canonical_uniform_seed_closure_compatibility
plain-language theorem explainer
For any nontrivial multilevel composition, seed-size law on the uniform normal form is equivalent to the golden constraint on the canonical base ratio; with growth orientation that law forces the base ratio to equal φ. The T5→T6 self-similarity bridge cites this certificate. Proof fills the two structure fields by applying existing seed-size and base-ratio lemmas.
Claim. Let $M$ be a nontrivial multilevel composition (positive level sequence with at least three levels). Then seed-size closure of the uniformized hierarchy of $M$ holds if and only if the canonical base ratio of $M$ satisfies the golden constraint. Moreover, if $M$ is growth-oriented and that seed-size law holds, the canonical base ratio equals $\varphi$.
background
The Unified Forcing Chain module derives T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. T5 pins the unique cost $J(x)=(x+x^{-1})/2-1$; T6 forces the self-similar fixed point $\varphi$. The bridge between them runs through discrete hierarchy dynamics.
A nontrivial multilevel composition is a positive real level sequence with at least three levels. Hierarchy forcing shows that absence of free scale parameters yields uniform adjacent ratios. Uniform closure is the normal form that enforces that uniformity; seed-size law is the discrete posting constraint at the seed surface of that normal form.
UniformSeedClosureCompatibility packages the precise link: the two normal forms commute at the seed-size surface exactly at the golden equation, so one need not assume that uniformization still preserves seed posting. The golden constraint is the fixed-point equation that characterizes $\varphi$ in the PhiForcing development.
proof idea
The proof is a structure constructor with two fields, each discharged by an existing lemma.
uniform_seed_iff_goldenis set touniformClosed_seed_size_law_iff_golden M, the biconditional between CanonicalSeedSizeLaw on the uniform-closed multilevel composition and the golden constraint oncanonicalBaseRatio M.base_ratio_phi_of_growth_seedis a short intro on growth orientation and seed law, thencanonicalBaseRatio_eq_phi_of_uniformClosed_seed M growth seed, which upgrades the golden side to equality with $\varphi$ under growth orientation.
No new algebra is done here; the certificate is the packaging.
why it matters
This certificate is the Lean replacement for an informal assumption that uniformizing a hierarchy still preserves seed posting. Downstream, t5_to_t6_bridge_holds consumes it as part of the theorem-backed T5→T6 self-similarity bridge: once J is unique (T5), internal hierarchy dynamics plus realized closed-scale normal forms force $\varphi$ (T6).
In the forcing chain this sits at the T5–T6 hinge: unique cost plus discrete self-similar ledger structure pin the golden ratio as the only admissible base ratio. Without uniform/seed compatibility, the hierarchy route to $\varphi$ would remain a compatibility claim rather than a forced equivalence. It does not itself prove T6 globally; it supplies the local normal-form glue the bridge theorem needs.
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