seedClosureEquiv_preserves_base_ratio
plain-language theorem explainer
Seed-closure equivalence between two nontrivial multilevel compositions leaves the base ratio (level 1 over level 0) unchanged. Hierarchy-forcing and the T5-to-T6 self-similarity bridge cite this invariance so that seed-closed replacements stay on the same forced ladder. The proof is a one-line rewrite from the equivalence's level-0 and level-1 equalities.
Claim. Let $M$ and $N$ be nontrivial multilevel compositions (positive real level functions with at least three levels). If $N$ is seed-closure equivalent to $M$ (seed-closed replacement of $M$, with $N_0=M_0$, $N_1=M_1$, and the seed level equal to the canonical additive closure $M_0+M_1$), then the base ratio is preserved: $N_1/N_0 = M_1/M_0$.
background
The Unified Forcing Chain module derives T0 through T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. The T5-to-T6 step needs self-similarity on a discrete multilevel hierarchy: unique $J$ forces $\varphi$ as the self-similar fixed point once adjacent level ratios are uniform and greater than one.
A nontrivial multilevel composition is a positive real-valued level map with at least three levels. Seed-closure equivalence is the forcing-relevant quotient between a hierarchy and its seed-closed replacement: non-seed levels agree, level 0 and level 1 are copied, and the seed level is the canonical additive closure of those two base levels. The base ratio $L_1/L_0$ is exactly the quantity fed to hierarchy forcing (no free scale parameters forces uniform adjacent ratios).
Upstream, the equivalence packages a seed-closed replacement together with the three level identities; this lemma isolates the ratio consequence of the two base identities.
proof idea
Term-mode one-liner. The seed-closure hypothesis supplies equalities $N_1=M_1$ and $N_0=M_0$. Rewrite the target ratio $N_1/N_0$ along those two fields; the goal becomes $M_1/M_0=M_1/M_0$ and closes. No arithmetic lemmas or positivity arguments are required beyond what the structure already records.
why it matters
This is the ratio-invariance hinge on the universal forcing spine between unique $J$ (T5) and forced $\varphi$ (T6). Downstream, the canonical seed-closed replacement theorem applies it directly to show the constructed replacement keeps $M$'s base ratio. The stronger identification theorem then lifts seed-closure equivalence to equality of forced hierarchy ratio fields once both sides carry uniformity and growth witnesses. That package is cited by the T5-to-T6 bridge theorem, which records that the self-similarity bridge is theorem-backed (internal hierarchy forces $\varphi$; realized closed scale forces $\varphi$). Without base-ratio preservation, seed closure could drift the ladder off the forced geometric progression and break the T6 fixed-point step.
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