minimalHierarchy_ratio_eq_phi
plain-language theorem explainer
Every minimal hierarchy carries geometric scale ratio exactly equal to the golden ratio φ. Anyone assembling T6 (φ forced) or the canonicality package for minimal hierarchies cites this. The proof is a one-line application of the hierarchy-forces-φ lemma from HierarchyMinimality.
Claim. If $H$ is a minimal hierarchy, then the common ratio of its geometric scale sequence equals $\varphi$, where $\varphi$ is the unique positive fixed point of self-similarity ($\varphi^2=\varphi+1$).
background
The Unified Forcing Chain module shows T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. T6 is the step that pins φ as the self-similar fixed point of the discrete ledger.
A minimal hierarchy is a scale structure forced by hierarchy-minimality constraints (from HierarchyMinimality): successive scales form a geometric sequence whose ratio is the only free continuous parameter once discreteness and self-similarity are imposed. The golden ratio φ arises as that unique positive solution of $x^2=x+1$, equivalently the fixed point of the self-similar rung map used throughout RS.
Upstream, hierarchy_forces_phi already proves that any such minimal hierarchy must have ratio φ. The shifted cost $H(x)=J(x)+1$ and the RCL d'Alembert form sit in the broader foundation but are not reopened here; this declaration only exports the ratio identity into the unified chain.
proof idea
One-line term wrapper: apply HierarchyMinimality.hierarchy_forces_phi to the given minimal hierarchy $H$. No local algebra; the content lives entirely in that upstream lemma.
why it matters
This is the T6 export into the unified forcing chain: every minimal hierarchy is forced to the same ratio φ. Downstream, minimalHierarchy_scale_eq_canonical rewrites each scale $k$ via this identity to match the canonical minimal hierarchy's geometric sequence. That pair assembles canonical_minimal_hierarchy_canonicality, which packages ratio equality and scale-sequence equality as the full canonicality witness.
In the primer landmarks, this is exactly T6 (φ forced as the self-similar fixed point). It does not itself derive the eight-tick octave or $D=3$; those are T7–T8. It closes the ratio half of hierarchy canonicality so later constant derivations (c, ℏ, G, α band) can treat φ as unique rather than a free parameter.
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