admissibleOrbitToRealizedHierarchy
plain-language theorem explainer
An admissible orbit on a closed observable framework is repackaged as a realized hierarchy: base state, positive orbit levels, first-step growth, constant adjacent ratios, and additive seed posting. Anyone bridging the newer admissibility reflection to the older hierarchy interface cites this. The construction is a field-by-field transfer of the orbit certificates.
Claim. Given a closed observable framework $F$ (state space $S$, dynamics $T$, positive observable $r$), a base state $b\in S$, and an admissible-orbit reflection at $b$ (first-step growth, self-similar adjacent ratios, additive seed posting), there is a realized hierarchy on $F$ with base state $b$, levels $\ell(k)=r(T^k b)$, $\ell(k)>0$, $\ell(1)/\ell(0)>1$, constant adjacent ratios, and the same additive posting.
background
The module UnifiedForcingChain aims at a complete inevitability chain from the cost foundation (Recognition Composition Law plus normalization and calibration) through T-1 to T8, including unique $J$, forced $\varphi$, the eight-tick octave, and $D=3$.
A closed observable framework supplies a state type $S$, an endomorphism $T$, and a strictly positive real observable $r$, with nontrivial observability and no external input. An admissible-orbit reflection at a base state is the missing exact data: the first orbit step grows, every adjacent ratio equals the previous one (self-similarity), and seed posting is additive along the orbit. That reflection is what turns a bare closed framework into $\varphi$-uniform normal form.
A realized hierarchy is the older RS-native interface for multilevel composition: levels are the orbit of $r$ under $T$, required positive, with first-step growth, self-similar ratios, and additive posting. This definition is the bridge from the newer admissibility package into that interface.
proof idea
Field-wise constructor, not a deep argument. Set the hierarchy base to the given base state. Level equality is definitional (rfl on the default orbit levels). Positivity of every level is F.r_pos at the iterated state. Growth is simpa from the reflection's first-step growth. Ratio self-similarity is copied verbatim from the reflection. Additive posting is simpa from the reflection's additive-posting field.
why it matters
Without this packaging, admissible-orbit certificates cannot feed the older hierarchy machinery used for uniform scale laws. Downstream, AdmissibleOrbitNormalFormReflection and the theorem canonical_admissible_orbit_normal_form_reflection build the canonical uniform-scale and growth certificates on top of the multilevel composition induced by this hierarchy. In the forcing chain this sits on the path from closed frameworks and ledger/recognition structure toward T6 ($\varphi$ as self-similar fixed point) and the discrete self-similar ladder. It is interface glue, not a new physical law, but it closes the gap between orbit admissibility and realized multilevel composition.
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