| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies number fields, rings of integers, valuations, ideals, units, class groups, Galois extensions, and local–global principles. Excludes transcendental structures unless algebraic approximations apply, and excludes analytic methods except where they reduce to algebraic forms. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates at the discrete–algebraic scale: extensions of ℚ, completions at primes, local fields, valuations, ideal factorization structures, and finite Galois groups. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Number fields, rings of integers, prime ideals, fractional ideals, units, class groups, Galois groups, valuations, completions, local fields, ramification data, norm/trace maps. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Degree of extensions, discriminant, ramification behavior, splitting of primes, ideal class numbers, unit rank, norm/trace values, residue field characteristics, integrality, local/global solvability. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Global fields, local fields, Dedekind domains, ideal classes, Galois extensions, ramified/unramified extensions, completions at primes, algebraic integers. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Field extension degrees, valuations, residue-field parameters, discriminant values, ideal factorizations, decomposition/ramification indices, norm/trace values, unit-rank parameters. |
| | Parameterization | How variables encode and represent the system’s state. | Encoded by minimal polynomials, embeddings into ℂ, prime factorizations in rings of integers, valuation data, Galois actions, local field expansions, and ideal-class structures. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Treating rings of integers as Dedekind domains; assuming unique factorization of ideals; ignoring analytic error terms; restricting to finite extensions; assuming tame ramification; using simplified local–global reductions. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Breakdowns occur in non-Dedekind domains, wild ramification, large discriminants, non-UFD behavior, infinite extensions, or settings requiring analytic or transcendental tools. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Assumes algebraic closure exists; integrality conditions hold; ideal factorization is unique; local fields are complete; valuations behave predictably; Galois correspondence is valid; local–global principles apply when assumed. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes algebraic structure captures arithmetic fully; splitting/ramification mirrors deep number-theoretic behavior; ideal theory replaces integer factorization; valuations encode arithmetic information completely. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Galois groups must match field extensions; ideal factorization must respect Dedekind structure; local computations must agree with global behavior; norm/trace must align with extension degrees. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires alignment among number fields, valuations, completions, ideal factorization, class groups, unit groups, and Galois theory; local and global viewpoints must integrate into one unified arithmetic picture. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Prime splitting in extensions, ramification patterns, discriminant size, ideal-factorization structure, residue-field behavior, norm/trace values, local–global solvability of equations. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Cannot detect analytic distribution of primes; factorizations become hard for large discriminants; local data alone may not determine global behavior; class group structure may be computationally inaccessible. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Field degree, discriminant, ramification index (e), residue degree (f), valuations (v_p), norm/trace values, class number, unit rank. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Minimal polynomials, factorization algorithms, valuation tools, p-adic expansions, Dedekind domain ideal theory, Galois-group computations, local-field structure tests. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Definitions of number fields, algebraic integers, ideals, fractional ideals, valuations, completions, units, class groups, norm and trace, ramification, splitting types, local fields. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Computing minimal polynomials; factoring primes in extensions; computing valuations; constructing completions; determining norm/trace; finding class group structures (when feasible); analyzing local/global solvability. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Generating prime decompositions in rings of integers; computing discriminants; gathering valuation data; constructing p-adic expansions; building Galois correspondence diagrams; assembling class-number data. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling primes in base fields; sampling algebraic integers; sampling extensions of small degrees; sampling residue fields; sampling ideals to measure class-group behavior. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Prime-decomposition tables, valuation tables, discriminant values, p-adic expansions, ideal-class lists, Galois-group diagrams, norm/trace matrices. |
| | Resolution | The granularity or precision with which data is captured. | Determined by precision of p-adic expansions, completeness of prime factorization, discriminant accuracy, refinement of ideal-class computations, and granularity of valuation data. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Validating prime factorizations; checking valuation consistency; verifying discriminant calculations; confirming norm/trace identities; cross-checking Galois-group computations; ensuring alignment between local and global data. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Misfactored primes; incorrect valuations; discriminant miscalculation; computational errors in class-group algorithms; incorrect splitting classification; mismatched local/global invariants. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Prime splitting laws; ramification patterns; norm–trace relations; unique factorization of ideals; behavior of units via Dirichlet structure; local–global principles; Frobenius element patterns in Galois extensions. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Discriminant, class number, unit rank, splitting type ((e,f)), ramification indices, residue degrees, norm and trace invariants, ideal-class representatives, regulator. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Field extensions generating splitting; valuations driving local structure; Galois actions structuring prime behavior; Dedekind factorization determining ideal structure; local completions causing local–global correspondences. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Tower-of-field extension pathways; ideal-factorization chains; ramification/decomposition pathways; p-adic completion pathways; class-group construction sequences; Galois correspondence pathways. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Number field, algebraic integer, ideal, fractional ideal, valuation, local field, discriminant, ramification, Frobenius element, class group, unit group, Galois group, norm/trace. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Ramified vs unramified primes; splitting types; Dedekind domains; global vs local fields; Galois vs non-Galois extensions; class-number types; unit-rank categories; tame vs wild ramification. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Norm and trace formulas; discriminant formulas; ideal-factorization formulas; ramification–decomposition–inertia relations; minimal-polynomial equations; valuation identities; local-field expansions. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Global number fields; local fields ((\mathbb{Q}_p), finite extensions); Dedekind domains; ideal-lattice models; Galois-action models; residue-field models; p-adic analytic models. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Quadratic fields; cyclotomic fields; Dedekind domains with simple class structure; tame ramification only; simplified Galois groups (cyclic, abelian); prime-power residue fields; basic p-adic examples. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Failure of unique factorization at element level; wild ramification; huge discriminants; large and intractable class groups; nonabelian Galois groups with unpredictable splitting; computational intractability. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Ideal-class theory; local–global principle; class field theory; Galois theory; valuation theory; algebraic integer theory; p-adic analysis as a local unifier; Artin reciprocity. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Links to algebraic geometry (schemes, number fields as global fields), topology (Galois cohomology), cryptography (factorization, discrete logs), harmonic analysis (adelic structures), and logic (decidability questions). |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Altering base fields, varying polynomial coefficients, adjusting ramification conditions, modifying valuations, and testing ideal behavior under extension to observe splitting, ramification, and class-group effects. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing natural arithmetic structure: watching how primes split in extensions, tracking valuations in completions, monitoring class-number behavior, observing Galois action patterns without modifying field structure. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing factorization in rings of integers; checking discriminant and ramification data; validating norm/trace identities; verifying ideal-class group and unit-rank computations; checking Frobenius elements in Galois extensions. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Recomputing factorizations of primes; repeating valuation calculations; recomputing discriminants via multiple methods; re-evaluating Galois groups; repeating class-group computations with alternative algorithms or bases. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Logical analogues: analyzing distribution of splitting types; comparing class-number growth; examining discriminant trends; evaluating local–global patterns; comparing ramification frequency across families of fields. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing number fields via discriminant, signature, class number, unit rank, ramification profile, and Galois group; comparing local fields by residue degree/ramification index; comparing completions at different primes. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Incorrect ideal factorizations; valuation errors; miscomputed discriminants; Galois-group misidentification; incorrect norm/trace calculations; computational failures in class-group algorithms; mismatched local/global data. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding selective field examples (e.g., only quadratic or cyclotomic fields); preventing over-reliance on small discriminants; ensuring representative sampling of primes; avoiding computational shortcuts that bias factorization or class-group outcomes. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Independent verification of factorization, discriminant, Galois-group, and class-number computations; reviewing ramification and splitting claims; checking norm/trace proofs; validating local–global arguments. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Updating ideal theory for new counterexamples; refining Galois-classification techniques; modifying local–global principles; revising computational algorithms; adjusting ramification theory to handle extreme or newly discovered cases. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Full disclosure of computational methods, field representations, choice of bases, valuation conventions, Galois-group algorithms, and assumptions behind class-group or discriminant computations. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Honest reporting of computational limitations; clear identification of conjectural vs. proved results; avoidance of overstating local–global correspondence; proper attribution of classical theorems and algorithms. |