# Pre-registration: the @37 test of the offset-correction family for the ~3.8 law *Written 2026-08-20 and committed alone, before any @37 census exists. The producer instrument already exists — `research/xchan-at29-01-segmented.js` runs any level by argument, and `research/attack-x-offset-02-profile.js` shares its kernel and gate — but neither has ever been run at @37, no @37 census of any kind exists in the corpus, and the level is priced only by extrapolation ("an overnight run", `xchan-at29.md` §7c). Everything below is fixed in advance: the candidate predictions, the σ model and its projection band, the scoring rule, and what each verdict does to TODO item X's offset clause.* ## 1. What is being tested `research/history/staging/xchan-at29.md` left `1 − J = 4S₂` the last law standing and NOT exact: absolute residuals Δ = +1.201e−4 (@29) and +1.185e−4 (@31), relative offset −0.41% / −0.48%, one detection (z = −2.32 at @31 on the slot-clustered σ) and one consistent non-detection (−0.43 at @29). `research/attack-x-offset-01-terms.js` (embedded) priced the finite-level correction family on those residuals; `research/attack-x-offset-02-profile.js` (embedded) measured the mechanisms available at the existing levels. The @37 aggregate census decides between the surviving aggregate descriptions. The statistic is unchanged and is the one the gate protects: J = obs/CRT with obs the mixed super-`W` triple census over the natal set and CRT = 6·N̄·Σ_{∏Q>W}1/∏Q, at x = 37, W = 7,420,738,134,810, N̄ = 145,286,237,250, K = 198,274 scour primes in (37, 2724079]. ## 2. The predictions, all from embedded arithmetic, none adjustable From `research/attack-x-offset-01-terms.js` (its OUTPUT block; sums exact, tail mass exact, fitted parameters frozen at their 2026-08-20 values): | candidate | 1 − J at @37 | mechanism class | |---|---|---| | **N1** `4S₂` | **0.021863** | no correction: the offset was a fluctuation | | **C1** `4S₂ − S₃` | **0.021785** | zero-parameter shrinking correction (POST HOC at @29/@31) | | C2 `4S₂ − 2S₃` | 0.021707 | zero-parameter, twice C1's term | | M-mult `4S₂(1 − 4.755e−3)` | 0.021759 | constant relative offset | | M-abs `4S₂ − 1.186e−4` | 0.021744 | constant absolute residual | | M-cS3 `4S₂ − 1.198·S₃` | 0.021770 | fitted S₃ multiple | | M-ln `4S₂ − 0.491·S₂/lnW` | 0.021772 | fitted 1/lnW-scale correction | C4, C8 (≥ 4S₃-size corrections) and the locality shape M-loc are already REFUTED at the existing levels and are not carried. **The error bar, fixed now.** σ_reg = √obs/CRT with CRT(37) = 6.1673e10 exact gives σ_reg ≈ 3.98e−6. The slot-clustered inflation is PROJECTED at 2.14, banded [2.12, 2.17] from the measured 2.078 → 2.106 → 2.124 (@23/@29/@31): **σ_slot(37) ≈ 8.52e−6**, band [8.44e−6, 8.64e−6]. The run's own printed σ_slot supersedes the projection; the projection exists so that the separations below cannot be re-derived after the fact. **The σ estimator is calibrated**: all seven known-truth control draws of `research/attack-x-offset-02-profile.js` (random masks at natal density, truth = the level's own J_line) landed inside 1.9σ, so the slot-clustered σ neither manufactures nor hides an offset at the scale those draws can see. **Separations at the projected σ.** N1 vs C1: 9.1σ. C1 vs C2: 9.1σ. C1 vs M-abs: 4.8σ. C1 vs M-mult: 3.1σ. C1 vs M-cS3: 1.8σ. C1 vs M-ln: 1.5σ. So @37 decides: (a) whether any correction is needed at all, (b) shrinking vs constant-absolute, and (c) C1's coefficient against C2's — but NOT between the members of the shrinking family (C1, M-cS3, M-ln, and M-mult only weakly), and no claim of that kind may be written from @37 alone. ## 3. The scoring rule, fixed now Compute z_c = ((1−J)_meas − pred_c)/σ_slot for every candidate, on the run's own slot-clustered σ_slot. - A candidate is **KILLED** by |z_c| > 3 and **ALIVE** otherwise; the family verdict names every survivor, sorted by |z_c|. - **If N1 survives** (|z_N1| ≤ 3), the offset's status reverts to "not established as structure": two sharp levels plus a third failing to reproduce it means the @31 detection does not generalize, and TODO item X's offset clause must say so. - **If N1 is killed and at least one shrinking-family member survives**, the offset is CONFIRMED as structure with a shrinking scale; C1's status upgrades from POST HOC to "survived one blind level" only if |z_C1| ≤ 3. - **If only M-abs survives**, the residual is a constant-absolute anomaly, no candidate mechanism on file produces it, and the item must carry that. - **J-band clause** (unchanged from `xchan-at29-prereg.md`): J(37) outside (0.94, 1.00] means the object changed; the report leads with that. - **Gate**: the producer must reproduce its own embedded reference levels before @37 is reported (the same gate the @29/@31 run used); a gate failure reports an instrument failure and no @37 value. ## 4. Secondary, registered but not decisive - **The line's lattice bias.** `attack-x-offset-02-profile.js` measured β = J_line − 1 = +9.88e−3, +7.44e−3, +5.38e−3, +4.03e−3 over @19..@31 (z = 2.3, 8.7, 34.7, 147.6): positive, decaying, and 16–24% of the natal deficit's own size. If the @37 run also runs the line ensemble (roughly 3–5× the natal cost, so it may be deferred), the registered reading is the continuation of that decay and of 1 − J/J_line (1.18× and 1.15× 4S₂ at @29/@31); no numeric band is fixed because no β model is on record. The sighting β/S₃ ≈ 39–43 over the four levels is recorded here so that, if a β model is ever fitted, this note is its provenance. - **P1**: aligned super-`W` obs = 0 at @37, an instrument check as always. ## 5. Scope Producer: `research/xchan-at29-01-segmented.js -- ... 37` or the profile instrument at @37; either must be embedded via `node research/qc/embed.js` with its gate green. Report: an extension of `research/history/staging/item-x-offset.md` or a successor staging note. No live document is edited by the @37 run; TODO item X's update is proposed in the report, not applied. This file is committed alone before any @37 number of any kind exists.