| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies solutions to polynomial equations using both algebraic geometry and number theory; includes rational points, integral points, Diophantine sets, reduction modulo primes, heights, arithmetic schemes, and global fields. Excludes analytic approaches unless used in service of algebraic–arithmetic structure; excludes purely geometric varieties with no arithmetic structure. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates from local fields (p-adic neighborhoods, reductions mod p) to global fields (number fields, function fields), to geometric scales (varieties, schemes), combining arithmetic and geometric data across dimensions. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Varieties over number fields, arithmetic schemes, rational points, integral points, primes and places of number fields, reduction maps, heights, Galois representations, ℓ-adic cohomology classes, Néron models. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Rational solvability, integrality, reduction types, good/bad reduction, height growth, local–global compatibility, behavior under specialization, Galois action on cohomology, Diophantine obstructions (e.g., Brauer–Manin). |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Rational points, integral points, local fields, global fields, abelian varieties, elliptic curves, curves of higher genus, arithmetic schemes, Selmer groups, Néron models, mod p reductions. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Field of definition, reduction prime p, height values, local solubility conditions, Galois action parameters, cohomology classes, Selmer ranks, discriminants, conductor values. |
| | Parameterization | How variables encode and represent the system’s state. | Encoded via equations over number fields, valuations at primes, reduction maps, height functions, mod-p fibers, Galois representations, cohomology groups, and geometric invariants (dimension, genus). |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Treating varieties as smooth; ignoring singular fibers; assuming good reduction; restricting to low dimension (curves, elliptic curves); assuming mild Galois behavior; using height bounds heuristically. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Break down for wild ramification, bad reduction, higher-dimensional pathologies, singularities, non-rational varieties, or when heights behave irregularly; failure of local–global principles. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Assumes arithmetic properties are reflected in geometry; local–global principles have meaningful content; reduction mod p preserves key structure; Galois actions encode arithmetic; heights meaningfully measure arithmetic complexity. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes geometric invariants predict arithmetic behavior; rational points form structured sets; Diophantine phenomena reflect cohomological obstructions; reduction mod p reveals deep global information. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Local data must align with global structures; reduction fibers must reflect original varieties; cohomology must be compatible with Galois actions; heights must behave coherently across embeddings and fields. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires compatibility among geometric invariants, number-field arithmetic, reduction maps, Galois representations, cohomological obstructions, and height functions within one unified arithmetic–geometric framework. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Rational and integral point patterns; reduction mod p behavior; splitting and ramification at primes; height growth; local solubility at completions; Galois action on torsion points; degeneration of fibers. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Local data insufficient to guarantee global solvability; reduction mod p may hide arithmetic structure; heights detect complexity only approximately; primes with bad reduction obscure geometry; cohomological obstructions difficult to detect computationally. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Height values; discriminant; conductor; norm and trace from number fields; reduction prime p; valuations; ranks of Mordell–Weil groups; Selmer ranks; local invariants of fibers. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Height functions; reduction maps; valuations; local-field solvers; point-search algorithms; Galois-representation computations; ℓ-adic cohomology machinery; ideal-factorization tools. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Rational/integral points; reduction modulo primes; heights; good/bad reduction; Néron models; Selmer groups; Mordell–Weil rank; Galois representations; local/global solubility. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Computing reductions mod p; evaluating heights; checking local solubility at completions; computing ranks and Selmer groups; constructing Néron models; factoring ideals in number fields; computing Galois actions on torsion points. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Sampling points on varieties; computing reductions across many primes; evaluating heights at chosen rational points; gathering local invariants; constructing cohomological data; computing torsion structures; scanning for integral/rational points. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling primes of good/bad reduction; sampling rational points across bounded height; sampling residue fields; sampling local completions; sampling fibers of arithmetic schemes; sampling torsion points. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Height tables; reduction tables mod p; local-invariant data; Selmer-rank tables; Mordell–Weil group presentations; ideal-factorization listings; Galois-representation matrices. |
| | Resolution | The granularity or precision with which data is captured. | Determined by height bounds, number of sampled primes, valuation precision, completeness of local-solubility tests, Galois-representation accuracy, and thoroughness of point searches. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Verifying height computations; cross-checking reductions; validating factorization results; checking local-global consistency; confirming Selmer computations; ensuring Galois-representation correctness across primes. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Incorrect height values; misclassified reduction types; factoring errors; false local-solvability conclusions; incorrect rank estimates; computational limits on Selmer groups; mismatches in Galois data across ℓ-adic levels. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Local–global principles (Hasse principle, weak approximation); reduction-mod-p behavior; height-growth relations; Galois-action patterns on torsion points; behavior of rational points under morphisms; Néron model compatibility. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Height invariants, discriminants, conductors, reduction types, ranks of Mordell–Weil groups, Selmer ranks, Galois-representation invariants, Tamagawa numbers, genus of curves, mod-p fiber invariants. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Reduction mechanisms mapping global points to local fibers; Galois-action mechanisms determining arithmetic structure; height mechanisms encoding Diophantine complexity; cohomology mechanisms generating obstructions; fibration mechanisms relating families of varieties. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Local-to-global lifting pathways; descent sequences; height-doubling pathways; Selmer-group filtration pathways; ramification/decomposition pathways; Galois-representation specialization pathways. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Rational point, integral point, reduction mod p, height, local field, global field, Selmer group, Mordell–Weil group, Galois representation, discriminant, conductor, fibration, Néron model. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Good vs bad reduction; additive/multiplicative reduction; rational vs integral points; curves by genus; varieties by dimension; abelian vs non-abelian Galois representations; torsion vs free components; local vs global obstructions. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Height formulas; reduction maps; norm/trace relations; discriminant and conductor formulas; local–global exact sequences; Galois-representation matrices; cohomological exact sequence equations (e.g., Selmer sequence). |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Arithmetic schemes; varieties over number fields; elliptic curves; higher-genus curves; abelian varieties; Néron models; local fiber models; Selmer-group models; cohomological diagrams. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Smooth projective curves; elliptic curves with good reduction; abelian varieties over ℚ; tame ramification only; simplified height functions; toy Galois representations; basic Selmer configurations. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Breakdowns for singular fibers; wild ramification; high-dimensional arithmetic schemes; failure of Hasse principle; insufficient height control; noncomputable class-group or Selmer behavior; undecidable Diophantine sets. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Diophantine geometry; Galois cohomology; height theory; local–global principles; Néron models; arithmetic deformation theory; relationship between geometry and arithmetic via schemes and morphisms. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Links to algebraic geometry (schemes, divisors), number theory (Galois, ramification), logic (undecidability), cryptography (elliptic curves), algebraic topology (cohomology), and differential geometry (Arakelov theory). |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Varying primes of reduction, altering height bounds, modifying field extensions, adjusting local conditions, changing models of varieties, or modifying coefficients to test rational/integral solvability and arithmetic behavior. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing rational/integral point patterns, monitoring reductions mod p, tracking Galois action on torsion points, observing height growth, watching Selmer group changes, examining local behavior at completions. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing local solubility; validating reduction type; confirming height formulas; verifying Galois-representation behavior; checking Selmer rank predictions; testing Hasse principle; validating Néron model compatibility. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Recomputing reductions at multiple primes; recomputing heights under alternate embeddings; repeating Selmer calculations; recalculating ranks with different point bases; re-evaluating Galois data with distinct primes ℓ. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Analyzing distribution of rational points across heights; studying density of good/bad reduction; assessing variation in Selmer ranks; comparing local invariants across primes; examining frequency of Hasse-principle failures. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing different models of a variety; comparing reductions across primes; comparing height functions; comparing Selmer groups and ranks; contrasting Galois representations; comparing arithmetic schemes. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Incorrect reduction classification; height miscomputations; factoring errors in number fields; incorrect Selmer/rank computations; misidentification of local obstructions; mismatched Galois data across primes. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding sampling only small primes; preventing selective height ranges; avoiding overrepresentation of curves with simple arithmetic; ensuring unbiased selection of embeddings and models; avoiding cherry-picked local conditions. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Independent checking of height computations, Selmer/rank arguments, reduction and ramification claims, Néron-model constructions, and Galois-representation accuracy. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Refining height theory; updating Selmer-group frameworks; modifying local–global principles; adjusting models of varieties; revising arithmetic obstructions; integrating improved Galois-cohomology techniques. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Full disclosure of height bounds, choice of primes, embeddings used, factorization methods, Selmer-group algorithms, and assumptions behind local/global tests. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Honest reporting of computational limits; distinguishing conjectural vs. proven results; proper attribution of theorems and algorithms; avoiding overclaiming based on partial data or selective primes. |