Two-hinge common divisors do not by themselves compress rooted atlases

Date: 2026-07-29
Status: algebraic theorem proved; finite cases exhaustively checked; hostile audit completed; physical WFL/D30 transport not proved.

1. Determination

The leading epi-GCD attack splits into a positive algebraic theorem and a negative compression theorem.

For the pinned finite Boolean two-hinge append/permutation algebra, every nonempty vector of substrate updates has a canonical greatest common left factor:

$$\gcd_L\{(p_j,A_j)\}_j = \left(1,\bigcap_j A_j\right),$$

unique up to permutation units. It is computed by mask intersection and is equivariant under relabeling of the carrier.

This common divisor is not sufficient for a sound rooted-atlas quotient. Look is noninjective, and equal current Look values can have different legal future activation rows. In selector-complete archive doctrines, every exact quotient must retain the whole reachable archive modulo the legal-observation kernel. The prefix-parity archive therefore still has $2^L$ grade-$L$ classes despite two residual-control states.

Thus a divisor may canonicalize the representation of substrate updates; it cannot by itself supply finite intrinsic compression.

2. Pinned algebra

Let the finite carrier be $X=\mathbb Z/n\mathbb Z$. Take the two reflection generators

$$r_0(i)=-i,\qquad r_1(i)=2-i\pmod n$$

and let $G=\langle r_0,r_1\rangle$. A Boolean mask $A\subseteq X$ records opened sites. Writes are append-only. An update is a pair

$$(p,A),\qquad p\in G,\ A\subseteq X,$$

acting on a current mask $S$ by $S\mapsto p(S)\cup A$. Sequential composition is

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

The complete fibers are the $G$-orbits, and the pinned final observation is

$$\operatorname{Look}(A)= \bigvee_{F\in X/G}\ \bigwedge_{i\in F}[i\in A]. \tag{2}$$

This is a deliberately small incidence algebra. It includes two reflection-induced bonds, append-only sites, complete fibers, and OR-of-AND Look. It does not include interval boundaries, partial physical folds, orientation, layer order, thickness, head motion, or causal reachability.

3. Canonical common-left-divisor theorem

Theorem 1

For updates $d=(p,A)$ and $v=(q,B)$,

$$d\mid_L v\quad\Longleftrightarrow\quad A\subseteq B. \tag{3}$$

Consequently, for every nonempty vector $v_j=(p_j,A_j)$, the common left factors are exactly the pairs whose masks lie in $\bigcap_jA_j$. Their maximal elements all have mask $\bigcap_jA_j$, differ only by right multiplication by a permutation unit, and admit the canonical representative

$$d_*=\left(1,\bigcap_jA_j\right). \tag{4}$$

Proof

If $(p,A)(r,C)=(q,B)$, equation (1) gives $B=A\cup p(C)$, hence $A\subseteq B$.

Conversely, suppose $A\subseteq B$. Put

$$r=p^{-1}q,\qquad C=p^{-1}(B\setminus A).$$

Then

$$(p,A)(r,C) = (q,A\cup p(p^{-1}(B\setminus A))) =(q,B).$$

This proves (3). A pair is therefore a common left factor of every $v_j$ exactly when its mask is contained in $\bigcap_jA_j$. Maximal common factors have the full intersection mask. Their permutation parts are units and hence differ from (4) by right association. $\square$

Corollaries

1. Computability: compute the divisor by one bitwise intersection.
2. Invariance: simultaneous relabeling sends the intersection mask to the relabeled intersection mask.
3. Residual construction: for $d=(p,A)\mid_L(q,B)$, one exact residual is $$\left(p^{-1}q,\ p^{-1}(B\setminus A)\right).$$
4. Scope: this proves a greatest common left factor in the pinned update monoid. Calling it an epi-GCD for physical WFL would require a typed category of physical carriers and a proof identifying its epimorphisms.

4. Smallest nontrivial future-row obstruction

The two reflections become genuinely distinct on $n=3$, where their group is transitive and the sole complete fiber is $\{0,1,2\}$. Consider

$$A=\varnothing,\qquad B=\{0\}.$$

Both have current Look value zero. Append

$$C=\{1,2\}.$$

Then

$$\operatorname{Look}(A\cup C)=0,\qquad \operatorname{Look}(B\cup C)=1. \tag{5}$$

So equality of current Look is not right invariant even in the smallest nontrivial carrier. The certificate is finite and checkable:

carrier: Z/3Z
fibers:  {0,1,2}
states:  A=000, B=001
current Look: 0, 0
legal append: C=110
future Look:  0, 1
failure type: successor/future-activation row

This is a minimality statement within the pinned cyclic model: $n=1$ has no distinct equal-Look masks, while $n=2$ makes the chosen reflection generators coincide and is not a nontrivial two-hinge instance.

5. Exact observational quotient

For a deterministic partial transition doctrine, let $D(c)$ be the legal continuation language, let $\tau(c)$ record legality type (including reachable head bonds), and let $O(c,w)$ contain every externally observable future activation, Look value, and output-cocycle component. Define

$$c\equiv_{\mathrm{obs}}d \iff \tau(c)=\tau(d),\quad D(c)=D(d),\quad O(c,w)=O(d,w)\ \text{for all }w\in D(c). \tag{6}$$

Under deterministic transitions, prefix-closed legality, and congruential legality types, (6) is a right congruence. For a finite reachable carrier it is the coimage of the complete future-row map, equivalently the quotient by the intersection of all future-row kernels. It is the unique coarsest exact quotient and is computable by labeled partition refinement.

This is standard deterministic behavioral minimization. The FPRD specialization lies in choosing the complete observation and legality type: current Look, scalar residual state, or the common divisor alone are insufficient.

6. Selector-complete archive theorem

Let $Z_L\subseteq\Gamma^L$ be the reachable archives at grade $L$. Suppose every coordinate $i\le L$ has a common legal late selector $s_i$ satisfying

$$O((q,z),s_i)=z_i. \tag{7}$$

Theorem 2

For fixed control and legality type, every exact observational quotient has at least $|Z_L|$ classes.

Proof

Distinct archives differ at some coordinate $i$; the common legal suffix $s_i$ distinguishes them by (7). $\square$

For the verified prefix-parity encoder, the emitted archive map is bijective on $\{0,1\}^L$, so the exact quotient has $2^L$ grade-$L$ classes. This does not merely refute residual-state minimization. It proves that every normalization must remain injective on the archive modulo the legal observation kernel.

7. Why the common divisor is not sufficient

If a future-update row is factored as $U(c)=g(c)\odot r(c)$, then:

• discarding $g(c)$ is sound only if it is inert under every common legal future observation;
• retaining $g(c)$ preserves correctness but may retain an unbounded archive;
• Boolean noncancellation can leave residuals nonunique even when the divisor is canonical.

The smallest scalar warning is

$$1=1\lor0=1\lor1.$$

The same extracted Boolean factor has two residuals. The append/permutation monoid avoids this ambiguity by storing masks and permutations, but projecting to Look reintroduces the loss of information shown in (5).

8. Comparator boundary

The closest primary comparator is Bianchini and Puppis, Minimization of Streaming Transducers (LICS 2026). Their typed Hankel rows, epi-GCDs, and residual vectors supply the right general proof template. Choffrut-style subsequential minimization, coalgebraic behavioral equivalence, weighted automata normalization, canonical BDDs, and Boolean matrix factorization provide adjacent standard machinery.

None automatically supplies the missing FPRD facts: a typed physical update category, preservation of reachable hinge/head bonds, complete observable archives, finite index, and effective physical residuals. No novelty claim is made from the comparator search.

9. Verification

check_two_hinge_epi_gcd.py exhaustively checks:

• the divisibility criterion;
• common-factor counts;
• maximality and uniqueness up to right units;
• all update pairs through $n=4$;
• all update triples through $n=3$;
• current-Look collisions and shortest distinguishing append masks.

The pair census covers $4,16,2304,4096$ vectors for $n=1,2,3,4$. The arity-three census covers 110,592 vectors at $n=3$.

10. Precise limitations and next attack

Proved

• canonical common left divisor for the pinned two-hinge Boolean update monoid;
• existence, uniqueness up to units, computability, and relabeling invariance;
• a smallest nontrivial current-Look/future-row obstruction at carrier size 3;
• the selector-complete archive lower bound;
• exact typed future observation is a right congruence.

Computationally verified

The full small-case census stated in Section 9 and the finite obstruction in Section 4.

Not proved

• that the algebra is the exact image of sequential physical two-hinge folds;
• an epi-GCD theorem in a typed physical WFL category;
• preservation of interval, orientation, layer-order, reachability, or D30 hold/advance head bonds;
• finite index for any unbounded archive-bearing subclass;
• a faithful WFL-to-D30 transport.

Exact next attack

Pin the smallest physical two-hinge carrier with material intervals, orientation/layer order, partial fold legality, and late selectors. Enumerate only reachable configurations. Refine typed future-row partitions while classifying every split as domain, successor, cocycle, archive, reachability, or geometry failure. Then either:

1. prove that the intersection-mask divisor lifts to a typed physical epi-GCD and that its residual quotient is complete; or
2. output the first reachable minimal core where an algebraically valid factor/residual violates fold legality, layer order, or future Look.