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Theorem · multiplicative chain rule and obstruction localization

FPRD-T170

Causal selection spectra compose multiplicatively through projection chains

Exact statement

Let Gamma -> Sigma -> Delta be finite letter projections, A a language over Gamma, and B=pi(A). If (m,s) lies in Sel(A,pi) and (n,t) lies in Sel(B,rho), then (mn,st) lies in Sel(A,rho composed with pi). The law holds at every finite horizon. Hence an empty composite selection spectrum forces an empty spectrum at some constituent stage, and escape lengths satisfy the corresponding minimum lower bound.

StatusSelf-contained product-state proof with exhaustive bounded composition audit
External reviewThe proof is a direct finite-state product and cover-composition argument. No external or specialist review of the selection-spectrum formulation is recorded.

Context

T167 handles one hidden layer and T169 gives each fixed resource budget a finite escape horizon. T170 makes the invariant compositional across any finite pipeline of hidden coordinates.

Hypotheses and scope

  • Both maps are letter-to-letter projections between finite alphabets.
  • The intermediate language is exactly B=pi(A).
  • All selectors are total deterministic synchronous finite-state transducers.

Proof or evidence

Pairwise selector composition produces mn selectors with product states. A final accepted word first selects an accepting intermediate lift and then an accepting top lift. Sequential use of T167 also preserves sharing and yields the sharper transport control bound q^(mn) s^(mn) t^n.

Causal selection chain rule →

Verification notes

The saved checker audits 512 one-letter top languages, 306,240 valid pairs of stage-cover families, 918,720 composed cover checks, and 983,040 product-state run comparisons over two-state selectors, using a restricted inner-machine fixture family.

Limitations

  • The product profile is sufficient and need not be Pareto-minimal.
  • The converse obstruction statement fails in general because later projection may erase an earlier difficulty.
  • The theorem does not cover asynchronous or unbounded-memory selectors.
  • An empty selection spectrum remains presentation-relative and is not alone an SCA or PEG lower bound.

Open work

Use the chain rule to factor complicated forgotten-coordinate maps and isolate the first projection stage whose selection spectrum becomes empty.