{ "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-12T09:33:28Z", "providers": [ "Codex" ], "breadth": "recalled by Codex without web access - no source was read", "missingProviders": [], "pass": 2, "cost": { "grok": { "seconds": 19.7, "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: 01a094f6-0218-78a0-8e39-5dc2f9f32a7e\n--------\nuser\nDescribe what is already known about one registered thing. Answer as JSON only, no prose around it.\n\nThing: Eulerian number\nSense to describe: (none recorded)\nDomain code: XCT.QTY\nAlso known as: (none)\n\n\nContext for this batch of 1001 things:\n# Batch 043: 1001", "usd": 0.0, "recall": true }, "codex": { "seconds": 63.7, "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: 01a094f6-020c-7d40-8ece-744edf25efdd\n--------\nuser\nYou are drafting a Vercy meta-model for one registered thing. Answer as JSON only, no prose around it.\n\nThing: Eulerian number\nRegistry id: vr.tr.eulerian-number\nPlane / domain: XCT / XCT.QTY\nRegistry definition: (none recorded)\nNames folded into this entry: (" } }, "enrichment": { "pass": 3, "engine": "claude", "producedAt": "2026-10-06T11:00:00Z", "ingestedAt": "2026-10-06T08:15:13+00:00", "fields": [ "agent_conduct", "ethics", "failure_modes_and_hazards", "owners" ] }, "gate": { "passed": true, "version": "1.0", "at": "2026-10-07T10:10:26+00:00", "evidence": [ "completeness 1.00", "sense: Wikidata description", "2 live sources", "review by codex: pass" ] } }, "metaModel": { "id": "THING-Q1373849", "registryId": "vr.tr.eulerian-number", "name": "Eulerian number", "version": "0.1.0-research.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QTY" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q1373849/", "model": { "registry_id": "vr.tr.eulerian-number", "name": "Eulerian number", "purpose": "Enable an agent to identify, compute, validate and apply Eulerian numbers without confusing their indexing conventions or counted permutation statistics.", "definition": "An Eulerian number of the first order, A(n,k), counts permutations of n distinct elements having exactly k descents, where a descent is an adjacent pair whose first element exceeds its second.", "scope_statement": "This model owns vr.tr.eulerian-number in XCT / XCT.QTY: the ordinary Eulerian number family and its indexed values, provisionally adopting the descent-count convention while leaving registry coverage of generalizations unresolved.", "in_scope": [ "The permutation statistic defining an Eulerian number", "Index meanings, notation translations and boundary conventions", "Exact values, computation methods and verification evidence", "Relations to Eulerian polynomials and permutation probabilities", "Recognition and routing of generalized Eulerian families" ], "out_of_scope": [ "Euler's constant e and the Euler-Mascheroni constant", "Euler numbers and Euler polynomials as separate mathematical families", "General permutation representations and manipulation", "Stirling numbers, binomial coefficients and their independent models", "Physical quantities, measurement units and unit conversions" ], "distinguishing_features": [ "Under the adopted convention, A(n,k) counts permutations of n distinct ordered elements with exactly k adjacent descents. [NIST DLMF](https://dlmf.nist.gov/26.14#i)", "Its identity requires both indices and their convention; an isolated integer or the name Euler is insufficient.", "The zero-based row for n=3 is [1,4,1], providing a concrete indexing check. [NIST DLMF table](https://dlmf.nist.gov/26.14.T1)", "A proposed interpretation must count adjacent descents rather than all inversions, descent-position sums or permutation cycles.", "Treat signed, multiset or higher-order variants as requiring an explicit family definition before accepting their values." ], "characteristics": [ { "name": "Permutation size", "kind": "measurement", "unit_or_values": "n: nonnegative integer; count of distinct elements", "why_it_matters": "Selects the permutation universe and Eulerian triangle row." }, { "name": "Statistic index", "kind": "measurement", "unit_or_values": "k: integer interpreted under the recorded convention", "why_it_matters": "Selects the counted class and prevents column shifts." }, { "name": "Counting convention", "kind": "category", "unit_or_values": "descents; ascents; excedances; runs with explicit translation; other explicitly defined statistic", "why_it_matters": "Determines how external notation maps to the adopted definition." }, { "name": "Eulerian value", "kind": "measurement", "unit_or_values": "Dimensionless nonnegative integer for an exact count", "why_it_matters": "Separates the mathematical count from a probability, approximation or modular residue." }, { "name": "Family interpretation", "kind": "category", "unit_or_values": "ordinary; explicitly defined generalization; unresolved", "why_it_matters": "Prevents importing values from a different Eulerian family." }, { "name": "Verification status", "kind": "state", "unit_or_values": "unevaluated; computed; independently checked; conflicting", "why_it_matters": "Records confidence in a representation without suggesting that the mathematical number changes." }, { "name": "Defining reference", "kind": "relation", "unit_or_values": "Read source with definition, notation and formula location", "why_it_matters": "Makes a claimed value and its index translation auditable." } ], "affordances": [ "Resolve notation and translate a documented indexing convention.", "Compute an exact indexed value or a finite row.", "Check a supplied table against boundary values and Eulerian identities.", "Derive descent probabilities after establishing uniform sampling of permutations.", "Extract or assemble Eulerian polynomial coefficients under an explicit exponent convention.", "Flag ambiguous family assignments or incompatible source values." ], "agent_conduct": { "may": [ "Compute and explain Eulerian numbers and their identities, showing working.", "Distinguish them clearly from Euler numbers and other similarly named sequences." ], "must_not": [ "Confuse Eulerian numbers with Euler numbers or the number e.", "Present unverified identities as proven.", "Use inconsistent indexing without stating the convention." ], "requires_human": [ "Publishing new results under someone else's name." ] }, "ethics": { "considerations": [ "Correct attribution and naming matter in mathematics, where similar names cause frequent errors." ], "affected_parties": [ "Students", "Mathematicians" ] }, "owners": { "steward": "Nobody owns them; they are held in common as mathematics.", "master_systems": [] }, "failure_modes_and_hazards": [ "Errors from mixing indexing conventions.", "Confusion with similarly named numbers." ], "what_it_is_for": [ "Eulerian numbers occur in enumerative combinatorics when counting permutations by their numbers of ascents or descents.", "They are used in permutation probability calculations and polynomial identities, with the indexing convention stated before values are compared." ], "facets_not_applicable": { "interfaces": "No standard, law or official classification governs Eulerian numbers as mathematical objects.", "recognition": "An Eulerian number is an abstract mathematical quantity without an intrinsic perceptible appearance." } }, "sources": [ { "id": "wikidata", "title": "Wikidata item Q1373849: Eulerian number", "url": "https://www.wikidata.org/wiki/Q1373849", "url_status": "live", "what_it_supports": "identity and sense of the item", "checked_by": "wikidata-api" }, { "id": "wikipedia-en", "title": "Wikipedia: Eulerian number", "url": "https://en.wikipedia.org/wiki/Eulerian_number", "url_status": "live", "what_it_supports": "general description of the item", "checked_by": "wikidata-api sitelink" } ], "structure": { "bundles": [ { "id": "counted-permutation-class", "name": "Counted permutation class", "description": "Establish what the number counts before interpreting any value.", "rationale": "Eulerian terminology alone does not establish the permutation universe or statistic.", "layers": [ { "id": "ordinary-family-definition", "name": "Ordinary family definition", "description": "Anchor the entry to permutations of distinct ordered elements.", "findings": [ { "id": "descent-count-anchor", "name": "Descent count anchor", "description": "Adopt the ordinary descent interpretation as the draft's working definition and record evidence for each imported interpretation.", "questions": [ { "text": "Does the source count permutations of n distinct ordered elements with exactly k positions i satisfying sigma(i) > sigma(i+1)?", "kind": "definition", "id": "descent-count-anchor-q01" }, { "text": "Which read definition establishes that the supplied notation denotes this count?", "kind": "provenance", "id": "descent-count-anchor-q02" } ] } ] }, { "id": "statistic-and-family-boundaries", "name": "Statistic and family boundaries", "description": "Resolve alternative statistics and proposed generalizations.", "findings": [ { "id": "interpretation-compatibility", "name": "Interpretation compatibility", "description": "Require an explicit translation or separate family declaration before reusing an imported count.", "questions": [ { "text": "If the source uses ascents, excedances or runs, what argument and index translation connect its count to the adopted convention?", "kind": "boundary", "id": "interpretation-compatibility-q01" }, { "text": "Does the source introduce repeated elements, signs or an order parameter that changes the counted family?", "kind": "boundary", "id": "interpretation-compatibility-q02" } ] } ] } ] }, { "id": "indices-and-boundaries", "name": "Indices and boundaries", "description": "Make each Eulerian triangle position unambiguous.", "rationale": "Off-by-one conventions and empty-permutation handling can invalidate otherwise correct calculations.", "layers": [ { "id": "notation-translation", "name": "Notation translation", "description": "Preserve source indices alongside their canonical interpretation.", "findings": [ { "id": "column-meaning", "name": "Column meaning", "description": "Record whether a column labels the statistic directly or uses an offset.", "questions": [ { "text": "What do the source's first and second indices mean, and where does column numbering begin?", "kind": "definition", "id": "column-meaning-q01" }, { "text": "Does translating the source's n=3 row recover canonical columns k=0,1,2 with values 1,4,1?", "kind": "measurement", "id": "column-meaning-q02" } ] } ] }, { "id": "support-and-empty-case", "name": "Support and empty case", "description": "Separate combinatorial support from input validation.", "findings": [ { "id": "boundary-contract", "name": "Boundary contract", "description": "Adopt A(0,0)=1; for n>=1, support is 0<=k<=n-1. Extend other integer k values by zero; reject negative or noninteger n. [NIST DLMF](https://dlmf.nist.gov/26.14)", "questions": [ { "text": "Does the implementation preserve the empty-permutation value separately from positive-size rows?", "kind": "boundary", "id": "boundary-contract-q01" }, { "text": "Will an out-of-support integer k return zero while malformed or unsupported indices produce an explicit error?", "kind": "action", "id": "boundary-contract-q02" } ] } ] } ] }, { "id": "exact-evaluation-and-checks", "name": "Exact evaluation and checks", "description": "Compute Eulerian counts and assess supplied representations.", "rationale": "Large integer counts require arithmetic and checks that preserve exactness.", "layers": [ { "id": "recurrence-evaluation", "name": "Recurrence evaluation", "description": "Connect each computed entry to initialized preceding rows.", "findings": [ { "id": "recurrence-contract", "name": "Recurrence contract", "description": "For n>=2, A(n,k)=(k+1)A(n-1,k)+(n-k)A(n-1,k-1). [NIST DLMF recurrence](https://dlmf.nist.gov/26.14.E8)", "questions": [ { "text": "Which base rows and zero-extension rules initialize the recurrence?", "kind": "definition", "id": "recurrence-contract-q01" }, { "text": "Can the selected arithmetic represent every intermediate count exactly, or must the result be labelled approximate or modular?", "kind": "action", "id": "recurrence-contract-q02" } ] } ] }, { "id": "independent-validation", "name": "Independent validation", "description": "Check individual values and complete rows using different evidence.", "findings": [ { "id": "row-invariants", "name": "Row invariants", "description": "For n>=1, require A(n,k)=A(n,n-1-k) and sum over k=0 through n-1 equal to n!. [NIST DLMF identities](https://dlmf.nist.gov/26.14#iii)", "questions": [ { "text": "Does the complete computed row satisfy symmetry and the factorial total?", "kind": "measurement", "id": "row-invariants-q01" }, { "text": "Which small-case enumeration or independently evaluated formula checks the claimed cell beyond these necessary row tests?", "kind": "provenance", "id": "row-invariants-q02" } ] } ] } ] }, { "id": "polynomial-and-probability-use", "name": "Polynomial and probability use", "description": "Control how Eulerian counts enter algebraic and probabilistic tasks.", "rationale": "Correct counts can still produce incorrect results when exponent conventions or sampling assumptions differ.", "layers": [ { "id": "polynomial-coefficients", "name": "Polynomial coefficients", "description": "Link counts to a declared polynomial representation.", "findings": [ { "id": "exponent-convention", "name": "Exponent convention", "description": "For this model's interface, define P_n(t)=sum_k A(n,k)t^k and require explicit translation from other normalizations.", "questions": [ { "text": "Does the referenced polynomial attach A(n,k) to t^k, t^(k+1) or another expression?", "kind": "definition", "id": "exponent-convention-q01" }, { "text": "What exponent shift or normalization must the agent apply before extracting or exporting coefficients?", "kind": "action", "id": "exponent-convention-q02" } ] } ] }, { "id": "descent-probabilities", "name": "Descent probabilities", "description": "Record the sampling assumption needed to normalize a count.", "findings": [ { "id": "uniform-sampling-condition", "name": "Uniform sampling condition", "description": "For a uniformly sampled permutation, the definition implies Pr(k descents)=A(n,k)/n!; record this as a derived probability.", "questions": [ { "text": "Are all n! permutations equally likely, or does the task impose weights or restrictions?", "kind": "boundary", "id": "uniform-sampling-condition-q01" }, { "text": "Should the derived probability remain an exact rational or be rounded with a stated precision?", "kind": "action", "id": "uniform-sampling-condition-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": [ "With the descent convention, A(0,0)=1; for n≥1 the recurrence is A(n,k)=(n−k)A(n−1,k−1)+(k+1)A(n−1,k), with out-of-range indices interpreted as zero.", "The unqualified name usually denotes first-order Eulerian numbers; the registry has no recorded sense, so confirm the intended order and indexing.", "This is a family of dimensionless combinatorial integers, not a measurement unit; statements here are recalled knowledge, not source-verified research." ], "questions": [ { "id": "evidence-confidence-notes-q01", "text": "Which of these check these first hold for the sense of Eulerian number 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": [ "Eulerian numbers of the first order", "Eulerian numbers of the second order" ], "questions": [ { "id": "evidence-kinds-q01", "text": "Which of these kinds and varieties hold for the sense of Eulerian number this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-identifiers", "name": "Identifiers and schemes", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ { "scheme": "OEIS", "value_or_pattern": "A008292", "note": "The triangle of first-order Eulerian numbers, read by rows." } ], "questions": [ { "id": "evidence-identifiers-q01", "text": "Which of these identifiers and schemes hold for the sense of Eulerian number 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": [ "Enumerating permutations by descents, ascents or excedances.", "Constructing Eulerian polynomials in enumerative combinatorics.", "Expressing generating functions for sums of powers.", "Studying descent statistics in uniformly random permutations." ], "questions": [ { "id": "evidence-real-world-use-q01", "text": "Which of these real-world use hold for the sense of Eulerian number 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": [ "Indexing conventions differ: some authors index by the number of descents, while others index by the number of ascending runs, which is one greater for a nonempty permutation.", "Confusing Eulerian numbers with Euler numbers or Euler zigzag numbers.", "Applying first-order formulas to second-order Eulerian numbers.", "Computing large values with fixed-width integers can cause overflow." ], "questions": [ { "id": "evidence-failure-modes-and-hazards-q01", "text": "Which of these failure modes and hazards hold for the sense of Eulerian number 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": "Eulerian polynomial", "difference": "A polynomial whose coefficients are Eulerian numbers, rather than an individual count." }, { "name": "Euler number", "difference": "A member of a different sequence associated with expansions of hyperbolic secant or related trigonometric functions, depending on convention." }, { "name": "Euler zigzag number", "difference": "Counts alternating permutations, imposing an alternating pattern of ascents and descents rather than specifying only their total number." }, { "name": "Stirling number of the first kind", "difference": "Its unsigned form counts permutations by number of cycles rather than by number of descents." }, { "name": "Binomial coefficient", "difference": "Counts subsets of a specified size rather than permutations with a specified descent count." } ], "questions": [ { "id": "evidence-neighbours-q01", "text": "Which of these neighbouring kinds and how to tell them apart hold for the sense of Eulerian number this model covers, and on what evidence?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Does the registry intend only ordinary Eulerian numbers, or also second-order and other generalized families?", "Which notation and column convention should become canonical across Vercy's existing mathematical publications?", "Does an existing world model already own this concept, requiring a registry link rather than another publication?", "Which independently checked tables and executable implementations should supply reference values and computation evidence?" ], "statistics": { "bundles": 5, "layers": 9, "findings": 14, "questions": 22 } }