realized_to_ladder
plain-language theorem explainer
realized_to_ladder converts a realized hierarchy on a closed observable framework into a uniform scale ladder. It feeds the hierarchy levels, positivity, self-similar ratios, and growth factor into the no-free-scale theorem. Researchers deriving the T5 to T6 bridge in Recognition Science cite it to replace external sensitivity and additive-composition hypotheses with native fields. The construction is a direct one-line wrapper around no_free_scale_forces_uniform.
Claim. Given a closed observable framework $F$ and a realized hierarchy $H$ on $F$, the uniform scale ladder is the structure whose levels are those of $H$, positivity holds by assumption, the scaling ratio is uniform by the no-free-scale theorem applied to the self-similar ratios, and the ratio exceeds one by the growth condition.
background
A closed observable framework consists of a state space $S$, deterministic dynamics $T: S → S$, and positive observable $r: S → ℝ$ with no continuous moduli. A realized hierarchy on such a framework is a structure whose levels are obtained by iterating $T$ from a base state and reading values with $r$; it carries the additional fields that adjacent ratios are identical (ratio_self_similar) and that level sizes satisfy the additive relation levels(2) = levels(1) + levels(0) (additive_posting). The uniform scale ladder is the structure that packages a positive sequence with a single ratio greater than one and the uniform scaling property levels(k+1) = ratio · levels(k). The module replaces earlier external bridge hypotheses on a zero-parameter ledger with these two fields derived from the closed framework.
proof idea
The definition is a one-line wrapper that applies no_free_scale_forces_uniform to the levels and positivity fields of the hierarchy, the uniform ratios obtained from realized_uniform_ratios, and the growth field of the hierarchy.
why it matters
This definition supplies the uniform scale ladder required by the T5–T6 bridge theorems bridge_T5_T6_internal and realized_hierarchy_forces_phi. It closes the earlier gap that required external sensitivity and HasAdditiveComposition hypotheses; both are now obtained from the ratio_self_similar and additive_posting fields of the realization. In the forcing chain it supplies the scale ladder at T6 once J-uniqueness (T5) has fixed the self-similar ratio to φ.
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