MinimalClosedScaleOrbit
plain-language theorem explainer
Packages orbit data in a closed observable framework that realizes a minimal closed geometric hierarchy: base state, positive amplitude, minimal hierarchy, and pointwise matching of the observable along the dynamical orbit to amplitude times the hierarchy scales. Downstream forcing lemmas cite this as the fixed-data certificate; growth and closedness are inherited from the hierarchy, not restated. Pure structure definition with no proof body.
Claim. Given a closed observable framework $F$ (state space $S$, dynamics $T:S\to S$, positive observable $r:S\to\mathbb{R}_{>0}$), a minimal closed-scale orbit consists of a base state $s_0\in S$, an amplitude $A>0$, and a minimal discrete hierarchy $H$, such that for every $k\in\mathbb{N}$, $r(T^{k}(s_0))=A\cdot\mathrm{scale}_H(k)$. Growth and closedness of the scale ladder are not extra data; they are those of $H$.
background
The ambient module is the Unified Forcing Chain: T-1 through T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. The local objects here sit between closed observables and the discrete geometric ladder that forces $\varphi$ (T6).
A closed observable framework supplies a state type $S$, an endomorphism $T$, and a strictly positive real observable $r$, with nontriviality ($r$ not constant) and the usual closure/finiteness package: no external input, countable states, no continuous moduli.
A minimal hierarchy is a geometric scale sequence closed under the first nontrivial composition step (the Fibonacci/self-similar closure). Its scales are the discrete ladder later identified with powers of $\varphi$. The structure below only records that some orbit of $T$ realizes that ladder up to a fixed positive amplitude.
proof idea
Definitional structure, not a proved theorem: five fields and no tactic or term proof. The nontrivial content is the single realization equation $\forall k,, r(T^{[k]}s_0)=A\cdot\mathrm{scale}_H(k)$. Positivity of $A$ is an explicit field; minimality, growth, and closedness are not duplicated here but come from the embedded MinimalHierarchy record (whose closure is the Fibonacci relation on the scale sequence).
why it matters
This is the fixed-data certificate that an orbit realizes a minimal closed geometric hierarchy inside a closed framework. Downstream, it feeds the growth lemma (the scale ratio is forced above 1, and via hierarchy-forces-$\varphi$ equals $\varphi$), the constructor from an isolated realization map, the passage to a realized closed-scale model, admissible-orbit reflection, and the bridge package that equates minimal closed-scale orbit, realized model, admissible orbit, and $\varphi$-uniform normal form.
In the forcing chain this is the data shape for the T6 self-similarity step: once a closed framework carries such an orbit, the hierarchy's minimal closure pins the golden ratio as the unique growth factor, linking ledger discreteness to the $\varphi$-ladder used for masses and constants. It does not itself close T6–T8; it is the interface those bridge theorems consume.
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