admissibleOrbit_canonical_base_ratio_phi
plain-language theorem explainer
Any closed observable framework carrying an admissible orbit reflection has canonical base ratio exactly equal to the golden ratio φ. Citation target for the T6 self-similarity step when an orbit already supplies growth, ratio self-similarity, and seed posting. The proof is a short term application of the uniform-seed base-ratio lemma to three orbit-to-canonical transfer theorems.
Claim. Let $F$ be a closed observable framework and $b$ a base state. If the orbit of $b$ is admissible in the reflection sense (first step grows, adjacent ratios are self-similar, and the additive seed-posting law holds), then the canonical base ratio of the multilevel composition built from that orbit equals $\varphi=(1+\sqrt{5})/2$.
background
The Unified Forcing Chain module derives T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. T6 is the self-similarity step that pins the discrete growth ratio to $\varphi$.
A closed observable framework supplies a state space $S$, a dynamics $T:S\to S$, and a positive observable $r:S\to\mathbb{R}$ with nontrivial range and no external input. An admissible-orbit reflection packages the exact data missing from a bare closed framework: the first orbit step grows, adjacent orbit ratios are constant (ratio self-similarity), and an additive seed-posting law holds. That package turns the orbit into a $\varphi$-uniform normal form.
Upstream, three transfer lemmas convert reflection data into the canonical interface: growth orientation, seed-size law, and uniformity of the multilevel composition. The base-ratio extractor then reads the common adjacent ratio of that composition.
proof idea
Term-mode one-shot application of canonicalBaseRatio_eq_phi_of_uniform_seed to the multilevel composition of the admissible orbit. The three side hypotheses are discharged by the orbit-to-canonical transfers already proved for the same reflection package: uniformity (admissibleOrbit_canonical_uniform), growth orientation (admissibleOrbit_canonical_growth), and seed-size law (admissibleOrbit_canonical_seed_size). No new arithmetic is done here; the $\varphi$ identification is inherited from the uniform-seed criterion once those three certificates are in hand.
why it matters
This is the base-ratio half of the admissible-orbit normal-form certificate. Downstream, canonical_admissible_orbit_normal_form_reflection assembles uniformity, growth, seed size, and this $\varphi$ equality into a single reflection object used by the T6 forcing step.
In the primer chain, T5 fixes the cost $J(x)=(x+x^{-1})/2-1$; T6 forces $\varphi$ as the self-similar fixed point of discrete ledger growth. The theorem states that whenever an orbit already carries the reflection data (growth + constant adjacent ratios + seed posting), the extracted canonical base ratio cannot be anything other than $\varphi$. That closes the self-similarity route from closed observables to the golden ratio without extra calibration hypotheses at this layer.
It does not by itself force existence of such an orbit; it only identifies the ratio once admissibility is given.
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