boxComplementLogic
plain-language theorem explainer
boxComplementLogic extracts the exponent component from a recovered divisor-exponent box point over LogicNat. Number theorists adapting Erdős-Straus square-budget results to the logic-native setting cite it when recovering the second factor of the pair. The definition is realized as a direct field projection with no additional computation.
Claim. Let $N$ be an element of LogicNat and let $(d,e)$ be a divisor-exponent box point for $N$, so that $d>0$, $e>0$ and $d·e=N·N$. The extracted complement is the exponent $e$.
background
LogicNat is the inductive type representing natural numbers forced by the Law of Logic, built from an identity element and a step constructor that iterates the generator. DivisorExponentBoxLogic N is the structure holding two LogicNat values d and e together with positivity proofs and the square-budget equation d·e = N·N. The module supplies a logic-native adapter for the Erdős-Straus square-budget box phase that later transports results to classical naturals via LogicErdosStrausRCL.
proof idea
The definition is a one-line field projection that returns the e component of the supplied box structure.
why it matters
This definition supplies the logic-native extraction of the exponent in the divisor box for the Erdős-Straus phase. It forms part of the interface that allows the combinatorial box-phase results to be stated over LogicNat before transport to the classical setting. No downstream uses are recorded in the current graph.
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