IndisputableMonolith.NumberTheory.LogicPhaseBudgetBridge
LogicPhaseBudgetBridge supplies a phase-budget engine native to LogicNat carriers for bounds and inputs. It transports finite gate witnesses to the classical Erdős-Straus surface through imported adapters. Researchers extending Recognition Science number theory bounds would cite this for maintaining representation consistency across logic and classical layers. The module composes the logic Erdős-Straus box phase adapter with the phase-budget to RCL chain.
claimThe module defines a logic-native phase-budget engine $E$ over carrier LogicNat such that finite gate witnesses transport to the classical Erdős-Straus box-phase theorem, returning results as LogicRat via the RCL closure.
background
This module sits in the NumberTheory domain and bridges logic-native structures to classical ones. It imports LogicErdosStrausBoxPhase, which supplies a logic-native adapter for the Erdős-Straus square-budget box phase with native combinatorial structures over LogicNat; the final theorem transports to the existing ℕ box-phase theorem and returns the result as a LogicRat representation via LogicErdosStrausRCL. It also imports PhaseBudgetToErdosStraus, which composes the phase-budget interface with the already-proved Erdős-Straus RCL closure chain, conditional on a PhaseBudgetEngine package that encodes the T1/RCL budget, uniform failure floor, and finite gate enumeration producing bounded subset-product phase hits for every residual trap.
proof idea
This is a definition module, no proofs. It structures the argument by importing the logic Erdős-Straus box phase adapter and the phase budget to Erdős-Straus composer, then exposing the bridged engine through sibling definitions such as PhaseBudgetEngineLogic, toClassicalEngine, and erdos_straus_residual_from_phaseBudget_logic.
why it matters in Recognition Science
The module earns its place by enabling logic-native computations to interface with the proved Erdős-Straus RCL closure chain. It feeds the parent results in the Recognition framework's number theory applications for phase-budget engines, particularly supporting the T1/RCL budget and finite gate enumeration setting. It closes the representation gap between LogicNat carriers and classical surfaces without introducing new combinatorial content.
scope and limits
- Does not introduce new combinatorial theorems beyond transport.
- Does not modify the underlying RCL composition or classical Erdős-Straus results.
- Does not extend to infinite or non-finite gate cases.
- Does not supply explicit numerical bounds or concrete computations.