Re-certifying the Viridis OS Decision Kernels: A Lean Proof of the Afforestation Seeding Core

Working paper, not peer reviewed. Nothing in this record is field-validated. Scope and novelty. Project P2 re-proves the formal cores of Viridis OS decision kernels so that anyone can re-check them with the current public Lean 4 toolchain. These are classical, elementary results, not new theorems: the first, Carbon Continuity, is the 2x2 case of the Collatz-Wielandt characterisation of the Perron root of a nonnegative matrix (A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM 1994, doi:10.1137/1.9781611971262). Each result holds only inside its stated model. It says nothing about coefficient values for any site, whether deployed kernel code implements the rule faithfully, or any carbon-credit or treatment outcome. Viridis OS runs decision kernels whose formal cores were checked by tools that can no longer be re-run. This program restates each kernel's formal core as a sealed, self-contained Lean 4 statement and re-proves it under a pinned toolchain (Lean v4.28.0, Mathlib 8f9d9cff). This revision treats the Afforestation kernel, which ranks seeding plans and site-preparation levers by seeding efficiency and scores how well a seed allocation aligns with the optimal one. We give a Lean proof of the sealed statement P2.Afforestation. Part 1 is the cubic establishment-efficiency law: with eff(a, sigma, Delta mu, C) = C (3 Delta mu / (2 a sigma))^3, scaling the drive Delta mu by t > 0 multiplies the efficiency by t^3. Part 2 is the alignment bound: in every real inner-product space, for nonzero n and m, the alignment eta = <n,m>^2 / (|n|^2 |m|^2) lies in [0,1]. The proof is a restructured port of the archived canon proof: part 1 is a ring identity and part 2 is the squared Cauchy-Schwarz inequality. We observe that the canon proof also replays verbatim under the pinned toolchain, and that none of the hypotheses in either part is needed. We set out what the theorem settles and what it leaves to field evidence: parameter values, the adequacy of the nucleation model for seeding, the equality case of the alignment score, the other canon results, and conformance of the deployed code. Earlier results (Carbon Continuity, Restoration) and the Harmonization proposal are retained. Verification. a Lean 4 proof of P2.Afforestation, a statement fixed before the proof attempt, machine-checked against Mathlib and replayed through the Lean kernel using only the standard axioms Run record, certification page and reproducible package: https://github.com/jdhart81/viridis-research/tree/main/runs/2026-11-01_P2 AI use: this version was drafted by an AI research agent (Claude, by Anthropic) running the Viridis research engine. Formal claims are checked by the Lean 4 kernel and labelled by an automated verifier. Justin D. Hart directs the research and is the accountable author. No AI system is listed as an author.