Optimised exterior extension: UB_c = (1/2)s^2|u|^2 exactly and UB_rot/UB_c = 1-theta

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: e2-optimised-uhlmann

Statement

In the NS-circle model (a = 2, Lg = 1, uv = (0.30,0.42), alpha = 3) with the exterior generator OPTIMISED (least squares over all anti-Hermitian Z_e supported on I^c, solved in closed form by the Qc-eigenbasis diagonalisation): UB(X) = (1/2) dist(X, Ext)^2 = (1/2)(||Pi X||^2 + dist((1-Pi)X, Ext)^2). Numerically dist((1-Pi)M, Ext)^2 and dist((1-Pi)v, Ext)^2 are at roundoff (|d^2| <= 1.3e-10 against ||(1-Pi)M||^2 = 2.45e-2, relative distance <= 1.2e-5 and usually exactly 0 in double precision) for L = 128, 192, 256, 384. Hence UB_c/((1/2)s^2|u|^2) = 1.00000000, 0.99999999, 1.00000000, 0.99999998 at L = 128, 192, 256, 384 -- this REPLACES the 4.3 of the fixed Gaussian-taper extension (numerics/optimality_all/uhlmann_rotated.py), a factor 4.3 that was entirely an artefact of a bad exterior extension. And UB_rot/UB_c = 0.70885, 0.71125, 0.71058, 0.70766, equal to 1 - theta = 0.70890, 0.71120, 0.71051, 0.70767 to 4-5 digits. UB is exactly quadratic in s, so the ratios are s-independent to machine precision (checked at s = 0.001, 0.002, 0.005). eps_* = -1.0000 in units of s, as predicted by <u,v_x> = ||v_x||^2 = <b,x>.

How to verify

run numerics/optimality_all/e2_dist_ub.py; each block prints 'UB_c/(1/2 s^2|u|^2) = 1.0000000x' and 'UB_rot/UB_c' matching 1-theta

Evidence

• numerics/optimality_all/e2_dist_ub.py — numerics
• rigor/exterior_extension_numerics.tex — document, at sec:ub

Related claims

• NS-circle model: Ext = N^perp exactly, with the dimension identity (L-|I|)^2 — numerical
• Is the compression beaten in EXACT fidelity? — numerical
• phase4-plan — done, no page of its own

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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