Reconciled hostile audit of the two-hinge observable epi-GCD

Scope and verdict

This audit uses exactly the finite Boolean incidence carrier pinned in README.md, but it tests semantic quotients rather than only raw masks.

The carrier is $X=\mathbb Z/n\mathbb Z$; the two named hinge symmetries generate a permutation group $G$; controlled sites form a finite append-only join-semilattice; and an update is

$$(g,a)(h,b)=(gh,\;a\vee g b). \tag{1}$$

There are two different results, separated by a load-bearing quantifier:

1. Raw updates and fixed-frame mask rows have GCDs.
2. Full future semantic update rows do not. The first failure in the pinned cyclic family occurs on five sites, where edge and diagonal target deficits have two nonassociated maximal common factors.

The apparent conflict is resolved by congruence direction. Complete future row equality is automatically congruential under appending a causal continuation, but need not be a two-sided congruence for the raw semidirect-product multiplication. Therefore its classes form a one-sided act, not the quotient monoid assumed in the first version of this audit.

This is not yet a WFL/THFL or restricted-D30 atlas-compression theorem. Physical folds are not merely units acting on a site semilattice, and legal heads, phases, layer order, interval boundaries, and causal reachability are absent.

1. Exact raw algebraic proof

Write $a\le b$ when $a\vee b=b$. In $M=G\ltimes L$,

$$(g,a)\mid_{\!L}(h,b) \quad\Longleftrightarrow\quad a\le b. \tag{2}$$

Necessity follows from (1). For sufficiency use the residual

$$(g^{-1}h,\;g^{-1}b),$$

because

$$(g,a)(g^{-1}h,g^{-1}b)=(h,a\vee b)=(h,b).$$

This proof is stronger than the powerset calculation using $g^{-1}(b\setminus a)$: it does not need relative complements and remains valid for every group-invariant append-only site semilattice.

For targets $(g_i,a_i)$, common left divisors are exactly the elements whose site component is a common lower bound of the $a_i$. A finite join-semilattice with bottom automatically has finite meets:

$$\bigwedge_i a_i = \bigvee\{c\in L:c\le a_i\text{ for every }i\}. \tag{3}$$

Therefore $(1,\bigwedge_i a_i)$ is a GCD. Every maximal common divisor has that same site component, and its group component differs only by a unit $(g,\bot)$. On the full Boolean mask lattice, (3) is ordinary intersection.

Computability is immediate for a finite explicit carrier: enumerate common lower bounds and join them. With a supplied meet operation, compute the iterated meet directly.

This proves the raw-mask result. It does not pass automatically through semantic equality of entire updates, because that equality entangles the root frame $g$ with the mask $a$.

2. Causal full semantic rows and the five-site obstruction

The composition orientation is load-bearing. Since $T_{(p,A)}(S)=A\cup pS$, the displayed product $xy$ represents $T_x\circ T_y$. Therefore a prefix update $x$ followed by continuation $y$ is the product $yx$, not $xy$.

For one transitive orbit $X$, define the causal complete row of a prefix update $x$ by

$$R_x(y)=\operatorname{Look}(yx).$$

Starting from the empty substrate, $x=(p,A)$ leaves exactly mask $A$; the prefix permutation is unobservable. For $y=(q,B)$,

$$R_x(y)=1 \quad\Longleftrightarrow\quad B\cup qA=X. \tag{4}$$

On a transitive fiber these rows distinguish every prefix mask $A$. The semantic factor order is

$$[A]\mid_R[C] \quad\Longleftrightarrow\quad \exists q\in G,\ qA\subseteq C. \tag{5}$$

Indeed, appending $y=(q,B)$ changes the prefix mask to $B\cup qA$, which equals $C$ exactly when $qA\subseteq C$ and $B=C\setminus qA$. The residual may independently reframe each candidate factor for each target. Greatest common factors are therefore maximal common subpatterns, modulo $G$.

On $X=\mathbb Z/5\mathbb Z$, let

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

Two maximal common factors are

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

Each target triple contains a dihedral translate of both $E$ and $F$. The targets are in different three-subset orbits because their complements are, respectively, an edge and a diagonal. Thus no three-site common factor exists. The edge and diagonal factors are maximal and nonassociated. No causal semantic GCD exists.

The factor witnesses are completely explicit. With identity-rooted target updates, append $\{2\}$ or $\{3\}$ after $E$ to obtain $C_1$ or $C_2$. Append $\{1\}$ after $F$ to obtain $C_1$. For $C_2$, first use $h_0$ to send $F=\{0,2\}$ to $\{0,3\}$, then append $\{1\}$. These are causal continuations acting after the factor.

Exhaustive enumeration finds no such pair for $n\le4$ and finds this failure at $n=5$. This proves computational minimality within the pinned cyclic two-reflection family.

The precise error in the positive quotient argument

For masks in one fixed root frame, equality of $C\mapsto\operatorname{Look}(A\cup C)$ is a join congruence. But an update row is tested against continuations that may independently reframe the prefix mask. Complete future equality is congruential only in the causal continuation direction, not necessarily on both sides of the raw semidirect-product multiplication. The semantic classes cannot be multiplied as $G\ltimes(L/{\sim})$.

If residual reframing is forbidden ($q=1$ in (5)), the obstruction disappears: the unique greatest common factor is the ordinary intersection $\bigcap_jC_j$. Independent residual alignment is therefore load-bearing.

3. Current Look equality is not a legal quotient

Let the physical fibers be the $G$-orbits and

$$\operatorname{Look}(A) = \bigvee_F [F\subseteq A]. \tag{4}$$

It is tempting to identify two masks with the same current Look bit and seek a GCD of those classes. This is ill-posed because the kernel of (4) is generally not a right congruence.

The smallest witness for the pinned cyclic hinges is $n=3$, where there is one three-site fiber:

$$A=\varnothing,\qquad B=\{0\},\qquad C=\{1,2\}.$$

Both $A$ and $B$ currently have Look $0$, but after the same legal append $C$,

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

Consequently “divide modulo current Look” does not define a quotient monoid. Any proposed computation using only the current Boolean output is unsound.

4. Fixed-frame complete future rows repair congruence

Define the complete future row

$$R_A(C)=\operatorname{Look}(A\cup C) \qquad(C\subseteq X). \tag{5}$$

Equality of these fixed-frame mask rows is stable under appending another mask and under the hinge group, so it is a $G$-invariant join congruence. Its quotient is again a finite join-semilattice, and the theorem above gives a fixed-frame GCD. This statement must not be substituted for the full semantic update result of Section 2.

The quotient size is exact. If the orbit fibers are $F_1,\ldots,F_t$, then

$$\boxed{ \#(2^X/{\equiv_R}) = 1+\prod_{j=1}^t(2^{|F_j|}-1). } \tag{6}$$

All masks already containing a complete fiber have the same constant-one future row, giving the single extra class. Otherwise the inclusion-minimal future masks that turn Look on are exactly the nonempty, pairwise-disjoint deficits $F_j\setminus A$. They recover $A$ uniquely, proving (6).

For a transitive hinge action, (6) equals $2^{|X|}$: complete future rows do not compress the site archive at all. This is the finite incidence version of the selector-complete archive obstruction.

5. Exhaustive certificate

check_two_hinge_observable_gcd_audit.py independently verifies:

• future-row equivalence is a join and group congruence;
• the exact class-count formula (6);
• fixed-frame left divisibility is precisely the quotient semilattice order;
• every fixed-frame pair has a greatest common left divisor;
• equality of current Look fails congruence at the first available carrier.
• the full-semantic deficit preorder has no failure through $n=4$ and first fails at $n=5$.

The complete census through $n=4$ is:

Every tested fixed-frame pair has a GCD up to units. The certificate records both the minimal current-Look congruence witness at $n=3$ and the first full-semantic GCD obstruction at $n=5$.

6. Boundary conditions and next hostile test

The negative theorem is already a semantic obstruction inside the finite Boolean incidence abstraction, but is not yet a physical atlas obstruction. In particular:

1. modeling both hinges as permutations makes every hinge component a unit and removes irreversible stacking information, yet independent residual reframing still suffices to destroy the semantic GCD;
2. permitting arbitrary residual append masks erases legal-head and bounded-role reachability;
3. complete future equality is a sound right congruence, not necessarily a quotient monoid congruence, and its index can remain exponential;
4. a canonical factor representation does not imply a finite stationary right congruence.

The exact next hostile test should replace the unit action by a typed category of irreversible, material-addressed two-fold incidence maps, with objects recording phase, interval extent, orientation/layer data, and reachable head bonds. One must then close only under legal typed composition. If that category lacks pullback-like common refinements, incomparable maximal epi factors may appear; if it retains them, the remaining obstruction is finite index, not GCD existence.

Status

• Proved: the raw semilattice/unit GCD theorem; current-Look noncongruence; the causal prefix-mask factor law; the five-site edge/diagonal semantic obstruction; fixed-frame congruence and exact class count.
• Exhaustively verified: every fixed-frame quotient multiplication and pairwise GCD through $n=4$; no full-semantic obstruction for $n\le4$; first full-semantic obstruction at $n=5$.
• Not proved: any epi-GCD theorem for irreversible physical folds, legal-head D30 updates, or the full rooted atlas; any finite-index theorem for archive-bearing causal presentations.