# PREREGISTRATION — Z2(p) in the unswept decade X in (1e11, 1e12], sealed before the ~1 h sweep runs on the incoming box *(2026-08-21, committed BEFORE any sweep past X = 1e11 exists anywhere in this repository. Model and every number below: `research/zonegap-03-model.js` (code-sha256 `94361acf23fa9c5a9265581f5cb499a4a7c3a33c37dc9785918057865657c59c`, out-sha256 `805cbcbeda6fdbfbb7ce522e1165492cb902cf7e9b5844fce25866c57787dffc`, embedded 2026-08-21), fitted on the X <= 1e11 sweep (`research/zonegap-01.js`, embedded, 27,292 zones) plus the ADOPTED tables only — A113274/A113275 (82 records, `research/a113274-gap-records.js`) and the Oliveira e Silva first-occurrence table (`research/tos-twin-gaps-1e16.txt`). Nothing here touches unpublished data, and no sweep beyond 1e11 was run, priced (315.7 s at 1e11, ~1 h at 1e12), or peeked at. Scoring document: the embedded OUTPUT of the stage-3 sweep at X = 1e12, no re-interpretation; misses stated first in the scoring note.)* ## The model this prereg comes from, in one paragraph zonegap-01's per-zone series Z2(p) is not statistically correlated with the record ladder — it IS the record ladder: Z2(p) = env(p) (the largest A113274 record wholly below p'^2) IDENTICALLY, a two-line theorem conditional only on the adopted ladder being the true running max. The premise (the env record's start exceeds p) holds at all 27,292 swept zones and, via s_k^2 > e_{k+1} at every ladder transition from record 3 on, at every zone of every sweep the published ladder covers (to e_82 = 7.05e16). The model reproduced every Z2 aggregate in zonegap-01's embedded output digit for digit — 204 equality assertions, zero free parameters, no twin data. Consequence, and the honest headline of this prereg: **every Z2 quantity the brief asked to predict is custody, not statistics — its sigma is 0** and a miss is a custody event (engine defect or a wrong published record), never model noise. The residual statistical objects are the record process (KW-null-calibrated in the model script, controls in-pass) and the boundary fields (head; run-correlation handled). Predictions are grouped accordingly. ## Group T — theorem-grade, sigma = 0 (adopted tables + primes, nothing else) - **T1. Zones.** 78,497 zones at X = 1e12 (p = 2 .. 999,979); 51,205 new zones, p in [316,223, 999,979]. Kill: any other count. - **T2. Records and envelope.** The running-max ladder at 1e12 equals A113274 records 1..49: exactly **8 new records** (42..49) enter, and the envelope value at the sweep top is **11,388**. The 8 new envelope-step rows are, exactly (p, Z2, gapStart, u, rec#): 366103 8994 134037421667 0.9999 42 445321 9312 198311685749 0.9999 43 472319 9318 223093059731 1.0000 44 594551 10200 353503437239 1.0000 45 696271 10338 484797803249 1.0000 46 799003 10668 638432376191 1.0000 47 885679 10710 784468515221 1.0000 48 891409 11388 794623899269 1.0000 49 Kill: any deviation in any field. Z2 > env at any zone = a missed published record (adoption void, report upstream); Z2 < env is impossible while s_42 = 1.34e11 >> p_max = 1e6. - **T3. Z2 = env fraction: 1.0000 at all 78,497 zones.** Sigma = 0 by the theorem. Kill: one zone off. - **T4. Whole-sweep staircase functionals at 1e12** (zones with p >= 100, n = 78,472): power-fit exponent **e = 3.332** (same estimator as zonegap-01; match to 3 decimals), mean u = **0.7913** (4 decimals), u deciles exactly **[0 0 1767 2105 1935 3878 8013 14607 23082 23085]**. Kill: any digit. - **T5. Bands at 1e12.** The `1e5-p.5` band is NOT an identical reprint: zone p = 316,223 joins it (17,701 zones; printed c3 stays 3.930 ± 0.219). New band [316228, 1e6): 51,204 zones, c3 = **4.182 ± 0.132**, mean u = 0.836, frac u>0.8 = 0.690, worst Z2/width = 1/1.24e+7. Full decade [1e5, 1e6): 68,905 zones, c3 = 4.117 ± 0.193, mean u = 0.808, frac u>0.8 = 0.631. (zonegap-01.js hard-codes its band edges at e = 5, so a 1e12 run of THAT producer prints no [316228, 1e6) row; these numbers score against whichever producer prints the band, else through T4.) Kill: any value off at the printed precision. ## Group S — statistical, blind, each with its sigma model and kill rule The head field is run-correlated (~10.1 consecutive zones share their first pair), so every sigma below uses n_eff = 5,076 pairs, not 51,205 zones — the correlation zonegap-01's misses paid for is priced in, not assumed away. - **S1. New-band head.** Mean head/ln^2 p over the new band [316228, 1e6): **0.7275 ± 0.0155** (deficit model (1/(2C2) - h)·ln p = 0.398 ± 0.150 fitted on bands 10^4 and 1e5-p.5; sampling on n_eff). Sealed band (3-sigma): **[0.681, 0.774]**. Kill: outside. - **S2. Whole-sweep head at 1e12** (p >= 100; scores against the printed HEAD line of the current producer): mean head/ln^2 p = **0.7259 ± 0.0101**, sealed 3-sigma **[0.696, 0.756]**. Kill: outside. - **S3. Worst head vs the 0.76 ln^3 p ceiling at height p.** New-zone max (Exp-head model, run-corrected): median 0.640, 95% band [0.520, 0.879]. Whole-sweep worst at 1e12 = max(0.7218, new-zone max): **P(stays 0.7218) = 0.80**; sealed: whole-sweep worst in **[0.7218, 0.90]** and the guard (< 1) holds. Kill: outside [0.7218, 0.90] kills the Exp-head model; >= 1 kills the guard (a publishable event on its own). - **S4. Twin pairs to 1e12: 1,870,593,490 ± 26,264** (two-point HL calibration, ratio 1.000046 at 1e10 -> 1.000032 at 1e11, extrapolated 1.000018). Sealed 4-sigma: **[1,870,488,435, 1,870,698,545]**. Honesty note: pi2(1e12) is published in the literature but sits in no adopted table of this corpus; this seal tests the sigma machinery, and scores only against the sweep's own printed count. Kill: outside. - **S5. Postulate, refined.** Minimum in-zone pair count stays 2 (at p = 2); single-pair zones stay 0. Conjecture-grade; a miss is mathematics news, not model noise. ## What is deliberately NOT predicted No blind record-process prediction exists below e_82 = 7.05e16: the published ladder covers the whole reachable range (that is Group T's whole point). The record-process statistics (rate 2.562 per ln x vs null 2.356 ± 0.304; trend-load location deficit A = 0.9295 vs null 0.9895 ± 0.0182; records-per-decade sigma 5.33 ± 2.11 with the observed 8 in (1e11, 1e12] at 1.3 null-sd) are calibration, validatable only beyond 1e17. The tail field has no per-zone data in the embedded corpus and is not predicted. ## Scoring rule T1-T5 and S1-S5 each score pass/fail against the stage-3 embedded OUTPUT at X = 1e12, no re-interpretation; conditional rows (T5's new band) score only if a producer prints them. Group T misses are CUSTODY events and trigger the named diagnosis paths, not band widening. Result goes in a `research/history/staging/zonegap-03-*.md` scoring note with misses first.