# Red team, 2026-08-20 evening: the two held empirical headlines attacked *(2026-08-20. Adversarial verification pass over the two staged headlines `perfold-error-model.md` and `item-x-offset.md`, before integration. Method: refuted-until-rederived; every decisive statistic recomputed on a different code path from the embedded artifacts; instruments calibrated on known-truth controls in the same pass. Scratch producers live outside the tree and are quoted inline. Grades: CONFIRMED / WEAKENED (with corrected sentence) / REFUTED (with counterexample). Bottom line: both headlines SURVIVE — every decisive number reproduced digit-for-digit on independent code, the blind window re-sieved and re-scored 33/37 with independently re-derived bands, the cofactor purity claim certified as a theorem — with two sentence-level weakenings that should be applied before integration: the anchor-direction "identical" in T1 is only bounded at ~6–10%, and the @31 σ "calibrated" claim in T2 outruns its controls. One sentence is refuted as stated (T2.x).)* ## T1. perfold-error-model.md ### T1.a Decisive statistics recomputed — CONFIRMED Both statistics were recomputed on a different code path, twice over. 1. **A fully independent sieve engine was written from the definition** (keys marked in increasing prime order with conditional writes where the producer marks decreasing/unconditional; X counted by a merge-on-equal monotone stack instead of kills−runs bookkeeping) and calibrated on the record first: it reproduces ΣX = 203165 (2·10⁷), 2031788 (2·10⁸), 20317943 (2·10⁹), 203176087 (2·10¹⁰) and **every per-fold X in [101, 1499] at all five re-sieved windows, 214/214 folds each**, against the embedded tables. 2. **Anchor-replicate χ²** = Σ(X_a−X_b)²/(X_a+X_b) over the 43 folds with X_a+X_b > 0: **50.2/43 = 1.167** — identical from my parse of the embeds AND from my own raw re-sieve of both 2·10⁹ windows (anchors 0 and 10¹⁰+2; the report's "anchors 0 and 10¹⁰" is off by the +2 the engine itself requires). Report's 1.17 CONFIRMED. 3. **Cross-window fold-factor deviance**, pooled M̂ per fold, cells λ ≥ 0.5: **G/df = 180.0/183 = 0.983**. Report's 0.98 CONFIRMED. The λ chain was rebuilt by the record's own recipe with my own parser (B2 verbatim for p ≥ 307, formula + m̄ chains below) and revalidated: decade sums at ratio 1.0001, clause replay 32 of 74 exact. 4. **The rival's readings are real, not tuned**: i.i.d. NB(k = 3) data pushed through MY implementations of the same two statistics gives anchor χ²/df ≈ 14.5 and G/df ≈ 74 (their seeded control C read 11.63 and 50.09; same order, both a factor ≥ 10 above the real data). The refutation of window-level mixing stands. ### T1.b Prereg custody and blind re-score — CONFIRMED - **Commit graph**: 199dd33 (10:38:46) contains ONLY the prereg; `attack-perfold-02-blindwindow.js` first exists at f0eb201 (10:50:44). The prereg file is byte-identical between 199dd33 and HEAD. - **The sealed bands had no tuning surface.** Both source embeds predate the prereg by half a day or more (`attack-foldL-06-scaling.js` at f7217ca, 2026-08-19 15:52; `foldL-window5-01-extinction.js` at f34e5c0 22:30, only comments added since). I recomputed every (Sx, Sλ, λ_h) pool myself from those committed embeds: **all 37 match the sealed table exactly**, and my own NB quantile code (different lgamma, different scan) reproduces **all 37 sealed band pairs integer-exactly**. So the bands are a deterministic function of data committed before the prereg — the only freedom was the model formula itself, declared in the same sealed note. - **Anchor freshness**: `git grep` at 199dd33 finds no prior use of 66,000,000,000 (the one hit is FOLD-PROFILE's unrelated 6.6·10⁴p²). - **Blind re-score from my own re-sieve of [6.6·10¹⁰, 6.6·10¹⁰+2·10⁹)** (my engine, my bands): **in-90% = 33/37, outside-99.73% = 0/37, old clause 29/37 = 78.4%** — digit-identical to the producer, misses at the same four folds (107, 113, 137, 257). HIT under the sealed rule CONFIRMED. - Residual, inherent to self-run preregs: git cannot exclude unrecorded pre-runs before 10:38. Priced: under the rival the HIT chance is < 2%, so anchor-shopping cannot cheaply fake the discrimination; low concern. ### T1.c Strongest confound: anchor/scale-dependent mixing — WEAKENED (one sentence) The strongest confound is not i.i.d. mixing (dead, §T1.a) but a **slowly anchor-varying field**: M_{p,anchor} = M_p·exp(δ), δ ~ N(0, σ_δ²) per (fold, anchor). Monte Carlo with the real (M̂_p, λ_p) grid, my own RNG (mulberry32, not the producer's sfc32), 4000 replicates per point: | σ_δ (per-anchor jitter) | mean χ²/df | P(χ²/df ≤ 1.167) | P(HIT-a at blind window) | |---|---|---|---| | 0.00 | 1.00 | 0.82 | 1.000 | | 0.06 | 1.16 | **0.55** | 1.000 | | 0.10 | 1.46 | **0.21** | 0.991 | | 0.15 | 1.99 | 0.03 | 0.915 | | 0.20 | — | — | 0.737 | So the observed 1.167 is fully typical under anchor-to-anchor variation of **up to ~6% relative sd, and only weakly disfavoured at ~10%**; the blind HIT itself tolerates ~15–20%. One anchor pair at one length is simply not enough replicates to pin "identical". **Across window lengths the bound IS tight**: per-window jitter ≥ 3% already makes G/df = 0.983 a ≤ 3% event (my table: σ_w = 0.03 → P(G/df ≤ 0.983) = 0.03; σ_w = 0.05 → mean G/df = 1.62). CORRECTED SENTENCE for the headline: *"M_p is constant in window length to within ~2–3% relative sd over 2·10⁷…2·10¹¹, and constant across the three anchors tested to within the ~6–10% resolution one replicate pair affords — 'deterministic' is the surviving model class, not yet a measured identity in the anchor direction."* Everything else in the note (the NB predictive, the mixing refutation, the blind protocol) survives this refinement untouched, because the predictive bands price fold identity, not anchor identity. ### T1.d The 8/8 import-stein unification — CONFIRMED (wording nit) Recomputed all ratios λ_derived/λ_model at 2·10⁹ against pooled M̂ with my own parse and pools: **8/8 agree, and the producer's +0.05·M̂ tolerance slack never fires — strict ±2se alone passes all 8** (p = 211: |2.189 − 2.225| = 0.036 < 2se = 0.089; worst case p = 331 at 0.60×2se). The tolerance choice is not load-bearing. Nit: import-stein §2.4 prints **14** folds, not 8; the producer's disclosed Sx ≥ 15 precision filter drops 401, 421, 457, 499, 601, 701. I recomputed the dropped six WITHOUT the filter: **all consistent too** (max 0.40×2se, error bars huge), so the filter hides nothing — but "at all 8 folds where §2.4 prints the derived first moment" should read "at all 8 of the 14 printed folds with pooled X ≥ 15 (the other 6 agree trivially)". ### T1.e Is retiring ±3√λ licensed? — CONFIRMED with a scope clause The clause was registered, scored, and failed twice independently: 32/74 = 43.2% at W = 2·10¹¹ (p ~ 10⁻¹⁹ against 90%) and now 29/37 = 78.4% at the blind window (P(≤29 | true 90%) ≈ 0.02, a marginal rejection on its own, and the model predicted 28.1). Retiring it as the per-fold error model of the law, and quoting NB fold-factor bands in future preregs, is licensed — with one scope clause the proposed §5 text should carry: the failure mechanism is the first-moment shift (M_p−1)√λ, which GROWS with window length; at Y ≤ 2·10⁸ the old clause still holds approximately (the P(old-clause) column's small-λ rows are ≈ 1.00) and was never separately refuted there. Since future window preregs run at larger Y, the retirement is licensed in exactly the direction it will be used; §6's honest √(1 + Y/ΣY′) widening already prices the sixth decade. No overreach found in the §7 consequences EXCEPT the T1.c sentence ("identical across anchors") which propagates into prop-thinning-null's proposed replacement text and should carry the corrected wording above. ## T2. item-x-offset.md ### T2.a The two exclusions recomputed — CONFIRMED My own sieve, my own tail pass (independent implementation of the exact U-test), measured (obs, CRT, σ_slot) cited from the xchan embed as the report does: - Prime sums and tail masses reproduce to the digit at all levels (4S₂ = 0.028943 / 0.024784; S₃ = 1.3114e−4 / 9.7575e−5; my rebuilt CRT = 55252747.16 / 1695051393.52, rel 6e−11 / 3e−13). - **C4 = 4S₂−4S₃: z = +1.45 / +5.33 — REFUTED at @31. C8 = 4Σ1/(q(q+2)): z = +3.17 / +12.40 — REFUTED at both.** Identical to the report. - **Locality shape: κ = (1−J)/T3avg recomputed = 120.37 → 228.04 → 371.10 → 571.55** over @19..@31. A constant-κ per-triple law is dead; the 120→572 quote is exact. - The four fitted members re-fit with my own weighted code: ε = 4.7550e−3 ± 2.0124e−3, A₀ = 1.1858e−4 ± 5.0169e−5, c₃ = 1.1978 ± 0.508, c_l = 0.4909 ± 0.208 — all digit-identical. ### T2.b The σ calibration controls — WEAKENED (corrected sentence) The seven control draws are genuine and pass as quoted (z = 0.01; 0.73, 0.25, 1.51; 0.50, 1.88, −0.23). Three limits the verdict sentence hides: 1. **No control exists at @31** — SEEDS = {19:[1], 23:[1,2,3], 29:[1,2,3], 31:[]} in the producer. The one level carrying the detection (z = −2.32) has zero known-truth draws; its inflation (2.124) is measured from the natal data itself. 2. **Seven draws bound the σ scale only to a factor ~[0.65, 2.0]** (Σz² = 6.71 ~ χ²₇; 95% interval on the true/reported σ ratio). A σ understated by 30% — fully consistent with these controls — moves the @31 detection from 2.32σ to ~1.8σ. 3. **Exchangeability is partial by construction.** The control statistic is J_ctrl − J_line: a random mask scored against the same window's own line truth, so any level-scale arithmetic fluctuation CANCELS in the control but does NOT cancel in the test statistic (natal J against an analytic formula). The controls therefore validate unbiasedness and the sampling-noise scale of the estimator, not the implicit null that the full-period arithmetic residual fluctuates at σ_slot scale. CORRECTED SENTENCE for verdict item 1: *"The σ estimator is validated as unbiased at random-mask truth and its scale is consistent within the factor ~2 that seven draws at levels ≤ @29 can resolve; no control was run at @31, and the offset remains one 2.3σ detection whose σ rests on the natal data's own clustering estimate."* This does not reopen the C4/C8 kills (5.3σ and 12.4σ survive any plausible σ miscalibration) but it does mean N1's death warrant cannot be signed at @29/@31 — which the report already says (z = −2.32 is its own §1 "one detection"). ### T2.c The cofactor-1 strip identity re-derived — CONFIRMED as derived-conditional; "exact" needs one word Re-derivation from scratch, in three parts: 1. **The purity claim is a THEOREM and I certify it.** For d = q₁q₂ in (W/2^{s+1}, W/2^s] with 2^{s+1} ≤ 16 < q_min: any cofactor m ≥ 2 has m < 2^{s+1} ≤ 16, so every prime factor of m is ≤ 13, i.e. in {2, 3, 5} ∪ base primes (7, 11, 13 are base at every level here, x ≥ 19). Even m leaves the odd line; 3 | m or 5 | m leaves {11, 17} mod 30; a base factor kills r-side natality; and any factor > x would be ≥ q_min > 16 > m. Hence m = 1. Verified exhaustively at @19 with my own code: **257,704 higher multiples scanned, 0 natal**. 2. **The enrichment factor is exact algebra**: P(natal | pair)/P_smooth = [¼·∏(p−2)/(p−1)] / [(1/15)·∏(p−2)/p] = (15/4)·∏_{7≤p≤x} p/(p−1) = ∏_{p≤x} p/(p−1), since 15/4 = 2·(3/2)·(5/4). The limit e^γ·ln x is Mertens' third theorem. Both CONFIRMED (my arithmetic: 5.8471, 6.1129, 6.3312, 6.5423 at @19..@31, e^γ ln 19 = 5.2443 — all match). 3. **The count law itself is an EXPECTATION under joint equidistribution of q₁q₂ over the unit classes mod 30·∏p — not a CRT counting identity.** The ¼ and the (p−2)/(p−1) are exact only if scour-pair products equidistribute; the producer discloses this ("WHAT THE UNIFORMITY ASSUMPTION IS") but the report's verdict item 5 ("is now DERIVED … derived exactly") compresses it away. I re-counted every strip at every level with my own enumeration: **digit-identical to the producer at all 16 strips** (e.g. @31: 505,212,797 pairs, 68,409,827 natal, dev +0.001%). The deviations are ~0.1–0.2× the Poisson scale of the natal counts (e.g. @31: 734 counts against √n ≈ 8300) — the equidistribution is sub-random-good, which is itself a striking measured fact, not a proof. CORRECTED WORD: "derived exactly, given pair-equidistribution, and the equidistribution measured to 0.001% at @31". ### T2.d POST HOC hygiene of 4S₂−S₃ — CONFIRMED Every occurrence of the candidate in the tree (item-x-offset.md lines 20, 88, 245, 300; xchan-at37-offset-prereg.md line 36) carries POST HOC or the found-against-the-known-residuals framing in the same sentence or row. No live document mentions it. The prereg's upgrade path ("survived one blind level" only if |z_C1| ≤ 3 at @37) is the correct promotion rule. Nothing leans on C1 as established. ### T2.e The @37 kill rule's σ — CONFIRMED Recomputed from scratch: miss(37) = 0.070749 (my own K = 198,274 tail pass), N̄(37) = 145,286,237,250, CRT(37) = 6.1673e10, σ_reg = 3.9826e−6, σ_slot = 2.14 × σ_reg = 8.5228e−6. All 21 pairwise separations digit-identical (N1 vs C1 = 9.1σ, C1 vs M-abs = 4.8σ, C1 vs C2 = 9.1σ). The separations use the SLOT-CLUSTERED σ (2.14× the naive one), i.e. the conservative choice; the inflation 2.14 [2.12, 2.17] is a projection from the measured 2.078 → 2.106 → 2.124 with decreasing increments (reasonable), and the prereg's supersede clause makes the run's own σ authoritative — so even a 2.3 inflation only shrinks 4.8σ to ~4.5σ. The kill threshold |z| > 3 on the run's own slot σ is the right instrument. One dependency to know: σ_reg uses obs ≈ (1−ω)·CRT with ω from C1 — candidate-dependence is O(0.1%), immaterial. ### T2.x Unrequested finding: the "each bin 100σ+" sentence is false at one bin item-x-offset.md §6a says the @31 octave bins "sit 92 to 283σ from the aggregate", verdict item 6 says "each bin hundreds of σ", and §10 says "each 100σ+". The producer's own embedded table has octave bin m = 9 at **(D_m−D)/σ_m = −7.47** and bin 0 at −92.6: eleven of twelve bins are ≥ 92σ, one is at 7.5σ. CORRECTED SENTENCE: *"eleven of the twelve octave bins sit 92–283σ from the aggregate; bin 9 sits at 7.5σ"*. The profile-shape conclusion is untouched (the bins collectively refute any constant-deficit reading at astronomical significance), but the "each" claims should not integrate as written. ## Verdict table | # | Claim | Grade | |---|---|---| | T1.a | anchor χ²/df = 1.17, G/df = 0.98, vs mixing's ~11.6 / ~50 | **CONFIRMED** (twice, once from raw re-sieve) | | T1.b | prereg custody + blind 33/37, 0/37, old clause 29/37 | **CONFIRMED** (bands derivable only from pre-committed data; re-scored from an independent sieve) | | T1.c | "M_p identical across window lengths and anchors" | **WEAKENED**: length-constancy bounded at ~2–3%; anchor-constancy only bounded at ~6–10% by one replicate pair — corrected sentence in §T1.c | | T1.d | 8/8 import-stein unification, p=211 2.189 vs 2.225 | **CONFIRMED** (passes strict ±2se; slack unused; 6 filtered folds also agree; "8 of 14 printed" wording fix) | | T1.e | retiring ±3√λ per §5 consequence | **CONFIRMED** for future (larger-Y) preregs; never separately refuted at Y ≤ 2·10⁸, add scope clause | | T2.a | C4 dead 5.3σ, C8 dead 12.4σ; locality κ 120→572 | **CONFIRMED** (digit-identical recompute) | | T2.b | "the σ behind the @31 detection is calibrated" | **WEAKENED**: no @31 control; 7 draws bound σ scale to factor ~[0.65, 2.0]; control cancels what the test statistic does not — corrected sentence in §T2.b | | T2.c | cofactor-1 layer law exact, enrichment ∏p/(p−1) → e^γ ln x | **CONFIRMED as derived-conditional**: purity is a theorem (certified + exhaustively verified), enrichment algebra and Mertens limit exact, count law is an equidistribution expectation measured to 0.001% | | T2.d | report does not lean on 4S₂−S₃ | **CONFIRMED** | | T2.e | @37 separations 4.8–9.1σ on slot-clustered σ | **CONFIRMED** (recomputed from scratch incl. miss(37); conservative σ family) | | T2.x | "each octave bin 100σ+" | **REFUTED as stated**: bin 9 sits at 7.5σ (11 of 12 are ≥ 92σ); one-sentence fix, conclusion survives | **Most load-bearing correction**: T1.c — the headline word "identical across … anchors" must become a bound (~6–10% from one 2·10⁹ replicate pair) before the model statement integrates or propagates into prop-thinning-null. Second: T2.b — the @31 detection's σ is uncontrolled at @31 itself and the verdict sentence "the σ … is calibrated" needs the §T2.b wording. ## Method and instruments All decisive numbers recomputed on different code paths: an independent streaming sieve (calibrated on the embedded [0, 2·10⁹) record, 214/214 folds at five windows, then run at the blind anchor), an independent NB quantile stack (Stirling lgamma), an independent tail-mass/T3 pass through @37, an independent strip enumerator with an exhaustive purity scan, and Monte Carlo confounds on my own RNG. Scratch producers: rt-t1-sieve.js, rt-t1-stats.js, rt-t1-confound.js, rt-t2-sums.js, rt-t2-cofactor.js in the session scratchpad (not part of the tree; every number above is restated here). ## Not reached - The xchan @29/@31 censuses themselves (obs, CRT, σ_slot) were cited from their embeds, as the producers do — not re-sieved (the @31 census alone is ~50 min). The two CRTs were rebuilt exactly from my own tail masses; obs was not independently re-counted. - Producer 02's line/marginal/profile tables (2948 s census) were accepted from its embed after internal-identity checks (Σ_b = Σ_m = aggregate, gate-vs-xchan PASS lines); its census kernel was not re-executed. The §3 and §5 derived ratios (1.15–1.18×, sign-erratic marginals) were verified arithmetically from the embedded numbers only. - The W = 2·10¹¹ window was not re-sieved (≈ 20+ min); its X entered T1 through the embedded C1 table, cross-checked against ΣX = 2,031,759,826. - The prereg's unscored columns (P(old clause), expected 34.6 / 28.1) were not re-derived; the scored objects (all 74 band integers) were. - embed.js --check was not re-run on any producer (concurrent gate work; judged on content per brief). --- *Adversarial verification note; grades and corrected sentences above are the deliverable. Nothing here edits the audited files.*