IndisputableMonolith.NumberTheory.ErdosStrausBoxPhase
The ErdosStrausBoxPhase module supplies the core combinatorial object for square-budget phase analysis in the Erdős-Straus setting. It encodes a divisor exponent-box point for N squared via a divisor d and its complement e, serving as the finite gate layer that RotationHierarchy isolates. Downstream modules import it to close finite sums and transport results to LogicNat and LogicRat representations.
claimA divisor exponent-box point for the square budget $N^2$, represented by a divisor $d$ and its complementary divisor $e$.
background
This module sits inside the NumberTheory domain and depends on ErdosStrausRotationHierarchy, which converts the RCL attack into a theorem-shaped skeleton by proving finite gate pieces and isolating prime phase separation. The supplied DOC_COMMENT defines the central object as the divisor exponent-box point for $N^2$ given by complementary divisors $d$ and $e$. Upstream work frames the hierarchy as supplying the combinatorial structures needed before analytic distribution theorems can be applied.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module feeds the four downstream consumers: LogicErdosStrausBoxPhase (which transports the box-phase theorem to LogicRat via LogicErdosStrausRCL), PhaseFailureCost (which accumulates positive unresolved phase from failed gates), PrimePhaseDistribution (which instantiates BoundedBalancedSearchEngine), and SubsetProductPhase (which converts supplied phases into unit-fraction constructions). It therefore closes the finite/gate layer required by the Erdős-Straus recognition rotation hierarchy before the remaining prime-distribution engine is invoked.
scope and limits
- Does not prove the full Erdős-Straus conjecture.
- Does not supply the prime phase distribution theorem.
- Does not construct unit fractions directly.
- Does not address infinite or non-square budgets.