PhiUniformClosure
plain-language theorem explainer
A φ-uniform closure certificate packages the normal-form properties of a multilevel hierarchy after φ-closure: uniform adjacent scaling, growth at the base step, seed-size additivity, base ratio exactly φ, exact level preservation iff the input already obeyed those three laws, and uniqueness among such normal forms sharing the base level. Hierarchy and T5–T6 bridge authors cite it when routing self-similarity through closed hierarchies. As a Prop structure it is definitional; certificates are propositionally unique via Subsingleton.
Claim. For a nontrivial multilevel composition $M$ (a map $\mathbb{N}\to\mathbb{R}_{>0}$ with at least three positive levels), a $\varphi$-uniform closure certificate asserts that the $\varphi$-uniform closed normal form $M^\varphi$ satisfies: (i) adjacent levels scale by $M^\varphi$'s own canonical base ratio; (ii) level $1$ strictly exceeds level $0$; (iii) seed posting $\mathrm{levels}(2)=\mathrm{levels}(0)+\mathrm{levels}(1)$; (iv) the canonical base ratio equals $\varphi$; (v) $M^\varphi$ agrees with $M$ at every level iff $M$ already obeys (i)--(iii); (vi) any other hierarchy $N$ with the same base level that obeys (i)--(iii) coincides with $M^\varphi$ levelwise.
background
The Unified Forcing Chain module derives T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. T6 is the step that forces φ as the self-similar fixed point of a discrete ledger hierarchy.
A nontrivial multilevel composition is a positive real sequence of levels with at least three entries. Three theorem-shaped certificates replace raw hypotheses: the uniform-scale law (every adjacent step multiplies by the hierarchy's own base ratio), growth orientation (level 1 exceeds level 0), and the seed-size law (posting levels 0 and 1 closes additively at the forced seed index 2).
The φ-uniform closed multilevel composition is the normal form obtained by imposing those laws with base ratio pinned to φ. This structure is the Prop bundle asserting that that normal form really is the φ-uniform closure of the input hierarchy.
proof idea
Definitional Prop structure: six fields, no proof obligations beyond inhabitation downstream. Uniform, growth, and seed restate the three canonical laws on the φ-closed normal form. Base-ratio pins the closed form's canonical ratio to φ. Exact-preservation-iff equates levelwise identity of the closed form with the input to the conjunction of the three laws on the input. Unique is a universal quantification over competing hierarchies sharing the base level and the three laws. The companion Subsingleton instance collapses any two certificates by rfl (all fields are Props).
why it matters
This certificate is the interface object for the T5→T6 self-similarity bridge: once a closed observable framework carries a realized hierarchy, the bridge routes through hierarchy dynamics and forces the scale ratio to be φ (primer landmark T6). Downstream, canonical_phi_uniform_closure inhabits the structure for every nontrivial multilevel composition, and ClosureNormalFormComposition uses it when comparing uniform-after-growth normal forms to the direct φ-uniform form.
In the forcing chain narrative, T5 supplies unique J; T6 needs self-similarity on a discrete ledger. Packaging uniform scale, growth, seed closure, base ratio φ, preservation, and uniqueness in one Prop keeps the bridge honest: bare closed-observable data do not smuggle hierarchy fields. The structure is the named place where those hierarchy obligations are discharged before φ is declared forced.
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