posting_extensivity_forces_phi
plain-language theorem explainer
A uniform scale ladder obeying additive closure at the base levels has scaling ratio exactly φ. Researchers tracing the Recognition Science derivation of the golden ratio from the Recognition Composition Law would cite this when linking posting potentials to self-similar scales. The proof is a direct one-line application of the hierarchy emergence theorem.
Claim. Let $L$ be a uniform scale ladder: a sequence of positive reals $ℓ_k$ with $ℓ_{k+1}=rℓ_k$ for ratio $r>1$. If the levels satisfy the additive closure $ℓ_2=ℓ_1+ℓ_0$, then $r=φ$.
background
The module derives additive scale composition from the Recognition Composition Law without assuming linearity of the recurrence. The RCL states $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$ where $J(x)=½(x+x^{-1})-1$. Posting potential is defined as $Π(x):=J(x)+1=½(x+x^{-1})$, which satisfies the d'Alembert identity leading to multiplicative composition for events at geometric scales.
proof idea
The proof is a one-line wrapper that applies hierarchy_emergence_forces_phi to the supplied uniform scale ladder L and the additive closure hypothesis.
why it matters
This supplies the posting-extensivity route to φ and feeds directly into bridge_T5_T6_via_posting, which connects realized hierarchies to the forced ratio. It closes Proposition 4.3 of the phi paper by showing the RCL forces additive scale composition for geometric sequences. The result links the RCL combiner to T6, the self-similar fixed point φ.
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