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Theorem · replay-width/residual-fan-out separation

FPRD-T151

Unit replay slices can have exponential residual fan-out

Exact statement

Let M contain the nonempty binary words whose middle symbol, at one-based position ceil(n/2), is 1. Every completed word has a one-source-cell replay certificate when length is public shape data. Nevertheless, for every r there are 2^r prefixes of length 2r and r continuations whose response profiles are all distinct. Thus M has at least 2^r residuals at that cut, and every exact deterministic one-pass recognizer needs at least 2^r configurations there.

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

Context

A small slice after one future is chosen need not mean that a prefix can forget most of its past. Different futures can select different cells and recover an entire stored vector one coordinate at a time.

Hypotheses and scope

  • Inputs are nonempty binary words and the selected middle position is ceil(n/2).
  • Completed word length is structural information; the replay certificate counts queried input cells.
  • The recognizer lower bound is deterministic, exact, one pass, and based on configurations after a fixed prefix length.

Proof or evidence

For p_b=0^r b and z_j=0^(2j-1), the middle position of p_b z_j is r+j, so the response equals b_j. The 2^r response vectors are therefore distinct. The checker passes 90,114 unit-slice selections and 90,114 single-bit mutations across all 8,190 prefixes through r=12.

Replay fan-out separation →

Verification notes

The proof was checked for one-based indexing, odd continuation length, prefix containment of the selected cell, response-vector injectivity, and the configuration lower-bound implication. The finite audit verifies every bit vector in its declared range.

Limitations

  • This is not a lower bound against scaffolding automata, PEGs, or persistent-pointer machines.
  • The theorem separates pointwise slice width from residual diversity; it does not yet quantify stationary address transport.
  • The residual proof is an elementary Myhill--Nerode/Index argument, and no literature novelty claim is made.

Open work

Use the explicit degree-two unary-query selector as a positive calibration, then study branching and merging demand families rather than a single backwards cursor.