canonical_uniform_posting_closure_forces_phi
plain-language theorem explainer
Canonical uniform scale on a multilevel hierarchy, together with posting closure and base ratio greater than one, forces the hierarchy ratio to equal φ. Anyone citing the T5→T6 self-similarity step or the complete inevitability chain needs this certificate. The proof is a one-line term applying the general posting-closure forcing lemma after the uniform law supplies the old no-free-scale hypothesis.
Claim. Let $M$ be a nontrivial multilevel composition (positive level sequence with at least three levels). Suppose every adjacent step is generated by $M$'s own base ratio $r$, that $1 < r$, and that the canonical posting closure holds: levels satisfy $\ell_0 + \ell_1 = \ell_2$ (and the derived additive recurrence). Then the forced uniform-scale ladder has ratio equal to $\varphi$, the golden ratio fixed point of self-similarity.
background
The Unified Forcing Chain module aims to show T0–T8 are forced from the cost foundation (Recognition Composition Law, normalization, calibration), not merely compatible with it. T6 is the claim that φ is forced as the self-similar fixed point of a discrete ledger hierarchy.
A NontrivialMultilevelComposition is a positive real level sequence with at least three levels. The older hierarchy theorem needed a raw all-pairs equal-ratio hypothesis ("no free scale"). CanonicalUniformScaleLaw replaces that: every adjacent step equals the hierarchy's own base ratio, so $\ell_{k+1} = r , \ell_k$ for a single $r$. From that certificate one derives the no-free-scale predicate used by hierarchy_forced.
CanonicalPostingClosure supplies the missing additive operation: composing levels 0 and 1 closes at level 2 ($\ell_0 + \ell_1 = \ell_2$), and from that the additive recurrence the hierarchy theorem consumes. Together with $1 < r$, these data build a UniformScaleLadder whose ratio is the object compared to φ.
proof idea
Pure term-mode wrapper. Instantiate the general lemma canonical_posting_closure_forces_phi on $M$, feeding it:
no_free_scale_of_canonical_uniform M uniform— the all-pairs equal-ratio predicate derived from the uniform-scale certificate;- the given
ratio_gt_onefact $1 < r$; - the given
CanonicalPostingClosurecertificate.
No extra algebra is performed here; the equality of the forced ladder ratio with $\varphi$ is inherited entirely from that upstream forcing lemma.
why it matters
This is a T6-facing certificate inside the complete inevitability chain: self-similarity plus discrete posting closure pins the scale ratio to φ (primer landmark T6; upstream uniqueness of $J$ is T5). It removes a free hypothesis by deriving no-free-scale from the canonical uniform-scale law, so callers need only theorem-shaped certificates.
Downstream, canonical_uniform_growth_posting_closure_forces_phi strengthens the package by also folding in canonical growth orientation. More importantly, t5_to_t6_bridge_holds cites this family when asserting that the T5-to-T6 self-similarity bridge is theorem-backed ("The T5-to-T6 self-similarity bridge is theorem-backed"), connecting unique $J$ to forced φ in the forcing chain. Without this style of closure, T6 would remain a compatibility claim rather than a forced step.
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