phiUniformClosedLevels
plain-language theorem explainer
Defines the φ-uniform normal form of any nontrivial multilevel hierarchy: keep the original base level and scale every higher rung by successive powers of φ. Hierarchy and forcing-chain arguments cite it as the canonical seed-closed level sequence. The body is a one-line function, base times φ to the k.
Claim. Given a nontrivial multilevel composition $M$ (positive real levels with at least three rungs), the $\varphi$-uniform closed level sequence is $\ell(k) = \ell_M(0)\,\varphi^{k}$ for $k\in\mathbb{N}$, where $\ell_M(0)$ is $M$'s base level and $\varphi$ is the unique self-similar growth ratio from $\varphi$-forcing.
background
The module UnifiedForcingChain derives the full T0–T8 inevitability chain from the Recognition Composition Law plus normalization and calibration. T6 pins $\varphi$ as the self-similar fixed point of the discrete ledger; later steps need a canonical multilevel scale built on that ratio.
A nontrivial multilevel composition supplies a positive sequence of real levels with at least three rungs. Adjacent ratios need not yet be uniform. The $\varphi$-uniform normal form freezes the base rung and replaces every higher rung by pure geometric growth with ratio $\varphi$, the unique uniform seed-closed growth factor in the forcing chain.
Upstream, NontrivialMultilevelComposition only records positivity and the three-level lower bound; the present definition overlays the forced geometric law without changing the base.
proof idea
Pure definition, not a proof. The level map is the anonymous function $k \mapsto M.\mathrm{levels}(0)\cdot\varphi^{k}$. No lemmas are applied; downstream positivity, base-ratio, and uniqueness theorems unfold this equation and use positivity of the base and of $\varphi$.
why it matters
This is the concrete level sequence behind the canonical $\varphi$-uniform multilevel composition. Downstream results use it to prove: the base ratio equals $\varphi$; growth orientation (each step strictly increases); positivity of every rung; the zero-rung simp identity; the seed-size and uniform-scale laws; and uniqueness—any positive uniform seed-closed hierarchy with the same base coincides with this normal form.
In the forcing chain it is the operational face of T6 ($\varphi$ forced by self-similarity): once $\varphi$ is unique, every seed-closed hierarchy has a single normal form with that growth ratio. Parent theorems such as phiUniformClosed_levels_unique and phiUniformClosed_base_ratio rest on this definition as the reference object.
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