Constraint-Rarity in Reduced-History Toy Models: A Finite-Dimensional Tube-Scaling Diagnostic

This working draft develops a restricted finite-dimensional mathematical-physics toy framework for studying rarity of regular constraint classes on spaces of reduced-history representatives. A reduced-history constraint is represented by a specified mismatch function on an admissible finite-dimensional representative space. Under explicit regularity assumptions—compact embedded zero set, assumed effective codimension, bounded reference density, and local distance comparability of the mismatch—the relative small-tolerance tube volume scales with the corresponding codimension. The tube-scaling statement used here is a standard finite-dimensional tubular-neighborhood estimate. The intended contribution is not a new theorem in differential geometry, but an assumption-explicit reduced-history diagnostic: reference-measure bookkeeping, bounded-reweighting robustness, macrostate-preimage criteria, comparison-family criteria, construction-order guardrails, and an explicit no-inference result. Restart compatibility is treated only as one structured example of such a reduced-history constraint, generated by early/late comparison data and an admissible comparison family. The paper does not claim that the universe satisfies a restart or closure condition. This draft deliberately makes no claim of physical selection from geometric rarity. It does not derive the Born rule, solve the measurement problem, define a physical measure on the full gravitational solution space, replace quantum mechanics or general relativity, or make laboratory or cosmological predictions. It is intended as a cautious working draft and technical narrowing of earlier broader complete-history-selection notes. This working draft is a narrowed technical follow-up to earlier broader Zenodo priority and roadmap records. It should not be read as a complete foundations theory or as a claim of physical history selection.