phiUniformClosed_uniform_scale
plain-language theorem explainer
Any nontrivial multilevel hierarchy has a φ-uniform normal form whose adjacent levels scale exactly by the golden ratio. Hierarchy and forcing-chain arguments cite this as the uniform-step half of the canonical φ-uniform closure certificate. The proof rewrites the base ratio to φ, unfolds the geometric levels, and finishes by pow_succ plus ring.
Claim. For every nontrivial multilevel composition $M$ (positive real levels with at least three levels), the associated $\varphi$-uniform normal form $M^\varphi$ with levels $\ell_k = M_0\,\varphi^k$ satisfies the canonical uniform-scale law: for all $k\in\mathbb{N}$, $\ell_{k+1} = r(M^\varphi)\,\ell_k$, where $r(M^\varphi)$ is the hierarchy's canonical base ratio (equal to $\varphi$).
background
The Unified Forcing Chain module derives T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. T6 is the step that forces $\varphi$ as the self-similar fixed point of discrete ledger scaling.
A nontrivial multilevel composition is a positive sequence of real levels with at least three entries. The canonical uniform-scale law replaces a raw all-pairs no-free-scale hypothesis: every adjacent step must be multiplication by the hierarchy's own base ratio. The $\varphi$-uniform normal form rebuilds any such hierarchy by freezing the original base level and growing geometrically: $\ell_k = M_0,\varphi^k$. An upstream lemma already shows that this normal form has canonical base ratio exactly $\varphi$.
The present result packages that geometric growth as an instance of the structure CanonicalUniformScaleLaw on the closed hierarchy, so later certificates can treat uniform scaling as a named Prop rather than an ad-hoc equation.
proof idea
Term-mode construction of the structure field uniform_step. Fix level index $k$. Rewrite the target base ratio via phiUniformClosed_base_ratio, which identifies it with $\varphi$. Change the goal to the explicit geometric statement $\ell_{k+1}=\varphi,\ell_k$. Unfold phiUniformClosedLevels to $M_0,\varphi^k$, apply pow_succ, and close by ring.
why it matters
This is the uniform half of the canonical $\varphi$-uniform closure certificate. Downstream, canonical_phi_uniform_closure assembles it with growth orientation and seed-size law into a full PhiUniformClosure for any nontrivial hierarchy. The companion equivalence phiUniformClosedLevels_eq_original_iff_uniform_growth_seed uses the same law to characterize when the normal form is already the original hierarchy: precisely when the original was uniform, growing, and seed-closed.
In the forcing chain this sits under T6 ($\varphi$ forced by self-similarity in the discrete ledger). It turns the geometric $\varphi$-ladder into a theorem-shaped scale law that later steps can cite without reopening free scale parameters. No scaffolding remains here; the claim is fully proved.
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