phiUniformClosed_base_ratio
plain-language theorem explainer
Any nontrivial multilevel hierarchy, after passage to its φ-uniform normal form, has adjacent-level base ratio exactly equal to the golden ratio φ. Hierarchy and T6 forcing arguments cite this to pin the scale of the closed normal form. The proof is a short term calculation: unfold the normal-form levels and the base-ratio definition, then clear the positive denominator.
Claim. Let $M$ be a nontrivial multilevel composition (a positive real level function with at least three levels). Form its $\varphi$-uniform normal form $M^\varphi$. Then the canonical base ratio of $M^\varphi$, namely $\mathrm{levels}_{M^\varphi}(1)/\mathrm{levels}_{M^\varphi}(0)$, equals the golden ratio $\varphi$.
background
The module UnifiedForcingChain assembles the complete inevitability chain T-1 through T8 from the cost foundation (Recognition Composition Law, normalization, calibration). T6 is the step that forces φ as the self-similar fixed point of the discrete ledger hierarchy.
A nontrivial multilevel composition is a positive map $\mathbb{N}\to\mathbb{R}$ with at least three levels. Its canonical base ratio is the adjacent ratio of the first two levels, $\mathrm{levels}(1)/\mathrm{levels}(0)$. The φ-uniform normal form rebuilds the hierarchy so that every adjacent step is generated by a single fixed scale; the present theorem identifies that scale.
Upstream, HierarchyForcing supplies the multilevel structure and the no-free-scale pressure toward uniform adjacent ratios. PhiForcing supplies φ itself. The base-ratio definition is local to this module and is the theorem-shaped stand-in for a free scale parameter.
proof idea
Term-mode calculation, not a deep lemma chase. Unfold three definitions: the base-ratio quotient, the φ-uniform multilevel composition, and the underlying φ-uniform level sequence. After unfolding, the claimed equality is an elementary field identity in the reals. Discharge the nonzero-denominator side condition with positivity of level 0 (from the multilevel structure). field_simp finishes.
why it matters
This is the scale-pinning fact for the φ-uniform normal form inside the T6 segment of the forcing chain. Downstream, phiUniformClosed_uniform_scale rewrites with it to obtain the canonical uniform-scale law on the normal form; that law is one of the three fields of the canonical φ-uniform closure certificate. The same identity feeds the characterization of when the normal form agrees with the original hierarchy (exactly when the original was already uniform, growing, and seed-closed).
In framework terms it locks the hierarchy's free scale to φ, the self-similar fixed point forced at T6, so later octave and dimension steps (T7, T8) inherit a unique geometric ratio rather than a free parameter. No scaffolding remains: the claim is fully proved.
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