P1 alone | | P3 | **per-divisor remainder control**, `\|r_d\| ≤ ω(d) ≤ 2^{ν(d)}`, again a function of P1 | | P4 | **the constant shift**: twin slots are `S₁ ∩ (S₁ − 2)`, so the two classes sit at distance exactly 2. Used by Fact A, Fact B, the Mirror-Sweep Lemma, the covering economy and the greedy | | P5 | **an L¹ count over positions**: the union bound over the period, the Möbius term count `3^{π(x)}`, the Level Ledger majorant | **What each buys, and where it stops.** - **P1–P3 alone cap at 4.26645 by construction.** The bound is class-blind: it proves the same exponent for any dimension-2 sieve with the same remainder control, whatever the two classes are, so it is really a bound on the free two-class adversary and pays that adversary's price. [PROVEN as a reading of the proof, §2a.] The corpus's own summary agrees: "any argument that uses only 'two classes per prime' is a dimension-2 sieve statement and caps at 4.26645 by construction" (`README.md` §Status). - **P4's three uses at the anchor are tested and all three return nothing.** Mirror symmetrisation gains **exactly 0** and the zero is structural, not empirical, since `σ(K_q(a)) = K_q(w−a)` is a one-line theorem; the ensemble does not localise; and the QR-immune classes at the quadratic point deliver exactly the guardrail's `r/(r−2)` and nothing past it, pre-registered with a matched control that voided a 6.5σ surplus. [`research/REFUTED.md`, rows on mirror symmetrization and the QR refinement.] - **P4 on the covering side dies at `x = 13`**, where `Σ 2/p` crosses 1, and `β_pure` diverges like `7.182 lnln x` (7.19 until the root was corrected 2026-08-29); the exact head reaches only `x₀ = O(ln x)`, so no exponent moves. [`research/REFUTED.md`; `sift-limit-attack.md` §7.] - **P5 is provably the sharp instrument, not a lossy one.** The `ℓ¹ → ℓ²√log` conversion **is** the sharp maximal law: `‖Θ·S_H‖₂ = rms(R_H)` exactly, and its true constant (0.56–0.92) sits below the TPC line (1.36–2.23). Generic chaining is worse: the entropy integral of the true metric already exceeds the union bound at every level, 1.04–1.10×, so chaining's ceiling sits below the union bound's floor. [`research/REFUTED.md`; `history/staging/import-l1l2.md`, `import-chaining.md`.] **What an argument would have to use beyond P1–P5.** Three constraints, each proven or measured somewhere in the corpus, and together they say what the shape of a winning argument is: 1. **It must survive the sieve's first discard point.** The entire distance from 4.2665 to 2 is a positivity problem and zero percent a distribution problem: distribution hypotheses (EH, GRH, bilinear inputs) enter the dimension-2 sieve at exactly one of its five discard points and the primorial formulation already saturates that point for free, so a perfect distribution oracle moves the exponent by nothing. The programme's own exact structure, the correlation function, the sub-Poisson window variance, the mod-30 rigidity, the mirror, the fusion identity, is quotiented away at the first discard point, where the sieve reduces the set to divisor counts. [Analysis of the method, marked as such at `paper/wall-note.md` §2 Face 4.] 2. **It must be positional, not ensemble.** The tile's transform vanishes nowhere and its dependency graph is complete, so no ensemble statement localises; one shared uniform residue draw defeats every local neighbourhood, which is what kills the whole concentration family and the whole local-lemma family. A measure-theoretic bound decides the anchored position only if `εW < 1`, and the corpus's best miss that by e^3025 at x = 19 in the window ensemble. [`README.md` §Status; `paper/wall-note.md` §2 Face 1; `research/REFUTED.md`.] 3. **It must not be an L¹ count over positions**, by the ℓ¹→ℓ² closure above. **And the one thing that is NOT proven, stated so it is not mis-cited.** There is no barrier theorem, in this corpus or in print, saying an axiom-only argument cannot do better inside `(2, 4.26645]`. `β(κ)` is unknown for every κ > 1/2 except κ = 1, no κ = 2 extremal example is in print, and the programme's one barrier statement (`β_interval(2) ≥ 2` on Granville's Siegel-zero hypothesis) sits **below** the band, at the point where it begins. [`paper/wall-note.md` §2 Face 4, corrected 2026-08-27; `history/staging/attack-barrier-kappa2.md`.] So killer 1 is a description of every attempt so far, not an impossibility result. ### 4b. The legal open set Statements about `G₂` that are (a) not proven, (b) not TPC-strength on the wrong-direction audit's axes, and (c) not in `research/REFUTED.md`. Verdicts use the audit's scheme: **(i)** strictly weaker than TPC, legitimate stepping stone; **(ii)** TPC-strength; **(iii)** undetermined. Only (i) and (iii) belong in a legal open set; the (ii) rows are listed at the end so they are not mistaken for members. **(i) Legal.** | # | statement | argument, one line | status | |---|---|---|---| | L1 | any unconditional `G₂(x#) ≪_ε x^{α+ε}` with `α ∈ (2, 4.26645)` | separation is exhibited rather than modelled: `α = 4.26645` is PROVEN while TPC is open, and both bridges transition at 2 and nowhere else | audit target 10, verdict (i); `TODO.md` item 0 | | L2 | `RML(α)`: `sup_y \|ρ̃_z(y)\| ≤ C·z^α` uniformly in z, any `α < β₂` | the pricing lemma is PROVEN except the law, and its win condition is `α < β₂`, not `α < 2`; the `ρ̃` form forfeits the lag-H cancellation, which is exactly why it is legal | `rho-maximal-law.md` §§1–2; audit §3.6(b) | | L3 | `Ĝ(2s) ≤ C₂·Ĝ(s)` **for all s**, any `C₂ < 19.2455` | custody forces `C₂ ≥ 5.2727`, so `log₂C₂ ≥ 2.3985 > 2` and the slice's own conclusion can never reach limsup < 2 | audit §3.3, verdict (i) as written | | L4 | (H-sub-pow) with an **unnamed** K, giving existence of `β = lim ln G₂/ln x` | `f(n) = 2 ln n` satisfies it at K = 0 and yields β = 2, on which TPC is undecided | audit §3.5; `TODO.md` 1d | | L5 | (H-sub-pow) with an **explicit** `K` in the legal zone | `K ∈ [1.3946, 11.3568)` trusted gives `β < β₂` with no TPC content; the floor sits at base 66 and the ceiling at base 82, and the single-base pairing `(ln 66 + K)/ln 16` that produced the superseded `[1.3555, 7.6394)` must not travel with the corrected zone (`hsubpow-explicit-K.md` §2; corrected per redteam-0829-objects-gb.md §2 E1). **The explicit-K route is CLOSED for three named mechanisms** (`hsubpow-explicit-K.md`); the statement stays legal and unproven by a fourth | `TODO.md` 1d | | L6 | **any** improvement of the lower bounds: refereeing the K–K substitution, running Maier–Pomerance's ledger at κ = 2 to a proof, extending the certificate ladder | a lower bound on `G₂` is compatible with TPC being false, so no statement in this family can be TPC-strength. This is the one wholly safe direction on the object | `two-class-lower-bounds.md` §§4b, 4c, 5 | | L7 | `G₂(x#) ≪ g(x#)·(ln x)^A` for some fixed A ≥ 0 | with Iwaniec/Vaughan's `g(x#) ≪ x²` (inexplicit constant) this would give `G₂ ≪ x²ln^A x`, above 2 and hence carrying no TPC content by Axis A, while being **below 4.26645** and therefore an improvement of the proven exponent. Three independent routes in the corpus say the second class is worth exactly one logarithm (`G2-STATE.md` §3d), which is the conjectural `A = 1` | **not asked**: no `Q-` id in `research/QUESTIONS.md` poses a `G₂`-to-`g` transfer, and `REFUTED.md` closes no such row | | L8 | the exact value `G₂(47#)`, or an exhaustive maximality certificate at any level 41 ≤ x ≤ 79 | a finite computation, decidable by enumeration, hence not an infinitude statement | `TODO.md` item 1 prices the arithmetic ceiling at `W = 1.308e16 > 2^53` | | L9 | whether `G₂ ≤ h₂` is strict from some level on, and the law of `h₂/G₂` | a comparison of two finite ladders; measured 1.63–1.81 at x = 47..73 with a low outlier 1.341 at x = 37 | `G2-STATE.md` §2 | | L10 | whether the corpus's `h₂` is the offset-max object or the free-per-prime object (defect (ii) of §1) | a definitional reading of two published definitions; no theorem depends on the answer, only the interpretation of the 1.57 control | **not asked**: no `Q-` id in `QUESTIONS.md` | **(iii) Undetermined.** | # | statement | why it is undetermined | |---|---|---| | U1 | whether `sup/sd` stays bounded in the two-class discrepancy channel `ΔΦ₂` | the corpus holds the ×2 and ×3 Möbius-count ceilings and the measured ×1.8356 per fold, but **no bridge from `ΔΦ₂` to `G₂` exists in the corpus**, so whether a bound there is TPC-strength cannot be read off. `G2-STATE.md` §9 item 7 poses it without the label | | U2 | whether `G₂(x#)/x²` **falls** (as against tends to 0) | falling monotonically is legal and decidable-looking; falling **to zero** is verdict (ii) by Axis A. `TODO.md` 1d and `Q-g2-falls-decision-rule` (PARTIAL) do not separate the two, and the measured slope is +0.05 ± 0.11 over eleven exact terms, consistent with flat | | U3 | whether the diagonal cancellation that makes `c` look constant is exact or a coincidence of the reachable range | if `c` is bounded above on the diagonal then `G₂ ≪ x ln²x`, exponent 1, which is far stronger than TPC, so the *upper* half of this is (ii) and the *lower* half and the mechanism are legal. `maxgap-law.md` §4 states the surface, not the bound | **(ii) TPC-strength, listed so they are not mistaken for legal targets.** - `x′²/G₂` unbounded, or `G₂ = o(x²)`, or `G₂ ≤ (1−ε)x′²` i.o., **including `ZONE-POSTULATE.md` §8 item 4 and `G2-STATE.md` §9 item 4 as written**, which currently carry no label. - the extreme-value law read as an **upper bound**: `G₂ ≤ C·m₂·(θ(x) − ln m₂)` gives exponent 1. - `Ĝ(2s) ≤ C₂Ĝ(s)` in the **eventual** form with `C₂ ∈ (3.1034, 4)`: the trap window, 0.254 nats, and no computation will ever close it since the next chain datum is `G₂(127#)` (audit §3.3). - (H-sub-pow) with explicit `K < 1.3946` trusted / `< 1.3555` custody (`TODO.md` 1d; note the audit §3.5 states the superseded numbers, §5 below). - the sharp Gaussian maximal law on `R_H` at the operative window (`phase1-T4-maximal-law.md` §2), which is additionally already false as literally stated, violated at z = 19 by 0.6%. ## 5. Inconsistencies found Seven items. Expected output: the corpus runs to roughly 420 notes and drift is normal. **None of them moves a proven statement**, and that is the first thing to say; five are staleness or label collapse inside a file that also carries the corrected number, one is a missing label on a live target, and one is a genuine definitional collision. The ranking is by what a reader would get wrong, not by size. **(1) A live target is TPC-strength and carries no label.** [Most consequential.] - `research/ZONE-POSTULATE.md:526`: "A proof that the window/G₂ ratio is unbounded, which is weaker than route A but would show the Gap Reformulation is not asymptotically self-defeating." - `research/G2-STATE.md:1212`: "**4. Prove that window/G2 is unbounded.** `ZONE-POSTULATE.md` §8 item 4. Strictly weaker than route A". Against `history/staging/attack-wrongdirection-audit.md:47` (Axis A, verbatim): "`G₂(w#) = o((log w#)²)` or `≤ (1−ε)(log w#)² i.o.` ⟹ TPC". `x′²/G₂` unbounded is exactly `G₂ ≤ ε·x′²` i.o., so the item implies TPC. Both files describe it as a modest sub-goal. The audit tested ten targets and this was not among them, so nothing has caught it. [PROVEN as an implication; §4 carries the one-line argument. Fix is a label, not a deletion: the item is still worth wanting, it is simply not cheap.] **(2) `h₂` is defined two incompatible ways in the live layer.** Detailed at §1 defect (ii). - `research/GLOSSARY.md:278`: "The adversarial version, where each prime may choose its class pair freely, is Ziller and Morack's h₂, OEIS A288815." Same reading at `research/U-FRAME.md:579-580` ("the ADVERSARIAL paired Jacobsthal, the maximum over ALL choices of two residues per prime") and `research/G2-STATE.md` §1a. - `research/two-class-lower-bounds.md:80`: "`h2(x#)` | Ziller-Morack paired Jacobsthal, **worst over even offsets**, A288815", with **free 2-class** listed on the next row (`:83`) as A072753, dominating `h₂`. - `research/covering-dive.md:56` reads ZM's Def. 2.2 from the PDF and gives the offset reading: `j₂(n)` quantifies over paired progressions `⟨a,b⟩_m` with `2 | (b−a)`, so the offset is one global even number, not a per-prime choice. OEIS states `A288815 = 6·A072753 + 6`, an exact renormalisation, so the corpus simultaneously says the two rows are one sequence and that one dominates the other. No downstream number moves; what is unresolved is what the 1.57 control exponent is an exponent of. [OPEN, definitional.] **(3) The certificate ladder's top rung is stale in three places inside `G2-STATE.md`, against `G2-STATE.md`'s own §5a.** - stale: `research/G2-STATE.md:421` "lower bound, best constructed | **356,712 at x = 4001**"; `:1026` "certified greedy ladder | ... | x ≤ 4001, **16 levels**"; `:1148` "the certified G2 lower-bound ladder to **x = 4001, sixteen levels**". - current, same file: `research/G2-STATE.md:919` "| 5003 | **479,339** | 0.01915 | 1.3206 | — (new rung) |". - source: `research/two-class-lower-bounds.md:975` "lower, best constructed here | `G2(5003#) >= 479,340`"; `:1100` "a certified `G2` lower-bound ladder to `x = 5003`, **seventeen levels**". [Staleness, not contradiction: the §3a row and the §8 row were not updated when §5a gained the rung. The 356,712/356,711 pair is not an error, one is the covering length, the other `+1` is the `G₂` floor it certifies.] **(4) The wrong-direction audit's 1d landing zone is superseded by TODO's.** - `history/staging/attack-wrongdirection-audit.md:82` and `:250`: legal zone `K ∈ [1.3555, 7.6394)`, trap floor 0.9694 to ceiling 1.3555, "0.386-nat sliver ... against a 6.28-nat legal band". - `TODO.md:366`: "the trusted legal zone is `K ∈ [1.3946, 11.3568)`, **not** the `[1.3555, 7.6394)` this item used to state" (CORRECTED 2026-08-28, `hsubpow-explicit-K.md`). The audit is a HELD staging note dated 2026-08-26 and TODO records the supersession explicitly, so the live layer is right; the note is not marked. The custody-versus-trusted split behind the two ceilings is set out at `history/staging/fekete-1d.md:114–116` and is not itself a defect: 1.3946 rests on Wang 2024's a(18) and 1.3555 on `G₂(13#) = 66`. [Staleness in a HELD note.] **(5) Three different shared-term counts for the same two pointwise relations.** - `research/G2-STATE.md:365`: "`G2 ≥ g(x#) = h(x#)` (PROVEN, VERIFIED at all **22** shared terms)"; `:375`: "`G2 ≤ h2(x#)` (VERIFIED at all **21** shared terms, to x = 73)". - `research/two-class-lower-bounds.md:84`: "**VERIFIED** at all **eleven** shared terms" for `G2 ≥ g`, with the eleven ratios printed. - `research/exponent-control.md:151`: "**G2 ≤ h2 pointwise**, verified at all **12** shared terms". The lower counts predate the 2026-08-20 adoption of A144311's 22 trusted terms and both are in live files. Nothing is wrong at any count, since a relation verified at eleven terms is verified at eleven terms; the defect is that a reader comparing the two live files sees two different evidence bases for one relation. [Staleness in the live layer.] **(6) `paper/beta2-note.md` is one ladder term behind the object.** Its §1 lists the exact terms "for `pₙ = 2, …, 41`" (thirteen) and its §5 sanity check is "at the largest computed level (`pₙ = 41`)", while `G2-STATE.md` §2's custody ladder runs to `G₂(43#) = 618`, fourteen terms. The note's §5 already carries one in-place correction for the 37 → 41 move and its numerical conclusion is unaffected (the slack widens further at 43). [Staleness in a paper draft under moratorium.] **(7) Two conjectural truths for `G₂`, in one file and propagated into `G2-STATE.md`.** - `research/two-class-lower-bounds.md:979`: "lower, conjectural truth | `x (log x)^{3+o(1)}` | CONJ". - `research/two-class-lower-bounds.md:401` (the §4c ledger, immediately after the table whose two-class total reads "**4 (CONJ); 3 unconditional**"): "the conjectural two-class ceiling is **`G₂(x#) = x (log x)^{4+o(1)}` (CONJ)**, one log above the unconditional floor". - propagated: `research/G2-STATE.md:423` "lower bound, conjectural truth | `x(log x)^{3+o(1)}`" and `:434` "conjectural shape | `G2(x#) = x(log x)^{3+o(1)}`". The reconciliation is that 3 is the ledger's unconditional total and 4 is the total with Maier–Pomerance's conjectural multi-kill row, so the rows labelled "conjectural truth = 3" have collapsed the two. Both are `x^{1+o(1)}` and nothing downstream turns on it, but a reader taking `G2-STATE.md` §3a at face value gets the unconditional number under a conjectural label. [Staleness / label collapse.] **Where the reading looked and found nothing.** The exponent readings 1.50 / 1.57 / 1.777 / 1.11–1.25 are quoted identically in `README.md` §Status, `research/G2-STATE.md` §§0, 3a, 6.1, `research/exponent-control.md` §5, `research/GLOSSARY.md` §G₂, `research/THE-DIALS.md` and `paper/beta2-note.md` §5, with the same n and the same control line in each. β₂'s twenty digits agree between `paper/beta2-note.md` §2 and `research/G2-STATE.md` §3a. The ladder's fourteen exact terms agree between `G2-STATE.md` §2, `GLOSSARY.md` §G₂ and `paper/beta2-note.md` §1 term for term. The doubling ratios agree across `attack-doubling-01.md` §1, `redteam-0820-night-proofs.md` §2b, `TODO.md` D and `attack-wrongdirection-audit.md` §3.3. `REFUTED.md` holds exactly 67 rows and `QUESTIONS.md` exactly 392 distinct `Q-` ids, matching `README.md` §Status and the brief. **No document-against-artifact check was run**: this note did not re-execute a producer to confirm that a quoted number matches its embedded OUTPUT block, so every item above is document-against-document only. ## 6. Cheap unknowns **What is wrong here first: none of these was run.** The brief permitted running only a measurement decisive for §4 or §5 and costing under about twenty minutes; the one candidate that is decisive for §4 (C1) needs a matched control to avoid a confound this corpus has already been bitten by once (the `M_p` field's raw twin contrast 0.155-vs-0.764 was confounded until non-twin pairs matched on predicted `M` were built, `G2-STATE.md` §0), and building that control is not a twenty-minute job. So this section is a specification, not a result, and nothing below carries a number that did not already exist. Ranked by what the answer would say about the wall. Each row names the grep that shows it has not run, and its verdict on the wrong-direction audit's scheme. **C1. Is `G₂` generic among two-class configurations at matched level?** [verdict **(i)**: a percentile carries no infinitude content.] The measurement §4a asks for. Killer 1 says every upper-bound argument discards the class *positions* and keeps only the class *count*. Nothing in the corpus measures what those positions are worth. Specification, with the confound named: by the CRT collapse `G₂(x#) − 1` is already a **maximum over the shift ensemble** `(a_p)`, so comparing it to a single random configuration is not a comparison at all. The matched control draws an offset vector `(d_p)` with each `d_p` even and `d_p ≠ 2`, takes the same maximum over shifts using classes `{a_p, a_p − d_p}`, and places `G₂ − 1` in the resulting distribution. The greedy oracle is the right instrument for the maximisation, exact at 14 of 14 sealed-scope terms to `x = 43` (`greedy-oracle-validation.md`), and its budget schedule is already tuned. - **What it decides.** If `G₂` sits at a typical percentile, the constant shift is worth nothing even to an adversary, and §4a's "what an argument must use beyond P1" has no target in the class positions, which would strengthen killer 1 from a description of the attempts into a property of the object. If `G₂` sits low in the tail, there is a coordinate and the corpus has never looked at it. - **Grep showing it is unasked.** `research/QUESTIONS.md` has zero hits for "random two-class", "free two-class", "class configuration". The nearest is `Q-discrepancy-two-class` (ANSWERED), which ran a random-class control on the **discrepancy** `ΔΦ₂` and refuted any twin-specific discrepancy law; it did not run one on the maximum gap. `Q-c2prime-drift` (PARTIAL) reports the `c₂′` drift as "class-count-blind", which is the closest thing on record and is a different statistic. - **Cost.** Minutes per draw per level with the existing oracle to `x ≤ 31`; a few hundred draws fits well inside four hours. Nothing needs the tile in memory. - **Not run**, and the reason is the confound above: a version without the matched offset control would reproduce the `M_p` mistake. **C2. Where does the maximum gap sit inside the tile, and how many positions attain it?** [verdict **(i)**: a position law at named levels is decidable by enumeration.] Killer 2 is a quantifier: worst position against almost all. The corpus has never measured whether the worst position is structurally distinguished. Two data points exist and they are isolated: `G₂(43#) = 618` is "at least position 830,330,079,152,051 and attained at **8 positions** of the period" (`Q-g2-43-term`, ANSWERED), and the Mirror-Sweep Lemma forces the gap multiset to be mirror-symmetric under `σ(s) = W − 2 − s`, so attaining positions come in pairs outside the fixed point. Specification: at every exact level, record the full argmax set, its multiplicity, its positions as fractions of `W`, and whether the multiplicity is exactly twice the number of mirror orbits. - **What it decides.** If the record positions are generic modulo the forced mirror pairing, the almost-all/worst-position gap is as wide as the corpus assumes and no positional argument has a target. If they cluster (near the origin, near `W/2`, near the mirror fixed point), a positional argument has one, and the mirror pairing is a free control that says whether any observed structure is more than the lemma already guarantees. - **Grep showing it is unasked.** Zero hits in `QUESTIONS.md` for "argmax", "position of the maximum", "where the maximum". `Q-gap-spectrum` (ANSWERED) covers the gap-length **distribution** to `x = 31` and the excess functional, not positions. `Q-records-placement` (ANSWERED) is about twin **prime** gap records above 2^53 landing inside stretches, a different object. - **Cost.** `research/maxgap-law.js --big` already walks exact periods to `29#` in 147 s in O(1) memory; recording the argmax set is a small change to an existing walk. Under twenty minutes to `x = 29`. This is the cheapest of the three. **C3. Does the free-per-prime two-class adversary beat the offset-max one?** [verdict **(i)**: a finite comparison of two ladders.] Settles §1 defect (ii) and §5 item (2), and prices `P4` from the adversary's side rather than ours. Specification: exhaustive or branch-and-bound maximum gap over **all** per-prime class pairs `{a_p, b_p}` at `x ≤ 17`, compared against A288815. At `x = 13` the configuration count is `3·10·21·55·78 ≈ 2.7·10⁶`, exhaustible; `x = 17` multiplies by 136 and needs the B&B the corpus already has for `advmin@11`. - **What it decides.** Whether ZM's `h₂` is the free adversary or a strictly weaker offset-max one, hence what the 1.57 control exponent is an exponent of; and whether the distance-2 constraint costs an adversary anything at all. - **Grep showing it is unasked.** Zero hits for "free two-class", "free-choice", "class configuration". `Q-advmin-1113` (CLOSED) is the free-class adversary against the **anchored Natal@5 comb**, a different object and a different functional (survivor count, not maximum gap). - **Cost.** Minutes at `x = 13`; hours at `x = 17` with the B&B. A page read of A072753's OEIS definition line beside ZM Def. 2.2 may settle it for free and should be tried first. **C4. Does the 1.50 exponent survive dropping the literature-grade terms?** [verdict **(i)**: a refit on a prefix.] The headline is measured on 22 trusted terms of which fourteen are custody-grade, six are Alekseyev 2009 and, decisively for the doubling chain, `a(17)–a(22)` are Wang 2024 single-effort. `Q-greedy-oracle` (ANSWERED) records that the corpus's own oracle matches the published optima at only 1 of 8 over `x = 47..79`. Specification: refit on custody-only (14 terms), custody+Alekseyev (16), and full (22), each against the control's bias at its own matched width, and report the three centrals. - **What it decides.** Whether the repository's most-quoted number about this object depends on eight terms it has never reproduced. It also prices `Ĝ(64)`, on which item D's eventual-form trap window rests. - **Grep showing it is unasked.** `Q-exponent-control` (PARTIAL) reports the 22-term reading and the bias; `Q-external-data-audit` (ANSWERED) adopts the external series and reports what adopting them changed; neither poses the provenance-block split. Zero hits for "custody-only refit" or an equivalent. - **Cost.** Seconds. `research/exponent-control.js` already carries the sliding-window control and the refit machinery. **C5. Not a cheap unknown, recorded so it is not mistaken for one.** `G₂(47#)` is priced by direct probe at **1.49 days per run** and stays out of session reach (`Q-g2-43-term`, ANSWERED). `L8` in §4b is legal but not cheap. ## 7. The conjectural true size of `G₂(x#)` **What is wrong here first: the corpus quotes two different conjectural truths for this object, `x(log x)^{3+o(1)}` and `x(log x)^{4+o(1)}`, in the same file.** The reconciliation is that 3 is the ledger's *unconditional* total and 4 is its total *with* Maier–Pomerance's conjectural multi-kill row; the rows that say "conjectural truth = 3+o(1)" have collapsed the two. §5 item (7) carries the file:line pair. Nothing in the shape below turns on which of the two is meant, because both are `x^{1+o(1)}`. ### 7a. What the one-class literature conjectures, at its own rungs - **Jacobsthal's order conjecture, `H(r) ≪ r²`:** OPEN, and it is about `H(r) = max_{ω(n)=r} j(n)`, not about the primorial function. Erdős problem #970. His *second* conjecture, that the maximum is attained at the primorial, is **PROVEN FALSE at r = 24** (Hajdu–Saradha, Math. Comp. 81 (2012)), so `H(r) = g(p_r#)` is a theorem with a counterexample and not a definition. [`covering-dive.md` §1.1, read at source.] - **Proven upper for one class:** `g(x#) ≪ x²`, inexplicit constant, Iwaniec 1971 Thm 2 / 1978 at primorials and Vaughan 1977 for general n. The elementary line is far weaker than exponent 2: Kanold `2^{√k}`, Stevens `k^{Θ(log k)}`, Paseman `k^{O(loglog k)}`. [PROVEN; `covering-dive.md` §1.1.] - **Proven lower for one class:** `g(x#) ≫ x log x logloglog x/loglog x`, Ford–Green–Konyagin–Maynard–Tao (JAMS 31, 2018) via Rankin 1938 and Pintz 1997. [PROVEN.] - **Conjectured truth for one class:** `g(x#) = x(log x)^{2+o(1)}`, i.e. exponent `1 + o(1)`, from Maier–Pomerance, TAMS 322 (1990) 201–237, p. 205, quoted verbatim in `two-class-lower-bounds.md` §2c. Their exponent decomposes as **1 + 1**: one log from the classical one-element-per-large-prime step, one more from the conjectural `(log x)^{1+o(1)}` elements per large prime, which they explicitly do not prove. A048670's own OEIS comment draws the same conclusion. Ford's 2018 Montreal slides record that even a uniform Hardy–Littlewood conjecture buys only `J(T) ≫ T(log T)^{1+c}`, not exponent 2. [CONJ.] ### 7b. What the two-class analogue reads The corpus's ledger, run for the first time in dimension 2 with Maier–Pomerance's own three-entry accounting (`two-class-lower-bounds.md` §4c; the qualifier "*their* ledger" is load-bearing, because Kalmynin–Konyagin run a dimension-2 Mertens ledger of their own on a different family, theirs has fixed centre and varying separation, ours varying centre and separation fixed at 2): | source of a `log` | one class | two classes | |---|---|---| | Erdős–Rankin base, one survivor per large prime | 1 | 1 | | survivor density after the small primes, `1/(log z)^κ` | 0 | **+1** | | the band-2 device, `M(f) = 2`, unconditional | 0 | **+1** | | Maier–Pomerance multi-kill, `(log x)^{1+o(1)}` per large prime | +1 (CONJ) | +1 (CONJ) | | **total exponent of `log x` above `x`** | **2 (CONJ)** | **4 (CONJ); 3 unconditional** | So the two-class analogue reads `G₂(x#) = x(log x)^{3 or 4 + o(1)}`, and in both readings the second class is worth **one logarithm, not a power of x**, a finding three independent routes in the corpus agree on (the Rankin accounting, the certificate ladder's `(Y2/Y1)/ln x` plateau at 2.18, and the Poisson law's `m₂/m₁ ~ 1.35 ln x`; `G2-STATE.md` §3d). [INFERRED, and the ledger's own unconditional rows are PROVEN or DERIVED-HERE; the multi-kill row is CONJ.] **Not in print.** No published statement about A144311's asymptotic growth was found, searched in the owning convention (`Q-a144311-vocabulary`, ANSWERED); no lower bound of any kind is attached to A288815 or A072753; and Ziller–Morack's Conjecture 6, `h₂(n) < p_n² − p_n`, is the only quantitative statement in the direct literature and is an upper bound with nothing proven about it. [ABSENT in the owning convention; `two-class-lower-bounds.md` §2 rows 7, 8; `covering-dive.md` §2.1.] Do not confuse any of this with Kourbatov's maximal twin-gap law: that is `0.76 log³p` for gaps between actual twin **primes**, a different object under a different owning convention (`SEARCH-CONVENTIONS.md` "maximal gaps between prime k-tuples, fitted against `log^{k+1}p`"), and it is CONJ. ### 7c. What the corpus's own measurements say against that - **The measured law is BELOW the conjectured one on everything computable.** `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 [MEASURED, `G2-STATE.md` §6.2]. Against `x ln³x` or `x ln⁴x`, the o(1) is strongly negative across the whole reachable range, which is exactly what the one-class control does too: over the whole available ladder `h(p#)` tracks `p·log p`, not `p·log²p`, and the frozen Maier–Pomerance shape is **refuted on this range** at rms 0.281 and maximum log residual 1.239, a factor of 3.5 [MEASURED, `exponent-control.md` §2]. - **The missing log is visible at a tenth of the required rate, and that is a measurement rather than a resolution limit.** `c₁ ~ (log p)^{0.12 ± 0.02}` on the top 47 exact one-class 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 [MEASURED, `maxgap-law.md` §6]. - **The two candidate shapes are not separable here in principle.** `x ln²x lnln x` and `x ln³x` cannot be told apart on this data, and attack E's §7 says so directly; the AICc figures 31.8 and 10.6 must not be quoted as exclusions [`G2-STATE.md` §6.2; `redteam-2026-08-18.md` §§1–2]. On the two-class side the same non-separability appears as `c·p^{1.847}` and `c·p·log^{2.448}p` differing by 0.1 AIC units while differing by `x^{0.85}` in the limit [`exponent-control.md` §2]. - **So the measured 1.50 is not evidence against exponent 1 plus logs.** The estimator returns 1.282 on 58 terms of an object whose exponent is 1, and the bias is upward in all 40 windows because a positive power of log inside the truth biases a finite-range power fit up. An object with three or four logs would be biased further up still. [MEASURED + INFERRED, `exponent-control.md` §§1, 3.] ### 7d. The shape, stated > **Conjectured `x·(log x)^{3 or 4 + o(1)}`, i.e. exponent 1 plus logs; > PROVEN upper `x^{4.26645+ε}`; NEEDED `x^{2}` with constant below 1; MEASURED > exponent 1.50 ± 0.05 stat on a demonstrably upward-biased estimator.** That is the shape the reading supports, with three qualifications that must travel with it. 1. **The proven lower bounds are nowhere near the conjecture either.** The free FGKMT floor is `x log x logloglog x/loglog x`, one log below the one-class conjecture and two or three below the two-class one. So "1-plus-logs" is conjectured at both ends and proven at neither: the proven bracket on `G₂` is roughly `x log x` to `x^{4.267}`, four orders of exponent wide. 2. **The gap that matters is 4.26645 against 2, and it is a proof gap rather than a truth gap**, provided the conjecture is right, which nothing here establishes. `research/sift-limit-attack.md` §2 finds nothing published that blocks the band and no κ = 2 extremal example in print. [`G2-STATE.md` §9 item 1.] 3. **"Consistent with" is doing real work in 7c and is not "implies".** Every measurement above is compatible with exponent 1 plus logs; none of them distinguishes it from exponent 1.2, and the estimator provably cannot at these sizes. What would distinguish them is not more ladder: 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 [MEASURED on 48 control cases, `G2-STATE.md` §9 item 6]. ## 8. Questions about `G₂` not in `QUESTIONS.md` **What is wrong here first: none of these is a route, and the base rate says each is probably worth less than it looks.** They are understanding questions about what the object is, posed because §4 could not answer them from the corpus. Each was checked against all 392 `Q-` ids of `research/QUESTIONS.md` and against all 67 rows of `research/REFUTED.md`; the grep that shows the absence is quoted. Ranked by what an answer would do for §4, not by cost. Verdicts use the wrong-direction audit's scheme. **Q1. Is `G₂` generic in the coordinate the sieve discards?** [(i)] Every upper-bound argument keeps the class count and discards the class positions (§4a, P1 against P4). Nobody has measured what the positions are worth for the maximum gap. The question is not whether they can be exploited but whether there is anything there: where does `G₂ − 1` sit in the distribution of the same maximisation run on offset vectors `(d_p)` with `d_p ≠ 2`? A typical percentile would say killer 1 describes the object and not just the attempts on it; a low percentile would say there is a coordinate nobody has looked at. Specification and the matched control it needs are §6 C1. *Grep:* zero hits in `QUESTIONS.md` for "random two-class", "free two-class", "class configuration". `Q-discrepancy-two-class` (ANSWERED) ran the random-class control on `ΔΦ₂` and refuted any twin-specific **discrepancy** law; it did not touch the maximum gap. Nothing in `REFUTED.md` closes it. **Q2. Where does the maximum sit inside the tile, and with what multiplicity?** [(i)] Killer 2 is a quantifier, and the corpus has never measured whether the worst position is distinguished. Two isolated facts exist: `G₂(43#) = 618` is attained at **8 positions** and one of them is recorded (`Q-g2-43-term`), and the Mirror-Sweep Lemma forces the gap multiset to be symmetric under `σ(s) = W − 2 − s`, so the multiplicity should be even outside the fixed point. Whether the positions are otherwise generic, and whether the multiplicity grows, is unknown. The multiplicity is the union bound's own coordinate: a maximum attained once over `W` positions is where an L¹ count over positions is most lossy, and §4a records that the ℓ¹→ℓ² conversion **is** the sharp maximal law, so how lossy it is at the extreme is a live descriptive question. *Grep:* zero hits for "argmax", "position of the maximum", "where the maximum". `Q-gap-spectrum` (ANSWERED) covers the gap-length distribution to `x = 31` and the excess functional `|B_N|`, not positions. `Q-records-placement` (ANSWERED) is twin **prime** gap records above 2^53 inside stretches, a different object. **Q3. Is there any structural reason `G₂/g` should stay polylog?** [(i)] `G₂ ≥ g` is PROVEN and one line; the reverse comparison has never been posed. Three independent routes say the second class is worth one logarithm (`G2-STATE.md` §3d), and the measured ratio drifts slowly up to 8.55 at `x = 79` with the maximum at the last term. What is absent is any argument, even heuristic, that bounds the ratio: `G₂ ≪ g·(ln x)^A` for a fixed A would give `G₂ ≪ x²ln^A x` from Iwaniec/Vaughan, which is **above** exponent 2 and so carries no TPC content, while sitting **below** 4.26645. That the corpus holds the trivial direction of a comparison and has never asked about the other is the gap; whether the reverse is provable is a route and is not asked here. *Grep:* zero hits for "one-class to two-class", "transfer from g"; no `Q-` id poses a `G₂`-to-`g` upper transfer; `REFUTED.md` closes no such row. [This is L7 of §4b, stated as an understanding question.] **Q4. Why do three independent instruments single out `x = 37`?** [(i)] `G2-STATE.md` §2 records that `x = 37` spikes `G₂/h` (8.00, the old peak), spikes `c₂′` (0.5939, the top of the eight-term range), and is the low outlier of the `h₂/G₂` column at 1.341 against [1.63, 1.81] at `x = 47..73`: "three instruments now point at `x = 37` as a `G₂`-side anomaly", and then does nothing with it. If the coincidence is arithmetic it is the only named structural irregularity on the ladder; if it is one draw seen three ways, saying so retires a standing observation. Note that the three readings are not independent: all three are ratios with `G₂(37#) = 528` in them, so a single high `G₂` at 37 produces all three signs at once, and the honest first move is to check whether any instrument **without** `G₂(37#)` in it also flags 37. *Grep:* `Q-delta37` (ANSWERED) asks what sources the `M_p` law's depth term and whether `x = 37` falls out under the Hardy–Littlewood lens, a different object (the extinction law's fold field). `Q-c2prime-drift` (PARTIAL) reports the `c₂′` drift as class-count-blind and does not treat 37 as a level. `Q-frontier37` (ANSWERED) is the fold-L instruments at fold 37. None poses the coincidence. **Q5. What is the `h₂` the corpus is controlling against?** [(i)] §1 defect (ii) and §5 item (2): `h₂` is defined as offset-max in one set of live files and as free-per-prime in another, and A288815 = 6·A072753 + 6 is asserted as an identity between two rows the corpus also separates by an inequality. The 1.57 control exponent is quoted in `README.md` §Status, `G2-STATE.md` §§0, 6.1 and `paper/beta2-note.md` §5, and nothing says which object it is an exponent of. *Grep:* zero hits for "free two-class", "free-choice". `Q-a144311-vocabulary` (ANSWERED) settles the *vocabulary* question for `G₂` and the sifting limit, not this one. **Q6. Does the headline exponent depend on terms this repository has never reproduced?** [(i)] Fourteen of the 22 trusted terms are custody-grade; six are Alekseyev 2009 and six are Wang 2024, and the corpus's own greedy oracle matches the published optima at only 1 of 8 over `x = 47..79` (`Q-greedy-oracle`, ANSWERED, which reads that as the oracle failing). `Ĝ(64) = 1080` rests on a(18) alone and item D's eventual-form trap window rests on `Ĝ(64)`. A provenance-block refit is seconds of work and nobody has asked for it. *Grep:* `Q-exponent-control` (PARTIAL) reports the 22-term reading and the bias; `Q-external-data-audit` (ANSWERED) reports what adopting the external series changed; neither splits the ladder by provenance. [§6 C4.] **Q7. How much of `G₂` is ever realised inside the zone?** [(iii), see the caveat.] `GLOSSARY.md` §"Zone gap" records that an interior gap of the zone is a gap of the tile's slot sequence, so the interior part of `Z₂` never exceeds `G₂(p#)`, and stops there. What is unknown is the other direction as a description: at each level, what fraction of `G₂(p#)` does the largest zone gap reach, and does the tile's record gap ever lie inside the zone? The caveat that makes this (iii) rather than (i): the zone is where a twin slot is a twin prime, so any statement about occupancy there is TPC-adjacent, and a version of this question phrased as "is the zone gap always below the width" is the Zone Postulate itself. The descriptive version, a measured ratio at named levels, is legal. *Grep:* the `Q-zonegap-*` family (model, reduction, witnesses, prior art, Z2) treats `Z₂`'s own law and its reduction; none poses the `Z₂`-realised fraction of `G₂`. **Flagged for the orchestrator:** a sibling agent owns `Z₂` in this wave, so this row may be answered on the other side. **Q8. Is there a level at which the mirror is the only structure, and does that matter?** [(i)] The Mirror-Sweep Lemma is PROVEN and the mirror gains **exactly 0** at the anchored cap, structurally rather than empirically (`REFUTED.md`). But the mirror is the one exact symmetry `G₂` has, and the corpus has only ever asked what it buys as an *improvement channel*. The descriptive question is what it constrains: which statistics of the gap sequence are forced by mirror symmetry alone, so that observing them is not evidence of anything further. That is the free control for Q2 and for any future positional measurement, and the corpus has run positional measurements without it. *Grep:* "mirror" has 8 hits in `QUESTIONS.md`, all under `Q-0c-holesweep` (ANSWERED, the sweep and the palindrome), `Q-anchored-unify`/`Q-history-dial` (the cap families) and the red teams; none poses the mirror as a null model. **Not proposed, and why.** The genealogy of the record gap (is the maximum born at its fold or inherited) is not listed: `attack-foldL-04-amortized.md` already records "births are maximal, not scarce", which answers the descriptive half, and the amortization route around it is CLOSED in `REFUTED.md`. Anything phrased as "prove `G₂ = o(x²)`", "prove `x′²/G₂ → ∞`", or "prove the extreme-value law as an upper bound" is omitted deliberately: all three are verdict (ii) by §4b and belong in the wall's ledger, not in a question list.