uniform_generator_eq_canonical_base_ratio
plain-language theorem explainer
Any constant scale factor that multiplies successive hierarchy levels equals the ratio of the first two levels. Hierarchy-forcing arguments cite this to collapse a free geometric generator onto the canonical base ratio of a multilevel composition. The proof is a short field calculation that evaluates the uniform-scale law at level zero and unfolds the base-ratio definition.
Claim. Let $M$ be a nontrivial multilevel composition: a sequence of strictly positive reals with at least three positive levels. If $\sigma \in \mathbb{R}$ generates the hierarchy uniformly, i.e. each successive level equals $\sigma$ times the previous level, then $\sigma$ equals the canonical base ratio of $M$ (the ratio of level $1$ to level $0$).
background
The Unified Forcing Chain module shows that T0 through T8 are forced from the cost foundation (Recognition Composition Law, normalization, and calibration), rather than merely compatible with it. Hierarchy structure sits on the path toward the self-similar fixed point $\varphi$ (T6) and the discrete octave (T7).
A nontrivial multilevel composition is a positive real level sequence with at least three strictly positive entries. Its canonical base ratio is the ratio of the first two levels. Upstream hierarchy forcing records that absence of free scale parameters forces uniform adjacent ratios; the present result identifies any such uniform generator with that canonical ratio, so the geometric scale is no longer an independent parameter.
Related scaffolding in the chain treats self-similar dressings, measure weights, and $\varphi$-powers as the forced scale law once uniformity is in hand.
proof idea
Specialize the uniform-scale hypothesis at $k=0$ to obtain $\mathrm{levels}(1)=\sigma\cdot\mathrm{levels}(0)$. Unfold the definition of the canonical base ratio. Positivity of level $0$ supplies a nonzero denominator. A two-line calculation then multiplies and divides $\sigma$ by level $0$ (field simplification) and substitutes the specialized hypothesis, yielding level $1$ over level $0$. No external lemmas beyond positivity and field arithmetic are required.
why it matters
Free geometric generators of multilevel hierarchies are pinned to a single observable ratio. That identification is the local content needed before self-similarity can force $\varphi$ (T6 in the forcing chain) and before discrete octave structure (T7, period $2^3$) and $D=3$ (T8) can be read off the ledger. The parent module's stronger claim is complete inevitability of T-1 through T8 from cost, not mere compatibility.
No downstream uses are recorded yet; the lemma is a local collapse step inside hierarchy forcing. It does not itself derive $\varphi$, but removes an otherwise free scale so later forcing steps can name the base ratio without an extra hypothesis.
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