realizedHierarchy_canonical_growth
plain-language theorem explainer
A realized hierarchy on a closed observable framework induces canonical growth orientation on its multilevel composition: level 0 is strictly below level 1. Hierarchy and φ-forcing arguments cite this when packaging realized ladders into the normal-form equivalence. The proof is a short rewrite that unpacks composition levels and applies the hierarchy's built-in growth inequality via the reciprocal form of one-less-than-ratio.
Claim. Let $F$ be a closed observable framework and let $H$ be a realized hierarchy on $F$. Then the multilevel composition associated to $H$ carries the canonical growth orientation: writing $\ell$ for its level map, one has $\ell(0) < \ell(1)$.
background
The Unified Forcing Chain module shows T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. Hierarchies enter near T6, where self-similarity on a discrete ledger forces the golden ratio $\varphi$ as the unique admissible base ratio.
A closed observable framework supplies a state space, a transition, and a positive real observable with at least two distinct values, closed under its own dynamics (no external moduli). A realized hierarchy on such a framework is a discrete level map $\ell:\mathbb{N}\to\mathbb{R}_{>0}$ with a growth certificate relating consecutive levels.
Canonical growth orientation is the order-level stand-in for a ratio-greater-than-one hypothesis: for a nontrivial multilevel composition $M$, it asserts only $M.\mathrm{levels},0 < M.\mathrm{levels},1$. The present theorem says every realized hierarchy, once packaged as a multilevel composition, automatically meets that orientation.
proof idea
The goal is the single field base_step_grows of the growth-orientation structure on the multilevel composition built from $H$. A change reduces that field to the concrete inequality $H.\mathrm{levels},0 < H.\mathrm{levels},1$. Rewriting with the reciprocal form of one-less-than-division (using positivity of level 0) converts the goal into the ratio form already stored on $H$, which is discharged by $H$'s growth field. No external lemmas beyond that positivity and the hierarchy's own growth certificate are required.
why it matters
This is a packaging lemma in the T6 corridor of the forcing chain: realized ladders must present the same growth orientation that the φ-uniform normal form assumes. Downstream, the canonical normal-form equivalence certificate takes this result as its growth field alongside the uniform certificate. The base-ratio-equals-$\varphi$ theorem and the levelwise equality with the φ-uniform composition both sit on the same multilevel packaging, so growth orientation is a prerequisite for identifying realized hierarchies with the self-similar φ-ladder forced by T6.
In the broader chain, once growth orientation and uniformity are in hand, the unique fixed point of the self-similar step is $\varphi$, feeding the eight-tick octave (T7) and the $D=3$ link (T8). The declaration itself is fully proved; it closes no sorry, it only normalizes data already present on realized hierarchies.
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