admissibleOrbit_canonical_uniform
plain-language theorem explainer
Any admissible orbit in a closed observable framework yields a canonical uniform-scale law on its multilevel composition: every adjacent level multiplies by the same base ratio. Cited when reducing orbit data to φ-uniform normal form in the forcing chain. The proof feeds equal adjacent ratios into the no-free-scale constructor.
Claim. Let $F$ be a closed observable framework with state space $S$, transition $T$, and positive observable $r$. Fix a base state $b\in S$ and an admissible-orbit reflection at $b$ (first step grows; adjacent orbit ratios are self-similar). Then the multilevel composition built from the orbit of $b$ satisfies the canonical uniform-scale law: for every $k$, the level $k+1$ equals the hierarchy's canonical base ratio times level $k$.
background
The module UnifiedForcingChain derives the full T-1 through T8 stack from the Recognition Composition Law plus normalization and calibration. Here the local object is a closed observable framework: countable states $S$, a transition $T:S\to S$, and a positive real observable $r$ with nontrivial range and no external input.
An admissible-orbit reflection at a base state supplies the missing dynamical data: the first orbit step grows ($r(Tb)/r(b)>1$), and adjacent ratios are self-similar along the orbit. From that orbit one builds a nontrivial multilevel composition whose levels are the successive $r(T^{[k]}b)$.
CanonicalUniformScaleLaw is the theorem-shaped replacement for a raw all-pairs no-free-scale hypothesis: every adjacent step is generated by one fixed base ratio of the hierarchy. That is the bridge from orbit self-similarity to the φ-uniform normal form used later in the chain (T6).
proof idea
Term-mode application of canonical_uniform_of_no_free_scale to the multilevel composition of the admissible orbit. The remaining obligation is the all-pairs equal-ratio statement: for every $j,k$,
$r(T^{[j+1]}b)/r(T^{[j]}b)=r(T^{[k+1]}b)/r(T^{[k]}b)$.
Both sides rewrite, via admissibleOrbit_ratio_eq_base, to the same base-step ratio of the orbit, so the equality is immediate. No further case splits.
why it matters
This is the first certificate that an admissible orbit is already in uniform-scale form, without assuming φ a priori. Downstream, admissibleOrbit_canonical_base_ratio_phi identifies that common ratio with φ; admissibleOrbit_levels_eq_phiUniform equates orbit levels to the φ-uniform closed multilevel composition; and canonical_admissible_orbit_normal_form_reflection packages uniform scale with growth into the full normal-form reflection.
In the forcing chain this sits at the T6 gate: self-similarity on a discrete ledger forces the golden ratio as the unique base of geometric growth. It converts the structure-level admissibility data into the scale law that later pins constants ($c$, $\hbar$, $G$, $\alpha$) once φ is fixed.
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