firstNontrivialClosureIndex_two

In the Unified Forcing Chain, every step from the absolute floor through T0–T8 is forced from the Recognition Composition Law plus normalization and calibration. Closure indices mark discrete levels at which a geometric scale sequence can close; the seed level is fixed at $1$.

FirstNontrivialClosureIndex n is the proposition that $n$ is the least natural strictly above that seed: $1 < n$, and every $m$ with $1 < m$ satisfies $n \le m$. This is pure order theory on $\mathbb{N}$, not yet a statement about $\varphi$ or the eight-tick octave.

Downstream, first-closure law packages on geometric scale sequences need a concrete, unique first index. Establishing that the index is $2$ supplies that pin before canonicality and equivalence-to-isClosed arguments run.

The structure has two fields. above_seed is discharged by norm_num on the literal inequality $1 < 2$. least_above_seed introduces an arbitrary $m$ with $1 < m$ and finishes with omega, which solves the linear arithmetic goal $2 \le m$. No external lemmas are required beyond the structure definition itself.

This pins the first nontrivial closure index to $2$, which is the concrete value fed into the canonical first-closure law package. canonical_first_closure_law_canonicality sets first_index := firstNontrivialClosureIndex_two and then proves uniqueness of that index. canonical_first_closure_law_iff_isClosed equates existence of a canonical first-closure law with the existing isClosed predicate on a geometric scale sequence.

In the forcing chain this sits under the discrete ledger and self-similarity steps that force $\varphi$ (T6) and the eight-tick structure (T7). It is arithmetic scaffolding for those later closures, not itself a physics claim: it only guarantees that “first nontrivial” means exactly the successor of the seed.