Hostile audit: rooted-atlas compression and the epi-GCD proposal

Status. The negative statements below are proved from explicitly stated abstract hypotheses. Their application to the derived-prefix WFL construction is immediate from the already verified selector semantics. No claim is made that they settle a separately pinned two-hinge or D30 doctrine: the present materials do not define one common category of substrate updates for those two models.

1. Executive finding

There are two distinct questions which must not be conflated.

1. Does a canonical factorization of future substrate updates exist?
2. Does that factorization induce a finite, computable, reachable right congruence preserving all legal observations?

Even a positive answer to (1) does not imply (2). In any append-only doctrine whose late legal queries can select every archived coordinate, observational equivalence has $2^L$ classes at archive length $L$. No state-only, aggregate-row-only, or common-factor normalization can reduce that number while preserving exact Look. The only sound quotient is the archive modulo the legal-observation kernel, together with control and legality type. In the verified prefix-parity instance that kernel is equality.

The phrase “epi-GCD” is presently under-specified: epimorphism depends on a category, and GCD depends on a divisibility preorder. Boolean Look is idempotent and noncancellative, so neither residual uniqueness nor a canonical choice follows from the output rows alone.

2. Typed partial transition model

The weakest adequate abstraction has:

• reachable configurations $C$;
• a finite action alphabet $A$;
• partial transitions $c\cdot a$, with legality language $D(c)=\{w\in A^*:c\cdot w\text{ is defined}\}$;
• observations $O(c,w)$ for $w\in D(c)$, including every externally observable Look row and, if required by the semantics, intermediate output;
• a legality type $\tau(c)$ recording material extent, phase, reachable head bonds, and other data which determine which one-step actions are admitted.

Define typed future observational equivalence by

$$c\equiv_{\rm obs}d \quad\Longleftrightarrow\quad \tau(c)=\tau(d),\quad D(c)=D(d),\quad O(c,w)=O(d,w)\ \text{for all }w\in D(c). \tag{1}$$

The domain-equality clause is essential. Comparing observations only on the intersection $D(c)\cap D(d)$ incorrectly identifies a state where an action is legal with one where it is illegal.

Proposition 1 (exact right-congruence criterion)

If legality is prefix closed, transitions are deterministic, and $\tau(c)=\tau(d)$ implies equality of the admitted one-step action set, then $\equiv_{\rm obs}$ is a right congruence:

$$c\equiv_{\rm obs}d,\ a\in D(c)\cap A \quad\Longrightarrow\quad c\cdot a\equiv_{\rm obs}d\cdot a.$$

Proof. Domain equality gives legality of $a$ at both states. For every suffix $w$, $$w\in D(c\cdot a)\iff aw\in D(c)\iff aw\in D(d) \iff w\in D(d\cdot a).$$ Likewise $O(c\cdot a,w)$ is the continuation component of $O(c,aw)$, and the same holds for $d$. If intermediate output on action $a$ is observable, it must be included in $O(c,a)$; equality then also preserves the transition-output cocycle. Legality types after $a$ must either be part of the observation or be required to update congruentially. $\square$

This proposition is standard deterministic observational equivalence machinery. Its FPRD content lies in correctly choosing $C,\tau,D,O$, not in the congruence proof.

3. Archive-kernel no-go theorem

Let $Z_L\subseteq\Gamma^L$ be the reachable archives at grade $L$. Suppose that for every coordinate $i\le L$ there is a common legal continuation $s_i$ such that

$$O((q,z),s_i)=z_i. \tag{2}$$

No assumption is made about how $z$ was emitted.

Theorem 2 (selector-complete archive obstruction)

For fixed control and legality type, every exact observational quotient has at least $|Z_L|$ classes at grade $L$. In particular, if $Z_L=\{0,1\}^L$, it has at least $2^L$ classes.

Proof. For distinct $z,z'$, choose a coordinate $i$ with $z_i\ne z'_i$. The common legal suffix $s_i$ distinguishes the two configurations by (2), so they cannot be observationally equivalent. $\square$

Corollary 2.1 (no finite intrinsic quotient for prefix parity)

In the verified two-state prefix-parity encoder, the archive map $$b_1\ldots b_L\longmapsto (b_1,b_1\oplus b_2,\ldots,b_1\oplus\cdots\oplus b_L)$$ is bijective on $\{0,1\}^L$, and every coordinate has a legal unary selector. Hence:

• the state quotient has two classes;
• exact future-Look equivalence has $2^L$ classes;
• quotienting by archive length, Hamming weight, current Look, final control, an aggregate OR row, or any other noninjective archive summary is unsound;
• no factor/residual construction can have uniformly finite index unless it removes some selectors, charges and bounds the archive, or weakens exact observation.

This is stronger than saying residual state alone fails: any proposed normalization must be injective on the archive modulo the legal observation kernel.

4. Why a common divisor cannot evade the theorem

Let an update vector $U(c)$ be factorized as $$U(c)=g(c)\odot r(c) \tag{3}$$ in some proposed carrier category or monoid. If the quotient records only a finite residual class $[r(c)]$, then exactness requires that the omitted factor $g(c)$ be observationally inert:

$$g(c)\sim g(d) \quad\Rightarrow\quad O(c,w)=O(d,w) \text{ whenever }[r(c)]=[r(d)] \tag{4}$$ for every common legal suffix $w$.

Under selector completeness, (4) forces the combined recorded factor/residual data to distinguish all $z\in Z_L$. Thus an epi-GCD may provide a canonical representation, but not compression. If the divisor is discarded, normalization is unsound; if retained, its value carries the unbounded archive.

This conclusion is independent of cancellation. Noncancellativity creates an additional problem: residuals in (3) need not be unique.

Small Boolean witness to nonunique residuals

In the Boolean join monoid $(\{0,1\},\lor,0)$, $$1=1\lor0=1\lor1.$$ After extracting the same factor $g=1$, both $r=0$ and $r=1$ are valid. For vectors and Boolean DNF rows this ambiguity occurs coordinatewise. Therefore “divide by the common Look contribution” is not an operation until a residual selection rule is specified and proved compatible with extension. A canonical GCD alone does not provide that rule.

5. Why “epi-GCD” is not yet well-posed

The following choices are mathematically load-bearing.

1. Objects. Raw sheets, rooted material-addressed sheets, fiber-incidence hypergraphs, Boolean functions, update programs, or reachable configuration transformers are different objects.
2. Morphisms. Quotient maps, embeddings, simulations, physical folds, and semantic row maps have different epimorphisms.
3. Product/composition. Archive concatenation, sequential physical update, Boolean OR of fiber terms, and function composition are not the same operation.
4. Divisibility. Left, right, or two-sided factorization differ in a noncommutative update system.
5. Equality. Syntactic equality, physical isomorphism, equality modulo gauge, and equality under all legal observations yield different GCDs.
6. Admissibility. A semantic residual may not correspond to a reachable legal sheet, legal head placement, or bounded-role transition.

Without fixing these, existence, uniqueness, computability, and invariance of an epi-GCD have no determinate truth value.

6. D30/WFL mismatch

The recovered atlas theorem is for a pinned trailing-head doctrine: configurations consist of a finite role and $k$ head positions, each head holds or advances by one cell, and observations are the symbols under the heads. WFL configurations instead contain growing material, site predicates, fold geometry, fiber incidence, layer orientation/order, and a global existential-of-conjunctions Look.

A D30 legal bond $p(ua)-p(u)\in\{0,1\}^k$ does not encode:

• whether a WFL hinge is geometrically available;
• whether material would be duplicated or leave the admitted interval;
• layer order/orientation after a fold;
• which sites cohabit a future fiber;
• whether an emitted site remains addressable by a late selector.

Consequently a theorem for “two hinges or restricted D30” needs either two separate specializations or an explicit faithful translation preserving reachability, legal suffix domains, and Look. Treating them as one smallest subclass risks proving a congruence for the wrong transition system.

7. Exact hypotheses for a valid computable enriched quotient

A positive theorem can be stated under the following sufficient hypotheses.

H1 — Pinned effective transition system. There is a recursive, finitely represented configuration space and decidable deterministic partial action.

H2 — Congruential legality type. A finite type $\tau$ determines one-step legality, and $(\tau,[c],a)$ determines the successor type. Reachable head bonds and phase/material boundaries are included.

H3 — Complete observation. The observation records final Look and every semantically observable intermediate output. The accumulated archive is retained modulo $$z\equiv_K z' \iff \text{all common legal future queries give the same answer}.$$

H4 — Effective factorization category. Objects, morphisms, composition, equivalence, and left/right divisibility are specified; composition is associative; admissible factors and residuals remain reachable; and the chosen divisor/residual pair is computable.

H5 — Extension invariance. Equal enriched states have equal legality domains, equal transition outputs, and equal enriched successors under every legal action. This must be proved, not inferred from current Look equality.

H6 — Finite index or bounded grade. The typed observation-kernel quotient has finite index on the domain of interest. Computability of pairwise equivalence alone does not imply a finite-state compression.

H7 — Effective minimization. Either the reachable enriched transition system is finite, allowing partition refinement, or the doctrine supplies a termination/well-quasi-order theorem for symbolic refinement.

Under H1–H7, Proposition 1 gives a sound right congruence and ordinary partition refinement (in the finite case) computes the coarsest one. The epi-GCD is optional implementation structure, not the source of correctness.

8. Minimal failure patterns to test computationally

Any search for a two-hinge counterexample should ask for one of these finite cores:

1. domain core: same proposed row summary, but an action legal on only one configuration;
2. successor core: same summary and common legal action, but successor summaries differ;
3. cocycle core: same normalized residual, but transition output differs;
4. archive core: same factor/residual summary, but a legal selector distinguishes archived sites;
5. reachability core: algebraically valid factor/residual whose residual is not physically reachable;
6. geometry core: same Boolean row but incompatible hinge center, orientation, or layer order.

Each is a two-state/two-configuration witness against a claimed quotient. Only after these tests pass is it meaningful to search for a larger minimal-unsatisfiable atlas core.

9. Strongest audited conclusion

Proved: selector-complete exact Look forces the quotient to retain the entire reachable archive modulo the legal-observation kernel; in the prefix-parity family this gives exactly $2^L$ distinguishable grade-$L$ histories. A canonical divisor cannot turn this into finite index.

Proved: typed full-future observational equivalence is a right congruence under deterministic partial transitions and congruential legality.

Not proved: existence of a useful epi-GCD for any pinned two-hinge WFL category; a faithful WFL-to-D30 translation; finite index for any nontrivial archive-bearing subclass; minimality of a physical two-hinge obstruction.

Exact next hostile test: pin one finite two-hinge carrier with material addresses, orientation/layer order, partial fold actions, and selectors. Enumerate reachable configurations, then refine partitions using the six failure patterns above. If the quotient stabilizes, report its typed classes; if it splits, output the shortest distinguishing suffix and classify it as domain, successor, cocycle, archive, reachability, or geometry failure.