uniformClosedMultilevelComposition_uniform_scale
plain-language theorem explainer
Any nontrivial multilevel hierarchy, after uniform closure, obeys the canonical uniform-scale law: each adjacent level is exactly the base ratio times the previous level. Hierarchy and forcing-chain work cites this as the normal-form certificate that closed ladders have no free scale parameters. The proof is a short term construction: rewrite by base-ratio preservation, then apply the closed-levels step lemma.
Claim. Let $M$ be a nontrivial multilevel composition (positive real levels $k\mapsto \ell_k$ with at least three levels). Form its uniform closure $\overline{M}$. Then $\overline{M}$ satisfies the canonical uniform-scale law: for every $k\in\mathbb{N}$, $\overline{\ell}_{k+1}=r_M\,\overline{\ell}_k$, where $r_M$ is the canonical base ratio of $M$.
background
This sits in the Unified Forcing Chain module, which derives T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. Hierarchies here are sequences of positive real levels; a nontrivial multilevel composition has at least three positive levels.
The canonical uniform-scale law is the theorem-shaped stand-in for a raw “no free scale” hypothesis: every adjacent step is generated by the hierarchy’s own base ratio, so $\ell_{k+1}=r_M\ell_k$ for all $k$. Uniform closure rebuilds a hierarchy so that this geometric progression holds by construction while retaining the original base ratio.
Upstream, uniformClosedLevels_step supplies the adjacent-step identity on the closed levels, and base-ratio preservation records that closure does not change $r_M$. Cosmology-side $\phi^k$ ladders and measure self-similarity are ambient RS scale language; this lemma is the abstract hierarchy fact those ladders instantiate.
proof idea
Build the CanonicalUniformScaleLaw structure on the uniform closure of $M$ by discharging its single field uniform_step.
Introduce an arbitrary level index $k$. Rewrite the goal with the lemma that uniform closure preserves the base ratio of $M$, so the multiplier in the closed hierarchy is still $r_M$. Finish by applying uniformClosedLevels_step, which states that adjacent closed levels already satisfy the geometric step $\overline{\ell}_{k+1}=r_M,\overline{\ell}_k$.
No case splits or induction: the closed hierarchy is defined so the step identity is immediate after the ratio rewrite.
why it matters
In the forcing chain, free scale parameters would break self-similarity (T6 $\phi$ fixed point) and the discrete ladder that feeds T7’s eight-tick octave and T8’s $D=3$. This theorem certifies that the canonical uniform-closed hierarchy is already in normal form under the uniform-scale law.
It is the uniform_normal_form field of canonical_uniform_closure_preservation, the package that also ties exact level preservation to the original hierarchy satisfying the same law and records idempotence of closure. The sibling idempotence theorem (applying closure twice leaves levels fixed) depends on the same closed object.
Together these results replace an open “no free scale” hypothesis with a constructed normal form, so later chain steps can quote a proved geometric ladder rather than an extra assumption.
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