| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Includes the development and application of advanced mathematical tools used to analyze, construct, and unify physical theories. Covers areas not tied to a single physical system, such as differential equations, variational principles, algebraic structures, integrable systems, and topological methods. Excludes purely empirical physics without deep mathematical formulation. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Not tied to any specific spatial or temporal scale; operates at the level of abstract mathematical structure. Applicable from microscopic to cosmological systems depending on context. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Mathematical structures such as functions, fields, operators, manifolds, algebraic systems, categories, differential equations, and geometric objects used to represent physical systems. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Properties include continuity, differentiability, symmetry characteristics, algebraic relations, boundary behavior, stability, and structural features defined by the mathematical model. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Categories include mathematical objects, equations, transformations, symmetry groups, geometric structures, topological classes, and algebraic frameworks used to describe systems. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Variables include fields, coordinates, functional values, system parameters, initial conditions, boundary conditions, and other quantities defining the mathematical state of a physical model. |
| | Parameterization | How variables encode and represent the system’s state. | System states are encoded using equations, coordinate choices, parameter sets, functional forms, operator definitions, and geometric or algebraic structures. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Simplifications include linearization, symmetry assumptions, smoothness assumptions, simplified boundary conditions, reduced dimensionality, and idealized forms of physical laws used for tractability. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Idealizations hold when nonlinearities are small, symmetry assumptions apply, scales are appropriate for continuum or differentiable models, and when neglected terms remain insignificant. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Assumes mathematical representations can fully describe physical behavior, continuity where needed, deterministic or stochastic rules depending on the chosen framework, and the applicability of abstract models to real systems. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes mathematical consistency maps to physical consistency, transformations preserve meaning, and advanced structures such as topological or algebraic objects faithfully represent aspects of physical reality. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Requires consistency of equations, compatibility of symmetry rules, well-defined boundary and initial value behavior, and no contradictions among mathematical structures used to define a physical framework. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires that variables, equations, geometric structures, algebraic rules, and assumptions fit into a unified and coherent mathematical system capable of describing physical phenomena. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Observables depend on the physical domain being modeled. Typical observables include field values, waveforms, trajectories, energy distributions, spatial patterns, and any measurable quantity reconstructed using mathematical models. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Limited by measurement resolution, noise, instrument sensitivity, and the ability to accurately relate mathematical quantities to experimental data across scales. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Uses all standard physics units including meters, seconds, mass units, energy units, and dimensionless parameters used in mathematical analysis. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Instruments include sensors, detectors, imaging systems, oscilloscopes, spectrometers, interferometers, timing devices, and computational tools for reconstructing physical signals from mathematical models. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Mathematical quantities such as field strength, potential, curvature, or flux are defined through associated measurement or reconstruction procedures used in the physical domain being modeled. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Procedures include collecting raw measurements, applying transformations, solving equations to infer physical quantities, and using standardized steps to relate mathematical variables to measurable outcomes. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Protocols include consistent sampling schedules, controlled environmental conditions, stable detector operation, and standardized methods for recording measurements used in mathematical modeling. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling rules specify how often measurements are taken, how spatial or temporal domains are covered, and how representative the samples are for the physical system modeled. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Data appears as time series, spatial maps, spectra, images, numerical tables, simulation outputs, and processed fields derived from mathematical transformations. |
| | Resolution | The granularity or precision with which data is captured. | Determined by instrument precision, sampling rate, spatial granularity, computational accuracy, and the quality of numerical approximations used to solve equations. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Calibration uses known standards, reference signals, stable sources, and repeated tests to ensure correct mapping between mathematical variables and measured quantities. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Errors arise from numerical rounding, finite precision, sensor noise, environmental influences, model assumptions, and uncertainties in solving mathematical equations. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Stable patterns include differential relationships, conservation rules, symmetry behavior, variational principles, and predictable responses described by mathematical equations across physical systems. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Invariants include conserved quantities, symmetry-preserved forms, topological invariants, geometric features that remain unchanged under transformations, and stable algebraic relations. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Mechanisms arise from mathematical structures that govern physical change, such as evolution equations, variational pathways, symmetry-driven constraints, and relations encoded by functional or algebraic rules. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Pathways include sequences of transformations, solution flows of differential equations, iterative steps of variational procedures, and structured mappings between states. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Core concepts include differential equation, boundary condition, operator, manifold, symmetry, functional, mapping, transformation, field, and topological structure. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Classifies systems by equation type, symmetry class, dimensionality, boundary conditions, stability properties, algebraic structures, and topological categories. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Formal constructs include partial differential equations, variational equations, stability equations, algebraic systems, and transformation rules that express physical laws mathematically. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Includes mathematical models of physical systems, computational simulations, variational models, topological models, integrable models, and any abstract representations used to analyze physical behavior. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Idealized structures include linear approximations, symmetry-reduced models, simplified geometries, low-dimensional reductions, and models with ignored nonlinear terms for analytical tractability. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Approximations hold when nonlinear contributions are small, when symmetry is exact or approximate, under weak perturbations, or when boundary or initial conditions allow simplified forms. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Provides unifying structures such as variational principles, symmetry frameworks, algebraic methods, and topological approaches that connect multiple physical theories under shared mathematical rules. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Links to physics, engineering, applied mathematics, computer science, geometry, topology, algebra, and any field requiring structured mathematical analysis of physical systems. |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Experiments are designed to test mathematical predictions by manipulating physical variables that correspond to terms in equations, symmetry assumptions, boundary conditions, or variational principles. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observational approaches rely on collecting natural physical data and comparing it to mathematical predictions, such as observing waves, fields, or motion patterns that match theoretical structures. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Tests whether physical measurements agree with solutions to equations, predicted symmetries, stability results, or other mathematically derived claims. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Requires independent reproduction of physical measurements, computational simulations, or analytical results using different methods or parameter choices. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Uses statistical tools to compare noisy data to mathematical predictions, estimate parameters in models, and determine how well equations describe observed behavior. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Compares models based on predictive accuracy, mathematical simplicity, stability, fit to data, computational efficiency, and robustness under parameter variation. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Identifies numerical errors, approximation errors, measurement noise, model simplifications, and uncertainties in solving differential or algebraic equations. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Minimizes bias through standardized procedures, repeated measurements, independent derivations, blind computational tests, and consistent calibration across models. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Mathematical results and physical interpretations are evaluated through proof checking, replication of simulations, publication review, and critique from mathematicians and physicists. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Models are revised when predictions fail, when new mathematical structures offer better explanations, or when inconsistencies appear among equations or assumptions. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Requires full disclosure of assumptions, derivation steps, numerical methods, boundary conditions, approximations, and limitations of mathematical modeling. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Requires accuracy in reporting equations, data, assumptions, and numerical methods, along with adherence to scientific and mathematical integrity in publication and analysis. |