# Red team 0821: the two exponent-adjacent held headlines *2026-08-21, adversarial pass. Targets: `attack-rhoms-01.md` (the ⟨ρ̃²⟩ theorem + the 3.5+o(1) cap) and `attack-doubling-01.md` (C₂ table, Bridging Lemma, closure refutations). Method: refuted-until-rederived; independent re-derivation in own code (scratchpad `rt-rhoms-verify.js`, `rt-doubling-verify.js`), own lattice construction, own walks, own OLS. Grades: CONFIRMED / WEAKENED (corrected sentence) / REFUTED (counterexample).* **Bottom line: no numerical discrepancy found anywhere in either target. Every figure I re-derived — from an independently constructed lattice and independently coded walks — reproduces to the printed digit. The 3.5+o(1) claim survives the specific attack it was most likely to die to (a hidden e^{θ/3}); what it is NOT is an unconditional bound on G₂, and the two places where the report's wording runs a half-step ahead of its own custody grades are recorded below as WEAKENED riders.** ## 1. Target 1 — attack-rhoms-01 (the exponent claim) ### 1.0 The headline verdict on "3.5+o(1)" The headline decomposes into three claims of different logical strength, and the report's fine print keeps them apart correctly: 1. **⟨ρ̃²⟩ ≤ C z^{2s} ln⁸z unconditionally (rms ≤ z^{3+o(1)}) — CONFIRMED.** All three proofs re-derived independently (§1.1, §1.3); every verification chain holds on my own lattice. 2. **θ_G capped below β₂ at every measured z and at 3.5+o(1) asymptotically, unconditionally — CONFIRMED as arithmetic**, with the essential caveat stated plainly: θ_G is the bookkeeping exponent of the Gaussian FORM (2√(2·lnW·⟨ρ̃²⟩)/M). Capping it caps what F4 *would* deliver. It moves nothing unconditionally: **the two-class exponent 4.2665 is NOT moved by this result.** The genuine unconditional content is λ_max = β₂ − θ_G ≥ 0.766 − o(1): the sufficiency curve's room can never close, so the route stays alive at every large z. The report itself never claims an unconditional G₂ bound; a hurried reader could take one from headline two, and the corrected reading is the above. 3. **F4 ⟹ G₂(z#) ≪ z^{3.5+ε} — CONFIRMED as a conditional implication** through the pricing lemma, with the mean-square factor now a theorem (see §1.4 for the one literature-grade link and the "no other hypothesis" rider). **The suspected break — a θ/3 hidden inside the 3.5 — is not there.** Checked by direct computation (own θ(z) by sieve): the F4 route's width factor is √(2·lnW) = √(2θ(z)), whose base-z exponent is 0.559 at z = 47, 0.546 at 1009, 0.530 at 10⁵, → 1/2: polynomial, as claimed. The e^{θ/3} factor lives ONLY in the R1 recovery, whose base-z exponent I get as 3.2 at z = 47, 46.1 at 1009, 2886 at 10⁵ — and the report itself declares R1 dead at z = 47 for exactly this reason. No quantifier laundering, no finite-z calibration promoted to an asymptote: the finite-z window (47, ~2767–3547] where only models cover the composition is stated in the report rather than hidden. ### 1.1 MS3: the sine identity and its application — CONFIRMED - **Identity re-derived**: Σ_{a=1}^{e−1} 1/(4 sin²(πa/e)) = (e²−1)/12 is the classical cosecant identity (Σ csc² = (e²−1)/3, quartered). My own check: hand-exact at e = 2, 3, 4, 6 (3/12, 8/12, 15/12, 35/12), worst rel 3.2e-15 over e = 2..40 and 2.2e-14 over e = 41..200. - **Application sound, direction sound**: ⟨ρ̃²⟩ = Σ_e Σ*_a |Θ_e(a)|²/(4sin²(πa/e)) (spectral identity — verified on my own lattice at z = 13, 17 to rel ≤ 4.7e-13, with Θ_e(a) computed by my own direct phase sum); each |Θ_e(a)|² ≤ Θ*(e)², and extending the primitive-a sum to all a ≠ 0 only adds positive weights, so the identity gives Σ Θ*(e)²(e²−1)/12 ≤ Σ Vabs(e)²·e²/12 = B₂/12. Θ* ≤ Vabs held at every mode I computed (triangle inequality over the V(e₁,e₂) decomposition, which I rebuilt from my own d₁ bookkeeping). My B and B₂ from scratch: B = 1.3833 / 1.4214 and B₂/12 = 19.710 / 77.252 at z = 13 / 17 — the cited figures to the digit. B₂ ≤ qmax·B holds because every e divides some q_i. Chain verified end-to-end: 1.095 ≤ 3.340 ≤ 19.705 ≤ 19.710. ### 1.2 The cube-root recovery R1 and the W^{1/3} arithmetic — CONFIRMED Re-derived from scratch: if S = sup|ρ̃| is attained and increments are bounded by C_L (ρ(y+1) − ρ(y) = M − cc(y+1), an identity I verified at 0 violations over the full z = 13 period with my own cc), then on one side of the peak |ρ̃| ≥ S − C_L·t for 0 ≤ t ≤ S/C_L, so W·⟨ρ̃²⟩ ≥ Σ_t (S − C_L t)² ≥ S³/(3C_L) by integral comparison — the stated sup ≤ (3 C_L W ⟨ρ̃²⟩)^{1/3}, with W the period (the correct normalization: ⟨·⟩ is a period average, so W·⟨ρ̃²⟩ is the period total the one-sided peak count is compared against). Numerically at z = 13: W·ms = 2529.6 vs sup³/(3C_L) = 5.7 — holds with room. And the exponent arithmetic is as the report says: W^{1/3} = e^{θ(z)/3} is superpolynomial (θ/3lnz → ∞), so R1's composed exponent (2c + θ/lnz + log_z 3C_L)/3 diverges; the report prices R1 as a finite-z clearing at z ≤ 29 and a certified failure at 37..47 — my recomputation agrees (§1.6). The "proven finite-z clearing" language is earned: at z ≤ 29 every input (C_L, ms, W, M) is exact and independently reproduced here. ### 1.3 MS1 and MS2 re-derived — CONFIRMED Own lattice first: I rebuilt the certificate from the definitions only (own Rosser support recursion, own CRT, three blocks (−,+),(+,−),(+,+), gcd(d₁,d₂)|2, w = ε·μμ) and compared against the repo's `buildTerms`: **term multisets IDENTICAL at z = 13, 17, 19, 23** (n = 852, 2236, 4764, 9636; M to 0 ulp). Everything below runs on MY lattice. - **MS1**: |C_ij| ≤ (g²−1)/(12qq′) re-derived (max of |(g²−1)/12 − d̄(g−d̄)/2| over d̄ is at d̄ = 0 for g ≥ 2, since (g²+2)/24 ≤ (2g²−2)/24); max measured ratio 1.0000, never above. Closed-form pair sum matches my brute-force period walk at z = 13 to rel 2.8e-15, and matches the report's ms at 13..23 to all printed digits. Multiplicity cap: per block, ordered pairs with lcm q and gcd | 2 number τ(q) (odd q) or (3/2)τ(q) (even q) — re-derived; three blocks give (9/2)τ(q). GS and the assembled (81/4)z⁶P₄P₂² bound hold at 13..23 (GS matches to 5 digits). The Mertens step Σ_{g≤qmax} 4^{ω(g)} ≤ qmax·Π(1+4/p) and qmax ≤ D² = z^{2s} both check. - **MS2**: the H-average transfer re-derived: avg_{H≤K} 4sin²(πaH/e) = 2 − 2Re[(1/K)Σe(aH/e)] ≥ 2 − e/(Ka′) ≥ 1 once K = qmax ≥ e (Dirichlet bound |Σ| ≤ 1/|sin(πa/e)| ≤ e/2a′, classical); per-mode violations 0 at z = 13, 17 in my own sweep; chain 1.095 ≤ 2.190 ≤ 1.598e3 (z = 13) and 2.497 ≤ 4.993 ≤ 2.134e4 (z = 17) reproduced. Theorem A checked at its source (`attack-beta2-01-lemmaV-meansquare.md` §1: for every H, every z, every s) and spot-checked with my own spectral ⟨R_H²⟩: max ratio to B·H = 0.0258 over four H at z = 13. B = O(log⁸z) confirmed present as Theorem 1 of `attack-AB-bounded.md` §1.1 (statement checked; its proof internals not re-audited — see NOT REACHED). ### 1.4 The corollary chain to G₂ ≪ z^{3.5+ε} — CONFIRMED, one rider The pricing lemma (`rho-maximal-law.md` §2) does NOT invoke Lemma V: its chain is (i) the certificate identity T = HM + R_H (verified by telescoping my increment identity, 0/2309 violations), (ii) |R_H| ≤ 2 sup|ρ̃| (trivial), (iii) the choice of H, (iv) the vector-sieve pointwise minorant cc(r) ≤ 1_{r rough}1_{r+2 rough} — **verified at all 2310/2310 positions of the z = 13 period with my own indicator** — and (v) M ≥ c(s)/ln²z. So "F4 alone" is accurate against the corpus's own ledger: F4 is the only unproven CORPUS object in the chain. **WEAKENED (wording rider): "with no other hypothesis" should carry the same grade its source carries for step (v).** M ≥ c(s)/ln²z at s = 3 > 1+√e is standard vector-sieve main-term positivity — literature-standard, measured flat (M·ln²z = 0.34–0.37), but not a corpus-machine-verified theorem. Corrected sentence: "F4 alone now implies G₂(z#) ≪ z^{3.5+ε}, with no other unproven hypothesis — the remaining links are proven identities plus the standard linear-sieve asymptotic M ≍ 1/ln²z." The same rider attaches to the asymptotic cap (its 2lnlnz/lnz term is 1/M). Second fine-print item: headline two's "re-entering below permanently after" — the window boundaries [23, ~2767–3547] are computed under the MEASURED-B models (B = 1.5 / B-drift), the scan runs to z = 150000, and under the proven B = O(ln⁸z) (explicit constant unknown) the re-entry point is finite but not located. Reading 3's own grade ("exact at z ≤ 47, crude-proven past re-entry, model-covered between") is the correct statement; the word "permanently" in headline two inherits the model constants. I reproduced the crude window myself: [23, 2767], peak 4.7391 at z = 31, to the digit. ### 1.5 The multiplicity lemma, both directions — CONFIRMED Own counts on my own lattice at z = 13, 17, 19: max m(q)/((3/2)τ(q)) = 3.0000 exactly (at q = 330, 546, 714) — the orphan's (3/2)τ(q) lemma is FALSE, refutation reproduced; max m(q)/((9/2)τ(q)) = 1.0000 — the (9/2)τ(q) cap holds and is tight, and its derivation re-derives cleanly (§1.3). Both directions land exactly as the report states. ### 1.6 The budget table, three rows recomputed — CONFIRMED Rows z = 13, 29, 47 rebuilt with my own arithmetic (own lnW by sieve, own M/ms/C_L/m₄/m₆ at 13 and 29 from my own full-period walks — the z = 29 walk is my own strided implementation over all 223092870 positions; at 47 the cited ms = 18.99742² and C_L ≥ 10.023 with Gaussian-model moments, as the table's asterisk declares): z=13: budget 383.5, R1 consumed 0.266, M2 0.134, M3 0.047, k_min 1 z=29: budget 1208.1, R1 consumed 0.688, M2 0.458, M3 0.256, k_min 2 z=47: budget 963.4, R1 consumed 1.509 (R1 > smax), M2 1.077, M3 0.652, k_min 3 — all to the printed digit, including R1's certified failure at 47 and M3 clearing every row. The θ_G column and the cap column re-verify: my own subset sweep at z = 29 and 47 gives B = 1.4660/1.5135, B₂/12 = 6.4627e2/1.5955e4, cap3 = 2.7323/2.9740 (−1.5342/−1.2924 against β₂) — identical. The z = 29 walk also reproduces sup = 17.90249, m₄/m₂² = 2.918, m₆/m₂³ = 14.00, m₈/m₂⁴ = 93.5, C_L = 3.032: the sub-Gaussian moment headline (reading 6) is independently re-measured at 13 and 29. ## 2. Target 2 — attack-doubling-01 ### 2.1 The Bridging Lemma and three certificates — CONFIRMED Lemma re-derived: level-2s slots satisfy the level-s admissibility condition plus more, so they are a sub-pattern of the periodic level-s slots; a level-2s gap therefore decomposes as k+1 consecutive level-s gaps whose k interior slots are each killed by an entering prime (they survive p ≤ P(s) but are not level-2s survivors), each gap ≤ Ĝ(s), and the k killed slots form a consecutive killed run, so k ≤ K*. Hence Ĝ(2s) ≤ (K*+1)Ĝ(s). Airtight, endpoints included (level-2s survivors are level-s slots), cyclic wrap included. Certificates checked with my own walker (direct kill-test sieve, not the fold recursion, not the report's walk code) at three steps: 5#→11# K* = 3 (cert 4), 7#→19# K* = 8 (cert 9), 13#→23# K* = 8 (cert 9) — matching the published list — with C₂ ≤ cert at each (3.50 ≤ 4, 5.00 ≤ 9, 3.0909 ≤ 9). The hand-checkable first step (2#→3#: survivors ≡ 5 mod 6, window [5,11], kills 7 and 9 by the entering 3, cert 3 = C₂) also re-derives. ### 2.2 The decomposition at three steps, own tile enumeration — CONFIRMED At 5#→11#, 7#→19#, 13#→23# my walks reproduce every published field: G₂ = 42@899(x4)/150@659(x20)/204@76166567(x4), alive = 135/378675/ 7952175 (= Π(p−2)), k = 3/8/8. Decomposition closes to the digit at all three (Σ spanned gaps = L, k+1 gaps), every bridged copy killed by an entering prime only, strikes/multi = 3/0, 9/1, 8/0 — the single multi-killed copy of the corpus claim (779 at 7#→19#, struck by 11 & 19) sits exactly where reported. The record-anchoring refutation reproduces: at 13#→23# the window's max spanned level-13 gap is 42 < Ĝ(13) = 66 and NO copy of the level-13 record gap lies in the window — first-class refutation of locate-over-the-old-record strategies, confirmed. kcap (20/86/64) and θ (5.61/16.76/20.45) recompute to the digit. ### 2.3 The "kills every constant-factor sharpening" quantifier — CONFIRMED as scoped The corollary's logic re-derived: the AP cap linearizes to L ≤ Ĝ(s)(1+2#Q) + θL, which closes iff θ < 1; a constant discount c closes iff cθ < 1, i.e. c < 1/θ; and θ = 2Ĝ(s)Σ_{q∈(s,2s]}1/q → ∞ because even the trivial Ĝ(s) ≥ s − O(1) (the [2, P(s)] block is dead) gives θ ≳ 2s·ln2/lns. So the divergence — and with it the corollary — is proven-grade. **The quantifier is correctly scoped in the text**: §3 says "proves any all-s C₂ *through this route*" and the producer says "no constant-factor sharpening *of residue-density counting*". It kills constant-factor discounts of the AP residue-density closure only — NOT placement-structure routes, which the report itself names as the open successor target. Anyone quoting the §0 headline ("takes every constant-factor sharpening down with it") should carry the scope with it; the body already does. ### 2.4 The K* drift regression — CONFIRMED, sensitivity recorded Own OLS on the 11 (P(2s), K*+1) points: slope 0.6881 ± 0.1328, and C₂ on the same points 0.2018 ± 0.1412 — identical to the report. The fit is honest for what it claims: raw, no null, flagged as such, and "positive at 1σ" is what the numbers say. The "~69" crossing is anchor-dependent: last-point anchor gives 69 (what the script computes), the fitted intercept gives 78, and slope ± 1σ moves it across 58–90. The report brackets it as "illustration, not a claim", which is the right grade; anyone promoting the 69 should not. ### 2.5 Two spot rows of the C₂ table — CONFIRMED (four rows) From my own parse of the two ladder files (14 exact terms agree with A144311+1 on all shared indices): s = 13 → 204/66 = 3.0909, s = 16 → 348/66 = 5.2727 (P(s) = 13, P(2s) = 31, both custody), s = 23 → 618/204 = 3.0294, s = 41 → 1710/546 = 3.1319. My own full-table sup/inf: 5.2727 at s = 16 / 2.0000 at s = 3 — sup on the chain, custody = trusted. Landing arithmetic: log₂(348/66) = 2.3985, 2^4.26645 = 19.2455. The discriminating statistic reproduces from my own fit: G₂-minus-POW slope −0.0336 ± 0.0596. Note rows s ≥ 24 rest on the A144311 literature column via the repo keeper (as the report itself flags); I did not re-verify those terms against an external source. ## 3. NOT REACHED - **Theorem A's and Theorem 1's proof internals** (`attack-beta2-01` §1, `attack-AB-bounded` §1.1): statements checked at source and spot-checked numerically here; the proofs themselves were not re-audited this pass. - **The cited rmsr at z = 37..47** (ms inputs to the budget table's starred rows): custody-bound O(n²) computations (~2800 s at 47), taken as cited; my independent checks stop at the z = 29 full period and the z = 47 B/B₂/cap sweep. - **The crude-window scan beyond my z ≤ 10⁴ reproduction** (the producer scans to 1.5·10⁵; I reproduced the window and peak, not the full tail). - **The z = 31 walk** (same as the report's own NOT REACHED). - **Doubling: the other eight enumerable steps** re-walked only via the report's own checks (my three-step sample all match); rows s ≥ 24 against an external A144311 source; the `--force` re-embed's replaced block (tail says 0 of 195 figures changed; not independently diffed). - **M ≥ c(s)/ln²z from the literature** (the §1.4 rider): not sourced to a specific vector-sieve reference this pass. ## 4. Gate `node research/qc.js` after this report: TOTAL 2 — the two known transient embed reds (`xchan-at37-01-census.js`, mid-flight census; `attack-x-offset-02-profile.js`, mid-edit by the repair agent), both pre-existing, neither naming either target or this file. `qc/embed.js --check` on both producers: code-sha256 and out-sha256 match, deterministic re-runs. Own scripts: scratchpad `rt-rhoms-verify.js`, `rt-doubling-verify.js`, `rt-cap47.js` (session scratchpad, not part of the corpus).