D-030 — Task 1.6: ξ′-world ceiling — certificate is window-saturated; adversary-side LP needs the paper's machinery

Date: 2026-08-18 | Session: 006 cont.5 | Task: 1.6 Status: complete. Question "is 0.86864 near the ξ′-world optimum?" answered on the attack side: YES — the quartic window is the window-class optimum to 1e-6. Adversary-side (data-class) ceiling: documented limitation.

1. Attack-side result (E-020b) [NUMERICAL-VALIDATED]

Rayleigh-quotient optimization of R_K(ψ) over even polynomial windows (degree ≤ 12, multi-start L-BFGS, trapezoid quadrature):

• CONTROL (kernel x): R_opt = 1.32749947 vs Montgomery–Taylor 1.32749930 — recovers the MT window to 1.7e-7 (machinery validated).
• TARGET (kernel D₁): R_opt = 1.13135921 vs R_quartic = 1.13136020. The paper's quartic window is within 1e-6 of the window-class optimum; certificate 2−R = 0.86864 is SATURATED on the attack side. (Optimizing coefficients reported in E-020b JSON; the optimal window is the quartic up to ~1e-6 coefficients.) Combined with D-022 §5 (0.86864 vs Wu's 0.86957 on-line-only): ξ′-world headroom ≈ 1e-3 from any direction. NOT a push target — confirms the R-002/T4 verdict quantitatively.

2. Adversary-side attempt (E-020) — documented limitation

The LawN256-mirror LP (min simple fraction over configuration mixes matching the datum law on rows) was implemented with a ~2300–6100 candidate pool (CUE-256 samples, lattice±doubles, adversarial corruptions, random):

• Control (law j): pool matches rows EXACTLY (ε* = 0 — LP with 2300 variables on 255 constraints cancels CUE fluctuations) but the min-simple mix gives 0.9959 vs the paper's 0.6818: the pool lacks POSITION-OPTIMIZED adversarial configs. Finite-pool bound is an upper bound, here vacuous.
• D₁ law (256·D₁(j/256)): ε* = 155.6 — the pool cannot realize the D₁-shaped form factor (dip at j = 128, top rows ~704) at all. Lesson: the adversary-side ceiling is an LP over CONTINUOUS-position configurations (the paper used exact-rational LP with tuned positions, achieving 1e-40 rows); a finite random/structured pool cannot approximate the extremal adversary. A meaningful cheap version would need position-optimized adversarial configs (Metropolis-tuned doubles) — logged as possible future work, not pursued: the attack-side answer (E-020b) already settles the task's question.

3. Verdict for task 1.6

• ξ′-world window-class optimum = 0.86864 to 1e-6 → the certificate is saturated; no window improvement available.
• By the control analogy (adversary 0.682 vs window-optimum 0.6725, gap 0.01), any adversary-side ξ′ ceiling exceeds 0.86864 by at most ~0.01 — and likely much less (D-022: mechanism is ξ′-specific).
• Task 1.6 CLOSED: the perfect-square node buys ~0 headroom beyond the paper's own window. Nothing transfers to ξ (R-002 stands).

4. Honesty notes

• E-020's failed checks (C1) are recorded in its JSON; the pool-LP limitation is machinery-level, not a bug (validated by eps* = 0 exact matching on control rows).
• E-020b windows: even polynomials degree ≤ 12 on [−1/2,1/2], the paper's window class; larger families (splines) could shift the 7th digit, not the conclusion.
• MT control value from E-001 (sympy + mpmath, 60-digit).