phiLatticePoint_zero
plain-language theorem explainer
The theorem states that the phi-ladder lattice point at rung zero equals the origin on the log scale. Number theorists and Recognition Science researchers cite this base case when indexing the discrete hierarchy or preparing Poisson summation over the lattice. The proof is a one-line wrapper that unfolds the lattice-point definition and simplifies.
Claim. Let $r$ be an integer rung and let the phi-lattice point at rung $r$ be defined by $r · log φ$. Then the phi-lattice point at rung zero satisfies $0 · log φ = 0$.
background
The phi-ladder lattice arises from RS theorem T6: self-similarity on the positive reals forces the golden ratio φ, so the multiplicative sequence {φ^r : r ∈ ℤ} becomes the additive lattice {r log φ} on the log scale. The module formalizes this lattice together with its reciprocal involution r ↦ -r. The upstream definition phiLatticePoint (r : ℤ) : ℝ := (r : ℝ) * Real.log phi supplies the concrete map used here.
proof idea
One-line wrapper that unfolds the definition of phiLatticePoint and applies simp to obtain the zero result.
why it matters
This supplies the zero-rung base case required by every subsequent lattice identity in the module, including the involution phiLatticeReciprocal_involutive and the cost reciprocity statements. It anchors the discrete hierarchy that supports T6 self-similarity and the eight-tick octave, while leaving the analytic content of Sub-conjecture A.2 (PhiLadderPoissonSummation) untouched.
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