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Theorem · parameter-uniform barrier for a cardinality method

FPRD-T154

Raw finite-horizon row counting cannot separate all scaffolds

Exact statement

For every finite input alphabet of size s, one fixed generic T153 parameter tuple, degree max(2,s), distance 2, one working label, and one control state, has a numerical abstract-view envelope at least as large as the maximum possible number of Boolean response rows over every continuation horizon h. Therefore a lower-bound proof using only finite-horizon response-row cardinality cannot exceed every T153 envelope and cannot by itself separate a language from all scaffolding automata. Any PEG transfer must account for reversal.

StatusSelf-contained elementary proof with exact and symbolic finite audit
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

T153 invited a direct strategy: make a language's finite-horizon response table larger than every fixed scaffold bound. T154 proves that the raw cardinality version of this strategy cannot work, because the continuation universe itself grows more slowly than one fixed abstract-view envelope.

Hypotheses and scope

  • The input alphabet is finite and nonempty.
  • The response coordinate for each selected continuation is Boolean.
  • The comparison uses the generic abstract-view bound V_r from T153, not a tighter reachable-view bound for a particular automaton.

Proof or evidence

There are at most 2^N response rows for N continuations. With d=max(2,s), k=2, one working label, and one control state, T153 has R=h+1 and V_R at least 2^(d^(h+1)), while the number of words of length at most h is at most d^(h+1). The checker verifies the exact recurrence on 40 cases and the exponent inequality on 16,448 cases.

Universal row-count barrier →

Verification notes

The proof was checked separately at horizon zero and for unary and nonunary alphabets. Mutants that reduce degree below the alphabet size or hold the readable radius at one fail in 8,510 and 16,383 audited cases respectively.

Limitations

  • The theorem is a no-go result for raw row cardinality, not a language-recognition theorem or a scaffold construction.
  • Abstract view counts can include unreachable views, so exceeding a tighter automaton-specific bound may still be useful.
  • Structured invariants involving cut advance, horizon coherence, algebra, allocation, or transport remain open.
  • The counting ingredients are elementary and no literature-priority or external-review claim is made.

Open work

Define a cut invariant that records advance compatibility, horizon coherence, or realizability structure rather than only the number of rows.