URB #517 — Lean 4 Formal Proofs for TI Sigma Mathematical Claims

We present formal Lean 4 proofs for the core mathematical claims of TI Sigma theory. Lean 4 is a dependently-typed theorem prover with the Mathlib4 library providing extensive real and complex number infrastructure. The proofs formalize: (1) the Genesis Identity — the derivation of √2 from i via the Four-Phase PK Protocol formula; (2) the i-Completeness chain — the sequence of derivations from i to each PRIMARY CONSTANT; (3) the C_EMERICK definition and key algebraic properties; (4) the LCC Unity Crossover threshold derivation; (5) the 6-element basis minimality claim. These constitute the first computer-verified proofs of TI Sigma's mathematical foundation, establishing that the formal apparatus is not merely intuitive but logically necessary.