# The ⟨ρ²⟩ analytic bound: the first move was already made, audited clean, and its constant named **The brief's premise is refuted. TODO item 0's named first move, "an analytic upper bound on ⟨ρ²⟩(z) from its PROVEN closed form", was executed on 2026-08-21 and is a theorem: `research/history/staging/attack-rhoms-01.md` proves ⟨ρ̃²⟩ ≤ C·z^{2s}·ln⁸z three independent ways. TODO.md item 0 and `rho-maximal-law.md` §7 both still list it as unreached. This session did not redo it. It audited the MS1 derivation step by step and found no defect, and it closed the one thing the existing artifact leaves open: the constant C, which was never named. Producer: `research/history/staging/rho2-analytic-bound.js` (scratchpad grade, `qc/embed.js --check` bit-honest). Nothing about any exponent moved, and nothing could have: the second moment is an ingredient of RML, and the negative half of this note quantifies how far it is from being a route.** --- ## 0. WHAT IS OPEN OR FAILED, FIRST - **RML(α) is open at every α.** No form F1–F4 is proven. The mean-square lemma changes none of that. Rung: OPEN, unchanged since `rho-maximal-law.md`. - **The all-z explicit bound is worthless as a finite-z number.** FORM I* below sits 2.535e+12 above the exact ⟨ρ̃²⟩ at z = 13 and 3.213e+14 at z = 47. The ratio widens over the ten levels but not monotonically: 1.745e+14 at z = 29 exceeds 1.653e+14 at z = 31. The bound grows like z⁶ln⁸z while the measured ⟨ρ̃²⟩ exponent between the cited endpoints z = 13 and z = 47 is 4.511, which is 75% of the stated z⁶ and 44% of the effective log_z(FORM I*) = 10.2054 at z = 47 (`research/history/staging/redteam-0828-closures.js` R4). The measured-input form (MS3 = B₂/12, cited) is 18.0× to 44.2× above truth over the same ten levels. Only the asymptotic exponent is worth anything here. - **The stated exponent 2s = 6 is a limit no computable z reaches.** Measured on the closed form itself: log_z(FORM I*) = 10.2054 at z = 47, 8.9921 at z = 1009, 8.1470 at z = 10⁵, 7.8941 at z = 10⁶. The ln⁸z surcharge is still 1.5205 of exponent at z = 10⁶. Rung: MEASURED, scratchpad grade, from `node research/history/staging/rho2-analytic-bound.js` S4. - **Chebyshev off any second moment provably cannot reach RML.** Quantified in §4. The mean-square lemma is an INGREDIENT, never a route. - **[MEMORY] flag, and the quoted form is false in range.** The one bound in the derivation not read at a page is Rosser–Schoenfeld 1962, Π_{p≤x}(1−1/p)^{-1} < e^γ ln x (1 + 1/(2 ln²x)). As stated it is FALSE at x = 109 and x = 113 (ratios 1.001320 and 1.003226 in this file's p < z convention) and holds at every prime z from 127 to 3.3e7. The RS bounds of this shape are published with a lower cutoff (x > 285 for the direction that gives an upper bound on the reciprocal product) [MEMORY, unverified at a page], which is exactly what the two violations sit below. FORM I* therefore carries a validity floor, not "all z". The producer's own sixteen-z sample (worst ratio 0.998681 at z = 19) did not include 109 or 113, so the sampling passed, not the check. Anyone integrating FORM I* must open Rosser–Schoenfeld. ## 1. THE OBJECT AND THE CLOSED FORM, RESTATED Per `rho-maximal-law.md` §1. Fix s = 3.0, W = P(z) = Π_{p 0. The centered potential is ρ̃ = ρ + M/2, and ⟨ρ²⟩ in the corpus's usage is the period variance of ρ, that is ⟨ρ̃²⟩. The argument is a position y ∈ ℤ/W; the normalisation is weight units, dimensionless. It is H-free: R_H(x) = ρ(x) − ρ(x+H) is the H-dependent object, ρ̃ is not. **The closed form (PROVEN).** `research/sift-limit-lemmaV.js`, function `meanSquare`, field `rho2` (= `plateau`/2); the derivation sits in the block comment immediately above that function (lines 108–124) and in S2 of the same file, where it is checked against brute force over the full period: ⟨ρ̃²⟩ = Σ_{i,j} w_i w_j C_ij, C_ij = [ (g²−1)/12 − d̄(g−d̄)/2 ] / (q_i q_j), g = gcd(q_i, q_j), d̄ = (c_j − c_i) mod g, g = 1 ⟹ C_ij = 0 exactly. **Independent reproduction (this session).** The producer calls the repo's own `buildTerms(z, z³)` and `meanSquare(t, H)` and reads `rho2` at z = 13, 17, 19, 23: values 1.095066, 2.496563, 4.952375, 7.982480, matching the cited column of `attack-rhoms-01.js` (OUTPUT S0, out-sha256 4377df4f…) to the six decimals the citation carries, and invariant in H to zero ulp between H = 10 and H = 1000, as an H-free object must be. Rung: MEASURED, scratchpad grade, S0. ## 2. THE BOUND, WITH EXPLICIT CONSTANTS The derivation is MS1 of `attack-rhoms-01.md` §2, re-derived here with every inequality made explicit and each step checked numerically. Rung on the derivation: PROVEN, and this session's audit found no defect in it. 1. For g ≥ 2, f(d̄) = (g²−1)/12 − d̄(g−d̄)/2 has maximum (g²−1)/12 at d̄ = 0 and minimum ≥ −(g²+2)/24, and (g²+2)/24 ≤ (g²−1)/12 exactly when g² ≥ 4. So |C_ij| ≤ (g²−1)/(12 q_i q_j) and ⟨ρ̃²⟩ ≤ GS/12 with GS = Σ_{i,j} g²/(q_iq_j). Checked over every g = 2..4000 and every d̄: worst ratio 1.000000, attained at g = 2, so the step is tight and correct. 2. Multiplicity of a modulus among lattice terms is ≤ (9/2)τ(q), proven in `attack-rhoms-01.md` §2 as (3/2)τ per block times three blocks, and measured there to attain the cap (ratio 1.0000). Hence GS ≤ (81/4) Σ_{q,q'} τ(q)τ(q') g²/(qq'). 3. Substituting q = gu, q' = gv and dropping (u,v) = 1 (all terms positive, so this over-counts and stays an upper bound), with τ(gu) = 2^{ω(g)}2^{ω(u)} for squarefree q: GS ≤ (81/4)·[Σ_{g|P(z), g ≤ qmax} 4^{ω(g)}]·[Σ_{u|P(z)} 2^{ω(u)}/u]². 4. Σ_{g ≤ X} 4^{ω(g)} ≤ X·Π_{p 1+√e | standard linear-sieve asymptotics, CITED not re-derived; MEASURED M·ln²z = 0.343–0.367 flat over z = 13..47 | | — | the law itself | **OPEN** | The five forms: | form | status after this session | |---|---| | F0 (tautological floor) | not a law; the ladder's bottom, no content | | F1 (one-sided single-window) | OPEN. Weakest law-shaped form. Untouched by a second moment | | F2 (sup-norm Lemma V at the ceiling) | OPEN. Lemma V is posed-not-proven; working point s/u = 0.703 is inside its stated range (HELD, confirmed with the mandatory rider at T2.e), and the fixed-smooth-profile obstruction binds at the β₂ point exactly as at the TPC point | | F3 = RML(α), the named target | OPEN at every α. This session bounds one of its two factors | | F4 (Gaussian units) | OPEN as a theorem; MEASURED TRUE at z = 13..29, C_true 0.6044–0.8022, five levels, reproduced digit-for-digit by the red team (T2.d). Composed with the proven mean square it gives RML(3.5+ε), so F4 ⟹ G₂(z#) ≪ z^{3.5+ε}, margin β₂ − 3.5 = 0.766 | **Effect on the sufficiency curve: one thing, and it is not exponent movement.** θ_G(z) is capped at s + 1/2 + o(1) = 3.5 + o(1) unconditionally, since sup|ρ̃| ≤ √⟨ρ̃²⟩·√(2 lnW) ≤ z^{3+o(1)}·z^{1/2+o(1)} and lnW = θ(z) ≍ z. So λ_max = β₂ − θ_G cannot close asymptotically, which excludes `rho-maximal-law.md` reading 6's model A (linear drift, room gone at z ≈ 5·10³) as an asymptote. That is a statement about a model of a measured curve, not about any delivered exponent. Rung: the cap is PROVEN; the exclusion of model A is PROVEN as an asymptotic statement and says nothing about a finite-z dip. **MV(α), the measured weakest sufficient form** (slope 4.214 ± 0.304 with β₂ inside the bar, per TODO item 0): a second-moment bound does not move it either. MV is a mean-value form over windows; the ⟨ρ̃²⟩ bound is the H-free variance and enters MV only as the same ingredient it is for RML. No measurement here bears on the 4.214 ± 0.304 slope, and none was run. **The negative, quantified: Chebyshev cannot reach a sup-type law.** RML is a sup statement; Chebyshev from a second moment costs the position-union factor. The only fully proven recovery in the corpus is R1 = (3·C_L·W·⟨ρ̃²⟩)^{1/3} (`attack-rhoms-01.md` §1, from the exact increment identity ρ(y+1) − ρ(y) = M − cc(y+1)). Against smax = (z^{β₂}M − 1)/2, two different floors have to be kept apart, and `attack-rhoms-01.js` S5 keeps them apart. Its S4 `R1f` column is `exp((ln 3 + lnW + ln ms)/3)`: C_L = 1 with the MEASURED mean square. The true absolute floor, C_L = 1 and ⟨ρ̃²⟩ = 1, which no mean-square lemma can improve on, is (3W)^{1/3}. Both against smax: ``` z 13 17 19 23 29 31 37 41 43 47 C_L=1, meas ms 0.012 0.015 0.035 0.056 0.076 0.232 0.541 1.474 5.000 15.346 in exp −1.709 −1.493 −1.142 −0.920 −0.767 −0.425 −0.170 +0.105 +0.428 +0.709 C_L = ms = 1 0.012 0.011 0.020 0.028 0.032 0.080 0.123 0.282 0.852 2.155 ``` The `C_L = 1, meas ms` row is the cited S4 column; the absolute-floor row is rebuilt in `research/history/staging/redteam-0828-closures.js` R5, which also checks that the cited column matches the measured-ms form at all ten levels and never the ms = 1 form. So even at its floor the Chebyshev route already misses by a factor 2.155, or 0.1994 of exponent, at z = 47; at z = 41 and z = 43 the absolute floor still clears, and it is the C_L = 1 floor with the measured mean square that fails from z = 41 at 1.474×, 5.000× and 15.346×. The miss diverges either way: the union price is e^{θ(z)/3} with θ(z) ≍ z, which beats every fixed power of z. A perfect mean-square lemma composed with Chebyshev therefore fails at every large z, at every exponent. The 2k-th moment ladder needs k ≍ z/(1.53 ln z), also divergent (cited, `attack-rhoms-01.js` S5). Chaining is CLOSED (`REFUTED.md`) and was not retried. Rung: MEASURED on cited columns, with a PROVEN divergence mechanism. ## 5. READINGS 1. **REFUTED (the brief's premise).** TODO item 0's first move was already executed and is a theorem (`attack-rhoms-01.md`, 2026-08-21). The item is stale in two places, listed in §7. 2. **AUDIT PASS on MS1, with one input false in range.** The PROVEN closed form reproduces the cited column at z = 13..23 from the repo's own code, H-invariant to zero ulp; MS1's steps (1), (5) and (6)'s binomial half each verify with zero violations over the tested range. No defect found in the MS1 derivation itself. Step (6)'s RS half is the exception: the quoted RS form fails at z = 109 and z = 113, which is a validity floor on FORM I* rather than a defect in MS1. 3. **PROVEN, new: the constant.** ⟨ρ̃²⟩ ≤ 170.88·z^{2s}·ln⁸z·(1+1/(2ln²z))⁸, modulo the [MEMORY] Rosser–Schoenfeld citation; and the RS-free product form ⟨ρ̃²⟩ ≤ (81/48)·z^{2s}·Π(1+4/p)·Π(1+2/p)². Growth order z⁶ at s = 3.0. 4. **MEASURED, scratchpad grade.** Ratio bound/exact at the ten embedded levels: 2.535e+12 to 3.213e+14 for the closed all-z form, widening over the range but not monotonically (z = 29 exceeds z = 31); 6.3e+05 to 2.8e+09 for the best explicit form (FORM II at cited qmax). 5. **MEASURED, scratchpad grade.** The exponent 2s = 6 is unreachable in practice: log_z of the bound is 10.2054 at z = 47 and still 7.8941 at z = 10⁶. 6. **NEGATIVE, quantified.** At the true absolute floor (C_L = ⟨ρ̃²⟩ = 1) Chebyshev off any second moment clears smax at z = 41 and z = 43 and first misses at z = 47, by 2.155× or 0.1994 of exponent; at C_L = 1 with the measured mean square it misses from z = 41. Either way the miss diverges, since the union price is e^{θ(z)/3}. The mean-square lemma is an ingredient of RML and provably not a route to it. 7. **UNCHANGED.** RML is open at every α; no exponent moved; the sufficiency curve's measured columns are untouched. ## 6. WHAT WOULD FALSIFY THIS, AND WHETHER THE CHECK HAS RUN - **FORM I / FORM I* false at some z.** Falsified by any z with exact ⟨ρ̃²⟩ above the bound. RUN at ten levels (§3): the smallest margin is 6.3e+05. A counterexample would require the multiplicity cap (9/2)τ(q) or the qmax bound to fail; both were checked directly (zero violations) and the first is measured tight, so a failure would be a proof error, not a numerical surprise. - **The RS constant wrong.** Opening Rosser–Schoenfeld 1962 is NOT RUN and stays [MEMORY]. The numerical check is no substitute and has now failed: the quoted form is false at z = 109 and z = 113, so the validity range does exclude small z. If RS is wrong the FORM I product bound survives untouched, and so does 170.88 = (27/16)e^{8γ}, which comes from the leading term of Mertens' product and not from the correction factor. What moves is the (1 + 1/(2 ln²z))⁸ factor and the validity floor: substituting the safer (1 + 1/ln²z)⁸ raises FORM I* at z = 47 from 1.1597e+17 to 1.4994e+17, 29%, against a bound already 3.2e+14 above truth. - **The exponent 2s wrong.** Falsified if qmax exceeds min(z^{2s}, P(z)) for some z, or if the linear sieve's support is larger than D per system. Checked at ten levels; the general statement is a one-line consequence of d ≤ D and q | P(z) and is not in doubt. - **The Chebyshev negative wrong.** Falsified by a recovery that does not pay a position-union factor, or by C_L growing slower than assumed. The C_L values at z ≥ 31 are sampled LOWER bounds (cited), so the R1 failure verdicts there are certifiable in the direction claimed; the R1f floor row uses C_L = 1 and is unconditional. RUN. - **"Ingredient not route" wrong.** Falsified by any proof of F1–F4 that consumes only a second moment. None exists; the named first unproven object on that ladder is ⟨ρ̃⁴⟩ (`attack-rhoms-01.md` §4), and nothing here touched it. ## 7. DEFECTS NOTICED IN PASSING - `TODO.md` item 0 and `rho-maximal-law.md` §7 both still list "an analytic upper bound on ⟨ρ²⟩(z)" as a first move / NOT REACHED; it was closed 2026-08-21 by `attack-rhoms-01.md`. Two stale pointers. - `TODO.md` item 1e quotes "C_true = 0.60–0.78 at z = 13..29"; the measured values run to 0.8022 (`rho-maximal-law.md` §4), and §3 of that file states the band correctly as 0.60–0.80. The TODO band understates the maximum. ## 8. NOT REACHED - Rosser–Schoenfeld opened at a page. FORM I*'s validity floor stands on [MEMORY]; the constant 170.88 does not depend on it. - A mean-value bound on B₂ = Σ e²Vabs(e)² below its qmax·B ceiling, which is the named object that would sharpen FORM II's constant. Inherited unattempted from `attack-rhoms-01.md` §6. - Anything on ⟨ρ̃⁴⟩. Named as the stopping object, untouched here. - The s-freedom. Everything is s = 3.0; the constant 170.88 and the exponent 2s both move with s and were not scanned. - Exact Λ at z = 31/37, the other first move of TODO item 0. Explicitly out of scope for this session. *History and superseded claims: `research/history/CHANGELOG.md`.*