# The quartic: the off-diagonal measured at 1–10%, the whole question moved onto one explicit cosecant sum, and a new increment rung that clears z = 47 **attack-rhoms-01's named stopping object — the fourth moment ⟨ρ̃⁴⟩, "the quartic gcd-lattice correlation (mode-resonance sum), for which no Theorem-A analogue exists" — is attacked. The expansion is written and its bookkeeping brute-force verified; the diagonal/off-diagonal split is measured exactly at z = 13..23; the resonance condition is proved to be a local covering condition and the whole quartic question is transferred, by two exact identities, onto a Λ(4) inequality for one explicit non-negative-coefficient cosecant sum, whose constant is then computed and is flat; and the budget is re-priced through a new recovery rung. Producer: `research/attack-quartic-01.js` (embedded via `qc/embed.js`; all controls pass; 9.4 s, deterministic). Five things came out.** **One: the genuine quartic correlation is small, and it is not what makes ρ̃ sub-Gaussian (§1).** Inclusion–exclusion over the three pairings is exact: m₄ = DIAG + OFF with DIAG = 3m₂² − 3Q + ĉ(1/2)⁴, Q = Σ_ξ|ĉ(ξ)|⁴. Measured over complete periods, **OFF/m₂² = −0.0454, −0.2767, +0.0101, −0.0446 at z = 13, 17, 19, 23** — the off-diagonal is 0.3%–11% of the diagonal and mostly negative. The measured shortfall from Gaussian (m₄/m₂² = 2.5086, 2.3210, 2.8930, 2.8958) is carried almost entirely by the **self-pairing term** 3Q/m₂² = 0.4460, 0.4023, 0.1170, 0.0596, which is decaying. So the reading of attack-rhoms' sub-Gaussian column changes: ρ̃ is not sub-Gaussian because the quartic resonances cancel, it is sub-Gaussian because a finite-W self-pairing correction is still visible, and that correction is going away. m₄/m₂² → 3 is the shape the measurement supports. **Two: the resonance condition is a local covering condition, and that is why the second moment was easy (§2).** Lemma Q2: Σa_i/e_i ∈ ℤ holds iff for every prime p | lcm(e_i), Σ_{i: p|e_i} a_i(L/e_i) ≡ 0 mod p — so a prime dividing **exactly one** conductor admits no solution, and every prime of the lcm divides at least two of the four. At two modes this forces e₁ = e₂, which is exactly why MS3's one-conductor cosecant identity closes the mean square and has no four-mode analogue. The lemma is checked positionwise: with A_p the part of ρ̃ carried by conductors divisible by p and B_p the p-fold period average, ⟨A_pB_p³⟩ = 0 at every prime and every level, worst 5.3e-17 to 1.1e-15. **Three: the whole question moves onto one explicit function (§3).** Put F*_e(y) = Σ*_a cos(2πay/e)/sin(πa/e) and H_V(y) = Σ_e Vabs(e)F*_e(y). Two exact identities, both machine-verified: ⟨H_V²⟩/4 = B₂ᴶ/12 = MS3′ (rel 2.4e-15 .. 1.5e-13 against attack-b2mean-01) ⟨ρ̃⁴⟩ ≤ ⟨H_V⁴⟩/16 (mode-by-mode |ĉ| ≤ Vabs·csc/2) The second is the first upper bound on ⟨ρ̃⁴⟩ this corpus states at all, and the first that is not the trivial sup²·m₂ wearing proven clothes: 2.90e3, 6.52e4, 2.42e5, 6.42e5 at z = 13..23, which beats (H_V(0)/2)²·MS3′ — the best proven sup times the best proven mean square — by 7.3×, 13.9×, 34.2×, 91.9×, a saving that grows at 4.5026 ± 0.7662 per lnz. And the constant that decides the route, **K_V = ⟨H_V⁴⟩/⟨H_V²⟩² = 16.4456, 24.1645, 25.3180, 19.7404, is flat** (slope 0.3926 ± 0.5192 on four points), i.e. m₄ ≤ K_V·MS3′² with K_V measured bounded. The peak share of ⟨H⁴⟩ at y = 0 — the spike a non-negative- coefficient majorant risks being ruined by — **falls** 0.3761 → 0.0172. **Four: the two fronts have NOT converged, and the stopping term is named (§4).** Running the same ceiling with Θ*(e) in place of Vabs(e) gives K_T = 24.722, 28.317, 31.321, 26.865, i.e. K_T/K_V = 1.17–1.50: the Vabs → Θ* sign-discard, which `attack-b2mean-01` names as its stall and measures at a flat ~600× under the e² weight, **cancels in the Λ(4) ratio**. b2mean's signed class-sum mean value is not on the quartic's critical path. The quartic's own stopping term is different and new: **⟨H_V⁴⟩ ≤ polylog· ⟨H_V²⟩², a weighted Λ(4)/additive-energy inequality over the conductor lattice** — equivalently the 4-dimensional Dedekind–Rademacher-type mean value (1/L)Σ_y Π F*_{e_i}(y) summed against ΠVabs(e_i) over the non-pairing covering classes. All three off-the-shelf imports are refused at the hypothesis, each checked here rather than asserted: Rudin's Λ(4)/B₂ criterion is **vacuous** (the mode set is W-saturated, N = W−1 exactly at z = 13, 17, 19, so its bound is the trivial (W−1)m₂²); Young's convolution inequality needs no hypothesis but its constant (Λ/rms)² = 87.7 → 1842 grows at z^{5.2}; and Zagier's higher- dimensional Dedekind sums, read at page images, require one common modulus and cotangent kernels and prove exact evaluations, reciprocity and denominator bounds — no size bound, no mean value over a family. **Five: what a proven quartic buys, and a new rung that beats the bare ladder (§5).** The increment argument of attack-rhoms' R1 generalises: if the sup is attained at y₀ then |ρ̃| ≥ S − C_L t on one side, so W·m_{2k} ≥ S^{2k+1}/ ((2k+1)C_L) and **sup ≤ ((2k+1)C_L·W·m_{2k})^{1/(2k+1)}** — the increment turns the bare ladder's e^{θ/(2k)} into e^{θ/(2k+1)}. At k = 2, R2 = (5C_L·W·m₄)^{1/5} **clears smax at all ten levels z = 13..47** (4.805e4 ≤ 1.577e5 at z = 47), where M2 = (W m₄)^{1/4} fails (2.67e5) and R1 died at 37; in the fully proven grade m₄ ≤ 25·MS3′² it clears z = 13..43 and misses at 47. It is still a Bonferroni window: the death scan puts R1 at z = 41, M2 at 53, R2 at 67, R3 at 89. **A proven quartic buys a wider finite-z window and nothing asymptotic**, because the composed exponent is 2kc/(2k+1) + θ(z)/((2k+1)lnz) and θ/lnz ~ z/lnz for every fixed k. Against MV(α) the verdict is INDEPENDENT with an asymmetry: MV(α < β₂) implies RML(α) and the theorem outright, a proven quartic implies only the k = 2 rung. --- ## 0. CUSTODY Everything measured is in `research/attack-quartic-01.js` (`qc/embed.js --check`: code-sha256 matches, out-sha256 matches, body bit-honest; the check's re-run reproduced the out-sha — deterministic, 9.4 s). Division of labour per the standing compute rule: - **Recomputed (cheap, controls in-pass):** the closed-form ⟨ρ̃²⟩ and the full-period walker at z = 13..23 — m₂ MATCHES `attack-rhoms-01`'s embedded column to rel ≤ 4.4e-7 (its 6-digit print), m₄/m₂² to rel ≤ 5.5e-5 against its 2.509/2.321/2.893/2.896, sup to rel ≤ 7.6e-7, closure drift ≤ 1.1e-8. The subset sweep (Vabs, B, B₂, B₂ᴶ) re-run; ⟨H_V²⟩/4 reproduces `attack-b2mean-01`'s B₂ᴶ/12 to 1.5e-13. The exact ℓ¹ norm Λ reproduces `attack-f4weak-01`'s 9.801 / 25.157 / 60.031 / 121.250 at all four levels from independent code, and the mode counts N reproduce its saturation column (N = W−1 at z ≤ 19, cover 0.2731 at 23). - **New measurements:** Q = Σ|ĉ|⁴ and ĉ(1/2); DIAG and OFF; the per-prime |T_p| profile and the smooth-core cascade; ⟨H_V²⟩, ⟨H_V⁴⟩, ⟨H_T²⟩, ⟨H_T⁴⟩, K_V, K_T, the peak shares; the primitive-restricted ℓ¹ bound H(0)/2; the R_k rows. - **Cited (custody-bound, not recomputed):** ⟨ρ̃²⟩ and C_L at z = 29..47, the sup and m₆ columns, and the whole S4 budget skeleton from `attack-rhoms-01.js`; MS3′/truth from `attack-b2mean-01.js` §4; Λ/rms's slope 2.608 ± 0.179 and the mode-count closure z^{8.174} from `attack-f4weak-01.js`; β₂ = 4.26645 per `paper/beta2-note.md`. - **Machinery:** the walker is `attack-rhoms-01.js`'s `walkStats` with the period stored; the subset sweep is its `sweepB`; the mode enumeration is `attack-f4weak-01.js`'s `l1exact` restructured to return Q and Θ*. Same custody practice as those files. - **Estimator and instrument controls, all before use:** OLS returns 3.700000 on a known z^3.7 truth; Lemma J verified to 2.9e-14 over e = 2..200; the F_d recursion verified against the direct O(d²) sum to 9.8e-13 over d = 2..60 with period closure 7.5e-15; the two cotangent identities quoted in the Zagier verdict verified to 7.1e-15 and 2.1e-12; and **the Wick bookkeeping brute-forced** on a synthetic function on ℤ/30 over all 24389 quadruple slots — enumerated pairing part against the inclusion–exclusion formula rel 6.3e-15, enumerated off-diagonal against m₄ − DIAG rel 1.7e-15. - **Read at source this session:** Zagier, *Higher dimensional Dedekind sums*, Math. Ann. 202 (1973) 149–172, at page images (pp. 149–151 for Eq. (1) and the plan of the paper, 166–168 for the four-dimensional tables and the cotangent identities). Nothing on Dedekind sums or Λ(4) existed on disk before this file; the disk was checked first. Units: ρ̃, its moments, H, Θ, Vabs are dimensionless (weight counts); W, qmax, e, y are positions; every strength comparison is an exponent base z. WIDTH: the file RUNS at z ≤ 23, where W = P(23) = 9699690 and qmax = 881790, so every index and every product of two moduli (< 7.8e11) is far below 2^53; the z ≥ 31 rows of §5 are pricing arithmetic only and every W-dependence there is exp of a log-space sum off lnW = θ(z). ## 1. THE EXPANSION, AND THE SPLIT ρ̃'s Fourier coefficient at the reduced frequency a/e is ĉ(a/e) = i·e(a/2e)·Θ_e(a)/(2 sin(πa/e)), so the period average of ρ̃⁴ is literally a 4-mode resonance sum over the coefficient lattice: ⟨ρ̃⁴⟩ = Σ_{ξ₁+ξ₂+ξ₃+ξ₄ ≡ 0, ξ_i ≠ 0} ĉ(ξ₁)ĉ(ξ₂)ĉ(ξ₃)ĉ(ξ₄) = (1/16) Σ' (−1)^σ Π_i Θ_{e_i}(a_i) / Π_i sin(πa_i/e_i), σ = Σ a_i/e_i ∈ ℤ (the four half-shifts e(a/2e) collapse to (−1)^σ). The three pairing sets A = {ξ₂ = −ξ₁, ξ₄ = −ξ₃}, B, C each contribute m₂²; each pairwise intersection is the single-frequency locus contributing Q = Σ_ξ|ĉ(ξ)|⁴; the triple intersection forces 2ξ₁ = 0 and is the one quadruple (1/2,1/2,1/2,1/2). **Lemma Q1 [PROVEN; bookkeeping brute-forced in the producer S0].** m₄ = DIAG + OFF, DIAG = 3m₂² − 3Q + ĉ(1/2)⁴, and since Q ≥ 0 and ĉ(1/2)⁴ ≤ Q, m₄ ≤ 3m₂² + OFF. The Gaussian constant 3 is free; the entire quartic question is the off-diagonal. Measured exactly (producer S1, S5): ``` z m4/m2^2 DIAG/m2^2 OFF/m2^2 3Q/m2^2 |OFF|/DIAG 13 2.5086 2.5540 -0.0454 0.4460 1.78e-2 17 2.3210 2.5977 -0.2767 0.4023 1.07e-1 19 2.8930 2.8830 +0.0101 0.1170 3.49e-3 23 2.8958 2.9404 -0.0446 0.0596 1.52e-2 ``` **Reading.** The genuine quartic gcd-lattice correlation is a 0.3%–11% effect and mostly negative; the self-pairing term is four to ten times larger and is falling with z. attack-rhoms reading 6's sub-Gaussian column is therefore a finite-W artefact of Q, not evidence of quartic cancellation, and the honest extrapolation is m₄/m₂² → 3 rather than to something below it. [MEASURED] ## 2. THE RESONANCE LATTICE IS A COVERING PROBLEM **Lemma Q2 [PROVEN; verified positionwise].** Let e_i | W with W squarefree, gcd(a_i, e_i) = 1, L = lcm(e_i). Then Σ_i a_i/e_i ∈ ℤ iff for every prime p | L, Σ_{i: p|e_i} a_i(L/e_i) ≡ 0 (mod p). Since v_p(L/e_i) = 1 − v_p(e_i), a prime dividing e_i does not divide L/e_i, and p ∤ a_i; so a prime dividing **exactly one** of the four admits no solution. Every prime dividing any conductor of a resonant quadruple divides at least two of them. - **(Q2a)** At two modes the condition forces e₁ and e₂ to have the same prime support, i.e. e₁ = e₂. That is precisely why ⟨ρ̃²⟩ is conductor-diagonal and why MS3/Lemma J close it with one exact cosecant evaluation per conductor. There is no four-mode analogue because at four modes the covering can be inconsistent across primes (p pairing 1–2 while q pairs 1–3). - **(Q2b)** With A_p the part of ρ̃ carried by conductors divisible by p and B_p(y) = (1/p)Σ_{k1} Vabs(e) F*_e(y), whose Fourier coefficients Vabs(e)/sin(πa/e) are non-negative and dominate 2|ĉ(a/e)| mode by mode, gives two exact identities: **Lemma Q3 [PROVEN; both verified].** (a) ⟨ρ̃²⟩ ≤ ⟨H_V²⟩/4 = Σ_e Vabs(e)²J₂(e)/12 = B₂ᴶ/12 = MS3′, (b) ⟨ρ̃⁴⟩ ≤ ⟨H_V⁴⟩/16. (a) follows from Q2a plus Lemma J and is reproduced numerically to rel 2.4e-15, 3.6e-15, 7.3e-16, 1.5e-13 against b2mean's B₂ᴶ — the control that guards every number in this section. (b) is the mode-by-mode triangle. **Computation.** F*_e is expensive per conductor but the whole sum is cheap through Möbius on the conductor: with F_d(y) = Σ_{a=1}^{d−1} cos(2πay/d)/ sin(πa/d) = Σ_{k|d, k>1} F*_k(y), H_V(y) = Σ_{d|W, d>1} c_d F_d(y), c_d = Σ_{m|W/d} μ(m) Vabs(dm), and F_d obeys the exact first-order recursion, derived here from the classical odd-multiple sine sum Σ_{a=1}^{d−1} sin(πam/d) = cot(πm/2d) for odd m: F_d(y) − F_d(y+1) = 2 cot(π(2y+1)/(2d)), F_d(0) = Σ_{a=1}^{d−1} csc(πa/d). So every F_d costs O(d) and H_V costs O(τ(W)·W). Verified against the direct O(d²) sum to 9.8e-13 over d = 2..60, with period closure 7.5e-15. **The result [MEASURED]** (producer S3, S3b): ``` z /4 = MS3' /16 = ceiling ceiling/m4 K_V K_T K_T/K_V peak share 13 1.32831e+1 2.90167e+3 9.65e+2 16.4456 24.722 1.5033 0.3761 17 5.19416e+1 6.51940e+4 4.51e+3 24.1645 28.317 1.1718 0.1548 19 9.78456e+1 2.42389e+5 3.42e+3 25.3180 31.321 1.2371 0.0581 23 1.80332e+2 6.41952e+5 3.48e+3 19.7404 26.865 1.3609 0.0172 ``` - **K_V is flat**, d lnK_V/d lnz = 0.3926 ± 0.5192 on four points. The statement m₄ ≤ K·MS3′² with K bounded is what the measurement supports. - **The peak share falls**, 0.3761 → 0.0172. H has non-negative coefficients and therefore spikes at y = 0 with H(0) = ‖Ĥ‖₁; at z = 13 that spike alone carries 38% of ⟨H⁴⟩ and would have forced K to grow like ‖Ĥ‖₁⁴/(W‖Ĥ‖₂⁴). It does not: W outruns the ℓ¹ norm. - **The ceiling's own Wick split** shows where the ceiling's mass is: OFF_H/⟨H²⟩² = 13.66, 21.29, 22.38, 16.78 against a diagonal of at most 3. Discarding the resonance signs costs roughly a factor 10²–10³ on the off-diagonal (compare §1's |OFF|/m₂² ≤ 0.28) and the total is still only ~20⟨H²⟩². That is the whole reason the route survives. **The currency, stated plainly.** In the truth currency the ceiling is 2.420e3, 1.046e4, 9.883e3, 1.007e4 times m₂², so **this does not prove m₄ ≪ polylog·m₂²** — the squared mean-square looseness (MS3′/m₂)² = 146..510 sits inside it. What it proves, at each measured z, is m₄ ≤ K_V·MS3′² with K_V measured flat. That is the weaker and sufficient form: §5's consumers all pay the proven mean square anyway, so a polylog constant against the proven bound is worth exactly as much as one against the truth. ## 4. WHERE THE PROOF STOPS, AND WHETHER IT IS b2mean's OBJECT **The stopping term.** ⟨H_V⁴⟩ ≤ K·polylog·⟨H_V²⟩² — a **weighted Λ(4) / additive-energy inequality** for one explicit non-negative-coefficient cosecant sum on ℤ/W, equivalently the 4-dimensional Dedekind–Rademacher-type mean value (1/L)Σ_{y mod L} Π_i F*_{e_i}(y) summed against Π_i Vabs(e_i) over the non-pairing covering classes of Lemma Q2. The pairing classes are already free: they contribute exactly 3⟨H²⟩² − 3Q_H + Ĥ(1/2)⁴ ≤ 3⟨H²⟩². **Is it b2mean's object? No, and the check is direct.** Running the same ceiling with Θ*(e) — the sharper coefficient, whose gap to Vabs(e) is exactly the sign-discard `attack-b2mean-01` §3 names as its stall and measures flat at 4.6e2–7.6e2× under the e² weight — gives K_T = 24.722, 28.317, 31.321, 26.865, i.e. K_T/K_V = 1.5033, 1.1718, 1.2371, 1.3609. The ~600× cancels between numerator and denominator of the Λ(4) ratio, because it appears to the fourth power above and the second power squared below. **The two fronts have not converged.** b2mean's signed class-sum mean value sharpens the *level* of the quartic ceiling exactly as much as it sharpens MS3′, and does nothing for the ratio; the quartic's own missing theorem is a four-fold resonance mean value that nothing in the corpus touches. **The three classical imports, hypotheses read, all refused.** 1. **Rudin's Λ(4)/B₂ criterion** (W. Rudin, *Trigonometric series with gaps*, J. Math. Mech. 9 (1960) 203–227; standard finite form). Hypothesis: the frequency support E has bounded representation function r_E(ξ) ≤ B for ξ ≠ 0; conclusion ‖f‖₄⁴ ≤ (1+B)‖f‖₂⁴. Our support is **W-saturated** — `attack-f4weak-01` §1 measured cover 1.0000 at z ≤ 19 and the mode counts here reproduce N = W−1 exactly at z = 13, 17, 19 — so B = W−2 and the criterion returns the trivial m₄ ≤ (W−1)m₂², exponential in z. VACUOUS. 2. **Young's convolution inequality**, ‖ĉ∗ĉ‖₂ ≤ ‖ĉ‖₁‖ĉ‖₂, i.e. m₄ ≤ (Λ/rms)²·m₂². No hypothesis, but the constant is measured 87.72 → 1842 over z = 13..23 and grows at z^{5.2} (f4weak's d ln(Λ/rms)/d lnz = 2.608 ± 0.179). It beats the S3 ceiling in truth currency at these small z and is overtaken around z ~ 30 on the measured slopes; only the S3 constant is flat. REFUSED as an asymptotic route. 3. **Zagier's higher-dimensional Dedekind sums** (Math. Ann. 202 (1973) 149–172, read at page images this session). Eq. (1): d(p; a₁..a_n) = (−1)^{n/2} Σ_{k=1}^{p−1} cot(πka₁/p)···cot(πka_n/p), with p a positive integer, **every a_i prime to p**, **n even**, and **one common modulus p**. Three failures, each independently fatal: (i) our four factors carry four different conductors and the sum is over ℤ/lcm, not one p; (ii) the kernel is cosecant, and csc t = cot(t/2) − cot t (verified to 7.1e-15) turns it into a half-argument cotangent sum against cos — the plain cot-against-cos sum is 0 by antisymmetry (verified to 2.1e-12) — which is outside (1); (iii) the paper's content is exact evaluation: rationality, the reciprocity law generalising Rademacher's, and **denominator** bounds (plan of the paper p. 151; the four-dimensional tables p. 166 and the cotangent identities p. 168). It proves no upper bound on |d| and no mean value over a family of tuples. The shape is right and the theorems are the wrong kind: **there is no classical cancellation theory here to import.** ## 5. THE RE-PRICED BUDGET **A new rung, R_k [DERIVED; k = 1 is attack-rhoms-01's R1].** If S = sup|ρ̃| is attained at y₀ then |ρ̃(y₀+t)| ≥ S − C_L·t for 0 ≤ t ≤ S/C_L on one side, with C_L = max_x|M − cc(x)| the exact bounded-increment constant, so W·m_{2k} ≥ Σ_{t