A Certified Bound and a Conditional Lean Theorem for a Bounded-Height Algebraicity Exclusion for Khinchin's Constant
Year: 2026
Type: preprint
Abstract: We deposit two complementary artifacts concerning Khinchin's constant K_0 ≈ 2.6854520010…: 1. **A certified Hermite/LLL bound `M_certified`** (578-digit integer; SHA-256 of decimal-string form `c51099378e8c199c70d4543fb6b69f4de80986af34b628f6d9619f2181f6ab8e`) emitted by an interval-arithmetic verification chain (FLINT/Arb) anchored against OEIS A002210 via the BBC (Bailey–Borwein–Crandall, 1997) series identity at precision 8663 decimal places (M1 top rung P_bits = 28712). 2. **A formally-verified `conditional` Lean 4 / Mathlib v4.14.0 theorem** (`no_euclidean_relation_conditional`) that, under three explicit hypotheses matching the certified-bound + GS-lower-bound supplied numerically by the bound emission chain, **rules out** any nonzero integer relation of squared Euclidean norm ≤ M_sq among the committed 15-element basis `B_D(C) = {1, K_0, K_0^2, …, K_0^6, log K_0, K_0·π, K_0·e, K_0·ln2, K_0·γ, K_0·ζ(2), K_0·ζ(3), K_0·G}`. A companion lemma (`c1_from_c2_zero_extension`, this session's R-0 deliverable) extends this to the degree-≤6 pure-power sub-block at the same bound, via a norm-preserving zero-extension. The Lean theorem no_euclidean_relation_conditional treats π, e, ln 2, ζ(2), ζ(3), Catalan's constant, the Euler–Mascheroni constant, K₀ and log K₀ as opaque real constants (9 axiom/opaque declarations), not Mathlib's Real.pi etc. They are linked to the true constants only through the stated numeric hypotheses. The Lean/ directory has no lakefile, so the README procedure fails as deposited (unknown module prefix 'M5'). With the files moved to the M5/ layout their imports expect and a minimal lakefile added, it builds at Lean v4.14.0, and #print axioms lists these 9 opaque constants, besides propext, Classical.choice and Quot.sound, for both headline theorems. The manuscript source shares its text with 10.5281/zenodo.20246708 (and carries that record's working title); this record adds the certified bound and the conditional Lean theorem.
Author: Papanokechi
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162212232