| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies the structural features of proofs and derivations independent of specific logical connectives; includes sequent structure, structural rules, cut-elimination, normalization; excludes purely semantic truth definitions or model-theoretic validity except where needed to justify structural results. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates at the level of formal derivations, inference-rule schemas, proof trees, sequent configurations, context manipulation, and structural transformations. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Sequents, contexts, structural rules (exchange, weakening, contraction), proof trees, derivations, cut steps, substitutions, structural transformations, inference schemas. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Structural positions, admissibility, derivability, proof height, proof width, normalization behavior, context sensitivity, permutation behavior of rules. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Sequent structures, structural-rule families, proof transformations, analytic vs. non-analytic steps, cut vs. cut-free derivations, normal vs. non-normal forms. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Active sequent, context configuration (Γ, Δ), presence/absence of structural rules, proof depth, branching structure, number of cut occurrences, rule-permutation state. |
| | Parameterization | How variables encode and represent the system’s state. | Representation via structural sequent formats (e.g., Γ ⊢ Δ), context-combinator rules, structural-rule specifications, permutation schemas, cut-rank measures, height and width metrics. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Idealizing contexts as multisets or sequences; treating structural rules as independent; assuming subformula property; ignoring resource sensitivity in fully structural logics; restricting to cut-free or analytic proofs. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Idealizations break down in substructural logics (linear, relevant, ordered), systems without contraction/weakening, modal calculi with non-local rules, logics lacking global normalization. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Proofs are discrete symbolic objects; derivations are finitary; structural rules determine core proof behavior; transformations preserve derivability; normalization is meaningful. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes rule schemas are well-formed; contexts are manipulable under structural constraints; cut-elimination or normalization is desirable or foundational; proof identity depends on structural invariants. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Structural rules cannot trivialize derivability; rule combinations must avoid collapse of logical distinctions; transformations must preserve correctness. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires alignment among sequent structures, structural rules, cut-elimination behavior, permutation principles, and the meta-theoretic framework governing admissibility and normalization. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Sequent transformations, rule applications, context rearrangements, structural rule effects (exchange, weakening, contraction), cut steps and their eliminations, derivation shapes, normalization sequences. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Limited by proof-search decidability, ability to compute cut-elimination, structural complexity (e.g., large contexts), and the computational cost of checking admissibility of structural rules. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Proof height, proof width, number of cut occurrences, count of structural-rule applications, normalization length, permutation depth, complexity class of derivability. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Automated proof checkers, sequent-calculus provers, structural rule analyzers, normalization engines, theorem provers (Coq, Lean, Isabelle), cut-elimination calculators, proof-graph tools. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Derivability defined by explicit structural-rule sequences; cut defined as a structural inference; cut-rank defined by formula complexity; normalization defined by elimination of non-analytic steps. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Running structural normalization, applying permutation conversions, collapsing contexts, verifying admissibility, checking cut elimination, generating sequent proofs, tracking proof metrics. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Standardized normalization runs, canonical sequent-construction procedures, controlled permutation experiments, systematic cut-elimination computations, benchmark derivation families. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Choosing representative sequents, typical derivation patterns, minimal and maximal proof forms, subsets with or without structural rules, normalized vs. non-normalized derivations. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Sequent derivations, context-structured proofs, normalization traces, permutation graphs, cut-elimination sequences, structural-rule application logs, proof trees. |
| | Resolution | The granularity or precision with which data is captured. | Determined by granularity of sequent encoding, specificity of structural-rule tracking, detail level in normalization traces, and the precision of cut-elimination steps. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Verifying correct implementation of structural rules, checking validity of permutation conversions, validating normalization algorithms, checking consistency of cut-rank computations. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Misapplied structural rules, incorrect context handling, failed normalization, non-terminating transformations, mistaken admissibility assessments, implementation errors in proof assistants. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Structural rule behavior (exchange, weakening, contraction), permutation conversions, cut–reduction laws, normalization relations, subformula constraints, analytic proof behavior. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Context invariance under exchange, preservation of derivability under structural permutations, cut-rank monotonicity, subformula property (in analytic systems), invariance of proof identity under rule permutations. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Structural-rule operations driving proof transformation, cut-elimination processes, permutation of rules producing normalization, context manipulation mechanisms, absorption and elimination of structural steps. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Normalization pathways (cut → reduced form → normal form), structural-rule chains (weakening → contraction → exchange variants), sequent-structure evolution, ordered permutations generating analytic derivations. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Sequents, contexts, structural rules, cut, cut-rank, normalization, permutation, analyticity, admissibility, proof height/width, proof identity, structural invariants. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Classical vs. intuitionistic systems, structural vs. substructural logics, sequent calculi (LK, LJ), calculi with/without structural rules, analytic calculi, deep inference systems, display calculi. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Cut-reduction equalities, permutation equations (e.g., commuting conversions), structural reflection principles, normal-form characterizations, equality of derivations modulo permutation. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Sequent-calculus proof trees, structural-rule transition graphs, normalization models, cut-free proof frameworks, deep-inference derivation structures, context-combinator models. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Analytic calculi with subformula property, cut-free systems, systems with reduced or idealized structural rules, simplified context operators, canonical normalized derivations. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Breakdowns in substructural logics (linear, relevant, affine), systems where contraction/weakening are disallowed, modal calculi with non-local rules, non-normalizing logics, calculi without global cut-elimination. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Cut-elimination as a unifying structural principle, Gentzen-style proof transformation theory, connection to Curry–Howard (structural correspondence), general proof-theoretic semantics. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Links to type theory, lambda calculus, category theory (e.g., monoidal categories for structural behavior), automated reasoning, computational complexity, structural semantics of programming languages. |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Manipulating structural rules (adding/removing contraction, weakening, exchange), altering sequent formats, restricting or enabling cut, modifying context-combinators to test effects on derivability and normalization. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing normalization behavior, monitoring cut-elimination steps, tracking structural-rule permutations, analyzing proof height/width changes, examining sequent evolution without altering the underlying calculus. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing admissibility of structural rules, testing whether cut-elimination holds, verifying normalization, determining analyticity, checking whether permutation conversions preserve derivability. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Reproducing derivations across different structural calculi, independently verifying cut-elimination, replicating normalization sequences, confirming structural-rule behaviors in multiple proof assistants. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Analyzing complexity of normalization, frequency of structural-rule usage, distribution of cut ranks, empirical behavior of proof-search algorithms under structural constraints. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing calculi by normalization strength, cut-elimination power, analytic vs. non-analytic derivations, structural-rule sensitivity, proof-size bounds, computational complexity. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Identifying incorrect context handling, misapplied structural rules, invalid permutations, incorrect cut reductions, failed normalization sequences, and implementation flaws in structural proof engines. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding heuristic bias in rule ordering, ensuring canonical derivations, controlling implementation-dependent structural transformations, and standardizing normalization strategies. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Cross-checking structural transformations, reviewing admissibility arguments, comparing normalization proofs, validating sequent configurations, and meta-theoretic critique of rule sets. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Updating structural rules, reformulating sequent formats, refining normalization procedures, strengthening or weakening structural assumptions, adjusting calculi to restore cut-elimination or analyticity. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Full disclosure of structural rule sets, sequent formats, normalization steps, permutation conversions, cut-elimination proofs, and implementation details of automated proof tools. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Ensuring accurate reporting of structural transformations, avoiding hidden assumptions in rule definitions, maintaining reproducible normalization workflows, and clearly documenting system limitations. |