# REC at its cheapest legal point: the saving asked is z^{1.031}, and the pointwise floor of the certificate says no accounting can supply it > **RIDER 2026-09-04 (orchestrator, from `redteam-0904-floor-growth-2.md` §5 > and `redteam-0904-item0.md`).** §4.2's p* window (4D^{1/27}, D^{1/9}/4) is > a convenience bound, neither necessary nor sufficient for family membership > at reachable levels: on exact integers the construction's onset is > z = 1.875e6 at this note's s = 2.698721 (the analytic 1.055e6 is early by > 1.78×) and 4.39e5 at s = 3.0. The floor's crossing of H = z^{u₀} depends on > both s and u₀ (5.6e31 is the (2.698721, 4.2165) cell; 10^15.3 at (3.0, β₂)). > The growth half survived a second adversarial pass at every admissible > level split (1643 splits, minimum margin 0.64); rung unchanged, DERIVED. > **RIDER 2026-08-30 (orchestrator; both halves red-teamed on independent > code).** EXACT half (`redteam-0830-floor-sign.md`): STANDS as PROVEN and is > stronger than stated in two places (E3's identity holds at every position, > 0 mismatches in 20.5M, not only at doubly non-rough points; sup|R₁| = Ω + M > exactly, so sup|ρ̃| ≥ (Ω + M)/2, not (Ω − M)/2); E2 needs the hypothesis > D ≥ z this note omits (it fails at s = 0.8; free in the legal band); Ω > reproduced at all 26 tabled values and extended to 285, 441, 684, 990, > 1155 at z = 79, 83, 89, 97, 101. Three sentences REFUTED as written: E1 > (L164) "every pair with gcd | 2 is realised" — exactly half are, the > missing half being the pairs where 2 divides one side only; the ledger's > and §0's "251 against z^{u₀} = 1.05e6" — 1.05e6 is HM(u₀), z^{u₀} at > z = 73 is 7.188e7, so the gap is 68.3× larger than stated; §4.4's reason > for λ⁺(P(z)) = 0 (0.24 exceeds 1/8; the closing figure is 0.1169–0.1199, > under 1/8 by 4–7%, a near-miss). The exact half constrains REC at NO u₀ by > itself, every exactly known log_z Ω being below 2. GROWTH half > (`redteam-0830-floor-growth.md`): six attacks, no break, stays DERIVED; > hypothesis s ≤ 3 added to the 16s/9 threshold; "0.012 → 0.35, rising" is > the wrong description (falls 2.5× first); q ≈ z^{5.00} not z^{4.8}; the > blind slope test tests the construction, not the law. Ruling: REC, F1, F2 > and RML below β₂ are false at rung derived-and-red-teamed-once, the route a > truth gap at that rung (`REFUTED.md`, 2026-08-30), its falsity beyond > z ≈ 5.6e31. *(2026-08-30 wave, staging. Producer: `research/history/staging/attack-0830-rec-cheapest.js`, embedded via `qc/embed.js`. No existing file edited, no git command run. Every number below is in the producer's OUTPUT block or is quoted from an embedded artifact cited by file and section. Calibration per claim: PROVEN, MEASURED, DERIVED (a written argument not yet adversarially checked), HEURISTIC, OPEN, REFUTED. HELD: no adversarial pass has run on §4.)* ## 0. What is open, first - **Nothing here lowers any exponent, and no route closes.** At the cheapest legal point the five routes of `attack-0829n-rml-proof.md` §4 die at the same statement they died at yesterday (§2), and the mixed strategy of the brief is worth z^{0} (§3). On the brief's kill rule this is PARTIAL. - **The new part is adverse to the item.** §4 shows the absolute-value accounting is sharp up to a polylog at one window that can be written down, so the saving REC asks for, z^{1.031} at the cheapest point and z^{2s − u₀} anywhere, is not a saving a proof has failed to find but a saving that is not there. The exact half of that (E1–E4 of §4.1) is PROVEN and machine-checked at z ≤ 73; the growth half (§4.2, exponent 16s/9 − o(1), and 2s − o(1) in the limit) is DERIVED and HELD for one adversarial pass. If it stands, REC(s, u₀) is false at every u₀ < 16s/9, which at s ↓ 1+√e is 4.7088 > β₂, so no legal point survives; F1, F2 and RML(α) at every α < β₂ go with it, and rider 2's floor th(nP) exceeds β₂ asymptotically. That is the event yesterday's §6 listed under "the floor crossing β₂: REC would be moot rather than false". - **At every computable z it remains a proof gap.** Ω(z), the pointwise floor, is exact to z = 73 and reads 251 against z^{u₀} = 1.05e6 (S3); the certified lower bound at z = 5e5 is 8.86e17 against 1.07e24 (S4b), a factor 1.2e6 short, with the deficit falling 3.01 → 1.65 exponents over 601..5e5 and needing to reach 0.58 before any window fails. No computation in reach falsifies REC; the falsity is a statement about the limit. - **What was not done.** No exact sup at s near 1+√e beyond z = 29 (S2), no exact Ω beyond z = 73, no six-prime chains in the certified family (they would raise the exponent toward 2s and are dormant below z ≈ 1e5), no literature search for the pointwise-size fact, which is standard. ## 1. The cheapest point, written out (deliverable a) Fix η > 0 and set s = 1 + √e + η, u₀ = β₂ − η, D = z^s, H = ⌊z^{u₀}⌋, W = P(z). REC(1+√e+η, β₂−η) reads: there are ε > 0 and z₀ with sup_{x ∈ ℤ/W} |R_H(x)| ≤ z^{(β₂−η)/2 − ε} · ⟨R_H²⟩^{1/2} for all z ≥ z₀, s and u₀ fixed before z, u₀ never a function of z (the quantifier discipline of `attack-0829n-rml-proof.md` §3, unchanged). Its truth gives G₂(z#) ≤ z^{β₂−η}, an exponent movement of exactly η. The point is legal for every η ∈ (0, β₂ − 2): u₀ − 2 = 2.2165 at the working η (S0), fixed and never sent to 2 (`attack-wrongdirection-audit.md` §3.10, row 10). It is the cheapest point in one currency and the dearest in the other, and both are read off S0 (constants 1+√e = 2.648721270700, β₂ = 4.26645028414864, difference of the doubles 2(1+√e) − β₂ = 1.030992): | η | s | u₀ | cap-currency ask 2s − u₀ | rms-currency ask u₀/2 | bought | |---|---|---|---|---|---| | 0 (limit, illegal: M → 0) | 2.6487 | 4.2665 | 1.0310 | 2.1332 | 0.00 | | 0.01 | 2.6587 | 4.2565 | 1.0610 | 2.1282 | 0.01 | | 0.05 (working point) | 2.6987 | 4.2165 | 1.1810 | 2.1082 | 0.05 | | 0.10 | 2.7487 | 4.1665 | 1.3310 | 2.0832 | 0.10 | | 0.35 (≈ yesterday's s = 3) | 2.9987 | 3.9165 | 2.0810 | 1.9582 | 0.35 | - **Cap currency.** Against any absolute-value accounting (C3-triv, C3-cap of yesterday's §2, both PROVEN at exponent 2s + o(1)) the saving asked is z^{2s − u₀ − o(1)} = z^{1.030992 + 3η}. At η → 0 this is the 2(1+√e) − β₂ gap of `sift-limit-attack.md` §4.5 and §7b (K_FH = 5.297442541400 against β₂), the brief's z^{1.031}: correct to the digits the brief gives. It is the smallest saving any legal point asks for, and it buys the least, η. - **Rms currency.** Against the PROVEN mean square (C1–C2) the recovery asked, sup/rms ≤ z^{u₀/2 − ε}, is LARGEST here: u₀/2 = 2.1332 − η/2 against 1.5 at u₀ = 3. Cheap in one accounting is dear in the other, and the routes of §2 live in different currencies, which is why they have to be priced separately. - **Main term at the point.** M ln²z is MEASURED 0.3332 → 0.2715 at z = 13..47 at s = 2.698721 (S2, S3; the s = 3 family reads 0.3359–0.3772, S1), falling: the linear-sieve main term ln(s−1)² − 1 vanishes at s = 1+√e, so the consumer's HM − 1 is itself thinner here than anywhere else in the band. ## 2. The five routes at a demand of z^{1.031} (deliverable b) Each route is re-priced at (s, u₀) = (2.698721, 4.216450) and at η → 0. 1. **Term-count cap (yesterday §4.1).** The cap delivers exponent 2s + o(1) by taking absolute values across the lattice, so its saving over itself is identically 1 at every (s, u₀). A demand of z^{1.031} is smaller than the z^{2.0} demand at (3, 4), but the route supplies zero saving at any positive demand, so it does not close and cannot close by lowering the demand. Measured at the point: u_cap is not re-tabled; §4 shows the cap is sharp up to a polylog, which is stronger than "not improved". 2. **Position union and moments (yesterday §4.2).** Price e^{θ(z)/q} at moment order q. Against the rms-currency ask z^{u₀/2}: with Gaussian-shaped moments ‖R‖_q ≤ C√q · rms the route needs e^{θ(z)/q}·√q ≤ z^{u₀/2 − ε}, i.e. q ≥ θ(z)/((u₀/2) ln z)(1 + o(1)). The cheapest point asks the largest u₀/2, so the moment order needed is the SMALLEST of the band, z/(2.1332 ln z) against z/(1.5 ln z) at u₀ = 3: a constant factor 1.42 on an order that still diverges. The only proven moment is q = 2 (I1); the failing step is a moment bound of unbounded order, the same step as yesterday. Against the cap-currency figure z^{1.031} the comparison is not even the right one: e^{θ(z)/q} ≤ z^{1.031} needs q ≥ z/(1.031 ln z) and then ‖R‖_q is unproven for every q > 2. 3. **Window smoothing and the P(y) lever (yesterday §4.3).** Polylog movers; at the cheapest point the reduced range of moduli (z^{u₀−δ}, z^{2s}] is only 1.181 + δ exponents wide, so there is less to remove, and the removed part was never the exponent. §4.3 below shows the counterexample window lives entirely inside the reduced range. 4. **The mean square (yesterday §4.4).** I1 is square-root cancellation at every level, s-uniform; not where the exponent is lost. Unchanged. 5. **The smooth-profile form (yesterday §4.5).** Disconnected at both ends; nothing at the point changes the hypothesis. Unchanged. **The single failing step, at the cheapest point:** a bound on the signed count of large-modulus lattice hits in one window, uniform in the window's position, by o(H/ln²z), with H = z^{β₂−η} and the moduli up to z^{2s}. That is the statement of yesterday's §4.3, the reduced arrow, at the point where the modulus range is narrowest. It is the same statement, so by the brief's kill rule the routes verdict is PARTIAL. §4 then says the statement is false. ## 3. The mixed strategy, with the arithmetic (deliverable c) Split the moduli at z^{u₀−δ}. Below it, the ℓ¹ accounting is PROVEN to sit under HM − 1 for large z (yesterday reading 4). Above it, the only two instruments are the cap (sup ≤ CAP_big ≤ z^{2s} polylog, PROVEN) and the mean square (⟨R²_big⟩ ≤ B_big H, PROVEN, almost-all only). A sup bound from the pair {sup ≤ A, rms ≤ B} is exactly A: with no moment above the second, the mean square contributes nothing to a bound at one position (Axis C of `attack-wrongdirection-audit.md` §1: the density of bad positions it controls is W·B²/λ², never zero). So every mixture "part from the mean square, part from a cap over a restricted range" reduces to the cap on the big moduli: saving = CAP_big / z^{2s} = z^{o(1)}, for every split point δ, because the ℓ¹ mass sits at the top of the modulus range (yesterday S3: the share of CAP above z^{3.5} at s = 3 reads 0.5684 at z = 37 and rising). The mixed strategy reaches z^{0}, not z^{1.031}. There is no arithmetic to do beyond this, and §4 makes the zero structural: the cap's exponent is the true exponent of the certificate at a planted window. ## 4. The pointwise floor of the certificate (deliverable e, the new part) Everything in §§2–3 dies at the same statement as yesterday. What is new is the reason: the absolute-value accounting is not loose. Its exponent, up to a polylog, is the true size of the certificate at one window, and the window can be written down. ### 4.1 Four exact facts Write λ^±(n) = Σ_{d | n, d ∈ D^±} μ(d) for the Rosser weights of level D (`sift-limit-lemmaV.js` `rosserSupport`, the pilot's own support), P = P(z), and for a position r let n₁ = gcd(r, P), n₂ = gcd(r+2, P). Then: - **(E1) cc(r) is a function of (n₁, n₂) alone**, cc(r) = λ⁻(n₁)λ⁺(n₂) + λ⁺(n₁)λ⁻(n₂) − λ⁺(n₁)λ⁺(n₂), and every pair (n₁, n₂) of divisors of P with gcd(n₁, n₂) | 2 is realised by some r in ℤ/W (CRT: r ≡ 0 mod n₁, r ≡ −2 mod n₂, r ≢ 0, −2 mod every other odd prime). PROVEN, one line. - **(E2) λ⁺(n) ≥ 0 and λ⁻(n) ≤ 0 at every n | P with n > 1.** The Buchstab boundary identity: for a prefix-closed support D, Σ_{d | n, d ∈ D} μ(d) = 1_{n = 1} − Σ μ(d₀)·1[(n, P(p_m)) = 1], the sum over first-exit chains d₀ = p₁ ⋯ p_m | n (p₁ > ⋯ > p_m, p₁ ⋯ p_{m−1} ∈ D, d₀ ∉ D). D⁺ exits only at odd m (its conditions are p₁ ⋯ p_{m−1} p_m³ ≤ D at odd m; the size cap is implied at even m), so every boundary term has μ(d₀) = −1 and λ⁺(n) is a count; D⁻ exits only at even m, so λ⁻(n) is minus a count. Checked exhaustively over every divisor of P(z) at z = 13..73 (S3, `viol` = 0 at every level, both s). PROVEN. - **(E3) At a doubly non-rough point, cc(r) = −(A₁A₂ + A₁B₂ + B₁A₂) with A_i = λ⁺(n_i) ≥ 0, B_i = −λ⁻(n_i) ≥ 0**, so cc(r) ≤ −A₁A₂. From (E1), (E2). PROVEN. (Where r+2 is rough, cc(r) = λ⁻(n₁) = −B₁; the S3 maximisers at z ≤ 19 are of that kind, B₂ printed as −1.) - **(E4) For every window (x, x+H] containing r: T(x) ≤ cc(r) + (H − 1)**, because cc(r′) ≤ θ(r′)θ(r′+2) ≤ 1 pointwise (Brüdern–Fouvry, PROVEN, `rho-maximal-law.md` §2 (iv)). PROVEN. Define **Ω(z, s) = −min_r cc(r) = max over splits of (A₁A₂ + A₁B₂ + B₁A₂)**. Then, for every H ≤ Ω(z, s): the window starting at r − 1 has T ≤ −1, so **nP(z) > Ω(z, s)** (rider 2's floor), **F1 fails at H**, and since R_1(x) = cc(x+1) − M, **sup|ρ̃| ≥ (Ω − M)/2**. Direction guard: every one of these is a lower bound on a remainder or an upper bound on the certificate, the adverse direction; none says anything about G₂ itself and none has TPC content. ### 4.2 The exit-chain count, and its size By the boundary identity, for n = P₁ with smallest prime p*, λ⁺(P₁) = #{ d′ ∈ D⁺ : d′ | P₁/p*, ω(d′) even, d′ > D/p*³ }, the chains that exit D⁺ exactly when p* is appended (the odd-m condition d′·p*³ ≤ D fails). Take the sub-family ω(d′) = 4, d′ = p₁p₂p₃p₄ with p₁³ ≤ D and p₁p₂p₃³ ≤ D (the two D⁺ conditions, both prefix conditions implied). At the extremal exponent pattern (s/3, s/3, s/9, s/9) the product reaches D^{8/9}, and it exits at any p* > (D/d′)^{1/3} ≈ D^{1/27}. Dyadic ranges p₁ ∈ (D^{1/3}/2, D^{1/3}], p₂ ∈ (D^{1/3}/4, D^{1/3}/2], p₃ ∈ (D^{1/9}/2, D^{1/9}], p₄ ∈ (D^{1/9}/4, D^{1/9}/2] satisfy both conditions identically, give d′ ≥ D^{8/9}/64, and exit at every p* > 4D^{1/27}. Split the primes in each range by index parity into two halves and give each half its own p*: two disjoint sets P₁, P₂, each with A_i ≍ D^{8/9}/ln⁴D by Chebyshev's bounds on the four dyadic prime counts, whence Ω(z, s) ≥ A₁A₂ ≫ z^{16s/9} / ln⁸ z (DERIVED, HELD). The same construction with 2k primes at the pattern (s/3, s/3, s/9, s/9, …, s/3^k, s/3^k) gives exponent 2s(1 − 3^{−k}), so **Ω(z, s) ≥ z^{2s − o(1)}** for fixed s as z → ∞, the o(1) being O(3^{−k}) at the cost of a polylog ln^{−4k} z. The 4-chain exponent already suffices for what follows. What this is not: it is not new mathematics. That combinatorial sieve weights take polynomially large values at smooth integers, and that absolute remainder accounting is therefore sharp for sifting short intervals, is the standard reason a level of distribution is needed at all. The corpus measured the absolute side of this on 2026-08-18 (`sift-limit-attack.md` §7b (3), the planted doubly-smooth n₀ at 0.978 of the trivial bound) and read the signed side as small because at z = 13 it is: Ω(13) = 1 (S3). The signed side grows, and the reason is (E2): at a planted point every term of cc has the same sign, because both weights are counts of exit chains. ### 4.3 What it does to REC, F1, F2, RML and the floor - **REC(s, u₀) is false for every u₀ < 16s/9, if §4.2 stands.** REC implies min_x T(x) ≥ 1 at H = z^{u₀} for all z ≥ z₀ (chain C3–C6, PROVEN), and (E4) with Ω ≥ z^{16s/9 − o(1)} gives a window with T ≤ −1 for all large z once u₀ < 16s/9. At s = 1+√e+η the threshold is 16s/9 = 4.7088 + 1.78η and the cheapest point u₀ = β₂ − η sits 0.4424 + 2.78η below it (S0); at s = 3 the threshold is 5.3333. So the whole legal band (2, β₂] is below the threshold at every admissible s, and the "cheapest" point is not cheaper in the only currency that turns out to matter. - **F1 and F2 (`rho-maximal-law.md` §1) fail at every H < Ω(z)**, including the per-position form ∀x ∃H: the window starting at r − 1 has T ≤ −1 for every H ≤ Ω. **RML(α) fails for every α < 16s/9** through sup|ρ̃| ≥ (Ω − M)/2, which is above β₂: the named legal target of item 0 cannot be true at any sufficient α. **Rider 2's floor th(nP) ≥ log_z Ω → ≥ 16s/9 > β₂**: the deliverable band [th(nP), β₂) is asymptotically empty. This decides the "two models, neither called" of `rho-maximal-law.md` reading 6 in the direction of the room closing, not by either model but by a construction. - **Where the counterexample lives.** The pairs (d₁, d₂) = (d′p*₁, d″p*₂) have q = d₁d₂ ≈ D^{16/9}·p*₁p*₂ ≈ z^{4.8}, inside yesterday's reduced range (z^{u₀−δ}, z^{2s}] and above the window: the reduced arrow of §4.3 there is exactly where the certificate is negative. - **What is NOT claimed.** Nothing about G₂(z#) itself moves: a certificate failing at one window says nothing about the count of admissible r in that window (the measured G₂ exponent stays ~1.5). Nothing about the mean square moves (planted windows are a vanishing fraction of the W positions; I1 stands). The Gaussian ρ law measured TRUE at z ≤ 37 (`rho-maximal-law.md` §4) is a finite-z fact in the regime where the chains are dormant (Ω(37, 3.0) = 21 against 2 sup|ρ̃| − M = 104.410, S3). ### 4.4 The numbers behind §4.2 **Exact, both s (S3).** Ω(z, 2.698721) = 1, 2, 3, 6, 10, 18, 22, 30, 45, 63, 86, 111, 134, 168, 205, 251 at z = 13..73, log_z Ω from 0 to 1.2878; the split enumeration reproduces the walked min cc at z = 13..29 (five MATCH lines), λ⁺(P(z)) = 0 at every level (the D/8 slab is empty: max d/D ≤ 0.24 everywhere, so the two-prime and four-prime chains cannot reach it at these z), and the maximisers are one band of middle primes with p* = 5 or 7 against the top three to five primes. At s = 3.0, Ω = 1, 2, 3, 3, 3, 9, 21, 36, 63, 100 at z = 13..47 and Ω − M ≤ 2 sup|ρ̃| holds at all seven cited levels (`ok` at 13..37), the control that ties the enumeration to the corpus's exact suprema. Against the consumer, Ω/HM reads 6.4e−5 to 3.8e−4 at s = 3 and 2.1e−4 to 4.0e−4 at the cheapest s: three and a half orders short of a failing window. **Hill-climb lower bounds (S4a).** Ω ≥ 6.840e2 at z = 89 rising to 2.108e6 at z = 601, log_z from 1.4543 to 2.2757, against z^{u₀}: ratio 3.7e−6 to 6.8e−6, no trend. The maximisers switch from band splits to interleaved splits at z ≥ 151 (A₁, B₁, A₂ all in the hundreds by z = 601). **Certified sub-family (S4b).** With the primes above 47 split by index parity and the best p* ≤ 47 per side, exact counting of 2- and 4-chains gives Ω ≥ 9.000e4 at z = 601 (control: the family count 300 equals the exact λ⁺ of the same set, so no chain is double-counted) and 8.863e17 at z = 5e5, log_z 1.7828 → 3.1493. Four-prime chains take over from z = 2e4 (1.56e5 of 2.29e5 at z = 2e4; all of it at 5e5, where the two-prime family is empty because p* is capped at 47). The deficit 16s/9 − log_z Ω falls 3.0149 → 1.6485 across 601..5e5, and the bound tracks z^{16s/9}/ln⁸z at 0.012 of it at z = 601 and 0.35 at z = 5e5, rising: consistent with the Chebyshev-count derivation and its ln⁸ z, and not decisive for it. **Fits (S5).** ln Ω on ln z: 3.2825 ± 0.1205 on the exact points z = 29..73, 4.1394 ± 0.0898 on the hill-climb points 89..601 (4.1033 ± 0.1272 on 251..601), estimator returning 2.500000 on a synthetic z^{2.5}. The slopes rise with z, as the corpus records for every sup-type instrument here (`rho-exact-z31-01.md` §0), and the certified family is what the slope is rising toward. At s = 3.0 the exact slope over 29..47 is 6.7327 ± 0.5818 on six points, above the cited sup|ρ̃| slope 2.766 ± 0.212 on 13..37: the pointwise floor is growing faster than the sup it sits under, as it must if it is to become the sup. ## 5. The arrow where it can be seen (deliverable d) **At the cheapest s, exact (S2).** s = 2.698721, H = ⌊z^{4.216450}⌋, full period, R_H walked directly (no factor 2): | z | H > W? | sup\|R_H\| | rms(R_H) | sup/rms | log_z(sup/rms) | allowance u₀/2 | log_z F | |---|---|---|---|---|---|---|---| | 13 | yes | 3.312 | 1.0567 | 3.134 | 0.4453 | 2.1082 | 2.5865 | | 17 | yes | 5.996 | 1.9121 | 3.136 | 0.4034 | 2.1082 | 2.4801 | | 19 | no | 9.649 | 2.4586 | 3.925 | 0.4644 | 2.1082 | 2.3114 | | 23 | no | 14.334 | 3.1119 | 4.606 | 0.4871 | 2.1082 | 2.2563 | | 29 | no | 27.971 | 5.8294 | 4.798 | 0.4657 | 2.1082 | 2.1679 | The recovery the arrow asks for is z^{2.1082}; the measured recovery is z^{0.40–0.49}, and min_x T runs 2517 → 41384 against HM 2520 → 41411: every window certifies, by a margin F = (HM−1)/sup of z^{2.59} falling to z^{2.17}. The z = 13, 17 rows have H > W and are calibration only. **At s = 3.0, cited (S1).** need − ms = 0.9200 → 1.0765 over z = 13..47 (the rms-currency ask at these z, smaller than the asymptotic 2.1082 because M and the polylogs bite), truth log_z(2 sup|ρ̃|) = 0.6458 → 1.2874 over z = 13..37, log_z F = 2.4457 → 1.9394. No exact sup exists at z = 41, 43, 47 (only ⟨ρ̃²⟩; `rho-exact-z31-01.md` §0 prices those walks at 37× to 65,232× the z = 37 cost), so the brief's "z = 13..47" is 13..37 for the sup and 13..47 for the rms. Margins fall with z at both s, by about 0.42 of exponent from 13 to 29 at the cheapest s and 0.51 from 13 to 37 at s = 3, the same drift as yesterday's table. ## 6. Proof gap or truth gap, and the falsifiers **Proof gap at every computable z; truth gap in the limit, on §4.2.** The two are not in tension: the exact Ω at z ≤ 73 is three and a half orders under HM, the certified bound at z = 5e5 is six orders under z^{u₀}, and the statement REC makes is about all z ≥ z₀, where the construction wins by a power of z once the deficit 16s/9 − log_z Ω (1.65 at z = 5e5, falling) drops below 16s/9 − u₀ = 0.5812. What would settle it, and whether it ran: - **(E2) wrong: a divisor n of P(z) with λ⁺(n) < 0 or λ⁻(n) > 0.** RUN exhaustively over all 2^{π(z)} divisors at z = 13..73 (cheapest s) and 13..47 (s = 3): zero violations. A failure at larger z would break §4.1 and the standard theory of the Rosser sieve with it. - **(E4) wrong: a window containing r with T > cc(r) + H − 1.** Cannot happen while Brüdern–Fouvry's pointwise inequality holds; checked implicitly by the S2 walks (min T ≥ Ω-consistent at five levels). - **The split enumeration wrong.** RUN: it reproduces the walked min cc at z = 13..29 and respects Ω − M ≤ 2 sup|ρ̃| at the seven cited exact levels; the certified family at z = 601 equals the exact λ⁺ of its own set. - **§4.2 wrong: the 4-chain count in the top slab not ≫ D^{8/9}/ln⁴D.** NOT proven here beyond the written argument; the certified counts rise from 0.012 to 0.35 of z^{16s/9}/ln⁸z over 601..5e5 (S4b), consistent and not decisive. An adversarial re-derivation of the dyadic-range construction (the p* > 4D^{1/27} exit, the two prefix conditions, the parity split) is the pass this note is HELD for. The claim needs only Chebyshev-type bounds on prime counts in dyadic ranges. - **A computable failing window.** Would need Ω(z) > z^{u₀}: out of reach at any z this producer or the box can hold (the deficit must fall by another 1.07 exponents from z = 5e5). NOT RUN and not runnable; this is why the verdict is asymptotic. - **Against the item's own instruments.** th(nP) measured 1.5963 → 1.8569 at z = 13..31 and ≥ 2.0106 at 47 (`rho-maximal-law.md` §2 rider 2) is consistent: nP > Ω ≥ 100 at z = 47, s = 3 (S3), far below the measured floor z^{2.0106} = 2.3e3, so the floor is not yet set by the chains. ## 7. Readings 1. **STATED (deliverable a).** REC(1+√e+η, β₂−η): cap-currency ask z^{1.030992 + 3η}, rms-currency ask z^{2.1332 − η/2}, exponent bought η; M ln²z measured 0.3332 → 0.2715 at z = 13..47 there (S0, S2, S3). The brief's z^{1.031} verifies. 2. **PARTIAL (deliverable b).** None of the five routes closes at the smaller demand: the cap supplies zero saving at any demand, the union route needs a moment of order z/(2.1332 ln z), a factor 1.42 smaller than at u₀ = 3 and still divergent, the rest are unchanged. The failing step is yesterday's reduced arrow, verbatim. 3. **ZERO (deliverable c).** A mixed mean-square-plus-cap strategy delivers the cap on the large moduli, z^{2s − o(1)}, at every split point: with no moment above the second the mean square adds nothing at one position. 4. **PROVEN, small, and machine-checked.** cc(r) at a doubly non-rough point is minus a sum of three products of exit-chain counts, every window containing r has T ≤ cc(r) + H − 1, and so nP > Ω, F1 fails below Ω, and sup|ρ̃| ≥ (Ω − M)/2; Ω exact at z ≤ 73 (251 at 73, cheapest s) and z ≤ 47 (100 at 47, s = 3), controls all pass (S3). 5. **DERIVED, HELD (deliverable e).** Ω(z, s) ≫ z^{16s/9}/ln⁸z by a dyadic four-prime construction, and z^{2s − o(1)} in the limit; then REC is false for every u₀ < 16s/9 (4.7088 at s ↓ 1+√e), the cheapest point and the whole band with it, RML(α) is false at every α < β₂, and rider 2's band is asymptotically empty. Certified lower bounds reach 8.863e17 at z = 5e5 (log_z 3.1493), 0.35 of the model and rising (S4b). 6. **MEASURED: proof gap at every computable z.** Ω/HM ≤ 4.0e−4 at z ≤ 73, Ω/z^{u₀} ≤ 8.3e−7 certified at z = 5e5; sup/rms at the cheapest s reads z^{0.40–0.49} against an allowance z^{2.1082} at z = 13..29 (S2). The slopes of ln Ω rise with z: 3.28 ± 0.12 exact (29..73), 4.14 ± 0.09 on the hill-climb range (S5). 7. **UNCHANGED.** No exponent moved; RML(α) stays OPEN as a question until the adversarial pass on §4.2 runs, at which point the item's named target is REFUTED rather than open. ## 8. Not reached, and corrections to the record Not reached: an exact Ω past z = 73 (3^{π(z)} splits; z = 79 is 3× the z = 73 cost and z = 89 27×); six-prime chains in the certified family (they carry the 2s − o(1) limit and are empty below z ≈ 1e5 at this s); the exact sup|R_H| at s near 1+√e beyond z = 29; a literature citation for the pointwise size of Rosser weights at smooth integers (standard, not searched under `SEARCH-CONVENTIONS.md`); the s-dependence of Ω beyond the two values run; any re-pricing of the u₀ = 3 and u₀ = 4 forms, which §4.3 covers without new numbers. Corrections to the record (proposed, no live file edited): - Brief, deliverable (d): "z = 13..47 where the data exist" — the exact sup|ρ̃| exists at z = 13..37 only; 41..47 carry ⟨ρ̃²⟩ alone (`rho-exact-z31-01.md` §0, §4.6). - Brief, deliverable (b): "the position-union route, whose price e^{θ(z)/q} may or may not beat z^{1.031}" compares a rms-currency price with a cap-currency ask; the union route's ask is z^{u₀/2}, largest at the cheapest point (§2 item 2). - `attack-0829n-rml-proof.md` §5 and reading 6, "proof gap, not truth gap": correct at every computable z, and to be read with §4 here for the limit. - `rho-maximal-law.md` reading 6, "two models, neither called": §4.3 calls the room closing, conditional on §4.2 surviving its pass. - `REFUTED.md`: a candidate row, HELD until the pass: "REC(s, u₀), F1, F2 and RML(α) below 16s/9 on the Rosser vector-sieve lattice — REFUTED (planted exit-chain window: cc(r) = −(A₁A₂ + A₁B₂ + B₁A₂), T ≤ cc(r) + H − 1, Ω ≥ z^{16s/9 − o(1)})". *Gate: `node research/qc.js --full` result recorded in the closing report; the producer's tail verifies under `qc/embed.js --check`.*