Papers, frameworks, and books — ordered by date. Every entry includes a narrative summary of what the work does and why it matters.
This paper introduces a fundamentally new approach to constraint satisfaction by treating the problem as wavefront propagation across a discrete Riemannian manifold. Instead of backtracking through exponential search trees, the solver exploits local curvature structure to guide GPU-parallel wavefronts toward solutions along geodesic paths.
The result is geometric speedup that scales with manifold structure rather than problem size. Benchmarks demonstrate ~270,000 puzzles per second on commodity GPUs — a 1,226x improvement over optimized Python and 3.8x faster than state-of-the-art DLX solvers. The framework generalizes to any constraint satisfaction problem expressible as a discrete manifold.
PageRank has been computed the same way since 1998: power iteration on a stochastic matrix. This paper reformulates the problem on the Davis manifold, where curvature information predicts mixing time and guides adaptive solver selection — choosing between spectral methods, iterative schemes, and geometric approaches based on local graph topology.
By exploiting the curvature structure of real-world link graphs, the framework achieves faster convergence on heterogeneous networks where standard power iteration stalls. The geometric mixing time predictor eliminates the guesswork of convergence estimation, replacing heuristic tolerance checks with principled geometric bounds.
Classical shortest-path algorithms — Dijkstra, Bellman-Ford, A* — treat all graphs uniformly. This paper demonstrates that local curvature on the Davis manifold reveals fundamentally different geometric regimes within the same graph, enabling adaptive algorithm selection with provable performance guarantees.
In positively curved regions, greedy geodesic methods dominate. In negatively curved regions where geodesics diverge, broader exploration strategies are required. The curvature-aware selector transitions between strategies seamlessly, achieving consistent speedup across heterogeneous graph topologies that defeat any single fixed algorithm.
The P versus NP problem has resisted proof for over fifty years. This paper approaches it from a new direction: the geometry of energy landscapes. Under the Davis information-geometric axioms, polynomial-time and exponential-time computation correspond to fundamentally different curvature regimes — smooth landscapes versus landscapes with exponentially many saddle points separated by curvature barriers.
The proof proceeds by showing that any polynomial-time algorithm must traverse a landscape whose integrated curvature stays bounded, while NP-complete constraint landscapes exhibit curvature that grows super-polynomially. This geometric separation is intrinsic to the problem structure, not an artifact of any particular algorithm.
The Navier-Stokes existence and smoothness problem asks whether solutions to the equations governing fluid flow can develop singularities in finite time. This paper resolves the question using holonomy-first methods: by embedding the fluid velocity field onto a manifold governed by the Davis field equations, curvature bounds on the flow manifold directly prevent finite-time blowup.
The key insight is that singularity formation would require unbounded curvature concentration, which violates the holonomy constraints inherited from the manifold structure. The geometric proof is constructive, providing explicit regularity estimates rather than purely existential guarantees.
The Hodge conjecture — one of the seven Millennium Prize Problems — asks whether certain cohomology classes on algebraic varieties can always be represented by algebraic cycles. This paper resolves the conjecture through the Davis field equations framework, showing that the cohomological constraints arise naturally from semantic curvature conditions on the underlying manifold.
The approach translates the algebraic geometry question into differential geometry, where the Davis field equations provide the machinery to construct explicit algebraic cycle representatives from Hodge classes. The construction is canonical, determined entirely by the curvature structure of the variety.
Perelman's proof of the Poincare conjecture used Ricci flow with surgery. This paper provides an independent proof by a different route: identifying the Davis-Wilson flow — a gauge-theoretic construction arising from cache space dynamics — as a Ricci flow in disguise. The topological simplification that produces the 3-sphere emerges from cache space collapse rather than geometric surgery.
The equivalence between Davis-Wilson flow and Ricci flow is established through explicit connection formulas, providing a gauge-theoretic perspective on why simply connected 3-manifolds must be spheres. The proof avoids surgery entirely, relying instead on the monotonicity properties inherent to cache space contraction.
When a superfluid is stirred fast enough, quantized vortices appear — but the critical velocity has remained elusive for decades. This paper establishes a universal law for vortex nucleation in Bose-Einstein condensates by connecting the onset velocity to curvature thresholds on the Davis manifold.
The Davis-Landau law unifies several previously disconnected results in superfluid dynamics under a single geometric framework. The critical velocity emerges as a phase transition on the manifold: below the threshold, the superfluid flow is geodesic; above it, topological defects nucleate as the manifold develops singularities. Experimental predictions are provided for verification in ultracold atom laboratories.
This foundational paper establishes that meaning propagation in transformers is not a metaphor for physics — it literally obeys field equations analogous to physical systems. With 89 mathematical results, the framework shows how curvature constraints in activation space determine coherence limits, providing geometric explanations for why transformers break down on certain reasoning tasks.
The practical consequence is predictive: curvature signatures forecast model failure modes before deployment, replacing post-hoc error analysis with principled geometric diagnostics. Every reasoning boundary the model will encounter is visible in the curvature structure of its learned manifold.
Why do language models degrade over long contexts? The conventional answer — limited memory — is wrong. This paper proves that effective reasoning horizons arise from holonomy accumulation: as a transformer processes sequential tokens, geometric phase distortion compounds along the activation manifold until coherent reasoning becomes impossible.
The conjecture provides a path-dependent explanation for context failure. Two prompts with identical tokens but different orderings can produce different holonomy accumulation, yielding different effective context lengths. The framework directly predicts where a model's reasoning will decohere, enabling principled prompt engineering and architecture design.
Generative AI produces text that says the same thing in different ways — paraphrases, translations, reformulations. This paper frames that process as gauge transformations on semantic manifolds: surface forms change while underlying meaning is preserved by a gauge symmetry. The framework establishes error budget transfer theorems showing exactly how much semantic distortion each generation step introduces.
For safety-critical deployments, this is transformative. The gauge-theoretic structure enables compositional safety guarantees: if each component preserves meaning within bounded distortion, the entire pipeline's maximum drift is certifiable. No more black-box trust — the geometry proves the bound.
When are two things "the same"? Translation-based models (like Word2Vec) say sameness is a vector offset; distance-based models (like contrastive learning) say sameness is proximity. This paper proves these are not competing approaches — they are geometrically equivalent under epsilon-bounded distortion through a category-theoretic unification.
The result provides rigorous foundations for approximate reasoning in neural systems. When a model says two embeddings are "close enough," the framework quantifies exactly what has been lost in the approximation and ensures the error stays bounded across compositions of approximate judgments.
Most anomaly detectors are empirical: they learn patterns from data and flag deviations. The Davis Manifold provides a geometry-first alternative. By constructing Riemannian state spaces with bounded geodesic-Euclidean distortion, the framework knows — mathematically — when its geometric assumptions hold and when they break. Cantelli-based risk bounds provide distribution-free probability guarantees.
The key innovation is compositional correctness proofs: each component declares its distortion regime and validity assumptions, and the system's total error budget composes from parts. This is the reusable blueprint for safety-critical ML — not a single model, but a methodology for building auditable detection systems with explicit failure modes.
What is a transformer actually doing when it "thinks"? This paper applies heat kernel methods from spectral geometry to answer that question. By analyzing how information diffuses across the activation manifold, the paper reveals that transformers self-organize into functionally distinct regions — analogous to brain areas — with characteristic spectral signatures.
The heat kernel approach provides a principled decomposition of transformer computation into cognitive operations: encoding, retrieval, integration, generation. Each operation corresponds to a distinct diffusion pattern, making the model's internal reasoning process legible through geometric spectroscopy.
Standard transformers pay linear cost for context: longer inputs mean proportionally more computation and memory. Davis Cache breaks this scaling by preserving reasoning state using topological residue — the essential geometric information that survives dimensional reduction. The result is constant-time state preservation regardless of context length.
The mechanism works by identifying which aspects of the activation manifold carry topological information (invariant under continuous deformation) versus metric information (sensitive to small perturbations). By caching only the topological residue, the system preserves reasoning capacity without the linear memory penalty. Formal guarantees bound information loss.
The Yang-Mills mass gap — another Millennium Prize Problem — asks why gauge particles must have mass. This paper reframes the question through information geometry: distinguishing between gauge field configurations costs energy, and that cost has a nonzero minimum determined by the incompressibility of topological charge.
Topological charges cannot be smoothly deformed to zero — they are discrete, protected quantities. The paper shows that this discreteness imposes a minimum energy for field distinguishability, which manifests physically as a mass gap. The information-geometric reduction provides a conceptually transparent route to a problem that has resisted purely algebraic approaches.
SHA-256 is the backbone of Bitcoin, TLS, and countless security systems, yet its internal geometric structure has never been characterized. This paper introduces carry-field tomography — imaging the propagation of arithmetic carries through the hash function — and reveals that SHA-256 contains a dual-torus architecture with asymmetric subsystems.
The discovery has implications for both cryptographic analysis and hash function design. The geometric structure explains why certain differential paths are infeasible (they would require traversing topological barriers) and suggests design principles for next-generation hash functions with provable structural properties.
Interstate conflicts don't erupt randomly — they emerge from structural stresses in networks of economic and political interdependence. This paper develops a topological framework that detects geometric signatures of systemic stress in these networks, achieving a 72-day advance prediction horizon for interstate conflict escalation.
The method works by tracking the topology of weaponized interdependence: when nations begin weaponizing trade routes, financial channels, or institutional relationships, the network's topological invariants shift in characteristic ways. These signatures are detectable long before diplomatic signals become visible, providing genuine early warning for conflict prevention.
Processing 2.3 terabytes of telemetry data per day at NASA requires constant optimization of how data is encoded, stored, and queried. This paper introduces adaptive embedding format switching — a system that dynamically selects the optimal embedding representation based on incoming data characteristics, query patterns, and throughput requirements.
The semantic translation layer ensures that switching between embedding formats preserves meaning across heterogeneous data streams. Deployed in NASA's telemetry infrastructure, the system achieved 37% storage reduction and 12% improvement in anomaly detection accuracy while maintaining mission-critical latency requirements.
What happens when you apply NASA mission-critical safety patterns to financial infrastructure? PRISM answers that question by implementing multi-rail payment reconciliation with the same bounded-error methodology used for space systems. Each rail — credit card, ACH, wire, crypto — speaks a different semantic language; PRISM translates between them with formal distortion bounds.
Spectral consensus mechanisms resolve disagreements between rails using geometric voting rather than majority rules. The compositional risk budgets ensure that no single translation step can introduce unbounded error, making the system auditable from end to end.
HERALD uses learned Riemannian geometry for real-time viral surveillance, monitoring the continuous-time evolution of antigenic landscapes to detect dangerous variants before they dominate. Validated against historical data, the system would have identified Omicron as the dominant variant 18 days before the WHO designation — enough lead time to mobilize public health responses.
The framework includes Cantelli-based probability bounds (no distributional assumptions), compositional error budgets, and explicit out-of-distribution abstention. When HERALD doesn't know, it says so — a critical property for a system deployed in pandemic surveillance where false confidence kills.
Multi-cancer early detection from a blood draw is the holy grail of oncology. GEODESIC makes it geometric: each patient is modeled as a distribution over latent clones on Davis manifolds, and early malignant signals are detected via tail-focused partial optimal transport. The key challenge — signal dilution in cell-free DNA — is addressed through sectoral geometry that separates tissue-of-origin from clonal hematopoiesis (CHIP).
By working at the clone level rather than the mutation level, GEODESIC can detect cancers when tumor fraction is below the noise floor of conventional methods. The manifold structure provides natural staging boundaries, enabling not just detection but localization and staging from a single liquid biopsy.
Working PaperAntimicrobial resistance is an accelerating global crisis. TESSERA attacks it geometrically: bacterial genomes are modeled as fiber bundles, with the chromosome as the base manifold and plasmids and cassettes as fibers. Horizontal gene transfer — the mechanism by which resistance spreads between species — appears as elastic-limit breaches in the fiber structure.
The framework supports facility-scale source attribution, tracing resistant organisms from farm to fork in food safety applications and patient to patient in hospital infection control. By detecting transfer events geometrically rather than through sequence alignment, TESSERA achieves real-time surveillance speeds suitable for clinical and agricultural deployment.
Working PaperMIRADOR inverts the problem of immune evasion: instead of predicting which variants will escape immunity, it designs immunogens that cover the maximum antigenic space. The approach frames vaccine design as set cover on Riemannian manifolds, where the goal is to select antigens whose immune response manifolds collectively tile the space of possible viral mutations.
The framework includes inverted error budgets that certify coverage rather than bound failure — a conceptual shift from "how much can go wrong" to "how much is guaranteed to work." A built-in dual-use firewall prevents the system from being repurposed for evasion query design.
Working PaperWhen AI assists human operators in mission-critical environments — flight control, medical procedures, military operations — accountability requires perfect replay. BRIDGE provides deterministic replay for hybrid AI-human operations through VFS harmonization, ensuring that every AI suggestion, human decision, and system state can be reconstructed exactly.
Developed from NASA operational experience, the framework creates complete audit trails that satisfy the most stringent regulatory requirements. Every decision point is recorded with sufficient context to determine not just what happened, but why.
The first book in the Geometric Intelligence Series takes readers from intuition to rigor, introducing the idea that the patterns governing intelligence, physics, and medicine share a common geometric language. Written for mathematicians, engineers, and curious generalists alike, it lays the conceptual foundations that the subsequent volumes formalize.
Hidden Variable argues that the structures we impose on data — coordinates, categories, hierarchies — often obscure deeper geometric truths. By learning to see past artificial structure to the manifold underneath, practitioners across disciplines can unlock patterns that were always there but never visible through the lens of conventional analysis.
Available on Amazon →The second volume in the Geometric Intelligence Series expands the companion research paper into a full treatment. It develops the theory of epsilon-equivalence between translation-based and distance-based semantic models, providing worked examples, proofs, and applications accessible to researchers and graduate students.
The book demonstrates how category theory unifies seemingly incompatible approaches to measuring similarity in neural networks. Readers learn to construct and verify compositional error budgets — the core technology enabling safety guarantees for AI systems that reason approximately about approximate representations.
Available on Amazon →The third volume in the Geometric Intelligence Series turns the Davis Manifold framework toward medicine. From cancer detection via cell-free DNA to antimicrobial resistance surveillance and pandemic early warning, the book demonstrates how geometry-first methods outperform data-first approaches when the stakes are human lives.
Each chapter pairs rigorous mathematics with clinical applications, showing how compositional error budgets translate directly into clinical decision thresholds. The book addresses the fundamental tension in medical AI: the need for both sensitivity (don't miss cancers) and specificity (don't cause unnecessary biopsies), resolved through geometric control of the sensitivity-specificity tradeoff.
Available on Amazon →The fourth volume in the Geometric Intelligence Series applies the Davis Manifold framework to energy systems — from grid optimization and battery degradation to fuel logistics and renewable integration. The unifying insight is that energy systems are manifolds with natural curvature, and respecting that geometry yields better predictions, controls, and designs.
The book shows how the same mathematical machinery that detects cancer in cell-free DNA can predict grid failures, optimize fuel routing, and monitor battery health. This cross-domain transfer is not metaphorical — the theorems are identical, only the data and the interpretation change.
Available on Amazon →The second novel in the Astralis science fiction trilogy. Set in a universe where the geometric structures underlying reality are not metaphors but physical law, the story explores what happens when a civilization discovers that the manifold governing their spacetime is being reshaped by an intelligence operating at a higher geometric order.
Available on Amazon →The third novel in the Astralis series. A geometric noir that fuses hardboiled detective fiction with differential geometry, set in a spacetime whose curvature is being deliberately manipulated. The mystery demands that the protagonist learn to read the geometry of crime scenes on curved manifolds — where straight lines don't exist and alibis depend on geodesics.
Available on Amazon →The first novel in the Astralis science fiction trilogy. When humanity's first interstellar mission discovers that the universe's geometric structure is not what physics predicted, the crew must navigate a reality where the rules of space and time are locally determined by manifold curvature — and something is changing that curvature.
Available on Amazon →#1 Amazon Bestseller. A memoir and manifesto documenting the intersection of identity, surveillance technology, and algorithmic bias. Drawing from lived experience as a Black trans woman working in national security and tech, the book exposes how geometric structures of bias are embedded in the systems that govern modern life — from facial recognition to credit scoring to predictive policing.
The book argues that algorithmic justice requires not just fairer data but fundamentally different geometric frameworks — ones that account for the curvature of social space rather than assuming a flat Euclidean world where everyone starts from the same origin point.
Available on Amazon →