Lean-Verified Chronos Yang–Mills Formalization: Gauge Covariance, Gauss Constraints, Regulated Hamiltonians, Exact Physical Spectrum, and Unitary Evolution — YM1–YM87-YM115-YM116

This release publishes the Chronos Yang–Mills Lean formal-verification chain through milestone YM87. The archive contains the complete merged theorem source, final cumulative and incremental milestone packages, verification scripts, axiom-audit sources, checksums, citation metadata, and a detailed theorem ledger.

The formal development includes reusable operator and spectral-gap theorems, bounded and dense-domain Hamiltonian interfaces, self-adjointness and closure machinery, regulator convergence, non-abelian gauge covariance, Gauss-law constraints, a gauge-physical state sector, finite-lattice electric and magnetic energy, a concrete regulated Hamiltonian, an exact physical spectral gap, exact spectrum and projectors, a constructive resolvent, unitary Schrödinger evolution, conservation laws, spectral probabilities, moments, variance, and energy-uncertainty classifications.

The final concrete results apply to the finite regulated Gauss-physical model formalized in the archive. The release also contains conditional continuum-transfer theorems, but it does not claim construction of the full interacting four-dimensional continuum Yang–Mills theory or resolution of the Clay Millennium problem.

The recorded verification used Lean 4.34.0-rc2 and the OpenAI NavierStokesAndEuler repository pinned at commit 8937a8f4cbc7abaab5e9e97d1cc7f5d2319d9538. The final milestone runs passed source scans, Lean kernel checks, and direct axiom audits. Audited declarations depended only on propext, Classical.choice, and Quot.sound. 

YM115 update — Parity–Ear Cascade Core v0.1: This record has been extended with a later supplemental Lean milestone from the continuing Chronos Yang–Mills formal-verification program. YM115 freezes a new algebraic and combinatorial core developed for the SU(2) strong-coupling cascade. The verified source includes the doubled-spin parity decomposition, finite-choice ear-recursion algebra, positivity results for the current scalar majorants, certified rational brackets for the local-ear critical equation and the six-orbit chain cubic, and a conditional scalar contraction interface for downstream Yang–Mills specialization.

YM115 was directly verified with Lean 4.34.0-rc2. The UTF-8 audit, static scan for sorry, admit, and sorryAx, direct Lean compilation, and axiom audit all passed. The audited declarations depend only on propext, Classical.choice, and Quot.sound; one of the audited parity declarations uses only propext and Quot.sound.

This YM115 archive is a supplemental later-stage milestone and does not by itself contain every intervening YM88–YM114 package. The original cumulative material on this record remains the formally verified chain through YM87, while YM115 documents a newer proof route being developed beyond that release. The remaining gauge-support, full parity-aware ear-decomposition, connected transfer-kernel, infinite-volume, and continuum/RG bridges remain explicit future certification targets. This update therefore does not claim construction of the full interacting four-dimensional continuum Yang–Mills theory or resolution of the Clay Millennium problem.

YM116 update — 3D Slice → 4D History Bridge v0.2: This record has been extended with a further Lean-verified milestone from the continuing Chronos Yang–Mills formal-verification program. YM116 imports the previously verified Chronos 4D History Core and YM115 Parity–Ear Cascade Core into one Lean dependency chain and adds a formal bridge between time-indexed spatial states and the four-dimensional history structure used by the developing Yang–Mills transfer framework.

The successful verification used Lean 4.34.0-rc2. The UTF-8 audit, static scan for sorry, admit, and sorryAx, compilation of the original 4D source, compilation of YM115, compilation of the YM116 bridge, native theorem-signature checks, and direct axiom audits all passed. The principal combined YM116 declarations depend only on propext, Classical.choice, and Quot.sound; present_agreement_future_separation depends only on propext.

YM116 formally joins the verified 3D spatial/contraction machinery to a verified 4D history framework, providing a dimensional and temporal backbone for the next transfer-matrix stage. It does not yet establish the complete connected transfer-kernel bound, infinite-volume spectral gap, continuum renormalization-group bridge, or the full four-dimensional continuum Yang–Mills construction required for the Clay Millennium problem.