seedClosed_phi_transfers_to_original_ratio
plain-language theorem explainer
If the canonical seed-closed replacement of a multilevel hierarchy is forced to the golden ratio phi as its uniform scale factor, then the original hierarchy has the same base ratio phi. Anyone proving T6 (phi forced by self-similarity) from an arbitrary hierarchy cites this transfer. The proof is a short rewrite: base-ratio preservation plus unfolding of the forced ladder.
Claim. Let $M$ be a nontrivial multilevel composition (positive levels, at least three). Let $M'$ be its canonical seed-closed replacement. Assume adjacent ratios of $M'$ are independent of level index, the base ratio of $M'$ exceeds $1$, and the uniform scale ladder forced from those data has ratio $\varphi$. Then $M_1/M_0=\varphi$.
background
In the Unified Forcing Chain, T0-T8 are derived as inevitabilities from the Recognition Composition Law plus normalization and calibration. T6 is the step that pins the golden ratio $\varphi$ as the unique self-similar scale factor of a discrete ledger hierarchy.
A nontrivial multilevel composition is a positive real level sequence with at least three levels. The no-free-scale hypothesis forces all adjacent ratios equal, so the forced hierarchy construction builds a uniform scale ladder whose single ratio is the common adjacent factor (and is required to exceed $1$).
Not every hierarchy is seed-closed. The seed-closed multilevel composition is the canonical seed-closed replacement $M'$ of an arbitrary $M$. Upstream, seed-closed replacement preserves the base ratio: $M'_1/M'_0 = M_1/M_0$. The present theorem packages that equality with the forced-ladder data so that $\varphi$ on $M'$ transfers to $M$.
proof idea
Term-style, three steps. First invoke base-ratio preservation for the seed-closed replacement of $M$ to obtain $M'_1/M'_0 = M_1/M_0$. Next unfold the forced-hierarchy construction in the hypothesis that the forced ladder of $M'$ has ratio $\varphi$, exposing that this ratio is exactly the base ratio of $M'$. Finally rewrite that equality along the base-ratio preservation identity, yielding $M_1/M_0 = \varphi$.
why it matters
This is a transfer lemma on the T5-to-T6 bridge: unique $J$ (T5) plus self-similar discrete hierarchy forces $\varphi$ (T6). Downstream it is consumed by the theorem that the T5-to-T6 self-similarity bridge is theorem-backed, including the realized-closed-scale path that forces $\varphi$.
Without the transfer, $\varphi$-forcing would apply only to hierarchies already written in seed-closed normal form. With it, any nontrivial multilevel composition whose seed-closed replacement satisfies the uniform-ratio and ratio-greater-than-one hypotheses inherits $\varphi$ as base ratio. That closes the gap between the canonical seed-closed objects used in the forcing arguments and the original hierarchies appearing in applications.
Framework landmark: T6 in the forcing chain ($\varphi$ as the self-similar fixed point on the discrete ledger).
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