| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies formal relationships of relative computability among sets/problems via reducibilities (Turing, many-one, truth-table, weak tt, bounded-Turing). Includes degree structures, completeness, jumps, and incomparable degrees. Excludes informal heuristic comparisons of difficulty not captured by effective reductions. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates at infinite-set, oracle-access, and infinite process scales: reducibility chains, jump hierarchies, equivalence classes, and structural relations across unsolvable problems. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Sets of naturals, decision problems, reducibility maps, oracle machines, degree elements, jump operators (A′, A″, …), equivalence classes under reducibility, minimal-pair constructions. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Relative computability, completeness, reducibility strength, degree equivalence, closure under reductions, monotonicity of jumps, existence of incomparable degrees, structural density or gaps. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Turing degrees, many-one degrees, truth-table and weak truth-table degrees, polynomial-time degrees (optionally), low/high degrees, minimal degrees, incomplete degrees, complete degrees, jump hierarchy categories. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Oracle state, reduction step index, approximation stage, encoding choice, current reducibility status, degree-invariant markers, jump iteration level. |
| | Parameterization | How variables encode and represent the system’s state. | Parameterized by reducibility type (≤ₘ, ≤ₜ, ≤{tt}, ≤{wtt}), encoding schemes, uniformity conditions, oracle-program specifications, stage-by-stage approximations in reducibility proofs. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Canonical encodings, idealized oracle access, simplified reduction forms, ignoring coding overhead, working with standard complete sets, abstracting away infinite-injury complexities unless needed. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Fail for non-effectively presented sets, higher-type domains, reducibilities not transitive/closed, encoding-sensitive anomalies, or contexts requiring semantic rather than syntactic comparisons. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Reductions faithfully capture relative computability; oracle computation is well-defined; degree structures form consistent partial orders; jumps strictly increase unsolvability; equivalence classes behave uniformly across encodings. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes Church–Turing framework; assumes well-founded reducibility definitions; assumes equivalence classes reflect real structural differences; assumes encoding choices do not distort essential relations. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Reducibility definitions must align; degree equivalence must not collapse distinctions; jumps must preserve monotonicity; complete sets must uniformly encode unsolvable problems. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires harmony among reducibility notions, oracle models, degree axioms, jump operations, invariance properties, and structural theorems describing the degree hierarchy. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Reducibility traces (Turing/m/tt/wtt), oracle-call patterns, stage-by-stage approximations to reductions, divergence or convergence of reduction attempts, behavior of jump operations, appearance of incomparable degrees in constructions. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Undecidability barriers prevent confirming reducibility or non-reducibility in general; inability to observe infinite computations; limits in detecting true convergence; reducibility relations can require infinitely many oracle queries. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Reduction steps, oracle-query counts, stage index of approximation, number of requirement satisfactions, injury counts (if priority used), jump level reached, complexity of encoding. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Oracle Turing machine simulators, reduction checkers, approximation loggers, priority-construction engines, degree-structure analyzers, jump-operator evaluators. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Reducibility defined via effective transformations; completeness defined by reducibility to canonical hard sets; degree defined as an equivalence class under reducibility; jump defined via relative halting computations; convergence defined as stabilization of approximations. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Running reducibility simulations, executing oracle computations, tracing approximation sequences, checking requirement satisfaction, computing jump outputs, testing equivalence under chosen reducibility notion. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Fixed reducibility tests with controlled encodings; systematic oracle-call tracing; structured approximation sampling; canonical benchmark sets (K, K₀, K⁽ⁿ⁾); standardized jump computations; uniform protocols for comparing degrees. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling across known degree types (minimal, low/high, incomplete, complete); sampling reducibilities (≤ₜ, ≤ₘ, ≤{tt}, ≤{wtt}); testing constructions under varied priority orderings; selecting representative r.e. and non-r.e. sets. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Reduction logs, oracle-query traces, degree diagrams, jump-output tables, priority-injury histories, approximation snapshots, equivalence-test results. |
| | Resolution | The granularity or precision with which data is captured. | Determined by granularity of approximation checkpoints, precision of oracle-call logging, detail of reducibility trace capture, and ability to track injury or stabilization across long running constructions. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Validating reductions against known complete sets, cross-checking oracle simulations, verifying jump computations, ensuring consistent encodings, replicating approximation runs across independent implementations. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Incorrect reduction implementation, miscounted oracle calls, premature convergence assumptions, misclassified degrees, encoding errors, requirement mismanagement, inconsistent jump evaluations. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Reducibility relations form stable partial orders; degree equivalence is transitive; jumps strictly increase computational strength; complete problems uniformly encode unsolvability; incomparable degrees arise via diagonalization; reducibility closure properties hold across encodings. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Degree invariants under encoding changes; invariance of completeness under many-one and Turing reductions; jump monotonicity (A <ₜ A′); structure of upper semilattice in Turing degrees; preservation of reducibility under oracle extension. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Oracle computation mechanisms generating relative strength; diagonalization producing incomparability; effective reductions transforming one decision problem into another; jump operator amplifying unsolvability; priority mechanisms organizing degree constructions. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Reduction pathway: encode A into B via computable transform; oracle pathway: A-oracle machine computes membership of another set; jump pathway: A → A′ → A″ … increasing degree height; priority pathway: requirement ordering → satisfaction → correction → stabilization. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Reducibility (≤ₜ, ≤ₘ, ≤{tt}, ≤{wtt}); degrees; complete sets; jump operator; incomparable degrees; minimal degrees; reducibility chains; equivalence classes; uniform vs. non-uniform reductions. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Turing degrees, many-one degrees, tt and wtt degrees; complete vs. incomplete degrees; low/high degrees; minimal degrees and minimal pairs; hyperimmune-free degrees; jump hierarchy levels (0, 0′, 0″, …). |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Reducibility equations (A ≤ₜ B ↔ ∃ oracle machine computing A from B); degree equivalence equations (A ≡ₜ B); jump equations (A′ ≡ₜ K^A); limit equations showing reducibility stabilization; diagonalization identities defining separation. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Degree-structure diagrams; oracle Turing machine models; reducibility graphs; jump-hierarchy trees; priority-construction schematics; limit-approximation models representing reducibility stabilization. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Canonical complete sets (K, K₀); simplified reducibility forms ignoring encoding overhead; single-injury priority models; minimal reducibility frameworks; toy oracle machines; simplified jump-hierarchy truncations (e.g., only 0, 0′, 0″). |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Simplifications fail for non-r.e. degrees, higher-type oracles, constructions requiring infinite injury, reducibilities that are not transitive or closed, or degree structures beyond classical recursion theory (hyperdegrees). |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Degree theory as global structure for unsolvability; jump hierarchy as stratification of computational power; uniform reducibility as unifying method; recursion theory grounding reducibility; diagonalization tying together incompleteness, separation, and hierarchy formation. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Links to recursion theory, computability, descriptive set theory (pointclass reducibility), complexity theory (reductions), algorithmic randomness (lowness/highness), and model theory (oracles as relativized structures). |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Manipulating reducibility types (≤ₜ, ≤ₘ, ≤{tt}, ≤{wtt}), altering oracle availability, varying encoding strategies, modifying priority-order schemes in constructions, adjusting approximation schedules, and testing degree relationships under structured transformations. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing reducibility behavior without intervention: tracking oracle-query patterns, monitoring convergence of approximations, detecting when reductions stabilize, recording injury-free or injury-prone behavior, and passively examining reducibility chains. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing whether A ≤ₜ B holds via simulation; testing completeness by reducing known hard sets to a candidate; testing incomparability with diagonalization; validating minimal-degree or minimal-pair constructions; checking jump relations (A <ₜ A′). |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Re-running priority constructions under identical requirement sequences; replicating reducibility simulations with independent encodings; replaying oracle computations; reproducing jump computations; verifying stabilization across multiple runs. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Measuring injury frequency distribution, estimating stabilization rates of approximations, comparing reducibility-step counts, analyzing degree-density patterns, evaluating convergence or divergence tendencies across sampled sets. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing reducibility notions (Turing vs. many-one vs. truth-table); comparing classical vs. oracle-enhanced reductions; comparing finite-injury vs. infinite-injury constructions; comparing degree-structure predictions from different recursion-theoretic frameworks. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Identifying misimplemented reductions, miscounted oracle calls, incorrect detection of convergence, encoding errors, incorrect requirement satisfaction, flawed diagonalization steps, and misclassified degree relations. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding arbitrary encoding choices that artificially simplify reductions; controlling for priority-order bias; removing bias in sampling r.e. sets; ensuring even-handed selection of reducibility types; avoiding confirmation bias in incomparability claims. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Reviewing reducibility proofs, auditing oracle-machine simulations, examining priority constructions, comparing independent degree computations, evaluating correctness of diagonalization arguments, and challenging claims of completeness or incomparability. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Revising reducibility definitions where needed, updating priority frameworks, repairing constructions producing inconsistent degrees, modifying encoding schemes, integrating new theorems on degree structure, and refining jump hierarchy formulations. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Releasing full reducibility scripts, oracle definitions, encoding functions, approximation logs, priority-order specifications, and diagonalization details; stating all assumptions clearly. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Accurate reporting of reducibility successes/failures, avoiding hidden assumptions, ensuring reproducibility of degree constructions, disclosing limitations of methods, and maintaining rigor in claims about unsolvability hierarchies. |