{ "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-12T10:55:11Z", "providers": [ "Codex" ], "breadth": "recalled by Codex without web access - no source was read", "missingProviders": [], "pass": 2, "cost": { "grok": { "seconds": 20.5, "error": "Reading additional input from stdin...\nOpenAI Codex v0.153.4\n--------\nworkdir: R:\\02_PROJECTS\\02_Meta_Models_Platforms\\Ver.cy\\current\\thing-registry-backlog\nmodel: gpt-6-astra\nprovider: openai\napproval: never\nsandbox: read-only\nreasoning effort: none\nreasoning summaries: none\nsession id: 01a09540-bcff-7e80-a2b0-5030d63d5bf3\n--------\nuser\nDescribe what is already known about one registered thing. Answer as JSON only, no prose around it.\n\nThing: Lebesgue-Stieltjes integration\nSense to describe: (none recorded)\nDomain code: XCT.REL\nAlso known as: (none)\n\n\nContext for this batch of 1001 things:\n# ", "usd": 0.0, "recall": true }, "codex": { "seconds": 69.1, "error": "Reading additional input from stdin...\nOpenAI Codex v0.153.4\n--------\nworkdir: R:\\02_PROJECTS\\02_Meta_Models_Platforms\\Ver.cy\\current\\thing-registry-backlog\nmodel: gpt-6-astra\nprovider: openai\napproval: never\nsandbox: read-only\nreasoning effort: none\nreasoning summaries: none\nsession id: 01a09540-bcfd-7b90-9c78-f2c5cfd01a3b\n--------\nuser\nYou are drafting a Vercy meta-model for one registered thing. Answer as JSON only, no prose around it.\n\nThing: Lebesgue-Stieltjes integration\nRegistry id: vr.tr.lebesgue-stieltjes-integration\nPlane / domain: XCT / XCT.REL\nRegistry definition: (none recorded)\nN" } }, "enrichment": { "pass": 3, "engine": "claude", "producedAt": "2026-10-06T12:00:00Z", "ingestedAt": "2026-10-06T08:46:56+00:00", "fields": [ "agent_conduct", "ethics", "failure_modes_and_hazards", "owners" ] }, "gate": { "passed": true, "version": "1.0", "at": "2026-10-07T10:10:30+00:00", "evidence": [ "completeness 1.00", "sense: Wikidata description", "2 live sources", "review by codex: pass" ] } }, "metaModel": { "id": "THING-Q3799278", "registryId": "vr.tr.lebesgue-stieltjes-integration", "name": "Lebesgue–Stieltjes integration", "version": "0.1.0-research.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.REL" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q3799278/", "model": { "registry_id": "vr.tr.lebesgue-stieltjes-integration", "name": "Lebesgue–Stieltjes integration", "purpose": "Enable an agent to recognise a Lebesgue-Stieltjes integral, record its generating function and conventions, and determine whether its evaluation or transformation is justified.", "definition": "Lebesgue-Stieltjes integration is Lebesgue integration with respect to a measure induced by a monotone function, with extensions to signed or complex measures induced by functions of bounded variation.", "scope_statement": "This model owns integration against measures generated by scalar monotone functions on the real line, with explicitly qualified extensions to signed or complex measures generated by functions of bounded variation.", "in_scope": [ "Construction of the measure associated with a generating function and a stated continuity convention", "Measurability, integrability and the status of a proposed integral", "Atomic, absolutely continuous and singular continuous contributions", "Endpoint conventions, restrictions of the integration domain and unbounded domains", "Conditions supporting convergence, integration by parts and comparison with Riemann-Stieltjes integration", "Probability expectations expressed as integration against a distribution function" ], "out_of_scope": [ "General Lebesgue integration independent of a Stieltjes generating function", "Riemann-Stieltjes integration as a separate theory", "Itô, Stratonovich and other stochastic integration theories", "Distributional derivatives and generalized functions beyond their relationship to the associated measure", "Vector-valued integration and operator-valued measures", "Multivariate Stieltjes constructions" ], "distinguishing_features": [ "The integrator determines a countably additive measure, and the integral is defined through Lebesgue integration against that measure.", "For a finite-valued, nondecreasing, right-continuous generator F, the convention μ_F((a,b]) = F(b) − F(a) fixes interval masses.", "A jump at x contributes an atom of mass F(x) − F(x−), so an integrand's value at that point can affect the integral.", "Replacing dF by F'(x) dx is sufficient only when the relevant measure is absolutely continuous; jumps and singular continuous mass require additional treatment.", "Existence does not depend on convergence of Riemann-Stieltjes partition sums, although the two integrals agree under suitable hypotheses." ], "characteristics": [ { "name": "Generating-function class", "kind": "category", "unit_or_values": "Nondecreasing real-valued; locally bounded variation; bounded variation; other explicitly specified extension", "why_it_matters": "Determines whether the construction yields a positive measure or requires signed or complex measure machinery and additional finiteness qualifications." }, { "name": "Continuity and interval convention", "kind": "category", "unit_or_values": "Right-continuous with (a,b]; left-continuous with [a,b); explicitly documented alternative", "why_it_matters": "Fixes how jumps are assigned to points and prevents incompatible endpoint formulas." }, { "name": "Associated measure", "kind": "relation", "unit_or_values": "Generator linked to a Borel measure and, where used, its completion", "why_it_matters": "Identifies the actual integration object and the measurable sets on which it is defined." }, { "name": "Integration domain", "kind": "category", "unit_or_values": "Specified Borel set; interval with explicit endpoints; whole real line", "why_it_matters": "Controls included atoms, local versus global assumptions and tail integrability." }, { "name": "Integral status", "kind": "state", "unit_or_values": "Not yet established; finite; positive infinity; negative infinity; undefined because both positive and negative parts diverge; inadmissible under the stated framework", "why_it_matters": "Separates meaningful extended integrals from undefined expressions and unsupported manipulations." }, { "name": "Absolute-integrability quantity", "kind": "measurement", "unit_or_values": "∫|f| dμ_F for positive measures, or ∫|f| d|μ_F| for signed or complex measures; value in [0,∞]", "why_it_matters": "Tests finite integrability and supports bounds and many interchange operations." }, { "name": "Measure-component profile", "kind": "category", "unit_or_values": "Atomic; absolutely continuous; singular continuous; mixture", "why_it_matters": "Determines whether evaluation requires a sum, a density integral, a singular contribution or a combination." }, { "name": "Generator equivalence", "kind": "relation", "unit_or_values": "Generators differing by an additive constant under the same interval convention generate the same measure", "why_it_matters": "Distinguishes a change of normalization from a change in the integral." } ], "affordances": [ "Construct or identify the measure generated by an admissible integrator.", "Check whether an integrand is measurable and whether its integral is finite, extended or undefined.", "Evaluate contributions from point masses and densities while retaining any singular continuous remainder.", "Translate a probability expectation into an integral against a cumulative distribution function.", "Approve or reject limit exchanges and integration-by-parts steps by checking their hypotheses.", "Compare alternative endpoint conventions or Riemann-Stieltjes formulations and explain any discrepancy." ], "agent_conduct": { "may": [ "Explain the theory with correct definitions, stating its assumptions.", "Check computations and proofs, showing each step." ], "must_not": [ "Present results that need conditions, such as bounded variation, as holding without them.", "Attribute theorems to the wrong mathematicians.", "Present a faulty proof as complete.", "Help students cheat in graded work." ], "requires_human": [ "Using results in financial or engineering models that affect people." ] }, "ethics": { "considerations": [ "Mathematical errors in applied models can mislead decisions about money and risk.", "Honest teaching respects learners' own work." ], "affected_parties": [ "Students", "Researchers", "People affected by applied models" ] }, "owners": { "steward": "Nobody owns mathematics.", "master_systems": [] }, "failure_modes_and_hazards": [ "Misapplied theorems in probability and finance models." ], "what_it_is_for": [ "Lebesgue-Stieltjes integration is used in measure theory, probability and analysis to integrate with respect to measures generated by suitable functions.", "In probability, it expresses expectations using a distribution function while accommodating discrete, continuous and mixed distributions.", "Its use requires explicit conventions for the generating function and interval endpoints when jumps contribute mass." ], "facets_not_applicable": { "interfaces": "No governing technical standard, law or official classification is required for this mathematical construction.", "recognition": "This abstract mathematical construction has no physical appearance that uniquely identifies it." } }, "sources": [ { "id": "wikidata", "title": "Wikidata item Q3799278: Lebesgue–Stieltjes integration", "url": "https://www.wikidata.org/wiki/Q3799278", "url_status": "live", "what_it_supports": "identity and sense of the item", "checked_by": "wikidata-api" }, { "id": "wikipedia-en", "title": "Wikipedia: Lebesgue–Stieltjes integration", "url": "https://en.wikipedia.org/wiki/Lebesgue%E2%80%93Stieltjes_integration", "url_status": "live", "what_it_supports": "general description of the item", "checked_by": "wikidata-api sitelink" } ], "structure": { "bundles": [ { "id": "generator-to-measure", "name": "Generator to measure", "description": "Record how the stated integrator determines the measure used by the integral.", "rationale": "The symbol dF does not by itself settle continuity conventions, admissibility or measure type.", "layers": [ { "id": "generator-admissibility", "name": "Generator admissibility", "description": "Establish the function class and locality of the construction.", "findings": [ { "id": "monotonicity-and-variation", "name": "Monotonicity and variation", "description": "Record whether F generates a positive measure or requires a qualified signed or complex extension.", "questions": [ { "text": "Is F finite-valued and nondecreasing, or is the construction based on bounded variation?", "kind": "definition", "id": "monotonicity-and-variation-q01" }, { "text": "Are the required variation and finiteness assumptions local to the integration interval or valid on the entire domain?", "kind": "boundary", "id": "monotonicity-and-variation-q02" } ] } ] }, { "id": "measure-convention", "name": "Measure convention", "description": "Specify the continuity representative, interval increments and measurable space.", "findings": [ { "id": "interval-increments", "name": "Interval increments", "description": "Make the correspondence between increments of F and interval masses explicit.", "questions": [ { "text": "Does the construction use right-continuous F with μ_F((a,b]) = F(b) − F(a), or another stated convention?", "kind": "definition", "id": "interval-increments-q01" }, { "text": "Which reference fixes this convention, and is integration performed on the Borel σ-algebra or its completion?", "kind": "provenance", "id": "interval-increments-q02" } ] } ] } ] }, { "id": "integrand-and-existence", "name": "Integrand and existence", "description": "Determine whether the proposed integral is meaningful and what kind of value it can have.", "rationale": "An integral expression must be distinguished from a justified finite or extended integral.", "layers": [ { "id": "measurability-and-representatives", "name": "Measurability and representatives", "description": "Check admissible integrands and the null sets governing equivalence.", "findings": [ { "id": "measure-relative-integrand", "name": "Measure-relative integrand", "description": "Assess measurability and equality almost everywhere with respect to the associated measure.", "questions": [ { "text": "Is f measurable for the chosen measurable space and measure completion?", "kind": "definition", "id": "measure-relative-integrand-q01" }, { "text": "Would changing f on a Lebesgue-null set alter its values on a set with positive μ_F mass?", "kind": "boundary", "id": "measure-relative-integrand-q02" } ] } ] }, { "id": "integrability-status", "name": "Integrability status", "description": "Record the tests distinguishing finite, extended and undefined integrals.", "findings": [ { "id": "positive-negative-and-absolute-parts", "name": "Positive, negative and absolute parts", "description": "Use the appropriate positive-part or total-variation tests for the selected measure framework.", "questions": [ { "text": "For a positive μ_F and real f, what are ∫f⁺ dμ_F and ∫f⁻ dμ_F, and do both diverge?", "kind": "measurement", "id": "positive-negative-and-absolute-parts-q01" }, { "text": "For a finite signed or complex measure, is ∫|f| d|μ_F| finite?", "kind": "measurement", "id": "positive-negative-and-absolute-parts-q02" } ] } ] } ] }, { "id": "mass-components-and-boundaries", "name": "Mass components and boundaries", "description": "Locate the mass that contributes to evaluation and specify which boundary points are included.", "rationale": "Ignoring jumps, singular mass or endpoint membership can change the value even when derivative-based calculations appear correct.", "layers": [ { "id": "atomic-and-endpoint-mass", "name": "Atomic and endpoint mass", "description": "Record jump contributions and the set denoted by an interval integral.", "findings": [ { "id": "jump-contributions", "name": "Jump contributions", "description": "Under the right-continuous convention, an included atom x contributes f(x)[F(x) − F(x−)].", "questions": [ { "text": "Which included points have nonzero jump mass, and what values of f are assigned there?", "kind": "measurement", "id": "jump-contributions-q01" }, { "text": "Does the notation ∫ from a to b denote integration over (a,b], [a,b], [a,b) or another explicitly defined set?", "kind": "boundary", "id": "jump-contributions-q02" } ] } ] }, { "id": "continuous-mass-components", "name": "Continuous mass components", "description": "Separate density-based mass from continuous mass singular to Lebesgue measure.", "findings": [ { "id": "density-and-singular-remainder", "name": "Density and singular remainder", "description": "Record the absolutely continuous contribution and any singular continuous remainder.", "questions": [ { "text": "What is the density of the absolutely continuous component with respect to Lebesgue measure?", "kind": "measurement", "id": "density-and-singular-remainder-q01" }, { "text": "What evidence establishes that a singular continuous component is absent or must be retained?", "kind": "boundary", "id": "density-and-singular-remainder-q02" } ] } ] } ] }, { "id": "valid-transformations", "name": "Valid transformations", "description": "Track hypotheses permitting limits and algebraic transformations of the integral.", "rationale": "Useful symbolic operations require conditions sensitive to the measure, integrand and jump conventions.", "layers": [ { "id": "limits-under-integration", "name": "Limits under integration", "description": "Justify convergence steps against a specified associated measure.", "findings": [ { "id": "convergence-hypotheses", "name": "Convergence hypotheses", "description": "Attach the relevant theorem and verified assumptions to a proposed interchange.", "questions": [ { "text": "For a fixed positive μ_F, do the integrands satisfy monotone convergence or dominated convergence hypotheses?", "kind": "action", "id": "convergence-hypotheses-q01" }, { "text": "If the generators or measures also vary, what separate convergence result justifies the claimed limit?", "kind": "boundary", "id": "convergence-hypotheses-q02" } ] } ] }, { "id": "integration-by-parts", "name": "Integration by parts", "description": "Control boundary terms and simultaneous jumps in product-based transformations.", "findings": [ { "id": "jump-aware-product-rule", "name": "Jump-aware product rule", "description": "Record regularity assumptions, one-sided evaluations and any required jump correction.", "questions": [ { "text": "Which integration-by-parts identity applies to the stated regularity and endpoint conventions?", "kind": "action", "id": "jump-aware-product-rule-q01" }, { "text": "Do the functions share jumps, and how does the selected identity account for their jump products?", "kind": "boundary", "id": "jump-aware-product-rule-q02" } ] } ] } ] }, { "id": "interpretations-and-neighbours", "name": "Interpretations and neighbours", "description": "Connect the construction to probability and related integration theories without conflating their assumptions.", "rationale": "The same notation appears in settings with different existence criteria and normalizations.", "layers": [ { "id": "probability-interpretation", "name": "Probability interpretation", "description": "Identify when the generator is a cumulative distribution function and the integral represents an expectation.", "findings": [ { "id": "expectation-against-a-law", "name": "Expectation against a law", "description": "Link a distribution function F_X to the law of X and the corresponding integral of g.", "questions": [ { "text": "Does F satisfy the monotonicity, right-continuity and limits at infinity required of a cumulative distribution function?", "kind": "definition", "id": "expectation-against-a-law-q01" }, { "text": "Under what integrability conditions does the proposed expectation E[g(X)] equal ∫g(x) dF_X(x)?", "kind": "action", "id": "expectation-against-a-law-q02" } ] } ] }, { "id": "comparison-with-riemann-stieltjes", "name": "Comparison with Riemann-Stieltjes integration", "description": "Identify the definition in use and justify any claimed agreement between theories.", "findings": [ { "id": "definition-and-agreement", "name": "Definition and agreement", "description": "Distinguish a measure-based integral from a partition-sum definition and document conditions for coincidence.", "questions": [ { "text": "Does the cited treatment define the integral through a generated measure or through tagged partition sums?", "kind": "provenance", "id": "definition-and-agreement-q01" }, { "text": "Which hypotheses guarantee agreement for this integrand and generator, and are their endpoint conventions compatible?", "kind": "boundary", "id": "definition-and-agreement-q02" } ] } ] } ] }, { "id": "evidence-and-external-alignment", "name": "Evidence and external alignment", "description": "What the world already says about this thing, gathered so the model can be checked against it.", "rationale": "A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.", "layers": [ { "id": "reported-evidence", "name": "Reported evidence", "description": "Findings from the breadth pass, kept separate from the structural claims.", "findings": [ { "id": "evidence-confidence-notes", "name": "Check these first", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ "A common convention uses a right-continuous nondecreasing F and defines μF((a,b]) = F(b) − F(a); left-continuous formulations use different interval conventions.", "Bounded variation on compact intervals supports local constructions; statements about integration on an unbounded domain require additional finiteness or integrability qualifications.", "Terminology varies over whether signed and complex extensions are included under the unqualified name; these are differences of scope and convention rather than competing mathematical theories." ], "questions": [ { "id": "evidence-confidence-notes-q01", "text": "Which of these check these first hold for the sense of Lebesgue-Stieltjes integration this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-kinds", "name": "Kinds and varieties", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ "Integration against a positive measure induced by a nondecreasing function", "Integration against a signed measure induced by a real function of bounded variation", "Integration against a complex measure induced by a complex function of bounded variation", "Absolutely continuous, discrete, and singular continuous cases" ], "questions": [ { "id": "evidence-kinds-q01", "text": "Which of these kinds and varieties hold for the sense of Lebesgue-Stieltjes integration this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-real-world-use", "name": "Real-world use", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ "Computing expectations and probabilities directly from cumulative distribution functions, including mixed discrete and continuous distributions", "Expressing weighted integrals that combine continuous contributions with point masses", "Representing linear functionals on spaces of continuous functions", "Studying functions of bounded variation and their distributional derivatives", "Integrating scalar functions against measures obtained from spectral resolutions of operators" ], "questions": [ { "id": "evidence-real-world-use-q01", "text": "Which of these real-world use hold for the sense of Lebesgue-Stieltjes integration this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-failure-modes-and-hazards", "name": "Failure modes and hazards", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ "Replacing dF by F'(x) dx loses jump and singular continuous contributions unless the induced measure is absolutely continuous.", "Ignoring endpoint conventions can include or exclude atoms incorrectly.", "An arbitrary integrator need not induce a positive measure; signed and complex extensions require appropriate bounded-variation assumptions.", "A signed integrand whose positive and negative parts both have infinite integral has no defined Lebesgue integral.", "Assuming that existence of a Lebesgue-Stieltjes integral guarantees existence of the corresponding Riemann-Stieltjes integral." ], "questions": [ { "id": "evidence-failure-modes-and-hazards-q01", "text": "Which of these failure modes and hazards hold for the sense of Lebesgue-Stieltjes integration this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-neighbours", "name": "Neighbouring kinds and how to tell them apart", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ { "name": "Lebesgue integration", "difference": "Lebesgue integration allows a general measure; Lebesgue-Stieltjes integration specifies a measure induced by a function, commonly on the real line." }, { "name": "Riemann-Stieltjes integration", "difference": "Riemann-Stieltjes integration uses limits of tagged sums, whereas Lebesgue-Stieltjes integration uses a measure; the latter accommodates a broader class of integrands." }, { "name": "Stieltjes measure", "difference": "A Stieltjes measure assigns masses to measurable sets; Lebesgue-Stieltjes integration integrates functions with respect to that measure." }, { "name": "Itô integration", "difference": "Itô integration handles stochastic integrators such as Brownian motion whose paths generally have infinite variation, so it is not generally pathwise Lebesgue-Stieltjes integration." } ], "questions": [ { "id": "evidence-neighbours-q01", "text": "Which of these neighbouring kinds and how to tell them apart hold for the sense of Lebesgue-Stieltjes integration this model covers, and on what evidence?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Which authoritative references should establish this registry entry's definition, notation and relationship to neighbouring integration concepts?", "Should signed and complex bounded-variation integrators be fully owned by this entry or represented as linked extensions?", "Which endpoint convention should Vercy adopt as its default, and how should alternative conventions be recorded?", "Should locally bounded-variation generators on unbounded domains be supported directly, and which global integrability framework should govern them?", "Which existing world model, if any, already covers this concept and should remain its single source of truth?" ], "statistics": { "bundles": 6, "layers": 11, "findings": 15, "questions": 25 } }