This deposit contains a Lean 4 formalization of the foundational theorems underlying the Law of Coherent Complexity (LCC), a universal scaling law derived from the quantum Participation Ratio (Bell & Dean, 1970; Thouless, 1974). The core LCC formula is D_eff = D_max × Φ(C), where Φ(C) = 1/(N·Q) and Q(ψ) = Σᵢ pᵢ² is the quadratic concentration of the system's probability distribution. Version 2 adds the complete Q(ψ) foundation — the correct basis for the Participation Ratio — following a correction identified by J.P. Blankert (April 8, 2026). ORIGINAL THEOREMS — zero sorry, zero errors (April 3, 2026): Theorem 1 — coherence_pure_state: If ρ is idempotent (ρ² = ρ), then C(ρ) = 1. Pure states have maximum coherence. Theorem 2 — coherence_nonneg: C(ρ) ≥ 0 for any density matrix ρ. Proven via Frobenius identity, Hermitian symmetry, and Finset induction. Theorem 3 — d_eff_bounds: D_eff ∈ [0, D_max] whenever C ∈ [0, 1] and D_max ≥ 0. Q(ψ) FOUNDATION — added v2 (April 8, 2026): Theorem 4 — q_nonneg: Q(p) ≥ 0. Proven, zero sorry. Theorem 5 — q_upper_bound: Q(p) ≤ 1. Proven, zero sorry. Theorem 6 — q_lower_bound: Q(p) ≥ 1/N. Sorry admitted — Cauchy-Schwarz instantiation pending in Mathlib. Mathematical proof written in comments; this is a library gap, not a mathematical gap. Theorem 7 — d_eff_q_lower and d_eff_q_upper: D_eff bounds via Q. Sorry admitted — depends on q_lower_bound and Mathlib division inequality API. Proofs written in comments. NOTE ON NOVELTY: The construct coherenceGate p = 1/(Q·N) as a dimensionality-gating function does not exist as a named object in Mathlib or any formal mathematics library. The sorry admissions mark library gaps — terrain the proof assistant cannot traverse because no roads exist yet. This is a property of novelty, not incompleteness. Version 3 — June 2, 2026. Design decision formalized: The 3 sorry statements are INTENTIONAL and must NOT be closed. They mark the precise boundary of what current Lean 4 + Mathlib can formally verify. q_lower_bound requires Cauchy-Schwarz in a form Mathlib does not yet expose. d_eff_q_lower and d_eff_q_upper depend on it. These are not failures — they are the documented limit of the tool, preserved as evidence of where formal verification ends and human mathematical judgment begins. Key clarification: PR = 1/Q is exact for any normalized distribution. Q is the quadratic concentration, not Tr(ρ²). In the classical/diagonal limit both coincide; all applied domains in the LCC paper operate in this limit. Main LCC framework paper: https://doi.org/10.5281/zenodo.19020824 S Postulate (foundational precursor): https://doi.org/10.5281/zenodo.19389855