# u_sup's level convention, adjudicated: `lemmaV-sup-extension.md` and `attack-0829n-rml-proof.md` §4.1 bound the same object, and the correction in passing stands against three sentences, not against the verdict *(2026-08-30, staging. Producer: `research/history/staging/verify-0830-usup-convention.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 by file and line. Calibration per claim: PROVEN, VERIFIED, MEASURED, HEURISTIC, OPEN, REFUTED. Abbreviations: LSE = `research/history/staging/lemmaV-sup-extension.md` with producer `research/lemmaV-sup-extension.js`; RML = `research/history/staging/attack-0829n-rml-proof.md` with producer `research/history/staging/attack-0829n-rml-proof.js`.)* ## 0. The ruling first, with what it does not do - **Nothing here moves an exponent, opens a route, or changes either note's ledger verdict.** RML's ledger verdict does not mention LSE; LSE's verdict ("plateau above β₂, or divergence, the same verdict") keeps its bottom line (§4 below). - **The suspected level-convention mismatch is REFUTED at the code.** Both notes bound the same object at the same level: the divisor-pair lattice of `sift-limit-lemmaV.js` `buildTerms(z, D)` at D = z³ (s = 3.0 fixed before z), the same moduli, the same weights V(e₁,e₂), the same Vabs(e), the same main term M. LSE's cited Ssat and u_sat reproduce on RML's lattice to every printed digit at z = 13, 17, 19, and do not reproduce at full level (§2). - **So RML §4.1's correction stands in substance, and is mis-aimed in its wording.** The PROVEN cap Ssat ≤ CAP(z) ≤ z^{2s}(2s ln z + 1)·6Π(1+2/p)² refutes three asymptotic sentences in LSE §4 (lines 485-486, 491-492 in its asymptotic clause, 508-516), and confirms the worry LSE itself put first under "where I am most likely wrong" (lines 690-695). It does not refute a "reading" of u_sup's growth as π(z)/ln z, because LSE's stated conclusion (THE VERDICT item 3, lines 50-56, and §3(3), lines 326-331) is that four rising shapes are indistinguishable on nine points; the π(z) shape is one of them, and the note's own text calls fixed C "not what the data show" (lines 455-456). §3 gives the sentence-level replacements on both sides. - **One brief error recorded.** The brief's parenthetical "moduli e | P(z) with e ≤ D" is not either note's modulus set: the moduli are e | qᵢ = [d₁,d₂] with d₁, d₂ ≤ D, so e ≤ min(D², P(z)), and at z = 13, 17, 19 the set is every divisor of P(z) (max e = P(z), OUTPUT S0). The brief's other numbers and quotations check at their records (§5). ## 1. The two objects, side by side, quoted by line | item | LSE | RML | same? (rung) | |---|---|---|---| | level | `S_LEVEL = 3.0` (`lemmaV-sup-extension.js`:93, "the level attack 1 measured at"); note lines 186-189: "e must divide some q_i = [d1,d2] with both d1, d2 ≤ D = z^s"; line 671: "Only s = 3.0 for u_sup" | `S = 3.0` (`attack-0829n-rml-proof.js`:64); note line 54: "Fix s = 3.0 ... D = z^s" | yes, VERIFIED: D = 2197, 4913, 6859, 12167 on both paths at z = 13..23 (OUTPUT S0) | | lattice | `lemmaV-parseval.js`:104-106, `REPO.buildTerms(z, round(z^s))` | `attack-0829n-rml-proof.js`:143, `L.buildTerms(z, Math.pow(z, S))` | same function, `sift-limit-lemmaV.js`:91-106; n = 852, 2236, 4764, 9636 and M equal on both paths (S0) | | modulus set | e \| P(z) carrying at least one V(e₁,e₂) ≠ 0 (`lemmaV-parseval.js`:117-130, the `v===0` skip); note line 418 writes "sum_{e \| P(z), e>1}" | every e \| qᵢ for some lattice term (`attack-0829n-rml-proof.js`:82-106) | equal on {Vabs(e) > 0}: sets equal at all ten levels z = 13..47 (S0, S3 "set=LSE recs"); RML's count exceeds LSE's from z = 29 (467 against 448; 6119 against 6035 at z = 47) and the excess is exactly-zero moduli, contributing 0 to Ssat and to CAP alike | | weights | V(e₁,e₂) = Σ_{i: e \| qᵢ, gcd(e,d₁)=e₁} wᵢ/qᵢ (note line 420); Vabs(e) = Σ\|V(e₁,e₂)\| (`lemmaV-parseval.js`:54, Theorem B) | the same V; Vabs via Theorem B (note line 218) | yes, VERIFIED: max\|ΔVabs\| = 3.469e-18, 6.939e-18, 1.735e-18, 1.735e-18 at z = 13..23 (S0) | | bounded quantity | Ssat = Σ_e Σ*_a \|Θ_e(a)\|/\|sin(πa/e)\| (note line 418); Ssup(H) with \|S_H(a/e)\| in place of 1/\|sin\| (`lemmaV-sup-extension.js`:17-18) | CAP(z) = Σ_e Vabs(e)·e·(ln e + 1) (note lines 215-216) | one chain: sup\|R_H\| ≤ Ssup(H) ≤ Ssat ≤ CAP, the last step by \|Θ_e(a)\| ≤ Vabs(e) and Σ_a 1/sin(πa/e) ≤ e(ln e + 1); Ssat ≤ CAP measured true at all ten levels (S1, S3), 0 frequencies with \|Θ\| > Vabs at z = 13..19 (S1) | | normalisation | u_sat = ln(Ssat/M)/ln z; u_sup = ln H_sup/ln z, H_sup the first window with H·M > Ssup(H) (`lemmaV-sup-extension.js`:21-22; note lines 201-211) | u_cap = log_z((CAP + 1)/M) (note line 230) | same M; u_sat recomputed here 2.2850, 2.4623, 2.7228 = the cited column, u_cap recomputed 3.4103, 3.5850, 3.7797 = RML's S2 column (S1) | So the level is D = z³ in both, with s fixed before z. The brief's alternative, "s there is ln P(z)/ln z", is what the S2 column of the producer instantiates, and LSE's numbers are not that column (§2, reading 3). ## 2. The measurement 1. **S0, the lattices are one.** Same D, n, M; modulus sets equal; per-modulus Vabs agreeing to 6.9e-18 or better. At z = 13, 17, 19 the modulus set is every divisor of P(z) (max e = 2.310e+3, 3.003e+4, 5.105e+5 = P(z), 31, 63, 127 moduli = 2^{π(z)} − 1); at z = 23 it is 243 of 255 divisors with max e = 8.818e+5 against P(23) = 9.700e+6 and D² = 1.480e+8. VERIFIED. 2. **S1, the objects evaluated on RML's lattice.** Ssat computed here from the raw terms, with no peel and no fast sine, is 1.9602e+1, 5.0314e+1, 1.2006e+2 at z = 13, 17, 19: equal to LSE's own `supBoundTable` path and to its cited column to every printed digit. CAP here is 350.4593 and 1209.9555 at z = 13, 17, RML's cited figures. Ssat ≤ CAP at every level, CAP/Ssat = 17.88, 24.05, 22.46. VERIFIED. 3. **S2, the full-level alternative, instantiated and rejected as either note's object.** With every Rosser condition vacuous (D_full = P(z)·z³, so both supports are all divisors of P(z)), s_full = ln P(z)/ln z reads 3.0196 and 3.6390 at z = 13, 17, and Ssat_full = 1.9544e+1, 4.5827e+1 against the cited 1.9602e+1, 5.0314e+1; u_sat_full = 2.2661, 2.4113 against the cited 2.2850, 2.4623. At z = 13 the two levels nearly coincide (D = 2197 against P(13) = 2310), which is why the figures are close there; at z = 17 they part. The cap holds at full level too (CAP_full = 4.7171e+2, 1.6998e+3), and its exponent there, 2 s_full = 6.0391, 7.2779, is indeed of order z/ln z as the brief supposed; that level is used by neither note. VERIFIED. 4. **S3, the cap against the H-free exponent at all ten levels.** u_cap ≥ u_sat at every level, 3.4103 → 4.2652 against 2.2850 → 3.4306 (z = 13..47); RML §4.1 had compared against u_sup to z = 37 only. The explicit bound z^{2s}(2s ln z + 1)·6Π(1+2/p)² sits 1.218e+9 to 2.583e+10 above the cited Ssat, which is why nothing in the measured range sees it. The modulus count is 6119 against 2^{π(z)} − 1 = 16383 at z = 47; 2^{π(z)} first exceeds z^{2s} at z = 223, and a fixed per-prime factor C = 2.0516 first breaches the proven cap at z = 331. MEASURED; the two crossings are arithmetic on the cited constant and the proven bound. ## 3. The ruling, sentence by sentence **Verdict: SAME object. The correction applies. Its substance stands; its attribution overreaches.** ### 3.1 RML §4.1, lines 241-246, and reading 3, lines 432-433 Quoted: "`lemmaV-sup-extension.md` reads u_sup's growth as "π(z)/ln z" from a per-prime factor ~2.05 on ten points. The lemma above caps u_sup at 2s + o(1) for every fixed s, so that growth cannot continue; the closure verdict there ("plateau above β₂, or divergence — the same verdict") is unaffected, since 2s ≥ 2(1 + √e) > β₂ at every admissible s." - "The lemma above caps u_sup at 2s + o(1) for every fixed s": STANDS, PROVEN. Exactly for u_sat (Ssat ≤ CAP, §1); for u_sup up to the grid protocol, since every H ≥ (CAP + 1)/M has H·M > CAP ≥ Ssup(H), so the protocol's first closing grid point is at most one grid ratio above it, o(1) in the exponent. - "reads u_sup's growth as "π(z)/ln z" ... on ten points": OVERREACHES as attribution. LSE line 390 lists u ~ (ln C)·π(z)/ln z as one of "two shapes the data cannot separate"; lines 455-456 say "C fixed is not what the data show either"; lines 690-695 put "the geometric-in-π(z) reading of Ssat" first under "where I am most likely wrong". The factor is 2.0516, the geometric mean of nine ratios over ten levels (line 442), and it is a factor on Ssat, hence on u_sat (line 452), not on u_sup directly. - "the closure verdict there ... is unaffected": STANDS (§4). Replacement for lines 241-246: "`lemmaV-sup-extension.md` §4 describes Ssat's growth over z = 13..47 as geometric in π(z) (per-prime factor 2.0516, its line 442), says the e-sum "has 2^π(z) terms" (lines 485-486), and writes the reachable theorem as Ssat ≤ A·C^{π(z)}, hence G₂(P(z)) ≤ exp((ln C)(1+o(1)) z/ln z), "superpolynomial in z" (lines 508-516). At fixed s the lemma above refutes the last two: the e-sum has at most z^{2s} moduli, Ssat ≤ CAP ≤ z^{2s+o(1)}, and the reachable theorem is G₂(P(z)) ≤ z^{2s+o(1)}, polynomial, the trivial-level exponent, which is what that note's §7 first bullet anticipated. Its measured tables stand as measurements and its closure verdict ("plateau above β₂, or divergence — the same verdict") is unaffected, since 2s ≥ 2(1 + √e) > β₂ at every admissible s; the divergence disjunct is now refuted at fixed s." Replacement for lines 432-433: "`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." ### 3.2 LSE, the sentences the cap refutes, each with its replacement - **Lines 485-486**, "the e-sum has 2^π(z) terms": REFUTED at fixed s as an asymptotic; true as a count at z ≤ 19 and within a factor 2.7 at z = 47 (S3). Replace with: "the e-sum runs over the divisors of some qᵢ ≤ D², at most min(2^{π(z)}, z^{2s}) moduli: every divisor of P(z) at z ≤ 19, 6035 of 16383 at z = 47, polynomially many from z = 223 on (`verify-0830-usup-convention.md` S3)." - **Lines 491-492**, "the moment it is only a mean value it is too weak by a factor exponential in π(z)": the asymptotic clause is REFUTED at fixed s (the loss Ssat/sup|R| ≤ CAP/sup|R| ≤ z^{2s+o(1)}, polynomial); the measurement it summarises (loss growing 1.35× to 1.5× per prime over z = 13..47, LSE §2, §5) stands. Replace "exponential in π(z)" with "measured to grow geometrically over z = 13..47 and bounded at fixed s by z^{2s+o(1)} (`attack-0829n-rml-proof.md` §4.1)". - **Lines 508-516**, "What is reachable as a theorem is a bound of the shape Ssat ≤ A·C^{π(z)} ... superpolynomial in z": REFUTED. This is the sentence the correction is actually about. Replace with: "What is reachable as a theorem at fixed s is Ssat ≤ CAP(z) ≤ z^{2s}(2s ln z + 1)·6Π_{p 1 gives u_sat ~ (ln C)π(z)/ln z ... which diverges": a correct conditional whose antecedent is now PROVEN false at fixed s (ln Ssat ≤ 2s ln z + O(ln ln z), so (ln Ssat)/π(z) → 0 and the average per-prime factor → 1). No replacement; a rider "(antecedent refuted at fixed s, `attack-0829n- rml-proof.md` §4.1)" suffices. - **Lines 471-475**, "prove that d ln Ssat/d ln z falls below β₂ and stays there ... 4.99 rising to 6.20": stands as measured. The cap gives the cumulative slope from any fixed z₀ as at most 2s + o(1) = 6 + o(1), which a local 6.20 over five points does not contradict; it does not give "below β₂". No edit. ### 3.3 What in LSE is untouched THE VERDICT items 1-4 (lines 36-66), the ladder (§2), the fits and the asymptote profile (§3), the two-representation comparison (§5), corrections C1-C6 (§6) and the §7 coverage bullets are measurements and extrapolations at z ≤ 47 and are not touched by an asymptotic cap. One consistency, HEURISTIC and one-sided: LSE §3(4)'s bounded-model asymptotes, A = 5.4635 for u_sup and 5.5649 for u_sat with "every limit compatible with the data to within a factor 3 in RSS ... between about 4.8 and 6.2" (lines 350-352), all sit under the proven cap 6 + o(1) at s = 3; the cap is an upper bound and says nothing about where under it the limit lies. ## 4. Ledger verdicts - **RML, `Q-rml-proof-0829n`: unchanged.** Its verdict line does not mention LSE; the parenthesis "term-by-term accounting is capped at 2s + o(1)" is the PROVEN lemma and is correct. - **LSE, `Q-lemmaV-usup`: bottom line unchanged, status ANSWERED stands.** Of its closing disjunction "plateau above beta2, or divergence", the divergence disjunct is REFUTED at fixed s (u_sat ≤ 2s + o(1)); the "plateau above β₂" disjunct remains MEASURED only (8.70× in RSS to force the limit to β₂, LSE line 342; no lower bound on the limit is proven anywhere). The route's verdict, that this instrument does not reach below β₂, is what the line says and it is the same after the cap. If the orchestrator sharpens it, the minimal edit is to append "(divergence since REFUTED at fixed s by `attack-0829n-rml-proof.md` §4.1)"; no edit is required by this adjudication. ## 5. The brief checked at its records - CAP(z) ≤ z^{2s}·(2s ln z + 1)·6·Π_{p 0 in one note's set and absent from the other's** at some z ≤ 47: RUN, none (S0, S3). At z > 47: NOT RUN, and the code gives no mechanism for one (both paths enumerate the divisors of the same qᵢ). - **LSE's cited Ssat failing to reproduce on RML's lattice**: RUN at z = 13, 17, 19, reproduces to every printed digit (S1). NOT RUN at z ≥ 23 (the direct sum over primitive frequencies is the cost LSE's peel removes; the S0 identity of the lattices makes it redundant). - **Ssat > CAP at some level**: RUN at all ten levels, none (S3). - **A reading of LSE under which its §4 asymptotics are at a level other than D = z³**: none found; every figure in LSE §2-§5 is produced at `S_LEVEL = 3.0`, and its §7 says "Only s = 3.0 for u_sup" (line 671). - Not reached: how loose the cap is asymptotically (CAP/Ssat = 17.88 to 24.05 at z = 13..19, the explicit bound 1e9 to 1e10 above Ssat; whether lim sup u_sat at s = 3 is 6 or something under it is OPEN in both directions); u_cap itself passes 4.2652 at z = 47, still under β₂ = 4.26645, and the level where it crosses was not computed, since it says nothing about the truth. *Gate: `node research/qc.js --full` result recorded in the closing report; the producer's tail verifies under `qc/embed.js --check`.*