# Z2(p) measured at every zone to 1e11: the record envelope in zone coordinates, a margin that runs away, and a prereg scored 2 of 5 with the misses naming the mechanism *(2026-08-21. Staging note; nothing here is integrated into a live document. Producer, formally embedded: `research/zonegap-01.js -- 1e11` (315.7 s; code-sha256 5b6814a5..., out-sha256 2be031a1...). The object and its framing are Chris's: in the zone (p, p'^2) tile survivors ARE primes, so the twin-slot gap structure is measurable from prime data alone, and Z2(p) — the largest gap between consecutive twin pairs inside the zone — is a first-class refinement of the Zone Postulate. Preregistration sealed before the sweep: `research/history/staging/zonegap-01-prereg.md`, commit cdd753c. Adopted data: `research/tos-twin-gaps-1e16.txt` (Oliveira e Silva; provenance in §1). Calibration marked per claim: PROVEN, VERIFIED by exact computation, MEASURED, REFUTED.)* ## 0. Conventions (stated once, the producer's header carries the same text) Pairs are named by opener a; a pair is in zone p iff p < a and a+2 < p'^2; gaps are opener-to-opener (the A113274 convention); Z2(p) is their max; head = a_first - p and tail = p'^2 - a_last are boundary slacks counted separately; u = (gapStart - p)/(p'^2 - p) locates the earliest maximal gap. Width audit: X = 1e11 < 2^53, no BigInt needed, asserted in-code for X <= 4e15. ## 1. Custody, and one attribution corrected [VERIFIED] - Engine: one segmented sweep; an independent brute-force engine recomputes all 167 zones below 1e6 (count, Z2, gap start, head, tail): IDENTICAL. - The sweep's running-max ladder == A113274 records 1..41 (all with p_end <= 1e11): EXACT. - **Adopted this pass under the series rule:** Tomás Oliveira e Silva, "Gaps between twin primes" (sweet.ua.pt/tos/twin_gaps.html, exhaustive to 1e16), copied verbatim to `research/tos-twin-gaps-1e16.txt`, sha256 78767cad d001b1b0 (gz 960e6ae3). Transcription guards in-code: 3,909 rows parsed, g monotone, g = 2 or 0 mod 6, and his 75 record-starred rows must equal A113274 records 1..75 — they do. The sweep then checks the FIRST OCCURRENCE of every gap size against his F(g), both directions: 1,116 gap sizes, 0 mismatches (one mismatch voids the adoption, by construction). - **The brief's pointer said Nicely's twin-gap tables.** Verified against the archived trnicely.net (the live domain is squatted spam): Nicely carries first-occurrence PRIME gaps and twin-prime COUNTS; the twin-gap first-occurrence table is Oliveira e Silva's. He is already trusted in this corpus (A113274 records 73-75 are his). - The envelope identity env(p) = largest published record wholly below p'^2 holds at ALL 27,292 zones: the object's whole extremal skeleton is record-exact against the literature. - `maxgap-law.md` §8's seven levels (M(x, x^2), x = 211..9973) reproduce here 7 of 7 EXACT — the two windows (x, x^2) vs (x, x'^2) happen to bind the same gap at all seven. ## 2. The postulate, refined [VERIFIED to 1e11] Every one of the 27,292 zones holds at least TWO twin pairs (minimum 2, at p = 2; zones with a single pair: 0). `window-check.js` verified one pair per zone; the gap sweep gets the second for free. ## 3. Z2 is the record envelope wearing zone coordinates [MEASURED] The deepest gap sits in the onset shell's deep end, emphatically: the last 15 envelope steps have u = 0.9992..1.0000, the max-position deciles are EMPTY below u = 0.2, 75% of the mass sits above u = 0.6, and in the top band (p in [1e5, 316228)) the fraction with u > 0.8 is 0.462. Mechanism: gap scale grows like ln^2 of height, and the top of the zone holds nearly all of its length, so for a large zone the binding gap is typically the youngest record just under p'^2. Consequence: the per-zone series Z2(p) is HEAVILY correlated across overlapping zones — one physical gap serves thousands of zones — and must never be fit as independent points (this is what broke three prereg bands, §6). ## 4. The law, control-calibrated [MEASURED, with the mandatory control] Naive power fit over 27,267 zones reads e = 3.192. Controls on the same grid, same estimator, matched noise (sigma = 0.064): truth ln^3 reads 3.000 ± 0.003, truth ln^2 reads 1.999 ± 0.004, and the deterministic E-form truth a·ln(w/a) (a = ln^2(p^2)/2C2, w = p^2 - p; the Kourbatov-family shape) reads 3.261. So the measurement is 58 control-sd away from constant·ln^3, 339 from ln^2, and sits BETWEEN constant·ln^3 and the E-form: Z2 lives inside the ln^3 family with a drifting constant, and no single exponent should be quoted (the one-class lesson replicated; at stage 1 the fit read 3.411 and at 1e8 it read 3.323 — the "exponent" is a range summary, exactly as `maxgap-law.md` found for c). Band constants c3 = Z2/ln^3 p: 3.426 ± 0.497, 3.681 ± 0.362, 4.022 ± 0.243, 3.930 ± 0.219 (10^2, 10^3, 10^4, top band). The ln^2·lnln candidate is REFUTED (its "constant" runs 11.60 -> 19.17). The published guard holds at every zone: max load Z2/(0.76 ln^3 s) = 0.7504 at the gap's own height, against the record ladder's worst 0.8434 (`research/a113274-gap-records.js`). ## 5. The margin and the G2 relation [MEASURED + one-line theorem] Whole-tile: window/G2 is FLAT at 3.2-4.5 over the 14 exact levels (`G2-STATE.md` §2). Zone-local: worst Z2/width per band falls 1/50 (10^2), 1/706 (10^3), 1/34,900 (10^4), 1/1,660,000 (top band) — Chris's "substantially better numbers" quantified: 5·10^5 times more slack than the tile's flat 3.5 at the sweep top, diverging like p^2/(3.9 ln^3 p). The only small ratio is the degenerate first zone, 1/3.50 at p = 2. Z2(p) <= G2(p#) is a one-line theorem (in-zone pairs are the tile's twin slots, certified prime below p'^2). Shared exact levels: EQUALITY at p = 2, 3, 5, 7; then G2/Z2 = 1.40, 2.20, 3.00, 2.08, 1.36, 1.72, 2.32, 3.52, 3.64, 4.12 at p = 11..43. Past level 7 the tile's extremal gap lives beyond p'^2: Z2 is NOT G2 restricted, and the zone maximum is strictly easier. **The honest logic.** Proving Z2 + head + tail < width for every p is exactly the strong Zone Postulate, which is TPC-strength; the weak form is TPC. The object reframes the wall, it does not evade it (first-moment argument closed; the max is the wall). What it buys: 27,292 measured levels against 14 exact tile terms; a classical object with a published guard and living literature (Kourbatov, A113274, Oliveira e Silva); and the anchored machinery's home turf. Route B (density advantage) stays closed; nothing here argues from it. ## 6. The head, and the prereg scored 2 of 5 — misses first [MEASURED] Head: mean head/ln^2 p = 0.7229 vs the HL mean-gap coefficient 0.7574 (4.6% under); worst head against the Kourbatov ceiling AT HEIGHT p is 0.7218 (at p = 659, where the record-8 gap of 150 IS the head). The head is a mean-scale (ln^2) object; Z2 is a max-scale (ln^3-at-p^2) object; the producer's printed "~8" hypothesis for Z2/head is refuted by its own row (16.6 -> 200.1). Prereg (`zonegap-01-prereg.md`, sealed commit cdd753c, stage-1 output sha-bound): - **P1 MISS**: top-band c3 = 3.930 below the sealed [4.05, 4.35]. - **P2 MISS**: mean u = 0.728 above [0.63, 0.71]; frac u>0.8 = 0.462 far above [0.21, 0.33]. - **P5 MISS with a sign flip**: whole-sweep e = 3.192 below [3.38, 3.48] and BELOW the E-form read (3.261) where stage 1 had it 0.130 ABOVE. - P3 PASS: head 0.725 in [0.64, 0.80]. P4 PASS: band max load 0.7272 in [0.66, 0.80], guard < 1 confirmed. The three misses share one mechanism, §3's correlation: stage-1 extrapolation treated band-to-band drift as independent, and the top band is dominated by a record staircase it shares with its neighbours. That is the finding: the prereg bought the exact failure mode a naive Z2 fit would have shipped as a law. Also disclosed: the prereg prose miscounted stage-1 zones (wrote 9,592; the sha-bound output says 9,591); predictions unaffected. ## 7. NOT REACHED - No sweep past 1e11 (records constrain the envelope to 1e19 via A113274, but per-zone structure beyond p ~ 3.16e5 is unmeasured). - No staircase-aware law fit: the right model treats Z2(p) as the record envelope at p'^2 plus a below-record correction field; neither the correction's law nor its variance was extracted. - Tail slack got a per-band mean (0.58-0.77 in ln^2(p^2) units) and no more; no extremal analysis of the tail. - The u-distribution's bimodality (dip at u ~ 0.4-0.5, deciles 1516/1424 against neighbours 2105/4071) is unexplained; no null model was scored against it. - The Oliveira e Silva table beyond 1e11 (to 1e16) is adopted but unused except through its guards; its F(g) shape against the E-form is an open cross-check. - Readings-traceability advisory: 12 figures in the READINGS are prose reformattings (1/7.06e+2 quoted as 1/706, sealed prereg bands quoted from the prereg file); accepted under the advisory's known floor.