FPRD-T154: The universal row-count barrier

Statement

Let $\Sigma$ be a finite input alphabet of size $s\ge 1$. For a language $L\subseteq\Sigma^*$, a set of prefixes $P$, and a continuation family $C_h\subseteq\Sigma^{\le h}$, write

$$\rho_h(p)=(\mathbf 1_L(pc))_{c\in C_h}.$$

Then

$$|\rho_h(P)|\le 2^{|C_h|} \le 2^{N_s(h)}, \qquad N_s(h)=\sum_{i=0}^h s^i.$$

Now choose the single scaffold parameter tuple

$$d=\max(2,s),\qquad k=2,\qquad |\Gamma|=1,\qquad |Q|=1.$$

Let $R_{k,h}$ and $V_r$ be the readable radius and abstract-view count from FPRD-T153:

$$R_{2,0}=0,\qquad R_{2,h}=h+1\ (h\ge1),$$

$$V_0=2,\qquad V_{r+1}=1+2V_r^d.$$

For every $h\ge0$,

$$2^{N_s(h)}\le V_{R_{2,h}}.$$

Consequently, a lower-bound argument that uses only the cardinality of a finite-horizon Boolean response table can never exceed every numerical upper envelope supplied by FPRD-T153. Such an argument cannot by itself separate a language from all scaffolding automata. Any transfer to PEGs must account for reversal.

Proof

A response row has one Boolean coordinate per continuation, so there are at most $2^{|C_h|}$ rows. Since $C_h\subseteq\Sigma^{\le h}$, $|C_h|\le N_s(h)$.

The view recurrence satisfies

$$V_{r+1}=1+2V_r^d\ge V_r^d.$$

Since $V_0=2$, induction gives

$$V_r\ge 2^{d^r}.$$

At $h=0$, $N_s(0)=1$, $R_{2,0}=0$, and $2^{N_s(0)}=2=V_0$.

Suppose $h\ge1$. If $s\ge2$, then

$$N_s(h)=\frac{s^{h+1}-1}{s-1}\le s^{h+1}\le d^{h+1}.$$

If $s=1$, then

$$N_1(h)=h+1\le2^{h+1}=d^{h+1}.$$

Because $R_{2,h}=h+1$, in both cases

$$2^{N_s(h)}\le2^{d^{h+1}}\le V_{h+1}=V_{R_{2,h}}.$$

This proves the claim.

What the theorem does not say

The recurrence $V_r$ counts abstract unfolded views. It can include views that are unreachable in a particular automaton. Therefore the inequality does not construct a scaffold, show that the chosen tuple realizes every response table, or imply that every language is a PEG language.

It proves a limitation of one proof pattern only. Comparing a semantic lower bound on the number of response rows with the generic T153 upper envelope cannot yield a parameter-uniform separation. A successful lower bound must use more structure, for example:

• compatibility of rows as the cut advances;
• coherence between successive horizons;
• algebraic restrictions on response tables;
• realizability constraints on unfolded views;
• the local cost of transporting, merging, and allocating persistent demands.

Relation to existing theory

The counting step is elementary and the response-table viewpoint is classical Myhill–Nerode and deterministic one-way communication. The contribution here is methodological: it closes the most direct cardinality route suggested by FPRD-T152 and FPRD-T153. No literature-priority claim is made.

Executable audit

verify_row_count_barrier.py checks the exact view recurrence on a tractable grid, checks the symbolic exponent inequality on a much larger grid, and records failures of two deliberately weakened parameter choices.