Causal full-future epi-GCD failure on the five-site two-hinge carrier

Status and scope

The theorem below is exact for the finite Boolean two-hinge incidence carrier defined here. It is a counterexample to an epi-GCD argument that uses only append-only Boolean sites, the two hinge permutations, and all causally subsequent Look rows. It is not yet a counterexample for interval WFL/THFL or restricted D30: those doctrines additionally require interval geometry, orientation and layer order, physical reachability, and legal head bonds.

The controller in this carrier is a singleton. The subset of opened sites is the persistent archive; it is not folded into the controller state.

1. Carrier and causal composition

Let

$$X=\mathbb Z/5\mathbb Z$$

and let the two hinge bonds be the reflections

$$h_0(i)=-i,\qquad h_1(i)=2-i\pmod 5.$$

They generate the dihedral group $G=D_5$. A Boolean substrate is a set $S\subseteq X$ of opened sites. An update is a pair

$$x=(p,A)\in G\ltimes 2^X$$

acting by

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

Here $A$ is an append-only write mask. Algebraic multiplication is

$$(p,A)(q,B)=(pq,A\cup p(B)), \tag{1}$$

so $T_{xy}=T_x\circ T_y$. To make temporal order explicit, write

$$x;y:=yx. \tag{2}$$

Thus $x;y$ means “execute prefix $x$, then continuation $y$.”

The two hinges generate a single fiber $X$, so

$$\operatorname{Look}(S)=1\quad\Longleftrightarrow\quad S=X. \tag{3}$$

For a causal prefix $x$, retain its complete future observation row

$$R_x(y)=\operatorname{Look}(T_{x;y}(\varnothing)) =\operatorname{Look}(T_{yx}(\varnothing)). \tag{4}$$

Define $x\equiv_Rx'$ iff $R_x=R_{x'}$, and define causal semantic prefix divisibility by

$$[d]\mid_R[v]\quad\Longleftrightarrow\quad \exists y\; R_{d;y}=R_v. \tag{5}$$

Equality therefore uses every legal causally subsequent Look row, not merely the current Look bit.

2. Exact archive representation

Lemma 1 (full rows are exactly opened-site archives)

For $x=(p,A)$ and $y=(q,B)$,

$$R_x(y)=1 \quad\Longleftrightarrow\quad B\cup q(A)=X. \tag{6}$$

Consequently,

$$x\equiv_Rx'\quad\Longleftrightarrow\quad A(x)=A(x'). \tag{7}$$

The prefix permutation $p$ is observationally dead, but the entire archive mask $A$ survives.

Proof. By (1), $yx=(qp,B\cup q(A))$, so (6) follows from (3). If $A\ne A'$, take $q=1$. If $A\not\subseteq A'$, the continuation $B=X\setminus A$ accepts $A$ and rejects $A'$; the other strict inclusion direction is symmetric. If neither contains the other, either choice separates them. Thus equality of all rows is exactly equality of masks. $\square$

Because (4) is a complete continuation row, $\equiv_R$ is a right congruence for temporal composition:

$$x\equiv_Rx'\Longrightarrow x;y\equiv_Rx';y \qquad\text{for every update }y. \tag{8}$$

Indeed $R_{x;y}(z)=R_x(y;z)$. On the five-site carrier this computable congruence has exactly $2^5=32$ classes, one for each archive mask. It retains the observable archive kernel; the singleton controller contributes no additional classes.

Lemma 2 (causal semantic factor order)

For archive classes $A=A(d)$ and $C=A(v)$,

$$[d]\mid_R[v] \quad\Longleftrightarrow\quad \exists q\in G\;\;q(A)\subseteq C. \tag{9}$$

Proof. If the residual is $y=(q,B)$, then the archive of $d;y=yd$ is $B\cup q(A)$. By Lemma 1 it is semantically equal to $v$ iff $C=B\cup q(A)$. This is possible exactly when $q(A)\subseteq C$, in which case take $B=C\setminus q(A)$. $\square$

Temporal units are the updates $(q,\varnothing)$. Appending such a unit sends $A$ to $q(A)$. Thus semantic factors are associated exactly when their masks lie in the same $G$-orbit.

3. Exact finite characterization

For target archives $C_1,\ldots,C_k$, define their common-subpattern family

$$\mathcal I(C_1,\ldots,C_k)= \{A\subseteq X:\forall j\ \exists q_j\in G,\ q_j(A)\subseteq C_j\}. \tag{10}$$

Theorem 3 (causal semantic epi-GCD criterion)

A greatest common causal semantic prefix factor exists up to units iff the inclusion-maximal members of $\mathcal I(C_1,\ldots,C_k)$ form one $G$-orbit. In that case any member of that orbit is a representative.

This criterion is computable on every finite carrier by enumerating $2^X$ and $G$, and invariant under simultaneous relabeling

$$(X,G,C_1,\ldots,C_k)\mapsto (\phi X,\phi G\phi^{-1},\phi C_1,\ldots,\phi C_k).$$

Proof. Lemma 2 identifies common factors with $\mathcal I$, and larger embeddable patterns are greater factors. A greatest factor exists up to units exactly when all maximal common subpatterns are associated. Enumeration proves computability. Conjugation preserves inclusion, orbits, maximality, and (9). $\square$

4. Small explicit failure

Take the target archives

$$C_1=\{0,1,2\},\qquad C_2=\{0,1,3\}. \tag{11}$$

The first is a three-consecutive-sites pattern; the complement of the second is a diagonal. Consider the two common factors

$$E=\{0,1\}\quad\text{(edge)},\qquad F=\{0,2\}\quad\text{(diagonal)}. \tag{12}$$

Both embed in both targets:

• $E,F\subseteq C_1$;
• $E\subseteq C_2$, while $h_0(F)=\{0,3\}\subseteq C_2$.

In update notation, take

$$v_1=(1,C_1),\quad v_2=(1,C_2),\quad d_E=(1,E),\quad d_F=(1,F).$$

Then

$$d_E;(1,\{2\})=v_1,\qquad d_E;(1,\{3\})=v_2,$$

and

$$d_F;(1,\{1\})=v_1,\qquad d_F;(h_0,\{1\})\equiv_Rv_2.$$

The last equivalence is semantic: the resulting archive is $\{0,1,3\}=C_2$, although the permutation component differs.

The target triples are not $D_5$-congruent. Their complements are respectively the edge $\{3,4\}$ and the diagonal $\{2,4\}$, and dihedral symmetries preserve cyclic distance. Hence no three-site pattern embeds in both: a three-site common factor would have to be congruent to each target.

Theorem 4 (five-site causal semantic epi-GCD obstruction)

The two targets $C_1,C_2$ have common causal semantic prefix factors but no greatest common factor, even up to units.

Proof. $E$ and $F$ are common factors. No common factor has more than two sites, by the noncongruence of the target triples. Therefore both two-site factors are maximal. They are not associated because an edge and a diagonal lie in distinct $D_5$-orbits. If a greatest common factor existed, it would have to be associated to both maximal factors, a contradiction. $\square$

5. Checkable finite obstruction certificate

The certificate is

$$(X,h_0,h_1; C_1,C_2;E,F)$$

with four finite checks:

1. $h_0,h_1$ generate $D_5$;
2. $C_1,C_2$ lie in distinct three-subset orbits because their complements are an edge and a diagonal;
3. $E,F$ each embed, up to $D_5$, in both targets;
4. $E,F$ lie in distinct two-subset orbits.

Checks 2–4 prove maximality and nonassociation without a floating-point solver or heuristic search.

Exhaustive enumeration of the same two-reflection family for $|X|\le4$ finds no pair with two nonassociated maximal common causal semantic factors. Thus five sites are computationally minimal within this pinned cyclic family. This is not a claim of minimality over all physical two-hinge geometries.

6. Consequence and exact boundary

The causal orientation matters. The raw intersection-GCD calculation applies to outer/postcomposition factors $xy=T_x\circ T_y$; it does not answer the prefix-followed-by-future question, whose temporal product is $x;y=yx$. Under the correct causal order, full-future semantic divisibility is movable subpattern containment, and the pentagon destroys the GCD property.

Therefore an intrinsic atlas compression theorem cannot assume an epi-GCD from Boolean fiber incidence alone. It must either:

• retain enough rooted geometry to canonically align archive patterns;
• restrict the hinge action so maximal common subpatterns have a unique orbit;
• retain the whole finite antichain of maximal factors rather than one divisor; or
• bypass factorization and minimize the typed full-future transition system directly.

The hinge-unit freedom is load-bearing. If residuals cannot reframe the rooted carrier (so $q=1$ in (9)), the unique greatest semantic prefix factor has archive $\bigcap_jC_j$. Likewise, forcing one common alignment for the entire target vector may restore a canonical divisor. The counterexample uses the ordinary componentwise convention in which each target has its own legal residual continuation.

Still unproved are a faithful embedding of this carrier into interval two-hinge WFL/THFL, preservation of legal layer order and head bonds, and the corresponding statement for restricted D30.