# The weakest sufficient form of F4: the mode-count check answered, the ℓ¹ frontier measured, the family charts drawn **The F4 follow-on of `attack-rhoms-01.md`, executed: with rms ≤ z^{3+o(1)} PROVEN, any recovery sup ≤ C·rms·z^ℓ with ℓ < β₂ − 3 = 1.26645 wins — this pass hunts the weakest such form that could be proven, treating ρ̃ as what it is: a deterministic explicit trig sum with computed coefficients. Producer: `research/attack-f4weak-01.js` (embedded via `qc/embed.js`; code-sha256 and out-sha256 match, body bit-honest; all controls pass). Four things came out. All new findings HELD for one adversarial pass per the standing rule.** **One: the first concrete check is answered — the mode-count/degree bound is NOT the theorem (§1).** The mode count N(z) = Σ_{e∈E} φ(e) is computed exactly at all ten z: the mode set is **W-saturated at z ≤ 19** (coverage N/(W−1) = 1.0000 — every divisor is a live conductor, so the degree bound √N·rms IS Chebyshev √W·rms there), detaches from W at z = 23 (coverage 0.2731, then 0.0936, down to lnN − lnW = −14.0 at 47) but still grows at d lnN/d lnz = 11.876 ± 0.328 — the divisor count is still doubling with each new prime. √N·rms grows at **z^{8.174 ± 0.249} against the z^{4.26645} ceiling**: it clears smax only at z ≤ 29 and is dead from z = 31 on; the no-measured-input variant √N·√(B₂/12) is dead from z = 19. Task (c)'s STOP condition is NOT met. **Two: a new PROVEN finite-z clearing — and it is the first to survive z = 47 (§2).** Λ_V(z) = Σ_e Vabs(e)·(e/2)(ln e + 0.307) is a fully proven sup bound (classical cosecant ℓ¹ bound + Θ* ≤ Vabs, no measured input anywhere) and **clears smax at ALL TEN levels, 1.498e5 ≤ 1.577e5 at z = 47 (5.0% margin)** — where R1, the previous best proven recovery, certifiably failed. But it is the Bonferroni window once more: its growth is 5.230 ± 0.186, at z = 47 it has converged onto the trivial (n+M)/2 = 1.470e5, and the collapse is structural — Σ_e e·Vabs(e) ≤ Σ_i σ(q_i)/q_i ≤ n·Π(1+1/p) = O(n ln z) (L4): absolute-value accounting cannot win asymptotically. **Three: the exact spectral ℓ¹ norm of ρ̃, measured for the first time — and the β₂ boundary passes through its error bar (§2, §5).** Λ(z) = Σ_{e,a}|Θ_e(a)|/(2 sin(πa/e)) = 9.801, 25.157, 60.031, 121.250, 261.555 at z = 13..29, exact (factorised vs direct-phase 1.9e-14; Parseval vs walker 1.2e-10; sup ≤ Λ holds with 3.7×→3.2× slack). **d lnΛ/d lnz = 4.214 ± 0.304 against β₂ = 4.26645** — the weakest law-shaped object this pass found, **MV(α): Λ ≤ C·z^α with α < β₂, a MEAN VALUE over modes with no position sup, implies RML(α) outright**, and it is the only form here whose truth the measurements do not already decide against (the a′≤3 model extension reads 5.649 ± 0.227 at 31..47, leaning low, and local slopes rise with the window throughout this corpus — quote no single power). The TRUTH's recovery loss is measured too: d ln(sup/rms)/d lnz = 0.891 ± 0.213 vs F4's √(2lnW) at 0.54–0.56 in z-exponent, vs the ℓ¹ instrument's 2.608 ± 0.179. **Four: the family charts (§3, §4).** sup ≤ C·rms·(lnW)^A wins iff A·lnθ/lnz < 1.26645 − o(1) — finite-z A_max runs 3.575 → 2.497 across 13..47 and 1.2666 at z = 999983; F4 (A = 1/2) carries a 5–7× margin in A units everywhere measured. Every fixed W^ε dies: z* = 13, 19, 23, 41, 71, 151, 419 at ε = 1/2..1/50. The moment ladder with Gaussian constants **rediscovers F4 exactly at its k-optimum** (1.0047–1.0218 × rms√(2lnW); synthetic control 1.0002, 1.0000 at θ = 1e3, 1e6): uniform sub-Gaussian moments ARE F4 at constant 1. Measured to m₁₆ for the first time: C_k/(2k−1)!! < 1 at every (z, k ≤ 8), falling in k, drifting up in z; the needed order is k(z) → 0.56·z/lnz. --- ## 0. CUSTODY Everything measured is in `research/attack-f4weak-01.js` (`qc/embed.js --check`: code-sha256 matches, out-sha256 matches, body bit-honest; deterministic — the check's re-run reproduced the out-sha). Division of labour per the standing compute rule: - **Recomputed (cheap, controls in-pass):** the subset sweep (B, B₂) — **MATCH against attack-rhoms-01's embedded columns at all ten z** (B to 6e-5, B₂/12 to rel 1.5e-3 of the 5-digit cited print); the closed-form pair sum at 13..23 (vs walker m₂, rel ≤ 2.5e-11); the full-period walker at 13..29 — sup MATCHES the cited column at all five levels, and the z = 29 moment ratios reproduce attack-rhoms S1 (2.918/14.00/93.5). - **New measurements:** N, #E, Σe, Vabs decay bins (S1); exact Λ, Λ₁, Λ₃, Θ*-form, nonzero-mode counts (S3); moments m₁₀..m₁₆ at z = 13..29 (S5). - **Cited (custody-bound):** rmsr at z = 31..47 (`theta-ladder.md` §2 via `rho-maxlaw-01-sufficiency.js`); sup at 13..29 (re-measured); B/B₂ lineage back to `lemmaV-parseval.js` / `attack-beta2-01`; β₂ = 4.26645 per `paper/beta2-note.md`. - **Proven inputs checked at source:** Θ* ≤ Vabs (Theorem B triangle); the spectral identity (verified here two independent ways at z = 13: factorised-vs-direct worst rel 1.9e-14 over ALL modes, Parseval vs walker 3.5e-13). - **Estimator control:** OLS returns 3.700000 on a known z^3.7 truth before any fit is quoted. The L2 cosecant bound is verified over e = 2..30000 (max ratio 0.6257, i.e. holds with ≥ 1.6× to spare); the mode-completeness identity Σ_{e|W}φ(e) = W at three W. Units: ρ̃ and moments dimensionless; W, qmax, N in positions/counts; strengths as exponents base z. WIDTH: W > 2^53 at z ≥ 31 — W appears as an integer only at z ≤ 29; all W-dependence at z ≥ 31 runs off lnW = θ(z); N and Σe stay below 2^53 at every level (max 3.5e10). --- ## 1. THE MODE-COUNT ARITHMETIC (the first concrete check), AND ITS VERDICT sup|ρ̃| ≤ Σ_modes|ĉ| ≤ √N·rms with N the number of distinct reduced frequencies a/e carrying a coefficient — the trig-polynomial "degree" bound the task asked to work exactly. Producer S1/S2, exact at all ten z: ``` z #E N cover N/(W−1) sqrtN·rms proven sqrtN·sqrt(B2/12) smax verdict 13 31 2.309e3 1.0000 5.03e1 2.13e2 1.58e3 Y|Y 19 127 5.105e5 1.0000 1.59e3 8.61e3 5.65e3 Y|n 23 243 2.649e6 0.2731 4.60e3 2.66e4 1.10e4 Y|n 29 467 2.089e7 0.0936 1.69e4 1.16e5 2.76e4 Y|n 31 827 8.207e7 lnN−lnW=−4.4 4.51e4 3.09e5 3.37e4 n|n 47 6119 1.115e10 lnN−lnW=−14.0 2.01e6 1.33e7 1.58e5 n|n ``` The task's hope — "if #modes grows polynomially in z, √(#modes)·rms may already beat the needed exponent" — fails on both halves. (i) At walkable z ≤ 19 the count is not polynomial structure at all: qmax = P(z), every divisor of W is a live conductor (coverage exactly 1), and √N·rms IS the Chebyshev bound M1. (ii) Where the count detaches from W (z ≥ 23) it is still in the transitional regime where each new prime doubles the divisor population: d lnN/d lnz = 11.876 ± 0.328 (12.531 ± 0.465 on 29..47), so √N·rms grows at z^{8.174 ± 0.249} — the √N factor alone delivers z^{5.94}, ~4.7× the entire 1.26645 recovery budget even before rms. The eventual polynomial ceiling (E ⊂ squarefree z-smooth integers ≤ qmax, so #E ≤ qmax ≤ z^{2s} and N ≤ #E·qmax) caps the exponent at 2s + o(1) per factor — an all-z bound of exponent ≥ 3 + s = 6 through this chain, nowhere near 4.26645. The 3,570 of the brief is not a mode count: it is the Rosser support size of the z = 101, D = z^{3.038} slice (`smoothness-front.md` §4.3, `redteam-0820-math.md` §2.3); the true conductor counts #E are 31..6119 across the ladder and the mode counts N are 2.3e3..1.1e10. **Verdict: the mode-count route is closed as an asymptotic route by measurement and by its own ceiling; a REFUTED.md row is proposed after the adversarial pass** (this file owns no shared-tree edits). ## 2. THE ℓ¹ / COEFFICIENT-DECAY ROUTE The classical maximal bound for a trig sum with coefficient decay is the ℓ¹ norm with the cosecant sum, and every link is proven (producer header L2/L3): sup ≤ Λ ≤ Λ_V ≤ CH = ((ln qmax + 0.307)/2)√(#E·B₂). The decay law, fitted first as asked: bin-rms Vabs(e) ~ e^{−0.723 ± 0.030} at z = 23, e^{−0.772 ± 0.018} at z = 47 (a pure triangle-scale law would be e^{−1}; the shortfall is the mass spreading up the conductors). ``` z Λ (exact) Λ/rms a′=1 ℓ¹ share Λ_V (proven) trivial (n+M)/2 smax Λ_V verdict 13 9.801 9.37 0.471 1.560e2 4.260e2 1.58e3 Y 17 25.157 15.92 0.355 5.524e2 1.118e3 4.17e3 Y 19 60.031 26.98 0.315 1.245e3 2.382e3 5.65e3 Y 23 121.250 42.92 0.269 2.462e3 4.818e3 1.10e4 Y 29 261.555 70.93 0.247 5.535e3 1.035e4 2.76e4 Y 31 — — — 1.041e4 1.793e4 3.37e4 Y 47 — — — 1.498e5 1.470e5 1.58e5 Y ``` Readings of the table. **Λ_V clears the β₂ ceiling at every measured level including z = 47** — the first proven recovery to survive 47 (R1 died there; attack-rhoms §3). The clearing is finite-z only: d lnΛ_V/d lnz = 5.230 ± 0.186, and at 47 Λ_V has converged onto the trivial bound — the ℓ¹ gain over trivial is 2.7× at z = 13 and gone by 47, because (L4, verified per z) Σ_e e·Vabs(e) ≤ Σ_i σ(q_i)/q_i ≤ n·Π_{p 0. Death points (first prime z with ε·θ(z) > 1.26645·lnz): z* = 13, 19, 23, 41, 71, 151, 419 at ε = 1/2, 1/3, 1/4, 1/6, 1/10, 1/20, 1/50. The admissible decay is ε*(z) = 1.26645·lnz/θ(z) (1.31e-1 at 47, 1.75e-5 at 1e6): **sub-polynomial-in-W is not enough; the boundary is exactly REC(ℓ): loss z^ℓ, ℓ < 1.26645.** ## 4. THE MOMENT LADDER sup ≤ (W·m_{2k})^{1/2k}. New measurement, moments to m₁₆ at z = 13..29 over complete periods (walker verified against cited sup, closed-form m₂, and attack-rhoms' z = 29 ratios): ``` C_k/(2k−1)!! k=2 k=3 k=4 k=5 k=6 k=7 k=8 z=13 0.8362 0.5731 0.3316 0.1659 0.0731 0.0287 0.0101 z=19 0.9643 0.8945 0.8121 0.7293 0.6431 0.5474 0.4426 z=29 0.9727 0.9335 0.8903 0.8441 0.7912 0.7264 0.6471 ``` Every ratio below Gaussian at every measured (z, k) — the sub-Gaussian window extends from attack-rhoms' k ≤ 4 to k ≤ 8 — falling in k at fixed z and drifting up in z (+0.22, +0.54, +0.79 per lnz at k = 2, 3, 4; the small levels 13, 17 sit low, 19..29 sit near 1). Five points cannot call the limit; what they support is stated in §5(iii). The chart: fixed k dies (k_min = 1,1,1,1,2,2,2,2,2,3 along the ladder, reproducing attack-rhoms S4 from independent code); the k-optimised Gaussian-model ladder lands at 1.0047–1.0218 × rms√(2lnW) (synthetic control 1.0002, 1.0000 at θ = 1e3, 1e6): **the ladder's optimum IS F4 at constant 1** — "uniform sub-Gaussian moments" and "the Gaussian maximal form" are the same hypothesis. The needed order with Gaussian constants is k_gauss(z) = 4.3 → 40435.9 across z = 47 → 1e6, i.e. → 0.56·z/lnz (the extra ½ln(2k/e) term costs ~0.5 of exponent — attack-rhoms' 1.53 denominator, re-derived). ## 5. THE WEAKEST SUFFICIENT FORMS, PRECISELY Ordered from cheapest hypothesis upward; each implies RML(α < β₂) and hence G₂(z#) ≪ z^{α}ln²z through the pricing lemma (`rho-maximal-law.md` §2, with its riders; the M ≍ 1/ln²z literature rider of `redteam-0821-exponent.md` §1.4 carries over verbatim). - **(i) MV(α), the mean-value form — NEW, the cheapest law-shaped object found: Λ(z) = Σ_{e,a}|Θ_e(a)|/(2 sin(πa/e)) ≤ C·z^α for some α < 4.26645.** No position sup anywhere: it is an ℓ¹ mean value of the mode amplitudes |Θ_e(a)| — the same family as Theorem A's and B₂'s mean values, the only grade of estimate this corpus has ever proven. Measured true-or-false status: **undecided by the window** — d lnΛ/d lnz = 4.214 ± 0.304 with β₂ inside the bar, the model extension (leaning low) at 5.649 ± 0.227, and the corpus-wide pattern that local slopes rise with z. What is measured true: Λ exact at five levels, sup ≤ Λ with 3.2–3.7× slack, Parseval closure to 1.2e-10. What would settle it: exact Λ at z = 31, 37 (see NOT REACHED), or an analytic mean value on Σ_e Σ_a |Θ_e(a)|/a′. - **(ii) REC(ℓ), the Gaussian-units form: sup ≤ C·rms·z^ℓ, ℓ < 1.26645.** F4 is ℓ = 1/2 + o(1) and is 2.5× stronger than needed; the measured truth is ℓ̂ = 0.891 ± 0.213 (first fit of sup/rms's own exponent — above F4's 0.54–0.56 finite-z reading but far inside the budget). Equivalently (lnW)^A with A < 1.26645. - **(iii) The moment form: m_{2k} ≤ C^k(2k−1)!!·m₂^k for k up to k(z) ~ 0.65·z/lnz** (any uniform C; with constants ignored 0.40·z/lnz). Measured true to k = 8 at every walkable level with C = 1. This form is F4 in disguise at its optimum (§4) — its value is that each finite rung is a mean value (a 2k-fold correlation sum), and the first unproven rung is still ⟨ρ̃⁴⟩ (attack-rhoms reading 5). What this pass ruled out: the degree/mode-count bound (§1, measured and ceiling-capped), every absolute-value route through Vabs (§2, L4: structurally n·polylog), sub-polynomial-in-W losses as a target class (§3: W^ε always dies — the target must be polynomial in z, not in W). Chaining stays CLOSED (`REFUTED.md`), untouched. ## 6. READINGS Producer readings 1–7 carry the *(grades in the script tail)*: 1 MEASURED (mode-count verdict), 2 PROVEN finite-z + MEASURED growth (Λ_V clears all ten, first to survive 47), 3 MEASURED new (exact Λ, β₂ through the error bar; sup/rms = 0.891 ± 0.213), 4 MEASURED (decay law, ℓ¹ vs ℓ² mass split, N_eff), 5 DERIVED + MEASURED (A/ε charts), 6 MEASURED new (C_k to k = 8; ladder⇒F4 equivalence), 7 INFERRED (the weakest-form triple). All HELD for one adversarial pass before anything moves to a live-layer surface. ## 7. NOT REACHED - **Exact Λ at z = 31, 37** — the decisive points for MV(α)'s measured slope (the 13..29 bar straddles β₂). Cost at 31 is ~1e10 table ops under this file's method (the per-conductor cos/sin tables cap at maxE = qmax); a factored-table or FFT-over-divisor-lattice pass would reach it. - **An analytic attempt on MV(α)** — the named object is the ℓ¹-with-weights mean value Σ_e Σ_{a′≤e/2}|Θ_e(a)|/a′; Theorem A's machinery bounds the ℓ² analogue; nothing here was attempted on the ℓ¹. - **The primitive-restricted cosecant sum** (Σ over gcd(a,e)=1 only) — would sharpen Λ_V by ~φ(e)/e per conductor; the full-a bound was used. - **The finite-z window past 47 for Λ_V** — its death point is just past 47 by the fit; the exact crossing was not located (needs the sweep at the next few primes, cheap but out of scope tonight). - **Whether s < 3 moves any chart** (inherited from attack-rhoms NOT REACHED; everything here is s = 3.0). ## 8. GATE `node research/qc.js` after producer + this report: TOTAL 1 — the standing census red (`xchan-at37-01-census.js`, mid-flight, pre-existing, does not name this producer or this file). `qc/embed.js --check` on the producer: code-sha256 matches, out-sha256 matches, body bit-honest (two runs, identical out-sha — deterministic); advisory: 2 readings figures are prose arithmetic over printed values (the known false-positive floor). *History and superseded claims: `research/history/CHANGELOG.md`.*