IndisputableMonolith.NumberTheory.PhaseBudgetToErdosStraus
This module supplies the phase-budget engine that delivers the bounded subset-product witness for the Erdős-Straus residual. Researchers tracing the Recognition Science derivation of number-theoretic statements cite it as the explicit computational bridge. It composes the effective prime phase input with the T1 phase budget bound into the required gate witness.
claimLet $\mathcal{E}$ be the phase-budget engine. Then $\mathcal{E}$ supplies a bounded subset-product witness for the residual Erdős-Straus statement, given the effective prime phase input and the T1/RCL budget bound.
background
The module resides in the NumberTheory domain. It imports EffectivePrimePhaseInput, whose doc states that the module supplies the exact prime-distribution input needed for the residual Erdős-Straus proof and proves that it implies PrimePhaseBoxDistribution. It also imports T1PhaseBudgetBound, whose doc states that a stable integer ledger with a finite T1/RCL budget cannot carry unbounded unresolved finite-phase cost.
The local setting treats the physical budget as an explicit interface. The module defines the phase-budget engine as the structure that supplies the actual bounded subset-product hit, completing the bridge from finite phase separation and failure-cost accumulation to an explicit gate witness.
proof idea
This is a definition module, no proofs. It assembles the imported prime-phase input and T1 budget bound into the engine definition together with the maps effectivePrimePhaseInput_of_phaseBudgetEngine, boundedBalancedSearch_of_phaseBudget, and erdos_straus_residual_from_phaseBudget.
why it matters in Recognition Science
The module supplies the phase-budget engine consumed by the Erdős-Straus residual proof chain. It feeds LogicPhaseBudgetBridge, which packages the engine behind recovered-natural inputs, and PhaseBudgetEngineFromRS, which converts the recovered-ledger visibility engine into this form. It completes the remaining physical bridge from the T1/RCL stable-budget interface to the explicit gate witness.
scope and limits
- Does not prove the Erdős-Straus conjecture outright.
- Does not import the bit-rotted PrimeDistributionBridge.
- Does not hide the T1/RCL budget assumption.
- Does not supply numerical verification of the engine.