canonical_uniform_growth_posting_closure_forces_phi
plain-language theorem explainer
Canonical certificates for uniform adjacent ratios, growth orientation, and posting closure force the hierarchy base ratio to equal φ. Anyone citing the T5→T6 self-similarity step or the complete inevitability chain uses this. The proof is a one-line term wrapper that feeds the growth-derived ratio inequality into the already-proved uniform-plus-posting-closure forcing lemma.
Claim. Let $M$ be a multilevel composition with at least three positive levels. Suppose every adjacent step is generated by $M$'s own base ratio, the first step grows ($M.{\mathrm{levels}}\,0 < M.{\mathrm{levels}}\,1$), and posting closes so that $M.{\mathrm{levels}}\,0 + M.{\mathrm{levels}}\,1 = M.{\mathrm{levels}}\,2$ (with the induced additive recurrence). Then the forced uniform-scale ladder has base ratio equal to $\varphi$, the golden ratio.
background
The module UnifiedForcingChain aims to derive the full T0–T8 chain as inevitabilities from the Recognition Composition Law plus normalization and calibration. T6 is the step that pins the self-similar fixed point $\varphi$ inside a discrete ledger hierarchy.
A NontrivialMultilevelComposition is a positive real level sequence with at least three levels. The older hierarchy theorem took two raw hypotheses: no free scale (all adjacent ratios equal) and ratio greater than one. Those are replaced here by theorem-shaped certificates. CanonicalUniformScaleLaw asserts every adjacent step is multiplication by a single base ratio. CanonicalGrowthOrientation asserts the first nontrivial level exceeds the base. CanonicalPostingClosure supplies the missing additive operation: composing levels 0 and 1 closes at level 2, and derives the additive recurrence the hierarchy theorem needs.
Upstream, hierarchy_forced builds the uniform scale ladder from those data; the present theorem identifies that ladder's ratio with $\varphi$.
proof idea
One-line term wrapper. From the growth certificate it derives the inequality $1 < M.{\mathrm{levels}},1 / M.{\mathrm{levels}},0$ via ratio_gt_one_of_canonical_growth, then applies canonical_uniform_posting_closure_forces_phi to $M$, the uniform-scale law, that ratio inequality, and the posting-closure certificate. No new algebra is done at this site; the work sits in the callee and in the two projection lemmas that turn canonical certificates into the older hierarchy inputs.
why it matters
This is a T6-facing link in the complete inevitability chain: self-similarity in a discrete multilevel ledger forces $\varphi$. Downstream it feeds canonical_uniform_growth_seed_forces_phi (the seed-posting variant) and t5_to_t6_bridge_holds, whose doc-comment states that the T5-to-T6 self-similarity bridge is theorem-backed. Together with T5 J-uniqueness ($J(x)=(x+x^{-1})/2-1$), it closes the passage from unique cost to the golden fixed point used for constants ($c=1$, $\hbar=\varphi^{-5}$, $G=\varphi^5/\pi$) and the mass ladder. It does not itself prove T7 or T8; those remain separate forcing steps (eight-tick octave, $D=3$).
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