window_opt: optimizing the window in the 7.5(g) distinct-zeros count

Exploratory study under hunts/rogue_frontier/. Per hunts/README.md, nothing here is a promoted result; every number below carries its rung on the repo's certainty ladder, stated inline and collected in section 8.

Subject. The 10 Aug 2026 preprint "More than two thirds of the zeros of the Riemann zeta function lie on the critical line" (single author, not peer reviewed) proves in its section 7.5(g), under RH, the distinct-zeros bound

Nd(T)/N(T) >= 1/2 + (2 m2 - m3)/18 + (4/9)(19/27) = 0.85082...

where m_k = m_k(1, v) are spectral moments of the windowed sine-kernel Gram matrix over the sine process, and the window is v(s) = cos(8s/5) on [-1/2, 1/2], for which the paper prints 2 m2 - m3 = 0.68524... The weight psi(m) is LP-optimal, but the window is a hand-picked guess. Any gain delta in F(v) := 2 m2 - m3 over an admissible window lifts the constant by delta/18. This study re-derives m2 and m3 as functionals of v, validates the derivation two independent ways, tests whether cos(8s/5) is optimal, and optimizes the window.

Headline. cos(8s/5) is not a critical point of F. The optimum over even v >= 0 is an interior critical point with

sup F = 0.6852870321770 (measured; three parametrizations agree)

versus 0.685243875537318 for the paper's window: a gain of 4.316e-5. The clean rational window

v*(s) = 1 - (1467/1000) s^2 + (1159/1000) s^4

achieves, in exact rational arithmetic,

F(v*) = 2245228120295149280/3276332462159207451 = 0.685287023288068,

within 8.9e-9 of the measured sup, and the RH-conditional constant becomes

1/2 + F(v*)/18 + (4/9)(19/27) = 50176758585216887915/58973984318865734118 = 0.850828702939872 (15 digits; exact rational)

against the paper's 0.850826305842608, a lift of 2.3971e-6 (enclosure-checked in both ball backends; see enclose.py output). The improvement is real but small: it moves the sixth decimal of the proportion.

1. Setup and derivation

Points x_j are the unit-density sine process, S(x) = sin(pi x)/(pi x), with determinantal correlations rho_2 = 1 - S^2 and rho_3 = 1 - S12^2 - S13^2 - S23^2 + 2 S12 S13 S23. The windowed Gram kernel is H_{jj'} = K_v(x_j - x_{j'}), K_v(x) = int_{-1/2}^{1/2} v(xi) e^{2 pi i x xi} d xi, and m_k(1, v) = lim E tr H^k / (N l1^k) with l1 = K_v(0) = int v.

Expanding E tr H^k over coincidence patterns of the index tuple and moving every distinct-point integral to the Fourier side (Parseval; the transform of the product S K_v is the convolution 1_B * v with B = [-1/2, 1/2]):

E tr H^2 / N = l1^2 + [int v^2 - int W^2], W := 1_B * v, E tr H^3 / N = l1^3 + 3 l1 [int v^2 - int W^2]

• [int v^3 - 3 int (tri*v) v^2 + 2 int W^3],

tri := 1_B * 1_B. The three rho_3 pieces map as: the constant term gives int v^3; the three -S^2 terms each give -int (tri*v) v^2 (equivalently -int (1_B*v)(1_B*v^2)); the +2 S12 S13 S23 term gives +2 int W^3. Hence with

D2 := (int v^2 - int W^2)/l1^2, D3 := (int v^3 - 3 int (tri*v) v^2 + 2 int W^3)/l1^3,

m2 = 1 + D2, m3 = 1 + 3 D2 + D3, F = 2 m2 - m3 = 1 - D2 - D3.

The m2 formula agrees with the coordinator's sketch; the m3 formula is the new derivation. Both are validated below. Everything reduces to the antiderivatives V0(x) = int_0^x v and P0(x) = int_0^x t v(t) dt through

W(eta) = V0(1/2) + V0(1/2 - eta) (0 <= eta <= 1), (tri*v)(s) = l1 - [2 s V0(s) - 2 P0(s) + 2 P0(1/2)] (|s| <= 1/2),

so for polynomial v every quantity is an exact rational.

Per-term anchor at v = 1: int v^2 = 1, int W^2 = int tri^2 = 2/3, int v^3 = 1, int (tri*v) v^2 = 2/3, int W^3 = 1/2, giving D2 = 1/3, D3 = 0, m2 = 4/3, m3 = 2: exactly the published sine-Gram moments m_k(1) = 1, 4/3, 2, 13/4 quoted in the paper's section 7.5(f).

2. Validation evidence

Four mutually independent routes agree (functional.py battery, all pass):

1. Float engine (vectorized Gauss-Legendre, exact for polynomial v up to rounding): cos(8s/5) gives m2 = 1.328018448999629, m3 = 1.970793022461940, F = 0.685243875537319. All printed digits of the paper's 0.68524... are reproduced.
2. Symbolic closed forms (sympy, antiderivative identities verified by differentiation): F(cos(8s/5)) = (878 cos(16/5) - 450 sin(8/5) - 495 sin(16/5) - 2960 cos(8/5) + 2082) / (1200 (cos(8/5) - 1)^2) = 0.685243875537318196018420884208 (30 digits), m2 = 1.32801844899962925344205522171, m3 = 1.97079302246194031086568955921.
3. Exact rational route (Fraction polynomial arithmetic, no sympy) equals the sympy route exactly on the rational window, and the float engine matches both to < 1e-14.
4. Windowed circular model, exact finite-N CUE (crosscheck_finiteN.py): using the exact joint trace moments E prod Tr U^{h_j} from ../sine_gram/exact_finite_N.py (read-only import), the windowed lattice sum E tr H^k = N^{-k} sum prod v(m_i/N) E prod Tr U^{h_j} at N = 21, 25, 29, 33, 37, 41, extrapolated in 1/N (degree-4 fit):

This model shares no step with the continuum derivation (no rho_3, no Parseval, no quadrature: integer lattice counts), so the agreement at two different windows is a genuine cross-check of the m3 formula, including its boundary structure at band lambda = 1.

Grade of the closed forms after this: hardened (independent routes agree). Had 0.68524 failed to reproduce, the protocol was to stop and report; it did not fail.

3. Euler-Lagrange test at cos(8s/5)

F is scale invariant (degree-0 homogeneous), so at a critical point over {v >= 0} the first-variation field g(s) = delta F / delta v(s) vanishes on the support of v, with no multiplier (the Euler identity int g v = 0 holds identically; the engine reproduces it to 4e-15, and g was verified against finite differences in four directions to 7 digits).

At v = cos(8s/5): max|g| = 5.64e-2, ||g||_2 = 2.01e-2. The residual is structured (positive at s = 0, negative near s = 0.35, positive at the endpoint), so cos(8s/5) is not a critical point of F. For contrast the optimizer's window has max|g| = 3.1e-9.

Within its own one-parameter family v = cos(c s): dF/dc at c = 8/5 equals +1.4892581773e-4 (exact symbolic derivative), so 8/5 is not even family-critical, but it is close: the family optimum is c* = 1.602374198452608 with F(c*) = 0.685244052480639, only 1.77e-7 above c = 8/5. The paper's guess was an excellent cosine but the cosine family itself is the binding restriction: the full-space gain is 244 times the in-family gain.

4. Optimization outcome

Three parametrizations, optimize.py:

• (a) Squared Chebyshev (v = w^2, w an even Chebyshev series, positivity free): BFGS with the analytic gradient, 13 starts (projection of the paper window + 12 random). All 13 converge to the same point, F = 0.685287032177, stable under basis growth J = 4..12 to 12 digits (J = 4: ...31978; J >= 6: ...32177).
• (b) Piecewise-linear v >= 0 on 41 knots with an exact lattice-aligned evaluator: L-BFGS-B, 8 starts, all converge to F = 0.685287031812, which is the same optimum up to the pw-linear representation bias (~4e-10). Cautionary note recorded: against the smooth-panel float quadrature instead of the exact evaluator, the optimizer "found" F = 0.685777 by exploiting kink quadrature error; the exact evaluator shows that window is actually worse than cos (F = 0.684814). Optimizing against a quadrature is only safe when the quadrature is exact for the search space.
• (c) Cosine family and low-degree even polynomials: quartic 1 + q1 s^2 + q2 s^4 reaches F = 0.685287023290 at q = (-1.46701891, 1.15904325); adding s^6 reaches 0.685287031922.

The optimum is an interior critical point of the cone {v >= 0}: strictly positive (v(1/2)/v(0) = 0.70539), strictly decreasing in |s|, even, and smooth; the positivity constraint is inactive. Normalized profile:

s 0 0.1 0.2 0.3 0.4 0.5 v*/v0 1 0.98541 0.94307 0.87726 0.79499 0.70539 cos ref 1 0.98723 0.94924 0.88699 0.80210 0.69671

sup F = 0.6852870321770 (measured; single basin found by every start in every parametrization; no claim beyond the searched classes).

5. The rational window and exact arithmetic

Rounding the quartic to q1 = -1467/1000, q2 = 1159/1000 costs 8.9e-9 of F. For v*(s) = 1 - (1467/1000) s^2 + (1159/1000) s^4, exact rational arithmetic (Fraction route, confirmed by sympy exactly, cross-checked at finite N):

m2 = 95695869320/72057314637 = 1.328052117985286 m3 = 6457052410893815440/3276332462159207451 = 1.970817212682504 F = 2245228120295149280/3276332462159207451 = 0.685287023288068

Exact admissibility facts (rational arithmetic, enclose.py): 2 q1 + q2 = -71/40 < 0, so v' = s(2 q1 + 4 q2 s^2) < 0 on (0, 1/2]: v is strictly decreasing in |s|; v(1/2) = 11291/16000 > 0: v is strictly positive; v is even with sup v = v(0) = 1; phi = sqrt(v) is smooth on the closed interval since v >= 11291/16000.

Strict improvement, enclosure-checked in both ball backends (Arb prec 200 and mpmath.iv dps 60, which agree and contain the 30-digit references):

F(v*) - F(cos(8s/5)) = 4.3147750749594648584e-5 (ball, both backends)

6. The new constant

Keeping the paper's LP-optimal weight psi(m) = m/2 + m^2/9 - m^3/18 + (4/9) 1_{m=1} (re-derived: the vertex is pinned by psi(1) = psi(2) = psi(3) = 1 plus the boundary admissibility beta = -2 gamma, so it does not depend on the window; an LP re-solve at the new (m2, m3) returns the same vertex), and the BHB simple-zeros input 19/27 unchanged:

Nd/N >= 1/2 + F(v*)/18 + (4/9)(19/27) = 50176758585216887915/58973984318865734118 = 0.850828702939872078905... (exact rational)

versus the paper's 0.850826305842608212536... The lift is

delta = 2.3970972638663694e-6 (ball, both backends).

At the measured sup the constant would be 0.8508287034337 (measured grade only). Both round to 0.85083 at five decimals, as does the paper's value at its sixth digit: the honest statement is that the headline "more than two thirds" and even "0.8508..." are untouched; the sixth decimal moves.

7. Admissibility analysis (task 4)

What the paper's proof imposes on the window, with sources (line numbers refer to the OCR text of the preprint; quotes reassembled where the OCR interleaves display math):

1. The window class, section 7.1 (lines 3191-3196): "Nothing in Sections 4-5 used that phi is flat-topped, only: phi in C^2_c, even, 0 <= phi <= 1, supp phi = [-L/2, L/2], phi nonincreasing in |u| (so ||phi'||_1 <= 2, ||(phi^2)'||_1 <= 2), and ||phi''||_1, ||(phi^2)''||_1 << 1." With "phi^2(u) = v(u/L)", the window v must be: even, nonnegative (structural: v = phi^2, and H must be PSD for the majorisation step), bounded (0 <= v <= 1, WLOG by scale invariance of F), nonincreasing in |s| (used only to bound the variation norms), with sqrt(v) of class C^2 and controlled second-derivative norms.
2. Endpoint vanishing is glossed by the paper itself. A strict reading of phi in C^2_c with supp phi = [-L/2, L/2] forces phi(+-L/2) = 0, i.e. v(+-1/2) = 0. But both of the paper's own windows violate this: section 7.1's maximiser "v*_lambda(s) = cos(sqrt(2) lambda s) > 0 (|s| <= 1/2)" (line ~3230, stated as attaining "c*lambda := sup{v>=0} c_lambda(v)"), and 7.5(g)'s "v(s) = cos(8s/5) 1_{|s|<=1/2}" with v(+-1/2) = cos(4/5) = 0.6967 > 0. Remark 4.3 (sharp cutoff: "some tapering is necessary") shows some smoothing at the edge is genuinely needed. The implicit repair is standard: taper over a width delta L (then ||phi''||_1 = O(1/(delta L)) -> 0 for fixed delta as L grows, so the class conditions hold), and let delta -> 0 after T -> infinity; F is continuous in the taper. Measured here (C^2 ramp, exact pw evaluator):

delta: 0.04 0.02 0.01 0.005 F*-F: 2.47e-2 1.20e-2 5.91e-3 2.93e-3 (slope ~0.59)

The loss is O(delta), so sup over the strictly admissible subclass equals the endpoint value; the bound survives in the lim inf form the theorem already carries. Our window needs exactly this argument and no more; it inherits the same status as the paper's own windows.

1. The band endpoint lambda = 1 and the triple-correlation input. 7.5(g) (line ~3819): "Under the Riemann hypothesis, the third trace tr G^3 (equivalently, the triple correlation of zeros in the band lambda < 1) is available as a theorem [Hej94, RS96]". The k = 3 test function built from v on [-lambda/2, lambda/2] has Fourier support of total spread <= 2 lambda, and the Hejhal / Rudnick-Sarnak theorems (under RH) cover the open range < 2, i.e. lambda < 1 strictly; at lambda = 1 the support touches the boundary (a null set, since v(+-1/2) != 0 puts mass at the edge of the closed band). The certificate is evaluated at m_k(1, v), i.e. at lambda = 1. Remark 6.1 states endpoint admissibility only for the Theorem 5.8 machinery ("A reader who prefers to keep a power saving everywhere may take lambda < 1 fixed and let lambda -> 1-; the limits ... are unaffected"); for the third trace the corresponding lambda -> 1- limit (rescale v_lambda(s) = v(s/lambda); m_k(lambda, v_lambda) -> m_k(1, v) by continuity, verified numerically by our band formulas) is used but not spelled out. The paper's cos(8s/5) at lambda = 1 needs this limiting argument; ours needs the identical one. Unconditionally the Rudnick-Sarnak range k lambda < 2 would cap k = 3 at lambda < 2/3 (7.5(e), line ~3786: "for lambda in (1/2, 1) this allows at most k = 3 (and only for lambda < 2/3)"); 7.5(g) is explicitly the RH-conditional branch, which is why the whole chain is stated under RH.
2. What was held fixed. The weight psi (LP vertex window-independent, re-checked), the BHB 19/27 input, and Proposition 4.4(iii) with the Schur-Horn majorisation of the admissible cubic (boundary case beta = -2 gamma). The paper notes its m2, m3 constants are verified from closed forms with interval arithmetic; our replacements carry exact rationals plus two-backend enclosures, which is the same standard or better.
3. Conclusion on admissibility. The optimization was run over even, nonnegative windows; the found optimum additionally satisfies, and the rational window provably satisfies, boundedness, monotonicity, smoothness of sqrt(v), and the variation-norm bounds. Within the class the paper itself uses (including its own two endpoint-nonvanishing cosines and the glossed lambda -> 1- limit), v* is admissible, and the improved constant is legitimate RH-conditional arithmetic inside the Prop 4.4(iii) + BHB chain.

8. Grading and caveats

Caveats, stated plainly:

• The source paper is a preprint dated 10 Aug 2026 and is not peer reviewed. Everything here inherits its Proposition 4.4(iii), its trace asymptotics, its use of Hejhal / Rudnick-Sarnak triple correlation under RH, and the Bui and Heath-Brown 19/27 simple-zeros proportion. If any of that fails, the constant fails with it; only the window mathematics (the m_k functionals and their optimization) is independently validated here.
• The whole statement is conditional on RH. Nothing in this directory is evidence for or about RH itself (docs/08 discipline).
• The improvement is quantitatively small (sixth decimal of the proportion). Its interest is structural: the hand-picked window is not optimal, the optimal window is characterized as an interior critical point of 1 - D2 - D3, and the remaining headroom within this certificate at k <= 3 is bounded by the searched sup, 0.68528703, so the paper's choice was already within 6.3e-5 of everything this branch can give.
• The lambda -> 1- and endpoint-taper limiting arguments are the paper's own glosses; this study measured the relevant continuity (section 7, items 2 and 3) but proves neither asymptotic; they remain inherited analytic debts of the source, shared equally by its window and ours.

9. Global optimality: landscape, exact slices, and a derived outer bound

Section 4's caveat said plainly that 0.6852870321770 was a local-search outcome and that global optimality over the admissible class was not claimed. This section settles as much of the global question as honest rigor allows. Scripts: global_bound.py (derived outer bound, exact rational arithmetic), global_landscape.py (EL landscape, measured), global_slice.py (exact quartic-slice supremum, measured higher slices); working log global_notes.md. Every script re-verifies the exact rational F(v*) of section 5 before computing anything, and the landscape script carries a freshly written mpmath Gauss-Legendre implementation of the closed forms (no shared engine with functional.py) that agrees with the exact rational to 50 digits (the gate demanded 25).

9.1 The outer bound: sup F <= 0.892744211644 (derived, exact rationals)

The chain runs through a central-moment identity. With l1 = 1 and m0 = m1 = 1, m2 = 1 + D2, m3 = 1 + 3 D2 + D3, the quantities D2 and D3 are the second and third central moments about 1 of the limiting spectral measure mu_v of A = H/l1, so

F = 2 m2 - m3 = 1 - int lam (lam - 1)^2 dmu_v(lam).

For every finite N the moments m_k^(N) = E tr A^k / N are moments of the expected empirical spectral measure, a positive measure on [0, inf) because H is Gram, hence PSD. Three consequences survive the limit that defines m_k:

1. F <= 1 (lam (lam-1)^2 >= 0 on [0, inf)): the concentration divergence and the flat-window 2/3 both sit under an unconditional ceiling.
2. F <= 1 - D2^2 pointwise in v (Cauchy-Schwarz (m2)^2 <= m1 m3).
3. sup F <= 1 - (inf D2)^2, since 1 - D2^2 is decreasing in D2 >= 0.

inf D2 is a quadratic problem solved with exact rational arithmetic: D2 = <v, (I - T) v>/l1^2 with T the tri-kernel operator on B (Fourier multiplier sinc^2 >= 0, tr T^2 = 1/2 exactly, so ||T|| <= 1/sqrt(2) < 1), and for any c and any window normalised to int v = 1,

<v, (I-T) v> >= 2 c - c^2 <1, (I-T)^{-1} 1>.

T maps polynomials to polynomials, so t_k = <1, T^k 1> are exact rationals; with K = 44 terms and the tail bounded by t_44 / (1 - 707107/1000000) (valid since 707107^2 >= 5 * 10^11 makes 707107/1000000 an upper bound for 1/sqrt(2) >= lam_max), the exact rational U = 3.053441685709720... satisfies U >= <1, (I-T)^{-1} 1>, and c = 1/U gives

inf D2 >= 1/U = 0.327499295198..., sup F <= 1 - (1/U)^2 = 0.892744211644411... (exact rational).

Grade: derived inequality chain, exact rational constants end to end, resting on the hardened closed forms of section 2 inside the source paper's framework. Cross-checks: the T-route D2 equals m2 - 1 from functional.py as an exact identity of Fractions at two windows; a trapezoid-grid discretization of (I-T) v = c converges to the same U with O(h^2); and the Neumann window w_K = sum_{k<=K} T^k 1 (nonnegative because T preserves nonnegativity) gives the exact upper bracket D2(w_24) = 0.327499296325..., so the cone infimum of D2 is pinned to a window of width 1.1e-9 and the grid minimizer is strictly positive (edge/center = 0.760), making the positivity constraint inactive for the D2 problem.

The bound is sharp for its inputs: among positive measures on [0, inf) with mean 1 and second central moment d, the maximum of int (2 lam^2 - lam^3) is exactly 1 - d^2, attained by (d/(1+d)) delta_0 + (1/(1+d)) delta_{1+d}. With inf D2 pinned to 1e-9, the value of this relaxation lies in [0.8927442109, 0.8927442116]: further progress on the outer bound requires excluding near-two-point spectral measures for actual windowed sine-Gram operators (a transition-layer statement in the spirit of time-band limiting theory), not more computation. That exclusion is the honestly-open part.

For calibration, the same chain pointwise at v* gives 1 - D2(v*)^2 = 0.892381..., against the true F(v*) = 0.685287...: essentially the whole gap of the method is the Cauchy-Schwarz step, which charges nothing for third-central-moment structure. The F-optimizer is close to but distinct from the D2 minimizer: at the (exactly evaluated) Neumann window w_12, D2 = 0.327499354, F = 0.684255566: the optimum spends 5.5e-4 of D2 to buy back 1.0e-3 of F through D3.

9.2 The first-order landscape (measured)

The EL stationarity system g = dF/dv = 0 (a quadratic integral equation; explicit form in functional.gradient_field) was SOLVED, not merely maximized, from random starts in three parametrizations (global_landscape.py), so saddles and boundary stationary points were findable, not just maxima:

• Cosine spectral, v = 1 + sum_{k<=J} c_k cos(2 pi k s), J = 6/8/10/12, 150 starts, all converged (Levenberg-Marquardt on the projected system P_k = int g cos(2 pi k s) = 0). At every J exactly ONE root has v > 0, and it is the known optimum; the restricted critical value climbs 0.685275 / 0.685282 / 0.685284 / 0.685285 with J (this basis carries an endpoint-kink truncation bias; its role here is root counting, not precision). Every other root (2 to 4 per J) is sign-changing, i.e. leaves the cone, with F <= 0.598 and 46/50, 45/50, 22/25, 19/25 of the starts landing on the positive root.
• Squared Chebyshev, v = w^2, J = 6/8/10, 120 starts, 118 converged (least squares on the analytic gradient dF/da). The strictly positive stationary point is unique at every J and identical to twelve digits, F = 0.685287032177 (59 hits across J). All other stationary points (12 to 16 per J) are windows TOUCHING ZERO at 1 to 8 isolated points; the best of them sits far below the optimum (0.6460 at J = 6, 0.6644 at J = 8, 0.6723 at J = 10), and the family decays with contact count down to strongly negative F (plus one degenerate l1 -> 0 artifact).
• Piecewise-linear cone with the exact lattice-aligned evaluator, n = 41 and 81 knots, 42 deliberately diverse starts (flat, cosine, quartic, three compact-support widths, bimodal, increasing profile, edge-heavy, randoms): every start converges to the SAME maximum, zero active knots (positivity inactive), KKT residual <= 5e-8, F = 0.685287031812 (n = 41) and 0.685287032154 (n = 81), the known pw-representation bias below the optimum. A separate Fischer-Burmeister stationarity solve (n = 21, 8 starts, 6 converged) finds exactly one KKT point, the optimum again: no stationary point with active positivity constraints was found anywhere.

Hessian at the optimum (cosine basis, 11 directions including the scaling ray): one numerically exact zero eigenvalue along the scaling ray (|H . ray| ~ 1e-10, eigenvalue 4e-12) and ten negative eigenvalues in [-2.06, -1.76]; signature 0 positive / 1 zero / 10 negative, i.e. a genuine local maximum modulo scale invariance, not a saddle.

Landscape verdict (measured): 336 stationarity, KKT, and maximization starts across three parametrizations and ten basis resolutions produced exactly one strictly positive stationary window, the reported optimum. Every other stationary object either leaves the cone (sign-changing) or touches v = 0 at isolated points with F at most 0.6723. The best F ever seen by anything in this study is 0.685287032177002 (float evaluation of the Chebyshev stationary point), i.e. the claimed sup was never beaten beyond 2e-15 of float noise.

9.3 Exact slice suprema and a two-sided witness bracket

The quartic slice v = 1 + q1 s^2 + q2 s^4 is settled exactly (global_slice.py). On it F is a rational function of (q1, q2) of degree 3/3, the admissible region is A = {psi(u) = 1 + q1 u + q2 u^2 >= 0 on [0, 1/4]}, and sup_A F is located by compactification: interior critical points, the two exact boundary families (endpoint zero v(1/2) = 0, i.e. q2 = -16 - 4 q1 with q1 >= -8; interior double root (q1, q2) = (-2/u, 1/u^2), u in (0, 1/4]), and the arc at infinity, where along admissible recession directions h = alpha s^2 + beta s^4 >= 0 the limit of F is F(h) by scale invariance and continuity (the denominator's leading form l1(h) = alpha/12 + beta/80 is strictly positive there). Results, all in exact arithmetic:

• The lex Groebner eliminant of the critical system factors as (cubic) * (quartic) with 5 real roots in q2; exactly ONE critical pair is admissible: q2 = 1.15904242554051... (root of 53574903041239 q2^4
• 7719414292779072 q2^3 - 126958598530649600 q2^2
• 34764906650381107200 q2 - 40111525465989120000), q1 = -1.46701866547..., with

F4* = 0.685287023289616136327866177020...

The four inadmissible critical branches carry F = 0.4675, -1.396, -371.9, -41041.9.

• Boundary: the endpoint-zero family peaks at F = 0.62175582 (its unique admissible critical point); the corner window (1 - 4 s^2)^2 gives exactly 610/3003 = 0.203130...; the double-root family stays below -0.21.
• Arc at infinity: everything is <= -0.195 (worst case -10.5 on the double-root family's concentration end).

So sup over the quartic slice = F4*, attained at the unique admissible critical point (derived, exact arithmetic; compactification argument as above). Consistency: the rational witness v* of section 5 sits 1.55e-12 below F4* (exact separation), and F4* sits 8.887e-9 below the full-space measured sup, exactly the gap section 5 already reported.

Higher slices (measured, 30 starts each): degree 6 reaches 0.6852870319222, degree 8 reaches 0.6852870321770 (all thirteen digits of the claimed sup), degree 10 the same to 1e-13. Nothing beat it. Polishing the degree-8 optimum and rounding to clean rationals gives an exactly admissible witness (Sturm count: psi has no root in [0, 1/4]):

v8(s) = 1 - (36773/25000) s^2 + (744/625) s^4 + (1341/25000) s^6

• (3397/6250) s^8, F(v8) = 8273865220036096559249598473358 / 12073576226528772158039668492375 = 0.685287032176620,

within 3.5e-13 of the measured sup. Pushing harder (Chebyshev J = 16, then exact rational evaluation of the resulting degree-44 dyadic- coefficient polynomial window, v = w^2 >= 0 automatic) gives the sharpest exact witness:

F = 0.685287032176998885912082211843 (exact rational, 208-digit numerator and denominator),

2.1e-15 below the float-measured sup. The global question is therefore bracketed by exact rationals on both sides:

0.685287032176998 <= sup F <= 0.892744211644412.

9.4 Verdict on the section 4 caveat

What changes:

What does not change: global optimality over the full admissible class remains open. The honest state is a factor-1.30 window [0.685287, 0.892744] whose upper end is the exact value of the (m1, m2)-moment relaxation; closing it requires spectral information beyond the second moment (ruling out near-two-point limiting spectral measures for windowed sine-Gram operators), which is research, not computation. Within everything searched or solved exactly, the reported optimum is the unique candidate: RF-C003's caveat can be upgraded from "local search outcome, global optimality not claimed" to "unique interior critical point across all parametrizations tried, exactly unique on the quartic slice, never exceeded anywhere, and globally capped by a derived bound 0.2075 above it; global optimality itself remains open".

Files

• functional.py: derivation notes, four evaluators (float GL, exact pw-linear, exact Fraction, sympy), validation battery.
• crosscheck_finiteN.py: exact finite-N windowed CUE cross-check (imports, read-only, from ../sine_gram/exact_finite_N.py).
• optimize.py: EL residual, cosine family, three optimization routes, writes best_window.json.
• enclose.py: two-backend ball enclosures, exact positivity and monotonicity, the strict inequality, the new constant.
• global_bound.py: the derived outer bound sup F <= 0.892744211644411 (exact rational chain; section 9.1).
• global_landscape.py: EL landscape, KKT search, Hessian, best-ever tracker, and the independent 50-digit evaluator gate (section 9.2).
• global_slice.py: exact quartic-slice supremum and degree 6..10 slice multistarts (section 9.3).
• global_notes.md: the working checkpoint log for section 9.

All four scripts are runnable standalone with .venv/bin/python from the repo root and re-produce every number in this file.