| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Includes phases of matter defined by topological properties rather than symmetry breaking, such as topological insulators, topological superconductors, quantum Hall states, Weyl and Dirac materials, and systems with protected boundary modes. Excludes ordinary phases described only by local order parameters or conventional symmetry breaking without topological character. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates from atomic and nanometer scales where band structure and topology emerge, to macroscopic scales where edge or surface states produce measurable responses. Time scales include fast electronic dynamics and slower transport processes. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Bulk bands, edge states, surface states, quasiparticles, topological defects, Berry curvature structures, nodes, domain boundaries, and external fields that probe topological response. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Topological invariants, band connectivity, robustness to disorder, edge conduction, chirality, quantized response properties, and protected boundary behavior. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Topological phases, bulk properties, boundary states, protected modes, nodal features, and response signatures such as quantized conductance. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Topological invariant values, carrier density, band gap size, chemical potential, magnetic field strength, symmetry class indicators, and edge or surface state occupancy. |
| | Parameterization | How variables encode and represent the system’s state. | States encoded by band structure configuration, symmetry class, topological invariant values, field strengths, and chemical or structural tuning parameters controlling transitions. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Treating systems as perfectly clean, assuming exact symmetries, using simplified lattice models, ignoring weak interactions, idealizing boundaries as sharp, and assuming perfect protection against disorder. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Idealizations hold when disorder is weak, symmetry is not strongly broken, interactions remain moderate, and temperature is low enough to preserve protected states. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Assumes bulk band topology determines boundary behavior, topological invariants are meaningful physical descriptors, and symmetry classes define allowable topological phases. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes band models capture essential topological features, boundary states persist under moderate perturbation, and topological classification reflects real material behavior. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Requires consistency between bulk topology, symmetry constraints, boundary state predictions, and measurable transport; no contradictions among band connectivity, invariant values, or edge behavior. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Entities, variables, and assumptions must jointly support unified topological descriptions linking bulk invariants, protected boundary modes, and quantized response properties. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Detectable signals include quantized conductance, robust edge or surface state transport, anomalous Hall responses, suppressed backscattering, band inversion signatures, nodal point features, and characteristic surface spectroscopy results. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Limited by low temperature requirements, sensitivity to weak edge signals, spatial resolution for surface state imaging, noise in transport measurements, and difficulty resolving small band inversions or tiny energy gaps. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Uses volts, amperes, ohms, meters, seconds, kelvins, teslas, electron volts, and momentum or energy units used in spectroscopy or scattering. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Instruments include angle resolved photoemission tools, scanning probe microscopes, transport setups, magnetometers, x ray and neutron scattering, cryogenic systems, and microwave or terahertz probes. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Quantities such as topological invariant signatures, edge state conductance, band inversion strength, and anomalous response terms are defined through measurement and analysis procedures specific to transport, spectroscopy, or scattering. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Procedures include field sweeps, temperature sweeps, current voltage measurements, surface spectroscopy scans, scattering pattern collection, and controlled symmetry breaking or restoring tests. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Data collected under low temperature, controlled magnetic fields, stable sample environments, defined crystallographic orientation, and repeated measurement cycles for consistency. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling rules include multiple field strengths, repeated spectra, surface scans across different regions, and systematic variation in temperature or doping to map phase behavior. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Data appears as conductance curves, Hall response curves, spectral maps, surface state images, scattering peaks, and band structure reconstructions. |
| | Resolution | The granularity or precision with which data is captured. | Determined by energy resolution of spectroscopy tools, spatial resolution of probes, stability of cryogenic temperatures, momentum or frequency resolution in scattering, and signal to noise ratio in transport. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Calibration uses reference materials with known topological features, standard conductance quantization values, magnetic field calibration, energy reference lines in spectroscopy, and repeated baseline checks. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Errors arise from sample disorder, thermal drift, magnetic noise, alignment in spectroscopy or scattering, contact resistance in transport, and finite resolution in reconstructing band inversion or surface states. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Stable patterns include quantized conductance, bulk boundary correspondence, band inversion relationships, robustness of edge or surface states, and predictable phase transitions driven by symmetry or topology changes. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Invariants include topological index values, winding numbers, Chern numbers, parity indicators, robustness of edge and surface modes, and stable bulk connectivity features. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Mechanisms arise from band inversion, symmetry protected boundary modes, Berry curvature effects, nodal point creation or annihilation, and collective electronic behavior determined by topological band structure. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Pathways include continuous deformation of band structure leading to topological transitions, creation of protected edge states, symmetry breaking or restoration sequences, and controlled modification of band connectivity. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Core terms include topology, band inversion, bulk boundary correspondence, Berry curvature, nodal point, winding number, protected state, topological invariant, and symmetry class. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Classifies materials into topological insulators, topological superconductors, quantum Hall systems, Weyl and Dirac materials, and symmetry protected phases, using symmetry and topological index criteria. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Uses equations describing band connectivity, energy dispersion rules, response terms, Berry phase relations, and mathematical formulations of topological index calculations. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Includes lattice models, band structure models, tight binding models, symmetry class models, effective Dirac or Weyl models, and continuum models capturing topological behavior. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Idealizations include perfect crystal symmetry, clean boundaries, simplified band structures, absence of disorder, and idealized two dimensional or one dimensional limits where topology is easier to characterize. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Models hold under weak disorder, stable symmetry conditions, low temperatures, and size scales large enough to support boundary modes but small enough to preserve coherence. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Unifying theories include symmetry based classifications, bulk boundary correspondence frameworks, topological band theory, and mathematical tools linking electronic structure and topological invariants. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Links to condensed matter physics, materials science, quantum information, mathematics, photonics, acoustics, and mechanical metamaterials that mimic topological behavior. |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Experiments vary magnetic field, temperature, strain, chemical composition, sample thickness, and symmetry breaking fields to test how these factors drive topological transitions, alter edge states, or affect quantized responses. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observational methods track natural fluctuations, spontaneous formation of edge states, domain evolution, or emergent boundary behavior without controlled manipulation. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Hypotheses tested by comparing measured quantized responses, surface state spectra, band inversion signatures, and anomalous transport behavior to predicted topological models and invariant based classifications. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Replication requires confirming quantized conductance, edge state imaging, band structure signatures, and phase transition points across different samples, instruments, and laboratories. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Statistical tools used to analyze noisy transport data, quantify deviations from quantization, extract surface state dispersion, evaluate scattering features, and determine uncertainty in topological invariant estimation. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Models compared based on predictive accuracy for edge state behavior, robustness under disorder, consistency with band inversion data, quantitative agreement with transport curves, and simplicity vs complexity tradeoffs. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Errors include sample disorder effects, thermal drift, alignment errors in spectroscopy, noise in transport measurements, field instability, and inaccuracies in reconstructing band topology from finite resolution data. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Bias minimized through controlled sample preparation, blind data analysis, multiple measurement geometries, repeated calibration, and cross verification with independent probes such as transport and spectroscopy. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Findings evaluated through scattering or spectroscopy replication, peer review, conference critique, and comparison with alternative topological or non topological explanations. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Theories revised when unexpected surface states appear, quantization breaks down, new topology linked phases emerge, or symmetry effects shift predicted invariant values. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Requires full disclosure of sample growth conditions, structural characterization, measurement settings, calibration routines, data processing, and assumptions used to extract invariants. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Requires accurate reporting of measurements, avoiding selective transport curves or spectra, proper treatment of sensitive topological materials, and adherence to standards for reproducibility and open communication. |