phiLatticeReciprocal_involutive
plain-language theorem explainer
The map sending each integer rung index r to -r is an involution on the phi-ladder lattice. Researchers formalizing discrete self-similarity and reciprocal symmetry in Recognition Science cite this result to confirm that the lattice-level reciprocal squares to the identity. The proof is a one-line algebraic reduction that unfolds the definition and applies the ring tactic.
Claim. Let $f:ℤ→ℤ$ be the map $f(r)=-r$. Then $f(f(r))=r$ for every integer $r$.
background
The phi-ladder lattice is the discrete hierarchy of complexity levels forced by Recognition Science theorem T6. On the log scale the positions form the additive lattice {r · log φ : r ∈ ℤ}. The function phiLatticeReciprocal implements the discrete analog of reciprocal symmetry by sending rung r to -r, matching the continuous map x ↦ 1/x on the multiplicative half-line. This module connects the lattice to the J-cost function via the reciprocal automorphism imported from CostAlgebra and the recognition-event reciprocal from LedgerForcing.
proof idea
The proof is a one-line wrapper. It unfolds the definition phiLatticeReciprocal (r : ℤ) := -r and then invokes the ring tactic, which reduces double negation on ℤ to the identity.
why it matters
This result feeds directly into the downstream theorem phi_ladder_certificate that collects the structural facts of the phi-ladder (positivity of φ, the golden-ratio identity φ² = φ + 1, and self-similarity). Within the Recognition Science framework it realizes the involutive character of reciprocal symmetry required by T5 (J-uniqueness) and T6 (phi fixed point), ensuring J(φ^r) = J(φ^{-r}) and supporting the eight-tick octave and D = 3. It closes a basic step toward the phi-Poisson summation hypothesis stated as a scaffolding structure in the same module.
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