The compression is not the minimiser of the second-order functional

Status: numerical — measured, with an error bar and a convergence order; not a theorem.

Area: Optimality of recovery channels (O8, Phases 3 and 3b) · Card type: result · Knowledge-base id: compression-not-second-order-minimiser

Statement

E_D u E_D != 0: composing the compression with a Bogoliubov rotation e^{-eps G*} inside B lowers g_Q by the factor 1 - theta, theta = 0.384 +- 0.003 (Phase 5 track F3, card theta-value, numerics/optimality_all/F3_RESULTS.md). The two published discretisations are reconciled, not in conflict: S10's spectral-model theta = 0.300(5) is a taper-suppressed LOWER bound (the smoothstep ultraviolet taper on the spectral derivative removes modes carrying ~20% of theta; there is no taper-free lattice limit because of the second Fermi point, and theta_circ rises linearly in log L, so the 1/L Richardson extrapolation used the wrong variable), while the modular Galerkin model's 0.29 -> 0.34 (increasing with resolution) extrapolates to the same 0.384 as the parameter-free modular Wiener-Hopf frame (agreement 3e-4 after extrapolation, measured order h^0.77). Hence min_F g_Q <= (1 - theta) f2 z^2 = (0.616 +- 0.003) f2 z^2, no dual certificate for f2 z^2 exists, and S8's 'consistent with optimal' is superseded at this level. theta > 0 is numerical, not proved.

How to verify

numerics/optimality_all/theta_galerkin_first.out, s10_theta.py outputs; rigor/optimality_all_channels.tex Sec. 5.2.

Evidence

• rigor/sdp_dual_certificate.tex — document, at sec:num
• rigor/sdp_dual_certificate.tex — document, at thm:value
• numerics/optimality_all/theta_galerkin.py — numerics
• numerics/optimality_all/s10_theta.py — numerics

Related claims

• Exact KKT certificate at the compression; Moebius rigidity — proved
• The first-order gain constant theta = 0.384 +- 0.003 (S10's 0.300 is a taper-suppressed lower bound) — numerical

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-A1.

The referee's report and the card's event log stay in the private knowledge base; only the verdict, the date and the referee are published here.

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