cft-cmi

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

Status: proved — a complete structured proof.

Area: Theorem A: exact recovery error of the free fermion · Card type: result · Knowledge-base id: rate-of-remainder

Statement

For the zero-collar compression of the free chiral fermion (f2 = g_B/8 = 1/(12 pi^2) per component): (1) group bound (lem:groupbound): for U_s = e^{sK} with [P_-, K] Hilbert-Schmidt, ||V_s||_2^2 <= s^2 ||P_- K P_+||_2^2 exactly, V_s = P_- U_s P_+; (2) global bound (thm:global with prop:rrr, unconditional): for K = D + i h' with h' in the admissible commutant class, Gamma(U_s)^* Omega represents the recovered state and Phi(zeta) <= -(1/2) log(1 - 2 f2^K zeta^2) with inf over h' of f2^K = g_B/8 = f2, hence Phi(zeta) <= -(1/2) log(1 - 2 f2 zeta^2) = f2 zeta^2 (1 + f2 zeta^2 + O(zeta^4)) for zeta < (2 f2)^{-1/2}, with no cubic term; (3) consequences (cor:c3sign): limsup_{zeta -> 0} (Phi - f2 zeta^2)/zeta^3 <= 0, so c3 <= 0 IF Phi = f2 zeta^2 (1 + c3 zeta + o(zeta)); and there is no remainder + b zeta^2/log^2(1/zeta) with b > 0, since Phi - f2 zeta^2 <= f2^2 zeta^4/(1 - 2 f2 zeta^2); this refutes the zeta^2 log^-2 remainder of Note 5 Cor. 4.2 (a double count). Index gap (rate_of_remainder.tex Correction C3 of 2026-09-24, l. 1026-1079): (1) and (2) use that the two off-diagonal blocks P_- U_s P_+ and P_+ U_s P_- of the unitary have the same singular values, which the printed proofs assert for every unitary (Thm 2.3 <1>4 and Lemma 6.1 <1>4 by the CS decomposition, l. 168-171 and 842-846; Thm 2.3 <1>1 and Lemma 6.1 <1>1 in the squared-norm form ||G_s||_2^2 = 2 ||V_s||_2^2, l. 156 and 828-832; C3, l. 1044-1050), but which holds only when P_+ U_s P_+ has index zero on P_+ H. The documents settle this while ||G_s||_2 < 1, G_s = P_+ - U_s^* P_+ U_s, which Lemma 6.1 <1>2 gives for |s| ||[P_-, K]||_2 < 1: so (1) is settled for |s| ||[P_-, K]||_2 < 1, the bound of (2) with f2^K for zeta < (4 f2^K)^{-1/2}, and the bound with f2 for zeta < (4 f2)^{-1/2} = pi sqrt(3) = 5.44 (the h' with 4 f2^K zeta^2 < 1 still have inf f2^K = f2, because inf f2^K = f2 < 1/(4 zeta^2), and prop:rrr <1>4 takes the infimum over them; for zeta >= pi sqrt(3) no h' qualifies, since f2^K >= f2). Beyond that, i.e. for |s| ||[P_-, K]||_2 >= 1 in (1), for (4 f2^K)^{-1/2} <= zeta < (2 f2^K)^{-1/2} in (2) and for pi sqrt(3) <= zeta < (2 f2)^{-1/2} = pi sqrt(6) = 7.70 in the bound with f2, they rest on the cards pushforward-blocks-equal-singular-values (heuristic) for the push-forwards W~_s = e^{sD} (h' = 0; thm:global with f2^chi) and group-index-zero-large-s (conjectural) for every other e^{sK}; (3) uses only small zeta and is unaffected. NOT claimed here: that Phi has an expansion in integer powers, and the values c3 = -0.99(1), c4 = +0.9(1) (card remainder-c3-c4-values, heuristic); every lower bound with a rate is conditional on a cutoff-uniform constant (summary table, l. 786-800).

How to verify

rigor/rate_of_remainder.tex lem:groupbound (l. 818), thm:global (l. 180-200), prop:rrr (l. 848-880; it removes the hypothesis f2* = f2 of the earlier version), cor:c3sign (l. 247-262), summary table (l. 786-800: upper bound proved, values numerical, lower bounds conditional); referee report rigor/referee_rate_of_remainder.tex prop:repair; the index gap and the settled ranges: Sec. 7 'Corrections of 2026-09-24', item C3 (l. 1026-1079; status of the uses, l. 1062-1079), which changed no proof text, so the summary table's 'proved' predates it. No numerics are needed for this card.

Evidence

Related claims

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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