toClassicalEngine
plain-language theorem explainer
The definition projects the classical phase-budget engine out of its logic-native wrapper structure. Number theorists applying phase budgets to the Erdős-Straus conjecture cite it when switching between recovered-natural and classical presentations of the engine. It is realized by direct field projection on the input structure.
Claim. Let $E$ be a logic-native phase-budget engine with bound map on logic naturals, classical engine component, and compatibility condition. Then toClassicalEngine($E$) equals the classical engine component of $E$.
background
The module supplies a logic-native wrapper for the Erdős-Straus phase-budget engine. PhaseBudgetEngineLogic is the structure whose bound and input carrier are LogicNat while the finite gate witness is transported to the classical engine surface. The classical PhaseBudgetEngine supplies a bounded subset-product hit for each residual trap, with the bound ensuring an admissible hard gate exists inside the subset-product phase hit.
proof idea
One-line definition that returns the classical_engine field of the input PhaseBudgetEngineLogic structure.
why it matters
This definition completes the bridge module by exposing the classical engine from the logic wrapper, enabling the recovered-rational reading of the residual theorem. It connects the logic phase budget to the eight-tick phases (kπ/4 for k in Fin 8) used in the engine construction and to the T7 octave periodicity in the Recognition framework.
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