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Theorem · finite-observer closure with explicit transport bounds

FPRD-T158

Finite future observers preserve stationary transport

Exact statement

Let (d,k,g,q) lie in Tr(L), let U be a nonempty finite set of fixed suffixes, and let b be any Boolean function of their membership answers. With R_(k,h)=0 for h=0 or k=0 and R_(k,h)=k+(h-1)(k-1) otherwise, put D_U=max over u in U of R_(k,|u|+1). Then the observer language {p : b((1_L(pu))_(u in U))=1} has transport tuple (d,D_U,g,q 2^|U|). The compiled machine uses exactly the original persistent graph. In particular, every fixed right quotient L/u preserves nonempty stationary transport spectrum.

StatusSelf-contained proof with compiled-control, locality, and sharpness audits
External reviewThe proof uses the locally established readable-cone theorem FPRD-T153; no external or specialist review of this resource-bound formulation is recorded.

Context

The residual tower describes all future answers, while the transport spectrum measures bounded persistent realizations. This theorem identifies a large family of semantic observations that can always be internalized without duplicating the persistent graph.

Hypotheses and scope

  • The source presentation is a finite deterministic scaffolding automaton with tuple (d,k,g,q).
  • The suffix family U is finite, nonempty, and fixed independently of the input prefix.
  • The observer applies an arbitrary Boolean function to the exact membership vector (1_L(pu)) indexed by U.

Proof or evidence

The new control stores the original state and the current future-answer vector. FPRD-T153 makes every next vector bit a finite function of the old control and radius R_(k,|u|+1) view, so one finite transition table updates all bits while appending exactly the original node. The audit checks 7,112 compiled runs, 21,336 vector coordinates, 1,820,672 Boolean outputs, 235,008 finite-scaffold future evaluations, and 24 sharp radius witnesses.

Finite future-observer theorem →

Verification notes

The checker covers empty suffixes, mixed suffix lengths, all 256 Boolean functions of three observer bits, unchanged graph evolution, same-view locality classes, distance zero in the radius formula, and the radius-minus-one chain mutation.

Limitations

  • The bound is sufficient and need not be Pareto-optimal for a particular language or observer family.
  • The sharpness statement is uniform over scaffold machines, not a lower bound for every individual observer language.
  • The theorem covers a fixed finite suffix family; growing, adaptive, or prefix-dependent future demands are not compiled here.
  • Closure under quotient-like operations is classical in many language families; no literature-priority claim is made beyond the explicit scaffold resource accounting.

Open work

Define and test observer families whose suffix sets grow with the cut, since every fixed finite right-context observer is now known to compile.