realizedClosedScale_canonical_uniform
plain-language theorem explainer
A realized closed-scale model on a closed observable framework yields the canonical uniform scale law: every adjacent hierarchy level is generated by one fixed base ratio. Hierarchy and forcing-chain authors cite this when normalizing multilevel compositions before pinning the base ratio to φ. The proof is a short term application of the no-free-scale-to-uniform bridge, discharging all-pairs adjacent-ratio equality via the model's constant step-ratio lemma.
Claim. Let $F$ be a closed observable framework (positive ratio observable $r$, dynamics $T$, nontrivial and closed) and let $H$ be a realized closed-scale model on $F$. Then the multilevel composition built from the $T$-orbit of $H$'s base state satisfies the canonical uniform scale law: for every $k$, the level at $k+1$ equals the composition's canonical base ratio times the level at $k$.
background
The ambient module is the Unified Forcing Chain: T-1 through T8 are forced from the Recognition Composition Law plus normalization and calibration. The present lemma sits in the hierarchy-realization layer that feeds T6 (φ forced as the self-similar fixed point).
A closed observable framework supplies a state space $S$, dynamics $T:S\to S$, and a positive ratio observable $r:S\to\mathbb{R}$ with nontrivial range and no external moduli. A realized closed-scale model on $F$ packages a base state whose successive $T$-images give a discrete scale hierarchy with a constant adjacent ratio (the closed-scale constraint).
The multilevel composition of that orbit is a nontrivial hierarchy object. The canonical uniform scale law is the theorem-shaped certificate that every adjacent step is generated by the hierarchy's own base ratio, replacing a raw all-pairs "no free scale" hypothesis. Upstream, the step-ratio identity for realized closed-scale models already equates each adjacent $r$-ratio along the orbit.
proof idea
One-shot application of canonical_uniform_of_no_free_scale to the multilevel composition of $(F,H)$. The remaining obligation is the all-pairs no-free-scale statement: for every $j,k$, the adjacent $r$-ratio at step $j$ equals that at step $k$. After unfolding the composition's level ratios, both sides rewrite by the upstream lemma realized_closed_scale_ratio_step, which makes every adjacent orbit ratio equal to the model's fixed scale factor. Equality of the two rewritten terms closes the goal; the uniform-step field of CanonicalUniformScaleLaw follows.
why it matters
This is the uniform half of the direct normal-form package for realized closed-scale models. Downstream, canonical_realized_closed_scale_normal_form_equivalence installs it as the uniform field alongside canonical growth. Two further corollaries depend on it: realizedClosedScale_canonical_base_ratio_phi pins the composition's base ratio to φ, and realizedClosedScale_levels_eq_phiUniform equates orbit levels with the φ-uniform normal form.
In the forcing chain this is infrastructure for T6: once uniformity holds, self-similarity plus the discrete ledger force the golden ratio as the unique admissible base. It does not itself derive φ or D=3; it clears the scale-law hypothesis so those later steps can fire without residual free scale parameters.
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