| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies geometric structures where distance is the primary invariant: metric spaces, geodesic spaces, length spaces, CAT(0)/CAT(k) spaces, Gromov–Hausdorff limits, curvature bounds via comparison geometry. Excludes smoothness assumptions unless additional structure (Riemannian, Finsler) is imposed. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates at local and global geometric scales: small-ball geometry, geodesic neighborhoods, large-scale (coarse) geometry, asymptotic invariants, and convergence behavior of spaces. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Metric spaces, length spaces, geodesics, distance functions, balls, curves, comparison triangles, isometries, Lipschitz maps, tangent cones (in nonsmooth contexts), Gromov–Hausdorff limits. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Distances, lengths, geodesicity, curvature bounds (via triangle comparison), diameters, Lipschitz constants, injectivity radius, doubling properties, bounded turning, completeness. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | General metric spaces, geodesic spaces, CAT(0)/CAT(k) spaces, Alexandrov spaces, length spaces, metric graphs, ultrametric spaces, coarse-geometric classes (e.g., hyperbolic spaces). |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Distances between sampled points, lengths of curves, curvature-comparison parameters, covering numbers, Lipschitz constants, diameter values, Gromov–Hausdorff distances. |
| | Parameterization | How variables encode and represent the system’s state. | Encoded via distance matrices, geodesic-parameter functions, local triangle data, covering radii, Hausdorff-approximation parameters, or Lipschitz norms. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Idealizing spaces as complete; assuming geodesic existence; assuming curvature bounds hold everywhere; approximating spaces by simpler polyhedral or graph models; ignoring measure-theoretic irregularities. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Break down in incomplete spaces, fractal or highly irregular spaces, metric spaces lacking geodesics, spaces without triangle-comparison validity, or in non-doubling or non-proper contexts. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Assumes metric satisfies symmetry, positivity, triangle inequality; geodesic or length-space structure when required; stable behavior under Gromov–Hausdorff limits; compatibility of distances under mappings. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes distance alone captures essential geometry; curvature can be expressed by comparison; coarse geometry retains qualitative structure; tangent cones provide meaningful local models. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Requires that distance, geodesic, and curvature notions align; triangle-comparison definitions must not contradict geodesic structure; Gromov–Hausdorff limits must preserve metric consistency. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires compatibility among distance functions, geodesics, curvature bounds, convergence notions, and large-scale invariants; Lipschitz mappings must preserve or control structure appropriately. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Distances, curve lengths, geodesic paths, triangle-comparison behavior, curvature bounds (CAT(k)), covering numbers, doubling properties, Gromov–Hausdorff convergence patterns. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Limits in resolving fine smooth structure; inability to detect differentiability; breakdown of triangle comparison near singularities; coarse invariants ignoring local geometry; instability in noisy or sparse sampling. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Distances, lengths, radii, diameters, Lipschitz constants, curvature bounds, covering radii, Gromov–Hausdorff distances. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Distance functions, metric balls, geodesic solvers, triangle-comparison tests, covering algorithms, GH-approximation tools, Lipschitz maps, polyhedral approximations. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Definitions of metric, geodesic, length space, CAT(k) space, Lipschitz/bi-Lipschitz maps, tangent cones, comparison triangles. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Computing pairwise distances, approximating geodesics, performing triangle comparisons, estimating covering numbers, computing GH-distances, constructing tangent-cone approximations. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Sampling point clouds, building distance matrices, multi-scale covering procedures, geodesic tracing, generating polyhedral/graph approximations, computing pointed GH-limits. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling points in metric balls, selecting representative geodesics, sampling across scales, choosing basepoints for pointed limits, selecting subsets for covering estimates. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Distance matrices, geodesic paths, curvature-comparison diagrams, covering-number tables, GH-convergence data, polyhedral models, tangent-cone samples. |
| | Resolution | The granularity or precision with which data is captured. | Determined by sampling density, numerical precision for distances, geodesic-approximation accuracy, covering refinement, GH-approximation granularity. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Checking triangle inequality, validating geodesic computations, confirming consistent distances, testing CAT(k) conditions, calibrating GH-approximation algorithms. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Distance-measurement noise, geodesic-integration error, covering-number misestimation, sampling bias, polyhedral-approximation artifacts, GH-convergence instability. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Triangle-comparison laws (CAT(k)); geodesic extension and uniqueness rules; metric inequality structures; behavior of distances under Lipschitz maps; coarse-geometric stability of large-scale invariants. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Diameter, curvature bounds, Lipschitz constants, doubling dimension, Gromov–Hausdorff invariants, asymptotic cones, quasi-isometry classes. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Geodesic mechanisms generating minimal paths; curvature-comparison mechanisms controlling shapes of triangles; covering mechanisms relating local to global geometry; limit mechanisms relating sequences of spaces to GH-limits. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Geodesic-flow pathways; triangle-comparison chains; multi-scale covering pathways; convergence pathways toward tangent cones or GH-limits; quasi-geodesic pathways in hyperbolic spaces. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Metric, geodesic, length space, CAT(k) space, Alexandrov curvature, quasi-isometry, Gromov–Hausdorff distance, doubling property, tangent cone, Lipschitz/bi-Lipschitz maps. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Geodesic vs. non-geodesic spaces; CAT(0)/CAT(k) spaces; Alexandrov spaces; hyperbolic spaces; ultrametric spaces; doubling spaces; coarse-geometric classes (e.g., quasi-isometry types). |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Triangle-inequality structures; curvature comparison inequalities; geodesic equations in metric form; GH-distance formulas; Lipschitz bounds; definitions of tangent cones via blow-up limits. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | CAT(0) model spaces; hyperbolic spaces; metric trees; polyhedral complexes; Riemannian manifolds viewed purely metrically; approximating graphs; GH-limit spaces. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Ideal triangles in CAT(k) comparison; metric trees (0-hyperbolic spaces); polyhedral approximations; simplified doubling spaces; canonical hyperbolic models; tangent-cone ideals. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Breakdown at singularities; spaces lacking geodesics; failure of CAT(k) conditions; loss of doubling property; non-existence of GH-limits; pathological or fractal metric structures. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Alexandrov geometry; comparison geometry; coarse geometry and quasi-isometry theory; Gromov’s metric viewpoint on curvature; geometric group theory’s metric framework. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Links to Riemannian geometry (metric curvature bounds), geometric group theory, topology (shape of spaces), computer science (metric embeddings, clustering), and analysis on metric spaces (Sobolev spaces, Poincaré inequalities). |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Varying metrics, altering curvature bounds, adjusting sampling density, modifying path constraints, or changing comparison models to test distance behavior, geodesic structure, and CAT(k) conditions. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing natural metric behavior without intervention: tracking geodesic patterns, monitoring triangle comparison, examining covering numbers, evaluating GH-limits, studying coarse invariants across scales. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing triangle inequality stability, verifying geodesicity, checking CAT(k) curvature bounds, validating quasi-isometry hypotheses, testing doubling or Poincaré properties, confirming GH-convergence. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Recomputing distances and geodesics across different samplings; repeating curvature-comparison tests; replicating GH-approximation sequences; verifying covering-number estimates under new sampling distributions. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Analyzing metric distributions; estimating curvature via comparison statistics; evaluating stability of approximated geodesics; examining scaling behavior of coverings; comparing metric invariants under perturbations. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing metric spaces by GH-distance, curvature bounds, doubling dimension, geodesic structure, quasi-isometry class, or distortion under embeddings; evaluating convergence of sequences of spaces. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Identifying distance-measurement noise, geodesic-approximation error, covering-number inaccuracies, curvature-comparison failures, GH-approximation instability, sampling bias, and discretization artifacts. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Ensuring uniform sampling; avoiding coordinate-system bias in embedded models; preventing overfitting from dense sampling in preferred regions; avoiding selective comparisons that force GH-convergence. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Independent verification of comparison-geometry results, GH-distance computations, curvature-bound checks, covering-number estimates, and geodesic-approximation methods by geometric analysts. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Revising curvature bounds; updating comparison models; modifying GH-approximation frameworks; adjusting regularity assumptions; redefining metric invariants when counterexamples appear. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Full disclosure of sampling procedures, distance algorithms, approximation methods, regularity assumptions, and curvature-test protocols. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Accurate reporting of geometric uncertainties; avoiding concealment of sampling bias; proper attribution of metric techniques; honest representation of approximation limitations and non-convergence cases. |