{
  "content": {
    "icon": "bi-signpost-split",
    "id": "the-mathematics",
    "intro": "The predicates are typed by these math types, each listed with its contracts and the domains of mathematics it draws on.",
    "subsections": [
      {
        "blocks": [
          {
            "entries": [
              {
                "description": "What members exist?",
                "term": "Question"
              },
              {
                "description": "[set-theory](/records/reasoning/math-domain-set-theory)",
                "term": "Fields"
              },
              {
                "description": "membership · cardinality · emptiness",
                "term": "Predicate family"
              },
              {
                "description": "set | boolean | ratio",
                "term": "Yields shape"
              },
              {
                "description": "[Evidence-Before-Generation](/records/algorithms/evidence-before-generation), [Capability Profile](/records/algorithms/capability-profile), [Scope Extraction](/records/algorithms/scope-extraction), [Domain Knowledge Base](/records/algorithms/domain-knowledge-base), [Existing Pattern Extraction](/records/algorithms/existing-pattern-extraction), [DSL Compliance Loading](/records/algorithms/dsl-compliance-loading), [Workspace Configuration Discovery](/records/algorithms/workspace-configuration-discovery), [Smell Taxonomy](/records/algorithms/smell-taxonomy), [Responsibility Boundary](/records/algorithms/responsibility-boundary), [Canonical Data](/records/algorithms/canonical-data), [Self-Description Manifest](/records/algorithms/self-description-manifest), [Runtime Discovery](/records/algorithms/runtime-discovery), [Canonical Semantics](/records/algorithms/canonical-semantics), [Self-Description and Discovery](/records/algorithms/self-description-and-discovery), [Concept Cluster Extraction](/records/algorithms/concept-cluster-extraction), [Capability Degradation](/records/algorithms/capability-degradation), [Automation Opportunity Detection](/records/algorithms/automation-opportunity-detection), [Capability Disclosure](/records/algorithms/capability-disclosure), [Detection Registry](/records/algorithms/detection-registry), [Orientation Stage](/records/algorithms/orientation-stage), [Authoritative Source Loading](/records/algorithms/authoritative-source-loading), [Skeptical Context Acquisition](/records/algorithms/skeptical-context-acquisition), [Explicit Termination](/records/algorithms/explicit-termination), [Context Initialization](/records/algorithms/context-initialization), [Early Success Exit](/records/algorithms/early-success-exit), [Trust Anchor Declaration](/records/algorithms/trust-anchor-declaration), [Duplicate Disposition Walk](/records/algorithms/duplicate-disposition-walk), [Lifetime Resolution](/records/algorithms/lifetime-resolution), [Coverage Workspace](/records/algorithms/coverage-workspace), [Surface Grid Walk](/records/algorithms/surface-grid-walk), [PAG Document Declaration](/records/algorithms/pag-document-declaration), [PAG Keyword Ontology](/records/algorithms/pag-keyword-ontology), [Analysis Workspace](/records/algorithms/analysis-workspace), [Registry Baseline](/records/algorithms/registry-baseline), [Semantic Domain Partitioning](/records/algorithms/semantic-domain-partitioning), [Cross-Class Pattern Detection](/records/algorithms/cross-class-pattern-detection), [Temporal Coupling Detection](/records/algorithms/temporal-coupling-detection), [Profile Compose](/records/algorithms/profile-compose), [Seed Composition](/records/algorithms/seed-composition), [Token Source-of-Truth](/records/algorithms/token-source-of-truth), [Custom Type Registration](/records/algorithms/custom-type-registration), [Taxonomy Jurisdiction](/records/algorithms/taxonomy-jurisdiction)",
                "term": "Contracts"
              }
            ],
            "kind": "glossary"
          },
          {
            "kind": "text",
            "text": "How it is checked"
          },
          {
            "entries": [
              {
                "description": "the yields check, which holds every algorithm's yields and every node's answer shape within the math type's shapes, the ontology resolution gate over every record that names the math type",
                "term": "Checked by"
              },
              {
                "description": "Every algorithm record, node, axis and substrate node that names the math type",
                "term": "Population"
              },
              {
                "description": "A verdict stands until the math type's shapes or a record that names it changes",
                "term": "Freshness"
              },
              {
                "description": "The gate fails on a yield or answer shape outside the type's shapes, or a reference to a type that does not exist",
                "term": "Refusal"
              },
              {
                "description": "None, because a math type is a family of answers, and nothing observes it while a run executes",
                "term": "Observation"
              },
              {
                "description": "Watched to fire and to accept: one suite plants a node with a dangling math type, another plants an answer shape outside the type's shapes, and the bundled data validates clean",
                "term": "Evidence"
              },
              {
                "description": "The math type's shapes, which every yield and answer shape conforms to",
                "term": "Authoritative side"
              },
              {
                "description": "Not answered",
                "term": "Depends on"
              },
              {
                "description": "Not answered",
                "term": "Shape it refuses"
              }
            ],
            "kind": "glossary"
          }
        ],
        "id": "reasoning-math-type-set-theory",
        "title": "set-theory"
      },
      {
        "blocks": [
          {
            "entries": [
              {
                "description": "Does it hold, and what follows from it?",
                "term": "Question"
              },
              {
                "description": "[logic](/records/reasoning/math-domain-logic)",
                "term": "Fields"
              },
              {
                "description": "boolean predicate",
                "term": "Predicate family"
              },
              {
                "description": "boolean",
                "term": "Yields shape"
              },
              {
                "description": "[Semantic Operation Boundary](/records/algorithms/semantic-operation-boundary), [Domain Cache Validation](/records/algorithms/domain-cache-validation), [Principle Extraction](/records/algorithms/principle-extraction), [Phase Validation Requirement](/records/algorithms/phase-validation-requirement), [Validation Strategy Composition](/records/algorithms/validation-strategy-composition), [Semantic Compliance Validation](/records/algorithms/semantic-compliance-validation), [Algorithmic Embodiment Validation](/records/algorithms/algorithmic-embodiment-validation), [Agent Generation Completion](/records/algorithms/agent-generation-completion), [Hybrid Workflow Orchestration](/records/algorithms/hybrid-workflow-orchestration), [Context Forking Configuration](/records/algorithms/context-forking-configuration), [Verb-Based Execution Classification](/records/algorithms/verb-based-execution-classification), [Workflow Type Document Selection](/records/algorithms/workflow-type-document-selection), [Handoff Signal](/records/algorithms/handoff-signal), [Orchestrator Action](/records/algorithms/orchestrator-action), [Workflow Principles Mapping](/records/algorithms/workflow-principles-mapping), [Workflow Validation Gate](/records/algorithms/workflow-validation-gate), [Anti-Pattern Inversion](/records/algorithms/anti-pattern-inversion), [Architecture Smell Record](/records/algorithms/architecture-smell-record), [Document Truth Alignment](/records/algorithms/document-truth-alignment), [Interface Contract](/records/algorithms/interface-contract), [Substitutability](/records/algorithms/substitutability), [Extension Point](/records/algorithms/extension-point), [Behavioral Dispatch](/records/algorithms/behavioral-dispatch), [Port Adapter](/records/algorithms/port-adapter), [Transaction Boundary](/records/algorithms/transaction-boundary), [Idempotent Side Effect](/records/algorithms/idempotent-side-effect), [Deterministic Core](/records/algorithms/deterministic-core), [Error Boundary](/records/algorithms/error-boundary), [Cache Correctness](/records/algorithms/cache-correctness), [Security Policy](/records/algorithms/security-policy), [Control Plane](/records/algorithms/control-plane), [Universal Architectural Concern Template](/records/algorithms/universal-architectural-concern-template), [Consumer Config SSOT](/records/algorithms/consumer-config-ssot), [Architectural Force Classification](/records/algorithms/architectural-force-classification), [Conflict and Tension Resolution](/records/algorithms/conflict-and-tension-resolution), [Violation Detection](/records/algorithms/violation-detection), [Enforcement Gate](/records/algorithms/enforcement-gate), [Modular Boundary Compliance](/records/algorithms/modular-boundary-compliance), [Contract Compatibility](/records/algorithms/contract-compatibility), [Runtime Extensibility](/records/algorithms/runtime-extensibility), [Event and Messaging Consistency](/records/algorithms/event-and-messaging-consistency), [State and Transaction Safety](/records/algorithms/state-and-transaction-safety), [Correctness Verification](/records/algorithms/correctness-verification), [Resilience Policy](/records/algorithms/resilience-policy), [Security Governance](/records/algorithms/security-governance), [Control Plane Coordination](/records/algorithms/control-plane-coordination), [Metaprogramming Safety](/records/algorithms/metaprogramming-safety), [Model Lifecycle Governance](/records/algorithms/model-lifecycle-governance), [Relationship Schema Validation](/records/algorithms/relationship-schema-validation), [Constraints Over Shortcuts](/records/algorithms/no-shortcuts), [Forward Compatibility Over Backward Compatibility](/records/algorithms/no-backward-compat), [Fail-Fast Over Fallback](/records/algorithms/no-fallback), [Explicit Removal Over Deprecation](/records/algorithms/no-deprecation), [Greenfield Over Legacy](/records/algorithms/no-legacy), [Single-Path Determinism Over Dual-Path](/records/algorithms/no-dual-path), [Immediacy Over Deferring](/records/algorithms/no-deferring), [Mandatory Over Optional](/records/algorithms/no-optional), [Now Over For-Now](/records/algorithms/no-for-now), [Observed Execution Over Unobserved](/records/algorithms/no-unobserved), [Compression Over Repetition](/records/algorithms/no-uncompressed), [Approved Evolution Over Unapproved](/records/algorithms/no-unapproved), [Enforced Feedback Over Ignored](/records/algorithms/no-ignored-feedback), [Single Owner Over Shared Ownership](/records/algorithms/no-shared-ownership), [Bounded Lifetime Over Unbounded](/records/algorithms/no-unbounded), [Enforced Symmetry Over Asymmetric Lifecycle](/records/algorithms/no-asymmetric), [Explicit Retention Over Implicit](/records/algorithms/no-implicit-retention), [Structural Release Over Discipline](/records/algorithms/no-discipline-release), [Immutable Data Over Mutable State](/records/algorithms/no-mutable), [Errors As Language Over Silent Errors](/records/algorithms/no-silent), [Explicit Invalidity Over Hidden](/records/algorithms/no-hidden-invalidity), [Event Emission Over Parent Callbacks](/records/algorithms/no-callbacks), [Monotonic Growth Over Retraction](/records/algorithms/no-retraction), [Semantic Addressing Over Location Addressing](/records/algorithms/no-location), [Ordinal Time Over Timestamps](/records/algorithms/no-timestamps), [Homoiconicity Over Separation](/records/algorithms/no-separation), [Bounded Complexity Over Unlimited](/records/algorithms/no-unlimited), [Computed Health Over Metric Health](/records/algorithms/no-metrics), [Secret Store Over Hardcoded Secrets](/records/algorithms/no-hardcoded-secrets), [Boundary Validation Over Unvalidated Input](/records/algorithms/no-unvalidated-input), [Least Privilege Over Broad Privilege](/records/algorithms/no-broad-privilege), [Config Externalization Over Env Fallback](/records/algorithms/no-env-fallback), [Profile-First Over Unmeasured Optimization](/records/algorithms/no-unmeasured-optimization), [Rule As Code Over Convention](/records/algorithms/no-convention-enforcement), [Design By Contract Over Implicit Contract](/records/algorithms/no-implicit-contract), [Versioned Evolution Over Breaking Change](/records/algorithms/no-breaking-change), [Schema-Validated Boundary Over Untyped](/records/algorithms/no-untyped-boundary), [Atomic Boundary Over Partial Commit](/records/algorithms/no-partial-commit), [Saga Compensation Over Distributed 2PC](/records/algorithms/no-distributed-2pc), [Async Events Over Synchronous Cross-Boundary](/records/algorithms/no-sync-cross-boundary), [Observable Signals Over Opaque Runtime](/records/algorithms/no-opaque-runtime), [Injected Dependency Over Hidden](/records/algorithms/no-hidden-dependency), [Convention Discovery Over Hardcoded Wiring](/records/algorithms/no-hardcoded-wiring), [Declarative Config Over Imperative](/records/algorithms/no-imperative-config), [Anti-Corruption Layer Over Cross-Context Leak](/records/algorithms/no-leaky-context), [Injected Nondeterminism Over Hidden](/records/algorithms/no-hidden-nondeterminism), [Pattern By Fit Over Speculative Pattern](/records/algorithms/no-speculative-pattern), [Runtime-Neutral Automation Boundary](/records/algorithms/runtime-neutral-automation-boundary), [Intentional Static Separation](/records/algorithms/intentional-static-separation), [Extension Interface Discovery](/records/algorithms/extension-interface-discovery), [Performance-Aware Discovery Design](/records/algorithms/performance-aware-discovery-design), [Dynamic Extension Architecture](/records/algorithms/dynamic-extension-architecture), [Cache Invalidation Strategy](/records/algorithms/cache-invalidation-strategy), [Dynamic Failure Isolation](/records/algorithms/dynamic-failure-isolation), [Measured-vs-Estimated Validation](/records/algorithms/measured-vs-estimated-validation), [Architecture Validation Before Persistence](/records/algorithms/architecture-validation-before-persistence), [Automation Completion Status](/records/algorithms/automation-completion-status), [Runtime-Agnostic Adapter Boundary](/records/algorithms/runtime-agnostic-adapter-boundary), [Pattern Classification](/records/algorithms/pattern-classification), [Refactor Intent Classification](/records/algorithms/refactor-intent-classification), [Architecture Compliance Targeting](/records/algorithms/architecture-compliance-targeting), [Existing Solution Conflict](/records/algorithms/existing-solution-conflict), [Additive Debt Gate](/records/algorithms/additive-debt-gate), [Pattern-Specific Validation](/records/algorithms/pattern-specific-validation), [Zero-Duplication Verification](/records/algorithms/zero-duplication-verification), [Completion Truthfulness](/records/algorithms/completion-truthfulness), [Trust Anchor](/records/algorithms/trust-anchor), [Intent & Directionality Normalization](/records/algorithms/intent-directionality-normalization), [Planning Stage](/records/algorithms/planning-stage), [Principle Activation](/records/algorithms/principle-activation), [Protocol Semantic Selection](/records/algorithms/protocol-semantic-selection), [Loop Class Labeling](/records/algorithms/loop-class-labeling), [Codebase Pattern Enforcement](/records/algorithms/codebase-pattern-enforcement), [Validator Coverage](/records/algorithms/validator-coverage), [Structured Observability Context](/records/algorithms/structured-observability-context), [Cross-Cutting Surface Coverage](/records/algorithms/cross-cutting-surface-coverage), [Legacy Elimination](/records/algorithms/legacy-elimination), [Validation Stage](/records/algorithms/validation-stage), [Semantic Debt Policy](/records/algorithms/semantic-debt-policy), [Evidence-Based Claim Verification](/records/algorithms/evidence-based-claim-verification), [Validation Suite Battery](/records/algorithms/validation-suite-battery), [Verification Execution](/records/algorithms/verification-execution), [Violation Classification](/records/algorithms/violation-classification), [Reverification Gate](/records/algorithms/reverification-gate), [Evidence-Gated Claim Verification](/records/algorithms/evidence-gated-claim-verification), [Environment Capability Verification](/records/algorithms/environment-capability-verification), [Behavioral Self-Test](/records/algorithms/behavioral-self-test), [Adversarial Input Testing](/records/algorithms/adversarial-input-testing), [Safe Arithmetic Contract](/records/algorithms/safe-arithmetic-contract), [Invocation Join](/records/algorithms/invocation-join), [Uncovered Gap Derivation](/records/algorithms/uncovered-gap-derivation), [Evidence Verdict](/records/algorithms/evidence-verdict), [Coverage Completion](/records/algorithms/coverage-completion), [PAG Handoff Gate](/records/algorithms/pag-validation-gate), [PAG Explicit Control Flow](/records/algorithms/pag-explicit-control-flow), [PAG Ambiguity Reduction](/records/algorithms/pag-ambiguity-reduction), [PAG Well-Formedness Validation](/records/algorithms/pag-well-formedness-validation), [Anti-Pattern Classification](/records/algorithms/anti-pattern-classification), [Abstraction Boundary Principle](/records/algorithms/abstraction-boundary-principle), [Base-Class Candidate Selection](/records/algorithms/base-class-candidate-selection), [Anti-Pattern Elimination Verification](/records/algorithms/anti-pattern-elimination-verification), [Pattern Distillation Completion Truthfulness](/records/algorithms/pattern-distillation-completion-truthfulness), [Delta Capture](/records/algorithms/delta-capture), [Plan Phase Verification](/records/algorithms/plan-phase-verification), [Canonical Config Resolution](/records/algorithms/canonical-config-resolution), [Machine Verdict Derivation](/records/algorithms/machine-verdict-derivation), [Mode Contract Validation](/records/algorithms/mode-contract-validation), [Type-Keyed Appearance](/records/algorithms/type-keyed-appearance), [Concern Classification](/records/algorithms/concern-classification), [Discovery Verification](/records/algorithms/discovery-verification), [Taxonomy Completion](/records/algorithms/taxonomy-completion), [Container Ladder](/records/algorithms/container-ladder), [Export Triage Ladder](/records/algorithms/export-triage-ladder), [Alignment Cadence](/records/algorithms/alignment-cadence)",
                "term": "Contracts"
              }
            ],
            "kind": "glossary"
          },
          {
            "kind": "text",
            "text": "How it is checked"
          },
          {
            "entries": [
              {
                "description": "the yields check, which holds every algorithm's yields and every node's answer shape within the math type's shapes, the ontology resolution gate over every record that names the math type",
                "term": "Checked by"
              },
              {
                "description": "Every algorithm record, node, axis and substrate node that names the math type",
                "term": "Population"
              },
              {
                "description": "A verdict stands until the math type's shapes or a record that names it changes",
                "term": "Freshness"
              },
              {
                "description": "The gate fails on a yield or answer shape outside the type's shapes, or a reference to a type that does not exist",
                "term": "Refusal"
              },
              {
                "description": "None, because a math type is a family of answers, and nothing observes it while a run executes",
                "term": "Observation"
              },
              {
                "description": "Watched to fire and to accept: one suite plants a node with a dangling math type, another plants an answer shape outside the type's shapes, and the bundled data validates clean",
                "term": "Evidence"
              },
              {
                "description": "The math type's shapes, which every yield and answer shape conforms to",
                "term": "Authoritative side"
              },
              {
                "description": "Not answered",
                "term": "Depends on"
              },
              {
                "description": "Not answered",
                "term": "Shape it refuses"
              }
            ],
            "kind": "glossary"
          }
        ],
        "id": "reasoning-math-type-logic",
        "title": "logic"
      },
      {
        "blocks": [
          {
            "entries": [
              {
                "description": "What connects what?",
                "term": "Question"
              },
              {
                "description": "[category-theory](/records/reasoning/math-domain-category-theory)",
                "term": "Fields"
              },
              {
                "description": "reachability · coverage · degree",
                "term": "Predicate family"
              },
              {
                "description": "edge-list",
                "term": "Yields shape"
              },
              {
                "description": "[Four-Dimensional Agent Graph](/records/algorithms/four-dimensional-agent-graph), [Anti-Pattern Relationship Record](/records/algorithms/anti-pattern-relationship-record), [Coupling Control](/records/algorithms/coupling-control), [Observability Trace](/records/algorithms/observability-trace), [Causality Ordering](/records/algorithms/causality-ordering), [Finite State Machine](/records/algorithms/finite-state-machine), [Statecharts](/records/algorithms/statecharts), [Petri Nets](/records/algorithms/petri-nets), [Architectural Relationship Record](/records/algorithms/architectural-relationship-record), [Architecture Knowledge Graph](/records/algorithms/architecture-knowledge-graph), [Dependency Closure](/records/algorithms/dependency-closure), [Reinforcement Propagation](/records/algorithms/reinforcement-propagation), [Observability and Auditability](/records/algorithms/observability-and-auditability), [Migration Action Mapping](/records/algorithms/migration-action-mapping), [Phase Decomposition](/records/algorithms/phase-decomposition), [Four-Dimensional Phase Graph](/records/algorithms/four-dimensional-phase-graph), [Dependency Linearization](/records/algorithms/dependency-linearization), [Ripple Chain Analysis](/records/algorithms/ripple-chain-analysis), [PAG Node Decomposition](/records/algorithms/pag-node-decomposition), [PAG Structure Declaration](/records/algorithms/pag-coordination-construct), [Relational Graph Duplication](/records/algorithms/relational-graph-duplication), [Causal Wiring Duplication](/records/algorithms/causal-wiring-duplication), [Stage Ordering](/records/algorithms/stage-ordering), [Path Role Walk](/records/algorithms/path-role-walk)",
                "term": "Contracts"
              }
            ],
            "kind": "glossary"
          },
          {
            "kind": "text",
            "text": "How it is checked"
          },
          {
            "entries": [
              {
                "description": "the yields check, which holds every algorithm's yields and every node's answer shape within the math type's shapes, the ontology resolution gate over every record that names the math type",
                "term": "Checked by"
              },
              {
                "description": "Every algorithm record, node, axis and substrate node that names the math type",
                "term": "Population"
              },
              {
                "description": "A verdict stands until the math type's shapes or a record that names it changes",
                "term": "Freshness"
              },
              {
                "description": "The gate fails on a yield or answer shape outside the type's shapes, or a reference to a type that does not exist",
                "term": "Refusal"
              },
              {
                "description": "None, because a math type is a family of answers, and nothing observes it while a run executes",
                "term": "Observation"
              },
              {
                "description": "Watched to fire and to accept: one suite plants a node with a dangling math type, another plants an answer shape outside the type's shapes, and the bundled data validates clean",
                "term": "Evidence"
              },
              {
                "description": "The math type's shapes, which every yield and answer shape conforms to",
                "term": "Authoritative side"
              },
              {
                "description": "Not answered",
                "term": "Depends on"
              },
              {
                "description": "Not answered",
                "term": "Shape it refuses"
              }
            ],
            "kind": "glossary"
          }
        ],
        "id": "reasoning-math-type-graph",
        "title": "graph"
      },
      {
        "blocks": [
          {
            "entries": [
              {
                "description": "How are parts arranged under laws?",
                "term": "Question"
              },
              {
                "description": "[algebra](/records/reasoning/math-domain-algebra)",
                "term": "Fields"
              },
              {
                "description": "composition · ordering invariant",
                "term": "Predicate family"
              },
              {
                "description": "ordered-structure",
                "term": "Yields shape"
              },
              {
                "description": "[Agent Sequence Definition](/records/algorithms/agent-sequence-definition), [Anti-Pattern Remediation Algebra](/records/algorithms/anti-pattern-remediation-algebra), [Structural Mediation](/records/algorithms/structural-mediation), [Architecture Refactoring Roadmap](/records/algorithms/architecture-refactoring-roadmap), [Atomic Refactor Phase](/records/algorithms/atomic-refactor-phase), [Hierarchical Numbering](/records/algorithms/hierarchical-numbering), [Technique and Invariant Selection](/records/algorithms/technique-invariant-selection), [Sequential Chain Duplication](/records/algorithms/sequential-chain-duplication), [Concrete-vs-Abstract Responsibility Split](/records/algorithms/concrete-vs-abstract-responsibility-split), [Template Method Lifecycle](/records/algorithms/template-method-lifecycle), [Migration Ordering](/records/algorithms/migration-ordering), [Cascade Layer Partition](/records/algorithms/cascade-layer-partition), [Assembly Composition](/records/algorithms/assembly-composition), [Name Projection](/records/algorithms/name-projection), [Dialect Resolution](/records/algorithms/dialect-resolution)",
                "term": "Contracts"
              }
            ],
            "kind": "glossary"
          },
          {
            "kind": "text",
            "text": "How it is checked"
          },
          {
            "entries": [
              {
                "description": "the yields check, which holds every algorithm's yields and every node's answer shape within the math type's shapes, the ontology resolution gate over every record that names the math type",
                "term": "Checked by"
              },
              {
                "description": "Every algorithm record, node, axis and substrate node that names the math type",
                "term": "Population"
              },
              {
                "description": "A verdict stands until the math type's shapes or a record that names it changes",
                "term": "Freshness"
              },
              {
                "description": "The gate fails on a yield or answer shape outside the type's shapes, or a reference to a type that does not exist",
                "term": "Refusal"
              },
              {
                "description": "None, because a math type is a family of answers, and nothing observes it while a run executes",
                "term": "Observation"
              },
              {
                "description": "Watched to fire and to accept: one suite plants a node with a dangling math type, another plants an answer shape outside the type's shapes, and the bundled data validates clean",
                "term": "Evidence"
              },
              {
                "description": "The math type's shapes, which every yield and answer shape conforms to",
                "term": "Authoritative side"
              },
              {
                "description": "Not answered",
                "term": "Depends on"
              },
              {
                "description": "Not answered",
                "term": "Shape it refuses"
              }
            ],
            "kind": "glossary"
          }
        ],
        "id": "reasoning-math-type-algebra",
        "title": "algebra"
      },
      {
        "blocks": [
          {
            "entries": [
              {
                "description": "How do states vary?",
                "term": "Question"
              },
              {
                "description": "[analysis](/records/reasoning/math-domain-analysis)",
                "term": "Fields"
              },
              {
                "description": "operation applied to state",
                "term": "Predicate family"
              },
              {
                "description": "operation",
                "term": "Yields shape"
              },
              {
                "description": "[Non-Destructive Domain Investigation](/records/algorithms/non-destructive-domain-investigation), [Risk Complexity Reversibility](/records/algorithms/risk-complexity-reversibility), [Event Messaging](/records/algorithms/event-messaging), [Streaming Dataflow](/records/algorithms/streaming-dataflow), [Measurement Normalization](/records/algorithms/measurement-normalization), [Breaking Point Calculation](/records/algorithms/breaking-point-calculation), [Scalability Projection](/records/algorithms/scalability-projection), [Behavioral Signature Extraction](/records/algorithms/behavioral-signature-extraction)",
                "term": "Contracts"
              }
            ],
            "kind": "glossary"
          },
          {
            "kind": "text",
            "text": "How it is checked"
          },
          {
            "entries": [
              {
                "description": "the yields check, which holds every algorithm's yields and every node's answer shape within the math type's shapes, the ontology resolution gate over every record that names the math type",
                "term": "Checked by"
              },
              {
                "description": "Every algorithm record, node, axis and substrate node that names the math type",
                "term": "Population"
              },
              {
                "description": "A verdict stands until the math type's shapes or a record that names it changes",
                "term": "Freshness"
              },
              {
                "description": "The gate fails on a yield or answer shape outside the type's shapes, or a reference to a type that does not exist",
                "term": "Refusal"
              },
              {
                "description": "None, because a math type is a family of answers, and nothing observes it while a run executes",
                "term": "Observation"
              },
              {
                "description": "Watched to fire and to accept: one suite plants a node with a dangling math type, another plants an answer shape outside the type's shapes, and the bundled data validates clean",
                "term": "Evidence"
              },
              {
                "description": "The math type's shapes, which every yield and answer shape conforms to",
                "term": "Authoritative side"
              },
              {
                "description": "Not answered",
                "term": "Depends on"
              },
              {
                "description": "Not answered",
                "term": "Shape it refuses"
              }
            ],
            "kind": "glossary"
          }
        ],
        "id": "reasoning-math-type-analysis",
        "title": "analysis"
      },
      {
        "blocks": [
          {
            "entries": [
              {
                "description": "What is admissible or best?",
                "term": "Question"
              },
              {
                "description": "validation · threshold · argmax",
                "term": "Predicate family"
              },
              {
                "description": "boolean | ranking | number",
                "term": "Yields shape"
              },
              {
                "description": "[Creation History Collision](/records/algorithms/creation-history-collision), [Adaptive Phase Boundary](/records/algorithms/adaptive-phase-boundary), [Replacement Safety](/records/algorithms/replacement-safety), [First-Time Initiation](/records/algorithms/first-time-initiation), [Architectural Style Boundary](/records/algorithms/architectural-style-boundary), [Verification Fitness](/records/algorithms/verification-fitness), [Performance Scaling](/records/algorithms/performance-scaling), [Architecture Selection Meta-Algorithm](/records/algorithms/architecture-selection-meta-algorithm), [Severity Policy](/records/algorithms/severity-policy), [Refactor Selection](/records/algorithms/refactor-selection), [Architecture Assessment](/records/algorithms/architecture-assessment), [Pattern Selection](/records/algorithms/pattern-selection), [Performance and Scalability](/records/algorithms/performance-and-scalability), [Architecture Evolution Governance](/records/algorithms/architecture-evolution-governance), [Architectural Recommendation](/records/algorithms/architectural-recommendation), [Architecture Decision Support](/records/algorithms/architecture-decision-support), [Static-to-Dynamic Readiness](/records/algorithms/static-to-dynamic-readiness), [Automation Operation Mode](/records/algorithms/automation-operation-mode), [Automation Priority Ordering](/records/algorithms/automation-priority-ordering), [Operation Mode Gating](/records/algorithms/operation-mode-gating), [Canonical Variation Selection](/records/algorithms/canonical-variation-selection), [Developer Decision Gate](/records/algorithms/developer-decision-gate), [Teleological Intent Gate](/records/algorithms/teleological-intent-gate), [Severity Assignment](/records/algorithms/severity-assignment), [Admissibility Constraint Gate](/records/algorithms/admissibility-constraint-stage), [Severity Failure Routing](/records/algorithms/severity-failure-routing), [Severity-Ordered Remediation](/records/algorithms/severity-ordered-remediation), [Phase-Separated Execution](/records/algorithms/phase-separated-execution), [Validation Gate](/records/algorithms/validation-gate), [Coverage Risk Prioritization](/records/algorithms/coverage-risk-prioritization), [PAG Invariant Record](/records/algorithms/pag-constraint-boundary), [Anti-Pattern Priority Matrix](/records/algorithms/anti-pattern-priority-matrix), [Boundary Reconciliation](/records/algorithms/boundary-reconciliation), [Phase Close Gate](/records/algorithms/phase-close-gate), [Bounded Cascade Termination](/records/algorithms/bounded-cascade-termination), [Layer Fitness Enforcement](/records/algorithms/layer-fitness-enforcement), [Reshape Risk Priority](/records/algorithms/reshape-risk-priority), [Vocabulary Admission Gate](/records/algorithms/vocabulary-admission-gate)",
                "term": "Contracts"
              }
            ],
            "kind": "glossary"
          },
          {
            "kind": "text",
            "text": "How it is checked"
          },
          {
            "entries": [
              {
                "description": "the yields check, which holds every algorithm's yields and every node's answer shape within the math type's shapes, the ontology resolution gate over every record that names the math type",
                "term": "Checked by"
              },
              {
                "description": "Every algorithm record, node, axis and substrate node that names the math type",
                "term": "Population"
              },
              {
                "description": "A verdict stands until the math type's shapes or a record that names it changes",
                "term": "Freshness"
              },
              {
                "description": "The gate fails on a yield or answer shape outside the type's shapes, or a reference to a type that does not exist",
                "term": "Refusal"
              },
              {
                "description": "None, because a math type is a family of answers, and nothing observes it while a run executes",
                "term": "Observation"
              },
              {
                "description": "Watched to fire and to accept: one suite plants a node with a dangling math type, another plants an answer shape outside the type's shapes, and the bundled data validates clean",
                "term": "Evidence"
              },
              {
                "description": "The math type's shapes, which every yield and answer shape conforms to",
                "term": "Authoritative side"
              },
              {
                "description": "Not answered",
                "term": "Depends on"
              },
              {
                "description": "Not answered",
                "term": "Shape it refuses"
              }
            ],
            "kind": "glossary"
          }
        ],
        "id": "reasoning-math-type-optimization",
        "title": "optimization"
      },
      {
        "blocks": [
          {
            "entries": [
              {
                "description": "What is preserved under change?",
                "term": "Question"
              },
              {
                "description": "[topology](/records/reasoning/math-domain-topology)",
                "term": "Fields"
              },
              {
                "description": "invariant / symmetry",
                "term": "Predicate family"
              },
              {
                "description": "boolean",
                "term": "Yields shape"
              },
              {
                "description": "[Domain Boundary](/records/algorithms/domain-boundary), [Portability Environment](/records/algorithms/portability-environment), [Fractal Scale Duplication](/records/algorithms/fractal-scale-duplication), [Placement Isolation](/records/algorithms/placement-isolation)",
                "term": "Contracts"
              }
            ],
            "kind": "glossary"
          },
          {
            "kind": "text",
            "text": "How it is checked"
          },
          {
            "entries": [
              {
                "description": "the yields check, which holds every algorithm's yields and every node's answer shape within the math type's shapes, the ontology resolution gate over every record that names the math type",
                "term": "Checked by"
              },
              {
                "description": "Every algorithm record, node, axis and substrate node that names the math type",
                "term": "Population"
              },
              {
                "description": "A verdict stands until the math type's shapes or a record that names it changes",
                "term": "Freshness"
              },
              {
                "description": "The gate fails on a yield or answer shape outside the type's shapes, or a reference to a type that does not exist",
                "term": "Refusal"
              },
              {
                "description": "None, because a math type is a family of answers, and nothing observes it while a run executes",
                "term": "Observation"
              },
              {
                "description": "Watched to fire and to accept: one suite plants a node with a dangling math type, another plants an answer shape outside the type's shapes, and the bundled data validates clean",
                "term": "Evidence"
              },
              {
                "description": "The math type's shapes, which every yield and answer shape conforms to",
                "term": "Authoritative side"
              },
              {
                "description": "Not answered",
                "term": "Depends on"
              },
              {
                "description": "Not answered",
                "term": "Shape it refuses"
              }
            ],
            "kind": "glossary"
          }
        ],
        "id": "reasoning-math-type-topology",
        "title": "topology"
      },
      {
        "blocks": [
          {
            "entries": [
              {
                "description": "How certain or likely is it?",
                "term": "Question"
              },
              {
                "description": "[probability](/records/reasoning/math-domain-probability)",
                "term": "Fields"
              },
              {
                "description": "confidence · distribution · counter",
                "term": "Predicate family"
              },
              {
                "description": "number[0,1]",
                "term": "Yields shape"
              },
              {
                "description": "[Knowledge Documentation Relevance](/records/algorithms/knowledge-documentation-relevance), [Evidence Grounding Validation](/records/algorithms/evidence-grounding-validation), [RAG Knowledge Boundary](/records/algorithms/rag-knowledge-boundary), [Queuing Theory](/records/algorithms/queuing-theory), [Convention Strength Analysis](/records/algorithms/convention-strength-analysis), [Research Guidance](/records/algorithms/research-guidance), [Validation Score](/records/algorithms/validation-score), [Tool Calibration](/records/algorithms/tool-calibration), [Recursive Self-Verification](/records/algorithms/recursive-self-verification), [Compliance Gap](/records/algorithms/compliance-gap), [Behavioral Inconsistency](/records/algorithms/behavioral-inconsistency), [Anomaly Outlier Detection](/records/algorithms/anomaly-outlier-detection), [Distillation Metrics](/records/algorithms/distillation-metrics)",
                "term": "Contracts"
              }
            ],
            "kind": "glossary"
          },
          {
            "kind": "text",
            "text": "How it is checked"
          },
          {
            "entries": [
              {
                "description": "the yields check, which holds every algorithm's yields and every node's answer shape within the math type's shapes, the ontology resolution gate over every record that names the math type",
                "term": "Checked by"
              },
              {
                "description": "Every algorithm record, node, axis and substrate node that names the math type",
                "term": "Population"
              },
              {
                "description": "A verdict stands until the math type's shapes or a record that names it changes",
                "term": "Freshness"
              },
              {
                "description": "The gate fails on a yield or answer shape outside the type's shapes, or a reference to a type that does not exist",
                "term": "Refusal"
              },
              {
                "description": "None, because a math type is a family of answers, and nothing observes it while a run executes",
                "term": "Observation"
              },
              {
                "description": "Watched to fire and to accept: one suite plants a node with a dangling math type, another plants an answer shape outside the type's shapes, and the bundled data validates clean",
                "term": "Evidence"
              },
              {
                "description": "The math type's shapes, which every yield and answer shape conforms to",
                "term": "Authoritative side"
              },
              {
                "description": "Not answered",
                "term": "Depends on"
              },
              {
                "description": "Not answered",
                "term": "Shape it refuses"
              }
            ],
            "kind": "glossary"
          }
        ],
        "id": "reasoning-math-type-probability",
        "title": "probability"
      },
      {
        "blocks": [
          {
            "entries": [
              {
                "description": "What is novel, and how compressible is it?",
                "term": "Question"
              },
              {
                "description": "[information-theory](/records/reasoning/math-domain-information-theory)",
                "term": "Fields"
              },
              {
                "description": "novelty · dedup · hash",
                "term": "Predicate family"
              },
              {
                "description": "hash | novelty-score | boolean",
                "term": "Yields shape"
              },
              {
                "description": "[Audit Artifact](/records/algorithms/audit-artifact), [Final Generation Report](/records/algorithms/final-generation-report), [Template Assembly](/records/algorithms/template-assembly), [Automation Session Report](/records/algorithms/automation-session-report), [Centralization Report](/records/algorithms/centralization-report), [Rendering Stage](/records/algorithms/rendering-stage), [Checklist Output Rendering](/records/algorithms/checklist-output-rendering), [Partial Success Reporting](/records/algorithms/partial-success-reporting), [Completion Report](/records/algorithms/completion-report), [Investigation Report](/records/algorithms/investigation-report), [Action Log](/records/algorithms/action-log), [Coverage Ledger](/records/algorithms/coverage-ledger), [Conceptual Duplication Detection](/records/algorithms/conceptual-duplication-detection), [Pattern Distillation History](/records/algorithms/pattern-distillation-history), [Version Provenance](/records/algorithms/version-provenance), [Living Plan State](/records/algorithms/living-plan-state), [Versioned Turn Provenance](/records/algorithms/versioned-turn-provenance), [Taxonomy Ledger](/records/algorithms/taxonomy-ledger)",
                "term": "Contracts"
              }
            ],
            "kind": "glossary"
          },
          {
            "kind": "text",
            "text": "How it is checked"
          },
          {
            "entries": [
              {
                "description": "the yields check, which holds every algorithm's yields and every node's answer shape within the math type's shapes, the ontology resolution gate over every record that names the math type",
                "term": "Checked by"
              },
              {
                "description": "Every algorithm record, node, axis and substrate node that names the math type",
                "term": "Population"
              },
              {
                "description": "A verdict stands until the math type's shapes or a record that names it changes",
                "term": "Freshness"
              },
              {
                "description": "The gate fails on a yield or answer shape outside the type's shapes, or a reference to a type that does not exist",
                "term": "Refusal"
              },
              {
                "description": "None, because a math type is a family of answers, and nothing observes it while a run executes",
                "term": "Observation"
              },
              {
                "description": "Watched to fire and to accept: one suite plants a node with a dangling math type, another plants an answer shape outside the type's shapes, and the bundled data validates clean",
                "term": "Evidence"
              },
              {
                "description": "The math type's shapes, which every yield and answer shape conforms to",
                "term": "Authoritative side"
              },
              {
                "description": "Not answered",
                "term": "Depends on"
              },
              {
                "description": "Not answered",
                "term": "Shape it refuses"
              }
            ],
            "kind": "glossary"
          }
        ],
        "id": "reasoning-math-type-information-theory",
        "title": "information-theory"
      },
      {
        "blocks": [
          {
            "entries": [
              {
                "description": "What procedure produces it?",
                "term": "Question"
              },
              {
                "description": "[computation](/records/reasoning/math-domain-computation)",
                "term": "Fields"
              },
              {
                "description": "algorithm / loop",
                "term": "Predicate family"
              },
              {
                "description": "procedure",
                "term": "Yields shape"
              },
              {
                "description": "[Portable Contract Composition](/records/algorithms/portable-contract-composition), [Adapter Rendering](/records/algorithms/adapter-rendering), [Agent Creator Kernel](/records/algorithms/agent-creator-kernel), [Agent Workflow File Modification Recovery](/records/algorithms/agent-workflow-file-modification-recovery), [Shared Document Workspace](/records/algorithms/shared-document-workspace), [Agent Document Responsibility](/records/algorithms/agent-document-responsibility), [Agent Activation Invocation](/records/algorithms/agent-activation-invocation), [Parallel Batch Execution](/records/algorithms/parallel-batch-execution), [Sequential Agent Execution](/records/algorithms/sequential-agent-execution), [Workflow Coordination Sequence](/records/algorithms/workflow-coordination-sequence), [Checklist Integration](/records/algorithms/checklist-integration), [Phase Documentation Template](/records/algorithms/phase-documentation-template), [Capability Invocation Protocol](/records/algorithms/capability-invocation-protocol), [Workflow Creation Kernel](/records/algorithms/workflow-creation-kernel), [Anti-Pattern Rule Compiler](/records/algorithms/anti-pattern-rule-compiler), [Architectural Contract Kernel](/records/algorithms/architectural-contract-kernel), [Construction Boundary](/records/algorithms/construction-boundary), [Saga Compensation](/records/algorithms/saga-compensation), [Declarative Metaprogramming](/records/algorithms/declarative-metaprogramming), [Manifest-Driven Documentation](/records/algorithms/manifest-driven-documentation), [Architecture Fitness Function Generation](/records/algorithms/architecture-fitness-function-generation), [Architecture Catalog Compiler](/records/algorithms/architecture-catalog-compiler), [Master Architecture Governance Kernel](/records/algorithms/master-architecture-governance-kernel), [Centralized Reference Resolver](/records/algorithms/centralized-reference-resolver), [Manual Fallback Preservation](/records/algorithms/manual-fallback-preservation), [Entry Point Migration](/records/algorithms/entry-point-migration), [Knowledge Capture](/records/algorithms/knowledge-capture), [Automation Kernel](/records/algorithms/automation-kernel), [Replacement Refactor](/records/algorithms/replacement-refactor), [Rollback-Centered Execution](/records/algorithms/rollback-centered-execution), [Centralization Kernel](/records/algorithms/centralization-kernel), [Dynamic Discovery Pattern Generation](/records/algorithms/dynamic-discovery-pattern-generation), [Compilation Stage](/records/algorithms/compilation-stage), [Verb Template Binding](/records/algorithms/verb-template-binding), [Task Atomization](/records/algorithms/task-atomization), [Checklist Creation Kernel](/records/algorithms/checklist-creation-kernel), [File-Scoped Fix](/records/algorithms/file-scoped-fix), [File Limit Remediation](/records/algorithms/file-limit-remediation), [Import Boundary Remediation](/records/algorithms/import-boundary-remediation), [Naming Convention Remediation](/records/algorithms/naming-convention-remediation), [Base-Class Compliance Remediation](/records/algorithms/base-class-compliance-remediation), [CSS Token Remediation](/records/algorithms/css-token-remediation), [DOM Factory Remediation](/records/algorithms/dom-factory-remediation), [Console Usage Remediation](/records/algorithms/console-usage-remediation), [Lifecycle Symmetry Remediation](/records/algorithms/lifecycle-symmetry-remediation), [Stylelint Post-Fix](/records/algorithms/stylelint-post-fix), [Codebase Verification Kernel](/records/algorithms/codebase-verification-kernel), [File Modification Recovery](/records/algorithms/file-modification-recovery), [Defensive String Normalization](/records/algorithms/defensive-string-normalization), [Advanced Tool Escalation](/records/algorithms/advanced-tool-escalation), [Contract-Based Verification Kernel](/records/algorithms/contract-based-verification-kernel), [Convergence Walk](/records/algorithms/convergence-walk), [Test Authoring](/records/algorithms/test-authoring), [Test Coverage Kernel](/records/algorithms/test-coverage-kernel), [PAG Semantic Operation](/records/algorithms/pag-tool-invocation), [PAG Authoring Kernel](/records/algorithms/pag-authoring-kernel), [Base Schematic Composition](/records/algorithms/base-schematic-composition), [Backup-Verified Migration](/records/algorithms/backup-verified-migration), [Registry Regeneration](/records/algorithms/registry-regeneration), [Anti-Reintroduction Gate](/records/algorithms/anti-reintroduction-gate), [Pattern Distiller Kernel](/records/algorithms/pattern-distiller-kernel), [Idempotent Merge](/records/algorithms/idempotent-merge), [Deterministic Merge Core](/records/algorithms/deterministic-merge-core), [Persistence Fork](/records/algorithms/persistence-fork), [Living Profile Kernel](/records/algorithms/living-profile-kernel), [Governed Autonomous Plan Loop](/records/algorithms/governed-autonomous-plan-loop), [Quality Governance Loop](/records/algorithms/quality-governance-loop), [Comment Normalization Remediation](/records/algorithms/comment-normalization-remediation), [Custom-Rule Derivation](/records/algorithms/custom-rule-derivation), [Composed Turn Contract](/records/algorithms/composed-turn-contract), [Loop-Owned Mode Selection](/records/algorithms/loop-owned-mode-selection), [Governed Construction Boundary](/records/algorithms/governed-construction-boundary), [Type-Migration Centralization](/records/algorithms/type-migration-centralization), [Container Reshape](/records/algorithms/container-reshape), [Taxonomy Kernel](/records/algorithms/taxonomy-kernel)",
                "term": "Contracts"
              }
            ],
            "kind": "glossary"
          },
          {
            "kind": "text",
            "text": "How it is checked"
          },
          {
            "entries": [
              {
                "description": "the yields check, which holds every algorithm's yields and every node's answer shape within the math type's shapes, the ontology resolution gate over every record that names the math type",
                "term": "Checked by"
              },
              {
                "description": "Every algorithm record, node, axis and substrate node that names the math type",
                "term": "Population"
              },
              {
                "description": "A verdict stands until the math type's shapes or a record that names it changes",
                "term": "Freshness"
              },
              {
                "description": "The gate fails on a yield or answer shape outside the type's shapes, or a reference to a type that does not exist",
                "term": "Refusal"
              },
              {
                "description": "None, because a math type is a family of answers, and nothing observes it while a run executes",
                "term": "Observation"
              },
              {
                "description": "Watched to fire and to accept: one suite plants a node with a dangling math type, another plants an answer shape outside the type's shapes, and the bundled data validates clean",
                "term": "Evidence"
              },
              {
                "description": "The math type's shapes, which every yield and answer shape conforms to",
                "term": "Authoritative side"
              },
              {
                "description": "Not answered",
                "term": "Depends on"
              },
              {
                "description": "Not answered",
                "term": "Shape it refuses"
              }
            ],
            "kind": "glossary"
          }
        ],
        "id": "reasoning-math-type-computation",
        "title": "computation"
      },
      {
        "blocks": [
          {
            "entries": [
              {
                "description": "What stable pattern arises, and has it converged?",
                "term": "Question"
              },
              {
                "description": "fixed-point · convergence · oscillation",
                "term": "Predicate family"
              },
              {
                "description": "boolean | counter",
                "term": "Yields shape"
              },
              {
                "description": "[Workflow Recovery Loop](/records/algorithms/workflow-recovery-loop), [Anti-Pattern Propagation Kernel](/records/algorithms/anti-pattern-propagation-kernel), [Resilience Control](/records/algorithms/resilience-control), [Recovery Deployment](/records/algorithms/recovery-deployment), [Iterative Variation Discovery](/records/algorithms/iterative-variation-discovery), [Repair Stage](/records/algorithms/repair-stage), [Bounded Repair Loop](/records/algorithms/bounded-repair-loop), [Verification Loop](/records/algorithms/verification-loop), [Iteration Bound](/records/algorithms/iteration-bound), [Recursion Control](/records/algorithms/recursion-control)",
                "term": "Contracts"
              }
            ],
            "kind": "glossary"
          },
          {
            "kind": "text",
            "text": "How it is checked"
          },
          {
            "entries": [
              {
                "description": "the yields check, which holds every algorithm's yields and every node's answer shape within the math type's shapes, the ontology resolution gate over every record that names the math type",
                "term": "Checked by"
              },
              {
                "description": "Every algorithm record, node, axis and substrate node that names the math type",
                "term": "Population"
              },
              {
                "description": "A verdict stands until the math type's shapes or a record that names it changes",
                "term": "Freshness"
              },
              {
                "description": "The gate fails on a yield or answer shape outside the type's shapes, or a reference to a type that does not exist",
                "term": "Refusal"
              },
              {
                "description": "None, because a math type is a family of answers, and nothing observes it while a run executes",
                "term": "Observation"
              },
              {
                "description": "Watched to fire and to accept: one suite plants a node with a dangling math type, another plants an answer shape outside the type's shapes, and the bundled data validates clean",
                "term": "Evidence"
              },
              {
                "description": "The math type's shapes, which every yield and answer shape conforms to",
                "term": "Authoritative side"
              },
              {
                "description": "Not answered",
                "term": "Depends on"
              },
              {
                "description": "Not answered",
                "term": "Shape it refuses"
              }
            ],
            "kind": "glossary"
          }
        ],
        "id": "reasoning-math-type-dynamical-systems",
        "title": "dynamical-systems"
      },
      {
        "blocks": [
          {
            "entries": [
              {
                "description": "quantity",
                "term": "Studies"
              },
              {
                "description": "What can be counted or measured?",
                "term": "Question"
              }
            ],
            "kind": "glossary"
          },
          {
            "kind": "text",
            "text": "How it is checked"
          },
          {
            "entries": [
              {
                "description": "the ontology resolution gate over the math types and foundations that name the domain",
                "term": "Checked by"
              },
              {
                "description": "Every math type and foundation layer that names the domain",
                "term": "Population"
              },
              {
                "description": "A verdict stands until the domain list or a record that names a domain changes",
                "term": "Freshness"
              },
              {
                "description": "The gate fails on a reference to a domain that does not exist",
                "term": "Refusal"
              },
              {
                "description": "None, because a math domain is a field of study, and nothing observes it while a run executes",
                "term": "Observation"
              },
              {
                "description": "Watched to fire and to accept: a suite plants a math type that names a missing domain, and the bundled references resolve",
                "term": "Evidence"
              },
              {
                "description": "The domain list, which every math type and foundation that names a domain cites",
                "term": "Authoritative side"
              },
              {
                "description": "Not answered",
                "term": "Depends on"
              },
              {
                "description": "Not answered",
                "term": "Shape it refuses"
              }
            ],
            "kind": "glossary"
          }
        ],
        "id": "reasoning-math-domain-number",
        "title": "number"
      },
      {
        "blocks": [
          {
            "entries": [
              {
                "description": "operations and structures",
                "term": "Studies"
              },
              {
                "description": "What rules govern transformations?",
                "term": "Question"
              }
            ],
            "kind": "glossary"
          },
          {
            "kind": "text",
            "text": "How it is checked"
          },
          {
            "entries": [
              {
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  "summary": "The predicates are typed by these math types, each listed with its contracts and the domains of mathematics it draws on.",
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