Session scratch log, updated incrementally. Nothing here is a result until it reaches RESULTS.md section 9 with a grade. No em dashes; the reserved word of zeta/rigor.py is not used in this directory.
0. Validation gate (done)
functional.pybattery: all validations pass (run 2026-08-17, this session).- Exact rational reproduced: F(v*) = 2245228120295149280/3276332462159207451 for v* = 1 - (1467/1000)s^2 + (1159/1000)s^4.
- TODO: independent mpmath evaluator must match to 25+ digits before any new number is trusted (gate inside global_landscape.py).
1. Reformulation used by the bound work (derivation notes)
With l1 = int v = 1 (scale invariance) and m_k the limiting spectral moments of A = H/l1 (the source paper's framework, hardened closed forms in functional.py):
m0 = 1, m1 = 1, m2 = 1 + D2, m3 = 1 + 3 D2 + D3
so D2 = m2 - 2 m1 + m0 and D3 = m3 - 3 m2 + 3 m1 - m0 are the second and third central moments of the spectral measure mu_v about 1:
D2 = int (lam - 1)^2 dmu, D3 = int (lam - 1)^3 dmu, F = 2 m2 - m3 = 1 - int lam (lam - 1)^2 dmu.
Since A is PSD, mu_v lives on [0, inf), so lam (lam-1)^2 >= 0 and F <= 1 (per-N: E nu_N is a positive measure with these moments; Cauchy-Schwarz and positivity hold at every N, then pass to the limit that defines m_k).
Cauchy-Schwarz (m2^2 <= m1 m3 for a positive measure on [0, inf)) gives
F <= 2 m2 - m2^2 = 1 - D2^2 (pointwise in v)
and 1 - D2^2 is decreasing in D2 >= 0, so
sup F <= 1 - (inf D2)^2.
D2 = <v, (I - T) v> / l1^2 with (T f)(xi) = int_B tri(xi - eta) f(eta) deta, tri(t) = 1 - |t| on |t| <= 1, B = [-1/2, 1/2]. T is PSD (Fourier multiplier sinc^2 >= 0), ||T||_HS^2 = tr T^2 = 1/2 exactly, so ||T|| <= 1/sqrt(2) < 1.
Lower bound for inf D2 via the quadratic dual: for any c, <v, (I-T) v> >= 2 c <1, v> - c^2 <1, (I-T)^{-1} 1>, so with U >= <1, (I-T)^{-1} 1> = sum_k t_k, t_k := <1, T^k 1>: inf_{int v = 1} D2 >= 1/U. t_k are exact rationals (T maps polynomials to polynomials, degree + 2). Tail bound for even K: sum_{k >= K} t_k <= t_K / (1 - lam_max) and lam_max <= 1/sqrt(2) <= 707107/1000000 (since 707107^2 >= 5*10^11).
Everything in the chain is exact rational. Status: derivation written, code next (global_bound.py).
2. Landscape plan (global_landscape.py)
- EL system: g(s) = dF/dv(s) = 0 on supp v (quadratic integral equation; explicit form in functional.gradient_field). Critical point finders: (a) cosine spectral basis v = 1 + sum c_k cos(2 pi k s), solve the projected system P_k(c) = int g cos(2 pi k s) = 0 by least squares from many random starts; classify roots by positivity of v. (b) squared Chebyshev v = w^2: solve grad_a F = 0 (finds saddles too). (c) pw-linear cone KKT (Fischer-Burmeister residual) plus L-BFGS-B multistart with diverse inits including compact-support ones.
- Hessian eigenvalues at the optimum (expect: one zero mode from scaling, rest negative).
- Best-F-ever-seen tracker across all runs.
3. Slice plan (global_slice.py)
- Quartic slice 1 + q1 s^2 + q2 s^4: exact critical system via sympy resultants, count real critical points in the admissible region, exact algebraic slice supremum.
- Degree 6..10 slices: high-precision multistart (measured).
Findings so far
F1 (prototype, 2026-08-17): exact rational bound chain works
t_k = <1, T^k 1> computed exactly (t_1 = 2/3, t_2 = 9/20 match hand derivation). Ratios t_{k+1}/t_k converge to lam_1(T) = 0.6755169434391893, comfortably below the exact HS bound 1/sqrt(2) used only for the tail. With K = 44: U = 3.0534416857, 1/U = 0.32749929520, sup F <= 1 - (1/U)^2 = 0.8927442116. Every constant is an exact rational. Note inf D2 = 0.3274993 sits close to D2(v*) = 0.3280521: the F-optimum is nearly the D2 minimizer.
F2 (prototype): cosine spectral landscape, J = 6, 40 starts
All 40 least-squares runs on the projected EL system converged; 3 distinct critical points: the known optimum (38 hits, v > 0, F = 0.685275 at this truncation), and two sign-changing solutions outside the cone (F = 0.5243, 0.4303). Unique interior critical point in this span. Truncation note: the cosine basis converges slowly for this problem (the optimum's periodic extension has endpoint derivative kinks), so the restricted critical value sits below 0.68528703 at small J; J up to 12 in the production run.
F4 (2026-08-17, global_bound.py run complete): derived outer bound
Gate: T-machinery agrees with functional.py exactly (D2(v*) = m2 - 1 as an identity of Fractions between two independent codes). Results:
- U = 3.053441685709720 exact rational (partial sum K = 44 plus tail bounded via lam_max <= 1/sqrt(2) <= 707107/1000000, tail <= 1.1e-7).
- inf D2 >= 1/U = 0.327499295198614; bracket from above by the explicit nonnegative Neumann window w_24: D2(w_24) = 0.327499296325274, so the bracket on inf-over-cone D2 has width 1.1e-9.
- sup F <= 1 - (1/U)^2 = 0.892744211644411 exact rational. Grade: derived chain (per-N Cauchy-Schwarz on the expected empirical spectral measure, plus the quadratic dual bound), on top of the hardened moment closed forms; all constants exact rationals.
- The 1 - D2^2 route is saturated: it cannot give better than 0.8927442109 whatever the numerics, since inf D2 is pinned to 1e-9.
- Sanity (measured): (D2 + D3) - D2^2 >= 0.2069 at 60 random admissible windows; at v* the Cauchy-Schwarz step gives away 0.892 vs 0.685, i.e. the slack of this route is the m3 >= m2^2 inequality, not the numerics.
F5 (2026-08-17, global_slice.py run complete): quartic slice EXACT
- Exactly ONE admissible interior critical point: q = (-1.46701866547, 1.15904242554), F4* = 0.685287023289616136327866177020 (isolated real algebraic number; q2 component is a root of 53574903041239 q2^4 - 7719414292779072 q2^3 - 126958598530649600 q2^2 + 34764906650381107200 q2 - 40111525465989120000, per the exact lex Groebner eliminant). Four other real critical branches, all inadmissible (v < 0 somewhere), with F = 0.4675, -1.396, -371.9, -41041.9.
- Boundary of A exact: family (a) endpoint-zero segment max 0.62175582 at its unique admissible critical point; corner (1-4s^2)^2 gives exactly 610/3003; family (b) double-root curve entirely below -0.21.
- Arc at infinity exact: max -0.19549 (all negative).
- Hence sup over the quartic slice = F4*, attained at the unique interior critical point. Grade: derived (exact arithmetic; compactification argument as in the file docstring).
- Consistency: F(v*) = 0.685287023288068 (the rational witness) sits 1.5e-12 below F4*, and F4* sits 8.887e-9 below the full-space measured sup 0.6852870321770, exactly as RESULTS.md section 5 predicted.
- Degree 6/8/10 slices (measured, 30 starts each): best F = 0.6852870319222 / 0.6852870321770 / 0.6852870321769. The degree-8 slice already attains the claimed full sup to all 13 digits and nothing beat it. Higher slices show a flat near-optimal ridge (many endpoints within 1e-6), consistent with a single basin plus soft directions.
F6: the relaxation value is itself pinned (both sides)
The outer bound is the exact value of a relaxation: among positive measures on [0, inf) with mean 1 and second central moment >= inf D2, the maximum of int (2 lam^2 - lam^3) is exactly 1 - (inf D2)^2, attained by mu = (d/(1+d)) delta_0 + (1/(1+d)) delta_{1+d}, d = inf D2 (the dual polynomial lam (lam - (1+d))^2 >= 0 touches at both atoms). With the inf D2 bracket [1/U, D2(w_24)] this pins the relaxation value to [0.8927442109, 0.8927442116]. So the derived bound cannot be improved by computing harder; it can only be improved by adding facts the relaxation does not know, i.e. by excluding near-two-point spectral measures for actual windowed sine-Gram operators. That exclusion is genuine spectral theory (the transition-layer behavior of band-limiting Gram operators) and is the honestly-open part of the global question.
F7: exact factor bookkeeping for the slice critical point
The admissible root q2 = 1.15904242554051 is a root of the quartic factor 53574903041239 q2^4 - 7719414292779072 q2^3 - 126958598530649600 q2^2
- 34764906650381107200 q2 - 40111525465989120000 of the eliminant (the
cubic factor's single real root 67.389 is inadmissible). Exact separation F4* - F(v*) = 1.5483e-12 > 0, and F4* < measured full sup by 8.887e-9.
F8: exact rational witness on the degree-8 slice (sup lower bound)
v8(s) = 1 - (36773/25000) s^2 + (744/625) s^4 + (1341/25000) s^6
- (3397/6250) s^8
is exactly admissible (Sturm: psi(u) = 1 + q1 u + ... has zero roots in [0, 1/4], psi(1/4) = 141077/200000 > 0, so v > 0 on B) and achieves, in exact rational arithmetic,
F(v8) = 8273865220036096559249598473358/12073576226528772158039668492375 = 0.685287032176620,
within 3.5e-13 of the measured sup 0.6852870321770. Together with F4:
0.685287032176620 <= sup F <= 0.892744211644411,
both ends exact rationals: the lower end by an explicit admissible witness, the upper end by the derived chain.
F9: exact evaluation of the recorded Chebyshev witness
The best_window.json Chebyshev coefficients define w exactly (dyadic rationals); v = w^2 is a degree-28 even polynomial, nonnegative by construction. Exact rational evaluation:
F = 0.6852870321769988738698323 (25 digits; exact fraction available)
so the two-sided bracket tightens to
0.685287032176999 <= sup F <= 0.892744211644412 (both ends exact).
The lower end is 2.0e-12 below the float-measured sup 0.6852870321770008.
F10 (2026-08-17, global_landscape.py full run complete): landscape
Gate: independent mpmath evaluator matched the exact rational to 50 digits; 30-digit cos reference reproduced to < 1e-28.
- Cosine spectral (J = 6/8/10/12, 150 starts, all converged): exactly one root with v > 0 at every J (the optimum; restricted value climbs 0.685275 -> 0.685285 with J); all other roots sign-changing, outside the cone, F <= 0.598.
- Squared Chebyshev (J = 6/8/10, 118/120 converged): the strictly positive stationary point is unique at every J, F = 0.685287032177 to twelve digits. Other stationary points touch v = 0 at 1..8 isolated points, best 0.6723 (J = 10), i.e. 1.3e-2 below the optimum.
- Piecewise-linear cone (n = 41/81, 42 diverse starts incl. compact support, bimodal, edge-heavy): every start converged to the same maximum, zero active knots, KKT residual <= 5e-8. Fischer-Burmeister stationarity solve (n = 21): one KKT point, the optimum. No stationary point with v = 0 on positive measure anywhere.
- Hessian (cosine basis + scale direction): signature 0 pos / 1 zero (scaling ray, eigenvalue 4e-12) / 10 negative in [-2.06, -1.76]: genuine local max.
- Best F ever seen across all 336 starts: 0.685287032177002 (float noise 2e-15 above the 13-digit claim); the pushed Chebyshev J = 16 window, degree-44 polynomial with dyadic coefficients, evaluates EXACTLY to F = 0.685287032176998885912082211843.
Final exact two-sided bracket: 0.685287032176998 <= sup F <= 0.892744211644412.
RESULTS.md section 9 written (landscape, bound, slices, verdict); Files list updated. Lexical scan of everything: clean.
F3 (prototype): quartic slice is exactly solvable
Symbolic F(q1, q2) rational of degree 3/3; sympy reproduces the exact rational F(v*) from functional.py. Critical system Groebner basis: lex, univariate element of degree 7 in q2 with 5 real roots: q2 = -62.573, 1.1590424255405073, 67.389, 87.446, 118.055. The second is the optimizer's basin. Admissibility of the others TBD in global_slice.py; boundary families and directional limits at infinity to be solved exactly (plan in section 3 above, expanded):
- admissible region A in (q1,q2): psi(u) = 1 + q1 u + q2 u^2 >= 0 on [0, 1/4]; cells C1 {q1 >= 0}, C2 {q1 <= 0 <= q2, q2 <= -2q1} (min at endpoint u = 1/4, contains the optimum, compact), C3 {q2 > 0, q1 < 0, q2 >= -2 q1, disc <= 0}, C4 {q2 <= 0}, all intersected with L = 16 + 4 q1 + q2 >= 0.
- recession cone of A: {alpha >= 0, alpha + beta/4 >= 0}; the denominator leading form alpha/12 + beta/80 is strictly positive there except at 0, so F extends continuously to the arc at infinity with limit F_inf(direction) = ratio of leading forms = F of the limit window h = alpha s^2 + beta s^4. Slice sup = max over finite exact list: interior critical values, boundary-family univariate critical values, arc-at-infinity univariate max.