| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies structures whose definable sets in one variable are finite unions of points and intervals; includes o-minimality, tame geometry, definable functions, cell decomposition. Excludes pathological or wild definability such as arbitrary subsets of ℝ. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates at the definability scale over ordered structures (especially expansions of (ℝ, <, +, ⋅)); concerns definable sets, functions, maps, and dimension theory. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | O-minimal structures, definable sets, definable functions, cells, dimensions, projections, fibers, definable groups, types over parameters, Skolem functions in tame settings. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Tameness, monotonicity, definability, stratification, cell decomposition behavior, continuity of definable functions, dimension invariants. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | O-minimal structures, weakly o-minimal structures, expansions of real closed fields, definable manifolds, definable groups, definable equivalence classes, cell complexes. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Variable assignments, definable maps, parameters defining subsets, dimension values, cell decomposition data, stratification data. |
| | Parameterization | How variables encode and represent the system’s state. | System state encoded via definable parameters, cell decompositions, dimension assignments, definable choices, and parameter-dependent definability. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Treat definable sets as cell complexes; assume clean decomposition; ignore analytic or measure-theoretic pathologies; idealize monotonicity and continuity properties. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Break down in non-o-minimal expansions, structures with dense independent sets, pathological definable sets, or when saturation and definable completeness fail. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Assumes first-order logic; well-ordered underlying structures; definable completeness; existence of cell decomposition; predictable dimension theory; monotonicity of definable functions. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes tameness across dimensions; definability behaves geometrically; types correspond to geometric features; definability interacts smoothly with projections and fiber structures. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Requires compatibility between definability, dimension theory, monotonicity, and cell decomposition; definable sets must align with o-minimal axioms. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires alignment among definable sets, maps, cell decompositions, dimensions, and order structure; definability must be preserved under projections, products, and parameter changes. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Definable sets in one variable (finite unions of points and intervals), monotonicity of definable functions, cell decomposition, definable continuity, dimension behavior. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | O-minimality prevents defining arbitrary discrete sets, fractal sets, wild oscillatory graphs, or sets of unbounded local complexity. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Dimensions, number of cells, sizes of definable partitions, quantifier complexity, lengths of definable chains. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Cell decomposition tools, projection maps, definable choice, monotonicity theorems, quantifier elimination in o-minimal expansions. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Definitions of o-minimality, definable sets, cells, dimension, definable completeness, monotone/continuous definable functions. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Running cell decomposition, computing dimensions, verifying monotonicity, constructing definable stratifications, applying quantifier elimination. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Constructing definable families, generating cell partitions, performing fiber analyses under projection, producing local stratifications. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling definable families over parameters, selecting representative cells, comparing definable slices by dimension, identifying generic fibers. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Cells, definable curves/surfaces, dimensional profiles, stratifications, monotonicity intervals, projection–fiber diagrams. |
| | Resolution | The granularity or precision with which data is captured. | Determined by fineness of cell partitions, dimensional granularity, precision of definable stratifications, expressive power of the language. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Verifying cell decomposition correctness, confirming definable continuity, validating dimension assignments, checking monotonicity, testing o-minimality under expansions. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Misassigned dimensions, incorrect cell boundaries, false monotonicity detection, definable incompleteness errors, projection/fiber misanalysis. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Cell decomposition theorem, monotonicity theorem, finiteness of definable partitions, o-minimal dimension laws, definable continuity patterns, tame growth constraints. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Dimension, number of cells in decomposition, definable connected components, monotonicity intervals, o-minimal rank-like invariants, invariance of definability under projections. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Mechanisms by which order and definability enforce tameness: cell decomposition, projection/fiber behavior, definable choice, monotone extension mechanisms, stratification processes. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Cell decomposition pathways, stepwise stratification building, projection–fiber analysis sequences, definable continuity pathways, inductive dimension computation sequences. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | O-minimality, definable sets, cells, dimension, definably complete, definable continuity, monotonicity, tame functions, definable manifolds, stratifications. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | O-minimal vs. weakly o-minimal, polynomially bounded vs. non-polynomially bounded o-minimal structures, expansions of real closed fields, tame vs. wild definable behavior. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Formal statements of cell decomposition, dimension axioms, monotonicity conditions, piecewise definability rules, projection formulas, quantifier-elimination schemas. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | O-minimal structures, expansions of (ℝ, <, +, ⋅), definable manifolds, tame geometric models, cell-complex models, piecewise-monotone definable function models. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Ideal cells (intervals × points), pure cell complexes, piecewise-linear definable sets, tame expansions with clean monotonicity, simplified geometric stratifications. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Breakdown in non-o-minimal expansions, expansions adding dense independent sets, definable discontinuities, non-cell-decomposable definable sets, loss of definability under non-tame functions. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | O-minimality as the unifying framework for tame geometry, real algebraic geometry, semialgebraic geometry, subanalytic geometry, and definability theory in ordered structures. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Connections to real algebraic geometry, subanalytic geometry, differential topology, combinatorics (VC-dimension), optimization, and dynamical systems with definable trajectories. |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Manipulating definable sets, expansions of the language, or choice of parameters to test monotonicity, dimension behavior, and cell decomposition structure. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing definable behavior in fixed structures: watching how fibers behave under projection, tracking monotonicity intervals, or monitoring dimensional stability without altering the theory. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing whether a structure is o-minimal, verifying cell decomposition existence, checking monotonicity of definable functions, validating dimension computations, testing tame behavior under expansions. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Re-running cell decompositions across models; reproducing dimension calculations; verifying monotonicity in definable families; replicating projection–fiber analyses with different parameter sets. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Logical analogues: analyzing distribution of cell counts, comparing dimensional spectra across definable families, assessing frequency of tame vs. non-tame behavior in expansions. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing o-minimal structures by cell complexity, definability strength, projection behavior, growth rates of definable functions, dimension theory robustness, and presence/absence of quantifier elimination. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Identifying misassigned cells, incorrect dimension values, definable discontinuities mistakenly labeled continuous, failures in cell decomposition, projection misanalysis, or erroneous conclusions from expansions. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding biased selection of definable samples, preventing over-enrichment of languages that trivialize tameness, ensuring neutral parameter choices in families, avoiding selective sampling of “easy” cells. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Reviewing cell decomposition proofs, dimension calculations, definability claims, monotonicity arguments, and tameness proofs among experts in model theory and tame geometry. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Modifying the underlying language or structure to regain o-minimality, refining definable completeness assumptions, revising cell decomposition steps, or adjusting stratification frameworks after counterexamples. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Full disclosure of language expansions, parameter choices, cell decomposition procedures, dimension rules, stratification methods, and all assumptions affecting tameness. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Honest reporting of non-tame behavior, correct attribution of decomposition tools, avoidance of hidden definability assumptions, and accurate representation of o-minimal limits. |