# Red team, 2026-08-20 night: mp-derivation and rho-maximal-law attacked *(2026-08-20. Adversarial verification pass over the two staged night headlines `mp-derivation.md` (empirical formula) and `rho-maximal-law.md` (stated law + pricing), before integration. Method: refuted-until-rederived; decisive statistics recomputed on independent code paths from the embedded artifacts; instruments calibrated on known-truth controls in the same pass; units stated before comparison. Scratch producers live outside the tree (`rt-night-*.js`) and are quoted inline. Grades: CONFIRMED / WEAKENED (with corrected sentence) / REFUTED (with counterexample). BOTTOM LINE: both headlines SURVIVE — every decisive number reproduced on independent code, the fresh window re-sieved with a different engine and all 37 X reproduced, the sup|rho| column re-walked with a different walker at all five levels — with one mandatory rider on T2's HELD s/u finding, one confound in T1's theta-claim evidence that a matched control closes, and one custody residual on the prereg timing. Nothing is refuted.)* ## T1. mp-derivation.md ### T1.a The W1 overlap derivation re-derived — CONFIRMED Brute enumeration (`rt-night-w1-brute.js`) over 182 (q, v) pairs, q = 5..31, all even v to 2q+6: the forbidden set {0, −2, −v, −v−2} mod q has size 2 iff q | v, 3 iff q | v±2, 4 otherwise; the three classes are mutually exclusive for q ≥ 5 (any collision forces q | 2); and the direct survival count over residues equals q − |S| in every case. So the per-fold factors (q−2)/(q−4), (q−3)/(q−4) are exact as pair-survival relative to a generic distance, and the (q−4) denominator is the correct base measure for that comparison (all four residues distinct). The base choice only moves a smooth-in-p normalization; the falsifiable content is the class-factor ratios, and the free exponent reads beta = 1.0313 on my own two-variable refit where the derivation says 1. The step "W1 as gap-count weight" remains a modeling step and the doc flags it as such. ### T1.b Prereg custody at 303711b + blind re-score — CONFIRMED, one residual - **Commit graph**: 303711b (17:16:41) contains ONLY the prereg. The anchor is genuinely fresh: `git grep` for 132000000000 / 1.32e11 / 132e9 at 303711b^ returns nothing. The producer-01 fingerprint pinned in the prereg (code-sha256 248c8c88…, out-sha256 856d0519…) matches the committed file; `embed.js --check` reports both producers bit-honest. - **Sealed bands re-derived**: from my own parse of the committed embeds, my own W1/mbar arithmetic, my own OLS refit and my own PLN quantile code (Simpson on [−6,6], N = 241, own lgamma — vs the producer's trapezoid [−5,5], N = 61), **all 37 sealed (mu, lam) values and all 74 band integers reproduce exactly**. - **The window re-sieved on an independent engine**: increasing-prime conditional-write marking and a merge-on-equal monotone stack counting adjacent pairs at level p (vs the record's decreasing unconditional writes and kills−runs bookkeeping). Calibration in-pass: 37/37 scored folds match the embedded [0, 2e9) record. At A = 1.32e11: **37/37 folds agree with producer-02's embedded X**. My own re-score: **34/37 in-90 (need ≥ 28), 0/37 outside 99.73% (allow ≤ 1), margin D4 − D0 = +25.4 nats. HIT stands.** - **Residual, custody**: producer-02 first exists in git at 254b689 (17:22:02), 5m21s after the seal. "The producer does not exist at the seal" is unverifiable from git, and at 15.8 s per run anchor-shopping was physically possible in that gap and before it. Mitigation, not proof: the bands are a deterministic function of artifacts committed half a day earlier plus the declared formula, and the outcome (34) sits AT the incumbent-truth expectation (34.6), not above it — a shopped anchor would be expected to overshoot. Recorded as a residual; future preregs should seal before the measuring producer is even drafted, with a longer gap. ### T1.c Model selection re-fit — CONFIRMED, no leak found My own OLS on the 37 pooled train rows (p ≤ 293) ONLY reproduces (k, delta) = (1.9468, 0.2771) exactly, s0 = 0.1767, beta = 1.0313 — so the two scalars are a function of the declared train split alone. My own PLN scoring reproduces every table entry to 0.1 nat: deep −128.2 / −118.3 / −116.2 / −122.7 / −107.8 / −88.6, blind −112.7; deep gaps D4 over D3/D1/D0/D2/raw = 19.2 / 27.6 / 29.7 / 34.1 / 39.6 nats, blind 10.2–20.8. Leak checks: the pooled Sx/Sl exclude the 6.6e10 blind window (Slam identity = six-window exposure 1.1211; producer-01's Stage 7 adds BLINDX separately for the incumbent), and the fresh anchor enters nothing fitted. One wording defect: producer-01's header says "weighted ln-regression"; the implemented fit is unweighted OLS in ln (I reproduced from the code's actual convention; results are internally consistent). ### T1.d The theta-not-p claim — CONFIRMED, but the doc's own evidence is confounded The cited contrast (twin rms 0.155 vs adjacent non-twin 0.764, both reproduced exactly) cannot by itself separate "function of theta" from "smooth function of p": among adjacent fold pairs, sharing theta and Delta p = 2 are perfectly collinear. The matched control closes it: - the 4 adjacent NON-twin pairs whose D4-predicted |dln M| ≤ 0.15 agree at rms 0.181 (noise-rms 0.135) — as well as the twins (0.155, noise 0.150); - across all 34 control pairs, sum(d² − noise) = 18.98 vs the D4-predicted theta-arithmetic sum 17.34 (ratio 1.09): the non-twin disagreement is the formula's own W1 + depth differences, nothing left for p; - cov(excess d², prime gap) = 0.03 ≈ 0 — adjacency in p adds nothing. Twin mean-squared Poisson pull = 2.23, confirming the doc's ~10–15% non-theta residual clause. **Corrected sentence for the doc**: *the 0.155-vs-0.764 contrast is confounded (theta-sharing ⟺ Delta p = 2); the claim rests on the matched control — non-twin pairs matched on predicted M agree like twins, and control-pair variance decomposes onto the theta-arithmetic at ratio 1.09.* ### T1.e The four flagged folds and 631's pricing — CONFIRMED, one convention note Under the producer's stated convention (deep Pearson (Sx−mu)/sqrt(mu)) the four flags reproduce exactly: 311 +3.10, 331 +3.21, 409 +3.28, 631 +3.36, and no train fold exceeds 2.5 s0. Note: these are Poisson pulls; under the formula's own PLN variance (s0 = 0.177) only 409 (+3.15) and 631 (+3.36) stay above 3 — the flag list is exposure-noise-flattered, worth one line. 631: P(≥2 events at 631 | D4) = 2.76e-3, trial-corrected P(≥2 at SOME p ≥ 521) = 0.170, E4 = 2.30 vs E0 = 9.60, observed 2, P(N ≤ 2 | D0) = 3.84e-3, top expected folds 541/547, W1(631) = 5.55, M_D4 = 0.338, pooled M̂ = 4.48 — all reproduce. The trial factor is priced correctly and stated. Secondary claims all reproduce on my parse: corr probe 0.6537, band means (measured 0.959/0.859/0.520/0.357 vs D4 0.946/0.872/0.423/0.233), deep delta MLE 0.252 vs train 0.277, import-stein 14/14, K = Psi/Phi² span 1.009 vs M̂ span 2.28 at 101/211/421 (the refutation of K stands). **T1 verdict: CONFIRMED.** The blind HIT, the fit, the bands, the sieve, the selection table and the pricing all survive independent rederivation; grade PARTIAL DERIVATION is right. Apply the T1.d corrected sentence, the T1.e convention note, and record the T1.b custody residual. ## T2. rho-maximal-law.md ### T2.a The pricing lemma chain — CONFIRMED Numerically at z = 13 over the full period (`rt-night-t2.js`): (i) the identity T = HM + rho(x) − rho(x+H) agrees with my own floor-sum evaluation to machine zero at 330 sampled positions; (ii) |R_H| ≤ 2 sup|rho~| is exact algebra (the mean cancels in the difference); (iii) H = ceil((2S+1)/M) gives HM − 2S ≥ 1 by construction — min T = 3.0 observed; (iv) the BF pointwise inequality holds at every sampled x (T ≤ #admissible in (x, x+H], 0 violations), and the delivered window H = 112 indeed covers the actual max admissible gap 42 in the z = 13 period; (v) M·ln²z measured 0.343–0.367 flat across z = 13..47 (my own M at 47 from buildTerms: 2.3166e-2), and 1+sqrt(e) is the 2f > F composition threshold, so M ≍ c/ln²z is standard as claimed. Composition: G2(z#) ≤ ceil((2Cz^a+1)/M) ≪ z^a ln²z — the ln²z factor is right, and ln²z ≪ z^e closes the "below 4.2665" clause. Only the law itself is unproven, as stated. ### T2.b The sufficiency curve — CONFIRMED by my own arithmetic Recomputed at z = 13 and 29 entirely from my own walker's period variance plus my own M = Σw/q and lnW, and at z = 47 from my own buildTerms (n = 293980, M = 2.3166e-2, lnW = 37.110 — all matching the cited row) plus the cited rmsr: theta_G = 1.94696 / 2.15889 / 2.48199, lambda_max = 2.31949 / 2.10756 / 1.78446, d_strict = 0.04467 / 0.00642 / 0.00079, 1/M = 17.9 / 31.4 / 43.2, alpha_max = 2.8712 / 3.0369 / 3.1085 — every column digit-for-digit. Floor arithmetic: th(nP) exact at all six z, z = 47 floor 2.0106, band width 2.2559, room consumed 0.47% ("0.5%"). ### T2.c The looseness factor — CONFIRMED: 47^(4.26645−2.48199) = 963.4. ### T2.d The direct sup|rho| measurement — CONFIRMED and extended My own walker (different blocking, own rho(0), own accumulation) reproduces sup|rho~| and C_true at ALL FIVE levels digit-for-digit (2.62013 / 4.33665 / 9.15247 / 12.10617 / 17.90249; C_true 0.6362 / 0.6044 / 0.8022 / 0.7554 / 0.7830), with period mean + M/2 ≤ 2.2e-7, closure drift ≤ 4.3e-7, and the measured period variance equal to the closed-form column at printed precision. Brute force from the definition at TWO levels (z = 13 AND 17, one more than the producer's own control) agrees to 1e-8. The Gaussian rho law is TRUE at every exactly measurable level; the C_true band 0.60–0.80 stands. z = 31 not walked (compute ceiling; unchanged). ### T2.e The HELD s/u = 0.703 finding — CONFIRMED, with a mandatory rider - **Arithmetic**: s/u = 3/4.26645 = 0.70316. Lemma V's stated range — verbatim at `research/sift-limit-attack.md` ("in some range s ≥ (0.63+δ)·u (θ_total ≥ 1.25) up to s = u − ε") — contains 0.703 for any δ < 0.073. Every delivered exponent u in (3, β₂) has s/u in (0.703, 1), inside the range; exponents ≤ 3 give s/u ≥ 1, outside. Edge precision: for FIXED ε the clean claim is u in [3/(1−ε), β₂). The TPC quote is faithful (`theta-ladder.md` §: s/u = 1.62–1.72 > 1, moot there). Lemma V's conclusion ≪ H/log³H is indeed strictly stronger than needed (log³H = u³ln³z ≫ ln²z ≍ 1/M). - **Pointer fix**: the sourced range lives in `sift-limit-attack.md`, NOT in `lemmaV-neighbours.md` (that file carries the DI/Maynard/Pascadi smoothness-hypothesis quotes); any integration should cite the former. - **THE RIDER (mandatory before this moves anywhere live)**: "inside the stated range" un-moots the corpus's own POSED lemma — Lemma V is needed-not-proven, so nothing "applies" in the theorem sense. And the fixed-smooth-profile hypothesis that killed the import of DI 6.12 / Maynard / Pascadi is a property of the Rosser weights (arbitrary functions of the whole modulus, no smooth factor), NOT of the working point: it binds at the β₂ point exactly as it did at the TPC point. What re-enters is therefore the ASK — a 0.1787-of-θ_total improvement (1.2090 needed vs 1.0303 delivered, re-verified) at a working point inside Lemma V's own range — not an import route. The doc's §5 wording is compatible with this; the rider should be explicit when the HELD block is resolved. The recorded Kloosterman pricing also remains R_H-configuration only (the doc already carries that caveat). **T2 verdict: CONFIRMED.** Statement, pricing, curve, measurement and position all survive; the HELD finding can be released WITH the rider and the pointer fix. ## Not reached - The z = 31 sup|rho~| walk (~500 s, barred by the compute ceiling) and the z = 37..47 conditional rmsr rows (O(N²), cited; their cheap fields n, M, lnW were re-derived here from my own buildTerms and match). - Producer-02's Stage A 11-of-11 engine replay (my own engine was calibrated on the same embedded record instead) and producer-01's two synthetic control gates (my refit reproduced the real-field numbers directly). - The incumbent-NB expected-score line (34.6 / +23.6) — taken from the sealed prereg, not re-derived. - The Kloosterman pricing at the rho-potential configuration (open in the doc's own NOT REACHED, unchanged). --- *This document states current understanding. Superseded claims, retired numbers and the reasons they changed are in [../CHANGELOG.md](../CHANGELOG.md), indexed by document.*