canonical_realized_hierarchy_normal_form_equivalence
plain-language theorem explainer
Any realized hierarchy on a closed observable framework is certified equivalent to the canonical φ-uniform normal form (uniform scaling, growth, seed, base ratio φ, level match, ladder ratio). Cited by the T5→T6 self-similarity bridge and sibling normal-form certificates. Proof is a term-mode assembly of six specialized component lemmas into the equivalence structure.
Claim. Let $F$ be a closed observable framework and $H$ a realized hierarchy on $F$ (carrier states observed by a positive ratio map $r$, generated by iterating dynamics $T$, with self-similar scaling). Then $H$ admits a full equivalence certificate to the canonical $\varphi$-uniform normal form: uniform scale law, growth orientation, seed closure, base ratio $\varphi$, levels identical to the $\varphi$-uniform ladder, and realized ladder ratio forced to $\varphi$.
background
The Unified Forcing Chain module aims to show T0–T8 are forced from the Recognition Composition Law plus normalization and calibration. T5 pins the unique cost $J(x)=(x+x^{-1})/2-1$; T6 forces the golden ratio $\varphi$ as the self-similar fixed point of discrete ledger scaling.
A closed observable framework supplies a state space $S$, dynamics $T$, and positive ratio observables $r$ with nontriviality and closure (no external input, finite description). A realized hierarchy on $F$ is the RS-native multilevel object: a base state whose iterated observables $r(T^{[k]}(\mathrm{base}))$ form a positive level sequence with growth ratio $>1$, self-similar scaling, and additive posting.
The target structure packages six properties equating that realized hierarchy route to the canonical $\varphi$-uniform normal-form route (uniform scale law, growth orientation, seed size, base ratio $\varphi$, level identification, and ladder ratio $\varphi$). Upstream, realized_hierarchy_forces_phi already shows a realized hierarchy forces the ladder ratio to $\varphi$.
proof idea
Term-mode construction of the six-field equivalence structure. Each field is discharged by a dedicated component lemma on the multilevel composition extracted from $H$:
- uniform scaling via
realizedHierarchy_canonical_uniform; - growth orientation via
realizedHierarchy_canonical_growth; - seed size via
realizedHierarchy_canonical_seed_size; - base ratio $\varphi$ via
realizedHierarchy_canonical_base_ratio_phi; - level identification with the $\varphi$-uniform ladder via
realizedHierarchy_levels_eq_phiUniform; - realized ladder ratio via the upstream
realized_hierarchy_forces_phi.
No extra algebraic work occurs at this site; the theorem is the packaging certificate.
why it matters
This certificate is the normal-form glue between realized hierarchies and the $\varphi$-uniform route inside the forcing chain. Downstream it feeds t5_to_t6_bridge_holds, whose doc states the T5-to-T6 self-similarity bridge is theorem-backed and lists realized-hierarchy/closed-scale normal-form equivalence among its witnesses. That bridge is the step from T5 (unique $J$) to T6 ($\varphi$ forced as self-similar fixed point).
It is also reused by canonical_realized_closed_scale_normal_form_equivalence (direct closed-scale models) and canonical_admissible_orbit_normal_form_reflection (admissible-orbit reflection). Together these close the hierarchy side of the Complete Inevitability Chain so $\varphi$-ladder physics (mass rungs, eight-tick structure) sits on forced self-similarity rather than an extra hypothesis.
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