canonical_posting_closure_forces_phi
plain-language theorem explainer
Canonical posting closure on a nontrivial multilevel composition, together with uniform consecutive ratios and base ratio greater than one, forces the hierarchy scale ratio to equal φ. Hierarchy and T5–T6 bridge arguments cite this as the certificate that additive level closure pins the golden ratio. The proof is a one-line application of the hierarchy-forcing lemma that additive recurrence yields φ.
Claim. Let $M$ be a nontrivial multilevel composition with positive level sequence. Suppose consecutive ratios are scale-free: $M.\mathrm{levels}(j+1)/M.\mathrm{levels}(j)$ is independent of $j$, and the base ratio satisfies $1 < M.\mathrm{levels}(1)/M.\mathrm{levels}(0)$. If $M$ admits canonical posting closure (primitive order $M.\mathrm{levels}(0)+M.\mathrm{levels}(1)=M.\mathrm{levels}(2)$, inducing the additive recurrence), then the forced uniform-scale ladder has ratio equal to $\varphi$.
background
The Unified Forcing Chain module shows T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. T6 is the step that pins φ as the self-similar fixed point of a discrete ledger hierarchy.
A NontrivialMultilevelComposition supplies only a positive level sequence. It does not carry a posting operation, so additive relations are not automatic. CanonicalPostingClosure is the theorem-facing certificate for that missing operation: primitive closure in the natural order $\mathrm{levels},0 + \mathrm{levels},1 = \mathrm{levels},2$, from which the additive recurrence used by hierarchy theorems is derived.
Upstream, hierarchy_forced builds a UniformScaleLadder from the multilevel data under zero-free-scale uniformity and base ratio $>1$. The companion lemma hierarchy_forced_gives_phi states that if the additive relation $\mathrm{levels},2 = \mathrm{levels},1 + \mathrm{levels},0$ also holds, the ladder ratio equals $\varphi$. That is exactly the algebraic content of self-similarity on a discrete scale ladder (Fibonacci-type recurrence with constant ratio).
proof idea
One-line term wrapper. Project the additive recurrence field from the CanonicalPostingClosure certificate (closure.additive_closure), then apply HierarchyForcing.hierarchy_forced_gives_phi to $M$, the uniform-ratio hypothesis, the base-ratio hypothesis, and that additive field. No extra algebra is done here; the φ identification lives entirely in the upstream hierarchy lemma.
why it matters
This declaration is the clean interface between posting-style ledger closure and T6 (φ forced). Downstream, canonical_uniform_posting_closure_forces_phi strengthens it by deriving the old free-scale hypothesis from a canonical uniform-scale law, so callers need only uniform law plus posting closure. It also feeds t5_to_t6_bridge_holds, which packages the T5-to-T6 self-similarity bridge as theorem-backed once J-uniqueness is available.
In the forcing chain, T5 uniquely fixes the cost $J(x)=(x+x^{-1})/2-1$; T6 then forces φ as the self-similar fixed point of discrete multilevel composition. Canonical posting closure is the natural-order additive seed ($\ell_0+\ell_1=\ell_2$) that makes that fixed-point argument applicable without smuggling the recurrence as a raw hypothesis. Constants downstream ($c$, $\hbar$, $G$, the $\alpha$ band) sit on the φ ladder, so pinning the ratio here is load-bearing for the complete inevitability claim.
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