# The @31 σ calibration gap closed: seven known-truth control draws at the level of the detection, both statistics run *(2026-08-21. Closes the two §T2.b findings of [redteam-0820-empirical.md](redteam-0820-empirical.md): the level carrying the offset's one detection (z = −2.32 at @31) had ZERO known-truth control draws (SEEDS 31:[] in the producer), and the control statistic (J_ctrl − J_line) cancels level-scale arithmetic fluctuation that the test statistic does not. Producer, formally embedded (5519.5 s): `research/attack-sigma31-01.js` — seven fresh random-mask draws AT @31, built by the identical construction and statistic pipeline as the ≤@29 draws of `research/attack-x-offset-02-profile.js`, scored with BOTH the cancelling and a non-cancelling statistic. Every figure below is from that file's OUTPUT block or one-line arithmetic on figures in it, inputs shown. The @37 pre-registration [xchan-at37-offset-prereg.md](xchan-at37-offset-prereg.md) is SEALED at `0a7dd73`: nothing here touches anything it scores — this pass calibrates the INSTRUMENT, and tensions are reported, not edited.)* ## Verdict **The slot-clustered σ at @31 is now calibrated at @31 itself: seven known-truth draws give σ_true/σ_reported = 1.127 at the point, 95% bracket [0.745, 2.293] (pooled with the ≤@29 draws: [0.772, 1.664]) — the registered factor-[0.65, 2.0] bound measured at the wrong levels is replaced by one measured at the right one. At the point calibration the @31 detection WEAKENS, z = −2.32 → −2.06, and it does not resolve inside the bracket (−1.01 to −3.12); a 3σ reading requires r < 0.775, which the bracket contains. And the red team's second flaw is now a measured number: the cancelling control form removes a level-scale arithmetic term of 82.2σ_slot, common to every draw, so the controls calibrate the SAMPLING σ only.** The offset stays exactly what `item-x-offset.md` §1 called it — one 2.3σ-scale sighting — now with its σ bounded at the level that carries it, and @37 remains the decider. Specifics: 1. **[MEASURED] The seven @31 draws: z_A = 0.75, 0.39, 1.37, −0.62, −0.82, −1.91, 1.25.** All inside 2σ of truth (E3 holds with margin), mean 0.058 (the estimator is unbiased at random-mask truth), sd 1.215, Σz² = 8.888 ~ χ²₇ — fully consistent with a correctly-scaled σ, and equally consistent with a σ understated by up to the bracket's edge. 2. **[MEASURED] The construction is certified identical, not similar**: all seven ≤@29 draws of producer 02 reproduce to the printed digit (N, J_ctrl, σ, z — EXACT, seven times), and the natal record reproduces at all four levels (obs = 1653241687 exactly at @31). Same seeds → byte-identical masks → identical statistics: the certification is mechanical. 3. **[MEASURED] Both statistics ran.** The non-cancelling form z_B = (J_ctrl − 1)/σ scores each draw against its analytic smooth truth — the natal test's reference class. z_B − z_A = 82.2 on every draw; sd(z_B) = 1.212 vs sd(z_A) = 1.215. The level-scale term is measured, and measured to be invisible to any within-window control ensemble. 4. **[MEASURED] Pricing surprise**: 716.5 s mean per draw (657.3–776.4) against the ~420 s brief price — 1.7× over (716.5/420) — and 0.8× the naive W-scaling price (~900 s); rss peaked at 408 MB. ## 1. The gap this closes `item-x-offset.md` §4 declared "the σ behind the @31 detection is calibrated" on seven control draws — all at @19/@23/@29. The red team (§T2.b) graded that WEAKENED on three counts: no control at the detection's own level; seven draws bound the σ scale only to ~[0.65, 2.0] (so a 30%-understated σ, fully consistent with those controls, moves 2.32σ to ~1.8σ); and the control statistic cancels the one component the test statistic keeps. This pass answers the first count directly (draws at @31), sharpens the second (the bracket is now measured at @31, and doubled in count by pooling), and turns the third from an objection into a measurement (§4 below). ## 2. The construction, certified identical The producer is producer 02's control kernel slimmed of bins, marginals and the line census — nothing in the mask construction or the statistic path is new. A control draw at level x, seed s: sfc32 keyed (0x9E3779B9^x, 0xC0FFEE, 0x20260820, s) drawn once per slot in segment order, mask at exact natal density P = N̄/(W/15); the mixed super-W triple census over the masked line; CRT = 6·N·miss with the exact integer U-test tail mass; σ_slot from the per-slot excess spread — the pooled-floor fix's estimator, unchanged. Truth for the cancelling statistic is the level's own line census, cited from producer 02's embed (obs_line = 5481871992 at @31) over this file's exact CRT_line (custody: reproduces the embedded 5459848120.06 to rel 5.02e−14). Certification is by reproduction, not by reading: the same seeds at @19/@23/@29 regenerate producer 02's seven embedded draws to the printed digit, and the natal gate reproduces the xchan record at all four levels (CRT to rel ≤ 5.5e−8, obs/J/σ to the digit). Everything the ≤@29 calibration was, this is, at @31. Seeds 1–7 at @31 had never been run (SEEDS 31:[] — that absence is the finding this file repairs). ## 3. The calibration | seed | J_ctrl | z_A = (J_ctrl−J_line)/σ_slot | z_B = (J_ctrl−1)/σ_slot | |---|---|---|---| | 1 | 1.004071 | +0.75 | 82.98 | | 2 | 1.004053 | +0.39 | 82.63 | | 3 | 1.004101 | +1.37 | 83.60 | | 4 | 1.004003 | −0.62 | 81.62 | | 5 | 1.003993 | −0.82 | 81.42 | | 6 | 1.003940 | −1.91 | 80.33 | | 7 | 1.004095 | +1.25 | 83.48 | **[MEASURED]** mean(z_A) = 0.058, sd = 1.215, Σz² = 8.888 ~ χ²₇. With r = σ_true/σ_reported estimated as √(Σz²/7): - **@31 alone (n = 7): r = 1.127, 95% bracket [0.745, 2.293]** (χ²₇ quantiles 1.690, 16.013, computed and self-tested in-file). - **Pooled with the seven ≤@29 draws (n = 14): r = 1.055, [0.772, 1.664].** The @31-alone bracket is WIDER than the registered [0.65, 2.0] at its top — that is arithmetic, not deterioration: seven draws resolve a factor ~3 whatever their level, and Σz² came out 8.89 where the ≤@29 seven had 6.71. What changed is what the bracket is ABOUT: it now bounds the σ at the level that carries the detection, where before it bounded a projection from levels that do not. P1 (aligned super-W ≡ 0) held on every draw, and every internal count identity passed. ## 4. Both statistics: what the control can and cannot bound The red team's third point was structural: J_ctrl − J_line cancels any whole-window arithmetic fluctuation, while the test statistic (natal J against an analytic formula) keeps it. Running the non-cancelling form on the same draws turns that from an assumption into a number: - **z_B − z_A = (J_line − 1)/σ_slot = 82.2 on every draw** — the window's arithmetic content stands 82 sampling-σ above the smooth model. - **sd(z_B) = 1.212 = sd(z_A) = 1.215 to the third digit**: the term is COMMON to every draw in the window, so the between-draw spread cannot see it — proof by measurement that no within-window control ensemble, however large, bounds its fluctuation. Only its size can be stated, and now is. So the calibration in §3 is a calibration of the SAMPLING σ. The natal z = −2.32 is a statement about (window arithmetic − analytic formula) in units of that sampling σ; its arithmetic component has no control at any fixed level, because each level owns exactly one window. That is not a defect of this pass — it is the measured reason the offset question can only be settled by a NEW level, which is what the @37 prereg is. ## 5. Verdict arithmetic and the @37 prereg **The @31 detection restated.** Registered: Δ = 1.1853e−4, σ_slot = 0.000051, z = −2.32. | calibration | z restated | |---|---| | point r = 1.127 | **−2.06** (weakens) | | bracket low r = 0.745 (σ overstated) | −3.12 | | bracket high r = 2.293 (σ understated) | −1.01 | A 3σ reading requires r < 0.775 and the bracket contains 0.775; a sub-2σ reading requires r > 1.16, which it also contains. **So the calibrated verdict is: the detection WEAKENS at the point estimate and stays a consistent-with; calibration at @31 excludes neither "fluctuation" nor "3σ structure".** One draw deserves its own sentence: seed 6, a pure random mask, scored z_A = −1.91 — an excursion of the detection's own order showed up once in seven null draws. The 2026-08-20 record's framing ("one detection plus one consistent non-detection, essentially one measured number") survives unchanged and slightly chastened. **What this does to the @37 prereg — reported only; it is sealed at `0a7dd73` and its scoring rule stands as registered.** The prereg's σ_slot(37) projection 8.5228e−6, READ at the @31 bracket, spans [6.350e−6, 1.955e−5]: - The primary separations survive the whole bracket: N1 vs C1 and C1 vs C2, registered 9.1σ, read 4.0σ at the bracket's weakest end — @37 still decides "is any correction needed" and "C1's coefficient vs C2's" under any σ miscalibration these draws allow. - The secondary separations do not: C1 vs M-abs registered 4.8σ reads 2.1σ–6.4σ across the bracket, and C1 vs M-mult 3.1σ reads 1.4σ–4.2σ. At the weak end, @37 separates {no correction | shrinking} but NOT {shrinking | constant-absolute}. If the @37 run's own printed σ_slot (which supersedes the projection by the prereg's own clause) lands near the projection while the true σ sits at the bracket's top, a 3σ kill under the registered rule corresponds to as little as 3/2.293 = 1.31σ of true significance — the registered |z| > 3 kills should be READ with this file's bracket beside them, exactly as the standing rule reads z = −2.32 with it. ## 6. Pricing Registered before the run: the brief priced an @31 draw at ~420 s; naive W-scaling of producer 02's @29 controls (~29 s × W₃₁/W₂₉ = 31) said ~900 s. Measured: **657.3–776.4 s, mean 716.5 s per draw**, rss ≤ 408 MB (the instrument's memory is level-free by construction; only the clock scales). The brief's price was optimistic by 1.7× (716.5/420); the W-scaling was pessimistic by 1.26× (900/716.5). Mechanism: a control draw pays the full line regardless of mask density — one rng call per slot (W/15 = 1.34e10 calls) plus the whole-line marking walk — so it costs more than a natal pass (422.9 s here) even at the same active count. For future preregs: an @31 control ensemble prices at ~12 min per draw, ~84 min for seven, on this instrument. One operational note for the session record: the first batch attempt was killed externally at four draws in; the producer is deterministic per seed, and the four landed draws reproduced byte-identically on the second, complete, bound run — nothing was lost and nothing had to be guessed. ## 7. Gate **Before:** `node research/qc.js` TOTAL = 3 — the standing `xchan-at37-01-census.js` embeds red (the @37 census running in another session; not this session's), an `uncited-script` on another session's `attack-c2drift-01.js`, and an expected `uncited-script` on this file's own producer before this report existed. **After:** TOTAL = **1** — only the standing census red remains; it belongs to the live @37 run and is not this session's to clear. The producer is formally embedded (`node research/qc/embed.js --streams both`, 5519.5 s, code-sha256 `e19d99d9…`, out-sha256 `1cb7292c…`), READINGS appended below the OUTPUT banner, and the static `embeds` check verifies the binding. ## 8. Not reached - **No control at @37.** Extrapolating this file's price by W₃₇/W₃₁ = 37 puts one @37 draw at ~7 h; a seven-draw ensemble there is a ~2-day compute decision nobody has taken. The @37 run's σ will rest on its own clustering estimate plus this @31 bracket. - **The level-scale arithmetic term is sized, not controlled** — §4 shows no within-window ensemble can do better; a cross-level null does not exist because each level owns one window. - **Seven draws resolve a factor ~3.** Narrowing the @31 bracket to, say, ±25% needs ~30 draws (~6 h); not run, and the pooled 14-draw bracket [0.772, 1.664] is the best current bound. - **Producer 02's 2948 s `--check` re-execution was not re-run**; its binding is verified statically, as `item-x-offset.md` §9 records. - **Nothing scored by the @37 prereg was touched, recomputed, or re-projected** beyond reading its frozen numbers against the measured bracket in §5.