Sewing-seam counterexamples and the missing hypotheses

Status: independent attack note. No novelty claim.

1. The smallest carrier-level counterexample

Let

$$L_{\square}=\{a^n:n\text{ is a perfect square}\}$$

over the one-letter alphabet $\{a\}$.

For every prefix grade $m$ and every continuation horizon $H$, the selected prefix family is the singleton $\{a^m\}$. Consequently its fixed-cut resolving field has a one-cell factorization with zero tracks:

$$X_{m,H}=\{a^m\},\qquad |C_{m,H}|=1,\qquad k=0.$$

The decoder is the square-indicator window

$$d_{m,H}(a^t)=\mathbf 1[m+t\text{ is a square}] \qquad(0\le t\le H).$$

These literal decoders have all the semantic compatibility one can ask of the target:

• restricting $H+1$ to $H$ gives the preceding horizon;
• moving one $a$ across the cut gives $$d_{m+1,H}(a^t)=d_{m,H+1}(a^{t+1});$$
• all restriction/action squares commute.

Thus every typed fixed-cut carrier has width one, uniformly in both parameters, and the literal cut field is diagonally compatible.

Nevertheless no finite online zero-track controller recognizes $L_{\square}$. A unary finite-state output sequence is ultimately periodic. The square indicator is not: the gaps between consecutive squares are $2n+1$, hence unbounded, while an infinite ultimately periodic set has bounded gaps among its accepted lengths.

This is minimal in alphabet size, output size, track count, and typed-cell width. It proves that:

Compatible typed fixed-cut factorizations of uniformly bounded marginal width do not automatically sew.

The missing information is cross-grade identity. The anonymous singleton at grade $m$ cannot be identified with the singleton at grade $m+1$, because their readout/future behavior drifts with the grade.

2. What detects the failure

Bundle all continuation lengths through $H$:

$$B_H(a^m)= \bigl(\mathbf1[m+t\text{ is a square}]\bigr)_{0\le t\le H}.$$

Then assemble across all prefix grades:

$$Q_H(L_{\square})=\{B_H(a^m):m\ge0\}.$$

This global at-most quotient is not bounded. In fact

$$|Q_H(L_{\square})|\ge H+2.$$

Choose a square separated from its neighboring squares by gaps greater than $H$. The $H+1$ windows in which its unique $1$ occupies each possible coordinate are distinct, and a window wholly inside a long gap supplies the all-zero row.

For a state-only doctrine, the exact missing condition is therefore the M26 global condition:

$$\sup_H |Q_H(L)|<\infty,$$

equivalently the existence of a genuine global plateau $$|Q_H(L)|=|Q_{H+1}(L)|$$ at some $H$. With the literal readout and exposure-one prefix actions retained, that plateau makes bounded future equality a finite-index right congruence and compiles the exact online DFA.

No condition stated only inside each typed cell $C_{m,H}$ can replace this cross-grade assembly.

3. A separate reachability failure with one supplied track

Carrier assembly is not the only seam. Supplied track positions may fail to be causally reachable.

On well-formed words

$$\lhd a^n\#a^t,$$

let the target accept exactly when $t=n$. At the cut after $\#$, supply one old-address track at the last $a$ of the first block (or at $\lhd$ when $n=0$). A fixed finite continuation controller recedes once per new $a$. It accepts exactly on the first node at which the track reads $\lhd$, and moves to a rejecting sink after another symbol. The same post-cut controller works for every continuation horizon.

If the online track doctrine applies the same no-birth/no-right rule while the first block arrives, this supplied initialization is unreachable. The required root address grows with $n$, while every live track’s absolute address is nonincreasing and an inactive track cannot be activated.

This example is conditional on that explicit upstream doctrine; without an upstream rule, “online reachability” is not yet a well-posed claim. Its point is exact:

A coherent post-cut observation process does not supply a causal initializer.

The necessary local bond is an upstream transition law $I$ satisfying, for every prefix $u$ and next symbol $a$,

$$I(ua)=\Delta_{\rm pre}(I(u),a,\text{permitted local observation}),$$

with the declared track actions. Arbitrary assignments $u\mapsto (q_u,\mathbf p_u)$ are advice, not reachability witnesses.

4. A precise sufficient sewing package

For a fixed resource budget, let $\mathcal K$ be one normalized presentation stratum containing:

1. one finite global role palette rather than separately grade-tagged role sets;
2. grade-independent readout and transition tables;
3. the declared finite sensor and track dynamics;
4. a causal upstream initializer satisfying the one-symbol bond above; and
5. all names needed to restrict a presentation from horizon $H+1$ to horizon $H$.

Let $F_H\subseteq\mathcal K$ be the presentations correct through horizon $H$. Exact restriction gives

$$F_{H+1}\subseteq F_H.$$

If $\mathcal K$ is finite, nonemptiness of every $F_H$ implies $\bigcap_HF_H\ne\varnothing$. More generally the same conclusion holds when $\mathcal K$ is compact and every $F_H$ is closed. Equivalently, in an inverse-system formulation, the finite-horizon presentations must live in one finitely branching restriction-closed tree; König’s lemma supplies an infinite compatible branch.

Each adjective is load-bearing:

• without a global palette, the square example escapes through grade tags;
• without readout/action naturality, a reused carrier name can change its meaning with the cut;
• without the upstream bond, supplied positions are arbitrary advice;
• without restriction closure, independently chosen $P_H$ need not form a branch; and
• without finiteness/compactness, exact presentations may escape every fixed core as $H$ grows.

Accordingly, the valid implication is not

$$\text{bounded compatible cut widths}\Longrightarrow \text{one online presentation}.$$

It is:

$$\boxed{ \begin{array}{c} \text{nonempty exact feasible sets inside one fixed compact presentation core}\\ +\ \text{closed restriction bonds}\\ +\ \text{local upstream reachability} \end{array} \Longrightarrow \text{one horizon-independent online presentation}. }$$