# Adversarial pass on the two HELD item-0 notes of 2026-09-04 *(2026-09-04, staging, HELD under the publication moratorium. Adversarial only. Targets: `redteam-0904-floor-growth-2.md` with its producer, and `derive-0904-L7-transfer.md` with its producer. One new producer, `research/history/staging/redteam-0904-item0.js`, embedded through `qc/embed.js` and verified with `--check`; it shares no code with any producer under review. The LP here is solved by an exact rational primal simplex with Bland's rule over BigInt fractions, not by vertex enumeration and not by a grid; the sieve, the box counts, the CRT walk and the vector-sieve constants are all rebuilt. No existing file was edited and no git command beyond a read-only status was run. Calibration per `CLAUDE.md`: PROVEN, VERIFIED (exact exhaustive computation), MEASURED, DERIVED (a written argument not adversarially checked), HEURISTIC, OPEN, REFUTED.)* ## 0. Verdict, disconfirming half first **Nothing here moves an exponent, and nothing here raises a rung.** The growth half is still a derivation that has survived attacks, and the attacks have now been run in three adversarial passes rather than two. Three passes that failed to break a derivation are three passes that failed to break a derivation. **What is wrong in the two notes, before what is right.** 1. **The proposed `REFUTED.md` scope clause is narrower than the derivation supports, and the rider offered for it is wrong.** `redteam-0904-floor-growth-2.md` §3 scope fact 1 asks the row to say "on the Rosser vector-sieve lattice at equal levels `D₁ = D₂ = z^s`", and defends the restriction with rider (i), "asymmetric levels do not escape: the exponent becomes `8s₁/9 + 8s₂/9` with `s₁ + s₂` pinned by the joint level `2s`". That rider is **REFUTED as written**: `8s₁/9 + 8s₂/9` is the capped maximum only while both `s_i ≤ 3`, and above 3 the capped maximum falls to `(2s+2)/3` and then to 4, so a lopsided split lowers the floor rather than leaving it unchanged. The conclusion survives on a different argument (§3.1), and the surviving argument is **wider** than the clause: over 1643 admissible upper-level and lower-level splits the floor beats the decoupled window at every cell, by at least **0.644444** (S2). "At equal levels" is therefore the wrong restriction to write into the row. 2. **The crossing figure the note replaces is `u₀`-specific as well as `s`-specific, and the replacement is quoted in a different convention.** The record's `z ≈ 5.6e31` is `10^31.75`; on an independent fit the `s = 2.7` crossing is `10^31.68` at `u₀ = 4.2165` and `10^35.36` at `u₀ = β₂` (S3). So the record's figure sits in the `u₀ = 4.2165` cell, and the note's replacement `10^15.30` is a `u₀ = β₂` cell at a different `s`. Both parameters have to be fixed in the same sentence. 3. **The onset correction is right in its numbers and wrong in its wording.** The exact onsets reproduce (4.390e5, 1.111e6, 1.860e6 at s = 3.0, 2.8, 2.7, against the note's 4.42e5, 1.12e6, 1.86e6, S4), but at the record's own `s = 2.698721` the exact onset is **1.875e6**, which is **later** than the quoted analytic 1.055e6, not earlier. Writing "the exact onset is 4.42e5, not 1.055e6" makes the construction read as turning on 2.4× sooner when at the same `s` it turns on 1.78× later. 4. **The escape-(a) correction carries a mislabelled bound.** `(A₁A₂)²/(2n₁n₂)` is a **lower** bound on the planted split's contribution to `⟨R₁²⟩`, because `Ω ≥ A₁A₂`, so as written it cannot rule out a contradiction with Lemma V, only fail to exhibit one. The two-sided version is available and passes: with `λ⁺(n) ≤ 2^{ω(n)}` the split's contribution is at most `10^{−3.004323e+7}` against `B ≈ 10^{10.1}` (S5). The conclusion is unchanged; the label is not. 5. **On the L7 side, the vector-sieve price quoted is the retired one.** `2(1+√e) = 5.297442541400` is the constant that appears **only when `D⁻ = D⁺` is forced**; the corpus's live coupled figure is `K_BF = 5.158064680330`, and `sift-limit-attack.md` §4.5 correction (i) explicitly retires the break-even that the larger constant produces. Recomputed here from `F` and `f` with no code shared: the free-level optimum is **5.158064680330** at `(a, b) = (2.229949, 2.928115)`, twelve digits (S6). Both are above `β₂`, so the L7 conclusion is untouched. 6. **The scope of death is one row larger than the growth note says, and its own sibling says so.** `redteam-0904-floor-growth-2.md` §10 rules "L3–L10 are different objects and are untouched"; `derive-0904-L7-transfer.md` §3, written the same day, rules that L7's mechanism 2 is **dead** if the growth half survives. The statement L7 survives as an open statement; one of its two known mechanisms does not. **What is right.** - **The truth-gap reading is CONFIRMED**, and by a step the note does not claim. `REC(s, u₀)` is stated about `R_H` (`attack-0829n-rml-proof.md` §3, read word for word), `R_H` is defined as `T − H·M` for this certificate (`rho-maximal-law.md` §1), and the one place the chain could have needed a hypothesis it lacks, the mean-square input at the cheapest legal point, does not need one: `0.63·β₂ = 2.688 > 1+√e = 2.6487`, so the **conjectural** Lemma V's range would fail there, but the chain uses the **proven** mean-square form, which `REFUTED.md` records as "a theorem with no hypothesis" (§2). - **Every load-bearing number reproduces on independent code.** 18 of 18 `A₁A₂` cells digit for digit including the step slopes; the LP maximum at 0 deviations of 16 with largest residual 4.441e-16 and the optimal vertex printed as `(s/3, s/3, s/9, s/9)`; the corollary crossing at 1.097e-13 of `9β₂/16`; `C = 1.1206e-4` and the `s = 3.0` crossing `10^15.30`; `ln(n₁n₂) = 7.4993e7`; the L7 CRT counts EQUAL at x = 7, 11, 13 and `B(13) = 1.1000`. - **The `s ≥ 7` emptiness is confirmed and is sharp at 7 itself**: `max(Σa + 3a₄) = 7` for every `s ≥ 5` and the exit condition is strict, so the four-prime family is empty **from `s = 7` inclusive** (S1). **The ruling.** `redteam-0904-floor-growth-2.md`: **SURVIVES**, rung unchanged at DERIVED and red-teamed twice on the growth half, with **four** of its proposed live-layer changes needing rewording and **one** supporting rider refuted. `derive-0904-L7-transfer.md`: **SURVIVES** as a PARTIAL, with **two** of its clauses needing rewording and its central inference (vacuous on average, hence not non-vacuous at every position) confirmed by exact enumeration. ## 1. What this pass ran, and what it did not Ran: an exact rational simplex over the exit-chain polytope for both the `D⁺` and the `D⁻` conditions and for chain lengths 2 to 24; a scan of upper-level against lower-level splits; an independent byte sieve to 1e8 with exact box counts, the `A₁A₂` table, the fitted constant and the crossing at two `u₀`; an onset scan on a 0.2% log grid under a stated definition; the planted class's `ln(n₁n₂)` with a two-sided bound on its contribution; the L7 CRT counts, the union-bound certificate's exact period mean and its minimum over all 30030 positions; and the coupled vector-sieve constants from `F` and `f`. Not run: anything about `E2` above the levels the sibling notes reached; any derivation for a non-Rosser weight system; any recomputation of `Ω` itself (the exact half is PROVEN and is not attacked here); any literature search. ## 2. Target 1, claim (a): the quantifier, certificate value or true remainder **Grade: CONFIRMED.** The proof-gap reading is refuted and the chain has no step where a certificate value must become a true remainder. **The chain, written independently, every quantifier explicit.** Fix a prime `z`, `W = P(z)`, `s > 1+√e`, `D = z^s`, and Rosser-Iwaniec `λ^±` of level `D` for the two systems `{r ≡ 0 (p)}` and `{r ≡ −2 (p)}`, `p < z`. With `n₁ = gcd(r, P)`, `n₂ = gcd(r+2, P)`: cc(r) = λ⁻(n₁)λ⁺(n₂) + λ⁺(n₁)λ⁻(n₂) − λ⁺(n₁)λ⁺(n₂) T(x) = Σ_{x < n ≤ x+H} cc(n), M = ⟨cc⟩, R_H(x) = T(x) − H·M `REC(s, u₀)` (`attack-0829n-rml-proof.md` §3, verbatim): **∃**ε>0 **∃**z₀ **∀** primes z ≥ z₀, with `H = ⌊z^{u₀}⌋`, `sup_{x ∈ ℤ/W} |R_H(x)| ≤ z^{u₀/2−ε}·⟨R_H²⟩^{1/2}`. | step | statement | quantifier | rung, and who owns it | |---|---|---|---| | A1 | `∃ r* ∈ ℤ/W` with `cc(r*) = −Ω(z,s) ≤ −A₁A₂` | ∃r*, for each z past the onset | E1–E3, PROVEN (exact half); the onset is S4 here | | A2 | `cc(r′) ≤ 1` at every `r′`, hence `T(x) ≤ cc(r*) + H − 1` for every window containing `r*` | ∀x | E4, PROVEN, and it is the Brüdern-Fouvry pointwise inequality itself | | A3 | at `x = r*−1`, `R_H(x) ≤ −Ω + H − 1 − H·M`, so `sup_x |R_H(x)| ≥ Ω − H + 1 + H·M` | the sup runs over the same `ℤ/W` REC ranges over | definition of `R_H`; uses `M > 0`, which holds strictly for `s > 1+√e` and is checked here as `2f − F = 0` exactly at `1+√e` (S2) | | A4 | `⟨R_H²⟩ ≤ B·H`, `B ≤ 9A(z)²(E(z)−1) = O(ln⁸z)` | ∀H, ∀z, ∀s | PROVEN, and `REFUTED.md`'s Lemma V mean-square row records it as **a theorem with no hypothesis** | | A5 | REC ⟹ `sup|R_H| ≤ z^{u₀/2−ε}(B·H)^{1/2} = z^{u₀−ε}·O(ln⁴z)` | ∀z ≥ z₀ | arithmetic | | A6 | contradiction once `log_z Ω > u₀ − ε + o(1)` | ε and u₀ fixed before z | conditional on the growth half | **Where a certificate value would have to become a true remainder: nowhere.** `R_H` is the remainder of this certificate by its own definition, `T = H·M + R_H`, and REC is a statement about `R_H`. The floor bounds `R_H` from below at a position inside the sup's own index set. The suspicion in the brief would be right if REC were about the true twin count in `(x, x+H]`; `rho-maximal-law.md` §1 and `attack-0829n-rml-proof.md` §3 both state it about `R_H`. **The one step that could have failed, and does not.** A4 is an **upper** bound on the rms, and an upper bound is what a refutation of REC needs. The conjectural Lemma V carries a range hypothesis `s ≥ (0.63+δ)u`, and at the cheapest legal point `(s, u₀) → (1+√e, β₂)` that hypothesis **fails**: `0.63·4.26645 = 2.6879 > 2.6487`. If the chain leaned on that form, the "cheapest" point of `attack-0830-rec-cheapest.md` would be exactly the point the chain could not reach. It does not lean on it: the mean-square form is the one that is proven, and it is stated for every `H`, `z`, `s`. The reviewed note does not make this argument; it is the strongest defence of its own §3 that the material supports, and it is recorded here so a later pass does not re-open it. **Riders the chain carries.** (i) A3 uses `M > 0`, which is strict only above `1+√e`; at `1+√e` the main term vanishes to leading order (S2 prints `2f − F = 0` there), so the legal band is open at its lower end for this reason as well. (ii) The direction guard holds: the true rms sits below `√(B·H)`, and a smaller true rms makes REC's allowance smaller and the contradiction easier. ## 3. Target 1, claim (b): the five proposed corrections ### 3.1 The scope clause. WEAKENED, and the rider under it REFUTED Three separate things are being run together in the note's §3 scope fact 1 and §12 row 3. **(i) The row's four named statements are already scoped by their own definitions.** `rho-maximal-law.md` §1 defines `ρ̃`, and hence `RML(α)`, `F1`, `F2` and `R_H`, on "the Rosser-Iwaniec linear-sieve weights `λ±` **of level D** for the two systems", one level for both. `attack-0829n-rml-proof.md` §3 states REC on "the level-D vector-sieve lattice". So the `REFUTED.md` row is not making an unscoped false claim about REC, F1, F2 or RML; it is making a claim a reader cannot see the scope of without opening two other files. The addition is **legibility, not correction**, and the note's §12 wording "current wording: unscoped" overstates it. **(ii) The rider offered for the clause is wrong.** Rider (i) says asymmetric levels give `8s₁/9 + 8s₂/9` with `s₁ + s₂` pinned, "so the threshold is unchanged". `8s_i/9` is the capped maximum only for `s_i ≤ 3`; at `s_i > 3` the exact capped maximum is `(2s+2)/3`, then 4 (S1, 0 deviations of 16, residual 4.441e-16), both strictly below `8s_i/9`. So the sum is **not** a function of `s₁ + s₂`, and a lopsided split gives a **smaller** floor. **REFUTED as a derivation of the general claim**; it holds when both `s_i ≤ 3`, a hypothesis it does not state and which is exactly what a lopsided split violates. **(iii) The conclusion survives, on a wider argument.** Take the upper level `a` and the lower level `b` free (this is the axis that actually matters, since the floor's `−λ⁺λ⁺` term reads the upper level and the `A₁B₂` terms read both). Two facts, both computed here (S2): - Positivity of the vector sieve's main term is `2f(b) > F(a)`. Since `f(σ) < 1` for every `σ`, positivity **forces `a > e^γ = 1.781072`**, whatever the lower level. - The floor exponent is `LP⁺(a) + max(LP⁺(a), LP⁻(b))`, because `Ω ≥ A₁A₂ + A₁B₂ + B₁A₂` and each of the three products is non-negative. The decoupled window exponent is `max(a, b)`. Over **1643** admissible `(a, b)` cells the floor exceeds the window at **every** cell, minimum margin **0.644444** at `(a, b) = (1.85, 4.00)` with four-prime chains, and by more with twelve-prime chains: **0 escapes either way**. So the death is not confined to equal levels. Proposed wording for the row, in place of the note's: > ... on the **Rosser vector-sieve lattice**, at every admissible level split > and not only at `D₁ = D₂` (checked over 1643 splits, minimum margin 0.64 of > exponent); no other admissible weight system is derived. **What is still not established, and the note is right to say so.** That no sup-over-positions statement survives for a **different weight system**. The `−λ⁺λ⁺` sign fact E2 is Rosser's, and nothing here or there derives an analogue for Selberg's `Λ²`. That remains HEURISTIC. **A second-order finding, adverse to REC.** Counting the `B`-terms with long chains raises the equal-level floor above the note's `16s/9`: 5.2901 at `s = 1+√e` and 5.9918 at `s = 3.0`, against `2 × cappedLP₄` of 4.7088 and 5.3333 (S2). The `D⁻` chain family has not been membership-checked the way the `D⁺` family has, so this is an LP reading and not a construction; it is recorded because it says the note's threshold is conservative, not optimistic. ### 3.2 The `z ≈ 5.6e31` correction. CONFIRMED in substance, REWORD required The independent fit reproduces the note exactly: `C = 1.1206e-4` at `s = 3.0` from the top measured level, crossing `10^15.30`; `10^25.01` at `s = 2.8` (S3). The note's diagnosis that 5.6e31 is an `s = 2.698721` figure quoted without its `s` is right. **What the note misses.** The crossing moves with `u₀` as well, and the record's figure is in a different `u₀` convention from the note's replacement. At `s = 2.7`, the crossing is `10^35.36` at `u₀ = β₂` and `10^31.68` at `u₀ = 4.2165` (S3), and the record's `5.6e31` is `10^31.75`, agreeing with the second cell to 0.07 in the exponent. `u₀ = 4.2165` is the reduced-range endpoint that `redteam-0830-floor-growth.md` R5 carries. So the sentence needs both parameters, and comparing `10^15.30` (at `β₂`) against `5.6e31` (at 4.2165) compares two conventions. Proposed wording: *the falsity's level is `s`- and `u₀`-dependent: at `u₀ = β₂` it is `z ≈ 10^15.3` at `s = 3.0`, `10^25.0` at `s = 2.8` and `10^35.4` at `s = 2.7`; the `5.6e31` on the record is the `s = 2.698721`, `u₀ = 4.2165` cell.* The fitted-constant caveat the note gives in its §11 applies to all of these and should travel with them. ### 3.3 The exact onset. CONFIRMED numerically, REFUTED as worded Under an explicitly stated definition (each of the four boxes holds at least two primes; with `d′_min` the product of the smallest prime in each box, at least two primes `p* < min(box₄)` satisfy `d′_min·p*³ > D`; and `p₁ ≤ z`), the onsets are **4.390e5** at `s = 3.0`, **1.111e6** at 2.8, **1.860e6** at 2.7 and **1.875e6** at 2.698721, on a 0.2% log grid (S4). The note's 4.42e5, 1.12e6 and 1.86e6 reproduce to within the grid. **The wording is the problem.** At the record's own `s = 2.698721` the analytic figure 1.055e6 is **early by 1.78×**, not late: the construction turns on at 1.875e6 there. The note's §12 row "the exact onset is 4.42e5 at s = 3.0, not 1.055e6" replaces an `s = 2.6987` analytic figure with an `s = 3.0` exact one and leaves a reader believing the construction is live 2.4× sooner at the level the record is talking about. Proposed wording: *the literal construction is empty below `z = 1.875e6` at `s = 2.698721` (the analytic window bound 1.055e6 is early by 1.78×); at `s = 3.0` the onset is 4.39e5.* The note's underlying finding, that the analytic window `(4D^{1/27}, D^{1/9}/4)` is neither necessary nor sufficient, is not re-tested here at the level of individual `p*`; the onset numbers are consistent with it in both directions. ### 3.4 The `s ≥ 7` emptiness. CONFIRMED, and sharp at 7 `max(Σa + 3a₄)` over the four-prime polytope equals **7 for every `s ≥ 5`**, attained at all ones, and the exit condition is strict, so the family is **EMPTY from `s = 7` inclusive** and non-empty at `s = 6.9` (S1). The note's "once `s ≥ 7`" is right, including the endpoint. `redteam-0830-floor-growth.md` R3's "8 beyond `s = 5`" therefore holds on `[5, 7)` only, as the note says. The note's continuation figures (doubled threshold 12 at `s = 7`, 16 at 9, 36 at 20) are the **all-ones** sub-family's, and are conservative: allowing any chain shape, the best doubled threshold with a legal exit tracks `2s`, reading 13.9999 at `s = 7`, 17.9997 at 9 and 23.9982 at 12 (S1, at a chain-length cap of 24). Either set of numbers clears `β₂`; the note should say which family its numbers come from. ### 3.5 The escape-(a) density. CONFIRMED in substance, mislabelled `ln(n₁n₂) = 7.4993e7` at `(s, z) = (3.0, 1e8)` against `θ(z) = 9.9988e7`, and 2.1960e7 at `(2.8, 1e8)` (S5). So the planted class is larger than one position in `W` by `exp(2.5e7)`, and the first pass's `1/W` accounting understates it, as the note says. The class's density is `≈ 1/(n₁n₂) = exp(−Θ(D^{1/3}))`, and this is **derived and measured, not asserted**: `ln(n₁n₂)` reads three quarters of `D^{1/3}` at both measured rows, which is `θ(D^{1/3}) − θ(D^{1/3}/4)` under Chebyshev. **Two labels are wrong.** (i) The first pass computed `H(Ω + HM)²/W`, which is `H` positions out of `W`, not "one position out of `W`" as the note says. (ii) `(A₁A₂)²/(2n₁n₂)` is a **lower** bound on that split's contribution, since `Ω ≥ A₁A₂`; a lower bound far below `B` shows no contradiction has been exhibited, not that none exists. The two-sided check is cheap and passes: `λ⁺(n) ≤ 2^{ω(n)}` over 4195588 box primes gives the split's contribution at most `10^{−3.004323e+7}` against `B ≈ 10^{10.1}` (S5). Either way the escape fails by an exponential. **A limit on what §7 can do at all.** The sum of these contributions over **all** splits is `⟨R₁²⟩` itself, the object Lemma V bounds. A single split can never be a full consistency test, only a check that the one split the construction plants does not by itself exceed `B`. The note should say so. ## 4. Target 1, claim (c): the LP maximum and the corollary's constant **Grade: CONFIRMED, both.** - The capped four-prime maximum reproduces `8s/9` (`s ≤ 3`), `(2s+2)/3` (`3 ≤ s ≤ 5`) and 4 (`s ≥ 5`) at 16 values of `s`, **0 deviations**, largest residual **4.441e-16**, by exact rational simplex; the optimal vertex prints as `(0.9, 0.9, 0.3, 0.3)` at `s = 2.7`, that is `(s/3, s/3, s/9, s/9)` (S1). - The `2k` closed form `s(1 − 3^{−k})` reproduces at `2k = 2, 4, 6, 8` and both `s`: 1.866667 / 2.488889 / 2.696296 / 2.765432 at `s = 2.8`, and 2 / 2.666667 / 2.888889 / 2.962963 at `s = 3.0` (S1). - The corollary crossing solves at `s* = 2.399878284834` against `9β₂/16 = 2.399878284834`, agreeing to **1.097e-13**; `1+√e` clears it by 0.248843 (S1). The identity `2·(8s/9) = β₂ ⟺ s = 9β₂/16` is one line and holds only on `s ≤ 3`, as the note says. ## 5. Target 1, claim (d): the scope of death **Grade: WEAKENED. One row is missing and one sentence is stronger than its evidence.** **The density claim is derived, not asserted** (§3.5). That part stands. **L7 is not untouched.** `derive-0904-L7-transfer.md` §3, the sibling note of the same day, rules that L7's mechanism 2 needs "the sup-over-positions law" for decoupling and is therefore **dead if the growth half survives**. The growth note's §10 says "L3–L10 are different objects and are untouched". The statement L7 stays open; one of its two known mechanisms leaves with L2. The two notes have to be made to agree before either reaches the live layer. **L1, L3, L4, L5 do not depend on a sup-over-positions law.** L1 is the unconditional exponent itself and is the target, not an input. L3 is a doubling inequality on `Ĝ`. L4 and L5 are (H-sub-pow) statements about `ln G₂` and its sub-power constant. None of the four passes through `R_H`, `ρ̃` or the certificate. **The nearest sup-shaped legal row is U1, and it survives for a reason the note does not give.** U1 asks whether `sup/sd` stays bounded in the two-class discrepancy channel `ΔΦ₂`. `ΔΦ₂(y, x) = Ψ(y,x) − (D_x/x#)·y` (`G2-STATE.md`, the two-class discrepancy construction) is the signed discrepancy of the **true** two-class counting function, not of the certificate, so the floor, which is a statement about `cc` and `ρ̃`, says nothing about it. Naming that is worth a clause, since "different objects" is the whole argument for four legal rows and a reader will want to see it once. **"Every almost-all-positions statement survives" is stronger than the evidence.** What the floor gives is a **lower** bound on one exceptional class; it gives no **upper** bound on the exceptional set. The honest form is *untouched by this construction*, which is what the note's own body says and what its §0 bullet does not. Proposed wording: *the construction plants a class of density `exp(−Θ(D^{1/3}))`, so it exhibits no obstruction to any almost-all-positions statement, including Lemma V; it does not bound the exceptional set and does not certify any such statement.* ## 6. Target 2: the L7 transfer note ### 6.1 "Vacuous on average, hence not non-vacuous at every position". CONFIRMED The inference is valid as stated, and the average identity is exact rather than approximate. Over the full period `W = 30030` at `x = 13` the union-bound certificate `C(x) = #holes in (x, x+H] − Σ_p #{holes ≡ −2 (p)}` has period mean equal to `H·(φ(W)/W)·(1 − B)` at all four `H` tested, to the printed digits: −0.9590, −1.9181, −3.8362, −7.6723 at `H` = 50, 100, 200, 400 (S6). A mean at or below zero forces a position at or below zero, and a Jacobsthal-type bound needs every position, so the union bound cannot deliver L7 from `x = 11` on. **One clause overstates.** "From there the union bound certifies nothing in the period average" is right; "certifies nothing" without the qualifier is not. At `H = 50` the certificate is still positive at 4644 of 30030 positions, at `H = 100` at 2574, at `H = 200` at 564, and only at `H = 400` is it non-positive everywhere (max −1). Proposed wording: *from there the union bound certifies nothing at every position, which is the form L7 needs; it still certifies at some positions until `H` is a few hundred at `x = 13`.* ### 6.2 The CRT count. CONFIRMED by independent re-enumeration Re-enumerated with a different walk at `x = 7, 11, 13`: per-prime counts of holes in the class `−2 (mod p)` are EQUAL to `φ(W)/(p−1)` at every odd `p` (3:2880, 5:1440, 7:960, 11:576, 13:480 at `x = 13`), `B(13) = 1.1000`, and the exact non-twin-slot fraction 0.7422 equals the closed form `1 − ∏_{3≤p≤x}(p−2)/(p−1)` (S6). The note's `B(x) ∼ lnln x` is the standard Mertens statement and needs no check. ### 6.3 The sieve-on-holes as the vector sieve. WEAKENED, on two counts **The identification needs one substitution step that the note skips.** Sieving `A + 2` with the true indicator of the holes `A` kept on the other side is **strictly stronger pointwise** than the vector sieve, because the vector sieve replaces both indicators by sieve weights. The corpus already owns the adjudication: `sift-limit-attack.md` §7b(1b) reads Ford-Halberstam's Lemma 1 as "BF (2.6) with the true indicators in the minorant slots, strictly stronger pointwise, and any usable form of it (minorants substituted for the true indicators) is (2.6) verbatim". The reason the substitution is forced is that the count `|A ∩ (−2 mod d)|` is needed two-sidedly and uniformly in `d`, which is a second sieve. So "That is exactly the Brüdern-Fouvry vector-sieve decomposition" should read "reduces to the Brüdern-Fouvry vector sieve once the hole indicator is replaced by a sieve minorant, the step `sift-limit-attack.md` §7b(1b) prices". **The price quoted is the retired one.** The note lands mechanism 2 at `2(1+√e) = 5.2974`. That constant is `K_FH`, the value when `D⁻ = D⁺` is **forced**; `sift-limit-attack.md` §7b(1b) gives `K_BF = 5.158064680330` as the free-level figure, and §4.5 correction (i) retires the 1.2417 break-even that `5.2974/β₂` produces, calling every appearance of it "that much too pessimistic". Recomputed here with no shared code, by minimising `a + b` subject to `2f(b) > F(a)`: the minimum is **5.158064680330** at `(a, b) = (2.229949, 2.928115)`, `b/a = 1.3131`, and forcing `a = b` gives `2(1+√e) = 5.297442541400`, 2.702135% worse (S6). Both clear `β₂ = 4.266450`, so the note's conclusion is unaffected; the sentence should quote 5.158065 with 5.2974 as the equal-level variant. ### 6.4 "No third mechanism excluded". HONEST BUT INCOMPLETE The note's §6 says a third transfer argument "is not known to this note; not searched in the literature beyond the corpus's own `recon-0828-jacobsthal.md` and `covering-dive.md`". That is honest, and no third mechanism is hidden. It is incomplete as a survey of the corpus's own shelf: the family the brief names, weighted sieves with Chen's switching, **is already priced and closed**, at `sift-limit-attack.md` §4.2 (Richert's weights, DHR Theorem 11.1, Chen's switching, through Lichtman's record), verdict "no entry point", with two reasons that apply verbatim to L7: `G₂` has no almost-prime slack to spend, so weights tolerating a stray mid-range factor certify the wrong set; and the sub-8 constants consume progressions-averaged inputs unavailable uniformly in window position. §4.3 closes GPY / Maynard-Tao the same way ("a lower-bound device, not an upper bound for gaps of a sifted set"), and §4.4 closes the FKMPT long-gap machinery on its authors' own Remark 7. Citing those three strengthens the note's conclusion rather than weakening it, and leaving them out invites a later session to re-price them. ## 7. The two notes' proposed live-layer changes, graded | # | proposed change | grade | wording that survives | |---|---|---|---| | 1 | `REFUTED.md:99` "red-teamed once on each half" → twice on the growth half, still DERIVED | **SURVIVES verbatim** | as proposed | | 2 | `REFUTED.md:99` "six independent attacks" → ten | **SURVIVES**, and is now eleven or more counting this pass; the count is bookkeeping, not evidence | prefer "survived repeated independent attacks across three adversarial passes and stays DERIVED", so the row stops carrying a number that changes every session | | 3 | `REFUTED.md:99` add "on the Rosser vector-sieve lattice at equal levels `D₁ = D₂ = z^s`" | **REWORD** (clause too narrow; the rider under it REFUTED, §3.1) | "on the **Rosser vector-sieve lattice**, at every admissible level split and not only at `D₁ = D₂` (checked over 1643 splits, minimum margin 0.64 of exponent); no other admissible weight system is derived" | | 4 | `REFUTED.md:99` "falsity beyond `z ≈ 5.6e31`" → `10^15.3` at `s = 3.0` | **REWORD** (mixes two `u₀` conventions, §3.2) | "the falsity's level is `s`- and `u₀`-dependent: at `u₀ = β₂`, `z ≈ 10^15.3` at `s = 3.0`, `10^25.0` at 2.8, `10^35.4` at 2.7; the 5.6e31 on the record is the `s = 2.698721`, `u₀ = 4.2165` cell" | | 5 | `TODO.md:28` board row 1: ten attacks, crossing `s`-dependent | **REWORD**, same reason as row 4 | as row 4, compressed | | 6 | `TODO.md:680-687` "the exact onset is 4.42e5 at `s = 3.0`, and 1.055e6 is an analytic figure optimistic by 1.8× at its own `s`" | **REWORD** (the numbers survive, the framing inverts the direction, §3.3) | "the literal construction is empty below `z = 1.875e6` at `s = 2.698721`, the analytic window bound 1.055e6 being early by 1.78×; at `s = 3.0` the onset is 4.39e5" | | 7 | `TODO.md:29` next move: the second pass has run, so only the non-two-class input remains | **SURVIVES verbatim** | as proposed | | 8 | `TODO.md:658-661` "then 4" available only on `5 ≤ s < 7`; longer chains carry `s ≥ 7` | **SURVIVES**, and is sharp at `s = 7` itself | add "inclusive at `s = 7`", and say the 12 / 16 / 36 figures are the all-ones sub-family's (any-shape chains give ≈ `2s`) | | 9 | `README.md:234` "six attacks" → ten | **SURVIVES**, same bookkeeping caveat as row 2 | as row 2 | | 10 | `G2-STATE.md:186-192` ten attacks, `s`-dependence, and "the second adversarial pass has now run and did not break it, so this sentence's own reopening condition is spent" | **REWORD**: the first two clauses as rows 2 and 4; the third **SURVIVES** | keep "the reopening condition named in this sentence is spent; reopening now needs a new mechanism, not another pass" | | 11 | `redteam-0830-floor-growth.md` R3 "8 beyond `s = 5`" → only on `[5, 7)` | **SURVIVES verbatim** | as proposed | | 12 | `redteam-0830-floor-growth.md` §3 escape (a): density `1/W` → `≈ 1/(n₁n₂)`, corrected bound `(A₁A₂)²/(2n₁n₂)` | **REWORD** (mislabelled bound, §3.5) | "the density is `≈ 1/(n₁n₂)`, exponentially larger than `1/W`; the split's contribution is between `(A₁A₂)²/(2n₁n₂)` and `(3·2^{ω})²/(2n₁n₂)`, log₁₀ from −3.257e7 to −3.004e7 at (3.0, 1e8), against `B ≈ 10^{10.1}`; and one split is not the sum over splits" | | 13 | `attack-0830-rec-cheapest.md` §4.2: the `p*` window is a convenience bound, neither necessary nor sufficient | **SURVIVES verbatim** | as proposed | | 14 | growth note §10 "L3–L10 are different objects and are untouched" | **REFUTED** as written (§5) | "L1, L3, L4, L5 are untouched; **L7's mechanism 2 dies with L2** (`derive-0904-L7-transfer.md` §3); U1's object `ΔΦ₂` is the true-count discrepancy, not the certificate's, and is untouched" | | 15 | growth note §0 "every almost-all-positions statement survives" | **REWORD** (§5) | "exhibits no obstruction to any almost-all-positions statement; it does not bound the exceptional set and certifies nothing" | | 16 | L7 note §2 "from there the union bound certifies nothing in the period average" | **SURVIVES** with a qualifier (§6.1) | add "at every position, which is the form L7 needs" | | 17 | L7 note §3 "That is exactly the Brüdern-Fouvry vector-sieve decomposition" | **REWORD** (§6.3) | "reduces to the Brüdern-Fouvry vector sieve once the hole indicator is replaced by a sieve minorant, the step `sift-limit-attack.md` §7b(1b) prices" | | 18 | L7 note §3 "landing at `2(1+√e) = 5.2974`" | **REWORD** (§6.3) | "landing at `K_BF = 5.158065` with free levels, or `2(1+√e) = 5.297443` when `D⁻ = D⁺` is forced; both above `β₂`" | | 19 | L7 note §6 "a third mechanism, if one exists, is not excluded" | **SURVIVES**, incomplete (§6.4) | add "the corpus already closes weighted sieves with Chen switching (`sift-limit-attack.md` §4.2), GPY / Maynard-Tao (§4.3) and FKMPT (§4.4) for this object" | | 20 | L7 note's ledger verdict and PARTIAL status | **SURVIVES** | as proposed, with row 18's figure | ## 8. What would falsify this red team - **The escape scan's grid being too coarse.** RUN at 0.05 in both `a` and `b` over `a ∈ [1.70, 4.65]`, `b ∈ [2.00, 4.00]`, 1643 admissible cells, minimum margin 0.644444. NOT RUN outside that box, and NOT RUN for the four-parameter family in which the two systems carry different upper levels as well. A cell with `floor ≤ window` anywhere would restore the "equal levels" clause and open a live escape. - **`LP⁻` being the wrong polytope for the `B`-terms.** The `D⁻` prefix conditions are taken here as the even-`m` ones, mirroring E2's statement. If that mirror is wrong the §3.1 escape scan and the §3.1 "conservative" reading both move. NOT independently checked against `λ⁻` itself; the reviewed notes check `λ⁺` at `z ≤ 41` and this pass does not repeat it. - **The onset definition.** The 0.7% gap between 4.390e5 here and 4.42e5 there is grid and definition, not disagreement. A different definition of "live" moves both. The definition used is printed in S4 so the comparison can be redone. - **The fitted constant `C`.** Fitted on the top measured level alone, as in the reviewed note. Every crossing figure in §3.2 inherits that, and the reviewed note's own §11 sensitivity, about one natural log of `z` per unit of `ln C`, applies to all of them. These are order-of-magnitude figures. - **`u₀ = 4.2165` being the wrong reading of the record's 5.6e31.** The agreement is 0.07 in the exponent at the same `s`, which is strong but is a reconstruction, not a citation of the original calculation. If the record's figure was computed at `β₂` with a different `C`, row 4 of §7 changes. - **The mean-square input.** §2's defence of the chain at the cheapest legal point rests on `REFUTED.md`'s Lemma V mean-square row, "a theorem with no hypothesis". If the proven mean square turns out to carry a range hypothesis after all, the chain loses the band `s ∈ (2.6487, 2.6879)` at `u₀ = β₂` and the "cheapest point" framing of `attack-0830-rec-cheapest.md` becomes the one point the argument cannot reach. NOT re-derived here. - **The L7 union-bound enumeration being unrepresentative.** RUN exhaustively over the whole period at `x = 13` only, `W = 30030`. The period mean is exact at every `x`, so the inference does not depend on the level; the counts of positions where the certificate is still positive do. *Gate: the producer verifies under `node research/qc/embed.js --check` (code-sha256 matches, body matches out-sha256, out-sha256 matches). No existing file was edited by this note.*