Legal reasoning---statutory interpretation, constitutional review, common-law adjudication---is currently conducted in natural language, where ambiguity is a feature, not a bug, and where ``arguing semantics'' is literally the professional activity. We propose : a mathematical framework in which the space of legal states is a weighted simplicial complex (the ), legal disputes are pathfinding problems on , and constitutionality is a topological invariant computable via path homology on directed graphs. The framework extends the Geometric Ethics programme---the Bond Invariance Principle (BIP), the Epistemic Invariance Principle (EIP), conservation laws, and gauge symmetry---from the moral domain to the legal domain. We define two legal invariance principles: the (LIP), requiring that legal judgments be unchanged under legally irrelevant transformations, and the (JBIP), requiring invariance under transformations that preserve Hohfeldian bond structure. We prove that equal protection is a gauge symmetry enforcing bilateral entitlement balance; that due process is the well-definedness of the evaluation map on the quotient space; that liability and damages are conserved within closed bilateral disputes; and that a new statute is constitutional if and only if it preserves the path homology of the constitutional subcomplex. We provide a concrete NLP pipeline for calibrating the complex's edge weights from case text. Three worked examples---a Fourteenth Amendment challenge, a statutory conflict, and a precedent-overruling scenario---demonstrate the framework's analytical power. The framework does not replace judicial judgment. It provides a that makes implicit reasoning explicit, detects hidden inconsistencies, and offers a formal criterion for constitutionality that is machine-checkable. Author preprint deposited for archival and citation. Draft — pending author review.
Andrew Bond (2026) studied this question.