BoundedVisibilityEngine
plain-language theorem explainer
The BoundedVisibilityEngine structure bundles a bound map, a cost function, and three properties that together guarantee bounded phase visibility for every non-identity reciprocal integer ledger. Number theorists working on residual closure for the Erdős-Straus problem inside Recognition Science cite it to obtain the visibility witness required by the phase-budget engine. The definition simply assembles the stable-budget and KTheta-floor hypotheses supplied by upstream ledger arithmetic without building the subset-product divisor.
Claim. A structure consisting of a function $b : ℕ → ℕ$, a cost function $c : ℕ → ℝ$, and proofs that for every positive integer $n ≠ 1$ possessing a reciprocal budget (i.e., $n$ divides some $N^2$), the cost $c$ yields a stable ledger budget, satisfies the uniform KTheta failure-floor condition on all failed gates, and supplies a witness that some admissible gate $c ≤ b(n)$ has a nonempty subset-product phase hit.
background
In the Bounded Phase Visibility module a recovered integer ledger is formalized as a NonIdentityReciprocal: a positive integer $n ≠ 1$ for which there exists $N$ with $n | N^2$. The KThetaFailureFloorHypothesis requires that every failed gate for such an $n$ carries cost at least KTheta. BoundedFiniteQuotientPhaseVisibility asserts the existence of an admissible gate below the supplied bound whose subset-product phase set is nonempty. The module premise is that a stable unresolved-phase budget together with the KTheta floor forces all phase invisibility to terminate within the bound.
proof idea
The declaration is a structure definition whose fields are the bound map, the cost function, and the three properties stable, floor, and visibility. The visibility field directly records the BoundedFiniteQuotientPhaseVisibility witness for each NonIdentityReciprocal. No tactics or lemmas are invoked; the structure is assembled from the floor and budget interfaces already supplied by the phase-budget engine.
why it matters
BoundedVisibilityEngine supplies the exact visibility package used by the theorem bounded_phase_visibility and by the construction phaseBudgetEngine_of_boundedVisibilityEngine that feeds the Erdős-Straus residual closure result. It thereby closes the loop from reciprocal ledger forcing and the eight-tick phase structure to finite phase-visibility bounds on the phi-ladder, ensuring that unresolved phases cannot persist for recovered ledgers. The definition touches the open question of explicit numerical bounds once concrete phase-budget engines are instantiated.
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