closure_forces_additive
plain-language theorem explainer
A geometric sequence of positive real levels with common ratio σ > 1, closed under the relation that the sum of the zeroth and first levels equals the second, satisfies the additive equality with swapped order. Recognition Science researchers working on scale hierarchies cite this to replace the HasAdditiveComposition axiom with an internal derivation from the ledger posting rule. The proof is a one-line application of linear arithmetic to the closure hypothesis.
Claim. Let ℓ : ℕ → ℝ be positive with ℓ(k+1) = σ ℓ(k) for σ > 1, and assume ℓ(0) + ℓ(1) = ℓ(2). Then ℓ(2) = ℓ(1) + ℓ(0).
background
The Posting Extensivity module derives additive scale composition from the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y) with J(x) = ½(x + x⁻¹) − 1. It introduces PostingPotential Π(x) := J(x) + 1 = ½(x + x⁻¹), which is multiplicative under RCL-governed composition, and shows that self-similar geometric ladders force additive level sizes. Upstream, the scale function is defined as phi^k, supplying the geometric ladder, while the uniform distribution provides the information-theoretic normalization for costs.
proof idea
The proof is a one-line wrapper that applies the linarith tactic directly to the closure hypothesis, using commutativity of real addition to obtain the conclusion.
why it matters
This result closes Proposition 4.3 of the phi paper by showing that geometric sequences closed under the first composition step obey additive composition without assuming linearity a priori. It replaces the HasAdditiveComposition axiom with an RS-internal justification from the ledger posting rule, where total recognition work sums. The theorem feeds the chain from RCL through posting extensivity to the golden fixed point and eight-tick octave.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.