growthClosedMultilevelComposition_idempotent_levels
plain-language theorem explainer
Growth closure on multilevel hierarchies is idempotent: a second pass leaves every level unchanged. Hierarchy and φ-forcing arguments cite this to treat the canonical φ-step orientation as a projection. The proof is a short term argument: the already-closed object satisfies the φ-step identity, so the iff lemma returns exact levelwise equality.
Claim. Let $M$ be a nontrivial multilevel composition (positive real levels with at least three occupied rungs). Write $G(M)$ for the canonical growth-closed reorientation of $M$, which replaces the first step by the $\varphi$-step above the base and leaves every other level fixed. Then for every index $k$, the levels of $G(G(M))$ equal the levels of $G(M)$.
background
In the Unified Forcing Chain, T0–T8 are forced from the Recognition Composition Law plus normalization and calibration. Near T6, self-similarity on a discrete ledger pins $\varphi$ as the unique growth ratio; multilevel hierarchies must therefore admit a canonical orientation whose adjacent step is that $\varphi$-step.
A nontrivial multilevel composition is a map $\mathrm{levels}:\mathbb{N}\to\mathbb{R}$ with every value positive and the first three rungs strictly positive. Growth-closed levels reorient such a hierarchy by setting level $1$ to $\mathrm{levels}(0)\cdot\varphi$ and copying every other level. The corresponding structure package is the canonical growth-closed multilevel composition $G(M)$.
An upstream iff states that $G(M)$ agrees with $M$ on all levels exactly when $M$ already has the $\varphi$-step at the first adjacent pair. That characterization is the lever for idempotence.
proof idea
Apply the reverse direction of growthClosedLevels_eq_original_iff_phi_step to the already closed object $G(M)$. It remains only to check that $G(M)$ itself satisfies the $\varphi$-step identity at the first rung: $\mathrm{levels}{G(M)}(1)=\mathrm{levels}{G(M)}(0)\cdot\varphi$. Unfolding the definition of growth-closed levels, level $0$ is the original base and level $1$ is base times $\varphi$, so the identity is immediate by simplification. The iff then yields levelwise equality of $G(G(M))$ with $G(M)$.
why it matters
Idempotence makes growth closure a genuine normal-form projection rather than an open-ended rewrite. Downstream, canonical_growth_closure_preservation packages this fact into the growth-closure preservation certificate (normal form, base ratio, and the exact-preservation iff). That certificate is the bookkeeping layer that lets the forcing chain treat $\varphi$-oriented hierarchies as stable under reapplication, which is required for the T6 self-similarity step: once the discrete ledger is forced to grow by $\varphi$, further closure passes must not move the rungs. In the broader T0–T8 narrative this is scaffolding for “$\varphi$ forced,” not a new physical constant derivation by itself.
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