boxComplement
boxComplement extracts the complementary divisor e from a divisor-exponent box point for square budget N squared. Number theorists closing the Erdős-Straus conjecture via finite phase conditions cite this selector to obtain the second leg of a balanced pair. The definition is realized as a direct field projection on the box structure.
claimLet $(d,e)$ be a divisor-exponent box for square budget $N^2$, so $d,e>0$ and $d·e=N^2$. The complementary divisor selected by the box is $e$.
background
The Erdős-Straus square-budget box phase module isolates the finite combinatorial part of the residual Erdős-Straus proof. A divisor exponent box is a structure holding a divisor d and its complement e with d·e=N² together with positivity and square-budget proofs. This representation encodes exponent choices in the prime factorization of N² without explicit factorization API overhead.
proof idea
The definition is a one-line field projection that returns the e component of the input box.
why it matters in Recognition Science
This selector is invoked by box_divisor_mul_complement to recover the square-budget identity and by box_phase_hit_gives_balanced_pair to build the balanced-pair support required by the RCL skeleton. It appears inside HitsBalancedPhase and in SubsetProductPhaseHit witnesses. It supplies the combinatorial closure step for the Erdős-Straus finite phase argument.
scope and limits
- Does not verify that the returned value multiplies with the divisor to recover N squared.
- Does not compute the complementary divisor from d and N alone.
- Does not enforce positivity or square-budget conditions on its own.
formal statement (Lean)
36def boxComplement {N : ℕ} (box : DivisorExponentBox N) : ℕ :=
proof body
Definition body.
37 box.e
38