boxDivisorLogic
This definition projects the divisor field from a recovered divisor-exponent box point over logic-native naturals. Number theorists adapting the Erdős-Straus square-budget phase to the logic setting cite it when splitting the pair (d, e) before RCL transport. The implementation is a direct field projection.
claimLet $N$ be a logic-natural number and let $(d,e)$ be a divisor-exponent box point for $N$, so that $d,e>0$ and $d·e=N^2$. The divisor component is then $d$.
background
The module supplies a logic-native adapter for the Erdős-Straus square-budget box phase. Native structures live over LogicNat; the final theorem moves to the classical box-phase result and returns a LogicRat via the RCL adapter. LogicNat is the inductive type forced by the Law of Logic: identity is the zero-cost unit and step iterates the generator, reproducing the multiplicative orbit {1, γ, γ², …}. DivisorExponentBoxLogic N is the structure holding d and e in LogicNat together with positivity witnesses and the square-budget equation d·e = N·N.
proof idea
One-line wrapper that returns the d field of the input DivisorExponentBoxLogic structure.
why it matters in Recognition Science
The definition supplies the divisor projection required by the logic-native box-phase adapter. It sits inside the Erdős-Straus development that ultimately feeds the RCL transport to classical naturals. Within Recognition Science it supports the number-theoretic side of the forcing chain by keeping the square-budget decomposition inside the LogicNat orbit.
scope and limits
- Does not prove existence of divisor-exponent boxes.
- Does not transport the result to classical natural numbers.
- Does not invoke the Recognition Composition Law.
- Does not address the full phase-hit or complement logic.
formal statement (Lean)
36def boxDivisorLogic {N : LogicNat} (box : DivisorExponentBoxLogic N) : LogicNat :=
proof body
Definition body.
37 box.d
38