boundedBalancedSearch_of_boundedVisibilityEngine
plain-language theorem explainer
Bounded visibility engines over recovered ledgers compose directly into bounded balanced search engines for the Erdős-Straus residual hierarchy. Number theorists working on the phase-budget formulation of the residual trap would cite this mapping. The definition is a one-line wrapper that composes the phase-budget extraction from the visibility engine with the balanced search construction from that phase-budget engine.
Claim. Let $E$ be a bounded visibility engine. The bounded balanced search engine obtained by first extracting the phase-budget engine from $E$ and then converting that engine to a balanced search engine is the image of $E$ under the given map.
background
BoundedVisibilityEngine is the structure whose fields are a bound map, a cost function from naturals to reals, and a stability predicate ensuring that non-identity reciprocal integers satisfy the stable integer ledger budget condition. BoundedBalancedSearchEngine is the stronger structure whose bound_ok field asserts that every residual trap admits an admissible hard gate $c$ below the bound that carries balanced pair phase support. The module converts a recovered-ledger visibility engine into the phase-budget engine already required by the Erdős-Straus residual proof chain, relying on the upstream extraction that supplies the bound and hit conditions directly from visibility data.
proof idea
The definition is a one-line wrapper that applies phaseBudgetEngine_of_boundedVisibilityEngine to the input engine and immediately feeds the result into boundedBalancedSearch_of_phaseBudget.
why it matters
This declaration closes the interface from recovered-ledger visibility to the bounded balanced search engine consumed by the Erdős-Straus rotation hierarchy. It sits between the visibility structure and the search engine required for residual trap proofs, completing the phase-budget path inside the NumberTheory layer. No open questions are discharged here, but the composition removes an explicit dependency on separate phase-budget construction.
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