uniformClosedLevels
plain-language theorem explainer
Canonically uniform-closes a multilevel hierarchy: keep the original base level and the ratio of the first two levels, then generate every level as a pure geometric sequence. Hierarchy-normalization and φ-forcing arguments cite this as the level map inside uniform closure. The body is a one-line geometric construction from the base level and canonical base ratio.
Claim. Given a nontrivial multilevel composition $M$ with positive real levels $\ell_k$, define the uniform-closed levels by $\ell^{\mathrm{uc}}_k := \ell_0 \cdot r^k$, where $r = \ell_1/\ell_0$ is the canonical base ratio of $M$.
background
The Unified Forcing Chain module derives T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. Inside that chain, multilevel hierarchies appear as positive real sequences with at least three levels (the structure NontrivialMultilevelComposition: a map $\mathbb{N}\to\mathbb{R}$ that is strictly positive and nontrivial at $0,1,2$).
The companion quantity is the canonical base ratio $r=\ell_1/\ell_0$. Upstream material records that absence of free scale parameters forces adjacent ratios to be uniform; the present definition is the constructive counterpart that rebuilds every level from that single ratio.
Locally this sits between raw multilevel data and the uniform-scale law used later to identify φ-uniform normal forms after growth closure.
proof idea
Pure definitional construction, not a proof. The level map is the geometric sequence with zeroth term equal to the original base level $\ell_0$ and common ratio equal to the canonical base ratio $r=\ell_1/\ell_0$. No lemmas are applied; positivity and step identities are proved separately as simp-friendly corollaries.
why it matters
This is the level sequence inside uniform multilevel closure. Downstream it feeds the package that rebuilds a hierarchy with the same base and ratio, the positivity and step lemmas, and the characterization that uniform closure fixes the original sequence if and only if the hierarchy already obeys the canonical uniform-scale law.
That characterization is the bridge to growth-then-uniform closure landing on the φ-uniform normal form, which is the hierarchy-side expression of T6 (φ forced as the self-similar fixed point) in the forcing chain. Without a canonical geometric rebuild, one cannot state that free scale parameters have been eliminated or compare an arbitrary hierarchy to the φ-ladder.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.