ledger_forces_phi
From a uniform scale ladder with additive composition at level 2, the theorem derives a minimal hierarchy whose scale ratio equals the golden ratio φ. Physicists deriving constants from zero-parameter ledgers cite this when showing that multilevel composition selects φ uniquely. The proof builds a geometric scale sequence from the ladder, confirms closure using the locality lemma, and applies the hierarchy forcing result.
claimLet $L$ be a uniform scale ladder (positive level sizes with uniform ratio $r>1$) such that the level sizes satisfy $L_2=L_1+L_0$. Then there exists a minimal hierarchy $H$ with scale ratio equal to the golden ratio φ.
background
The module proves hierarchy emergence from zero-parameter scale closure in a comparison ledger. A UniformScaleLadder is a sequence of positive real level sizes equipped with a uniform scaling ratio greater than one. The locality theorem states that additive composition at the next level depends only on the two preceding levels and forces the minimal recurrence with positive coefficients. The hierarchy forcing theorem packages a closed geometric sequence satisfying the resulting quadratic into a minimal hierarchy.
proof idea
The proof constructs a GeometricScaleSequence S from the ratio and positivity data of L. It proves S.isClosed by unfolding the ledgerCompose and scale definitions, invoking locality_forces_additive_composition to obtain the quadratic relation, and closing with nlinarith. It then applies hierarchy_forces_phi to the resulting closed sequence to extract the minimal hierarchy.
why it matters in Recognition Science
This combined emergence theorem completes the derivation of φ from ledger primitives. It realizes the module's fourth step: minimal nondegenerate closure forces the Fibonacci recurrence, hence σ²=σ+1 and σ=φ. It connects directly to the self-similar fixed point in the T0-T8 forcing chain. No downstream uses are recorded yet.
scope and limits
- Does not apply to ladders lacking a uniform ratio.
- Does not derive the hierarchy without the additive closure hypothesis.
- Does not address ladders with non-positive level sizes.
- Does not extend the result beyond the minimal hierarchy package.
formal statement (Lean)
114theorem ledger_forces_phi
115 (L : UniformScaleLadder)
116 (additive_closure : L.levels 2 = L.levels 1 + L.levels 0) :
117 ∃ H : MinimalHierarchy, H.scales.ratio = φ := by
proof body
Tactic-mode proof.
118 let S : GeometricScaleSequence :=
119 { ratio := L.ratio
120 ratio_pos := lt_trans (by norm_num) L.ratio_gt_one
121 ratio_ne_one := by linarith [L.ratio_gt_one] }
122 have h_closed : S.isClosed := by
123 unfold GeometricScaleSequence.isClosed
124 unfold ledgerCompose
125 unfold GeometricScaleSequence.scale
126 have hrec := locality_forces_additive_composition L additive_closure
127 nlinarith [hrec]
128 exact ⟨⟨S, h_closed⟩, hierarchy_forces_phi ⟨S, h_closed⟩⟩
129
130end HierarchyEmergence
131end Foundation
132end IndisputableMonolith