# The proof attempt on RML: the chain from the proven mean square to an exponent below β₂, and the one arrow that does not close > **RIDER 2026-08-30 (orchestrator, from `redteam-0830-rml.md`: 12 claims > stand, 14 fail at sentence level, none inflating any result — the > load-bearing one errs in the CONSERVATIVE direction).** PROVEN under > independent code: the whole CAP lemma of §4.1, including the sine-sum step > this note checked only numerically (Σ 1/sin(πa/e) ≤ e·H_{⌊(e−1)/2⌋} + 1 ≤ > e(ln e + 1), violated at no e ≤ 20000), the slope 2.7660 ± 0.2120 with its > 1.10 and 5.82 se placements, and the reduction. WHAT FAILS: (i) §2's > `ms_m`/`ask` column is log_z √(B·H) with B the PROVEN Lemma V constant, i.e. > the exponent of the proven BOUND on the rms, not of the rms ⟨R_H²⟩^{1/2} > that REC names; the two part by 36.53× to 200.45× because ⟨R_H²⟩ saturates > once H passes the correlation length while B·H grows linearly, so §5's > reading "at u₀ = 3.0 and 3.5 ask does not exceed gauss" REVERSES at every > row when both sides are measured against the same mean square; (ii) "gauss > → 0" and "sup/rms ≤ √(2 lnW)·O(1) = z^{o(1)}" are both wrong (the limit is > 1/2, measured 0.5645 at z = 1e7) and contradict this note's own §0; (iii) > "the recovery exponent … 0.99–1.29" quotes the `truth` column, whose real > value is 0.56–0.67; (iv) "REC would be moot rather than false" should read > false; (v) "u_cap → 2s" and "n ~ z^{2s}/polylog" are asserted, not proven > (`verify-0830-usup-convention.md` §6's "OPEN in both directions" is the > better-calibrated reading); (vi) the 1.509-octave correction is still > unapplied in three places. NEW: u_cap crosses β₂ at z = 53, reaches 4.4891 > at z = 79, and passes u_triv from z = 53. *(2026-08-29 night wave, staging. Producer: `research/history/staging/attack-0829n-rml-proof.js`, embedded via `qc/embed.js`. No existing file edited, no git command run. Every number below is either in the producer's OUTPUT block or a quotation of an embedded artifact cited by file and section. Calibration per claim: PROVEN, MEASURED, HEURISTIC, OPEN, REFUTED.)* ## 0. What is open, first - **RML(α) is open at every α, and so is every form F1–F4. Nothing here lowers any exponent.** The deliverable is a location, not a theorem: the chain from Lemma V's PROVEN mean-square bound to an exponent below β₂ = 4.26645 has one open arrow, written down in §3 with its quantifiers, and none of the five attempts in §4 closes it. - **The open arrow is Lemma V's sup form (F2 of `rho-maximal-law.md` §1) with the proven mean square factored out.** It is not a new statement. What is new is the accounting: which part of F2 is proven (the rms), what the remainder must supply (a recovery of the sup from the rms by a factor z^{u₀/2−ε}), what every proven recovery actually costs (either the position-union price e^{θ(z)/q} or the term-count cap z^{2s+o(1)}), and what the truth measures (the sup exceeds the rms by about √(4θ(z)), i.e. z^{1/2+o(1)} if the Gaussian form persists). - **Finite-z sufficiency is calibration, not a result.** At z ≤ 37 even the term-count cap reads below β₂ (u_cap 3.4103 → 4.0593, OUTPUT S2), which is the Bonferroni-x ≤ 227 phenomenon of `rho-maximal-law.md` reading 4 at yet another instrument. No row of any table below is evidence for an asymptotic statement; the rows are there to show the arrow is true where it can be seen, and by how much. - **Asymptotically the arrow is undecided in both directions.** The truth's measured slope, 2.7660 ± 0.2120 on seven points spanning less than one octave of z (OUTPUT S5), leaves 1.23 exponents of room under u₀ = 4 and 0.23 under u₀ = 3; the corpus's standing observation that local slopes here rise with z (`rho-maximal-law.md` reading 6; `rho-exact-z31-01.md` §0) applies. The u₀ = 3 form could be false; the u₀ = 4 form is 5.8 se clear on a fit that cannot refute an asymptotic statement. - **What this note did not do.** It ran no new sup over positions (the exact points at z = 31, 37 were bought on 2026-08-29 by `rho-exact-z31-01.md`), it did not price the smooth-profile form again (CLOSED, `REFUTED.md` "manufacturing the smooth profile"), and it did not touch ⟨ρ̃⁴⟩, the named stopping object of `attack-rhoms-01.md` §3. ## 1. The objects and the direction of every inequality (wrong-direction guard) Fix s = 3.0 (the measured family; any s > 1 + √e = 2.6487 gives M > 0), a prime z, W = P(z), D = z^s, the divisor-pair lattice (block, d₁, d₂) with q = [d₁, d₂], w = ε μ(d₁)μ(d₂) ∈ {±1}, c the CRT class, n the term count and M = Σ w/q > 0 (`rho-maximal-law.md` §1). The potential ρ(y) = Σ_j w_j ψ((y − c_j)/q_j), the centred ρ̃ = ρ + M/2, and for a window H the certificate T(x) = H·M + R_H(x), R_H(x) = ρ(x) − ρ(x + H), with T(x) ≤ #{twin-admissible r ∈ (x, x+H]} pointwise (Brüdern–Fouvry, PROVEN, `rho-maximal-law.md` §2 (iv)). The consumer is min_x T(x) ≥ 1, which gives G₂(z#) ≤ H. Every inequality this note uses or asks for, with its direction and why it is legal under `attack-wrongdirection-audit.md`: | # | inequality | direction | rung | legality | |---|---|---|---|---| | I1 | ⟨R_H²⟩ ≤ B(z,s)·H, B ≤ 9A²(E−1) = O(ln⁸z) | upper bound on a period mean | PROVEN (`lemmaV-parseval.js` L1–L5; `redteam-0829-theorem1.md`) | no position quantifier; gives only almost-all statements at any H, so it carries no TPC content even at H < z² | | I2 | sup_x \|R_H(x)\| ≤ H·M − 1 at H = z^{u₀}, ONE fixed u₀ ∈ (2, β₂) | upper bound on the remainder, the direction that bounds G₂ from above | OPEN (= F2 at (s, u₀)) | legal by audit row 10: the band (2, β₂] is exhibited legal, α = β₂ being PROVEN with TPC open. TPC-strength if u₀ ≤ 2 (Gap Reformulation, constant < 1 at exponent 2) or if stated uniformly in u down to 2 (Axis A). This note fixes u₀ and never lets it move | | I3 | REC(s,u₀): sup_x \|R_H(x)\| ≤ z^{u₀/2−ε}·⟨R_H²⟩^{1/2} at H = z^{u₀} | upper bound on a sup by a multiple of an rms | OPEN, the arrow | implies I2 through I1 and M ≍ 1/ln²z; same legality as I2, same trap at u₀ = 2 (there it would read sup ≤ z^{2−ε}·polylog < z²/ln²z, i.e. o(z²), which is TPC) | | I4 | sup\|R_H\| ≤ Σ_e Vabs(e)·e·(ln e + 1) =: CAP(z) for every H | upper bound, absolute values across moduli | PROVEN here (§4.1) | harmless: delivers exponent 2s + o(1) ≥ 2(1+√e) = 5.2974 | | I5 | M ≥ c(s)/ln²z at s > 1+√e | lower bound on the main term | standard, CITED; MEASURED M·ln²z = 0.3359–0.3772 at z = 13..37 (OUTPUT S0) | no twin content | | I6 | the floor: no correct law delivers H below nP(z); th(nP) ≥ 2.0106 at z = 47 | lower bound on any deliverable window | MEASURED (`rho-maximal-law.md` §2, rider 2) | the reverse direction; if it ever exceeded β₂ it would kill the item, and it is undecided asymptotically | Two things the table makes explicit. I1 is the strongest possible statement about the remainder on average: at every level D and every H the mean square is √H·polylog, square-root cancellation with no level restriction, and it is proven. The consumer needs something far weaker in size, H/ln²z, but pointwise. The whole distance between I1 and I2 is the quantifier over x, which the corpus has priced before (`sift-limit-attack.md` §7e: "the price is exactly the quantifier"); what §3 does is write that price as one inequality with its exponent. ## 2. The chain, each arrow graded The vector sieve consumes one statement, min_x T(x) ≥ 1 at some H = z^{u₀}. Read from the proven end: ``` (C1) <= B(z,s) H PROVEN L1-L5 (lemmaV-parseval.js); every H, z, s (C2) B(z,s) <= 9 A(z)^2 (E(z)-1) = O(ln^8 z) PROVEN redteam-0829-theorem1.md; no hypothesis => rms(R_H) <= z^{u0/2} (ln z)^4 O(1) PROVEN composite of C1, C2 at H = z^{u0} (C3) sup_x |R_H(x)| <= z^{u0/2 - eps} rms OPEN REC(s,u0), the arrow, sec.3 (C4) M >= c(s)/ln^2 z (s > 1 + sqrt e) CITED standard linear sieve; MEASURED 0.3359-0.3772 (S0) => sup_x |R_H(x)| <= H M - 1, z >= z0 PROVEN from C3 + C2 + C4 (the polylogs are absorbed by z^eps) (C5) T(x) = H M + R_H(x) PROVEN exact identity (C6) T(x) <= #{admissible r in (x, x+H]} PROVEN Bruedern-Fouvry pointwise, pilot-verified => G2(z#) <= H = z^{u0}, z >= z0 PROVEN from C4-C6: exponent u0 < beta_2 ``` Beside the open arrow stand two proven arrows that reach the consumer without it, and both land above β₂: ``` (C3-triv) sup|R_H| <= 2 sup|rho~| <= n + M PROVEN triangle inequality over the lattice => exponent u_triv -> 2s + o(1) at s = 3: 6; at s -> 1+sqrt e: 5.2974 (C3-cap) sup|R_H| <= CAP(z) = sum_e Vabs(e) e (ln e + 1) PROVEN sec.4.1, this note => exponent u_cap -> 2s + o(1) same asymptote; 3.4103 -> 4.0593 at z = 13..37 (S2) (C3-cheb) #{x : |R_H(x)| >= HM-1} <= B H W/(HM-1)^2 PROVEN Chebyshev off C1 => almost-all exponent 0, sup exponent none attack-beta2-01 item 2; rho2-analytic-bound.md sec.4 ``` At numbers, s = 3.0, from OUTPUT S1 (exponents base z; rows with H > W dropped): | u₀ | z | need = log_z(HM−1) | ms_m = log_z √(B_meas H) | ask = need − ms_m | union = θ(z)/(2 ln z) | truth = log_z(2 sup\|ρ̃\|) | |---|---|---|---|---|---|---| | 4.00 | 19 | 2.9033 | 2.0613 | 0.8420 | 2.2319 | 0.9873 | | 4.00 | 29 | 2.9763 | 2.0568 | 0.9195 | 2.8544 | 1.0626 | | 4.00 | 37 | 3.0104 | 2.0551 | 0.9553 | 3.6036 | 1.2874 | | 3.00 | 19 | 1.9021 | 1.5613 | 0.3408 | 2.2319 | 0.9873 | | 3.00 | 37 | 2.0102 | 1.5551 | 0.4551 | 3.6036 | 1.2874 | | 4.25 | 37 | 3.2604 | 2.1801 | 1.0803 | 3.6036 | 1.2874 | Read across a row: the consumer at (3.0, 4.0), z = 37 needs sup|R_H| under z^{3.0104}; the proven mean square hands over an rms of z^{2.0551}; so the arrow must recover the sup from the rms within a factor z^{0.9553} (asymptotically z^{u₀/2 − o(1)} = z^{2−o(1)}, the finite-z figure being smaller because M and the polylogs bite); the only fully proven recovery, a union over the W positions, charges z^{3.6036} and rises like z/(2 ln z); and the truth, certified from the exact sup|ρ̃|, sits at z^{1.2874}. The arrow is the inequality truth ≤ ask, and at every measured level it holds by a wide margin (§5); the union price exceeds the ask at every level from z = 19 on and diverges. **Which arrow is "the" open one.** C3 is the only arrow in the first block that is not PROVEN or CITED-standard. C4's citation is standard linear-sieve asymptotics at s > 1+√e and is not in doubt; C6 is Brüdern–Fouvry's pointwise inequality, verified. So (a) of the brief is answered: one open arrow, C3, and everything else in the chain is a theorem. ## 3. The open arrow, isolated, with its quantifiers and its payoff **REC(s, u₀).** Fix s > 1 + √e and u₀ ∈ (2, β₂). There exist ε > 0 and z₀ such that for every prime z ≥ z₀, with D = z^s, H = ⌊z^{u₀}⌋, W = P(z), the signed sawtooth remainder of the level-D vector-sieve lattice satisfies sup_{x ∈ ℤ/W} |R_H(x)| ≤ z^{u₀/2 − ε} · ⟨R_H²⟩^{1/2}, the mean on the right taken over one full period. Quantifier discipline: the sup is over ALL W positions; s and u₀ are fixed before z; ε may depend on (s, u₀) but not on z; u₀ is never a function of z and is never sent to 2. **Equivalent forms, so the arrow can be recognised in other clothing.** - In sieve currency: a level of distribution for the signed two-class Rosser comb in short intervals, uniformly in the interval's position. The joint level is D² = z^{2s} = H^{θ_total} with θ_total = 2s/u₀; at (3.0, 4.0) that is θ_total = 1.5, and the break-even the corpus has already priced is θ_total > 1.2090 for Brüdern–Fouvry's asymmetric split (`sift-limit-attack.md` §7b) and 5.297442541400/β₂ = 1.241651 for the symmetric lattice used here (§7b (1b)). REC is that level statement with the remainder's size measured against its own rms rather than against H. - In the F-ladder of `rho-maximal-law.md` §1: REC(s, u₀) is F2 at window exponent u₀ with the mean square factored out. F2 ⟸ REC, and F3 = RML(α) with α < u₀ ⟹ REC (through |R_H| ≤ 2 sup|ρ̃|). F4 (the Gaussian form, λ = 0) ⟹ REC with room to spare: it supplies sup/rms ≤ √(2 lnW)·O(1) = z^{o(1)}, against an allowance of z^{u₀/2 − ε}. - The weakest law-shaped form beneath it is per-position in H: ∀x ∃H ≤ z^{u₀} with T_H(x) ≥ 1. It is implied by REC and it is exactly "every window of length z^{u₀} contains a certified survivor"; no instrument in the corpus distinguishes it from F1, and it is listed so the ladder has its bottom. **What its truth gives, quantified.** REC(s, u₀) ⟹ G₂(z#) ≤ z^{u₀} for all z ≥ z₀ (chain C4–C6), an unconditional two-class exponent u₀. At u₀ = 4.0 the movement below β₂ is 0.266 of exponent; at u₀ = 4.25 it is 0.016; at u₀ = 3.0 it is 1.266 [ARITHMETIC: β₂ − u₀ on the cited β₂], and the u₀ = 3.0 form is the one the data leave least room for (§5). Every u₀ here is above the measured floor th(nP) ≥ 2.0106 (z = 47, I6), so none is excluded by the floor at any measured level. **What it costs, in the corpus's own currencies.** Against the trivial accounting (C3-triv, C3-cap) REC asks for a saving of z^{2s − u₀ − o(1)} over the triangle inequality, uniformly in x: z^{2.0} at (3.0, 4.0), z^{1.75} at (3.0, 4.25), and z^{1.031} at the cheapest legal point (s → 1+√e, u₀ → β₂) [ARITHMETIC: 2s − u₀, and 5.2974 − 4.2665 on the cited constants], which is the 2(1+√e) − β₂ gap the corpus has carried since `sift-limit-attack.md` §4.5. Against the proven mean square it asks for a recovery exponent below u₀/2, where the proven recoveries cost θ(z)/(2k ln z) at moment order k (`attack-rhoms-01.md` §3), i.e. diverge. The measured truth of the recovery exponent is log_z(2 sup|ρ̃|/rms): about 0.99–1.29 in the table of §2, and if the Gaussian form persists it is 1/2 + o(1). **Kill checks, run.** (i) Is REC on a REFUTED row? No. The rows nearest it are "Lemma V's mean-square form as the missing factor" (CLOSED because B was never binding, which is the statement C1–C2 are proven and irrelevant to C3), "u_sup as an unconditional worst-position bound" (CLOSED; C3-cap is that route with its asymptote named, not the arrow), "the ℓ¹ → ℓ²√log conversion" (CLOSED because that conversion IS the sharp maximal law; REC asks for ℓ² × z^{u₀/2}, not ℓ² × √log), "generic chaining" (CLOSED; not retried), and "the Kowalski–Michel–Sawin branch" and "manufacturing the smooth profile" (CLOSED; §4.5 says why they are not REC). (ii) Is REC TPC-strength? No, at fixed u₀ > 2 (§1, I3). (iii) Is it already listed ANSWERED? No: QUESTIONS.md carries Q-rho-maximal-law as PARTIAL with the law itself open, and this note is the proof attempt that item calls for, not a re-run of its measurements. ## 4. The attempt, and where it dies Five routes were tried on REC. One produced a small theorem (4.1, an asymptote for a closed route); none produced the arrow. Each is recorded with the exact place it stops. ### 4.1 Term-by-term accounting is capped at 2s + o(1), and the cap is the trivial-level exponent **Lemma (PROVEN here).** For every H, sup_x |R_H(x)| ≤ CAP(z) := Σ_{e | P(z), e > 1} Vabs(e)·e·(ln e + 1), and CAP(z) ≤ z^{2s}·(2s ln z + 1)· 6·Π_{p β₂ at every admissible s; the divergence disjunct is now refuted at fixed s. ### 4.2 Position-union routes, including hypercontractivity, are dominated by 4.1 Chebyshev and the moment ladder are on record: sup ≤ W^{1/2k}·‖R_H‖_{2k} pays e^{θ(z)/2k} and needs k ≍ z/(1.53 ln z) (`attack-rhoms-01.md` §3; `rho2-analytic-bound.md` §4). The one route not on record is the product structure of the positions: x ∈ ℤ/W ≅ Π_{p 2s + log_z (#moduli): strictly worse than 4.1. Rung: the negative is elementary; no hypercontractive theorem was cited because its best case is already dominated. **Where it dies:** the atom √p, i.e. one residue class has no cancellation by itself; the saving must come from the signs w_j across terms, which no per-term or per-coordinate bound sees. ### 4.3 Window smoothing and the P(y) lever move polylogs, not the exponent Averaging T_H(x) over H ∈ [H₀, 2H₀) is legitimate for the consumer (an average ≥ 1 forces some T_H(x) ≥ 1, hence a survivor in (x, x + 2H₀]) and it replaces |S_H(a/e)| ≤ 1/(2‖a/e‖) by a Fejér-type kernel ≤ c/(H₀‖a/e‖²), which cuts the per-modulus ℓ¹ mass from e ln e to about e²/H₀ for e < H₀ and does nothing for e > H₀. Choosing H ≡ 0 mod P(y) kills every modulus all of whose primes are < y exactly (V1). Measured: at u₀ = 4 the moduli above the window carry 0–27.26% of CAP over z = 13..37 (S3), so at computable z the smoothing would remove most of the ℓ¹ mass, and the P(y) lever at y = ⌈4 ln z⌉ removes 31.64% of CAP at z = 13 falling to 1.713% at z = 37 (S4). Neither touches the asymptote: **the part of CAP below z^{u₀ − δ} is at most z^{u₀ − δ}·polylog (PROVEN, from the second inequality of 4.1 applied to e ≤ z^{u₀ − δ}), so it is below HM − 1 for large z at any fixed δ > 0, and REC needs proving only for the moduli in (z^{u₀ − δ}, z^{2s}]**. At (u₀, δ) = (4, 0.5) that part reads 0.3948, 0.3884, 0.5067, 0.5346 of HM − 1 at z = 19, 29, 31, 37, already below 1 but rising (the polylog dominates the z^{−δ} at these z), and the remaining moduli carry 24.45–56.84% of CAP (S3). **Where it dies:** the moduli above the window are exactly the terms whose class hits the window at most once, so the reduced arrow is a bound on the signed count of large-modulus hits, Σ_{r ∈ (x, x+H]} Σ_{d₁ | r, d₂ | r+2, [d₁,d₂] > z^{u₀−δ}} w(d₁,d₂) = o(H/ln²z) for every x — the definition of the problem with the easy part removed, and no instrument for it. ### 4.4 The mean square is not improvable into the arrow I1 is already square-root cancellation at every level, uniform in s (B ≤ 9A²(E−1) with no s in it, `redteam-0829-theorem1.md` §1). Improving B to O(1) (measured flat 1.3833–1.4883, `attack-AB-bounded.md` §1.4) would change ms_m by O(ln ln z/ln z) and nothing in the ask column of §2. The corpus's row "Lemma V's mean-square form as the missing factor — CLOSED" is confirmed from this side: C1–C2 are a theorem, and they are not where the exponent is lost. ### 4.5 The fixed-smooth-profile form is not the arrow, from both ends Written in the form the Kloosterman-fraction instruments prove: for smooth compactly supported g₁, g₂ and arbitrary 1-bounded α, β, Σ_{d₁, d₂} g₁(d₁/D₁) g₂(d₂/D₂) α(d₁) β(d₂) e(−2h d̄₁/d₂) e(−hx/(d₁d₂)) ≪ (D₁D₂)^{γ}·H^{o(1)}, uniformly in x, with γ ≤ 0.827142 needed against Bettin–Chandee's 0.970624 (`rho-maximal-law.md` §5; `sift-limit-attack.md` §7b). It is disconnected from REC at both ends. (i) The instruments need a smooth factor in the modulus and the inverted variable; the Rosser weight μ(d)·1_{S^±}(d) has none and cannot be decomposed into any: its low-frequency share sits at the random-sign noise floor at 31 of 31 slices (`smoothness-front.md` §4.3; `REFUTED.md` "manufacturing the smooth profile", CLOSED). (ii) The consumer cannot take a smooth g either: Brüdern–Fouvry's pointwise inequality needs λ⁻ ≤ 1_{rough} ≤ λ⁺, which Möbius-signed Rosser supports supply and a smooth profile does not. (iii) Granting the smoothness anyway, the published bound prices at exponent 5.0907 > β₂ and holds for x ≪ H^{1.21} against a period of exp(H^{0.2344}) (`rho-maximal-law.md` §5). So "at a fixed smooth profile" the statement is a different statement with a closed hypothesis, not REC in other clothing. Rung: CITED throughout; nothing re-priced here. ## 5. Proof gap or truth gap: the arrow against the data **Against the sharp form first, as the brief asks.** The TPC-implying sharp law is sup|R_H| ≤ √⟨R_H²⟩·√(2 ln W) at the operative window H ≈ 0.5 z², with constant 1; it fails at z = 19 by 0.6% and on 528 of 640 windows at z = 31 (`rho-maximal-law.md` §4; `rho-exact-z31-01.md` §4.5). REC is at H = z^{u₀} with u₀ > 2 and allows a factor z^{ask} in place of √(2 ln W): in the table of §2 the allowance ask exceeds the Gaussian factor gauss = log_z(2√(2 ln W)) at every level from z = 19 at u₀ = 4.0 (0.8420 against 0.7905 at z = 19, 0.9553 against 0.7392 at z = 37) and at u₀ = 4.25, while at u₀ = 3.0 and 3.5 it does not (0.3408–0.4551 and 0.5918–0.7053 against 0.7392–0.7905). So at computable z the u₀ ≤ 3.5 forms ask for less than Gaussian recovery and are the forms a sharp-law failure could touch; the u₀ ≥ 4.0 forms sit above the Gaussian line there, and the sharp law's death at z = 19 says nothing about them. Asymptotically ask → u₀/2 − o(1) while gauss → 0, so every form ends above the Gaussian line; the finite-z ordering is a statement about where the polylogs bite. **At every computable level the arrow holds, with a certified margin.** Since sup|R_H| ≤ 2 sup|ρ̃| (exact), F(z, u₀) = (HM − 1)/(2 sup|ρ̃|) is a lower bound on the true margin. From S1 (rows with H < W only): | u₀ | F at z = 19, 29, 31, 37 | log_z F | |---|---|---| | 3.00 | 1.478e+1, 2.166e+1, 1.548e+1, 1.360e+1 | 0.9148, 0.9134, 0.7977, 0.7228 | | 3.50 | 6.462e+1, 1.168e+2, 8.625e+1, 8.276e+1 | 1.4157, 1.4137, 1.2980, 1.2230 | | 4.00 | 2.819e+2, 6.290e+2, 4.803e+2, 5.035e+2 | 1.9160, 1.9137, 1.7980, 1.7230 | | 4.25 | 5.885e+2, 1.460e+3, 1.133e+3, 1.242e+3 | 2.1660, 2.1637, 2.0480, 1.9730 | The margin in exponent falls with z at every u₀, by 0.19 from z = 19 to 37 at u₀ = 3.0 and 4.0 alike: in the §2 table the truth column rises from 0.9873 to 1.2874 while need rises only from 2.9033 to 3.0104 (u₀ = 4.0), and u₀ enters need as an additive constant, so the fall is the same at every u₀. Rung: MEASURED on the cited exact suprema; the factor 2 in |R_H| ≤ 2 sup|ρ̃| is up to 2× conservative and was not tightened (sup|R_H| at H = z^{u₀} was not walked). **Proof gap, not truth gap, at every z where the question can be put.** No computable level contradicts REC at any u₀ ≥ 3, and the sharp-form failure is at a window two exponents below the legal band. Asymptotically: the truth's slope is 2.7660 ± 0.2120 (S5, seven points, estimator returns 2.500000 on a synthetic z^{2.5} truth), leave-one-out 2.6180 ± 0.2601, and the room under u₀ reads 0.2340 (1.10 se) at u₀ = 3.0, 0.7340 (3.46 se) at 3.5, 1.2340 (5.82 se) at 4.0, 1.4840 (7.00 se) at 4.25. The F4 envelope 2√⟨ρ̃²⟩√(2 ln W) itself has slope 2.5111 ± 0.1775 on the same points (S5), so the measured truth is tracking the Gaussian envelope, not outrunning it. Two honest limits on that reading: seven points under one octave decide no asymptote, and the corpus records local slopes here rising with z (`rho-exact-z31-01.md` §0). The u₀ = 3.0 form is therefore the one that could be false; the u₀ = 4.0 form is measured true with 1.2 exponents to spare and no known mechanism against it. Neither is a theorem, and the distance from measured truth to proof is the whole of §4. ## 6. What would falsify the arrow, and whether it has run - **REC(3.0, u₀) false at a computable z**: an exact sup|ρ̃|(z) with 2 sup|ρ̃| ≥ HM − 1 at H = z^{u₀}, or, sharper, a walked sup|R_H| at that H above HM − 1. RUN at z = 13..37 through the cited exact suprema (S1): no failure at any u₀ ≥ 3.0; margins in §5. NOT RUN for sup|R_H| directly at H = z^{u₀} (the factor 2 stands in), and NOT RUN at z ≥ 41 (the z = 31, 37 walks cost 6928 s, `rho-exact-z31-01.md`; z = 41 is a period of 7.4e12 positions, well above the four-hour rule, so it is an ask). - **The truth's slope reaching u₀**: the seven-point slope 2.7660 ± 0.2120 (S5) against u₀. RUN. Decides nothing asymptotically; u₀ = 3.0 is 1.10 se away and u₀ = 4.0 is 5.82 se away. The next exact point moves this more than anything else here. - **A worst-position construction**: a position x with a large-modulus hit count of order H/ln²z (the §4.3 reduced form). The corpus's closest test is `sift-limit-attack.md` §7b (3): at a CRT-planted doubly-smooth position the absolute sum reaches 0.978 of the trivial bound while the signed sum stays about H (max 6 over 20,000 positions at pair count 588). RUN at one level; it passed; it is one level. - **The floor crossing β₂**: if th(nP(z)) ever exceeded β₂, no law of any strength could deliver and REC would be moot rather than false. MEASURED ≥ 2.0106 at z = 47 (prefix bound); undecided asymptotically (`rho-maximal-law.md` reading 6, two models, neither called). - **The cap lemma of 4.1 wrong**: a modulus e with Σ*_a |Θ_e(a)||S_H(a/e)| above Vabs(e)·e(ln e + 1), or u_cap below the cited u_sup at some z. RUN at z = 13, 17 over every primitive frequency (no violation) and at seven levels against the cited column (u_cap ≥ u_sup at all seven, S2). - **The reduction of 4.3 wrong**: CAP_{≤ z^{u₀−δ}} not below HM − 1 for large z. The inequality is a one-line consequence of 4.1's second bound and cannot fail asymptotically; at (4, 0.5) it reads 0.39–0.53 and rising over z = 19..37 (S3), so at computable z it is the polylog, not the exponent, that is being watched. ## 7. Readings 1. **STATED, one open arrow.** From Lemma V's proven mean square to an exponent below β₂ the chain is C1–C6 of §2; C1, C2, C5, C6 are PROVEN, C4 is standard and measured (M ln²z = 0.3359–0.3772 at z = 13..37, S0), and C3 = REC(s, u₀) is the only open arrow. Every proven substitute for C3 lands at exponent 2s + o(1) or pays e^{θ(z)/q}. 2. **ISOLATED.** REC(s, u₀): for fixed s > 1 + √e and fixed u₀ ∈ (2, β₂), sup_x |R_H(x)| ≤ z^{u₀/2 − ε}·⟨R_H²⟩^{1/2} at H = z^{u₀} for all large z. Its truth gives G₂(z#) ≤ z^{u₀}. It is F2 with the rms factored out, a level-of-distribution statement at θ_total = 2s/u₀ measured against the remainder's own rms, legal at fixed u₀ > 2 and TPC-strength at u₀ = 2 or uniformly down to it. It is on no REFUTED row (§3, kill checks). 3. **PROVEN, small, an asymptote for a closed route.** Every absolute-value accounting across the lattice is capped by CAP(z) ≤ z^{2s}(2s ln z + 1)· 6Π(1 + 2/p)², so u_sup ≤ 2s + o(1), which at s ↓ 1 + √e is 5.2974, the trivial-level exponent. Measured u_cap 3.4103 → 4.0593 against the cited u_sup 2.0617 → 3.0125 (S2); the cap is loose by 28.62–36.94× against the exact Fourier ℓ¹ sum at z = 13, 17. `lemmaV-sup-extension.md`'s asymptotic prose at lines 485-486 and 508-516 (2^{π(z)} moduli; a C^{π(z)} theorem as the reachable end) cannot hold at fixed s; its verdict does (`verify-0830-usup-convention.md` §3). 4. **PROVEN, a reduction.** REC needs proving only for moduli in (z^{u₀ − δ}, z^{2s}]; the rest is ℓ¹-controlled below HM − 1 for large z. At (4, 0.5) the controlled part reads 0.3948–0.5346 of HM − 1 over z = 19..37, rising (S3). The reduced arrow is a bound on the signed count of large-modulus lattice hits in every window, with no instrument. 5. **NEGATIVE, each at a named place.** Term-by-term accounting dies at the term count (4.1); position-union and hypercontractive routes die at the atom √p and the union price e^{θ/q} (4.2); window smoothing and the P(y) lever move 0–27% and 1.7–32% of CAP at computable z and no exponent (4.3, S3, S4); B is not where the exponent is lost (4.4); the smooth-profile form is disconnected from the consumer at both ends and priced at 5.0907 even if granted (4.5). 6. **MEASURED: proof gap, not truth gap, at every computable z.** Certified margin F = (HM − 1)/(2 sup|ρ̃|) at u₀ = 4: 2.819e+2, 6.290e+2, 4.803e+2, 5.035e+2 at z = 19, 29, 31, 37 (log_z F 1.9160 → 1.7230); at u₀ = 3.0: 1.478e+1 → 1.360e+1 (log_z F 0.9148 → 0.7228). The truth's slope 2.7660 ± 0.2120 sits 1.10 se under u₀ = 3.0 and 5.82 se under u₀ = 4.0 (S5); the F4 envelope's slope is 2.5111 ± 0.1775 on the same points. Seven points under one octave; asymptotically undecided, with the u₀ = 3.0 form the one the data could yet refute. 7. **UNCHANGED.** RML(α) open at every α; no exponent moved; the item's named riders ("Lemma V re-enters as ASK not import; fixed-smooth-profile binds") stand, and §4.5 says why the second is not a property of the working point. ## 8. Not reached - sup|R_H| walked directly at H = z^{u₀} (the factor 2 in |R_H| ≤ 2 sup|ρ̃| stands in; at most 2× conservative). - Any exact point at z ≥ 41 (period 7.4e12; above the four-hour rule). - ⟨ρ̃⁴⟩, the named stopping object of `attack-rhoms-01.md` §3; a sub-Gaussian fourth moment (measured m₄/m₂² = 2.32–2.92 there) would cut the union price to e^{θ/4}, which still diverges, so it was not pursued. - The s-freedom: everything is s = 3.0. The cap's asymptote 2s says a smaller s is cheaper for the trivial accounting; whether REC is easier at s near 1 + √e was not measured. - The u₀-dependence of the reduced arrow's mean square (the sub-lattice B_big ≤ 9A²(E−1) holds by the same proof, since dropping terms only lowers T(e); its measured size was not tabled). - Re-pricing the Kloosterman imports at the ρ-configuration (carried over from `rho-maximal-law.md` §5 with its caveat). *Gate: `node research/qc.js --full` result recorded in the closing report; the producer's tail verifies under `qc/embed.js --check` (code-sha256 matches, out-sha256 matches, body bit-honest).*