realizedHierarchy_canonical_uniform
plain-language theorem explainer
A realized hierarchy on a closed observable framework yields the canonical uniform-scale law: each adjacent level equals the previous level times one fixed base ratio. Hierarchy and forcing-chain authors cite this when reducing multilevel compositions to a single scale factor. The proof is a one-line term application of the realization's no-free-scale certificate.
Claim. Let $F$ be a closed observable framework and $H$ a realized hierarchy on $F$. Then the multilevel composition extracted from $H$ obeys the canonical uniform-scale law: for every level index $k$, $\mathrm{levels}(k+1) = \rho\,\mathrm{levels}(k)$, where $\rho$ is that composition's own canonical base ratio.
background
The ambient module is the Unified Forcing Chain: T-1 through T8 are derived as inevitabilities from the Recognition Composition Law plus normalization and calibration. The present lemma sits in the hierarchy-realization layer that feeds the self-similarity step (T6, $\varphi$ forced).
A closed observable framework supplies a state space, a dynamics, and a positive-valued observable ratio with nontrivial range and no external moduli. A realized hierarchy on such a framework packages discrete levels together with the structural facts needed to treat them as a nontrivial multilevel composition.
The target predicate, the canonical uniform-scale law, is the theorem-shaped stand-in for a raw all-pairs no-free-scale hypothesis: adjacent levels are generated by one base ratio belonging to the composition itself, i.e. $\mathrm{levels}(k+1)=\rho,\mathrm{levels}(k)$ for all $k$. Upstream, realized hierarchies already carry a uniform-ratios certificate; this declaration promotes that certificate to the canonical law form.
proof idea
One-line term proof. Apply the general bridge lemma that turns a no-free-scale fact on a nontrivial multilevel composition into a CanonicalUniformScaleLaw instance, feeding it the multilevel composition extracted from the realized hierarchy together with the upstream certificate realized_uniform_ratios already proved for every realized hierarchy.
why it matters
This is the uniform half of the normal-form package for realized hierarchies. Downstream, canonical_realized_hierarchy_normal_form_equivalence installs it as the uniform field of the equivalence certificate between a realized hierarchy and the $\varphi$-uniform normal form. Two sibling corollaries then pin the base ratio to $\varphi$ and identify the level sequence with the $\varphi$-uniform closed multilevel composition.
In the forcing chain this is infrastructure for T6: once levels advance by a single canonical ratio and that ratio is forced to $\varphi$, self-similarity of the discrete ledger is no longer an extra assumption. It does not itself derive $\varphi$ or the eight-tick octave; those land in the base-ratio and T7/T8 steps that consume the normal form.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.