realizedHierarchy_levels_eq_phiUniform
plain-language theorem explainer
Any realized hierarchy over a closed observable framework is levelwise identical to the φ-uniform normal form of its multilevel composition. Hierarchy and normal-form arguments in the unified forcing chain cite this equality. The proof is a short term application of level-uniqueness for φ-uniform closed compositions, fed by the realized hierarchy’s canonical uniformity, growth, and seed-size facts.
Claim. Let $F$ be a closed observable framework and $H$ a realized hierarchy over $F$. Then for every level index $k$, the $k$-th level of $H$ equals the $k$-th level of the $\varphi$-uniform closed multilevel composition built from the multilevel composition of $H$.
background
The module UnifiedForcingChain assembles the complete inevitability chain T-1 through T8 from the cost foundation (Recognition Composition Law, normalization, calibration). Here the focus is the hierarchy layer that sits above a closed observable framework: a structure with positive-valued observables, a ratio interface, nontrivial observability, closure (no external input), and finite description.
A realized hierarchy packages a discrete multilevel organization of that framework. The φ-uniform closed multilevel composition is the normal-form route that forces self-similar growth by the golden ratio φ (the T6 fixed point of the discrete ledger). Level maps record the successive shells of that hierarchy.
Upstream, uniqueness of levels for φ-uniform closed compositions is the comparison engine: two such compositions with matching uniformity, growth, and seed size agree levelwise. Companion facts already show that the multilevel composition of any realized hierarchy is canonically uniform, has canonical growth, and has canonical seed size.
proof idea
Term-mode one-shot application of the uniqueness lemma for levels of φ-uniform closed multilevel compositions. Both composition arguments are the multilevel composition of the given realized hierarchy (so the identity witness is rfl). The three side conditions are discharged by the realized-hierarchy canonical-uniformity, canonical-growth, and canonical-seed-size lemmas. No further case analysis.
why it matters
This is the levelwise half of the bridge between the realized-hierarchy route and the canonical φ-uniform normal form. The sole downstream consumer is the equivalence certificate canonical_realized_hierarchy_normal_form_equivalence, whose doc-comment calls it “the canonical equivalence certificate between realized hierarchies and the φ-uniform normal form.”
In the forcing chain this pins the hierarchy data to the same normal form that carries T6 (φ forced as the self-similar fixed point) and the discrete octave structure feeding T7/T8. Without level equality, the two routes could diverge in shell content even if they shared abstract growth laws. The result closes that gap inside the Complete Inevitability Chain narrative of the module.
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