# Red team 2026-08-20 — the three held mathematical headlines, adversarial pass *2026-08-20. Adversarial verifier, branch `opus-try`. Brief: break, before integration, (1) `fekete-1d.md`, (2) `smoothness-front.md`, (3) `import-hypergraph.md` (+ its prereg at 469aaa3 and producers). Method: refuted-until-rederived; every checked claim graded CONFIRMED / WEAKENED (with the corrected sentence) / REFUTED (with the counterexample). Independent re-derivations live in this session's scratchpad as `rt0820-*.js`; published math re-read from rendered page images only (pages re-rendered by this pass from the PDFs of record already on disk; identities of the PDFs re-checked with `pdfinfo`). This file is the pass's only repo write. Nothing committed, nothing pushed.* --- ## 0. Verdict summary **The single most load-bearing correction:** `import-hypergraph.md` §0.3/§5(3) — "the engine can never carry a ln-power; leftover floor 1/log₂x" is **REFUTED as a universal claim**: (4.1) with bounded-size edges permits depth `m ≈ 0.434·log₂x` and a leftover floor `(ln x)^{−log₁₀5} ≈ (ln x)^{−0.699}`; the printed floor is correct only for Cor 4 (its own hypothesis (4.21)) and for FGKMT-sized edges (`r ≍ log x`). The row's verdict survives on the corrected floor (0.699 of a log is still short of Maier–Pomerance's +1 log and two logs short of the K–K reading, and the C-window wall is intact), but the sentence must change before integration (§3.2 has the corrected wording). Scoreboard (details in the numbered sections): | # | claim | verdict | |---|---|---| | 1a | fekete threshold/freeze arithmetic (S(16), caps, flip, sensitivity) | **CONFIRMED** (all re-derived from scratch; §1.1) | | 1a′ | trusted A144311 note: freeze survives, but riser has FIRED at trusted grade — operative threshold 1.3946 at b=66, trap window widens to [1.0761, 1.3946) | **WEAKENED** — corrected sentence in §1.2 | | 1b | bounded-defect Fekete lemma, 8-line proof + (H-mono) discharge | **CONFIRMED** (§1.3) | | 1c | "1.3946 literature-conditional on Wang 2024" | **CONFIRMED** + trust rider (§1.4) | | 1d | boundedness instrument null design | **CONFIRMED** (§1.5) | | 2a | H^0.818235 exponent chain, frontier 0.354437→0.393922 | **CONFIRMED** (independent implementation; §2.1) | | 2b | Y_N-axis exhausted; Theorem A third-term reading | **CONFIRMED at source** — (1.5), Assumption 14, Thm 2 envelope, Cor 18's ℐ² all verbatim (§2.2) | | 2c | smooth-profile impossibility + Fourier null | **CONFIRMED**, one nit: k=0 mode omitted from the null (measured ≤ 2.8e−3 of mass — verdict unchanged; §2.3) | | 2d | KMS prime-modulus quote | **CONFIRMED verbatim** (§2.4) | | 3a | C-window closed form (5/4)ln5 vs 2ln(5/3) | **CONFIRMED at source** (§3.1) | | 3b | yield cap 10^{m+2} ⟹ floor 1/log₂x | **REFUTED as stated / row verdict survives** (§3.2) | | 3c | held theorem G₂(x#) ≫ x ln x | **CONFIRMED** — every ingredient at source, chain re-derived, finite cover independently rebuilt and replay-clean (§3.3–3.4) | | 3d | verdict-vs-prereg grading | **CONFIRMED no overgrade**, one asymmetric-label nit on N3 (§3.5) | ## 1. TARGET 1 — fekete-1d.md ### 1.1 Threshold and freeze arithmetic, re-derived from scratch — CONFIRMED `rt0820-t1-threshold.js`, a from-scratch implementation sharing no code with the producers (data: custody values + A144311+1, parity-guarded). Every number reproduces: - `S(16) = ln(256/66) = 1.35552`, and it IS the custody max over all integer bases `b ∈ [2,47)` (runner-up `b = 28` at 1.34629 — not close to the caps). - Window caps from the greedy floors (floors are corpus certificates, CITED): `ln(2704/705) = 1.34429`, `ln(3364/870) = 1.35239`, `ln(3600/954) = 1.32803` — all `< 1.35552`, so the freeze is real: since `S(b) = ln(b²/Ĝ)` falls as `Ĝ` rises and the enumeration can only return `G₂ ≥ floor`, **no enumeration outcome at 47, 53 or 59 can raise the custody threshold.** The refutation of 1c's "each new ladder x raises 1d's TPC threshold" stands, and the certification logic (floor ⟹ upper cap on the window's best S) is valid as an inference. - Riser condition: `S(66) > S(16) ⟺ G₂(61#) ≤ ⌊66³/256⌋ = 1123`. Confirmed. - Sensitivity: `ln(708/705) = 0.00425`; sup `R = 2.93333` at `(4,10)` on the custody, `st ≤ 79` and `st ≤ 82` domains alike (pair counts 50/104/111, matching); flip value `G₂(47#) ≥ 740` (A144311 low by 32 = 4.5%); the 47-window's ten pairs reproduce to every printed digit, including `D(4,12) = 1.0330` and `(7,7) = −0.2400`; TPC window `ln C ∈ [1.07614, 1.35552)`, width 0.27938; the floor-7 domain-cap repair (`1.5533` at `(8,9)`, `st ≤ 79` → `1.9000` at `(8,10)`, `st ≤ 82`) exact. One wording nit, not a defect: §2's parenthetical names p = 37 and p = 71 around the 47-window's rank; the actual #2 window is **p = 23 at 1.0415** (71's 1.0202 is #4). "Ranks 3rd of 21" itself is correct (re-derived). ### 1.2 The trusted ladder moves the operative threshold — WEAKENED (freeze survives; the frozen VALUE does not) Since the report was written, the full A144311 ladder (22 terms to x = 79) was adopted as trusted (`research/a144311-full-ladder.js`, hard-gated on the 14-term custody overlap). Re-derived consequences: - `S(61) = ln(3721/1080) = 1.23703` — the base 61 itself raises nothing; the riser base is 66, in the window `[61,67)` where `Ĝ = G₂(61#)`. - `S(66) = ln(4356/1080) = 1.39459 > 1.35552`: the published 1080 sits under the fire line 1123, so **at trusted grade the riser has FIRED**. Full-ladder threshold: 1.39459 at b = 66 (top-5: 66:1.39459, 78:1.38039, 82:1.36919, 65:1.36406, 16:1.35552) — re-derived, matches the producer's cross-check. - **The freeze claim survives verbatim** — 47/53/59 certified unable, first riser x = 61 — but the report's correction (ii), "the custody-grade threshold is 1.3555 and is **effectively frozen there**", is now misleading as a statement about the repo's operative number. Corrected sentence: *the custody-grade threshold is 1.3555 and no in-reach enumeration can move it; at trusted grade (series rule 2026-08-20) the operative threshold is 1.3946 at b = 66, so the trap window is `ln C ∈ [1.0761, 1.3946)`, 0.3185 nats wide, its top 0.039 nats resting on trusted-not-custody terms.* - The §3 sentence "with no literature term anywhere in it" (about the TPC window) survives only for the custody window `[1.0761, 1.3555)`; the trusted window's extension is literature-borne by construction. - Robustness worth recording: beating custody's 1.3555 does not hang on a(18) alone — b = 78 (a(21), `G₂(73#) = 1530`) gives 1.38039 and b = 82 (a(22)) gives 1.36919. All three risers are Wang-2024 terms, so the custody/trusted split is really a custody/Wang split. ### 1.3 The bounded-defect Fekete lemma — CONFIRMED (proof correct, hypotheses honestly discharged) The eight-line proof in `attack-fekete-1d-02-lemma.js` was re-derived line by line, not re-read: - limsup: `f(b^{k+1}) ≤ (k+1)f(b) + kK` by induction on (H-sub); for `n ∈ [b^k, b^{k+1})`, (H-mono) gives `f(n) ≤ f(b^{k+1})` and `ln n ≥ k ln b`, so `f(n)/ln n ≤ ((k+1)f(b)+kK)/(k ln b) → (f(b)+K)/ln b`; taking inf over integer bases b ≥ 2 (which is the same index set as the inf defining L) gives limsup ≤ L. liminf ≥ L − K/ln n → L from the definition of inf. Correct; `f ≥ 0` is used only for `L ≥ 0`; no circularity. - (H-mono) discharge: the one-line argument is sound. Slots of `T_next` mod the old period satisfy a superset of the old kill conditions, so the new slot set is contained in the periodized old one; a subset's maximal circular gap is ≥ the superset's (the superset's maximal gap contains no subset point); the set is nonempty (`x#−1` is always a twin slot). The seven-fold computational check is corroboration, not the proof's load. - The two equivalences of §6 re-derived exactly: `β < 2 ⟺ ∃n: S(n) > K` (strictness both ways checks), and `(ln 66 + ln 2.93333)/ln 16 = 1.8992` reproduces. - (H-sub) is correctly carried as OPEN and correctly identified as the entire remaining content; the report never uses it unconditionally. The BGT remark (no split needed) is accurate: the proof never decomposes an integer. Not re-run: the POW/LOG finite-inf calibration (+2.9%/+53%) — control-only, not load-bearing for any verdict. ### 1.4 "1.3946 is literature-conditional on Wang 2024" — CONFIRMED, with the trust-adoption rider The value 1.3946 needs `Ĝ(66) = G₂(61#) = 1080` = A144311 a(18), which is a Wang-2024 term (provenance read at the OEIS entry per `a144311-full-ladder.js`). Correct as written. Rider from §1.2: after the series-rule adoption the corpus's own grading of that number is now "trusted", so the report's custody/literature dichotomy should be restated as custody/trusted — and the weaker claim "the threshold exceeds 1.3555" is robust across three independent Wang terms (a(18), a(21), a(22)), while the exact 1.3946 is a(18) alone. ### 1.5 The bounded-consistent instrument — CONFIRMED as designed and as bounded-CONSISTENCY only `rt0820-t1-instrument.js`, independent re-implementation: slopes and contrasts reproduce to all printed digits (G2-full +0.1661 ± 0.0637 vs POW +0.2075 ± 0.0558, contrast −0.0414 ± 0.0846; custody contrast −0.0188 ± 0.1466; LOG null −0.2415 ± 0.0554, so the instrument is genuinely not pinned positive). The null design is like-for-like (same primes, same step-function construction, same windows, same pass), which is exactly the discipline the step-sampling artifact demands. The verdict text correctly claims only consistency-with-bounded at 1 s.e. and §4's blind-spot paragraph correctly concedes the diagonal. Nothing here is overclaimed; the [MEASURED] grade is apt. (Minor: the custody LOG null slope −0.1433 ± 0.0999 is unquoted in the report; immaterial.) ## 2. TARGET 2 — smoothness-front.md ### 2.1 The exponent chain H^0.818235 — CONFIRMED (independent implementation, all five calibration rows exact) `rt0820-t2-frontier.js` implements the block bound from the report's printed `pre`/`T1`/`T2` formulas and, separately, a DI form `J² = CS(RS+N)(C+DR) + C²DS√((RS+N)R) + D²NR`, with a fresh scan (10⁻⁵ grid in `a′`, bisection in `σ`). Everything lands on the printed digit: - Calibration: Pascadi `Y_N = 1` frontier 0.354437; full-rough worst block 1.818235; all-smooth 0.867941 (Pascadi) and 0.962157 (DI); DI frontier 0.151372. All five match `lemmaV-neighbours.md` §5. - The new rows: Vaaler-only frontier **0.354437 — unchanged**, exactly as claimed; Theorem-A-third-term-only and both: **0.393922**; shortfalls 0.857720 → **0.818235**. Binding at `a′ = 0` (`T2 = 2.000000` there, slack at every other block), exceptional factor dead there for `σ > 1.424314/4 = 0.356079`, and the `Y_N`-free ceiling `1.212157 + 2σ ≤ 2 ⟹ σ ≤ 0.3939215` reproduces — the "impossible past 0.393921 whatever `Y_N`" sentence is exact arithmetic, not rhetoric. - The ridge claim checks: total affordable rough mass stays 0.393922 for `ρ ∈ [0, 0.145]`, collapses by `ρ = 0.2`. - The KMS-fantasy row: `(MN)^{−3/16}q^{11/64}` = `H^{0.028652 − 0.1875a′}`, ≥ 1 for `a′ < 0.152812`, `H^{−0.011127}` at `a′ = 0.212157`; and the Kloosterman-completion prices `(D⁻)^{3/2}M₁^{1/2} = H^{1.318236}` = trivial × `H^{0.106079}` — both re-derived. - Downstream percentages: 0.393922/1.212157 = 32.50%, 0.354437/1.212157 = 29.24% — the §6(1) correction figures are right. ### 2.2 The Y_N axis and the Theorem A third term — CONFIRMED at source (pages re-rendered and read by this pass) Read from page images rendered by this pass from the PDF of record (`pdfinfo` title/author verified; arXiv:2404.04239v3): - **Theorem A, p. 3:** (1.4) is exactly as the report renders it, and the range (1.5) is verbatim `X ≪ max(1, q/N, q²/N³)` — the third term is real, unconditional, for arbitrary complex sequences, any `q ∈ ℤ₊`. (P. 6's (2.3) even shows its mechanism: Weil + Cauchy–Schwarz.) The derived move `Y_N := max(1, q/N, q²/N³)/max(1, q/N)` (= `max(1, q/N²)` for `q ≥ N`) is airtight: (1.4)'s sequence is arbitrary, so the twist `e(nξ/N)` absorbs at the same ℓ²-norm, and Assumption 14's `(1+|ξ|²)` divisor only shrinks the required range. Re-derived, not just re-read. - **Assumption 14, p. 31:** verbatim as quoted, including the range `X ≪ max(1, q/N)·Y_N/(1+|ξ|²)`, the `Y_N = 1` example-tuple sentence, and (5.18)/(5.19). A tuple certified at `Z = 1` also serves any `Z ≫ 1` (the RHS `(qNZ)^ε` only grows), so feeding Cor 18's `Z` is legitimate. - **Theorem 2, p. 4:** the envelope clause is verbatim: "*The same result holds if `e(nα)` is multiplied by `Φ(n/N)`, for any smooth function `Φ : (0,4) → ℂ` with `Φ^{(j)} ≪_j 1`*", range (1.7) as quoted; the parity split at `α ∈ {0, ½}` with `T_N(½) ≤ 2` checks. - **Corollary 18, pp. 37–38:** hypothesis list matches the report's item 1–8 rendering, including the load-bearing derivative asymmetry (the printed condition is `∂_x^j∂_y^k∂_z^ℓ Φ_q ≪_{j,k,ℓ,ε} Z^{jε}` — only `j` in the exponent) and the norm remark on p. 36 (norms over `(r,s)`, "but not `n ∼ N`"). The conclusion (5.35) with `ℐ² = D²NR + (1 + C²/(R²SY_N))^{2θ_max} CS(C+DR)(RS+N)` is verbatim, so the `T1`/`T2` translation is faithful, and since `Y_N` enters ℐ² ONLY in the exceptional factor, the "Y_N axis exhausted at 0.393922; the binding object is the regular-spectrum main term `CS(C+DR)(RS+N)`" claim is a theorem about Cor 18's printed shape, airtight. (It is, as the report itself scopes, a statement about this corollary, not about all possible spectral inputs.) ### 2.3 The Fourier-flatness impossibility and its null — CONFIRMED, one instrument nit (k = 0 excluded) - **The "necessary" quote, p. 32:** verbatim, including the word "here": "*While the smooth weight in the `c` variable is necessary here (stemming from Proposition E), the smooth weight in `n` only confers additional flexibility.*" The report quotes it in full and uses it only for what it says (necessity as input to the Kuznetsov step of THIS machinery); the impossibility verdict rests on the Fourier argument, not on overreading the word. The completion mechanism (p. 32's proof of Cor 15, `∂_y^k Ψ̂ ≪ Z^{O(ε)}/(1+ξ⁴)`) matches the report's §4.1 reading. - **The measurement, re-derived from the stated definition alone** (`rt0820-t2-rosser.js`, fresh enumeration of `𝒟⁺(z=101, D=z^{3.038})`): support 3,569 elements (+ d = 1 = the reported 3,570); aggregated-block low-mass fractions 5.65e−3 / 2.29e−3 / 1.43e−3 against flat nulls 7.81e−3 / 3.91e−3 / 1.95e−3 — the printed 5.6e−3/2.3e−3/1.4e−3 vs 7.8/3.9/2.0e−3 reproduce; exactly **31 slices** qualify, and the low-mass/null ratios come out **0.57–1.15, median 0.82** — the printed numbers to the digit. The flat null `K/C̃` is the correct expectation for a sign-flat sequence (K of C̃ modes), and the LOG/random-sign benchmarks behave as claimed. - **The nit:** the instrument sums `1 ≤ k ≤ 64` and omits `k = 0`, the mode where a smooth DC component would live. Measured here: the DC share is up to 2.75e−3 of the mass (2^14 block) — ~45× its flat per-mode share, i.e. a real small bias the instrument was blind to — but still two orders below the ≳0.4 mass share a usable smooth component needs. Corrected sentence for §4.3, if adopted: *including k = 0, every mode's share stays ≤ 2.8e−3 of the factor's mass, two orders under a usable smooth component; the k ≥ 1 rows are at the flat noise floor as printed.* The verdict is unchanged; the null design should say so explicitly. - The §4.2 chain (admissible ⟹ low-frequency up to A-decay tails; Davenport ⟹ `max_k |γ̂| ≪_A (log C)^{−A}` for the μ-model; remainder keeps `1 − o(1)` mass and is exactly as inadmissible) re-derives cleanly; the §4.4 completion Parseval (`Σ_b|λ̃|² = d₂·Σ|λ|²`, no wraparound since `M₁ ≤ d₂`) and the Weil pricing re-derive (2.1). The scope guards in §4.5(i)–(ii) prevent the overread this pass went looking for. ### 2.4 KMS prime-modulus quote — CONFIRMED verbatim KMS 2020 (arXiv:1802.09849v5, `pdfinfo` title/author verified), p. 2, read at a page image rendered by this pass: "*When the modulus `q` is composite, a number of techniques exploiting the possibility of factoring `q` (starting with the Chinese Remainder Theorem) become available, and results exist in fair generality. In this paper, we will only consider the case where `q` is a prime, and when `K` is a trace function*". Pascadi p. 6's pricing is also verbatim: "*an extension of the work of Kowalski–Michel–Sawin [29] to general moduli should improve Theorem A in the critical range `q ≈ N²`, but even then the final numerical savings would be relatively small.*" ## 3. TARGET 3 — import-hypergraph.md ### 3.1 The C-window closed form — CONFIRMED at source FGKMT arXiv:1412.5029 (v3 PDF on disk; pp. 12, 15, 16 re-rendered and read by this pass). Corollary 4, p. 15, is verbatim as the report lists it: `P′, Q′` are ABSTRACT sets ("Let `P′, Q′` be sets with `#P′ ≤ x` and `#Q′ > (log₂x)³`") — so instantiating `P′` as any ≤ x primes is inside the statement, not a stretch; (4.16)–(4.21) all match, including (4.17) `P(q ∈ e_p) ≤ x^{−1/2−1/10}`, (4.19) `(5/4)log 5 ≤ C ≪ 1`, and the hypothesis (4.21) `m ≤ log₃x/log 5`. P. 16 confirms the deduction: (4.22) `|ℐ_j| = 5^{1−j}log5/C` DISJOINT in [0,1] — which needs `C ≥ log5·Σ_{j≥1}5^{1−j} = (5/4)log5`, exactly the report's "forced by disjointness" reading — and (4.23) `δ := x^{−1/20}`. The closed form re-derives: exact marginal `2/q ≤ x^{−3/5}` forces `q ≥ 2x^{3/5}`; the sup of `Σ 2/q` over ≤ x primes above that line is the x smallest such primes, largest `(1+o(1))x ln x`, giving `C → 2 ln((ln x)/((3/5)ln x)) = 2 ln(5/3) = 1.021651` against `(5/4)ln5 = 2.011797`: short by 1.9692, at every scale. On the FGKMT-shaped range `(x/2, x]`: `2 ln(1/(1−ln2/ln x)) = (2ln2+o(1))/ln x → 0`. Both CONFIRMED as closed forms. (Also corroborated: `ps89.pdf` on disk is an HTML bot page, not a PDF — the report's "PS 1989 NOT REACHED" is true.) ### 3.2 The yield-cap argument — WEAKENED: the 1/log₂x floor is r-dependent, not a consequence of polynomial marginals alone Theorem 3's (4.1)–(4.9) were read verbatim at p. 12. Re-deriving the cap: polynomial marginals with polynomial block sizes force polynomial `δ` ((4.3): `δ ≥ marginal·(#I_j)^{1/2}`), so `ln(1/δ) ≍ ln x`; (4.1) then needs > `10^{m+2}·(A·ln(1/κ) + AD + ln C₀) ≤ ln(1/δ) ≍ c·ln x`, with `A ≥ 2rm+1` > (Cor 2's covering depth) and `κ ≍ 5^{−m}`, i.e. `10^{m}·r·m² ≲ ln x`. - At FGKMT's own `r ≍ log x·log₃x/log₂²x` this gives `10^m ≲ (log₂x)²` ⟹ `m ≍ log₃x` and floor `≍ 1/log₂x` — the report's arithmetic, and (4.21) says the same inside Cor 4. CONFIRMED for that regime. - **But for an "ideal" family with bounded a.s. edge sizes (`r = O(1)`) the same quantifiers allow `10^m ≲ ln x`, i.e. `m ≤ (1+o(1))·log₂x/ln 10`, floor `5^{−m} = (ln x)^{−log₁₀5+o(1)} = (ln x)^{−0.699+o(1)}` — a genuine ln-power.** So §0.3's "it can never carry a `ln`-power" and §5(3)'s "flooring the leftover fraction at `1/log₂x` … ONE factor of lnln, ever" are REFUTED as universal statements about the engine; they are true of Cor 4 (by hypothesis (4.21)) and of FGKMT-sized edges, not of (4.1) plus polynomial marginals in general. Corrected sentence: *(4.1) caps the depth at `10^m ≲ ln x`; with FGKMT's `r ≍ log x` that is `m ≍ log₃x` (leftover floor `1/log₂x`), and even with ideal bounded-size edges it is `m ≲ 0.434·log₂x` (leftover floor `(ln x)^{−log₁₀5} ≈ (ln x)^{−0.699}`) — still short of the one full log the Maier–Pomerance target needs and two logs short of the K–K reading, so the row verdict stands, on a weaker floor.* The verdict-relevant conclusions (K–K residual not dischargeable; the priced THEOREM mispriced) survive because they also rest on §5(1)'s C-window arithmetic and the named missing concentration input, and because even the corrected ceiling `(ln x)^{0.699}` is below both targets. - Two subsidiary defects in §5(2), safe-direction but worth the record: (i) "raw traces do satisfy its (4.2)–(4.4) over ranges `(x^θ, x]`" glosses (4.2): at `q ≍ x^{1/2+ε}` a trace meets `≍ x^{1/2−ε}/polylog` survivors, so `r` is polynomial and (4.1) is then UNSATISFIABLE for any `m ≥ 1` — Theorem 3 cannot actually run on untruncated raw traces over that range, making the claimed "gain ×0.52" an over-credit to the machine (the true yield is smaller, so the "constant, not a log" conclusion holds a fortiori). (ii) The interior numbers `e^{−2.05}` and `×0.52` have no producer artifact and did not re-derive here (a base-5 packaging of a 1.386 budget actually yields leftover ≥ 0.31, WORSE than independent `e^{−1.386}`; a base-2 packaging can do better; the report does not say which it priced). Constant-vs-log is unaffected; the specific numbers are unbacked. ### 3.3 The held theorem chain `G₂(x#) ≫ x ln x` — CONFIRMED, every ingredient verified at its source Treated as a claimed theorem at the door. Ingredients: 1. **K–K Corollary 1** — re-read by this pass at a page image of arXiv:2302.00459v2 p. 4, fetched fresh and md5-matched to the corpus's recorded artifact (`b5d7d2a2…`). Verbatim: "*Let κ, z, g(d), V(z) and X be as above. Suppose that for any p ≤ z the set Ω_p ⊂ ℤ/pℤ contains g(p) elements. Let S(X, Ω) be the number of n ≤ X such that n mod p ∉ Ω_p for all p ≤ z. Then S(X, Ω) ≪ XV(z).*" "As above" = Lemma 1's hypotheses: `g` multiplicative, `g(p) ≤ κ`, `g(p) < p` for ALL primes, `z ≪ X`, constant `≪_κ`. The instantiation `Ω₂ = {0}` (g(2) = 1 < 2), `Ω_p = {0, −2}` (g(p) = 2 < p, classes distinct for odd p), `κ = 2`, `z = √y ≪ y = X` discharges every hypothesis. Lemma 1 is proved by citation to Halberstam–Richert Thm 2.2 inside a refereed paper (Izv. Math. 88:2 (2024)) — "published theorem consumed as a theorem" is the right grade. 2. **Mertens step** — elementary and re-derived: `(1−2/p) = (1−1/p)²·[(1−2/p)/(1−1/p)²]` with the bracket convergent, so `∏_{2