# Red team 2026-08-28: the five certificate-engine notes, adversarial pass *(2026-08-28. Adversarial verifier, branch `opus-try`. Brief: break, before anything reaches a live document, `thm-mod30-tail.md`, `thm-capK-bv.md`, `thm-buchstab-transfer-shallow.md`, `thm-sharp-sieve-range.md` and `comb-discrepancy-tight.md`. Method: refuted-until-rederived, per `redteam-0820-math.md` §0. Every re-derivation lives in this pass's own producer, `research/history/staging/redteam-0828-engine.js`, which shares no code with any of the five target scripts and is written from the definitions in `paper/staircase-note.md` §1/§2/§4/§7 and `research/natal-cap-28-analytic-certificate.js` RESULT 1. This file and that script are the pass's only writes. No existing file was edited. No git command was run. `research/qc.js` was not run. Under the house publication moratorium.)* --- ## 0. Scoreboard **The single most load-bearing correction is not on the one-class factor, which survives.** It is that **`thm-capK-bv.md` §0 and its ledger verdict give the wrong level for the fundamental lemma's own threshold.** §0 says "the first primorial level at which `s` clears 10.82 is `x = 239`, where `W ~ 10^96`". The level at which `s` clears 10.82 is **`x = 263`** (`W ~ 10^105.6`); `s(239) = 10.0936`, `s(257) = 10.7075`, `s(263) = 10.9132`. The note's own §4 sentence, "First `s >= 10` at `x = 239`", is right; §0 restates it against the wrong threshold, and `research/QUESTIONS.md` now carries the wrong number through the ledger verdict line ("first at `x = 239`"). `thm-sharp-sieve-range.md` inherits it in §0's summary table and in §5. Corrected sentence in §3.2. The direction is against the notes: emptiness is worse, not better, so the cluster's verdict is unaffected. **The one-class factor survives, twice, and is stronger than either note says.** Independent recount reproduces both exact-count tables digit for digit, and the proof needs only `P^-(m) >= q > q_i`, not `m` prime, so the correction holds at every scour prime and not only in the prime regime. It needs one scope clause before it enters a live document, or it will read as contradicting `thm-buchstab-transfer-shallow.md`'s `kappa = 2`. §1. | # | claim | verdict | |---|---|---| | 1a | one class per freshness prime, `PROD(1 - 1/(q_i - 1))`; survey's `PROD(1 - 2/(q_i - 1))` refuted | **CONFIRMED and strengthened** (§1.1, §1.2) | | 1b | `GLOSSARY.md` "the two forbidden freshness residues" | **WEAKENED**: wrong as a dimension count, sound as a cap (§1.3) | | 1c | `staircase-note.md` §7's two conditions per `q'` | **CONFIRMED as a definition**, redundant not wrong (§1.3) | | 1d | `bv-import-survey.md` §3.2 "dimension 2 in the range `(x, q_K]`" | **WEAKENED** for `q_i < q` (§1.3) | | 2a | S1(a) classification, hypotheses, and the injection cited in `staircase-note` §2/§4 | **CONFIRMED at source and exhaustively** (§2.1) | | 2b | S1(b) `cap30 >= fresh`, and D2 (survey's loose form is not below `cap1`) | **CONFIRMED**, 5 of 29 and 16 of 105 (§2.2) | | 2c | S1(c) constant `(ln 2)/2`, factor exactly 4 | **CONFIRMED**, re-derived and re-measured (§2.3) | | 2d | D3, the third class, `(3/4) ln 2`, tail factor `8/3` | **CONFIRMED** (§2.4) | | 2e | D4/D5, PNT-in-APs at fixed modulus 30; `M << (ln W)^K`; SW with `A = K + 1` | **CONFIRMED** (§2.5) | | 3a | `K_dim = 1.2000`, `s >= 10.82 / 11.52 / 15.43` | **CONFIRMED and strengthened** to `z1 <= 2e8` (§3.1) | | 3b | "the first primorial level at which `s` clears 10.82 is `x = 239`" | **REFUTED**: it is `x = 263` (§3.2) | | 3c | Step 3, BV handles the `q`-dependent excluded class by `max_a` | **CONFIRMED**; but the `pi(q-1)` term needs BV's `max_{y<=T}` form: **WEAKENED** (§3.3) | | 3d | Step 2's `=` for a lemma that states two inequalities; §4's `D = T^{1/2}` against the proof's `T^{1/3}`; `z = x` against Step 2's `z = q_K + 1` | **WEAKENED**, all three wording, none moves a verdict (§3.4) | | 4a | three-object separation (`B` dim 1, survey's pair-Buchstab dim 2, certificate's two-class two-depth) | **CONFIRMED** (§4.1) | | 4b | `(1 - 1/p)(1 - 1/(p-1)) = 1 - 2/p` | **CONFIRMED**, exact identity (§4.1) | | 4c | `K = 7/5`, `s >= 22.06 / 23.67 / 25.97`, `s_head` table | **CONFIRMED and strengthened** to `z1 <= 2e8` (§4.2) | | 4d | band empty until level 131 (`delta = 0.5`), 151 (`delta = 0.01`), coverage 3.24/6.22/7.34/8.50% | **CONFIRMED** exactly (§4.3) | | 4e | the two `s` definitions (`z = q` and `z = q_K`) stated and not conflated | **CONFIRMED** in all three notes (§4.4) | | 5a | DH Thm 9.1 carries no written constant | **CONFIRMED** against the repo's page record (§5.1) | | 5b | bare-shape 1.05/1.66/4.32 (`kappa=1`), 3.66/4.82/9.22 (`kappa=2`); peaks 8.66/19.49; crossings 2.1e11/8.8e19 | **CONFIRMED** (§5.2) | | 5c | `V*F1 = 1.19001 > 1` at @29 and the whole @13..@29 row | **CONFIRMED** (§5.3) | | 5d | `f2(4.7088) = 0.4056`, `F2 = 1.4878`, `f2(4.5)`, `f2(5.0)` | **CONFIRMED** by an independent solve (§5.4) | | 5e | "first non-empty `kappa = 1` level is @53" | **REFUTED**: it is **@37** (§5.5) | | 5f | "first non-empty `kappa = 2` band is @23" | **CONFIRMED** (§5.5) | | 5g | §0 finding 1 pairs `s >= 21.37` with level 131 | **WEAKENED**: 131 is the 22.06 figure; 21.37 gives 127 (§5.6) | | 6a | Theorem A: sup over windows `= max G - min G`; `D_x` at @7..@23 | **CONFIRMED**, proof and every value (§6.1) | | 6b | Theorem B: grouping by divisor returns `2*3^k` exactly | **CONFIRMED** (§6.2) | | 6c | slack 28.99 @23; 53.64 @29 | **CONFIRMED** @23; @29 **NOT RE-DERIVED** (§6.2) | | 6d | §3's `rho * g_max` lower bound (1.429 / 2.338 / ... / 7.699) | **WEAKENED**: off by one gap; correct values in §6.3 | | 6e | §5's @17 row: 1 -> 4 certified primes, 9.69% -> 28.14%, per-term constant 17.4 | **CONFIRMED structurally**, two label corrections (§6.4) | | 7 | output custody of the five producers | **CONFIRMED as each note declares** (§7) | **The cluster's shared verdict, "theorems with empty computable range", stands.** Two of its four thresholds move. The `kappa = 1` fundamental-lemma threshold moves *away* from reach (239 to 263). The `kappa = 1` sharp-sieve limit statement moves *toward* reach (53 to 37), which strengthens `thm-sharp-sieve-range`'s confirming half without touching its reading (F), since DH Theorem 9.1's `O`-constant is still unwritten and its bare shape still exceeds 1 at every level in the tables. --- ## 1. The one-class freshness factor: the claim that reaches live documents ### 1.1 Re-derived from the definitions From `paper/staircase-note.md` §7, verbatim: the depth-`K` freshness conditions are imposed "for each of the first `K` entries `q'` of `M` with `q' < q`", and `#A_K` counts cofactors `2 <= m <= floor((W-1)/q)` **with `P^-(m) >= q`** satisfying them. Take such an `m` and such a `q'`. Then `v = qm ≡ 0 (mod q')` forces `q' | q` or `q' | m`. The first fails because `q'` and `q` are distinct primes. The second fails because `P^-(m) >= q > q'`. So the condition `v !≡ 0 (mod q')` excludes no residue class of `m` at all, and only `v !≡ -2 (mod q')` (A-side) bites, one class, a unit since `q' ∤ 2q`. The surviving share is `(q' - 2)/(q' - 1)`, that is `1 - 1/(q' - 1)`. **This is stronger than either note states.** `thm-mod30-tail.md` §5(i) and `thm-capK-bv.md` §1 both run the argument through "`m` is prime with `m >= q > q_i`", which is available only in the prime regime `q^3 > W + 1`. The argument above uses only `P^-(m) >= q > q_i`, which is part of the definition of `#A_K` at **every** scour prime. So the correction holds in the head and the middle as well, not only in the tail. `thm-buchstab-transfer-shallow.md` §1 has this right ("`P^-(m) >= q > q_i` is impossible") and the other two do not. The same conclusion is what `natal-cap-28-analytic-certificate.js` already codes: line 138-141, `F = 1/(q-1)` and `P[j+1] = P[j]*(1 - F[j])`, and `dP = 0.25 * mids.reduce((a,p) => a*(1 - 1/(p-1)), 1)`. Its line 184 does test `t === 0` as well as `t === Q[j]-2`, guarded by `if(lpf[m] < q) continue`, so the test is present and never fires. The code is right and the survey is the outlier, exactly as `thm-mod30-tail.md` D1 says. ### 1.2 Re-measured, independently `redteam-0828-engine.js` SEC A counts, from scratch, the A-side admissible prime cofactors over the prime-regime scour primes with the mod-30 classes and the depth-`K` freshness conditions, and compares against both candidate density factors applied to the exact `K = 0` base. Tail primes 105 at @17 and 396 at @19, base 5336 and 90501, all four figures matching `thm-capK-bv.md` §5's independently written cross-check. | level | K | exact | `PROD(1-1/(q'-1))` | ratio | `PROD(1-2/(q'-1))` | ratio | |---|---|---|---|---|---|---| | @17 | 1 | 5 037 | 5 039.56 | 0.99949 | 4 743.11 | 1.06196 | | @17 | 2 | 4 802 | 4 810.48 | 0.99824 | 4 311.92 | 1.11366 | | @17 | 4 | 4 474 | 4 484.06 | 0.99776 | 3 737.00 | 1.19722 | | @17 | 8 | 4 043 | 4 059.11 | 0.99603 | 3 054.42 | 1.32366 | | @19 | 1 | 86 390 | 86 387.32 | 1.00003 | 82 273.64 | 1.05003 | | @19 | 2 | 83 313 | 83 302.06 | 1.00013 | 76 396.95 | 1.09053 | | @19 | 4 | 78 294 | 78 288.51 | 1.00007 | 67 342.49 | 1.16262 | | @19 | 8 | 71 478 | 71 492.21 | 0.99980 | 56 038.31 | 1.27552 | Every entry reproduces `thm-mod30-tail.md` §6's D1 table to the digit it prints. The survey's factor is off by 5.00% to 32.37%, worsening with `K`, exactly as claimed. **CONFIRMED.** ### 1.3 Wrong as a dimension count, sound as a cap: the corrected sentences The distinction the brief asks for is real and it is the whole scope question. - **`paper/staircase-note.md` §7 is a definition, and it is correct.** The set it defines with two conditions equals the set defined with one, because the second is void on the domain `P^-(m) >= q`. No `cap_K` value moves. It is redundant, not wrong. Corrected sentence, one added parenthetical: > "A-side: `v !≡ 0` and `v !≡ -2 (mod q')` [else `q'` strikes `r`]. (The first > is void: `q' ≠ q` and `P^-(m) >= q > q'`, so only `v !≡ -2` excludes a class, > and the per-prime surviving share is `1 - 1/(q'-1)`.)" - **`research/GLOSSARY.md`:354 is a dimension count, and it is wrong.** It reads "cap_K adds the two forbidden freshness residues modulo the first K scour primes". Corrected sentence: > "cap_K adds, for each of the first K scour primes `q' < q`, the **one** > forbidden freshness residue that can fire (`v !≡ -2` on the A side, `v !≡ +2` > on the B side); the companion `v !≡ 0 (mod q')` is void under `P^-(m) >= q`, > so the density factor is `PROD(1 - 1/(q'-1))`, one class per prime. In a > formulation that *sifts* for `P^-(m) >= q` rather than assuming it, `m !≡ 0` > is a real second class and the dimension is 2." The second sentence is not optional. Without it the glossary line will read as contradicting `thm-buchstab-transfer-shallow.md` §2(c), whose `kappa = 2` table is correct in its own formulation, and the two notes will look like they disagree when they do not. - **`research/bv-import-survey.md` §3.1**, the density factor `PROD(1 - 2/(q_i - 1))`: **REFUTED**, measured, §1.2. Replace with `PROD(1 - 1/(q_i - 1))`. - **`research/bv-import-survey.md` §3.2**, "two excluded classes per large freshness prime, dimension 2 in the range `(x, q_K]`": **WEAKENED**. For `q_i < q` the dimension is 1 on the sequence of `q`-rough cofactors. It is 2 only if the roughness itself is being sifted rather than assumed, or if `q_i > q`, and in the latter case `v ≡ 0 (mod q_i)` selects the single value `m = q_i`, not a class. `thm-capK-bv.md` §7 already says this and correctly marks it NOT AUDITED; that flag is right and this pass does not lift it. --- ## 2. `thm-mod30-tail.md` ### 2.1 S1(a), and the injection it cites `staircase-note.md` §2 Lemma 1 says what the note says it says: the A/B dichotomy, the ranges `2 <= m <= floor((W±1)/q)`, `P^-(m) >= q`, and `m >= q`, all four in the printed statement. §4 Theorem 3(i) says what the note says it says: under `q^3 > W+1` one has `t <= (W+1)/q < q^2`, a composite `m` with `P^-(m) >= q` satisfies `m >= q^2 > t`, hence `m` is prime, and the printed text adds "The structural statement is Lemma 1 with `m` prime". **CONFIRMED at source.** SEC B marches `N_x` from the definition, finds each member's smallest scour striker, and checks every non-self fresh victim of every prime-regime scour prime: `q | v`, `v/q` prime, `v/q >= q`, and `v mod 30` in `{11,17}` on the A side and `{13,19}` on the B side. Census asserted against `N = 2*PROD(p-2)`. | level | census `N` | prime-regime victims | violations | tail primes | |---|---|---|---|---| | @13 | 990 | 312 | **0** | 29 | | @17 | 14 850 | 3 888 | **0** | 105 | Both victim counts and both tail-prime counts match the note's 312 and 3 888, 29 and 105. **CONFIRMED.** ### 2.2 S1(b) and D2 `cap30` in the sharp form is asserted against `fresh(q)` at every tail prime by `assert`, so a failure aborts. It passes at all 29 and all 105. `Sum cap1` over the tail comes out **2 720** at @13 and **42 895** at @17, matching `staircase-note.md` §6 digit for digit, computed here from Theorem 3(i)'s own `pi(A) + pi(B) - 2*pi(q-1) + s(q)`. Tail `fresh/cap1` comes out 0.11654 and 0.09097, matching the note's 0.117 and 0.091. D2, the survey's loose form (the `pi(q-1;30,.)` terms dropped) exceeding `cap1`: **5 of 29** at @13 and **16 of 105** at @17, matching the note exactly. **CONFIRMED.** ### 2.3 S1(c) The measured cap ratios reproduce: `Sum cap30(sharp) / Sum cap1` = **0.25699** at @13 and **0.25138** at @17 against the note's 0.2570 and 0.2514; the loose form gives **0.37390** and **0.33824** against 0.3739 and 0.3382. The analytic step re-derived: each of the four class counts contributes `Li(T)/8`, so the four contribute `(Li(A) + Li(B))/4` against Theorem 6's `pi(A) + pi(B)`, a ratio of exactly `1/4`; and `INT_{1/3}^{1/2} dθ/(θ(1-θ)) = [ln(θ/(1-θ))] = ln 2`, so the constant is `2 ln 2 / 4 = (ln 2)/2`. **CONFIRMED.** One wording nit: the note writes the Theorem 6 shape as "`2(pi(A)+pi(B))/2`", which is `pi(A)+pi(B)`; harmless, but it reads as a typo. ### 2.4 D3 Over the full mod-30 twin census the slot classes are `r ≡ 11, 17, 29`, so 3 of the 8 unit classes per side rather than 2. The tail factor is `1/(3/8) = 8/3` and the constant is `2 ln 2 * 3/8 = (3/4) ln 2`. **CONFIRMED** arithmetically. The hypothesis `r ∈ N_x` is required and is stated in S1's hypothesis block. ### 2.5 D4 and D5 - PNT in APs at the fixed modulus 30 is what S1 needs, and Siegel-Walfisz is strictly more. **CONFIRMED**: the modulus never grows, the classes `q^{-1}a` move but SW-type uniformity in the class at a fixed modulus is the standard statement. - D5: `PROD_{7<=p<=P0} p` is a constant independent of `x`, so `M <<_{P0,K} q_K^K << (ln W)^K`. Siegel-Walfisz applies to moduli `M <= (log T)^A`, so any `A > K` serves, and `A = K + 1` is enough for large `T` since `M <= C(P0,K)(log T)^K <= (log T)^{K+1}` eventually. **CONFIRMED.** --- ## 3. `thm-capK-bv.md` ### 3.1 `K_dim`, and a stronger search than the note ran `K_dim` is the sup over `z1 >= w >= 2` of `PROD_{w<=p= 7` and `0` below. SEC C reproduces **1.2000000000**, attained at `w = 7, z1 = 7^+`, where the product is `(1 - 1/6)^{-1} = 6/5` and the log ratio is 1. The note's search ran to `z1 <= 1e5`; this pass ran the binding base `w = 7` out to `z1 <= 2e8` and the sup is still exactly `6/5` at `z1 = 7^+`. The tail behaviour is the reason: `PROD_{7<=p= 10.82322` at `δ = 1`, `11.51636` at `1/2`, `15.42839` at `1/100`. **CONFIRMED**, all three digits. ### 3.2 The `x = 239` claim: REFUTED With `s = ln D / ln z`, `D = T^{1/2}`, worst tail case `T = sqrt(W)`, `z = x` at `K = 0`, so `s = ln W / (4 ln x)`. SEC C reproduces the note's whole §4 table to the digit it prints: 0.8075, 1.0049, 1.1597, 1.3659, 1.5327, 1.6772, 4.5756, 8.9058, 15.7128, 34.8128 at @11 through @1009. The crossings: | threshold | first level | `s` there | |---|---|---| | `s >= 10` | `x = 239` | 10.0936 | | `s >= 10.8232` (the actual factor-below-1 threshold) | **`x = 263`** | 10.9132 | | `s >= 12` | `x = 283` | 12.0161 | `s(241) = 10.3283`, `s(251) = 10.5023`, `s(257) = 10.7075`. So §4's two printed sentences are both right, and §0's is not. Corrected sentence for `thm-capK-bv.md` §0, and for its ledger `verdict:` line, and for `research/QUESTIONS.md`'s two rows for `Q-capK-bv`: > "The first primorial level at which `s` clears the lemma's own > factor-below-1 threshold 10.82 is **`x = 263`** (`W ~ 10^105.6`); `s` first > reaches the lemma's floor `9κ + 1 = 10` at `x = 239`." `thm-sharp-sieve-range.md` §0's summary table row "κ = 1 (`thm-capK-bv`) first non-empty level: `x = 239` `[CITED]`", its §0 finding 1, and its §5 table row "@239 | ... | EMPTY (FL crosses)" all inherit the error and take the same correction. ### 3.3 Step 3, the BV step: CONFIRMED with one WEAKENED clause The `max_a` answer to the moving excluded class `ω_p = -2q^{-1} (mod p)` is correct and is the right answer: BV bounds `max_{(a,k)=1}`, so a class that moves with `q` costs nothing, and the moduli `30d`, `d <= D` squarefree and coprime to 30, are distinct and at most `30D`, so the sum over `d` is a sub-sum of the BV sum at level `30D`. With `D = T^{1/3}` the level `30 T^{1/3}` is inside `T^{1/2}/(ln T)^B` for large `T`. **CONFIRMED.** **The weakened clause.** The remainder in Step 3 is `r_d = E(T;30d,a_d) - E(q-1;30d,a_d) - φ(d)^{-1}[...]`, and the note calls "the first two terms ... sub-sums of the Bombieri-Vinogradov sum at level `30D`". The second term is evaluated at `q - 1`, not at `T`. Bombieri-Vinogradov stated at its own endpoint `q - 1` permits level `(q-1)^{1/2}/(log)^B` only, and `30 T^{1/3}` exceeds that on the whole tail: `q > W^{1/3}` and `T < W^{2/3}` give `T^{1/3} < W^{2/9}` against `q^{1/2} > W^{1/6}`, and `2/9 > 1/6`. The repair is one clause, because the standard form of the theorem carries an inner maximum over the argument, and `q - 1 < T` holds under (H2). Corrected sentence: > "the first two terms are sub-sums of the Bombieri-Vinogradov sum **in its > `max_{y<=T}` form** at level `30D <= T^{1/2}/(ln T)^B`; the `pi(q-1)` term is > covered by that inner maximum, since `q <= T/2` by (H2), and is not a > BV statement at the endpoint `q-1`, where the level `30T^{1/3}` would be > inadmissible." Rung after the repair: unchanged, PROVEN at short-note grade. The note's own falsification table already flags this step as the one with no second reader and the `max_a` clause as quoted rather than read at a page; that flag stands, and this pass adds the `max_y` requirement to it. ### 3.4 Three wording defects, none moving a verdict - Step 2 writes `Sum_{d|P} λ^±_d h(d) = (1 ± e^{9κ-s} K_dim^{10}) V` with an equals sign. FI Lemma 6.8(iii), as recorded verbatim at `research/dhr-verification.md` §4.1 (from arXiv:2301.07679 p. 33), states two one-sided inequalities. The sandwich in (ii) makes the inequalities sufficient, so the proof goes through; the display should read `<=` and `>=`. - §4's `s` table is computed at `D = T^{1/2}`, which BV does not permit; the proof's own choice is `D = T^{1/3}`, at which every tabled `s` is `2/3` of the printed value. The direction is conservative (emptiness is worse under the proof's own `D`), but §4 should say which `D` it is tabling and why. - Step 2 sets `z = q_K + 1` while §4's table is computed with `z = x` at `K = 0` (checked: `ln W / (4 ln 17) = 1.1597` matches the printed 1.16, while `ln W / (4 ln 18) = 1.1368` does not). The note should state `z = x` at `K = 0` where `thm-sharp-sieve-range.md` §1 states it for it. --- ## 4. `thm-buchstab-transfer-shallow.md` ### 4.1 The three-object separation, and the identity `(1 - 1/p)(1 - 1/(p-1)) = ((p-1)/p)((p-2)/(p-1)) = (p-2)/p = 1 - 2/p`, exactly, for every `p >= 3`. **CONFIRMED**, and it does what the note uses it for: the engine's product `(N/W) PROD_{x κ = 1` structure is already inside the engine's arithmetic and only `B` is imported. The three-object separation is sound and is the note's best content. The reason the engine's `κ = 1` `B` and this note's `κ = 2` ensemble are both right is the scope clause of §1.3 above: `capK-bv` sifts a sequence of primes, where `m !≡ 0 (mod p)` is free, and this note sifts a sequence of integers, where it is not. **CONFIRMED.** ### 4.2 `K = 7/5` and the thresholds SEC D reproduces `K = 1.400000` at `w = 7, z1 = 7^+`, where `(1 - 2/7)^{-1} = 7/5` and the squared log ratio is 1, and this pass extended the binding base to `z1 <= 2e8` with the sup unchanged; the tail ratio `PROD_{7<=p= 9κ + 10 ln(7/5) + ln(1/δ) = 18 + 3.36472 + ln(1/δ)` gives **22.05787** at `δ = 0.5`, **23.66731** at `0.1`, **25.96989** at `0.01`, and **21.36472** at `δ = 1`. All four reproduce. `s_head = ln(W/q0)/ln q0` reproduces at every level the note tables: 2.63896, 3.46372, 4.13080, 4.70876, 5.57847, 10.01903, 17.14219 at @13, @17, @19, @23, @29, @53, @97. **CONFIRMED.** ### 4.3 The band Least level with `ln q0 <= ln W/(1 + s*)`: **`x = 131`** at `s* = 22.06`, **`x = 151`** at `s* = 25.97`, `x = 23` at `s* = β2`, `x = 7` at `s* = 1`. Coverage of `[ln x, ln sqrt(W)]`: **0.00%** at @97, 0.35% at @131, **3.24%** at @199, **6.22%** at @499, **7.34%** at @1009, **8.50%** at @10007. Every figure the note prints. **CONFIRMED.** ### 4.4 The two `s` conventions `thm-buchstab-transfer-shallow.md` §3.1 states `z = q` and `D = T^{1-ε}`; `thm-capK-bv.md` §3 Step 2 states `z = q_K + 1` and `D = T^{1/3}`; `thm-sharp-sieve-range.md` §0 finding 5 and §1 state the difference explicitly and its §3 and §4 keep the two columns in separate tables. **CONFIRMED: stated in all three, conflated in none.** The one residue is §3.4's third bullet, `z = q_K + 1` against the table's `z = x`, which is internal to `capK-bv` and worth 0.02 on `s`. --- ## 5. `thm-sharp-sieve-range.md` ### 5.1 DH Theorem 9.1 carries no written constant `research/history/staging/lit-pdf-halberstam-richert.md` §6.1 transcribes p. 104 from an `attestation/` photograph. The note's §2 quotation matches that record word for word, including the hypothesis "Suppose that κ >= 1 and that 2κ is an integer", the factor 2, the weight `4^{ν(m)}`, the range `m|P(z), m 1` `V(z) = PROD_{7<=p<=x}(1 - 1/(p-1))` and `F1(s) = 2e^γ/s` on `[1,3]`: `V*F1` = 2.43706 / 1.97968 / 1.58746 / 1.35043 / **1.19001** at @13 / @17 / @19 / @23 / @29, against the trivial `S <= X`. **CONFIRMED**, and the brief's own example is refuted in the brief's own terms: `F1(1.6772) = 2.12386`, but the bound is `X V F1`, and `V(29) = 0.56030`. ### 5.4 `f2(4.709) = 0.4056` Re-derived without the repo's solver. From Booker-Browning Theorem 3.1 as quoted verbatim at `research/dhr-verification.md` §1.1, `σ_2(u) = c u^2` on `(0,2]` with `c = 1/(2(2e^γ)^2)`, and on `(2,4]` the defining ODE integrates in closed form to `σ_2(u) = c u^2 (1 - 2 I(u))`, `I(u) = ln u + 4/u - 2/u^2 - (ln 2 + 3/2)`; above 4 it is one quadrature. Then `F_2 = 1/σ_2` on `(0,α2]`, and `u^2 f_2(u) = 2 INT_{β2}^u t F_2(t-1) dt` on `(β2, α2]`. | quantity | this pass | note / cited artifact | |---|---|---| | `F2(1)` | 25.37775 | 25.37775 | | `F2(2)` | 6.34444 | 6.34444 | | `F2(1.7829)` | 7.98361 | 7.98361 | | `F2(2.317)` | 4.73619 | 4.73619 | | `F2(4.7088)` | 1.48778 | 1.4878 | | **`f2(4.7088)`** | **0.40559** | 0.405614 | | `f2(4.5)` | 0.240279 | 0.240280 (`attack-beta2-04` OUTPUT §3c) | | `f2(5.0)` | 0.578997 | 0.578997 (same) | | naive `2(2e^γ/u)^2` at `u = 2.317` | 4.72717 | below the true `F2`, as §2 says | **CONFIRMED**, including the note's §2 claim that `F2 = 1/σ2` exceeds the naive formula above `u = 2`, and its bracket width `F2/f2 = 3.668` at @23. ### 5.5 "First non-empty κ = 1 level is @53": REFUTED The note's §3 table samples @11, @13, @17, @19, @23, @29, then jumps to @53. The five levels it skips are where the crossing happens. Sweeping every prime level from @11 to @103 on the note's own conventions (`s = ln W/(4 ln x)`, `V = PROD_{7<=p<=x}(1-1/(p-1))`, `F1 = 2e^γ/s` on `[1,3]`, `f1 = 2e^γ ln(s-1)/s` on `[2,4]`): | level | `s` | `V(z)` | `F1` | `V*F1` | `f1` | verdict on the note's own criterion | |---|---|---|---|---|---|---| | @29 | 1.6772 | 0.56030 | 2.12388 | 1.19001 | 0 | EMPTY, worse than trivial | | @31 | 1.8946 | 0.54163 | 1.88014 | 1.01833 | 0 | EMPTY, worse than trivial | | **@37** | **2.0518** | **0.52658** | **1.73612** | **0.91421** | **0.08765** | **BRACKETED** | | @41 | 2.2451 | 0.51342 | 1.58665 | 0.81461 | 0.34778 | BRACKETED | | @43 | 2.4666 | 0.50119 | 1.44413 | 0.72379 | 0.55306 | BRACKETED | | @47 | 2.6597 | 0.49030 | 1.33933 | 0.65667 | 0.67851 | BRACKETED | | @53 | 2.8292 | 0.48087 | 1.25908 | 0.60545 | 0.76031 | BRACKETED | @31 is genuinely empty, by a hair (`V*F1 = 1.018`). @37 is bracketed on both tests: the upper bound beats the trivial one and the lower bound is positive. Corrected sentence for `thm-sharp-sieve-range.md` §0's table, §3's first reading, §7's buys-item 1, and its confidence row: > "First non-empty κ = 1 level: **@37** (`s = 2.0518`, `V*F1 = 0.914`, > `f1 = 0.0877`), the first level above the `V*F1 < 1` crossing at @31 and above > `f1`'s positivity threshold `s = 2`. Scope moves from `W ~ 10^105.6` (the > fundamental lemma at `x = 263`) to `W ~ 10^12.9`." The bracket at @37 is `F1/f1 = 19.8`, which is a wide bracket and not an asymptotic; the note's @53 row remains the first *narrow* one at `1.656`. If §3's intended criterion was "bracket narrower than some width" rather than "non-empty", the note does not say so, and its own §5 verdict table uses BRACKETED as the non-empty label at @53. The `κ = 2` side needs no correction: `s_head` crosses `β2 = 4.26645` between @19 (4.1308) and @23 (4.7088), which are consecutive prime levels, so **@23** is right. **CONFIRMED.** ### 5.6 §0 finding 1's pairing Finding 1 says the fundamental lemma's factor "crosses 1 at `s >= 10.82` (κ = 1) and `s >= 21.37` (κ = 2), which the siblings locate at reachable levels `x = 239` and `x = 131`". Two slips. `x = 239` is the `s >= 10` figure (§3.2 above). And level 131 is the sibling's `δ = 0.5` figure at `s* = 22.06`; at `s* = 21.365` the least level with a non-empty band is **`x = 127`**. **WEAKENED**, wording; the verdict does not move. --- ## 6. `comb-discrepancy-tight.md` ### 6.1 Theorem A The proof is one line and it is right: `#(C_x ∩ [s,s+ℓ)) - ρℓ = G(s+ℓ) - G(s)`, `G` is `W`-periodic because `G(0) = G(W) = 0`, so the sup of the absolute difference over integer `s, ℓ` is `max G - min G`, attained by taking `s` at the argmin and `s+ℓ` at the argmax (or the reverse), with `ℓ <= W`. SEC F recomputes `D_x` by a single ascending pass over `[0,W)`, asserting the census and `G(W) = 0`: | level | k | `2*3^k` | **`D_x`** | `2*3^k / D_x` | `sup|G| / D_x` | |---|---|---|---|---|---| | @7 | 1 | 6 | **2.238095** | 2.6809 | 0.51064 | | @11 | 2 | 18 | **4.194805** | 4.2910 | 0.50464 | | @13 | 3 | 54 | **8.164835** | 6.6137 | 0.50202 | | @17 | 4 | 162 | **13.102133** | 12.3644 | 0.50111 | | @19 | 5 | 486 | **29.618855** | 16.4085 | 0.50044 | | @23 | 6 | 1 458 | **50.295195** | **28.9889** | 0.50024 | Every value matches the note's table, including the `28.99` slack at @23 and the `sup|G|/D_x -> 1/2` regularity to four digits. @29 is outside this pass's compute budget and the note's **53.64** is **NOT RE-DERIVED** here. **CONFIRMED at @7..@23.** ### 6.2 Theorem B `Sum_{d|M} 2^{ω(d)} = PROD_{p|M} (1 + 2) = 3^k`, so `Sum_{d|M} 2^{ω(d)+1} = 2*3^k`: bounding each block's range by its point count returns the standing lemma exactly. **CONFIRMED.** The bound `D_x <= Sum_d range(A_d)` is the range of a signed sum bounded by the sum of the ranges, which holds regardless of the signs. **CONFIRMED.** ### 6.3 §3's `rho * g_max`: WEAKENED, off by one gap The note writes `D_x >= ρ * g_max` and prints 1.429 / 2.338 / 2.967 / 4.538 / 5.622 / 7.699 at @7..@23. The justification given is "a run of length `g` with no comb member gives a window with count 0 against share `ρg`". The `g_max` used is the **member-to-member gap**, and a window of that length contains one member, not none. At @7 the comb is `{11, 17, 41, 71, 101, 107, 137, 167, 191, 197}`, the maximal gap is 30 and the maximal empty run is 29. The valid bound is `D_x >= ρ * (maximal empty run) = ρ * (g_max - 1)`: | level | note prints | correct | |---|---|---| | @7 | 1.429 | **1.381** | | @11 | 2.338 | **2.299** | | @13 | 2.967 | **2.934** | | @17 | 4.538 | **4.509** | | @19 | 5.622 | **5.596** | | @23 | 7.699 | **7.676** | The alternative reading of the same window, one member against share `ρ g_max`, gives `|1 - 1.429| = 0.429` at @7, which is weaker still. The printed inequality happens to hold numerically at every level, because the overstatement is one `ρ` and the slack is large, so no downstream number moves; the derivation and the percentages do. "The maximal gap explains 64% of `D_x` at @7" becomes **61.7%**. The note's verdict, that this avenue is dominated, is unchanged. ### 6.4 §5's @17 row SEC F reproduces the engine's certification at @17 from `natal-cap-28-analytic-certificate.js` RESULT 1's own test, with exact `cap2` counts and with sharp per-dilation ranges computed by walking the dilated comb `{m : q d m mod W ∈ N_x}` over a full period: | | this pass | note | |---|---|---| | `Sum cap2` at @17 | 16 135 | 16 135 (`staircase-note` Thm 8) | | certified primes, old | 1 (`q = 19`) | 1 | | share, old | 9.693% | 9.69% | | certified primes, new | **4** (`q = 19, 23, 29, 31`) | **4** | | share, new | **28.212%** | 28.14% | | walk stopped on | a genuine failure at `q = 37` | a genuine failure | | max single-dilation range used | **18.369** (at `q = 31`) | 17.4 | Per-prime detail: `cap2 = 1564 / 1227 / 926 / 835` at `q = 19 / 23 / 29 / 31` against `main = 1564.13 / 1219.56 / 923.23 / 832.86`; new error terms 32.34 / 73.64 / 117.01 / 249.37 against the half-main tests 781.57 / 609.78 / 461.62 / 416.43; and 536.14 against 337.26 at `q = 37`, which is the failure. **CONFIRMED structurally.** Two label corrections: - **The "share of `Sum cap2`" column is computed on main-term mass, not on `cap2` mass.** `comb-discrepancy-tight.js` PART 5 accumulates `cap = main + s` with `main = (A+B)*NW*Pj`, which is the analytic main term, not the exact `cap2`. On the exact `cap2` mass the new share at @17 is **28.21%**, not 28.14%. The old column is unaffected to two digits (9.693% either way), which is why the custody assert against cap-28's 9.69% passes and hides the difference. Corrected column header: "share of `Sum cap2`, main-term mass". - **"per-term constant, max over used dilations" is the maximum over certified primes of the *mean* per-dilation constant, not the maximum per-dilation constant.** The producer computes `errNew/2^{j+1} - 1`, an average over the `2^{j+1}` terms. At @17 that maximum is 17.41, attained at `q = 23`; the maximum single dilation range is 18.369, at `q = 31`. Both are far below `2*3^4 = 162` and the reading is unchanged. Corrected column header: "mean per-term constant, max over certified primes". Neither correction moves the note's §5 reading: the certified head roughly quadruples in count at @17 and lands near 28%, not past 50%. ### 6.5 One item the note flags and this pass confirms `certificate-engine.md`'s status table cell, "every Legendre term is off its share by at most `2*3^k`": each term is off by less than 1 and the sum of the `2*3^k` terms is off by at most `2*3^k`. **CONFIRMED as a defect**, wording, one cell. --- ## 7. Output custody `node research/qc/embed.js --check` on each of the five producers: | producer | result | |---|---| | `thm-mod30-tail.js` | code-sha256, out-sha256 and body all **match** | | `thm-capK-bv.js` | code-sha256, out-sha256 and body all **match** | | `thm-buchstab-transfer-shallow.js` | **no OUTPUT banner**, outside output custody | | `thm-sharp-sieve-range.js` | **no OUTPUT banner**, outside output custody | | `comb-discrepancy-tight.js` | **no OUTPUT banner**, outside output custody | The last three are exactly as their notes declare (each states SCRATCHPAD-GRADE and "not embedded, not qc-gated" in its header), so this is a confirmation and not a defect. It does mean that **no figure from those three may be quoted outside its own note** until it is re-derived inside an embedded producer under `research/`, which is what each note already says. The one hash a note publishes, `thm-sharp-sieve-range.md`'s code-sha256 `92a22c21cf1bca1eb4b112f0929bdc3a76af10ee9c782be228f2692fac9e41e8`, matches the file. This pass's own producer carries no banner either and is SCRATCHPAD-GRADE on the same terms. --- ## 8. Corrected sentences, collected For the live layer, in load order. 1. `research/QUESTIONS.md` (`Q-capK-bv`, both rows) and `thm-capK-bv.md` §0 and its ledger `verdict:` line: "first at `x = 239`" becomes "the lemma's floor `9κ+1 = 10` is first reached at `x = 239`; the factor-below-1 threshold 10.82 is first cleared at `x = 263`". 2. `research/GLOSSARY.md`:354, the full replacement in §1.3, including the sifted-versus-assumed clause. 3. `research/bv-import-survey.md` §3.1: `PROD(1 - 2/(q_i - 1))` becomes `PROD(1 - 1/(q_i - 1))`. 4. `research/bv-import-survey.md` §3.2: "dimension 2 in the range `(x, q_K]`" becomes "dimension 1 wherever `q_i < q`, on a sequence that already carries `P^-(m) >= q`". 5. `paper/staircase-note.md` §7: add the void-condition parenthetical of §1.3. 6. `thm-capK-bv.md` §3 Step 3: add "in its `max_{y<=T}` form" (§3.3). 7. `thm-sharp-sieve-range.md` §0 table, §3 reading 1, §7 buys-item 1: "@53" becomes "@37" (§5.5); §0 finding 1's "`x = 239` and `x = 131`" becomes "`x = 263` and `x = 131` at `δ = 0.5`, or `x = 127` at `δ = 1`". 8. `comb-discrepancy-tight.md` §3: `ρ * g_max` becomes `ρ * (g_max - 1)`, values in §6.3, and "64% of `D_x` at @7" becomes 61.7%. 9. `comb-discrepancy-tight.md` §5: both column headers, §6.4. 10. `certificate-engine.md` status table cell, §6.5. Nothing in this list moves an exponent, opens a route, or bears on the twin prime conjecture. Two of the ten are arithmetic errors in numbers that had already reached `research/QUESTIONS.md`; the rest are labels and scope. --- ## 9. What would falsify this, and whether that check has run | claim of this pass | what would falsify it | has the check run | |---|---|---| | the one-class factor, `PROD(1-1/(q_i-1))` | an admissible cofactor `m` with `P^-(m) >= q` and `q_i | qm` for some `q_i < q`, or the exact depth-`K` counts favouring the two-class predictor | **RUN**, independently: 8 exact counts at @17 and @19, `K ∈ {1,2,4,8}`, corrected factor within 0.40%, survey's factor 5.00% to 32.37% out. The proof is two lines and needs only `P^-(m) >= q > q_i` | | the one-class factor's scope clause (dimension 2 when roughness is sifted) | a formulation in which `m !≡ 0 (mod p)` is simultaneously free and sifted | **NOT RUN as a check**; it is a definitional statement, and it is what reconciles `thm-capK-bv` with `thm-buchstab-transfer-shallow` | | S1(a) at @13 and @17 | one prime-regime non-self fresh victim with composite cofactor, cofactor below `q`, or `v mod 30` outside its pair | **RUN**, exhaustive, 312 + 3 888 victims, **0** violations. NOT run at @11, @19, @23, @29 | | `Sum cap1` = 2 720 and 42 895, ratios 0.2570 and 0.2514, D2's 5 and 16 | any of them differing from `staircase-note` §6 or from the note | **RUN**, all reproduce | | `s` clears 10.82 first at `x = 263` | a different `(D, z, T)` convention than the note's own §4 table, which this pass reproduced to the digit at ten levels | **RUN**. The convention is pinned by matching all ten printed `s` values. Under the proof's own `D = T^{1/3}` the crossing is later still, not earlier | | the BV `max_{y<=T}` requirement in Step 3 | a form of Bombieri-Vinogradov at endpoint `q-1` admitting level `30T^{1/3}` | **NOT RUN as a literature check.** No source was read at a page in this pass. The exponent arithmetic (`2/9 > 1/6`) is elementary and is in §3.3 | | `K_dim = 1.2000` and `K = 7/5` are the true suprema | a `(w, z1)` pair outside `w <= p_800`, `z1 <= 2e8` with a larger ratio | **RUN to `z1 <= 2e8`** at the binding base and to `p_800` in `w`, plus the tail limits 0.984 and 0.910. Still not a proof; the sup at `z1 = 7^+` is elementary and the tail is measured | | first non-empty `κ = 1` level is @37 | a different non-emptiness criterion than the note's own (`V*F1 < 1` and `f1 > 0`), or a different `V(z)` or `s` convention | **RUN** on the note's own conventions, every prime level @11..@103. The note's `V` and `s` columns reproduce exactly at the seven levels it prints, which pins the convention | | `D_x` at @7..@23, and Theorem A | a window beating `max G - min G`, or a census mismatch | **RUN**, census and `G(W) = 0` asserted at all six levels; Theorem A's proof is one line and is checked above. @29's 53.64 **NOT RE-DERIVED** | | `ρ * g_max` is off by one gap | a window of length `g_max` containing no comb member | **RUN**: the maximal empty run is `g_max - 1` at all six levels by direct pass, and the @7 comb is small enough to list | | the @17 certificate row, 1 -> 4 primes | a different certification test than cap-28's `errC <= 0.5(main - s)`, or a per-dilation range below the sharp one | **RUN** with exact `cap2` counts and sharp full-period dilated ranges. The 0.07-point share gap and the 17.4-versus-18.37 gap are both traced to the producer's conventions, in §6.4 | | the cluster's shared verdict, "empty computable range" | a finite-level statement at any reachable level from either instrument | **NOT RUN as a search.** DH Thm 9.1's constant is unwritten (page record), and the fundamental lemma's own crossing is at `x = 263` and level 131. Both remain limit statements | | DH Thm 9.1's transcription | the book's p. 104 differing from `lit-pdf-halberstam-richert.md` §6.1 | **NOT RUN.** The book was not opened in this pass; the page record's photograph was read 2026-08-14 and is what carries it | | `f2(4.7088) = 0.4056` | a solver error in either implementation | **RUN**, by a closed form on `(0,4]` plus one quadrature, sharing no code with `attack-beta2-04-loss-budget.js`; nine figures agree to 5 decimals or better |