R12 Holonomy State No-Go

Status: rejected as an R12 invention. Loop holonomy is a valid gauge-orbit observable and can identify a declared finite-dimensional connection up to conjugacy. It does not identify the current causal state, and complete finite signatures reduce to ordinary operator recurrence or PSR machinery.

1. Candidate object

For a typed event graph, assign every edge e:v->w an invertible residual transport

T_e : F_v -> F_w.

For history h=e_t...e_1, let T_h=T_(e_t)...T_(e_1). A loop l:v->v has holonomy H_l=T_l, and two paths p,q:v->w have defect

C_(p,q) = T_q^-1 T_p.

Under hidden-coordinate changes g_v,

T'_e = g_w T_e g_v^-1,
H'_l = g_v H_l g_v^-1.

Traces, spectra, characters, and other conjugacy-class functions are therefore gauge invariant. They identify properties of the orbit; by definition they cannot select a hidden gauge.

2. Strong finite survivor

For a connected finite graph and compact matrix group K subset U(d), choose a spanning tree and gauge every tree edge to identity. The remaining

m = |E|-|V|+1

chord transports are fundamental-loop holonomies. They determine the connection up to one simultaneous global conjugation, and every unseen history is a word in those m operators. Finite joint trace-word invariants can separate simultaneous unitary-conjugacy orbits for fixed d,m.

The two-generator SU(2) case is explicit. Write

A = x I + i a.sigma,
B = y I + i b.sigma.

The three signatures

x = tr(A)/2,
y = tr(B)/2,
z = tr(A B^-1)/2

recover the norms and inner product a.b=z-xy; their Gram matrix determines (A,B) up to simultaneous conjugation. Under nondegeneracy margin kappa, a signature error epsilon yields generator error on the order of epsilon/kappa^3 and length-L word error on the order of L epsilon/kappa^3.

Static storage is O(md^2 b) bits, dynamic operator state is O(d^2 b), and each event costs O(d^3). This is exponentially shorter than a table of all words but exactly matches matrix recurrence and observable-operator controls.

3. State obstruction

Holonomy describes the connection, not the current point in its fiber. Two future-distinguishable causal states under one connection have identical state-independent loop signatures. Adding state-dependent probe responses creates continuation/query prediction coordinates, which is a PSR/OOM.

The smallest example uses one observable vertex, hidden fiber {1,2,3}, gauge group S_3, and events a,b:

M_0: A=B=(12)
M_1: A=(12), B=(13).

Both individual permutation traces equal one. The composite has trace three in M_0 and zero in M_1, so joint loops recover relative operator orientation. Once recovered, unseen behavior is ordinary permutation multiplication. No loop trace reveals which hidden point is currently occupied.

Local loops are also incomplete: a flat U(1) connection on a noncontractible cycle can have zero local defect and nontrivial global holonomy. Unrestricted nonlinear actions admit compactly supported off-probe perturbations that preserve every finite loop test and alter a later composition.

4. Exact collapse test

For any finite proposal:

1. enumerate transport tuples modulo vertex gauge;
2. map each orbit to the proposed finite signature;
3. reject if one signature fiber contains two orbits differing on a target word/query;
4. repeat on joint (transport,current_state) orbits;
5. if every fiber is singleton, reconstruct a canonical tuple and run the ordinary recurrence U_(t+1)=A_(e_t) U_t;
6. reject novelty if this matched recurrence is exact;
7. under noise, use minimum signature separation Delta_n; vanishing Delta_n forces at least Omega(Delta_n^-2) samples.

Incomplete signatures fail identification; complete signatures reconstruct an established operator model.

5. Prior-art boundary and verdict

Connection reconstruction from loop holonomies, periodic-orbit cocycle identification, gauge-equivariant computation, synchronization, cycle consistency, and PSR/OOM operator learning already occupy every surviving case. Loop-based state correction additionally requires redundant state-bearing measurements and becomes synchronization, error correction, or denoising.

No CPU falsifier is authorized. Reconsideration requires naturally available loop observations that identify the joint action-state orbit with a uniform margin, correct runtime noise, and beat equally informed operator recurrence, PSR, synchronization, and ECC controls.