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Theorem · exact cut factorization and capacity transfer

FPRD-T152

Cut responses factor exactly through their distinct rows

Exact statement

For every finite cut-response system, the minimum number of exact deterministic summary states equals the number of distinct response rows. Hence the fixed-length one-way cost is ceil(log2 R) bits for R rows. For linear observers a·x over F_q^n whose span has dimension r, the exact minimum is q^r states, or ceil(r log2 q) bits. Any resource class with at most B(s) cut presentations can realize the system only if R<=B(s).

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

Context

The theorem identifies the exact information bottleneck at a cut before any machine-specific local-update restrictions are imposed.

Hypotheses and scope

  • The past set, continuation set, and answer alphabet are finite for the cardinality statement.
  • Summaries and decoders are deterministic and exact on the selected cut system.
  • The linear specialization uses the standard nondegenerate pairing over a finite field.

Proof or evidence

Summary fibres must refine equality of response rows, while the row itself is a matching summary. In the linear case the response-map kernel is the annihilator of the observer span. The checker covers 74,954 Boolean matrices, 4,066,394 undersized encodings, and 65,812 binary observer families.

Cut-response factorization →

Verification notes

The proof was checked for unused summary symbols, non-power-of-two row counts, zero observers, dependent observers, the annihilator dimension, and the distinction between capacity and local update costs. Mutants based on largest support and observer count fail in 32,028 and 64,365 audited families respectively.

Limitations

  • The theorem is a standard deterministic one-way communication fact plus rank-nullity, organized here as an FPRD bridge; no literature novelty is claimed.
  • A large response dimension does not imply large scaffold degree, descriptor distance, allocation, or per-symbol update work.
  • Applying the capacity transfer requires an independently proved upper bound B(s) for the model in question.

Open work

Use FPRD-T154 to replace row cardinality by a structural response invariant or a tighter reachable-presentation capacity theorem.