Cyclic Boolean two-hinge carrier

Exact statement

For $X_n=\mathbb Z/n\mathbb Z$ , the two reflections (h_0(i)=-i) and (h_1(i)=2-i), and append-permutation updates $T_{(p,A)}(S)=A\cup p(S)$ , the observable is completion of a hinge-group orbit. Complete future rows identify causal prefixes, and $[d]\preceq_R[v]$ means that some temporal continuation carries the future-row class of (d) to that of (v).

StatusDefined self-containedly and used by the proved obstruction

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The model separates persistent archive data from controller state and compares prefixes by every possible future observation.

Definitions

Temporal composition is (x;y=yx), so a causal prefix is the inner factor in the raw semidirect product.
The future row is $R_x(y)=\operatorname{Look}(T_y(T_x(\varnothing)))$ .
The factor order is $[d]\preceq_R[v]$ when some continuation (y) satisfies ([d;y]=[v]).

Proof or evidence

The definition is instantiated explicitly for every finite carrier size and supports the exact row-count and five-site nonexistence theorem.

Carrier and causal order →

Verification notes

Temporal order, orbit completion, future-row equality, temporal units, and the distinction between semantic factors and typed epimorphisms were rechecked against the proof record and executable model.

Limitations

This is a finite Boolean incidence carrier, not an interval-folding or physical material model.
No category of legal typed physical updates or epimorphism class is included.

Open work

Determine which parts of the carrier survive in a typed irreversible incidence category with physically legal compositions.

Notes

This is a finite Boolean incidence carrier, not interval WFL/THFL or restricted D30. No category of legal typed physical updates or epi class is part of the definition, so its semantic factors must not be called epis. The archive mask is persistent substrate history, not controller state.