# Red team, 2026-08-21 wave 2 (structural/empirical): kstar-01, delta37-01, c2drift-01 *2026-08-21. Adversarial verifier, independent re-computation throughout — own engines in scratch (DFS inclusion–exclusion, own tile sieve, own parses, own RNG for nulls), never the producers' code paths; both published sources re-read at page images this session. Default was refuted-until-rederived. Targets: `attack-kstar-01.md` (+ prereg at `552143c`), `attack-delta37-01.md`, `attack-c2drift-01.md`, all HELD in staging.* ## 0. Verdict summary **All three targets survive. Every load-bearing number re-derived independently reproduced exactly — including four full N_k curves recomputed with a different I–E implementation (validated against brute-force period enumeration), the delta37 credit arithmetic from an independent parse, and the c1 column from a b-file fetched fresh from OEIS.** Two sentences weakened, no claim refuted: - **WEAKENED** (kstar §1): "cert/C₂ widens monotonically in scale" — the five new values run 2.81, 3.41, 4.34, 3.98, 4.67; 17#→31# (4.34) exceeds 19#→37# (3.98), and 13#→23# (2.91) exceeds 13#→29# (2.81). Corrected: *"the slack cert/C₂ trends wider with scale — 1.00 at the first three steps to 4.67 at 19#→41# — though not monotonically (4.34 at 17#→31# against 3.98 at 19#→37#)."* The producer's READINGS block already states it correctly (lists the five values without "monotonically"). - **WEAKENED** (c2drift, commit message `b19d77d` only): "the one-class column drifts identically" — the shared facts are direction and shape; the rates are 0.15–0.22 (c1) vs 0.23–0.43 (c2') per ln ln x, equal only within ~1.6σ. The document's own §1 ("same signature... same direction, same order, wide errors") is the supported statement and needs no change. ## 1. TARGET 1 — attack-kstar-01 (K* ladder 11 → 16, I–E certificate engine) ### 1a. Prereg custody at 552143c — CONFIRMED, one nuance `552143c` (2026-08-21 09:52:26) touches exactly one file, the prereg; the producer + report land in `fa087d3` at 10:15:04 — a 23-minute gap consistent with the 345 s run executing after the commit (git cannot prove the walks did not run earlier; that limit is inherent to any local prereg). The prereg was never modified after commit. The blemish arithmetic checks: applying the prereg's own hist[ℓ] = N_ℓ − 2N_{ℓ+1} + N_{ℓ+2} to its own committed tables gives 36 (not 6) maximal 10-runs at 13#→29# and 6 (not 2) maximal 13-runs at 17#→31# — exactly what the scans found — while the third derived figure (2 maximal 17-runs) is correct. The slips contradict the prereg's OWN committed table under its own stated formula, so they are derivation slips, not failed model predictions; the scoring-rule section (fixed before measurement) binds only K*, the N_k cells, and M1's ±2, none touched. Nuance: "illustrative prose" is slightly generous — the sentence uses binding language ("the scan must find exactly...") — but the disclosure in §4 is complete and accurate. ### 1b. The I–E identity — CONFIRMED (re-derived analytically) Slot x_j alive against Q iff t avoids the ν_q(J) = |{x_j, x_j+2 mod q}| residues; CRT over the entering primes (coprime to the base width) makes the copy index sweep all M residue combinations exactly once, so per shape N_k = Σ_J (−1)^{|J|} Π_q (q − ν_q(J)) is an exact integer identity — the HL 2|J|-tuple local product truncated to Q, as claimed. The pair case ν = 4 − 2[q|d] − [q|d−2] − [q|d+2] holds for all q ≥ 5 (no double collisions possible), which is w1-singular-series' table one level up. Cost D·2^k·|Q| and period-independence follow by construction. ### 1c. The period-free certificates — CONFIRMED by full independent re-derivation Own engine: direct-sieve 19# tile (D = 378675 reproduced) + DFS inclusion–exclusion with incremental residue counts (no Gray code), first validated against my own brute-force full-period enumeration at 5#→11#, 7#→13#, 7#→17# (every N_k cell and K* exact). Then: - **19#→37#: the full 14-value N_k curve reproduced digit-exact, including N_14 = 0** (9.6 s, ~1.44e8 subsets) — K* = 13, C₂ ≤ 14 re-derived. - **19#→41#: the full 17-value curve reproduced digit-exact, including N_17 = 0** (79 s, ~1.04e9 subsets) — K* = 16, C₂ ≤ 17 re-derived. The report's caveat "rests on the I–E route alone" can now cite a second, independent engine. - **All three prereg tables (13#→29#, 13#→31#, 17#→31#) reproduced digit-exact** tile-only (1.3 s) — so K* = 10/17/13 are theorems of two independent engines, and the prereg's binding cells are independently certified, not merely scan-matched. ### 1d. G2 custody gates — CONFIRMED `exact-g2-ladder.js` rows: 258 @ 1205437109n (×2), 348 @ 8813641451n (×4), G2(37#) = 528, G2(41#) = 546 — matching every quoted C₂. Own arithmetic: both argmax windows verified locally (endpoints alive at 29#/31#, zero alive positions strictly inside, spans exactly 258 and 348). ### 1e. The near-linear verdict — CONFIRMED, one honesty note Own regression on the 16-point ladder: 0.6881 ± 0.1328 / 0.8041 ± 0.1098 / 0.8184 ± 0.0908 — all three exact; I_eff range [0.55, 2.56] and trend 0.3676 ± 0.1400 exact. The instrument's logic checks: a log-type K* anchored at 13#→23# (K* = 8) forces I_eff ≈ 3.53 by P(2s) = 43 (×1.74); measured I_eff at 19#→41# is 1.88 — flat. Honesty note: the 16 points are not independent (7#→19# and 11#→19# share a target period, as do the two 31# steps; Q sets nest), so the ±0.0908 understates the true slope uncertainty — the report's own framing ("raw, no null exists for K*", verdict [MEASURED]) already carries this. The 31#→61# band is labeled illustration and stays one. cert = 18 vs 19.2455 at s = 16 (93.5%) is exact arithmetic on now-independently-proven K*. ## 2. TARGET 2 — attack-delta37-01 (δ decomposition, interior comb, x = 37) ### 2a. Interior 6-tuple comb — CONFIRMED (construction re-proven, numbers reproduced) The conditional ratio r_q(j) = [(q−ν₆)/(q−ν₄)]·[q/(q−2)] is the correct endpoint-conditioned survival ratio, and mean-one holds exactly: Σ_j (q−ν₆) = (q−ν₄)(q−2) (proven by the (t, j) double count; verified numerically at the producer's q × θ grid) — so the whole effect is indeed the finite interior range. Own parse of the mp-derive Stage-6 embed (102 rows) + own ell implementation (boolean-array residue counting): ell values match to 4 decimals (0.5509 at p = 101, 0.4651 at 709); **Pearson 0.907 / Spearman 0.842 with the D4 train residuals reproduced exactly**; tilt −0.0208 / +0.0033 / φ_pred 0.0241 exact; not comb-collinear. ### 2b. Combined-credit arithmetic — CONFIRMED, all cells Own fits (train OLS, own ternary-search deep MLE, own Simpson PLN): base 1.9485/0.2773/0.1715/0.2526/φ 0.0247; interior credit 0.2565/0.2374/sd 0.0830; both credits **0.1964/0.1879/φ 0.0085/sd 0.0834**; flags 3.10→2.63, 3.21→2.22, 3.28→2.31, 3.36→2.45, deep |pull|>3 count 4→0; bands [300,500) 0.480 vs 0.520 and [500,710) 0.351 vs 0.357; deep PLN −88.0 → −84.3 (+3.6 nats). Every number exact. ### 2c. Finite-size refutation — CONFIRMED Factorization audited: p | θ+2η always (θ = 2(p−η)), no prime > p divides θ/2, so only the (θ±2)/2 sides contribute — correct. Worst case re-derived by hand: p = 101, (θ+2)/2 = 103 prime, dln = ln(98/97)+ln(100/99) = 2.03e-2 ✓; window-edge (1416+4254)/2.2422e11 = 2.53e-8 ✓. Tilt −0.0024 vs needed −0.277 is 2.05 orders. Nit (producer prose only): the script's VERDICT line "predicts |s| ≤ ~1e-4 ... REFUTED by ≥ 3 orders" overstates its own measured −2.44e-3 (2 orders); the report's table says "2 orders short" correctly and the §0 header's "2–5 orders" spans the edge term. ### 2d. Leave-one-out instrument at x = 37 — CONFIRMED, and conservative Own parse of both ladder embeds, own LOO: G2 +2.39, h +0.14, h2 −0.10 — exact. Sensitivity: dropping the noisy head (x ≥ 11 instead of ≥ 5) makes the G2 reading **+5.01** — the producer's convention is the conservative one; the spike-real verdict strengthens under the variation. ### 2e. The Wolf negative — CONFIRMED for both arXiv sources; 909/96 not re-verified I re-read **all 10 pages of math/0105211 at page images this session**: the characterization is accurate, in places nearly verbatim — m(d,N) defined (eq. 4), Fig. 1's oscillations plainly visible, champions d = 30 then d = 210 named with "It will be discussed in more detail in the forthcoming paper" (p. 7), the only derived separation formulas are μ(s,N) ~ A(N)e^{−B(N)s} with A ≈ c₂²N/ln³N, B ≈ c₂/lnN (eqs. 7, 13), s_max ~ (1/c₂)ln²N (eq. 16) vs arithmetic maximal twin gaps ~ ln³N, and the closing sentence is exactly "the change in the 'measuring sticks' removes oscillations from Fig.1 and leaves pure exponential decrease." **No divisor-modulated twin-separation formula appears anywhere** — the NOT-FOUND stands on my independent read. MS math/0409258 pp. 1–4 also re-read at page images: Theorem 1 (eq. 15) and Theorem 2 verbatim as quoted, unconditional; the ψ normality explicitly "Assuming a strong form of the Hardy-Littlewood conjecture (1)"; nothing indexes by fold depth. NOT re-verified by me: IFTUWr 909/96 (no local copy, publication page not fetched this pass) and the "no champions follow-up on his list" check — both rest on the producer's session. ## 3. TARGET 3 — attack-c2drift-01 (c1 drifts with c2', frame property) ### 3a. c1 column + post-dip slope — CONFIRMED from a fresh OEIS b-file b048670.txt fetched from OEIS this session: 64 terms, **identical to the producer's corpus-58 + Bozek-6 array**. Own frames: c1(11) = 0.4712, min 0.3359 at x = 59, c1(311) = 0.3756; window slopes 0.1553 ± 0.0261 (x ≥ 59), 0.1524 ± 0.0182, 0.2187 ± 0.0442 — all exact. Own 4000-replicate nulls with a DIFFERENT RNG (xorshift vs mulberry32): rank-p = 0.0000 under both noise models; largest null slope seen 0.119 (logn) / 0.074 (EV) vs observed 0.155. ### 3b. Instrument calibration — CONFIRMED Own synthetic-truth recovery on the real grid: flat truth reads b_PLNX = −0.0140 (exact match), Gumbel truth reads −0.1341 (exact) — the bias direction (down, so measured drifts conservative) verified; Gumbel b_FS recovery +0.5164 vs theory +0.577 (rounding attenuation, as printed). The 92.8%/100% power figures were not re-run (same machinery, low stakes). ### 3c. Gumbel finite-size sign-refutation — CONFIRMED Re-derived: EV theory gives c = c∞(1 + γ/lnD), slope +γ on the 1/lnD regressor, i.e. a high head and a FALLING column. Both columns rise post-dip; fitted b_FS: −0.597 (c2' full), −3.522 (c1 x ≥ 59) — both reproduced exactly, both negative. Implied empirical-mirror limits c1∞ = 0.3799, c2'∞ = 0.5345 reproduced. ### 3d. The 1d algebra — CONFIRMED G2/x² = c2'·(m·lnD/x²) is definitional; measured decomposition −0.2365 = +0.0540 + (−0.2905) exact (residual < 1e-12); drift share 18.6%; threshold −dF/dlnx = 0.147; b_PLND = 0.0525 (36% of threshold) — all reproduced. The asymptotic threshold 1 − 2·lnln x/ln x → 1 follows from m·lnD → 2.40·x·ln²x; every fixed (ln x)^s keeps G2/x² → 0 — sound. ### 3e. Shape-indistinguishability — proven by calibration ON-RANGE; the extension is asserted The two cross-talk rows are real calibration facts, reproduced exactly with my own generator: a true (lnD)^0.12 reads b_PLNX = 0.4969, a true (ln x)^0.30 reads b_PLND = 0.0699; corr(ln lnD, ln ln x) = 0.9956 on the c1 grid. So the on-range indistinguishability — the only form the verdict table uses — is demonstrated, not asserted. The further sentence "no reachable ladder length fixes it" extrapolates beyond the grids the calibration ran on; plausible from the collinearity mechanism, but it is an argument, not an in-pass demonstration. ## 4. What this pass did NOT reach - The three period WALKS themselves (6.47e9, 2.006e11 ×2) were not re-run; their custody is the two G2 window checks (§1d), the ladder anchors, and the fact that the scans' claimed censuses equal N_k curves I re-derived independently. A full walk replay stays the outstanding (expensive) check. - kstar §5's record-anatomy details (ρ ladder, strikes/kills 86/84 across fourteen steps, inherited shares) — not re-derived. - delta37: the THN r_null formula was re-evaluated as given (tilts match) but not re-derived from import-thinning's own derivation; the 5-figure Sl = 1.1211·λ₁₁ upgrade was replayed, not re-derived; IFTUWr 909/96 and the Wolf publication-list negative (§2e). - c2drift: AICc tables beyond the key fits, the power figures, and the ledger deep-window AICc win were arithmetic-spot-checked (5.5× overshoot, 13.2% vs 3.1% rise re-derived by hand) but not fully re-run; the single-witness status of the c2' tail and Bozek terms stands as the report already flags it. - The null ensembles everywhere treat ladder rows as independent; the rows are separate exact computations per level, so this is reasonable, but no dependence-aware null was constructed here either. --- *This is a history/staging record: a process document, superseded-in-place by later work.*