growthClosedMultilevelComposition
plain-language theorem explainer
Given any nontrivial multilevel composition (a positive real level sequence with at least three levels), the growth-closed composition replaces the first adjacent step by the canonical φ-step above the base and leaves every other level unchanged. Hierarchy and φ-forcing arguments cite this as the canonical growth orientation of a discrete ladder. The body is a structure package: positivity and the three-level witness are inherited from the positivity lemma for the rewritten level map.
Claim. For any multilevel composition $M$ with positive levels $\ell_k>0$ and at least three levels, the growth-closed composition $M'$ has levels $\ell'_k=\ell_0\cdot\varphi$ when $k=1$ and $\ell'_k=\ell_k$ otherwise, and $M'$ remains a nontrivial multilevel composition (all levels positive, with a three-level witness).
background
In the Unified Forcing Chain, T0–T8 are derived as inevitabilities from the Recognition Composition Law with normalization and calibration. On the hierarchy side, discrete scale ladders are packaged as multilevel compositions: maps $\mathbb{N}\to\mathbb{R}$ that stay strictly positive and open with at least three levels.
A nontrivial multilevel composition is exactly that package (positive levels plus a three-level witness). Growth closure is the canonical reorientation that forces the adjacent base step to be multiplication by $\varphi$, the self-similar fixed point forced at T6, while freezing every other rung. The rewritten sequence is the growth-closed level map; its positivity is already proved separately.
This definition re-wraps that sequence as a full multilevel composition so later normal-form and preservation lemmas can treat growth closure as an endomorphism of the hierarchy type rather than a bare function $\mathbb{N}\to\mathbb{R}$.
proof idea
Structure construction, not a deep argument. The levels field is set to the growth-closed sequence (base rung times $\varphi$ at index 1, original values elsewhere). Positivity of every level is the already-proved positivity theorem for that sequence. The three-level witness is obtained by specializing that positivity theorem at indices 0, 1, and 2 and packaging the three inequalities into the required conjunction.
why it matters
Growth closure is the first normal-form step toward the $\varphi$-uniform hierarchy used in the complete inevitability chain. Downstream it feeds the base-ratio identity (canonical base ratio equals $\varphi$), the growth-orientation certificate, preservation of every non-index-1 level, the fixed-point characterization (growth closure leaves $M$ unchanged iff the original first step was already the $\varphi$-step), and idempotence of a second application.
Those facts assemble into GrowthClosurePreservation and the ClosureNormalFormComposition certificate, which record that after growth (and then uniform) closure the hierarchy sits in the direct $\varphi$-uniform normal form. In framework terms this is the operational packaging of T6: once $\varphi$ is forced as the self-similar fixed point, every discrete ladder can be canonically reoriented so its first step is exactly that fixed point, without disturbing the rest of the rung data.
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