Chapter 22: Geometric Jurisprudence — Algorithmic Justice and Legal Manifolds
RUNNING EXAMPLE — Priya’s Model
HealthBridge’s legal team invokes the principle that algorithmic decisions are not discriminatory if no protected category is explicitly used as input. Priya recognizes this as the jurisprudential version of scalar collapse: legality is evaluated on a single axis (explicit use of protected features) while the full Hohfeldian structure—the rights of patients, the duties of the platform, the liabilities of deployment—requires the octad.
This chapter applies the Geometric Ethics framework to legal reasoning and algorithmic justice. The central thesis is that law — the operating system of civilization — has an implicit mathematical structure that natural language obscures but geometry makes explicit. Legal reasoning is pathfinding on a directed simplicial complex (the judicial complex), constitutional review is a topological computation (path homology preservation), Hohfeldian jural relations form a gauge group (the octahedral group D_4 x| D_4), and conservation laws characterize valid legal transformations. The framework formalizes what the legal system already demands — equal treatment, procedural regularity, constitutional conformity — and makes those demands computationally checkable.
22.1 Law as Geometric Structure
Legal reasoning has resisted formalization for centuries. Hart's positivism, Dworkin's interpretivism, and the law-and-economics movement each capture aspects of legal structure but none provides a complete mathematical framework. The geometric approach identifies the source of this difficulty: law operates on a multi-dimensional space (not a scalar), the space is directed (precedent flows forward in time and downward in hierarchy), and the space has topological structure (constitutional constraints are not metric but topological — they concern which paths exist, not how long they are).
The law-and-economics movement attempted scalar reduction: legal outcomes should maximize economic efficiency. This is the legal analog of Homo economicus — and it fails for the same reason. The Scalar Irrecoverability Theorem (Chapter 15) applies: no continuous function from the legal decision space to a scalar can be injective. Legal decisions that are "efficient" may be unjust, autonomy-violating, or trust-destroying. The scalar projection loses the information needed to evaluate these dimensions.
22.2 The Judicial Complex
Definition 22.1 (Judicial Complex). The judicial complex K is a weighted, directed simplicial complex with the following structure:
Vertices (0-simplices): Each decided case c_i carries an attribute vector v(c_i) in R^8 encoding eight legal dimensions, court level metadata l(c_i) in {trial, appellate, supreme}, and decision date τ(c_i).
Directed edges (1-simplices): c_i -> c_j exists when case c_j cites case c_i. This encodes temporal precedence (τ(c_i) < τ(c_j): cases cannot cite future decisions) and hierarchical authority (binding precedent has low edge cost; persuasive authority has high cost).
Higher simplices (k-simplices): A k-simplex exists when cases form a doctrinal cluster — a set of mutually citing cases that collectively establish a legal doctrine.
The eight legal dimensions are:
d_1: Entitlement structure — the configuration of Hohfeldian positions (rights, duties, liberties, no-rights, powers, immunities, liabilities, disabilities).
d_2: Factual nexus — causal, evidentiary, and material connections between parties and events.
d_3: Procedural posture — standing, jurisdiction, timeliness, exhaustion of administrative remedies.
d_4: Statutory authority — the legislative basis and statutory framework governing the dispute.
d_5: Constitutional conformity — respect for fundamental rights, structural provisions, separation of powers.
d_6: Precedential constraint — the weight of stare decisis, binding versus persuasive authority.
d_7: Remedial scope — available relief (compensatory damages, injunctive relief, specific performance, declaratory judgment).
d_8: Public interest — societal implications, third-party effects, regulatory impact.
Remark. The legal complex uses eight dimensions rather than the moral manifold's nine because the Hohfeldian entitlement structure (d_1) subsumes some moral dimensions that are separately tracked in the ethical context. The framework's validity does not depend on this specific count; the theorems hold for any finite-dimensional complex.
Definition 22.2 (Legal Edge Weights). The weight of a directed edge c_i -> c_j in K is:
w(c_i -> c_j) = ΔvT Σ−1 Dv + β * 𝟙[regime boundary crossed] + h(l(c_i), l(c_j))
where Dv = v(c_j) - v(c_i), Sigma is the 8x8 legal-dimensional covariance matrix, β is the regime-boundary penalty, and h(l_i, l_j) is the hierarchical cost: 0 if l_i > l_j (binding: higher to lower court), eta (persuasive penalty) if same level, and infinity if reverse hierarchical direction (lower courts cannot overrule higher courts).
22.3 Legal Invariance Principles
Definition 22.3 (Legal Invariance Principle, LIP). A legal judgment procedure J_law is epistemically well-posed if and only if:
J_law(τ(x)) = J_law(x) for all τ in T_irrelevant
where T_irrelevant includes: renaming parties, changing protected-class attributes (race, gender, religion, national origin), rephrasing legal arguments without changing logical content, changing order of evidence presentation, and translating between languages. The LIP formalizes what the Equal Protection Clause, Due Process Clause, and Rule of Law already demand in prose.
Definition 22.4 (Judicial Bond Invariance Principle, JBIP). Legal evaluation must be invariant under transformations preserving Hohfeldian bond structure:
J_law(τ(x)) = J_law(x) for all τ in T_bond-preserving
The JBIP is stronger than the LIP: it specifies the mathematical structure (Hohfeldian relations) whose preservation defines legal equivalence.
Definition 22.5 (Legal Bond Index). The Legal Bond Index is:
LBI = (1/|S|) Sum_{s in S} ||J_law(τ(s)) - J_law(s)||
where S is a test set of cases and τ ranges over bond-preserving transformations. LBI is computable from existing case databases and provides a quantitative measure of judicial consistency.
22.4 The Hohfeldian Octad and Gauge Theory
The Hohfeldian system of jural relations — developed by Wesley Hohfeld in 1913 and discussed in Chapter 3 — consists of eight fundamental positions organized in two squares:
First square (first-order relations): Right <-> Duty (correlatives), Liberty <-> No-right (correlatives), Right <-> No-right (opposites), Liberty <-> Duty (opposites).
Second square (second-order relations, governing legal change): Power <-> Liability (correlatives), Immunity <-> Disability (correlatives), Power <-> Disability (opposites), Immunity <-> Liability (opposites).
Theorem 22.1 (Octahedral Gauge Group). The symmetry group of the full Hohfeldian octad is the semi-direct product:
G_Ho = D_4 x|_phi D_4
where the first D_4 acts on first-order relations (right-duty-liberty-no-right), the second D_4 acts on second-order relations (power-liability-immunity-disability), and the homomorphism phi: D_4 -> Aut(D_4) encodes how second-order operations act on first-order positions.
Proof. Each Hohfeldian square admits D_4 symmetry: two correlative pairs generate the dihedral group of the square (two reflections plus rotations). However, the squares are not independent — second-order positions (powers, immunities) actively transform first-order positions (creating and destroying rights and duties). This asymmetric coupling is precisely the structure of a semi-direct product. The multiplication rule (g_1, g_2) * (h_1, h_2) = (g_1 * phi(g_2)(h_1), g_2 * h_2) encodes that the first-order transformation h_1 is "twisted" by the second-order transformation g_2. []
Remark. In the parent framework (Chapter 12), the gauge group was D_4 acting on the first four Hohfeldian positions. Law requires the full octad — both first-order relations (which govern static legal positions) and second-order relations (which govern legal change). The extension from D_4 to D_4 x| D_4 is the mathematical content of the distinction between static and dynamic legal structure.
Definition 22.6 (Legal Wilson Loop). A legal Wilson loop is the holonomy around a closed path in the judicial complex:
W(γ) = P exp(integral_gamma A_mu dx^mu)
where A_mu is the legal connection encoding how Hohfeldian positions change under parallel transport through case law.
Proposition 22.1 (Wilson Loop Test for Consistency). A body of case law is internally consistent if and only if all Wilson loops are trivial. Non-trivial Wilson loops identify specific circuits of legal reasoning that produce contradictions — "bugs" in the legal system that are detectable algorithmically rather than waiting for a case to expose them.
22.5 Topological Constitutionality
Constitutional review — determining whether a statute is constitutional — is the most consequential legal computation. The geometric framework recasts it as a topological problem.
Definition 22.7 (Constitutional Subcomplex). The constitutional subcomplex C is a subcomplex of K consisting of all simplices satisfying constitutional constraints:
C = {sigma in K | Phi_k(sigma) = true, k = 1, ..., K}
where the Boolean predicates Phi_k encode constitutional provisions: Phi_EP for Equal Protection invariance, Phi_DP for Due Process well-definedness, Phi_1A for First Amendment protection, Phi_SoP for Separation of Powers.
Because K is a directed complex (edges encode temporal and hierarchical order), the appropriate homology theory is path homology (Grigor'yan-Lin-Muranov-Yau), not standard simplicial homology. An elementary p-path is a sequence of vertices (c_{i_0}, ..., c_{i_p}) connected by directed edges. The path homology groups are:
H_n^path(K; Z) = ker partial_n / im partial_{n+1}
Theorem 22.2 (Topological Constitutionality). A statute ell is constitutional if and only if it preserves the path homology of the constitutional subcomplex:
H_n^path(C; Z) = H_n^path(C_ell; Z) for all n >= 0
Equivalently: ell is constitutional if and only if it does not create new non-trivial directed cycles in C.
Proof. A non-trivial directed cycle in C_ell that does not exist in C represents a directed sequence of legal reasoning that returns to the starting vertex with altered Hohfeldian structure — a legal contradiction. Since the cycle is directed, each step respects temporal and hierarchical order, making it a genuine inconsistency rather than an artifact of undirected analysis. Preservation of path homology prevents such contradictions. Conversely, if a new non-trivial cycle exists, a sequence of valid legal steps creates a contradiction, and the statute that enabled the cycle is the source. []
This theorem converts constitutional review from a semantic dispute ("what does equal protection mean?") into a topological calculation on a finite directed graph. The computation is finite: path homology groups are finitely generated abelian groups computable via Smith normal form on the directed path complex.
22.6 Conservation Laws in Legal Systems
Theorem 22.3 (Liability-Damages Conservation). In a closed bilateral dispute between plaintiff A and defendant B within a fixed legal framework F:
L(A) + L(B) = 0
where L(X) is the net liability of party X. Every dollar of damages to A imposes exactly one dollar of liability on B. Every right created for A imposes a correlative duty on B.
Proof. By Hohfeldian correlative structure: (1) every right of A is a correlative duty of B; (2) every liability of B is a correlative power of A; (3) adjudication recognizes or assigns positions within the fixed framework F — it does not create positions ex nihilo. Therefore, at every step, Delta E(A) + Delta E(B) = 0. []
Remark (Conservation Breaking). Three operations break liability-damages conservation: (a) Legislation — creates causes of action ex nihilo (symmetry-breaking analogous to gauge symmetry breaking in physics); (b) Constitutional amendment — alters the topological constraint space C; (c) Third-party intervention — adding a new party breaks closure (conservation is restored by extending the system to include all parties).
Theorem 22.4 (Equal Protection as Gauge Symmetry). Equal Protection (14th Amendment) requires that legal evaluation be invariant under protected-class transformations. In any closed bilateral dispute, this invariance implies entitlement balance: the signed sum of Hohfeldian positions is invariant under adjudication.
Theorem 22.5 (Due Process as Well-Definedness). Due Process requires that J_law be well-defined on the quotient space K / T_irrelevant. Failure to factor through the quotient means legal outcomes depend on representation (the specific description of the case) rather than substance (the equivalence class under legally irrelevant transformations). This is the formal content of procedural due process.
22.7 Legal Disputes as A* Pathfinding
A legal dispute is pathfinding on K. The plaintiff starts at vertex c_0 (current legal position) and seeks a path to a goal region G (vertices where similarly situated plaintiffs prevailed). The path gamma represents the chain of doctrinal steps through case law — the legal argument.
Definition 22.8 (Legal Friction). The legal friction of a path gamma is:
BF_law(γ) = Σ w(c_i, ci+1) + Σk βk * 𝟙[crosses regime k] + Σj ωj * δj(γ)
with three components: (1) argument complexity — total edge weight, measuring doctrinal distance; (2) boundary penalties — cost of crossing jurisdictional, statutory, or constitutional regime boundaries; (3) burden of proof — cost of establishing facts, with ωj weighting evidentiary requirements.
Theorem 22.6 (Optimal Legal Argument). The optimal legal argument is the minimum-cost path:
gamma* = arg min_gamma BF_law(γ) subject to gamma(0) = c_0, gamma(end) in G
This is A* search with f(n) = g(n) + h(n), where g(n) is accumulated legal friction from c_0 to n (the argument built so far) and h(n) is the doctrinal heuristic estimating remaining friction to G (legal judgment about remaining burden).
Theorem 22.7 (Admissibility of Doctrinal Heuristics). A doctrinal heuristic h_D is admissible if the doctrine correctly identifies claim elements and assigns non-negative costs not exceeding actual costs. If h_D is admissible, A* search using h_D finds the optimal legal argument.
Examples of doctrinal heuristics: The prima facie case in tort law estimates the cost of establishing duty, breach, causation, and damages — an admissible decomposition of total legal friction. Burden-shifting frameworks (McDonnell Douglas in employment discrimination) provide a three-step heuristic for evaluating discrimination claims. Strict scrutiny assigns high estimated cost to the government's path and low cost to the challenger's. Stare decisis is a pattern database of pre-computed optimal paths: precedents provide near-optimal heuristics for similar cases.
Adversarial Pathfinding. Litigation is a two-player game. The plaintiff minimizes friction to G_plaintiff; the defendant either blocks the path, increases plaintiff's friction above the burden threshold, or establishes a shorter path to G_defendant (dismissal). The minimax value V(c_0) = min_{gamma_d} max_{gamma_p} [BF_law(gamma_d) - BF_law(gamma_p)] determines the outcome: V(c_0) > 0 implies plaintiff prevails; V(c_0) <= 0 implies defendant prevails.
Settlement as Shortcut. A settlement is a vertex c_s both parties prefer to the expected litigation outcome. A non-empty settlement region exists when the combined litigation costs exceed the distance between the parties' goal regions. This explains why most disputes settle: path lengths through case law nearly always exceed the distance between the parties' positions.
22.8 Precedent as Weight Deformation
A binding precedent P at vertex c_P with holding H_P modifies edge weights throughout its neighborhood:
w(c_i -> c_j) -> w(c_i -> c_j) + Dw(c_i -> c_j; P)
The deformation concentrates near c_P and decays with graph distance: |Dw| ~ w_P / d(c_i, c_P)^n, where w_P is precedential weight (Supreme Court > Circuit > District).
Overruling as Phase Transition. Overruling precedent P by P' is graph surgery:
w(c_i -> c_j) -> w(c_i -> c_j) - Dw(c_i -> c_j; P) + Dw(c_i -> c_j; P')
This is a legal phase transition: the filtration structure changes, vertices cross regime boundaries discontinuously, and the shortest-path structure alters abruptly. Stare decisis creates a cost barrier around precedents — a form of legal inertia. The barrier height equals the accumulated reliance interest (the total weight of subsequent cases decided under the precedent). Courts overrule when the topological benefit (removing constitutional inconsistency, reducing Wilson loop complexity) exceeds the transition cost.
Example 22.1 (Brown v. Board of Education). Overruling Plessy v. Ferguson (1896) by Brown v. Board (1954) is a phase transition in the equal-protection region of K. Under Plessy, "separate but equal" vertices had finite weight to the constitutional region. Under Brown, those edges acquire infinite weight — segregation itself violates equal protection. Previously "legal" vertices become unreachable by valid constitutional paths. The transition cost was enormous (decades of subsequent litigation), but the topological benefit was a drastic simplification of the equal-protection homology.
22.9 Empirical Calibration Pipeline
The judicial complex K is not merely a theoretical construct — it is empirically constructible from existing legal databases. The calibration pipeline proceeds in five steps:
Step 1 (Embedding): Each judicial opinion t_i is embedded via LaBSE (Language-agnostic BERT Sentence Embedding) into e(t_i) in R^768. The language-invariance of LaBSE ensures that the same legal concept in different languages maps to the same embedding region — critical for cross-jurisdictional analysis.
Step 2 (Dimension Scoring): Eight linear probes f_k: R^768 -> [0,1] (logistic regression classifiers trained on labeled legal texts) produce the attribute vector v(c_i) = (f_1(e), ..., f_8(e)). The predecessor moral probes achieved F_1 = 0.74-0.91; legal probes target the same performance.
Step 3 (Covariance Estimation): From N scored cases, estimate the 8x8 covariance matrix Sigma_hat. This captures cross-dimensional dependencies (statutory authority correlates with remedial scope; procedural posture gates merits access).
Step 4 (Edge-Weight Computation): For each citation c_i -> c_j, compute the Mahalanobis-weighted directed edge cost plus boundary penalties and hierarchical cost.
Step 5 (Outcome Calibration): Fit a regularized logistic model from attribute vectors to case outcomes; the fitted weight matrix provides outcome-calibrated inverse covariance. The effective metric blends corpus-derived and outcome-derived structure: Sigma_eff^{-1} = lambda * Sigma_hat^{-1} + (1-lambda) * W, with lambda set by cross-validation.
Critical property: the pipeline is deterministic. LaBSE weights are frozen; linear probes are logistic regression (no sampling, no temperature). The same input always produces the same vector. This contrasts with LLM-based scoring, where outputs vary across runs. For legal applications, determinism is non-negotiable.
22.10 Worked Examples
The preceding sections developed the geometric jurisprudence framework abstractly. This section applies it to three landmark cases — spanning constitutional law, criminal procedure, and property law across two legal systems — to demonstrate that the framework produces genuine analytical insight rather than mere re-description.
Example 22.2 (Obergefell v. Hodges, 2015 — Topological Constitutionality in Practice). The Supreme Court held 5-4 that the Fourteenth Amendment requires states to license and recognize same-sex marriages. This decision is analyzable as a topological computation on the judicial complex.
Analysis on the eight legal dimensions: d_1 (entitlement structure): Before Obergefell, same-sex couples lacked the Hohfeldian right to marry — and correlatively, states had no duty to issue marriage licenses to them. Obergefell created a right-duty pair. d_5 (constitutional conformity): The majority applied strict scrutiny: marriage is a fundamental right under Due Process (d_5-preserving), and denying it based on gender or orientation violates Equal Protection (d_5-preserving). The dissenters argued Obergefell created a new cycle in the constitutional subcomplex by extending substantive due process beyond historical bounds. d_6 (precedential constraint): The decision rested on Loving v. Virginia (1967, interracial marriage), Lawrence v. Texas (2003, intimate conduct), and Windsor v. United States (2013, federal recognition). These precedents defined the neighborhood weights. The geodesic through these cases was the shortest path from the ban to constitutional recognition. d_8 (public interest): The majority weighed the societal interest in equal dignity; the dissent weighed federalism and democratic process.
Topological analysis (Theorem 22.2): The pre-Obergefell constitutional subcomplex C contained a non-trivial directed cycle: [Loving (marriage is a fundamental right) -> Romer v. Evans (anti-gay animus is unconstitutional) -> Lawrence (intimate conduct is protected) -> Windsor (federal marriage recognition required)] — yet state marriage bans persisted, creating a cycle that returned to the starting entitlement position with inconsistent Hohfeldian structure. Obergefell resolved this cycle by completing the doctrinal path. H_1^path(C_post) was simpler than H_1^path(C_pre).
The dissent (Roberts, CJ) is analyzable as a competing topological claim: that the marriage bans did NOT create a non-trivial cycle because marriage and intimate conduct occupy different doctrinal clusters (different simplices in K). The majority's holding prevailed because the Loving-Lawrence-Windsor path was connected — the graph distance between the cases was small enough to support a doctrinal chain.
Example 22.3 (Miranda v. Arizona, 1966 — Gauge Invariance in Criminal Procedure). The Supreme Court held that the prosecution may not use statements from custodial interrogation unless the defendant was informed of the right to remain silent and the right to counsel (Miranda warnings).
Legal Invariance analysis: Before Miranda, the admissibility of confessions depended on the "totality of the circumstances" — a case-by-case evaluation that was inherently gauge-variant. The SAME confession, described differently by different courts (emphasizing different contextual factors), could be admissible in one jurisdiction and inadmissible in another. This was a systematic LIP violation (Definition 22.3): the legal outcome depended on the description of the interrogation, not on its underlying Hohfeldian structure.
Miranda imposed gauge invariance by fiat: a bright-line rule that any custodial statement is inadmissible unless preceded by specific warnings. This is a mechanical gauge condition — a procedure that ensures the legal evaluation is representation-independent.
The framework's analysis: d_3 (procedural posture): Miranda standardized the procedure, eliminating the gauge-variant "totality" test. d_5 (constitutional conformity): The Fifth Amendment (self-incrimination) and Sixth Amendment (counsel) rights were given operational content through the warning requirement. Wilson Loop Test (Proposition 22.1): Before Miranda, the judicial complex contained non-trivial Wilson loops in the confession admissibility region — circuits where transporting a confession through different courts produced different Hohfeldian outcomes. Miranda trivialized these loops by making the legal transformation path-independent.
The subsequent narrowing of Miranda (e.g., New York v. Quarles, 1984 — "public safety exception"; Berghuis v. Thompkins, 2010 — invocation must be unambiguous) represents re-introduction of gauge variance. Each exception creates a new region where the bright-line rule does not apply, and the "totality of circumstances" test reappears — reopening non-trivial Wilson loops.
Example 22.4 (Mabo v. Queensland (No. 2), 1992 — Precedent as Phase Transition in Australian Law). The Australian High Court overturned the doctrine of terra nullius ("land belonging to no one") that had been the legal foundation for British colonization of Australia since 1788. The court recognized native title — Aboriginal and Torres Strait Islander peoples' rights to land based on traditional connection.
Phase transition analysis (Section 22.8 framework): Before Mabo, terra nullius meant Aboriginal peoples had no Hohfeldian rights (d_1 = 0) to their traditional lands. Crown sovereignty extinguished all pre-existing title. The overruling was a legal phase transition: the entire weight structure of Australian property law changed. Previously valid paths (crown grants on "unoccupied" land) acquired new boundary penalties (native title must be considered). Previously unreachable vertices (Aboriginal land rights) became accessible.
The transition cost was enormous: decades of subsequent litigation (Wik Peoples v. Queensland, 1996; Native Title Act 1993; multiple amendments), uncertainty in mining and pastoral leases, and political backlash. The topological benefit: the pre-Mabo constitutional subcomplex contained a non-trivial cycle between Australian law's commitment to equality (Racial Discrimination Act 1975) and its simultaneous denial of Aboriginal property rights. Mabo resolved this topological inconsistency.
d_8 (public interest): The decision transformed Australian national identity by acknowledging historical injustice — parallel to Brown v. Board's effect on American racial jurisprudence. This is the most dramatic demonstration of precedent overruling as phase transition: a 204-year-old legal doctrine was overturned, fundamentally restructuring the judicial complex.
22.11 Falsifiable Predictions
Prediction 1 (Legal Bond Index Correlates): Courts with higher LBI scores (lower invariance under bond-preserving transformations) should have higher reversal rates on appeal. Falsified if: LBI is uncorrelated with reversal rates.
Prediction 2 (Dimensional Factor Structure): Factor analysis of scored judicial opinions should recover approximately eight independent legal dimensions. Falsified if: the factor structure is consistently lower-rank (fewer than six) or higher-rank (more than ten).
Prediction 3 (Precedent Decay): The influence of a precedent on edge weights should decay with graph distance from the precedent vertex, with binding precedent showing slower decay than persuasive authority. Falsified if: precedential influence is distance-invariant.
Prediction 4 (Constitutional Cycle Detection): Non-trivial directed cycles in the constitutional subcomplex should correlate with statutes subsequently struck down by courts. Falsified if: cycle presence is uncorrelated with judicial invalidation.
Prediction 5 (Settlement Region): The existence and size of the settlement region should be predictable from the parties' graph positions and expected path costs. Falsified if: settlement rates are independent of computed path costs.
Prediction 6 (Cross-Jurisdictional Invariance): The framework's structural features (gauge group, conservation laws, topological constitutionality) should hold across legal traditions (common law, civil law, Islamic law, customary law) at the structural level, even as the specific dimensions and metric vary. Falsified if: the framework applies only to common-law systems.
22.12 Connection to the Framework
The Geometric Jurisprudence program extends the parent framework in five directions:
1. Chapter 3 introduced Hohfeldian relations as the discrete structure of moral bonds. This chapter elevates them to a gauge group (D_4 x| D_4) and shows that the semi-direct product structure captures the asymmetric coupling between static legal positions and dynamic legal change.
2. Chapter 11 established pathfinding as the formal model of moral reasoning. This chapter shows that legal reasoning is adversarial pathfinding — a two-player variant where plaintiff and defendant simultaneously optimize on K, with doctrines serving as domain-specific heuristic functions.
3. Chapter 12 established conservation laws from invariance. This chapter derives liability-damages conservation from Hohfeldian correlative structure and shows that conservation-breaking operations (legislation, constitutional amendment, third-party intervention) correspond to gauge symmetry breaking.
4. Chapter 15 established scalar irrecoverability. This chapter applies it to the law-and-economics program: scalar efficiency measures (Kaldor-Hicks) lose the seven non-monetary dimensions of the legal decision space.
5. The directed nature of the judicial complex (temporal, hierarchical) requires path homology rather than standard simplicial homology — a mathematical upgrade that reflects the irreversibility of legal time and the asymmetry of legal authority.
22.13 Summary
This chapter has shown that the geometric ethics framework, when applied to law, yields:
1. The judicial complex K: a directed, weighted simplicial complex constructible from case law databases.
2. The octahedral gauge group D_4 x| D_4: the full symmetry of Hohfeldian jural relations, extending the D_4 of the parent framework to include second-order legal change.
3. Topological constitutionality: constitutional review as path homology preservation — a finite computation on a directed graph.
4. Conservation laws: liability-damages conservation from Hohfeldian correlative structure, with legislation as symmetry breaking.
5. Legal disputes as adversarial A* pathfinding, with doctrines as heuristic functions and settlement as shortcut.
6. An empirical calibration pipeline (LaBSE + linear probes + Mahalanobis weights) that makes the judicial complex constructible from existing data.