Research proof index

bv-pin bridge (over ℕ): bv ⌜fixitr 0 (fvSup χ) ▹ χ⌝ = fvSup χ. - ≤ is immediate from quote_univCl_eq + bv_qqAlls (closing fvSup quantifiers reaches a sentence, whose bv is 0). - ≥ is by level-factoring: were the body an IsSemiformula j for some j < fvSup, IsSemiformula.sound + castLE-invariance would re-express χ as `γ ⇜ ![...

Crucial inducion for formalized $\Sigma_1$-completeness.

fvarVec is the code of the typed substitution vector fun i ↦ ^&i (over a standard length).

Inversion of the bounded-∀ clause: a ^∀-headed 𝚺₁ code is a qqBall.

termBShift only grows codes: t ≤ termBShift t for well-formed terms. The ^#z → ^#(z+1) bvar shift grows, ^&x is fixed, and functions recurse componentwise. Bounds the ∃ t guard in the bounded-∀ clause.

Closure inversion at the code level. Substituting the free-variable atoms &0 … &(m-1) back into the fixitr-image recovers ⌜φ⌝. This is the DECODE direction: the recognizer can recover ⌜succInd ψ⌝ (hence ψ) from the freevar-free closure body using the already-proven internal subst, with no need for an internal fixitr. Meta witness: `sub...