hierarchy_emergence_forces_phi
Given a uniform scale ladder with additive composition at the second level, the scaling ratio must equal the golden ratio φ. Researchers deriving self-similar structures from minimal ledger composition cite this unconditional bridge. The tactic proof builds a geometric sequence from the ladder, confirms closure via the locality lemma, and applies the closed-ratio uniqueness result.
claimLet $L$ be a uniform scale ladder consisting of positive level sizes with uniform ratio $σ > 1$. If the levels satisfy the additive closure $L_2 = L_1 + L_0$, then $σ = φ = (1 + √5)/2$.
background
UniformScaleLadder is a structure of positive real level sizes with uniform scaling ratio σ > 1 obeying levels(k+1) = σ · levels(k). The module proves hierarchy emergence from zero-parameter scale closure: multilevel composition induces a ladder, no free scales force uniformity, locality forces a finite recurrence, and minimal closure yields the Fibonacci relation hence σ = φ. Upstream locality_forces_additive_composition derives the quadratic σ² = σ + 1 from additive closure; closed_ratio_is_phi then shows the unique positive solution is φ.
proof idea
Constructs a GeometricScaleSequence S with ratio L.ratio, verifies positivity and inequality via lt_trans and linarith. Proves S.isClosed by unfolding ledgerCompose and GeometricScaleSequence.scale, applying locality_forces_additive_composition to obtain the recurrence, then nlinarith. Concludes with exact closed_ratio_is_phi S h_closed.
why it matters in Recognition Science
This is Bridge B1, feeding directly into realized_hierarchy_forces_phi, posting_extensivity_forces_phi, and bridge_T5_T6. It completes the step from minimal nondegenerate closure to φ, realizing T6 where phi is forced as the self-similar fixed point from J-uniqueness and the eight-tick octave in the unified forcing chain.
scope and limits
- Does not assign physical units or dimensions to the levels.
- Does not extend the result to non-uniform or parameter-dependent ratios.
- Does not address stability under small perturbations of the closure.
- Does not derive the numerical value of φ from external constants.
Lean usage
theorem use_site (L : UniformScaleLadder) (h : L.levels 2 = L.levels 1 + L.levels 0) : L.ratio = φ := hierarchy_emergence_forces_phi L hformal statement (Lean)
95theorem hierarchy_emergence_forces_phi
96 (L : UniformScaleLadder)
97 (additive_closure : L.levels 2 = L.levels 1 + L.levels 0) :
98 L.ratio = φ := by
proof body
Tactic-mode proof.
99 let S : GeometricScaleSequence :=
100 { ratio := L.ratio
101 ratio_pos := lt_trans (by norm_num) L.ratio_gt_one
102 ratio_ne_one := by linarith [L.ratio_gt_one] }
103 have h_closed : S.isClosed := by
104 unfold GeometricScaleSequence.isClosed
105 unfold ledgerCompose
106 unfold GeometricScaleSequence.scale
107 have hrec := locality_forces_additive_composition L additive_closure
108 nlinarith [hrec]
109 exact closed_ratio_is_phi S h_closed
110
111/-- Combined emergence theorem: from ledger primitives (uniform scale
112ladder + additive composition), derive the `MinimalHierarchy` package
113and conclude `φ`. -/