{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.centered-cube-number","code":"thing-q190893","url":"https://ver.cy/models/thing/q190893/","name":"centered cube number","alternateNames":[],"kind":"thing","status":"published","version":"0.1.0-research.1","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QTY"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"A centered cube number is a figurate integer counting a central point together with successive cubically arranged layers of points, with terms indexed here by C(n) = n^3 + (n+1)^3 for integers n >= 0; the equivalent one-based convention is C_n = n^3 + (n - 1)^3 for integers n >= 1.","purpose":"Enable an agent to recognize centered cube numbers, verify their indices and point-count representations, and generate or compare them using exact arithmetic.","scope":{"in":["Centered cube numbers defined by C(n) = n^3 + (n+1)^3 for nonnegative integers n","Sequence indices, exact values, and equivalent indexing conventions","Finite cubic point arrangements that realize the defining count","Exact generation, membership verification, and arithmetic consistency checks"],"out":["Ordinary cube numbers as a separate figurate-number family","Centered square numbers and other centered figurate-number families","Physical crystal structures, material properties, and lattice spacing","Volume measurements, cubic units, and unit conversions","Arbitrary sums of two cubes whose bases need not be consecutive"],"boundaries":[]},"distinguishingFeatures":["A member must equal n^3 + (n+1)^3 for an integer n >= 0; resemblance to a cubic growth pattern is insufficient.","The initial values are 1, 9, 35, 91, and 189, as recorded in [OEIS A005898](https://oeis.org/A005898).","The defining sum uses two consecutive cube bases, distinguishing this family from unrestricted sums of cubes.","A finite point realization combines (n+1)^3 grid vertices with n^3 cell centers; it counts points rather than physical volume.","Being odd is necessary but insufficient: 3 is odd but lies strictly between C(0) = 1 and C(1) = 9."],"structure":{"bundles":[{"id":"sequence-identity","name":"Sequence identity","description":"Establish the mathematical family and normalize the index used to identify each member.","layers":[{"id":"defining-membership","name":"Defining membership","description":"Bind centered cube number claims to a specific integer-domain formula.","findings":[{"id":"consecutive-cubes-definition","name":"Consecutive cubes definition","description":"Use C(n) = n^3 + (n+1)^3 for n >= 0, following [OEIS A005898](https://oeis.org/A005898). Record an index witness for each claimed member.","questions":[{"text":"Which nonnegative integer n makes the candidate equal to n^3 + (n+1)^3?","id":"consecutive-cubes-definition-q01","kind":"definition"},{"text":"Does the supplied reference identify centered cube numbers specifically, or merely sums of two cubes?","id":"consecutive-cubes-definition-q02","kind":"provenance"}]}]},{"id":"index-normalization","name":"Index normalization","description":"Preserve term identity when sources number the first term differently.","findings":[{"id":"equivalent-index-origins","name":"Equivalent index origins","description":"By substituting m = n+1, the canonical expression becomes m^3 + (m-1)^3 for m >= 1. Both conventions assign the value 1 to their first term.","questions":[{"text":"Does the supplied index begin at n = 0 or m = 1?","id":"equivalent-index-origins-q01","kind":"definition"},{"text":"What index translation preserves the imported term's value?","id":"equivalent-index-origins-q02","kind":"action"}]}]}]},{"id":"finite-point-realization","name":"Finite point realization","description":"Make the geometric count explicit through a reproducible finite construction.","layers":[{"id":"vertices-and-cell-centers","name":"Vertices and cell centers","description":"Specify two disjoint point sets whose sizes yield the defining expression.","findings":[{"id":"coordinate-count-witness","name":"Coordinate count witness","description":"A direct construction uses all integer triples in {0,...,n}^3 and all triples (i+1/2,j+1/2,k+1/2) with 0 <= i,j,k < n. These disjoint sets contain (n+1)^3 and n^3 points respectively.","questions":[{"text":"Are both the integer-grid vertices and one center per elementary cell included?","id":"coordinate-count-witness-q01","kind":"definition"},{"text":"Does counting the two disjoint point sets reproduce the claimed member value?","id":"coordinate-count-witness-q02","kind":"measurement"}]}]},{"id":"finite-boundary-accounting","name":"Finite boundary accounting","description":"Control which points belong to the finite arrangement and what the count represents.","findings":[{"id":"whole-point-boundary-rule","name":"Whole-point boundary rule","description":"In this construction every distinct boundary vertex counts once. Fractional atom contributions used in crystallographic unit-cell accounting answer a different counting question. At n = 0, the arrangement contains one vertex and no cell centers.","questions":[{"text":"Are boundary points counted as distinct whole points rather than weighted unit-cell contributions?","id":"whole-point-boundary-rule-q01","kind":"boundary"},{"text":"Does the construction correctly reduce to one point when n = 0?","id":"whole-point-boundary-rule-q02","kind":"measurement"}]}]}]},{"id":"exact-term-operations","name":"Exact term operations","description":"Generate terms and decide membership without numerical approximation.","layers":[{"id":"term-generation","name":"Term generation","description":"Use algebraic identities derived from the defining formula.","findings":[{"id":"polynomial-and-step","name":"Polynomial and step","description":"Expansion gives C(n) = 2n^3 + 3n^2 + 3n + 1. Subtraction gives C(n+1) - C(n) = 6(n+1)^2 + 2, allowing exact incremental generation from C(0) = 1.","questions":[{"text":"Do direct evaluation and incremental generation produce the same requested term?","id":"polynomial-and-step-q01","kind":"measurement"},{"text":"Can the chosen integer representation hold the generated values without overflow?","id":"polynomial-and-step-q02","kind":"action"}]}]},{"id":"inverse-membership","name":"Inverse membership","description":"Recover an index or prove that no admissible index exists.","findings":[{"id":"monotone-integer-search","name":"Monotone integer search","description":"The positive step proves strict increase. Exact integer search therefore finds at most one index witness; a positive candidate strictly between adjacent terms is a verified nonmember.","questions":[{"text":"Does exact substitution verify the index proposed for the candidate?","id":"monotone-integer-search-q01","kind":"measurement"},{"text":"If no equality exists, which adjacent terms bracket the positive integer candidate?","id":"monotone-integer-search-q02","kind":"action"},{"text":"Was a noninteger or nonpositive input rejected before searching the sequence?","id":"monotone-integer-search-q03","kind":"boundary"}]}]}]},{"id":"arithmetic-boundaries","name":"Arithmetic boundaries","description":"Use derived properties to check claims while keeping the family distinct from neighboring concepts.","layers":[{"id":"factorization-checks","name":"Factorization checks","description":"Derive constraints from the sum-of-cubes identity.","findings":[{"id":"oddness-and-compositeness","name":"Oddness and compositeness","description":"Factoring the definition gives C(n) = (2n+1)(n^2+n+1). Both factors are odd; for n >= 1 both exceed 1. Thus every member is odd, every member above 1 is composite, and the initial member 1 is neither prime nor composite.","questions":[{"text":"Does the candidate satisfy these necessary parity and factorization checks?","id":"oddness-and-compositeness-q01","kind":"measurement"},{"text":"Has passing a necessary check been incorrectly treated as sufficient evidence of membership?","id":"oddness-and-compositeness-q02","kind":"boundary"}]}]},{"id":"neighboring-family-separation","name":"Neighboring family separation","description":"Keep numerical overlaps separate from equivalence of mathematical families.","findings":[{"id":"shared-values-do-not-merge-families","name":"Shared values do not merge families","description":"A value can have other figurate-number classifications without changing this entry's definition. Ordinary cube membership asks for a single integer cube; centered cube membership asks for a sum of consecutive nonnegative integer cubes.","questions":[{"text":"Is another classification being attached to an individual value or incorrectly asserted for the entire family?","id":"shared-values-do-not-merge-families-q01","kind":"boundary"},{"text":"Should a neighboring mathematical concept be linked through a relation rather than absorbed into this model?","id":"shared-values-do-not-merge-families-q02","kind":"action"}]}]}]},{"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.","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.","questions":[{"text":"Which of these check these first hold for the sense of centered cube number this model covers, and on what evidence?","id":"evidence-confidence-notes-q01","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.","questions":[{"text":"Which of these real-world use hold for the sense of centered cube number this model covers, and on what evidence?","id":"evidence-real-world-use-q01","kind":"provenance"}]},{"id":"evidence-measurements","name":"Typical measurements","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these typical measurements hold for the sense of centered cube number this model covers, and on what evidence?","id":"evidence-measurements-q01","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.","questions":[{"text":"Which of these failure modes and hazards hold for the sense of centered cube number this model covers, and on what evidence?","id":"evidence-failure-modes-and-hazards-q01","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.","questions":[{"text":"Which of these neighbouring kinds and how to tell them apart hold for the sense of centered cube number this model covers, and on what evidence?","id":"evidence-neighbours-q01","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Compute and check centered cube numbers from the formula n^3 + (n - 1)^3.","Cite the sequence with its starting index stated, since sources differ on where it begins."],"mustNot":["Confuse centered cube numbers with ordinary cube numbers or other centered figurate families.","Present a term as verified when it was not computed or checked against a trusted source.","Attribute discoveries about the sequence to a person without evidence."],"requiresHuman":["Correcting a published table or reference value that others rely on."]},"ethics":{"considerations":["Small errors in reference sequences propagate into teaching materials and software tests."],"affectedParties":["Students and teachers","Users of integer sequence references"]},"owners":{"steward":"Nobody owns the concept; it belongs to mathematics and is held in common.","roles":[],"masterSystems":[]},"relations":[{"target":"Q14469439","type":"parent","note":"registry parent class"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.centered-cube-number","Wikidata: Q190893 (https://www.wikidata.org/wiki/Q190893)"]},"properties":{"applicability":"not-applicable","items":["Exact member value: Positive integer; dimensionless point count","Canonical sequence index: Integer n >= 0 with C(n) = n^3 + (n+1)^3","Indexing convention: Zero-based n; one-based m = n+1; unresolved imported convention","Membership verification: Unverified; verified member; verified nonmember","Consecutive cube decomposition: Ordered bases (n, n+1), with their cubes summing to the member value","Point realization convention: Finite vertex-and-cell-center construction; another documented construction; unspecified"]},"recognition":{"applicability":"optional","items":["Drawn as a central point with cubic shells of points around it, like a body-centred cubic lattice.","Terms run 1, 9, 35, 91 and grow roughly as twice a cube."]},"capabilities":{"applicability":"required","items":["Generate an exact term or bounded sequence segment from nonnegative integer indices.","Verify membership by finding an integer index and recomputing its defining sum.","Reject a candidate using exact bounds between adjacent terms.","Translate between zero-based and one-based formulas while preserving values.","Construct and count a finite vertex-and-cell-center point arrangement.","Check proposed terms against derived recurrence and factorization identities."]},"hazards":{"applicability":"required","items":["An off-by-one index shift silently gives the wrong term.","Mixing it with plain cube numbers corrupts calculations and tests."]},"interfaces":{"applicability":"not-applicable","items":[]},"context":{"applicability":"required","items":["Centered cube numbers occur in number theory and mathematics education as integer counts associated with cubically arranged points.","They are used to connect geometric representations with exact arithmetic, generate sequences, and test whether an integer has a valid sequence index."]}},"sources":[{"title":"Wikidata item Q190893: centered cube number","url":"https://www.wikidata.org/wiki/Q190893","note":"identity and sense of the item"},{"title":"Wikipedia: Centered cube number","url":"https://en.wikipedia.org/wiki/Centered_cube_number","note":"general description of the item"}],"openQuestions":["Does the existing Vercy catalogue already contain an authoritative model of this family that vr.tr.centered-cube-number should reference?","Which historical or primary mathematical source should anchor the geometric terminology beyond the sequence reference?","Which concentric-shell constructions are documented for centered cube numbers, and how do their shell indices correspond to the finite coordinate construction?","Should downstream registry records represent the number family, individual members, or both through explicitly distinguished roles?"],"resources":{"spec":"/models/things/publications/thing-q190893/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q190893/spec.json"],"providers":["Codex"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T08:30:46Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"not-applicable","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T12:16:30+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties.","interaction.interfaces":"No specific standard, law or official classification applies to this mathematical sequence."},"score":1.0}}