| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies definable sets of reals in Polish spaces; includes Borel sets, analytic/coanalytic sets, projective hierarchies, regularity properties (measurability, Baire property), classification by definability and complexity. Excludes arbitrary subsets of reals unless considered relative to axioms extending ZFC (e.g., determinacy). |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates at the definability and topological scale: Polish spaces, σ-algebras, pointclasses, hierarchies (Borel, projective), Wadge reducibility, determinacy levels. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Reals, Borel sets, analytic sets, projective sets, Polish spaces, continuous maps, trees, pointclasses, equivalence relations, scales, norms, determinacy games. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Definability level, measurability, Baire property, perfect set property, complexity (Borel rank, projective level), reducibility degrees, regularity properties under determinacy. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Borel hierarchy, projective hierarchy, Wadge degrees, pointclasses (Σ, Π, Δ levels), equivalence relation complexity classes, definable sets under determinacy axioms. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Definability parameters, ranks of sets, codes for trees, game positions, Wadge degrees, equivalence relation classes, norms/scales on sets. |
| | Parameterization | How variables encode and represent the system’s state. | Encoding states via Borel codes, projective levels, Wadge degrees, tree representations, definability predicates, determinacy game lengths. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Treat Polish spaces as canonical; treat definability as absolute under continuous reductions; assume standard hierarchies; ignore pathologies from arbitrary sets of reals unless considering determinacy extensions. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Break down when choice is too strong (destroying regularity), when determinacy is absent at higher levels, in non-Polish spaces, or in models without appropriate regularity properties. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Assumes ZFC (or ZF + DC + determinacy in stronger contexts), classical topology of Polish spaces, definability hierarchies, closure under continuous preimages, basic regularity principles. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes definability reflects genuine structural complexity; hierarchies behave monotonically; determinacy (when used) provides canonical regularity; topological structure informs logical classification. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Requires consistent interaction among definability hierarchies, regularity properties, reducibility relations, and determinacy assumptions; hierarchies must not collapse arbitrarily. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires alignment of Borel/projective ranks, Wadge reducibility, measurable/Baire properties, determinacy levels, and coding systems across all definable sets and Polish spaces. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Borel ranks, projective levels, Wadge degrees, definability behavior in Polish spaces, regularity properties (measurability, Baire property, perfect set property), game outcomes under determinacy. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Limits of first-order ZFC in classifying projective sets, inability to detect arbitrary non-definable sets, failure of regularity properties under full Choice, expressive limits of reducibility frameworks. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Borel rank, projective level, Wadge degree, tree rank, norm lengths, scale complexity, descriptive complexity. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Borel codes, trees, continuous reductions, Wadge tests, equivalence-relation reducibility tools, determinacy games, scale-construction machinery. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Definitions of Borel/analytic/coanalytic/projective sets, pointclasses (Σ, Π, Δ), Wadge reducibility, equivalence-relation reducibility, scales and norms. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Constructing Borel codes, forming analytic sets via projections, computing ranks, running infinite games, building scales, performing Wadge reductions, analyzing canonical Polish-space encodings. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Generating hierarchy levels, constructing tree representations, building reductions, testing definability across Polish spaces, exploring determinacy consequences (when allowed). |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Selecting representative definable sets at each hierarchy level, sampling equivalence relations, testing reducibility behavior on canonical examples, examining definable families under parameter variation. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Borel codes, trees, Wadge diagrams, definability hierarchies, scale sequences, norm data, equivalence-relation complexity tables, determinacy game trees. |
| | Resolution | The granularity or precision with which data is captured. | Determined by granularity of Borel rank, depth of projective hierarchy, precision of Wadge degrees, fidelity of tree codings, and strength of determinacy assumptions used. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Verifying correctness of Borel codes, checking well-foundedness of trees, validating reducibility results, confirming rank computations, ensuring determinacy rules match payoff sets. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Misranked sets, incorrect tree encodings, faulty reductions, non-well-founded trees, misidentified Wadge degrees, determinacy misapplications, incorrect complexity classification. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Borel and projective hierarchy laws; closure properties under continuous preimages; regularity patterns (measurability, Baire property, perfect set property); stability of Wadge reducibility; determinacy-driven structural regularities. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Borel rank, projective level, Wadge degree, equivalence-relation complexity class, tree rank, scale invariants, classification invariants under continuous reductions. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Tree representation mechanisms; projection mechanisms generating analytic sets; game-theoretic mechanisms producing determinacy outcomes; continuous-reduction mechanisms creating Wadge order; scale-construction mechanisms under determinacy. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Borel hierarchy progression, projective hierarchy steps, iterative Wadge reduction paths, determinacy game progressions, scale-refinement pathways, tree refinement processes. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Borel, analytic, coanalytic, projective, Polish spaces, pointclasses, Wadge reducibility, equivalence-relation reducibility, trees, norms, scales, determinacy, regularity properties. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Borel hierarchy, projective hierarchy, Wadge hierarchy, equivalence-relation complexity classes, definability classes (Σ/Π/Δ), determinacy-based classifications of definable sets. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Rank equations for Borel sets; projective recursion equations; Wadge reduction formulas (A \leq_W B); definitions via tree projections; scale inequalities; hierarchies expressed as ordinal-indexed sequences. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Polish-space models; tree models of analytic sets; hierarchy models; determinacy-model frameworks (e.g., (AD)-based universes); Wadge-order models; equivalence-relation complexity models. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Canonical trees for analytic sets; idealized Borel codes; simplified Wadge chains; toy determinacy games; stripped-down Polish spaces illustrating definability phenomena. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Collapse of regularity properties under AC; breakdown of determinacy at high projective levels without additional axioms; non-well-founded trees; inability to classify arbitrary sets of reals in ZFC. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Borel and projective hierarchy theory; determinacy as a unifying framework for regularity; Wadge theory unifying reducibility; scale and norm theory unifying projective structural behavior. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Links to topology (Polish spaces), real analysis (regularity properties), logic (determinacy, large cardinals), computability theory (degree structures), and ergodic theory (measurability structures). |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Varying definability parameters, modifying codes or representations of sets, altering topological structures (within Polish limits), manipulating reduction frameworks (e.g., switching reductions from continuous to Borel) to test definability and complexity. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing Borel/projective behavior without altering axioms; tracking Wadge degrees, determinacy game outcomes, tree well-foundedness, or equivalence-relation complexity in natural settings. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing whether a set is analytic, coanalytic, or projective; checking Borel rank; verifying reducibility relations; determining correctness of tree representations; validating determinacy-induced regularity. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Reproducing tree codings; re-testing reducibility using multiple reductions; recomputing Borel/projective ranks; repeating determinacy games; verifying equivalence-relation classifications across models or Polish spaces. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Logical analogues: analyzing distribution of definability levels, comparing Wadge-degree frequencies, assessing regularity prevalence under determinacy, evaluating complexity spectra of equivalence relations. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing definability hierarchies across Polish spaces; comparing models with/without determinacy; evaluating differences under large cardinal assumptions; comparing reducibility frameworks and their structural strength. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Detecting mis-coded Borel sets, incorrect projective classification, faulty tree constructions, misapplied reductions, incorrect determinacy assumptions, errors in equivalence-relation complexity assessments. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding selective sampling of “nice” definable sets; preventing bias toward low-level Borel structures; ensuring non-biased choice of reductions; avoiding cherry-picking canonical representations to force desired complexity. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Community evaluation of definability proofs, rank computations, reducibility arguments, determinacy results, equivalence-relation complexity claims, and tree or scale constructions. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Revising pointclass definitions, adjusting reducibility frameworks, updating determinacy assumptions, refining scale theory, modifying tree constructions, or redefining equivalence-relation hierarchies after counterexamples. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Full disclosure of coding methods, reducibility assumptions, determinacy reliance, regularity hypotheses, game definitions, and topological constraints. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Honest reporting of definability failures, accurate treatment of determinacy limitations, avoidance of overstated complexity claims, and correct attribution of classification results and game-theoretic techniques. |