cft-cmi

Corner calculus: sequential one-sided recovery = Schwarzian point masses at the junctions

Status: proved — a complete structured proof. Conditional. The statement is not unconditional: it rests on H, quoted below. Read the badge only together with them.

Area: Lattice, thermal, massive and gapped geometries (O7, O9, O10) · Card type: theorem · Knowledge-base id: corner-calculus

Statement

One-sided zero-collar protocol on adjacent intervals A_1..A_n: step m adjoins A_new to the block J_{m-1} conditioning on a sub-block A_B adjacent to it, corner p^(m) = the end of A_B not shared with A_new, parabolic compression toward p^(m) with sigma_m = s_m/(l_B+l_new), s_m in [lambda_m, 2 lambda_m] (each step is the VACUUM Petz map of its own inclusion F(A_B) subset F(A_B u A_new), applied to whatever state the earlier steps produced); composite point map Phi = k_1 o ... o k_N (first step OUTERMOST). (1) GENERAL RULE, no hypothesis (Thm 5.2): Phi is C^1 piecewise-Moebius and S(Phi) = -2 sum_m sigma_m G_m'(x_m) delta_{x_m} -- with the DERIVATIVE G_m'(x_m), not G_m(x_m) -- where G_m := k_{m+1} o ... o k_N and x_m := G_m^{-1}(p^(m)); terms with p^(m) outside the range of G_m are absent (the corner is swallowed). (2) UNDER (H) (no earlier corner lies in the interior of a later step moving part A_B u A_new) (Thm 5.3): S(Phi) = -2 sum_k kappa_k delta_{p_k} over the INTERIOR junctions of J_N, kappa_k = sum of the sigmas of the steps whose corner is p_k; in particular S(Phi) is order independent. Sufficient for (H): (a) every A_B is a SINGLE chain interval AND the starting block has >= 2 chain intervals; (b) all corners coincide (VWZ 1(B)); (c) N = 1. CORRECTION (G1a): single-interval conditioning ALONE does not imply (H) -- Example 5.6, chain (1,1,1) started from the single interval A_2, add A_3 on A_2 then A_1 on A_2: the first corner is swallowed (not in the range of k_2), S(Phi) = -2 sigma_2 delta_{p_3} and Phi = (global Moebius) o k_2, so the first step leaves the state unchanged. (3) The composite algebra map is a normal *-isomorphism onto its range (Note 4 Thm 7.1 per step; Phi itself is NOT a single zero-collar map) and equals beta_{V_Phi} (V_{k_1}V_{k_2} = V_{k_1 o k_2}); the recovered state depends on Phi ONLY through S(Phi), because equal Schwarzians means Phi_2 = g o Phi_1 with g a global Moebius map (without pole on the OPEN interval Phi_1(J); the closure version is false) and omega o Ad U_g = omega (Thm 6.3). For a single corner the state, hence Phi_A, depends only on zeta = sigma beta_I(p) and not on the side of the compression (Cor 6.5). (4) RIGIDITY (Thm 7.3): omega o beta_Phi = omega on F(I) iff Phi is Moebius iff S(Phi) = 0; since all Schwarzian masses are negative and the last step is never swallowed, exact recovery never occurs for a protocol WHOSE LAST STEP KEEPS A NON-EMPTY PART (A_B strictly inside the block) -- unconditionally, without the second-order law. The hypothesis is needed: a degenerate step conditioning on the WHOLE block puts its corner on the boundary of I, where beta_I = 0, and is a pure Moebius compression that recovers the vacuum exactly (Rem 7.4; it is not a recovery in the Petz sense, since it keeps nothing fixed). (5) Consequences: VWZ 1(B) = 1(A) LITERALLY as point maps and for every common rotation t (2c/(b(b+c)) + 2d/((b+c)(b+c+d)) = 2(c+d)/(b(b+c+d))); VWZ Protocol 3 in either order is literally the same point map (parabolic L and R at the same point commute) and equals, up to post-composition with a global Moebius map, the over-compressed Theorem-A step with kept block AB, conditioning interval C, new interval D and s = (sigma_1+sigma_2)(c+d) > 2 lambda = 2d/c STRICTLY, hence Phi^(3) = Phi_A(zeta_eff), zeta_eff = (sigma_1+sigma_2)(a+b)(c+d)/(a+b+c+d); protocols with the same corner measure give the same state; and with single-interval conditioning the corner measure -- hence the holonomy -- depends only on the STARTING BLOCK, so the 2^{n-2} protocols starting from an adjacent pair fall into exactly n-1 classes, distinct pairs giving distinct measures (Prop 9.1 + Lem 9.2; tables for n = 4 and n = 5).

Hypotheses

This result is conditional: at least part of the statement holds only where the hypotheses below hold. The sentences are quoted from the statement above; read the status badge only together with them.

Hypothesis labels used: H.

(2) UNDER (H) (no earlier corner lies in the interior of a later step moving part A_B u A_new) (Thm 5.3): S(Phi) = -2 sum_k kappa_k delta_{p_k} over the INTERIOR junctions of J_N, kappa_k = sum of the sigmas of the steps whose corner is p_k; in particular S(Phi) is order independent.

Sufficient for (H): (a) every A_B is a SINGLE chain interval AND the starting block has >= 2 chain intervals;

CORRECTION (G1a): single-interval conditioning ALONE does not imply (H) -- Example 5.6, chain (1,1,1) started from the single interval A_2, add A_3 on A_2 then A_1 on A_2: the first corner is swallowed (not in the range of k_2), S(Phi) = -2 sigma_2 delta_{p_3} and Phi = (global Moebius) o k_2, so the first step leaves the state unchanged.

How to verify

To be proved in rigor/network_corner_calculus.tex (track G1a) and checked numerically by G2 (1(B) = 1(A) identity; Protocol 3 against paper 1's Phi_A table).

Evidence

Related claims

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-B1.

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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