| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Includes the rigorous mathematical structures used to define quantum mechanics: operators, Hilbert spaces, linear transformations, probability rules, functional analysis, and axiomatic reconstructions. Excludes heuristic or purely operational quantum models lacking formal mathematical grounding. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates at the quantum scale, dealing with systems at atomic and subatomic levels, but the mathematics itself is scale-independent and applies to any system described by quantum principles. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Abstract state vectors, operators, observables, probability measures, transformation rules, density operators, and mathematical structures such as algebras and spaces. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Linearity, completeness of state spaces, operator domains, spectral properties, probability rules, and transformation behavior under symmetry or measurement operations. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | States, observables, transformations, measurement outcomes, operator classes, and structural relations such as commutation and algebraic rules. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Quantum states, operator values, probability distributions, expectation values, and parameters specifying physical or mathematical configurations. |
| | Parameterization | How variables encode and represent the system’s state. | States encoded as vectors or density operators; observables encoded as operators; probabilities encoded through inner products or trace rules; transformations encoded as linear or algebraic maps. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Idealized Hilbert spaces, perfectly defined operators, simplified measurement models, ignoring environmental entanglement, and assuming exact symmetries or isolated systems. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Idealizations hold when measurement imperfections are small, systems remain isolated, operators remain well-defined, and environmental interactions do not dominate or distort the quantum formalism. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Assumes linear structure, probabilistic interpretation, continuity of state evolution, discrete or continuous spectra, and existence of well-defined mathematical spaces governing system behavior. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes mathematical formalisms accurately represent physical systems; measurements correspond to operator rules; probability assignments are complete; and the underlying space is sufficient for all quantum descriptions. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Mathematical structures must avoid contradictions, operator rules must align with probability rules, and transformations must preserve consistency of states and observables. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | States, observables, operators, and measurement rules must fit together into a unified formal system that preserves linearity, probability laws, and allowed transformations. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Observable phenomena include measurement outcomes such as energy levels, position readings, spin results, interference patterns, and probability distributions reconstructed from repeated trials. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Detection limited by instrument resolution, noise, decoherence effects, and the inability to measure certain quantum properties simultaneously with high precision. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Uses standard units such as meters, seconds, energy units, and probability values to express measurement outcomes and operator expectations. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Instruments include interferometers, detectors, sensors, spectrometers, timing devices, and systems capable of resolving quantum-level events or distributions. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Quantities such as position, momentum, spin, and energy are defined through specific measurement procedures linked to operators and measurement rules. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Procedures involve repeated measurement, controlled preparation of states, analysis of outcome frequencies, and application of rules that map measurement results to operator expectations. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Protocols require stable preparation of quantum states, consistent detector operation, fixed sampling intervals, and standardized collection of measurement outcomes. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling rules involve repeated trials, ensemble measurements, or time-series data collection to reconstruct probability distributions from limited quantum outcomes. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Data appears as probability distributions, outcome counts, time-series records, interference images, and reconstructed density matrices. |
| | Resolution | The granularity or precision with which data is captured. | Determined by instrument precision, clock stability, spatial granularity, noise levels, and detector efficiency. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Calibration uses known quantum standards, reference measurements, stable sources, and repeated checks to ensure accurate mapping from measurement to operator interpretation. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Errors arise from statistical uncertainty, detector noise, state preparation imperfections, decoherence, and limits of measurement precision. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Regularities include linearity of evolution, stable relationships between operators and measurement outcomes, rules linking probabilities to inner products, and predictable spectral patterns of operators. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Invariants include probability conservation, norms of state vectors, symmetry-based quantities, operator relationships preserved under evolution, and structural properties of algebras. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Mechanisms arise from linear transformations, state evolution rules, operator actions, and probability assignments that generate observable patterns from underlying mathematical structure. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Pathways include sequences of state preparation, transformation, measurement, and post-measurement updating that form the causal chain linking mathematical objects to empirical outcomes. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Core concepts include state, observable, operator, spectrum, expectation value, superposition, measurement rule, density operator, transformation, and algebra. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Classifies objects into states, operators, observables, transformations, measurement types, operator classes, and algebraic types such as commutative vs noncommutative structures. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Formal constructs include operator rules, spectral decompositions, linear evolution rules, probability assignments, and algebraic relationships among operators and states. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Includes Hilbert space models, algebraic models, density matrix models, operator algebra constructions, and axiomatic reconstructions that represent quantum behavior. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Simplified forms include finite dimensional models, two-level systems, idealized measurement models, noise-free evolution, and symmetric operator setups. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Approximations hold in isolated systems, when noise is negligible, when operators are well-defined, or when finite-dimensional truncations remain accurate for prediction. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Provides unified mathematical structures such as operator algebras, functional analytic frameworks, and axiomatic approaches linking diverse quantum systems under a single theory. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Links to information theory, probability theory, functional analysis, operator algebra, mathematical physics, and computational modeling of quantum systems. |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Experiments are designed to test whether measurement outcomes follow the mathematical rules of quantum theory, such as probability assignments, operator relationships, and predicted spectral values. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observational approaches collect measurement results without direct manipulation, such as repeated passive detection of quantum events or natural fluctuations that reveal operator structures. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Tests whether observed frequencies, correlations, and outcomes match the predictions derived from operators, spectral rules, and state transformations. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Requires independent reproduction of quantum measurement results across different detectors, laboratories, or state-preparation procedures. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Uses statistical tools to infer probabilities, reconstruct states, estimate operator expectations, and analyze noise in quantum data. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Evaluates competing mathematical models based on consistency with measurement data, simplicity of operator structure, predictive accuracy, and robustness under repeated trials. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Identifies errors from detector noise, decoherence, imperfect state preparation, statistical variability, and limits of operator-domain definitions. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Reduces bias by using blind data-processing methods, standardized measurement rules, repeated calibration, and independent cross-checks of state reconstruction. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Mathematical and empirical claims are evaluated through formal proof checking, replication of experiments, publication review, and comparison with alternative operator or algebraic models. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Occurs when inconsistencies are discovered in operator rules, when new data contradicts mathematical predictions, or when alternative axioms produce clearer or more accurate formulations. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Requires explicit disclosure of assumptions, operator definitions, data-processing rules, approximations, and limitations of the mathematical model. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Requires accurate reporting of proofs, measurements, uncertainties, assumptions, and adherence to rigorous standards of mathematical and scientific integrity. |