phiUniformClosed_levels_unique
plain-language theorem explainer
Any nontrivial multilevel hierarchy that is uniform-scale, growth-oriented, and seed-closed, and that shares a base level with a reference hierarchy M, agrees levelwise with the φ-uniform normal form built from M. Hierarchy and φ-forcing arguments cite this uniqueness. The proof pins the base ratio to φ, then inducts on the level index using the uniform step law.
Claim. Let $M$ and $N$ be nontrivial multilevel compositions (positive real level sequences with at least three levels). Suppose $N_0 = M_0$, every adjacent step of $N$ multiplies by $N$'s canonical base ratio, the first step of $N$ grows ($N_0 < N_1$), and the seed posting law $N_2 = N_0 + N_1$ holds. Then for every $k \in \mathbb{N}$, $N_k$ equals the $k$-th level of the $\varphi$-uniform closed multilevel composition of $M$.
background
The Unified Forcing Chain module derives T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. Here the local object is a nontrivial multilevel composition: a positive map $\mathbb{N}\to\mathbb{R}$ with at least three positive levels, the discrete hierarchy on which scale laws act.
Three certificates replace raw free-scale hypotheses. Canonical uniform scale says each step multiplies by the hierarchy's own base ratio. Canonical growth orientation requires the first step to increase. Canonical seed size law closes the seed at index 2 by additive size: level 2 equals level 0 plus level 1. The φ-uniform closed multilevel composition of a reference $M$ is the normal form whose levels are $M_0\cdot\varphi^k$.
Upstream, canonicalBaseRatio_eq_phi_of_uniform_seed already forces that base ratio to equal $\varphi$ once uniform scale, growth, and seed closure hold. That is the T6 self-similarity input this uniqueness theorem consumes.
proof idea
First apply canonicalBaseRatio_eq_phi_of_uniform_seed to $N$ with the three certificates, obtaining that $N$'s canonical base ratio equals $\varphi$.
Induct on the level index $k$. At zero, the shared-base hypothesis and the definition of the φ-uniform closed form give equality immediately.
At the successor, rewrite the next level of $N$ by the uniform step law, substitute the ratio $\varphi$, and apply the inductive hypothesis. The goal reduces to $\varphi\cdot(\varphi$-uniform levels of $M$ at $k)$ equaling the $(k+1)$-st φ-uniform level; unfold that definition, use pow_succ, and finish by ring.
why it matters
This is the uniqueness half of the φ-uniform normal form for hierarchies inside the complete inevitability chain (T6: φ forced by self-similarity on the discrete ledger). Once a hierarchy is uniform, growing, and seed-closed, matching a single base level forces every level onto the geometric φ-ladder built from that base.
Downstream it feeds the symmetric preservation statement that the normal form leaves an already-closed hierarchy unchanged, the canonical φ-uniform closure certificate, and the level-equivalence theorems for realized hierarchies, realized closed-scale models, and admissible orbits. Those results identify physical and logical scale towers with the unique φ-ladder rather than an arbitrary geometric sequence.
In the primer landmarks this sits at T6 (φ as self-similar fixed point), with seed index 2 and additive seed size preparing the discrete ladder that later supports the eight-tick octave (T7) and $D=3$ (T8).
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