canonicalBaseRatio_eq_phi_of_uniform_seed
plain-language theorem explainer
Uniform adjacent scaling, growth orientation, and additive seed closure force a multilevel hierarchy's canonical base ratio to equal the golden ratio φ. Anyone citing T6 (φ forced by self-similarity) or uniqueness of φ-uniform normal forms will use this. The proof reduces the three certificates to the quadratic r² = 1 + r with r > 0, then invokes uniqueness of the positive self-similar ratio.
Claim. Let $M$ be a nontrivial multilevel composition (positive real levels, at least three). Suppose every adjacent step multiplies by the canonical base ratio $r_M$, the first step grows ($M_0 < M_1$), and the seed closes additively ($M_2 = M_0 + M_1$). Then $r_M = \varphi$, where $\varphi$ is the unique positive solution of $x^2 = x + 1$.
background
The Unified Forcing Chain module derives T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. Landmark T6 is that φ is forced as the self-similar fixed point of a discrete ledger hierarchy.
A NontrivialMultilevelComposition is a positive real sequence of levels with at least three entries. The canonical base ratio is the growth factor between level 0 and level 1. Three certificates replace raw free hypotheses: the uniform-scale law says every adjacent step multiplies by that same base ratio; growth orientation says the first step is strictly increasing; the seed-size law says posting levels 0 and 1 closes at the forced seed index 2 with additive size $M_2 = M_0 + M_1$.
Upstream, phi_unique_self_similar states that any positive $r$ satisfying the golden constraint $r^2 = r + 1$ equals φ. The present theorem is the bridge that turns hierarchy certificates into that algebraic input.
proof idea
Write $r$ for the canonical base ratio. Uniformity gives $M_1 = r M_0$ and $M_2 = r M_1$. Seed closure rewrites as $M_2 = M_0 + M_1$. Substitute the two uniform steps to obtain $M_0 \cdot r^2 = M_0 \cdot (1 + r)$. Positivity of $M_0$ cancels, yielding $r^2 = 1 + r$. Growth orientation plus the first uniform step imply $r > 0$ (via ratio_gt_one_of_canonical_growth). Finish by PhiForcing.phi_unique_self_similar, matching the golden constraint up to add_comm.
why it matters
This is the local T6 engine inside the complete inevitability chain: once a hierarchy is certified uniform, growing, and seed-closed, its scale ratio cannot be free; it must be φ. Downstream wrappers apply it verbatim to admissible orbits (admissibleOrbit_canonical_base_ratio_phi), realized hierarchies, and realized closed-scale models, each concluding that the corresponding multilevel composition has base ratio φ. It also feeds phiUniformClosed_levels_unique, which identifies any such hierarchy with the φ-uniform normal form sharing the same base level.
In the primer landmarks this is exactly T6 (φ forced as self-similar fixed point), sitting after T5 J-uniqueness and before T7 eight-tick / T8 D = 3. Constants (c, ℏ, G, α) and the mass ladder all ride on φ, so pinning the hierarchy ratio here is load-bearing for the rest of the chain.
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