Classification Theory

				Formal Sciences
				Logic
				Model Theory
Element	Scope Category	Sub-Item	Definition	Classification Theory
1. Domain	1.1 Scope of the Domain	Boundaries	The range of phenomena the science includes and excludes.	Studies dividing lines among first-order theories (stable, superstable, simple, NIP, NSOP, o-minimal, etc.); includes ranks, independence relations, and behavior of types. Excludes empirical classification or semantic categories outside model-theoretic structure.
		Scale	The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic).	Operates at the theory–model–type scale: formulas, types over sets, ranks, independence relations, and saturation levels.
	1.2 Ontological Commitments	Entities	The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.).	Theories, models, types, formulas, definable sets, ranks (Morley rank, U-rank), forking/dividing relations, independence relations, saturated models.
		Properties	The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.).	Stability, simplicity, NIP/SOP status, rank values, definability of types, canonicity of independence, saturation degrees, homogeneity patterns.
		Categories	The basic ontological types used to classify domain elements (substances, processes, relations, structures).	Stable theories, superstable theories, ω-stable theories, simple theories, NIP theories, NSOP theories, classifiable theories, o-minimal theories.
	1.3 State-Variables	Variables	The measurable or definable properties that describe system conditions.	Types over parameters, independence configurations, rank assignments, forking patterns, definability patterns, saturation cardinalities.
		Parameterization	How variables encode and represent the system’s state.	State encoded by chosen base sets, realized/omitted types, rank values (e.g., RM, U), forking diagrams, and cardinalities used to define saturation.
	1.4 Admissible Idealizations	Simplifications	Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases).	Assume saturated “monster model,” clean independence relations, well-behaved forking symmetry, definable types, and availability of prime or limit models.
		Validity Conditions	The limits and contexts in which idealizations hold or break down.	Failures occur in unstable theories (SOP, TP), non-elementary classes, insufficient saturation, or when independence is non-symmetric or not well-defined.
	1.5 Domain Assumptions	Structural Assumptions	Background ontological stances such as determinism, continuity, randomness, discreteness.	Assumes first-order logic, compactness, existence of saturations, robust type spaces, elementary embeddings, and classical model-theoretic independence frameworks.
		Implicit Commitments	Unstated but necessary assumptions that shape the field’s conceptual structure.	Assumes ranks track structural tameness; forking reflects genuine independence; definability aligns with stability; type spaces accurately represent model-theoretic geometry.
	1.6 Internal Coherence Requirements	Consistency	The demand that domain concepts do not contradict one another.	Requires non-contradictory interaction among ranks, independence relations, definability, and saturation; dividing lines must be logically compatible.
		Compatibility	The requirement that entities, variables, and assumptions fit together into a unified descriptive framework.	Requires alignment between types, ranks, independence relations, definability criteria, saturation, and classification-theoretic dividing lines across all models.
2. Evidence Layer	2.1 Observable Phenomena	Observables	The aspects of the domain that can produce detectable signals accessible to measurement.	Stability behavior, forking/dividing patterns, type multiplicities, rank values (Morley rank, U-rank), independence configurations, saturation behavior, classification dividing lines.
		Detection Limits	The boundaries of what can be resolved or sensed by current instruments or methods.	Limits arising from expressiveness of first-order logic: inability to detect unstable patterns via low-rank formulas, limits in distinguishing theories with similar type spectra, compactness constraints, cardinality barriers in saturation detection.
	2.2 Measurement Systems	Units	Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison.	Rank values (RM, U), cardinalities measuring saturation, multiplicity of types, forking depth, dividing chains, independence dimensions, definability degrees.
		Instruments	Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements.	Forking/dividing tests, rank computations, Morley sequences, indiscernible sequences, type-space topology, EF-game–like independence diagnostics, saturation checks, prime-model constructions.
	2.3 Operational Definitions	Definitions	Terms defined by specific measurement procedures, ensuring empirical clarity.	Formal definitions of stability, simplicity, NIP, forking, dividing, rank definitions (e.g., RM, U), definability of types, indiscernibility criteria, independence axioms.
		Procedures	The explicit steps required to perform a measurement in a reproducible way.	Calculating ranks, checking forking/dividing behavior, building Morley sequences, constructing indiscernibles, testing NIP via VC-dimension analogues, verifying existence of NF (non-forking) extensions.
	2.4 Data Acquisition	Protocols	Formal processes for gathering data under controlled or standardized conditions.	Building saturated models, forming type spaces S(A), generating independence configurations, constructing indiscernible sequences, computing rank spectra across models.
		Sampling	Rules determining which subset of the domain is measured and how representative it is.	Selecting representative types, choosing base sets for forking tests, sampling indiscernible sequences, examining definable families, isolating key formulas causing instability.
	2.5 Data Character & Format	Data Types	The form raw evidence takes (time series, spectra, images, counts, qualitative records).	Types, rank distributions, forking diagrams, indiscernible sequences, Morley sequences, dividing chains, independence trees, saturation profiles.
		Resolution	The granularity or precision with which data is captured.	Fineness of rank discrimination, granularity of type-space distinctions, cardinality resolution in saturation, precision of dividing/forking detection, sensitivity to instability patterns.
	2.6 Reliability & Calibration	Calibration	Adjustment procedures ensuring instruments produce accurate results.	Ensuring ranks computed consistently; verifying independence properties; checking that forking matches dividing; calibrating stability results across different models and cardinalities.
		Error Characterization	Identification and quantification of noise, uncertainty, bias, and measurement error.	Miscalculated ranks, misclassified stability/simplicity status, false identifications of forking or dividing, incorrect independence assumptions, saturation errors, type miscounting.
3. Structural Layer	3.1 Patterns & Regularities	Laws / Relations	Stable, repeatable patterns governing how observables behave across conditions.	Stability criteria, dividing/forking relations, rank monotonicity, type regularity, existence of Morley sequences, symmetry/transitivity properties of independence, tameness dividing lines.
		Invariants	Quantities or properties that remain constant under transformations (symmetries, conservation laws).	Rank invariants (Morley rank, U-rank), type invariants, saturation levels, independence invariants, definability of types, behavior of indiscernible sequences under automorphisms.
	3.2 Causal Architecture	Mechanisms	Underlying processes or structures that produce the observed regularities.	Mechanisms by which forking/dividing detect instability; independence generating tree-like structures; rank computation mechanisms; embeddings producing or eliminating instability; indiscernible collapse mechanisms.
		Pathways	Organized sequences of interactions forming a causal chain or network.	Forking/dividing chains, Morley sequence construction, indiscernible generation pathways, saturation-building chains, rank-refinement sequences, extension pathways for types.
	3.3 Theoretical Vocabulary	Concepts	Core terms that encode the domain’s structure (force, gene, equilibrium, field).	Stability, simplicity, NIP, NSOP, rank (RM, U), forking, dividing, indiscernibles, saturation, types, Morley sequences, independence relations, tameness, Shelah’s dividing lines.
		Classifications	Taxonomies, categories, or typologies that organize entities and relations.	Stable vs. unstable, superstable vs. stable, ω-stable vs. superstable, simple vs. non-simple, NIP vs. IP, NSOP vs. SOP, o-minimal vs. unstable expansions.
	3.4 Formal Representations	Equations	Mathematical constructs expressing laws, relations, or mechanisms.	Rank inequalities (e.g., RM(a/A) ≥ RM(a/AB)), forking equivalences, dividing formulas, independence axioms, characterization statements for stability/simplicity/NIP.
		Models	Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena.	Saturated models, homogeneous models, prime models, limit models, models witnessing stability/simplicity, Morley sequence structures, independence trees.
	3.5 Idealized Structures	Simplified Models	Purposeful abstractions that capture essential dynamics while omitting irrelevant detail.	Monster model (ℭ), pure-indiscernible arrays, stable fragments of unstable theories, simplified rank-1 theories, clean independence frameworks.
		Limit Conditions	Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear).	Instability (SOP, TP), absence of non-forking extensions, rank divergence, failure of saturation at certain cardinals, non-definability of types, breakdown in non-first-order contexts.
	3.6 Integrative Frameworks	Unifying Theories	Higher-order structures that connect disparate laws or mechanisms under a coherent whole.	Shelah’s classification theory, stability theory, simplicity theory, NIP theory, geometric stability theory, o-minimality, model-theoretic tameness frameworks.
		Interdisciplinary Links	Points where the theory connects to adjacent sciences or larger explanatory systems.	Links to algebraic geometry (Zariski geometries), real analysis (o-minimal structures), combinatorics (VC-dimension, indiscernibles), group theory (definable groups), and topology (type-space topologies).
4. Method Layer	4.1 Inquiry Design	Experimental Design	Structured plans for manipulating variables to test causal claims.	Manipulating base sets, cardinalities, or model constructions to evaluate stability, simplicity, NIP/NIP, and rank behavior; altering formulas to test forking/dividing response.
		Observational Design	Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments).	Observing forking, dividing, and independence behavior in existing models; studying type-space topology; monitoring rank changes without altering the language or theory.
	4.2 Testing & Validation	Hypothesis Testing	Procedures for evaluating whether evidence supports or contradicts specific claims.	Testing whether a theory is stable, simple, NIP, or NSOP; verifying symmetry/transitivity of independence; checking whether rank assignments behave predictably; identifying dividing formulas.
		Replication	The requirement that results be independently reproducible under similar conditions.	Repeating rank computations across saturated and unsaturated models; reproducing forking/dividing results with different parameter sets; re-testing independence relations across embeddings.
	4.3 Inference & Evaluation	Statistical Inference	Rules for drawing conclusions from noisy or incomplete data.	Logical analogues: evaluating distribution of types, comparing rank spectra, analyzing frequency of forking, measuring complexity of dividing chains, identifying extremal type configurations.
		Model Comparison	Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models.	Comparing theories by their stability class, rank complexity, independence behavior, saturation profiles, definability of types, and robustness under model constructions.
	4.4 Error Management	Error Analysis	Identification and quantification of random and systematic errors.	Identifying miscalculated ranks, incorrectly classified stability/simplicity/NIP status, mistaken forking/dividing diagnoses, saturation errors, and false independence assumptions.
		Bias Control	Methods for minimizing subjective, instrumental, or procedural biases.	Avoiding biased selection of models or base sets; preventing rank overfitting through artificially enriched languages; ensuring neutrality in choice of witnessing types and indiscernible sequences.
	4.5 Adjudication & Revision	Peer Scrutiny	Collective evaluation of claims through critique, review, and debate.	Reviewing independence proofs, rank computations, dividing analyses, and classification claims; cross-checking with alternative constructions or canonical witnesses.
		Theory Revision	Procedures for modifying, replacing, or discarding models based on new evidence.	Adjusting axioms, modifying languages, refining independence frameworks, recalibrating rank definitions, altering type-space assumptions in response to new counterexamples.
	4.6 Integrity Conditions	Transparency	Requirements to disclose methods, data, assumptions, and limitations.	Full disclosure of rank calculations, forking/dividing criteria, independence assumptions, saturation parameters, and model-building methods.
		Ethical Standards	Norms ensuring responsible conduct in experimentation, data handling, and publication.	Honest representation of instability or complexity; avoidance of hidden assumptions in independence claims; precise attribution of dividing lines and classification theorems.