uniformClosed_after_growthClosed_eq_phiUniform
plain-language theorem explainer
Uniform closure after growth closure of any nontrivial multilevel hierarchy equals the direct φ-uniform normal form (base times φ to the level index). Hierarchy-forcing and T6 arguments cite this to collapse the composite closure path onto the φ-ladder. The proof rewrites the growth-closed base ratio as φ, then unfolds the level maps.
Claim. Let $M$ be a nontrivial multilevel composition (positive real levels, at least three occupied). For every level index $k\in\mathbb{N}$, the $k$-th level after growth-closing $M$ and then applying uniform closure equals the $k$-th level of the $\varphi$-uniform normal form of $M$, i.e. equals $M_0\cdot\varphi^k$.
background
This sits in the Unified Forcing Chain module, which derives T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. Multilevel compositions are maps $\mathbb{N}\to\mathbb{R}$ with strictly positive levels and at least three occupied rungs; they package discrete hierarchy data before scale is forced.
Growth closure reorients a hierarchy so the first step is the canonical $\varphi$-step above the base while leaving other levels alone: level $1$ becomes $M_0\cdot\varphi$. The companion lemma records that the resulting canonical base ratio is exactly $\varphi$. The $\varphi$-uniform normal form keeps the original base and fills every rung by pure geometric growth: level $k$ is $M_0\cdot\varphi^k$.
Uniform closure is the operator that forces adjacent ratios equal once a base ratio is fixed. The claim is that composing uniform closure after growth closure lands on the same ladder as the direct $\varphi$-uniform construction.
proof idea
Term-style tactic proof. Introduce the level index $k$, then change the goal to equality of the underlying level functions: uniform-closed levels of the growth-closed hierarchy versus $\varphi$-uniform levels of $M$. Unfold both level definitions. Rewrite with the lemma that the growth-closed hierarchy has canonical base ratio $\varphi$. Finish by simplifying through the definitions of growth-closed multilevel composition and growth-closed levels, which substitutes the forced first step and collapses the uniform ladder onto $M_0\cdot\varphi^k$.
why it matters
Identifies the two-step composite (growth then uniform) with the direct $\varphi$-uniform normal form, so later certificates need not track intermediate hierarchies. Downstream, canonical_closure_normal_form_composition packages this equality as the uniform_after_growth field of the closure normal-form certificate. The sibling seedUniformGrowthClosed_eq_phiUniform extends the path by one more seed closure and again lands on the same $\varphi$-ladder.
In the forcing chain this is T6 infrastructure: once self-similarity pins $\varphi$ as the unique discrete growth ratio, every admissible hierarchy normalizes to the $\varphi$-ladder. That ladder is the discrete skeleton behind the eight-tick octave (T7) and the mass/rung formulas built on $\varphi$-powers. No open scaffold remains here; the result is fully proved.
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