Public evidence

math.geometry.solid

Machine-checked verification for calculation version 1.19.0 · tier candidate

Open calculator API Capability Download JSON CVP Manifest MCP Review policy

CVP status

20 right Euclidean models, positive normal binary64 inputs/required measures, strict decimal aliases, safe-integer n and exact normalized endpoints. Ellipsoid axis ratio>=1e-100; bounded2-D figures; no oblique solids or CAS.

Declared partition coverage: PASS · 21/21 declared partitions (sphere, cylinder, cone, torus, frustum, prism, pyramid, tetrahedron, octahedron, icosahedron, dodecahedron, regular-prism, regular-pyramid, regular-frustum, ellipsoid, spherical-cap, spherical-zone, spherical-wedge, spherical-lune, spherical-sector, invalid-domain)
Versions: Calculation 1.19.0 · CVP protocol 1.0.0-proposed · Evidence 2026-10-29.math-uuce-fields-v1
Verification revision: 2026-10-29.math-uuce-fields-v1 · 4/4 property · digest de157af6f244

Verification

Legacy regression suite: 115/118 calculation tests · Production surface contract 5/5 · 3 failed

Three distinct suites: educational examples on the calculator page, this legacy regression count (115/118 golden + boundary), and the CVP verification suite below. They are not the same number. Property, metamorphic, round-trip, O3, and interface statuses/checks are recorded in the CVP Manifest . SKIP (0/0) means that suite has no published vectors.

This calculator CURRENT · Public schema 1.19.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓

CVP verification suite

Protocol checks from the Evidence Manifest. Distinct from the legacy regression suite above. Verification revision 2026-10-29.math-uuce-fields-v1. Calculation version 1.19.0. CVP 1.0.0-proposed. 188/188 golden · 27/27 CVP boundary · 96/96 invalid · 4/4 property · 4/4 metamorphic · 8/8 cross-interface · 2/2 cross-calculator · 5/5 CVP contract. digest de157af6f244.

Golden (oracle-backed) 188 / 188
CVP boundary 27 / 27
Invalid input 96 / 96
Property 4 / 4
Metamorphic 4 / 4
Cross-interface 8 / 8
Cross-calculator 2 / 2
CVP contract 5 / 5
Interfaces PASS · UI (SSR) / REST / MCP
Interface claim ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 2/2 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Verified scope all20 published modes; full secondary-output review and raw input/range partitions
Assurance Core

release artifacts aligned PASS
schema output required declared PASS
schema valid input executes PASS
engine success output validates schema PASS
rest success payload validates schema PASS

Fingerprint sha256:5642a443c845e36aa1877f1675f0fe8cca99e24e8dc5f8145585381d9ee0db39

Input schema sha256:567cb075bc4cfb15a18b1e07650bfc398ac4837d087d63cd9e94f8042e9d0e63 · Output schema sha256:1ca59b7fd14cb18c509c0237d92c9654d309fd28d42faf073fd84aa885f09bc1 · Build-time SHA-256 digests prove published artifacts agree with each other (release_integrity.scope=workspace). Production attestation GETs canonical URLs twice (public/cached view and origin), including Capability HTML, via npm run attest:production --write (published at /.well-known/calculatorx-production-attestation.json) and fails if the public Capability page advertises a different calculation_version. This is artifact consistency, not an independent immutable release log.

Build & scope

Engine identity and declared limitations.

Tool: solid-geometry
Engine: solid-geometry
Engine build: 2026-10-29
Evidence core: 2026-10-29T08:55:15.563Z
Rendered: 2026-11-10T05:18:42.121Z
Last reviewed: —

Assumptions

Euclidean 3-space
The Result Card includes an engine-linked 2-D generating section or silhouette of the published lengths. Not a 3-D mesh. Same pack as REST/MCP.
Right solids; not oblique
Ring torus R > r
Not the circle-only page
Not a vector
Not ISO 5167-5
Not a sphere page for ellipsoid
Not CAS

Limitations

Right solids only; not oblique
Not the circle-only page — 2-D circle stays on math.geometry.circle
Not a vector — math.vector is R²/R³ algebra
Not ISO 5167-5 cone meter (physics.fluid.iso_cone)
Torus is a ring torus: major R > tube r; horn/spindle out of scope
Right circular frustum needs R,r,h>0; r=R recovers a cylinder; not a cone page
Right rectangular prism needs a,b,h>0; a=b=h is a cube; not a frustum page
Right rectangular pyramid needs a,b,h>0; V=abh/3; apex above the base center; a=b is a square base; not a prism page
Regular tetrahedron needs a>0; V=a³√2/12, A=a²√3; not a rectangular pyramid page
Regular octahedron needs a>0; V=(√2/3)a³, A=2√3 a²; not a tetrahedron page
Regular icosahedron needs a>0; V=(5/12)(3+√5)a³, A=5√3 a²; not an octahedron page
Regular dodecahedron needs a>0; V=(1/4)(15+7√5)a³, A=3√(25+10√5) a²; not an icosahedron page
Regular n-gon prism needs integer n≥3 and s,h>0; base from math.geometry.plane; n=4 matches a square rectangular prism; not /prism; not an n-gon pyramid
Regular n-gon pyramid needs integer n≥3 and s,h>0; V=A h/3 with A from math.geometry.plane; n=4 matches a square rectangular pyramid; not /pyramid; not a tetrahedron; not /regular-frustum
Regular n-gon frustum needs integer n≥3 and s,t,h>0; V=h/3(A+a+√(Aa)) with A,a from math.geometry.plane; s=t recovers a regular prism; lateral faces from the plane isosceles trapezoid; not /frustum; not /regular-pyramid
Ellipsoid needs a,b,c>0; V=4/3 πabc; surface is the exact elliptic-integral / spheroid model numerically evaluated (modulus k, not m=k²); axes sorted internally; a=b=c is a sphere; not /sphere; not the 2-D ellipse
Spherical cap needs r>0 and 0<h≤2r; V=(1/3)πh²(3r−h); h=r is a hemisphere; h=2r is a sphere; not /sphere; not a spherical zone
Spherical zone needs a,b,h>0; V=(πh/6)(3a²+3b²+h²); r recovered; a=b is symmetric; not /spherical-cap; not /sphere; not a spherical wedge
Spherical wedge needs r>0 and 0<θ≤360°; V=(2/3)r³θ with θ in radians; θ=180° is a hemisphere; θ=360° is a sphere; not /spherical-cap; not /spherical-zone; not /sphere; not a spherical lune
Spherical lune needs r>0 and 0<θ≤360°; A=2r²θ with θ in radians; θ=180° is a hemisphere surface; θ=360° is a sphere; not /spherical-wedge; not /spherical-cap; not /spherical-zone; not /sphere; not a spherical sector
Spherical sector needs r>0 and 0<h≤2r; V=(2/3)πr²h; a=√(h(2r−h)); A=πr(2h+a); h=r is a hemisphere; h=2r is a sphere; not /spherical-cap; not /cone; not /sphere; not /spherical-wedge; not /spherical-lune
unknown mode (including cas, vector, iso-cone) → INVALID_MODE

Sources

Standards and references supporting this tool specification. Methods are listed separately — they are algorithms, not bibliographic sources.

ISO 80000-2:2019 Quantities and units — Mathematics
Supports: Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.
NIST DLMF §19.2 Incomplete elliptic integrals — 19.2(ii) Legendre’s integrals F(φ,k) and E(φ,k)
Supports: Ellipsoid triaxial surface uses incomplete F and E with modulus k, not parameter m = k²
Euclid’s Elements — XIII.13: Tetrahedron
Supports: Construction and edge-to-circumsphere geometry of the regular tetrahedron. Face areas are obtained from triangle or regular-pentagon geometry; volume is the sum of pyramids from the center to the faces. These area and volume expressions are derived from the construction and the separately cited pyramid-volume method. Mathematical reference only; not validation or endorsement of this implementation.
Euclid’s Elements — XIII.14: Octahedron
Supports: Construction and edge-to-circumsphere geometry of the regular octahedron. Face areas are obtained from triangle or regular-pentagon geometry; volume is the sum of pyramids from the center to the faces. These area and volume expressions are derived from the construction and the separately cited pyramid-volume method. Mathematical reference only; not validation or endorsement of this implementation.
Euclid’s Elements — XIII.16: Icosahedron
Supports: Construction and edge-to-circumsphere geometry of the regular icosahedron. Face areas are obtained from triangle or regular-pentagon geometry; volume is the sum of pyramids from the center to the faces. These area and volume expressions are derived from the construction and the separately cited pyramid-volume method. Mathematical reference only; not validation or endorsement of this implementation.
Euclid’s Elements — XIII.17: Dodecahedron
Supports: Construction and edge-to-circumsphere geometry of the regular dodecahedron. Face areas are obtained from triangle or regular-pentagon geometry; volume is the sum of pyramids from the center to the faces. These area and volume expressions are derived from the construction and the separately cited pyramid-volume method. Mathematical reference only; not validation or endorsement of this implementation.
NIST DLMF — 19.33 Triaxial Ellipsoids — 19.33(i), equations 19.33.1–19.33.3
Supports: Ellipsoid volume V = 4πabc/3 and exact surface-area expressions in symmetric and Legendre elliptic integrals. An approximate displayed area, where offered, must be distinguished from these exact identities. Mathematical reference only; not validation or endorsement of this implementation.
Gilbert Strang, Calculus — Chapter 8: Applications of the Integral (MIT OpenCourseWare) — §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §8.5 Masses and Moments, exercises 24–29 §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area
Supports: Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Pappus’s volume and surface-area theorems, with a torus volume exercise. For a disk of radius r whose center is distance R>r from the axis, V = 2π²Rr²; for its boundary circle the surface theorem gives S = 4π²Rr². These expressions apply the stated centroid theorems. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
Gilbert Strang, Calculus — Chapter 1: Introduction to Calculus (MIT OpenCourseWare) — §1.5 A Review of Trigonometry
Supports: Right-triangle definitions of sine, cosine and tangent, the sine/cosine identities, and the law-of-cosines and parallelogram exercises 31–32. Chord length follows by bisecting the radii triangle. Regular-polygon apothems and areas follow by splitting the polygon into equal triangles. The law of sines follows by equating a triangle’s altitude computed from either adjacent side; SSA can give two triangles. These applications are derived from the stated relations. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
Gilbert Strang, Calculus — Chapter 14: Multiple Integrals (MIT OpenCourseWare) — §14.4 Cylindrical and Spherical Coordinates
Supports: Spherical coordinates and volume element ρ² sin(φ) dρ dφ dθ. Restricting azimuth to α gives a wedge α/(2π) of a ball and the corresponding surface lune. Radial integration to r with polar angle φ₀ and h=r(1−cos φ₀) gives spherical-sector volume 2πr²h/3. These formulas are derived applications; cap and cone surfaces use the separate surface reference. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Methods

Algorithms this engine uses. Not external references.

Right solids: closed Euclidean identities in IEEE-754 binary64
Ellipsoid exactness: volume closed identity; surface exact elliptic-integral / spheroid model, numerically evaluated
Ellipsoid algorithm: sort p≥q≥r; sphere 4πp²; oblate artanh; prolate arcsin; else Carlson RF/RD → F(φ,k), E(φ,k) with modulus k
Ellipsoid singularity_policy: a=b=c uses 4πa² so 1/sinφ is never evaluated at φ=0
Ellipsoid independent_oracle: 2-D parametric surface quadrature (O2) plus Legendre-integrand F,E for the k vs m convention

Verification disclosure

Aggregate outcomes, scope and artifact commitments are public. Complete test inputs, expected values, execution records and oracle infrastructure are retained privately. This summary does not independently reproduce the full verification run.