# The mean value of B₂: the mass located, the primitive-mode repair, and a proven qmax ln⁷z bound **attack-rhoms-01's NOT-REACHED item — a mean-value bound on B₂ = Σ e²Vabs(e)² below its trivial qmax·B ceiling — is executed: the mass of B₂ is located (primorial-core smooth e, mid-scale, nowhere near qmax), one exact repair is proved and priced (the primitive-mode weight J₂(e)/12, a flat 1.48× cut of MS3 at every level), and the analytic ceiling is lowered by one logarithm (B₂ = O(qmax·ln⁷z) against the trivial O(qmax·ln⁸z)), with the finite forms computed exactly at z = 13..47 and the proven all-z re-entry point of the cap located for the first time. Producer: `research/attack-b2mean-01.js` (embedded via `qc/embed.js`; all controls pass; 6.4 s, deterministic). Four things came out.** **One: the decay law of Vabs is the envelope shape times a decaying sign-survival factor (§1).** At fixed ω(e) the slope d lnVabs/d lne is −0.97 ± 0.15 at ω = 1 (exactly Theorem 1's envelope 3Aρ(e) ~ 2^ω/e) and steepens to −1.24..−2.08 at higher ω: Vabs runs BELOW its proven envelope by a factor that itself decays in e. The statistic φ₂ = e·Vabs/2^ω is bounded (sup 0.39–0.45, attained at the odd-primorial-core e = 105, 1155, 2310) and its B₂-weighted rms FALLS with z (0.362 → 0.247 over 13..23) — the opposite sign of drift from B's sup-statistic (`attack-AB-bounded.md` §1.4), so a uniform φ₂ ≤ 1/2 is not excluded by these levels, though only measured. **Two: B₂'s mass sits at smooth mid-scale e, and it spreads (§1).** The top contributor is always the primorial core (2·3·5·7·11 at z = 13..19, growing to 2·3·5·7·11·13·17·19 at z = 47); the ω-profile peaks two below the maximum (ω = 7 carries 47.0% at z = 47); e > qmax/8 carries only 9–16% at z ≥ 23; and 75–78% of the mass has 6 | e at every level. Concentration DILUTES: the top mode's share falls 27.1% → 0.8% and N₉₀ (the e count holding 90% of B₂) grows 8 → 1233 over z = 13..47. So the e² weight is priced by many smooth moduli of intermediate size — not by a few e near qmax, which is exactly why the trivial qmax·B shape wastes a growing factor (13× → 1764× measured). **Three: two proven reductions, and their price (§2, §3).** (i) **Lemma J + Theorem MS3′**: the primitive cosecant sum is Σ*_a 1/(4sin²(πa/e)) = J₂(e)/12 exactly (Möbius inversion; verified to 2.9e-14 over e = 2..200), so the MS3 chain tightens to ⟨ρ̃²⟩ ≤ B₂ᴶ/12, B₂ᴶ = Σ J₂(e)Vabs(e)² — a strict, hypothesis-free sharpening. Measured gain B₂/B₂ᴶ = 1.479–1.487, FLAT across all ten z, at 90–92% of its own ζ(2) ceiling (because the mass is 6-divisible): **MS3's 18–48× over truth becomes 12.1–32.0×.** (ii) **Theorem MV**: through the verified steps Vabs ≤ T ≤ 3Aρ(e) plus e ≤ qmax and a divisor-trick count (one log below Rankin), B₂ ≤ 9A²S(z) ≤ 9A²·qmax·Π_{p 2^53 and are floats entering float products only (rel ~1e-16, stated); all W-dependence runs off lnW = θ(z). ## 1. THE STRUCTURE OF Vabs AND THE MASS OF B₂ **The law.** The single-power fit d lnVabs/d lne = −0.51..−0.83 (z = 13..23) is not the law — lne and ω(e) are collinear, and the two-regressor fit reads b = −1.30..−1.57 with c/ln2 = 2.08..2.54 (rmse 0.54–0.82, a factor-2 scatter). The fixed-ω slopes decide: z=23: ω=1: −0.97±0.15 ω=2: −1.26 ω=3: −1.49 ω=4: −1.64 ω=5: −1.81 z=47: ω=1: −0.99±0.07 ω=2: −1.24 ω=3: −1.28 ω=4: −1.30 ω=5: −1.58 ω=6: −1.48 ω=7: −1.44 ω=8: −2.08 Reading: at ω = 1 the decay is exactly the proven envelope's e⁻¹ (T(e) ≈ 3Aρ(e) is within 0.7–3.4% of its ceiling, §3); at higher ω the measured Vabs falls a further e^{−0.2..−1.1} below the envelope — the sign cancellation inside the e₁-classes deepens with e at fixed ω. Support is dense, not sparse: 6119 of the 9003 admissible divisors of P(47) below qmax carry mass (68%; 100% at z ≤ 19). So the correct model is **Vabs(e) ≈ φ₂(e)·2^{ω(e)}/e with a bounded, slowly-decaying survival factor φ₂** — sup φ₂ = 0.39–0.45 (at e = 105, 1155, 2310), B₂-weighted rms falling 0.362 → 0.247 over z = 13..23. [MEASURED] **The mass.** Share of B₂, all ten z [MEASURED]: - top e is the primorial core at every level (2·3·5·7·11 through 2·3·5·7·11·13·17·19), holding 27.1% at z = 13 down to 0.8% at z = 47; - N₉₀ grows 8 → 1233: the mass spreads over ever more moduli; - ω-profile at z = 47: ω = 7 carries 47.0%, ω = 6 30.7%, ω = 8 13.9% — the peak sits two below the attainable maximum ω = 9; - e > qmax/8 carries 9.9–16.2% at z ≥ 23 (57% at z = 13 where qmax is tiny): **the mass is NOT near qmax**, which is why B₂ ≤ qmax·B wastes a measured factor 13.5 → 1764 (growing) and why an e-aware mean value can beat it; - 6 | e carries 74.7–77.8%, flat — the fact the ζ(2) repair of §2 monetizes. This answers the concentration question the a′ = 1 fact posed: the variance lives at the lowest frequency of MANY smooth mid-scale moduli, not at few large ones. The e²-weight is a faithful price for each such modulus (its a′ = 1 weight is ~e²/4π²), and the remaining slack is in Θ* vs Vabs and in the a-profile, quantified in §2. ## 2. LEMMA J AND THEOREM MS3′ — THE PRIMITIVE-MODE REPAIR **Lemma J** [PROVEN; verified to 2.9e-14 over e = 2..200]. For e > 1, Σ_{01} Φ(f) with Φ the primitive sum; Möbius inversion gives Φ(e) = Σ_{f|e} μ(e/f)(f²−1)/12 = J₂(e)/12 (the −1 dies since Σ μ(e/f) = 0). **Theorem MS3′** [PROVEN; chain verified exactly at z = 13, 17, 19]. ⟨ρ̃²⟩ = Σ_e Σ*_a |Θ_e(a)|²/(4sin²(πa/e)) ≤ Σ_e Θ*(e)²J₂(e)/12 ≤ B₂ᴶ/12, B₂ᴶ = Σ J₂(e)Vabs(e)². MS3 extended the primitive sum to all a ≠ 0 to buy the (e²−1)/12 identity; those imprimitive modes are already counted at their true denominators, and Lemma J refunds them. The chain holds with room at every verified level (e.g. z = 19: 4.952 ≤ 33.443 ≤ 97.846 ≤ 145.235). **The price of what remains, measured mode-exactly** (z = 13 / 17 / 19): max_a step costs 2.08× / 6.04× / 6.75× after the repair (was 3.05 / 8.94 / 9.97); the Θ* → Vabs triangle costs 5.84× / 3.44× / 2.93×; total 12.13× / 20.81× / 19.76× (was 18.00 / 30.94 / 29.33). Gain B₂/B₂ᴶ = **1.479–1.487 at all ten z, flat**, against the per-z ζ(2) ceiling 1.608–1.638: the repair realizes 90–92% of everything a totient-type weight can give, because the mass is 6-divisible (§1). [VERIFIED / MEASURED] **A route killed on measurement before anyone walks it.** The ℓ² alternative to the max_a step — bound Σ*_a|Θ_e(a)|² per e and pay the worst (a′ = 1) weight — would deliver 55.8× / 665× / 5594× truth at z = 13/17/19, diverging: |Θ_e(·)|² carries substantial mass at high a′ where the true weight is tiny, so trading the weight profile for an ℓ² norm is ruinous. A residue-class ℓ² mean value is NOT a useful missing object; the useful one is the signed class-sum bound below. [MEASURED] ## 3. THEOREM MV — THE MEAN-VALUE LADDER ON B₂ With T(e) = Σ_{i: e|q_i}|w_i|/q_i and Theorem 1's verified steps (Vabs ≤ T ≤ 3A(z)ρ(e), zero violations through z = 47; ρ(e) = Π_{p|e} r(p), r(2) = 3/5, r(p) = 2/(p+2), A = (5/2)Π_{21.5e5], peak 5.695 at z = 31 — the window is real and wide in proven currency. The Rosser–Schoenfeld formula scan (upper envelope, control-verified against the exact scan) locates the re-entry: **z\* = 2.77e12 at c_M = 0.35; 2.39e13 at c_M = 0.1; 1.12e15 at c_M = 0.01** — the cap then stays below β₂ forever (→ 3.5). The redteam rider's sentence — re-entry finite but not located under proven B — is retired: located, with stated sensitivity to the one literature-standard constant (a factor 35 in c_M moves lnz\* by 6.0). Between z = 47 and z\* the composition remains model-covered, exactly as before, but now with a proven far end. [VERIFIED, with the flagged M rider] - Nothing changes on the asymptote: cap → s + 1/2 = 3.5 either way; the F4 margin 0.766 is untouched (the √(2lnW) factor owns the asymptotics; B₂'s improvements move constants and the finite window only, exactly as `attack-rhoms-01` §2 predicted). ## 5. READINGS 1. **PROVEN.** Lemma J (primitive cosecant sum = J₂(e)/12) and Theorem MS3′: ⟨ρ̃²⟩ ≤ B₂ᴶ/12 = Σ J₂(e)Vabs(e)²/12, a strict hypothesis-free sharpening of MS3; chain verified exactly at z = 13, 17, 19. Producer S0, S2. 2. **PROVEN.** Theorem MV: B₂ ≤ 9A²S(z) ≤ 9A²qmax·Π(1+h/p) = O(qmax·ln⁷z), one log below the trivial qmax·B's proven O(qmax·ln⁸z); finite forms verified at all ten z, 2.1–3.4× below the trivial proven product already at walkable z. Theorem 1's steps re-verified with zero violations through z = 47 (new at 43, 47). Producer S3. 3. **MEASURED.** The decay law: Vabs ≈ φ₂(e)·2^{ω(e)}/e with sup φ₂ = 0.39–0.45 and fixed-ω slopes −0.97 (ω = 1, the envelope) steepening to −2.08; B₂'s mass at primorial-core smooth mid-scale e (ω peak two below max, 9–16% beyond qmax/8, 76% at 6|e, N₉₀ 8 → 1233). The trivial qmax·B wastes a measured 13.5× → 1764×. Producer S1. 4. **VERIFIED / MEASURED.** The re-priced caps: MS3′/truth = 12.1–32.0× (was 18–48×), gain 1.479–1.487 flat = 90–92% of the ζ(2) ceiling; cap3′ margins −1.83..−1.34; the fully-analytic capMV1J < β₂ at z = 13..37. Producer S4. 5. **VERIFIED (with the flagged M rider).** The proven all-z re-entry is located: z\* ≈ 2.8e12 (c_M = 0.35), sensitivity stated; window peak 5.695 at z = 31; window open at the 1.5e5 exact-scan end. Producer S4. 6. **MEASURED, negative.** The residue-class ℓ² route (Σ*_a|Θ_e(a)|² × the a′ = 1 weight) reads 55.8× → 5594× truth, diverging — closed before proposal. The useful missing object is not an ℓ² norm bound; it is the signed class-sum mean value of reading 7. Producer S2. 7. **INFERRED, the named stall.** The whole remaining proven-to-true gap in the B₂ currency is the sign-discard step Vabs ≤ T, measured flat at 4.6e2–7.6e2× under the e² weight — `attack-AB-bounded.md` §1.4's irreducibly-mean-value object. No proof attempted here. Producer S3. ## 6. NOT REACHED - **The signed class-sum mean value** (reading 7) — the ~600× flat factor between MV0 and B₂; the one theorem that would make the analytic ladder meet the measured B₂. Same family as the B = O(1) question; not attempted. - **The max_a residual** (2.1–6.8× after the repair): needs a proven a-profile for |Θ_e(a)| beyond the triangle; no corpus tool owns one. - **The exact-qmax cap on 47 < z < z\*** — still model-covered; MV1ᴶ needs buildTerms (for qmax and M) beyond z = 47, deferred under the compute ceiling exactly as in `attack-rhoms-01` §6. - **Mode-exact decomposition at z ≥ 23** (Σφ(e) ~ P(z) modes; z = 19 cost seconds, z = 23 would cost ~100× more) — the step-cost columns stop at 19. - **Any re-pricing of the moment-ladder budget rows** (M2/M3 use measured moments, not B₂; untouched) and **the s-freedom** (everything s = 3.0, inherited). - **A uniform φ₂ bound** (sup φ₂ ≤ 1/2 measured at 13..23 with the weighted rms falling): would give B₂ ≤ qmax·Σ4^ω/e-type constants directly; only measured here, and `attack-AB-bounded.md`'s experience with sup-statistics (its φ rose linearly) warns against optimism without the fixed-ω decay being understood first. ## 7. GATE `node research/qc.js` after the producer and this report: TOTAL 2, both findings on files other agents of this session are mid-writing (`xchan-at37-01-census.js`, the known standing census red; `attack-delta37-01.js`, minutes-old, its embed banner mid-repair by its author — the count fluctuated 2 → 4 → 2 during this attack as the fleet landed files). Neither names this producer or this file, and both of this attack's files are clean in every check. `qc/embed.js --check` on the producer: code-sha256 matches, out-sha256 matches, body bit-honest (6.4 s run, deterministic). *History and superseded claims: `research/history/CHANGELOG.md`.*