phiUniformClosedLevels_zero
plain-language theorem explainer
At index zero, the φ-uniform closed level sequence of any nontrivial multilevel composition equals that composition's base level. Anyone normalizing a hierarchy onto the golden-ratio ladder cites this as the base-case identity. The proof is a one-line simp: unfold the definition and use φ^0 = 1.
Claim. Let $M$ be a nontrivial multilevel composition (positive real levels $\ell_M:\mathbb{N}\to\mathbb{R}$ with at least three positive values). Define the $\varphi$-uniform closed levels by $\ell_\varphi(k)=\ell_M(0)\cdot\varphi^k$. Then $\ell_\varphi(0)=\ell_M(0)$.
background
The module UnifiedForcingChain derives the full T0–T8 inevitability chain from the Recognition Composition Law plus normalization and calibration. Landmark T6 forces $\varphi$ as the unique self-similar fixed point of the discrete ledger; the present definition packages that ratio into a concrete level sequence.
A nontrivial multilevel composition supplies a positive map $\ell_M:\mathbb{N}\to\mathbb{R}$ with $\ell_M(0),\ell_M(1),\ell_M(2)>0$. The φ-uniform normal form keeps the original base and multiplies by successive powers of $\varphi$: $\ell_\varphi(k)=\ell_M(0)\cdot\varphi^k$. This is the seed-closed growth ratio forced when free scale parameters are forbidden.
The theorem is the $k=0$ evaluation of that normal form, the base case any later comparison of $\ell_\varphi$ against $\ell_M$ needs.
proof idea
One-line wrapper: simp [phiUniformClosedLevels]. Unfolding gives $\ell_M(0)\cdot\varphi^0$, and the power reduces to $1$, so the product is $\ell_M(0)$. No external lemmas beyond the definition and the simp database for real powers.
why it matters
In the forcing chain this anchors the φ-ladder at the composition's native base level, so later steps can compare ratios without an extra scale factor. It sits under T6 (φ forced as self-similar fixed point) inside the Complete Inevitability Chain: once φ is the unique uniform seed-closed growth ratio, the normal-form sequence must start at $\ell_M(0)$. No downstream theorems currently depend on it in the graph; it is a simp convenience for hierarchy-normalization arguments and for any proof that rewrites φ-uniform levels at the origin. It does not itself force φ or close T6; those live in PhiForcing and the surrounding chain.
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