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Theorem and exact frontier

FPRD-T69

Exact fixed-cut snapshot–track factorization

Exact statement

At one selected cut, the minimum number of complete snapshots in a (k)-track factorization equals the minimum number of cells partitioning the past set such that each cell is resolving for the positions readable from some (k)-track layout. For (d=2^r) independent bits spaced (r+1) apart and an (r)-bit selector, the exact number is (2^{max(d-k,0)}).

StatusProved and independently rechecked
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The theorem splits information available at one cut between a finite snapshot label and symbols that future one-way tracks can read.

Definitions

  • A cell is resolving for a readable set A when equal data on A imply equal future answers throughout that cell.
  • The factorization converse starts from a supplied cut snapshot and supplied track placement.

Hypotheses and scope

  • The cut, horizon, track count, and labelled layouts are fixed.
  • Late-INDEX data sites are spaced r+1 apart and the selector has r future steps.

Proof or evidence

Deterministic induction maps a protocol to resolving snapshot cells. Conversely, a supplied cell label and layout let the continuation store the cone symbols and answer the future query. In late INDEX each track fixes at most one independent bit, so every cell has size at most 2^k; tracking k bits and snapshotting the rest attains the bound.

Fixed-cut factorization →

Verification notes

The archived verifier passed 17,844,572 decoder checks; the independent audit checked 38 exact frontier cases and the spacing-r hostile example.

Limitations

  • The converse is a strict fixed-cut continuation, not an online method for producing the snapshot or track layout from the prefix.
  • Track-only projection is vacuous when arbitrarily many snapshots are free; the meaningful object is the joint frontier.

Open work

Find compatibility conditions that sew optimal factorizations across multiple cuts or prove that no bounded compatible family exists.

Notes

Produced by the review lane; captured with attribution and independently verified here. The head-motion geometry matches mirrored one-way multi-head automata and trailing heads. The converse, however, begins from a snapshot and layout already supplied at the cut and is not itself an online 1DFA realization. The partition lemma is classical one-way communication complexity; the consequence shape occurs in Schweikardt (STACS 2009, Theorem 4.3) for a different language. The additive form is a comparison of two pinned regimes, not a universal law for all query-dependent probes.