IndisputableMonolith.NumberTheory.LogicErdosStrausRCL
This module supplies logic-native Erdős-Straus rational representations over LogicNat and LogicRat. It bridges the classical RCL algebraic reduction to the logic setting via equivalence maps. Researchers developing foundational logic versions of the Erdős-Straus conjecture cite it for transport between domains. The module consists of definitions and equivalence theorems rather than a single proof.
claimThe module defines a logic-native predicate asserting existence of positive rationals $x,y,z$ satisfying the residual equation $c/N = 1/y + 1/z$ (with $c=4x-n$, $N=nx$) after the first denominator choice, together with equivalence theorems to the classical rational representation.
background
The module operates inside the logic foundations of number theory. It imports RationalsFromLogic to obtain LogicNat and LogicRat structures, and ErdosStrausRCL whose doc states: "After choosing the first denominator x, the residual equation is c / N = 1 / y + 1 / z, with c = 4x - n and N = nx."
This supplies the logic-native counterpart of the Erdős-Straus representation predicate and the transport maps between logic and classical formulations.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
This module feeds LogicErdosStrausBoxPhase, whose doc states it "transports to the existing ℕ box-phase theorem, then returns the result as a LogicRat representation via LogicErdosStrausRCL." It supplies the required logic-native rational representation step for the Erdős-Straus RCL ledger reduction inside the Recognition framework.
scope and limits
- Does not prove the Erdős-Straus conjecture.
- Does not contain the box-phase algorithm itself.
- Does not extend the representation beyond positive rationals.
- Does not address numerical verification or explicit solutions.