lean

Crossed Cosmos — Lean 4 formalization

Toolchain : Lean 4.29.1 + lake 5.0.0 (leanprover/lean4:v4.29.1). Status : first PROVED theorems landed 2026-05-20 (no mathlib dependency).

Build

cd /root/cc-private/lean
lake build              # build library + executable
lake exe crossed        # run the demo executable

Layout

• lakefile.toml — Lake project (library Crossed, executable crossed).
• lean-toolchain — pinned to leanprover/lean4:v4.29.1.
• Crossed.lean — library root (imports Crossed.Basic).
• Crossed/Basic.lean — 9 PROVED theorems, 0 sorry.
• Main.lean — demo executable that prints the verified identities.
• EciM142.lean — STUB for M142 (α₂ = 1/12 for LMFDB 4.5.b.a). NOT in the Lake build yet (requires mathlib + heavy CM-modular-form theory). 6 sorrys remain.

First PROVED theorems (Crossed/Basic.lean)

All closed by decide or omega — no axioms, no sorry, kernel-checked.

Theorem	Statement	Tactic
xi_star_denominator	(2 : ℕ) + 1 = 3	decide
xi_star_numerator	1 * 2 = 2 ∧ 1 * 3 = 3	decide
c_DW_denominator	(9 : ℕ) + 1 = 10	decide
FN_two_numerator	9 * (2² + 1) = 45	decide
FN_two_denominator	10 * 2² = 40	decide
FN_two_gcd	Nat.gcd 45 40 = 5	decide
three_adic_split	D % 3 = 2 → ¬ 3 ∣ D (over ℤ)	omega
two_pow_two_eq_four	2 ^ 2 = 4	decide
four_lt_two_pow_three	4 < 2 ^ 3	decide
chained_identities	conj of three	decide

These are the algebraic substrates of the Crossed Cosmos universal ratios ξ* = 2/3, c_DW = 9/10, the F(N) genus-expansion identity, the 3-adic split rule, and the Nat.log 2 4 = 2 building block used in CR_prime_N4 of the Theorem CR' formalisation.

Next steps

1. Add mathlib dependency to lift the Nat identities into the actual ℚ statements ((1 : ℚ) / (1 + 1/2) = 2/3, (9 : ℚ) / 10 = 9/10, Nat.log 2 4 = 2, etc.). Adds ~30 min initial mathlib download.
2. Move EciM142.lean into the mathlib-enabled Lake project, then address the 6 remaining sorrys in order of decreasing tractability.
3. Bring in CRtheorem.lean + CRtheorem_helpers.lean from /root/notes/lean/ (16 sorrys combined, 4 mathlib gaps documented).

See /root/cc-private/notes/LEAN_FORMALIZATION_STATUS_2026-05-20.md for the full roadmap and DS V4 Pro dispatch report.