uniformClosedMultilevelComposition_preserves_base_ratio
plain-language theorem explainer
Uniform closure of a nontrivial multilevel hierarchy leaves its base ratio unchanged: level-1 over level-0 after closure equals the original ratio. Hierarchy and forcing-chain arguments cite this when showing the closed form still steps by the same scale factor. The proof is a one-line simplification unfolding the base-ratio and uniform-closure definitions.
Claim. For any nontrivial multilevel composition $M$ (a positive level sequence with at least three levels), writing $M'$ for its uniform closure, the base ratio is preserved: $\mathrm{levels}_{M'}(1)/\mathrm{levels}_{M'}(0) = \mathrm{levels}_M(1)/\mathrm{levels}_M(0)$.
background
In the Unified Forcing Chain, multilevel hierarchies encode discrete scale ladders forced from cost. A nontrivial multilevel composition is a map $\mathbb{N}\to\mathbb{R}$ of strictly positive levels with at least three levels present. Its base ratio is the adjacent ratio of the first two levels, $\mathrm{levels}(1)/\mathrm{levels}(0)$.
Uniform closure rebuilds such a hierarchy so that every adjacent step is generated by a single scale factor (the canonical uniform-scale law). The module's claim is that T0–T8 are forced from the Recognition Composition Law plus normalization and calibration; hierarchy uniformity sits on the discrete self-similarity path toward $\varphi$ (T6) and the eight-tick structure (T7).
Upstream, the base-ratio definition is exactly levels 1 over levels 0 on a nontrivial multilevel composition. Positivity of levels makes the ratio well-defined and positive.
proof idea
Term-mode proof by simplification. Unfold the base-ratio definition (levels 1 over levels 0) on both sides together with the uniform-closure constructor. After unfolding, both sides reduce to the same ratio drawn from the original hierarchy, so the equality is definitional. No intermediate lemmas are required beyond those two unfolds.
why it matters
This is the ratio-invariance step inside the uniform-scale certificate for the closed hierarchy. Downstream, the uniform-scale theorem rewrites with it to show every adjacent step of the closed form equals the preserved base ratio. That result feeds the canonical uniform-closure preservation certificate, which bundles normal form, exact preservation iff, and idempotence of closure on levels.
In the forcing chain, a single free-of-parameters scale on discrete levels is the self-similarity ingredient later used to pin $\varphi$ as the fixed point (T6) and to support the octave structure (T7). Without base-ratio preservation, closure would introduce a new free scale and break the no-free-parameters forcing that the chain relies on.
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