Index zero for e^{sK} beyond ||G_s||_2 < 1: for a unitary group e^{sK} with [P_-, K] Hilbert-Schmidt the two off-diagonal blocks have the same singular values also where |s| ||[P_-, K]||_2 >= 1 (open; no document contains an argument)
Status: conjectural — believed, with partial evidence; not proved.
Area: Theorem A: exact recovery error of the free fermion · Card type: conjecture · Knowledge-base id: group-index-zero-large-s
Statement
For every one-parameter unitary group U_s = e^{sK} on H = L^2(R) with K^* = -K and [P_-, K] Hilbert-Schmidt (the setting of rate_of_remainder.tex Lemma 6.1, lem:groupbound; in particular K = D + i h' with h' in PA's class A', Prop. 6.2), the operator P_+ U_s P_+ on P_+ H has Fredholm index 0, so that P_- U_s P_+ and P_+ U_s P_- have the same singular values, also for the real s with ||G_s||_2 >= 1, G_s = P_+ - U_s^* P_+ U_s, where the documents' argument (both blocks of norm < 1, so P_+ U_s P_+ is invertible) does not apply; by Lemma 6.1 <1>2 such s have |s| ||[P_-, K]||_2 >= 1, i.e. |s| >= (4 f2^K)^{-1/2} for K = D + i h' (f2^K = (1/2) ||P_- K P_+||_2^2).
How to verify
rigor/rate_of_remainder.tex Sec. 7 (Corrections of 2026-09-24), C3, l. 1026-1079: the mechanism, l. 1034-1043 (the singular values of the two blocks agree except at 1, whose multiplicities are dim ker X^* and dim ker X; the truncated winding of numerics/kb-checks-2026-09-23/refb9_winding.py has squared block norms 0.0071 and 1.0071 at R/b = 5, index 1); the affected steps, l. 1044-1057 (Lemma 6.1 <1>1, l. 828-832, and <1>4, l. 842-846; Prop. 6.2 <1>3, l. 871-874; Cor. 2.9 <1>3); 'Status of the uses' (ii), l. 1069-1079: settled while ||G_s||_2 < 1, which Lemma 6.1 <1>2 gives for |s| ||[P_-, K]||_2 < 1, with ||[P_-, K]||_2^2 = 2 ||P_- K P_+||_2^2 = 4 f2^K; 'The rest needs a standard Fredholm-index continuity argument that no document contains, and is open': Lemma 6.1 for |s| ||[P_-, K]||_2 >= 1 and Prop. 6.2 for (4 f2^K)^{-1/2} <= zeta < (2 f2^K)^{-1/2}. The same claim, with P and P^perp for P_+ and P_-, underlies rigor/exact_fidelity_upper_half.tex Lemma 6.1 (lem:group) <1>1, l. 662-667, whose Correction C3 (l. 1156-1205) settles the uses there at small s.
Evidence
- rigor/rate_of_remainder.tex — document, at
corr0924:C3
Related claims
- Remainder of Theorem A: global upper bound Phi <= -(1/2) log(1 - 2 f2 zeta^2); c3 <= 0 as a limsup; no positive log^-2 term —
proved pushforward-blocks-equal-singular-values—heuristic, no page of its own- (UH) proved: optimised Uhlmann bound = g_Q of the defect, for every isometric quasi-free channel —
proved
Review
Review record: passed 2026-09-24 by REF-CARD-ROR.
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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