canonical_phi_uniform_closure

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 $\varphi$ as the self-similar fixed point of a discrete ledger hierarchy.

A nontrivial multilevel composition is a positive sequence of level sizes with at least three levels. The $\varphi$-uniform closure certificate packages the normal-form properties of that hierarchy after the canonical closing operations: uniform adjacent ratios (no free scale parameters), growth orientation at the base step, and seed closure (seed composite as the additive posting of levels 0 and 1).

Upstream hierarchy-forcing work already shows that absence of free scale parameters forces uniform adjacent ratios. Posting-potential semantics connect the shifted $J$-cost to d'Alembert composition; the seed-size law is the bridge from that surface to concrete level sequences.

Tactic-mode structure construction: each field of the certificate is discharged by a named specialized lemma on $M$.

• uniform from the uniform-scale law for the $\varphi$-closed multilevel composition.
• growth from the growth-orientation lemma.
• seed from the seed-size law.
• base_ratio from the base-ratio lemma.
• exact_preservation_iff from the equivalence between original levels and the closed sequence under uniform growth plus seed.
• unique by introducing a competitor $N$ with matching base and the three laws, then applying the levels-uniqueness lemma.

No new algebra is done here; the proof is a field-wise assembly of prior hierarchy-closure results.

This certificate is the reusable normal-form handle for $\varphi$-closed hierarchies inside the forcing chain. Downstream, canonical_closure_normal_form_composition builds the full closure-composition certificate from it (uniform-after-growth and seed idempotence). More critically, t5_to_t6_bridge_holds uses realized closed-scale normal-form equivalence as one arm of the T5-to-T6 self-similarity bridge: unique $J$ plus internal hierarchy dynamics force $\varphi$.

In primer terms this sits at the T5→T6 step: once $J$ is unique, self-similarity on a discrete multilevel ledger pins the golden ratio as the only consistent scale. Without a packaged uniform-closure certificate, later normal-form and bridge theorems would re-prove the same six properties inline.