IndisputableMonolith.NumberTheory.LogicErdosStrausBoxPhase
This module supplies divisibility lemmas for naturals recovered from logic representations inside the Erdős-Straus square-budget box phase. Researchers extending logic-native proofs of the conjecture cite it to transport combinatorial box structure through recovered arithmetic. It consists of targeted definitions and transport lemmas that adapt the classical divisor-exponent pairs to the logic layer.
claimFor a recovered natural $N$, a divisor-exponent box is a complementary pair $(d,e)$ satisfying $d e = N^2$; divisibility relations over recovered naturals hold for such pairs and are preserved under the logic-to-classical transport.
background
The Erdős-Straus Square-Budget Box Phase isolates the finite combinatorial part of the residual proof. For a square budget $N^2$, a divisor exponent box is represented by a complementary pair $(d,e)$ with $d e = N^2$. This logic module adapts that structure using recovered naturals supplied by ArithmeticFromLogic and the rational adapter from LogicErdosStrausRCL.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
This module feeds LogicPhaseBudgetBridge, the logic-native wrapper for the Erdős-Straus phase-budget engine. It packages the box phase behind recovered-natural inputs so the final residual theorem can be read as a recovered-rational theorem.
scope and limits
- Does not prove the full Erdős-Straus conjecture.
- Does not treat non-square budgets or infinite cases.
- Does not connect to the phi-ladder or physical constants.
- Does not supply computational search procedures.