posting_extensivity_forces_phi
The theorem shows that any uniform scale ladder whose level sizes satisfy additive closure at the first two steps must have scaling ratio exactly equal to the golden ratio φ. Researchers deriving the emergence of φ from the Recognition Composition Law would cite it when closing the loop from RCL through posting potentials to self-similar scales. The proof is a direct one-line application of the hierarchy emergence theorem that already forces φ under the same additive hypothesis.
claimLet $L$ be a uniform scale ladder with positive level sizes satisfying the uniform scaling $ℓ_{k+1} = σ ℓ_k$ for ratio $σ > 1$. If the levels obey the additive closure $ℓ_2 = ℓ_1 + ℓ_0$, then $σ = φ$, where $φ = (1 + √5)/2$.
background
The Recognition Composition Law states that J(xy) + J(x/y) = 2 J(x) J(y) + 2 J(x) + 2 J(y) with J(x) = (x + x^{-1})/2 - 1. Posting potential is the shifted form Π(x) := J(x) + 1 = (x + x^{-1})/2, which satisfies the multiplicative d'Alembert identity Π(xy) Π(x/y) = 2 Π(x) Π(y) under extensive posting. A UniformScaleLadder is a sequence of positive real level sizes with uniform scaling ratio σ > 1 and the property that each level is obtained by multiplying the previous by σ.
proof idea
The proof is a one-line wrapper that applies hierarchy_emergence_forces_phi to the supplied ladder L and the additive closure hypothesis.
why it matters in Recognition Science
This result completes the internal derivation of additive scale composition from the RCL, closing Proposition 4.3 of the phi paper without external linearity assumptions. It feeds directly into bridge_T5_T6_via_posting, which routes the T5-T6 connection explicitly through the posting-extensivity layer. In the forcing chain it supports T6, where φ emerges as the self-similar fixed point of the scale recurrence.
scope and limits
- Does not prove existence of a uniform scale ladder from the RCL alone.
- Does not extend additive closure beyond the first composition step.
- Does not derive the RCL or the J-cost function.
- Does not produce numerical values for physical constants.
Lean usage
posting_extensivity_forces_phi (realized_to_ladder F H) (additive_closure_from_realization F H)formal statement (Lean)
176theorem posting_extensivity_forces_phi
177 (L : UniformScaleLadder)
178 (closure : L.levels 2 = L.levels 1 + L.levels 0) :
179 L.ratio = PhiForcing.φ :=
proof body
Term-mode proof.
180 hierarchy_emergence_forces_phi L closure
181
182/-! ## Posting-Derived Recurrence Data
183
184Instead of assuming `HasAdditiveComposition` or `HasDiscreteAdditiveComposition`,
185we provide the recurrence data directly from posting extensivity. -/
186
187/-- The additive closure gives recurrence coefficients (1, 1). -/