Corner data of all one-sided protocols for n = 4 and n = 5: the holonomy is the starting block

Status: proved — a complete structured proof. Stated under hypotheses H, quoted below.

Area: Lattice, thermal, massive and gapped geometries (O7, O9, O10) · Card type: result · Knowledge-base id: protocol-corner-data-n4-n5

Statement

For a complete one-sided protocol with SINGLE-interval conditioning, starting block J_0 = A_i..A_j (j >= i+1) and common rotation parameter t (theta_t = (1+e^{-2 pi t})/2, theta_0 = 1 the ordinary Petz value), hypothesis (H) holds and the corner measure is kappa = sum_{k<i} sigma^L_k delta_{p_{k+2}} + sum_{k>j} sigma^R_k delta_{p_{k-1}} with sigma^L_k = theta_t 2 l_k/(l_{k+1}(l_k+l_{k+1})) and sigma^R_k = theta_t 2 l_k/(l_{k-1}(l_{k-1}+l_k)) -- INDEPENDENT of the order of the steps. Hence all binom((i-1)+(n-j), i-1) orders for a given starting block give the same recovered state, the 2^{n-2} protocols with a starting PAIR fall into exactly n-1 classes (one per pair), the left corners occupy p_3..p_{i+1} and the right corners p_j..p_{n-1}, and they overlap in exactly one point (a double corner at p_{i+1} = p_j) iff j = i+1. n = 4 (4 protocols, 3 classes): start A_1A_2 (RR) -> {p_2: sigma^R_3, p_3: sigma^R_4} = VWZ Protocol 2; start A_2A_3 (LR and RL, identical) -> {p_3: sigma^L_1 + sigma^R_4} = VWZ Protocol 3; start A_3A_4 (LL) -> {p_3: sigma^L_1, p_4: sigma^L_2}. Equal lengths l = (1,1,1,1): zeta^(I) = (0.75 at p_2, 1 at p_3), (2 at p_3), (1 at p_3, 0.75 at p_4). n = 5 (8 protocols, 4 classes): start A_1A_2 (RRR) -> p_2,p_3,p_4 simple; start A_2A_3 (LRR, RLR, RRL) -> double corner at p_3 plus p_4; start A_3A_4 (LLR, LRL, RLL) -> p_3 plus double corner at p_4; start A_4A_5 (LLL) -> p_3,p_4,p_5 simple. Union-conditioning variants for n = 4: VWZ 1(A) (one corner at p_2) and 1(B) (two corners at p_2, same measure by the identity 2c/(b(b+c)) + 2d/((b+c)(b+c+d)) = 2(c+d)/(b(b+c+d))) satisfy (H); the variant 'add D on C, then A on BC' does NOT (REF-P6-0's counterexample geometry: the mass is displaced off the junction).

Hypotheses

The statement is made under named hypotheses. The sentences are quoted from the statement above; read the status badge only together with them.

Hypothesis labels used: H.

For a complete one-sided protocol with SINGLE-interval conditioning, starting block J_0 = A_i..A_j (j >= i+1) and common rotation parameter t (theta_t = (1+e^{-2 pi t})/2, theta_0 = 1 the ordinary Petz value), hypothesis (H) holds and the corner measure is kappa = sum_{k<i} sigma^L_k delta_{p_{k+2}} + sum_{k>j} sigma^R_k delta_{p_{k-1}} with sigma^L_k = theta_t 2 l_k/(l_{k+1}(l_k+l_{k+1})) and sigma^R_k = theta_t 2 l_k/(l_{k-1}(l_{k-1}+l_k)) -- INDEPENDENT of the order of the steps.

Union-conditioning variants for n = 4: VWZ 1(A) (one corner at p_2) and 1(B) (two corners at p_2, same measure by the identity 2c/(b(b+c)) + 2d/((b+c)(b+c+d)) = 2(c+d)/(b(b+c+d))) satisfy (H); the variant 'add D on C, then A on BC' does NOT (REF-P6-0's counterexample geometry: the mass is displaced off the junction).

How to verify

rigor/g1a_calculus.py section 5 enumerates all protocols for n = 4, 5, checks that they fall into n-1 classes, and measures every Schwarzian mass of the composite map by one-sided finite differences (agreement to 3e-3 relative).

Evidence

• rigor/network_corner_calculus.tex — document, at prop:startblock
• rigor/network_corner_calculus.tex — document, at tab:n4
• rigor/g1a_calculus.py — numerics
• rigor/ref_p6_3_calculus.py — numerics

Related claims

• Corner calculus: sequential one-sided recovery = Schwarzian point masses at the junctions — proved
• VWZ four-interval protocols: Phi(1):Phi(2):Phi(3) -> 1:2:4; middle-out is the worst order — conjectural
• phase6-plan — active, no page of its own

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-B2.

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.

Documentation map · All claims by area · What is open · Symbols · Definitions