realizedClosedScale_levels_eq_phiUniform
plain-language theorem explainer
Any realized closed-scale model is level-equivalent, orbit by orbit, to the φ-uniform normal form of its multilevel composition. Hierarchy and scale-forcing arguments cite this to identify physical ranks with the canonical φ-ladder levels. The proof is a short uniqueness application: same composition twice, reflexivity, plus the three canonical certificates (uniform, growth, seed size).
Claim. Let $F$ be a closed observable framework and $H$ a realized closed-scale model on $F$. Write $r$ for $F$'s observable rank, $T$ for its tick map, and $s_0$ for $H$'s base state. Then for every $k\in\mathbb{N}$, $$r(T^{[k]}(s_0)) = L_k,$$ where $L_k$ is the $k$-th level of the $\varphi$-uniform closed multilevel composition induced by the multilevel composition of $(F,H)$.
background
The ambient module is the Unified Forcing Chain: T-1 through T8 are forced from the Recognition Composition Law plus normalization and calibration. Here the local objects are closed observable frameworks (state space, tick dynamics $T$, and an observable rank $r$) and realized closed-scale models, which package a base state together with a discrete scale hierarchy compatible with that framework.
The φ-uniform closed multilevel composition is the normal form that flattens such a hierarchy onto the geometric φ-ladder (levels scaling as powers of the golden ratio, the T6 fixed point). Upstream, the shifted cost $H(x)=J(x)+1$ rewrites RCL as d'Alembert form, and the scale map $k\mapsto\varphi^k$ is the geometric skeleton those levels must match.
Three prior certificates on the same pair $(F,H)$ are assumed available: canonical uniformity, canonical growth, and canonical seed size. They pin the realized composition to the φ-uniform class so that level uniqueness can fire.
proof idea
Term-mode application of phiUniformClosed_levels_unique to the realized closed-scale multilevel composition of $(F,H)$ against itself. The identity witness is rfl. The three side hypotheses are exactly the already-proved canonical certificates: uniformity, growth, and seed size for that same realized model. No further case analysis; uniqueness of φ-uniform levels does the rest.
why it matters
This is the orbit-level identification step that lets a realized closed-scale hierarchy be replaced by its φ-uniform normal form without changing ranks. Downstream it is assembled into canonical_realized_closed_scale_normal_form_equivalence, the direct equivalence certificate bundling uniform, growth, seed, and this level equality.
In the forcing chain it sits after T6 (φ forced as self-similar fixed point) and supports the discrete scale ladder used for mass rungs and large-scale structure. Without level equality, “realized” and “canonical φ-uniform” would only be loosely related; with it, physical ranks on the tick orbit are literally the φ-ladder levels.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.