A New Mathematical Framework
Senior Member, IEEE · San Jose State University
Moral evaluation is not a number. It is a tensor on a stratified manifold.
Why ethics needs geometry
Traditional ethics reduces moral evaluation to a single number — a score, a utility, a ranking. This scalar collapse destroys the geometric structure that matters most: which values are at stake, where uncertainty concentrates, and where the rules change discontinuously.
Moral evaluation is a tensor on a stratified manifold — a structured object in a space with dimensions, distances, curvature, and boundaries. The mathematical structures physicists use to describe nature are also the right structures for describing moral reasoning.
塞翁失马 — Sai Weng Shi Ma
Neighbors: "Bad fortune!" Old man: "Maybe."
Neighbors: "Good fortune!" Old man: "Maybe."
Neighbors: "Bad fortune!" Old man: "Maybe."
The broken leg saves his life.
The old man's "maybe" is not epistemic humility. It is a recognition that scalar evaluation is the wrong tool for the job. The loss is bad along the wealth dimension while saying nothing about health, family, or political dimensions. The uncertainty has shape. The evaluation depends on which regime the world occupies.
Ethics has shape, and the shape matters.
Core mathematical objects at a glance
The "space" of morally relevant situations
9-dimensional Whitney-stratified space with smooth strata and semantic gates at boundaries
What an agent must do, with direction and magnitude
Tangent vector on M (rank-1 tensor)
What a patient needs, measuring obligations
Cotangent vector on M (rank-1 tensor)
How well obligations meet interests
S = Iμ Oμ — covector × vector → scalar
The "exchange rate" between moral dimensions
Symmetric rank-2 tensor encoding trade-off structure
Moral evaluations must not depend on mere labeling
Gauge symmetry: E(d) = E(d') for admissible re-descriptions
Reducing a tensor to a scalar, and what is lost
T(n) → S via iterated contraction; R captures discarded structure
Quantitative alignment score for AI systems
Scalar measure of structural × invariance × residue compliance
Derived from a 3×3 scope/mode grid
The outcomes of actions — who benefits, who is harmed, and by how much. The utilitarian dimension.
The deontological constraints — what agents owe to others regardless of consequences.
The distributional structure — how benefits and burdens are allocated across persons.
The capacity for self-determination — freedom to choose, informed consent, non-manipulation.
Informational self-determination — control over personal information and surveillance boundaries.
Collective and ecological obligations — commons stewardship, intergenerational duty.
Character and relational obligations — what a good person would do, care ethics.
Process fairness — whether decisions follow legitimate procedures and institutions.
Knowledge conditions — what is known, what is uncertain, and the ethics of belief.
From scalars to the full moral tensor — each level reveals structure the previous conceals
30 chapters across 7 parts, from philosophical motivation to engineering implementation
Read the Full Book Online →The shape of the problem. Three failures of "Flatland" in AI alignment, policy analysis, and moral philosophy itself. The parable of the old man and his horse.
Read full chapter →Why rank-0 ethics is insufficient: no directional information, uncertainty has shape, paths cross boundaries. What geometric structure provides.
Read full chapter →Aristotle's mean as calibration. Kant's imperative as invariance. Ross's duties as vectors. Rawls's veil as symmetry. Sen and Nussbaum. Hohfeld's jural relations.
Read full chapter →Manifolds, tangent bundles, covectors, tensors, metrics, connections, curvature, fiber bundles, stratified spaces, gauge theory. The complete geometric toolkit.
Read full chapter →The 9-dimensional moral space. What are the points? The 3×3 derivation. Coordinates, admissible transformations, stratification, singularities, topology.
Read full chapter →From scalars to tensors. Obligations as vectors. Interests as covectors. The fundamental contraction: satisfaction. The moral metric. The full moral tensor.
Read full chapter →A single medical allocation case analyzed at each tensor level. Six claims that require tensors. The pedagogy of accumulation.
Read full chapter →Whitney stratification. Moral boundaries: thresholds, phase transitions, absorbing strata, forbidden regions. Semantic gates. The geometry near a boundary.
Read full chapter →Discovery, construction, or governance? Realist, constructivist, expressivist, and governance accounts. Constraints on admissible metrics. Pluralism and the meta-metric.
Read full chapter →The moral connection. Parallel transport of obligations. Holonomy: path-dependence of moral evaluation. Geodesics: paths of least resistance. Gradient flows.
Read full chapter →A* search on the moral manifold. Obligation vectors as heuristic functions. Computational intractability of exact moral reasoning. Deontology as pre-compiled heuristics.
Read full chapter →The Bond Invariance Principle as continuous symmetry. The moral Lagrangian. The conservation of harm. Four consequences: euphemism doesn't reduce harm, harm is auditable, re-description can't redistribute harm, moral debt persists.
Read full chapter →Superposition as deliberation. Moral observables and measurement. Interference between framings. The density matrix. The stratified Lagrangian. Entanglement. The moral Schrödinger equation.
Read full chapter →Aggregation, emergence, and shared obligation. The collective agency tensor. Distributed responsibility. Institutional geometry. AI systems as collective agents.
Read full chapter →The moment of choice. Non-commutativity. Information lost in contraction. Moral residue. Deferred contraction. Contraction in AI systems.
Read full chapter →Three types of uncertainty. Decision under moral uncertainty. The intertheoretic comparison problem. Robust obligations. Residual indeterminacy. The modesty of the framework.
Read full chapter →The Dear Abby corpus analysis. BIP experiments. The Dear Ethicist game. Quantum cognition predictions. Cross-lingual invariance. The philosophy engineering code corpus (2.77M lines).
Read full chapter →Tensor-valued objectives. Invariance as alignment. The No Escape Theorem. Escape route analysis. Implementing geometric AI ethics. The alignment problem as contraction mismatch.
Read full chapter →ErisML: a modeling language for geometric ethics. The translation layer. DEME: the ethics engine. The Bond Index. The separation principle. The Norm Kernel. The Grand Unified AI Safety Stack.
Read full chapter →The Bond Geodesic Equilibrium. Behavioral game theory as manifold geometry. Prospect theory. The 2008 financial crisis as manifold failure.
Read full chapter →Triage, consent, and the clinical geodesic. The QALY Irrecoverability Theorem. Mathematical theory of moral injury. Geometric informed consent.
Read full chapter →Law as geometric structure. The Hohfeldian octad and gauge theory. Topological constitutionality. Legal disputes as A* pathfinding.
Read full chapter →Market microstructure on the decision manifold. The flash crash as dimensional collapse. Option pricing as scalar projection.
Read full chapter →Religious reasoning on the moral manifold. The Euthyphro dilemma as gauge ambiguity. Theodicy as dimensional projection. Genesis 3:22.
Read full chapter →Climate, commons, and intergenerational obligation. The discount rate as dimensional collapse. Species extinction as irreversible boundary crossing.
Read full chapter →The moral geometry of algorithmic systems. Alignment as geodesic preservation. Algorithmic bias as scalar projection. The paperclip maximizer as dimensional collapse.
Read full chapter →Population-level ethics. CRISPR as irreversible manifold modification. Research ethics and the double consent condition. Neuroethics as autonomy curvature.
Read full chapter →War and constrained pathfinding. Proportionality as multi-dimensional cost-benefit. The doctrine of double effect as dimensional decomposition. Moral injury as manifold damage.
Read full chapter →The empirical program for moral curvature. The moral field equation. Torsion in moral space. Tensorial interpretability for AI. Scalability of structural containment. What would falsify the framework.
Read full chapter →Return to the border. What the old man knew. The arc of the argument. What the framework provides. Ethics is not a number. A final "maybe."
Read full chapter →Seven central theorems, each conditional on stated assumptions
The space of admissible moral metrics admits a partial order that is not total: genuine moral disagreement is geometrically irreducible.
If the moral Lagrangian is invariant under re-description (BIP), then harm is a conserved Noether charge. Euphemism does not reduce harm.
The maximal gauge group consistent with Hohfeldian structure and bounded harm is D4 × U(1)H.
Under canonicalization, grounded evaluation, structural audit, and verification integrity, an AI system cannot circumvent moral constraints by re-description.
The order in which moral dimensions are contracted affects the outcome: contraction paths are not interchangeable.
Core moral heuristics never overestimate true cost to equilibrium. Deontological rules are admissible A* heuristics.
Exact moral geodesic planning is intractable in manifold dimension. This is why evolution pre-compiled heuristics into our cognitive architecture.
Nine established domains, each with worked examples and falsifiable predictions
Bond Geodesic Equilibrium
Triage & Clinical Geodesic
Legal Manifolds & Gauge
Market Microstructure
Religious Reasoning
Climate & Commons
Alignment & Containment
CRISPR & Population Ethics
Constrained Pathfinding
The Fundamental Equation of Moral Reasoning
Moral reasoning is A* pathfinding on the moral manifold. Obligation vectors are gradient vectors of heuristic functions. Classical moral rules — "do not kill," "keep your promises" — are admissible heuristics: they never overestimate true cost to moral equilibrium.
Noether's Theorem applied to ethics
Relabeling "killing" as "end-of-life transition facilitation" does not change the conserved Noether charge.
Because harm is conserved, it can be tracked and verified regardless of how a situation is described.
You cannot move harm from one dimension to another merely by changing the coordinate system.
Unresolved harm carries forward. The conservation law is the formal basis for moral accountability.
Choose your journey through the book
The core argument: why scalar ethics fails, geometric precursors, the governance account, contraction and residue, the limits of the framework.
The formal development: moral manifold, tensor hierarchy, stratification, connection, curvature, Noether theorem, quantum extension.
Direct application: tensor-valued objectives, invariance testing, the No Escape Theorem, DEME architecture, ErisML, empirical validation.
The accessible argument: geometric structure of ethics, why scalar AI governance fails, the governance account, the mandate question.
The core argument in five chapters. Why geometry, why not scalars, one case at five levels, from tensor to decision, geometric AI ethics.
From theory to implementation: the Grand Unified AI Safety Stack
Every claim is tagged with its epistemic status
Stipulated structures — the 9 dimensions, manifold topology, Hohfeldian strata. Architectural decisions chosen for explanatory and engineering utility. They could be otherwise.
Mathematical results that follow rigorously from stated premises. The theorem itself is not empirical; what is empirical is whether the premises obtain.
Findings supported by data. "Preliminary" = single study. "Robust" = replicated or survived deliberate falsification attempts. The distinction is marked honestly.
Ideas that cannot yet be supported with proof or data. The Orch-OR connection, the moral field equation — flagged as speculative. They suggest research directions, not defended claims.
Learn geometric ethics hands-on with Jupyter notebooks and live code
Your first ethics evaluation. Create EthicalFacts objects, instantiate the RightsFirstEM module, and compare two options.
View SourceTest whether moral evaluations are invariant under re-description. Apply BIP to real scenarios and measure violation scores.
View SourceBuild a complete geometric ethics pipeline for emergency triage: obligation vectors, interest covectors, contraction, and residue analysis.
View SourceExplore the dihedral group structure of Hohfeldian jural relations: obligation, claim, liberty, no-claim. Verify the D4 symmetry computationally.
View SourceEvaluate large language models using the Bond Index. Measure structural compliance, invariance, and residue across AI moral judgments.
Open NotebookFive complete pipelines: Emergency Triage, Whistleblower's Dilemma, Autonomous Vehicle, Algorithmic Hiring, Climate Policy. Full tensor computation with Bond Index.
View Sourcepip install erisml
# Run the hello world
python -m erisml.examples.hello_deme
# Run the full Appendix D pipeline
python -m erisml.examples.appendix_d_pipeline
# Launch the Jupyter notebook
jupyter notebook scripts/bond_index_llm_evaluation.ipynbAn advice column game that measures the mathematical structure of your moral reasoning
Testing quantum-like structure in moral reasoning — SQND experimental program
Stratified Quantum Normative Dynamics (SQND) tests whether moral reasoning exhibits quantum-like features: superposition, interference, and entanglement.
The order in which moral dimensions are presented affects the final judgment. Non-commutativity is empirically observed.
Marginal probabilities in moral judgment do not always sum classically — consistent with interference terms.
The CHSH test applied to moral correlations. Does the Dear Ethicist game data exceed the classical bound S = 2?
Every Dear Ethicist game session generates data for the SQND experiment. Your moral judgments help test whether ethics has quantum structure.
Ten volumes applying differential geometry to the human sciences
Across domains — ethics, law, economics, cognition, medicine, communication — the standard methodology compresses multi-dimensional structure into scalar numbers and then wonders why the numbers behave badly. The Geometric Series develops a unified mathematical framework, grounded in Riemannian geometry, gauge theory, and topological data analysis, that recovers what the scalars destroy.
Andrew H. Bond · San Jose State University · Spring 2026
Coordinate on Discord and GitHub Discussions
Real-time discussion, research coordination, experiment design, and community support.
Join Discord →Long-form research questions, feature requests, philosophical debate, and open problems.
Start a Discussion →Submit your Dear Ethicist session data. Every game contributes to the Bell test and moral structure research.
Submit Data →Reach out to andrew.bond@sjsu.edu for collaboration, speaking, or research partnerships.
Send Email →This book argues that the mathematical structures physicists developed to describe nature — manifolds, tensors, metrics, connections, curvature, conservation laws — are also the right structures for describing moral reasoning. Not metaphorically. Mathematically.
The methodology is inductive, not axiomatic. The structures were discovered in data and then formalized. The framework makes predictions, and the predictions are confirmed by data.
"The window for this mandate is finite. The mathematics is ready. The question is whether we will use it."