Cut-response system and response dimension

Exact statement

A finite cut-response system records the answer beta(p,c) for each past p and possible continuation c. Its response dimension is log2 of the number of distinct response rows, separating the information needed for readiness across a continuation family from the replay support used by any one completed continuation.

StatusDefined self-containedly and connected exactly to residual rows

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Replay support is pointwise, whereas a prefix must remain ready for an entire family of continuations. The response table isolates exactly the information shared across that family.

Definitions

The response row of a past p is the vector of beta(p,c) over the selected continuation family.
Response dimension is log2 of the number of distinct response rows; the unrounded value is useful even when the row count is not a power of two.
For languages, beta(p,c)=1_L(pc); allowing every continuation recovers full right-residual equivalence.

Proof or evidence

The public proof note gives the exact factorization theorem, the language specialization, and a finite audit over every Boolean response matrix up to four by four.

Cut-response definition →

Verification notes

The definition was checked against finite Myhill--Nerode tests, deterministic one-way messages, T150 replay support, and T151 response profiles.

Limitations

The invariant depends on the chosen cut families when they are finite restrictions.
It measures retained answer information, not update radius, allocation, degree, or access latency.
No novelty claim is made for the communication-matrix formulation.

Open work

Extend the finite-horizon scaffold capacity bound to richer stationary frontier presentations.