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Theorem · exact support-quotient factorization

FPRD-T155

Replayable memory factors through support and quotient

Exact statement

For a map F on a finite product source, the essential-coordinate set E(F) is the unique least sufficient coordinate support; the minimum exact deterministic summary has |F(X)| states; and |F(X)| is at most the product of the alphabets on E(F), with equality exactly when the induced map is injective. For a family of futures, essential support is the union of component supports. Every stable adaptive replay envelope contains E(F).

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

Context

FPRD-T150 gives per-execution replay supports and FPRD-T152 gives exact future-response quotients. This theorem supplies the missing semantic coordinate layer and proves how all three compare.

Hypotheses and scope

  • The source is a finite product of nonempty coordinate alphabets.
  • The demanded answer can be scalar or a finite vector of future responses.
  • Replay supports used in the operational comparison are stable under agreement on their named coordinates.

Proof or evidence

Sufficient coordinate sets are exactly the supersets of the essential coordinates. The answer map itself gives the minimum quotient. The checker exhausts 65,814 scalar Boolean maps and 65,812 two-component maps, for 131,626 maps total and 1,576,046 support-subset checks.

Support-quotient theorem →

Verification notes

The proof was checked for empty essential support, redundant coordinates, vector-valued answers, equality in the carrier bound, over-querying traces, parity compression, and tight coordinate selectors.

Limitations

  • The theorem compares static memories at one cut; it does not bound the work needed to update them over time.
  • Semantic support is coordinate-dependent, while quotient memory is invariant under source recoding.
  • The ingredients are standard finite-function support and quotient facts, organized here as an FPRD bridge; no literature-priority claim is made.
  • No external or specialist review is recorded.

Open work

Use D45/T157 and the later normal-form and selector results to study the surviving structural cost of stationary transport across cuts.