Research proof index

Closure inversion (forward keystone). A freevar-free level-m formula β whose internal bv is m and which substitutes back to succInd γ is exactly the fixitr-image, so its m-fold closure is (succInd γ).univCl'. Mirror of bv_quote_fixitr's ≥-direction inversion; the genuine remaining math.

(⟹) A 𝚺₁-recognized code is the code of a 𝚺₁ formula. Meta-induction on the formula: atoms are 𝚺₁ unconditionally; ∧/∨/∃ recurse; the ^∀ case is forced into the bounded shape by the recognizer (IsSigma1.of_all), and the bound is a bShift-image (positivity via termBV_termBShift_le), so Hierarchy.ball applies.

Gödel's first incompleteness theorem

(⟸) Every 𝚺₁ formula has a 𝚺₁-recognized code. By sigma₁_induction'.

RHS of chSigma1_mem_iff reduced to a clean ∃ψ (with the 𝚺₁ side condition).

RHS of chUniv_mem_iff reduced to a clean ∃ψ over the syntactic universal closure.