cft-cmi

The C^{1,1} extension of the zero-collar map is implementable on Fock space (uhlmann_upper_bound.tex Cor. 2.7, Prop. 3.1)

Status: proved — a complete structured proof.

Area: Theorem A: exact recovery error of the free fermion · Card type: theorem · Knowledge-base id: c11-extension-implementer

Statement

Geometry of rigor/uhlmann_upper_bound.tex (eq:geom, l. 72-74): A = (-infinity, 0), D = (0, 1), I = AD = (-infinity, 1), I' = (1, infinity), zeta = s (a = infinity, L = 1). Let k~_s be the flow of the vector field -chi d/dx, with chi = 0 on x <= 0 and chi(x) = x^2 beta(x-1) on x > 0 (def:ext). (k~_s) for s real is a one-parameter group of C^{1,1} increasing bijections of R, equal to the identity outside (0, 2). For s >= 0, moreover, k~_s(x) = x/(1+sx) = k_s(x) on [0, 1] (lem:ext (E1), (E2)); the document states (E1)-(E4) only for every s >= 0 (l. 122). (E2) does not extend to s < 0. At s = -1, x = 0.9, x/(1+sx) = 9 lies outside [0, 2], which k~_s maps onto itself. The flow value depends on beta, which the document fixes only by its properties (def:ext, l. 105-106): 1.756 for the standard smooth step, 1.629 for beta^2, 1.905 for beta = 1 on [0, 1/2] (refb6_flow.py). For every allowed beta, (E2) fails for all s <= -1/2; for -1/2 < s < 0 it depends on beta (with beta = 1 on [0, 1/2] it holds for -1/3 <= s < 0; with the standard step it fails for every s < 0). Let W~_s be the unitary push-forward of k~_s on half-densities. Then P_- W~_s P_+ is Hilbert-Schmidt for every s. prop:HS gives an explicit bound when ||id - k~_s||_inf <= 1/2 (l. 286-287). Every other s follows from W~_s = (W~_{s/n})^n, since products of unitaries with Hilbert-Schmidt off-diagonal parts have Hilbert-Schmidt off-diagonal parts. The other block is P_+ W~_s P_- = (P_- W~_{-s} P_+)^*, Hilbert-Schmidt by the same argument. Until 2026-09-24 the document omitted that step, although its 'every 0 <= s <= 1' (l. 293) and cor:implementer's 'every s in R' used prop:HS beyond its hypothesis: with the standard step, ||id - k~_s||_inf = 0.61 at s = 0.5 and 0.95 at s = 1 (refb6_flow.py). It now takes the step, for both blocks (cor:implementer <1>1, l. 373; the box, l. 359 and l. 362; Correction C2), and restricts prop:HS's explicit bound to 0 <= s <= 1 with s ||chi||_inf <= 1/2 (l. 293, l. 327, l. 333, and the same range in prop:expansion, l. 760; Correction C3). So there is a unitary Gamma(W~_s) of Fock space, unique up to a phase, with Gamma(W~_s) a(f) Gamma(W~_s)^* = a(W~_s f) for all f (cor:implementer). The same holds for W~_s(1_{L^2(I)} (+) W'), with W' a unitary of L^2(I') such that P_-(1 (+) W')P_+ and P_+(1 (+) W')P_- are both Hilbert-Schmidt. The Shale-Stinespring criterion needs both off-diagonal blocks; an example is W' = e^{ish'} with h' self-adjoint, as in def:admissible, l. 745-754. Until 2026-09-24 the document assumed only the first condition, which is not enough (DOCUMENT ERROR note); it now requires both blocks in the box headline (l. 351), in cor:implementer (l. 371, with <1>2 at l. 374) and in the later uses lem:commutant (l. 417), def:admissible (l. 753) and the Ruijsenaars/Berezin box (l. 593) (Corrections C1, C4). For s >= 0, Gamma(W~_s) X Gamma(W~_s)^* = beta_s(X) on the CAR generators of F(I), and Psi_s = Gamma(W~_s)^* Omega is a vector representative of the recovered state omega~_s = omega_{J_s} o beta_s on F(I) (prop:vector; (C4), l. 76-80). The Shale-Stinespring criterion (and Araki 1970) is cited, not obtained ('Screenshot: NOT OBTAINED', l. 360).

How to verify

rigor/uhlmann_upper_bound.tex Def. 2.1 (def:ext, l. 104), Lemma 2.2 (lem:ext), Prop. 2.5 (prop:HS, l. 284), cited box before Cor. 2.7 (Shale-Stinespring, l. 348-363), Cor. 2.7 (cor:implementer, l. 367-376), Prop. 3.1 (prop:vector, l. 392-414); for the lines changed on 2026-09-24, the section Corrections of 2026-09-24 (Sec. 12, l. 1522-1717), items C1-C4 (l. 1532-1635).

Evidence

Related claims

Review

Review record: passed 2026-09-24 by REF-CARD-CFT.

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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