IndisputableMonolith.NumberTheory.ErdosStrausResidualClosed
This module establishes conditional residual closure for the Erdős-Straus representation problem: bounded phase visibility on recovered integer ledgers implies every trapped n admits such a representation. Researchers extending Recognition Science to classical Diophantine questions would cite it to close the residual step in the number-theory chain. The module imports the phase-budget engine and applies it directly via its sibling declaration.
claimConditional on bounded phase visibility for recovered integer ledgers: every residual trapped integer $n$ admits an Erdős-Straus representation of $4/n$.
background
The module belongs to the NumberTheory domain of the Recognition Science development. It imports Mathlib together with PhaseBudgetEngineFromRS, whose documentation states that it 'Turns the recovered-ledger bounded visibility engine into the phase-budget engine already consumed by the Erdős-Straus residual proof chain.' The central object is therefore the conditional closure property that finishes residual handling once the visibility budget is supplied.
proof idea
This is a module with no internal proofs. It imports the phase-budget engine from the upstream PhaseBudgetEngineFromRS result and exposes the closure statement through the sibling declaration erdos_straus_residual_closed.
why it matters in Recognition Science
The module supplies the conditional closure step required by the Erdős-Straus residual proof chain. It feeds any downstream theorem that consumes the phase-budget engine to handle trapped integers under the recovered-ledger visibility constraints of the Recognition Science framework.
scope and limits
- Does not prove the full Erdős-Straus conjecture without the visibility hypothesis.
- Does not remove or weaken the bounded phase visibility assumption.
- Does not supply explicit constructions or bounds, only existence.