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Theorem · exact prescribed-size global overhead

FPRD-SH-T18

Disjoint-support total overhead has an exact finite-size formula

Exact statement

For a one-support-per-letter policy with pairwise-disjoint active supports, let NaN_a be the number of useful states in the synchronous block advanced by letter aa, let rr be the number of active letters, and let M=∏aNaM=\prod_aN_a. Then Δ(E)=∑a(Na−1)∏b≠aNb−(M−1)=(r−1)M−M∑aNa−1+1\Delta(E)=\sum_a(N_a-1)\prod_{b\ne a}N_b-(M-1)=(r-1)M-M\sum_aN_a^{-1}+1. Pure shuffle is the singleton-support specialization.

StatusSelf-contained Cartesian-factorization proof; strict-gap scope corrected and independently audited
External reviewNo documented external or specialist review of these FPRD results is recorded.

Context

Independent support blocks give the first exact global answer beyond the local clique theorem and include arbitrary-arity pure shuffle.

Proof or evidence

The useful product factorizes across blocks. Signature a targets exactly the tuples with a noninitial a-block, and the target sets cover every tuple except the all-initial state.

Disjoint-support overhead theorem →

Verification notes

All 3,905 state-count vectors through arity five, comprising 809,710 noninitial product states, satisfy the formula and exact clique-bound gap.

Limitations

  • Active supports must be pairwise disjoint; overlapping supports couple reachability and fall outside the proof.
  • For at least two active blocks and at least one block of size greater than one, the clique bound is strictly larger than the exact finite overhead. With one active block, both quantities are zero.
  • Novelty is plausible but unconfirmed; no external review is recorded.

Open work

Determine the exact prescribed-size maximum when one-support events overlap.