IndisputableMonolith.NumberTheory.PhaseBudgetToErdosStraus
The PhaseBudgetToErdosStraus module supplies the phase-budget engine that delivers the bounded subset-product hit for the Erdős-Straus residual. Upstream modules establish finite phase separation and T1/RCL budget stability; this module translates those facts into an explicit gate witness via definitions such as PhaseBudgetEngine and the connecting lemma erdos_straus_residual_from_phaseBudget. It consists of interface definitions and one-line wrappers that consume the prime-phase input and budget bound without new analytic content.
claimA phase-budget engine $E$ (with bounded search and finite T1/RCL cost) yields the Erdős-Straus residual witness via the effective prime-phase input, where the input implies PrimePhaseBoxDistribution and the budget satisfies the stable-ledger condition of no unbounded unresolved finite-phase cost.
background
The module operates in the NumberTheory domain and imports EffectivePrimePhaseInput, which states the exact prime-distribution input needed for the residual Erdős-Straus proof and proves that it implies PrimePhaseBoxDistribution while deliberately avoiding the bit-rotted PrimeDistributionBridge. It also imports T1PhaseBudgetBound, whose doc-comment states that a stable integer ledger with a finite T1/RCL budget cannot carry unbounded unresolved finite-phase cost, keeping the physical budget as an explicit interface.
proof idea
This is a definition module. It assembles PhaseBudgetEngine and the sibling lemmas effectivePrimePhaseInput_of_phaseBudget, boundedBalancedSearch_of_phaseBudget, and erdos_straus_residual_from_phaseBudget as one-line wrappers that apply the upstream prime-phase input and T1 budget bound directly to produce the residual gate witness.
why it matters in Recognition Science
The module feeds LogicPhaseBudgetBridge, which packages the engine behind recovered-natural inputs to read the residual as a recovered-rational theorem, and PhaseBudgetEngineFromRS, which converts the recovered-ledger visibility engine into the phase-budget engine consumed by the Erdős-Straus chain. It supplies the exact remaining physical bridge from the T1/RCL stable-budget interface to the explicit gate witness in the Recognition Science number-theory development.
scope and limits
- Does not prove the full Erdős-Straus conjecture.
- Does not import or rely on the bit-rotted PrimeDistributionBridge.
- Does not hide the T1/RCL budget assumptions behind the interface.
- Does not supply computational search implementations beyond the bounded definitions.