seedClosedLevels_one
plain-language theorem explainer
Seed-closing a multilevel composition leaves level 1 untouched: the closed sequence at index 1 equals the original level-1 value. Hierarchy and φ-forcing arguments cite this as a simp fact when they rewrite after posting the adjacent seed (levels 0 and 1) into the canonical post slot. The proof is a one-line simp unfolding of the seed-closure definition and the fact that the post index is 2.
Claim. Let $M$ be a nontrivial multilevel composition (a positive real level sequence with at least three positive levels). If $\mathrm{seedClosed}(M)$ is the sequence that replaces level $2$ by $M_0+M_1$ and leaves every other index unchanged, then $\mathrm{seedClosed}(M)_1 = M_1$.
background
The Unified Forcing Chain module derives T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. Hierarchy forcing supplies the discrete level ladder on which self-similarity (T6, φ) and the eight-tick structure sit.
A nontrivial multilevel composition is a map $\mathbb{N}\to\mathbb{R}$ with strictly positive values and at least three positive seed levels. Seed-closure is the local second-order step that posts the adjacent seeds: the canonical post index is $2$, and the closed sequence equals the original sequence except at that index, where it becomes level $0$ plus level $1$.
This lemma records the trivial side of that definition at index $1$: the post slot is $2$, so level $1$ is not rewritten.
proof idea
One-line wrapper. Unfold seedClosedLevels (the if-then that writes $M_0+M_1$ only when the index equals the canonical post index) and canonical_seed_post_index (definitionally $2$). The branch $1=2$ is false, so simp reduces the goal to reflexivity on $M.\mathrm{levels},1$.
why it matters
In the complete inevitability chain, discrete hierarchy ratios are forced once free scale parameters are banned; seed-closure is the local operation that installs the first composite level from the two seed rungs. Having a simp lemma that level $1$ is invariant under that rewrite keeps later equalities about adjacent ratios and φ-self-similarity free of case noise.
The declaration sits in Foundation.UnifiedForcingChain beside the T−1 through T8 forcing stack (unique $J$, φ fixed point, eight-tick octave, $D=3$). No downstream consumers are recorded yet; it is infrastructure for hierarchy rewrites rather than a named paper proposition. It does not itself force φ or the octave; it only stabilizes the seed bookkeeping those steps rely on.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.