uniformClosedMultilevelComposition_idempotent_levels
plain-language theorem explainer
Applying the canonical uniform-closure operator twice to a nontrivial multilevel hierarchy leaves every level value unchanged. Hierarchy and forcing-chain arguments cite this as the idempotence half of the uniform-closure preservation certificate. The proof is a one-line application of level-fixity for hierarchies already on the uniform-scale law, using that the once-closed hierarchy satisfies that law by construction.
Claim. For every nontrivial multilevel composition $M$ (a positive real sequence of levels with at least three positive entries) and every index $k\in\mathbb{N}$, $$\mathrm{levels}(U(U(M)))(k)=\mathrm{levels}(U(M))(k),$$ where $U$ is the canonical uniform-closure operator on multilevel compositions.
background
The ambient module is the Unified Forcing Chain: it derives the full T-1 through T8 stack as inevitabilities from the Recognition Composition Law plus normalization and calibration. Multilevel hierarchies appear as positive real sequences levels : ℕ → ℝ with at least three strictly positive entries (the structure NontrivialMultilevelComposition).
Uniform closure rebuilds such a hierarchy so that adjacent level ratios obey a single canonical scale law. The operator uniformClosedMultilevelComposition produces that closed form; a companion lemma records that the result satisfies the canonical uniform-scale law by construction. A further upstream result states that if a hierarchy already obeys that law, then uniform closure fixes every original level pointwise.
Idempotence of levels under a second application is the natural fixed-point corollary of those two facts.
proof idea
Term-mode one-liner. Instantiate the upstream lemma that uniform closure preserves levels whenever the input already satisfies the canonical uniform-scale law, taking the input to be the once-closed hierarchy U(M). Discharge the scale-law hypothesis by the construction theorem that U(M) obeys the canonical uniform-scale law. The resulting equality is exactly levels(U(U(M))) = levels(U(M)) at every index.
why it matters
This lemma supplies the idempotent field of the canonical uniform-closure preservation certificate, which packages uniform normal form, exact preservation iff the scale law, and level-idempotence into one structure. Downstream, that certificate is the clean interface for arguing that uniform closure is a true normal-form operator on multilevel hierarchies inside the forcing chain.
In the broader RS picture the chain forces T0–T8 from cost (unique J, φ as self-similar fixed point, eight-tick octave, D = 3). Hierarchy scale rigidity is part of the discrete ledger and self-similarity story that pins φ and the octave; an idempotent uniform closure means there is a single canonical scale skeleton rather than an infinite regress of renormalizations. No open scaffold remains here: the claim is fully proved.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.