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Theorem · exact inverse-image closure and unchanged-graph resource bound

FPRD-T166

Finite-state causal preimages preserve stationary transport

Exact statement

Let T be a deterministic synchronous Mealy transducer with s states. If (d,k,g,q) lies in Tr(L), then (d,k,g,sq) lies in Tr(T^{-1}(L)). The compiled presentation builds exactly the source persistent graph and adds only the transducer state to finite control. Hence nonempty transport spectrum is invariant under finite-state synchronous causal coordinate changes with finite-state inverses.

StatusSelf-contained product proof with exhaustive finite-state audit
External reviewThe construction is a finite-state product argument. No external or specialist review of the transport-spectrum formulation is recorded.

Context

FPRD-T164 shows that forgotten-action projection can leave PEG and bounded stationary transport. This theorem isolates what cannot be responsible: deterministic finite-state preprocessing of the observed input.

Hypotheses and scope

  • T is total, deterministic, synchronous, and emits exactly one output letter for every input letter.
  • The source presentation is a finite deterministic scaffolding automaton with tuple (d,k,g,q).
  • For the two-way invariance corollary, the word bijection has a finite synchronous inverse.

Proof or evidence

The compiled control pairs the transducer state with the source scaffold control. Each input letter produces one source letter, after which the compiler copies the source label and every port descriptor exactly. Induction gives identical persistent graphs and source controls on the transduced word.

Finite-state causal preimage theorem →

Verification notes

The checker compares 5,898,240 product runs across all 256 two-state binary Mealy machines and 1,536 finite local-machine/environment families. It also checks 16,256 word roundtrips across all 64 statewise-invertible binary machines in the enumerated family.

Limitations

  • The theorem is an inverse-image result; it does not preserve direct images or existential projections.
  • It covers synchronous one-letter output, not erasing, expanding, delayed-output, nondeterministic, or two-way transductions.
  • The sq control bound is sufficient and need not be minimal.
  • Product closure under deterministic finite-state recoding is a standard automata idea; no literature-priority claim is made.

Open work

Use the theorem as the deterministic baseline when classifying which existential hidden-history operations destroy bounded stationary transport.