Enterprise agents increasingly operate in environments where claims, actions, and policies must remain valid under model substitution, shifting evidence, and evolving operational constraints. In such settings, capability is not sufficient: promotion of conclusions and actions must be governed by explicit regimes, admissible transforms, falsifiers, and budgeted verification. Our work presents MATHINE as a field-first kernel that turns governance into computation through Math Machines — verification-oriented instruments that keep solver behavior separate from proof-layer promotion. This kernel thesis builds on MINUANO and PAMPA as method exemplars and on the broader “Math Machines” separation between solvers and proof-layer decisions. The kernel is modular by design: it evolves by adding and refining methods with narrow, explicit responsibilities rather than expanding a monolithic system. This stance follows classic arguments for decomposing complex systems into modules with stable interfaces and contracts. Each method contributes a distinct governance contract — intake normalization, regime coherence, drift corridors, identity canonicalization, receipt bundling, promotion ladders, dispute replay templates, standards mining, repeated-run protocols, proof harnesses, cost ledgers, integrity chains, and minimization gates — while the kernel composes these contracts into a replayable pipeline. In this design, progress is measured by how well methods make obligations explicit, transportable, and checkable under declared budgets, so new capabilities can be introduced without silently changing what counts as promotable. This “fail-closed” posture aligns with dependable and secure computing principles: when obligations cannot be verified within scope, the kernel holds rather than promotes. Verification cost is treated as a first-class constraint because some obligation families reduce to computationally hard search or proof tasks; the kernel therefore uses explicit budgets and HOLD states rather than implicit promotion under uncertainty. We define an adversary model (Solvers vs Math Machines) and specify a To-Be architecture that composes evidence intake, closure methods, integrity gates, and publication controls into a single governance pipeline suitable for enterprise agent settings. Intake modules transform raw external material into typed case objects with clear provenance boundaries; closure methods convert those objects into regimes, motifs, obligations, and verdict-ready bundles; integrity gates attach lineage and enforce append-only replay; and publication controls ensure that outward-facing artifacts are both audit-friendly and minimization-preserving. This draws on established patterns for ordering and replay in distributed systems, transparency and append-only integrity, and provenance models that make derivations explicit.