Models of computation

Causal factors in finite persistent carriers

A complete future observation can preserve an entire finite archive even when the controller is trivial. Once each target may choose its own continuation frame, greatest common factors can fail to exist: the smallest example in this cyclic family is a pentagon.

FPRD-D32 · finite semantic model

The cyclic Boolean two-hinge carrier

Let $X_n=\mathbb Z/n\mathbb Z$ and use the two reflections

h_0(i)=-i,\qquad h_1(i)=2-i\pmod n.

Their group is $G_n=\langle h_0,h_1\rangle$ . A persistent archive is a mask $S\subseteq X_n$ , and an update $x=(p,A)$ acts by

T_x(S)=A\cup p(S).

Raw multiplication satisfies $T_{xy}=T_x\circ T_y$ . Temporal composition is written in execution order, $x;y=yx$ , so after prefix $(p,A)$ and continuation $(q,B)$ the archive is $B\cup q(A)$ .

If $F_1,\ldots,F_t$ are the group orbits, the observable is true exactly when some entire orbit has been archived. The complete future row of a prefix is

R_x(y)=\operatorname{Look}(T_y(T_x(\varnothing))).

Equality of complete rows is a right congruence for temporal extension: if $R_x=R_{x'}$ , then $R_{x;y}=R_{x';y}$ for every continuation $y$ .

The future-row quotient has an exact size

Every mask containing a complete orbit has the constant-one future row. Otherwise, the inclusion-minimal accepting suffixes are the nonempty deficits $F_j\setminus A$ , which recover the mask independently on every orbit. Therefore

\#(2^{X_n}/{\equiv_R}) =1+\prod_{j=1}^{t}\bigl(2^{|F_j|}-1\bigr).

When the action is transitive, the quotient has $2^n$ classes and retains every archive bit. At five sites the action is transitive, so a row class is exactly one archive mask. Transitivity is not automatic: at four sites the two orbit fibres produce ten classes rather than sixteen.

Causal factors are movable subpatterns

Define the causal factor preorder by

[d]\preceq_R[v] \quad\Longleftrightarrow\quad \exists y\;[d;y]=[v].

At five sites, where rows are masks, this becomes

[A]\preceq_R[C] \quad\Longleftrightarrow\quad \exists q\in G_5:\ q(A)\subseteq C.

For targets $C_1,\ldots,C_k$ , a greatest common factor exists up to temporal units exactly when the inclusion-maximal masks that can be moved into every target all lie in one $G_5$ -orbit. This is a finite, computable criterion.

FPRD-T81 · finite obstruction

An edge and a diagonal prevent a greatest factor

On the pentagon take

C_1=\{0,1,2\},\qquad C_2=\{0,1,3\}.

Their complements are respectively an edge and a diagonal, so the targets lie in different three-subset orbits. The masks

E=\{0,1\},\qquad F=\{0,2\}

are both common factors: $E$ is already contained in both targets, while a reflection moves $F$ into the second. Neither can be enlarged to a three-site common factor, because that would make the two targets congruent. Yet an edge and a diagonal are not associated under the dihedral action.

Thus $E$ and $F$ are two nonassociated maximal common factors. No greatest common causal semantic factor exists, even up to temporal units.

Exhaustive quotient enumeration finds no failure for one through four sites and this failure at five. The minimality statement is only for this pinned cyclic two-reflection family.

Raw and fixed-frame factorizations still have meets

In the raw semidirect product, left divisibility depends only on archive inclusion. Boolean masks therefore have ordinary intersection as a greatest common divisor. Fixed-frame complete-row classes also form a finite join-semilattice and have meets.

The obstruction appears only when each target may choose an independent residual frame. If the alignment is pinned, or one frame must serve the entire target tuple, the order reduces again to aligned inclusion and intersection supplies the meet.

Scope and open boundary

This theorem concerns semantic factors in a finite right action. It does not define a typed epimorphism class and does not contradict GCD results for typed streaming updates. It also omits interval extent, material addresses, layer order, irreversible stacking, and legal head bonds.

The next question is whether the pentagon certificate survives in a typed irreversible incidence category. If legal composition permits the two independent alignments, the obstruction persists; if it forces one alignment, that condition becomes a repair theorem.