Finite-horizon fitting is closure membership

Exact statement

If matching observation cylinders form a neighborhood basis, then a target $x$ is feasible at resource $r$ through every finite horizon exactly when $x\in\overline{S_r}$ . Consequently a compact semantic stratum in a Hausdorff behavior space sews all uniformly bounded finite-horizon fits into one exact fit. For nested compact natural-number strata, exact demand is the supremum of finite-horizon demand.

StatusProved and internally audited

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

Context

The theorem gives the exact topological meaning of the quantifier pattern 'for every horizon there exists a bounded fit'. Compactness is a sufficient closedness criterion, not part of the closure equivalence itself.

Hypotheses and scope

Matching cylinders form a neighborhood basis at every target.
For the compact sewing conclusion, X is Hausdorff and the fixed stratum S_r is compact.
The scalar demand identity additionally uses nested compact strata indexed by natural numbers.

Proof or evidence

A point lies in the closure of a stratum exactly when every basic observation cylinder meets it. In a Hausdorff space a compact stratum is closed, so closure membership becomes exact membership.

Closure and compact sewing →

Verification notes

The proof was reconstructed independently and checked against the M33 proof note and closure audit. No material repair was required.

Limitations

Compactness is not automatic for a resource doctrine.
The theorem makes no novelty claim for its classical topological ingredients.

Open work

Identify useful presentation classes whose semantic resource strata are closed or compact, and document their topology explicitly.

Notes

The compactness and finite-intersection ingredients are classical and no novelty is claimed for them. Compactness, Hausdorffness, observation separation, the cylinder basis, and empty or unattained minima remain explicit proof obligations. Residual-size strata alone are calibration; nontrivial T60 needs checkable presentation-class hypotheses.