{ "protocol": { "name": "CVP", "version": "1.0.0-proposed", "revision": "2026-09-13", "spec_sha256": "f62725968356c92930531cd6085db76e216da06e95be53d636b0d2cf3218b148", "spec": "https://www.calculatorx.com/developers/cvp/spec/1.0" }, "document_status": "proposed", "calculator": { "id": "special-functions", "version": "1.10.6", "title": "Special Functions Calculator", "formula_family": "special_functions", "capability_id": "math.special", "verification_revision": "2026-10-19.special-full-review-v8" }, "profile": "core", "verified_scope": "all published modes", "status": "VERIFIED", "verification_status": "VERIFIED", "production_freshness": "CURRENT", "overall_trust": "VERIFIED", "runner": { "name": "cvp-runner", "version": "1.0.0" }, "environment": { "runtime": "node", "runtime_version": "22.20.0", "numeric_model": "IEEE-754 binary64", "os": "darwin" }, "suite": { "id": "special-functions", "version": "1.10.6", "uri": "https://www.calculatorx.com/evidence/math.special/1.10.6.json", "verification_revision": "2026-10-19.special-full-review-v8" }, "obligation_set": [ "CVP-PROV-01", "CVP-MODEL-01", "CVP-SCOPE-01", "CVP-UNIT-01", "CVP-REF-01", "CVP-GOLD-01", "CVP-BND-01", "CVP-INV-01", "CVP-NUM-01", "CVP-COV-01", "CVP-CONTRACT-01", "CVP-DEP-01", "CVP-ART-01", "CVP-PROD-01", "CVP-DIM-01", "CVP-PROP-01", "CVP-META-01", "CVP-XCAL-01", "CVP-IFACE-01" ], "provenance": { "mode": "direct", "sources": [ { "label": "ISO 80000-2:2019 Quantities and units — Mathematics", "kind": "standard", "reference": "Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.", "url": "https://www.iso.org/standard/64973.html" }, { "label": "Abramowitz and Stegun, Handbook of Mathematical Functions — erf, Gamma, Bessel, incomplete beta", "kind": "other", "reference": "Error function 7.1.26; Γ(n)=(n−1)!; integer-order J_n/Y_n/I_n/K_n; incomplete beta integral" }, { "label": "NIST DLMF §7.2 — Error Function Definitions", "kind": "other", "reference": "§7.2", "url": "https://dlmf.nist.gov/7.2" }, { "label": "NIST DLMF §5.2 — Gamma Function Definitions", "kind": "other", "reference": "§5.2", "url": "https://dlmf.nist.gov/5.2" }, { "label": "NIST DLMF §5.12 — Beta Function", "kind": "other", "reference": "§5.12", "url": "https://dlmf.nist.gov/5.12" }, { "label": "NIST DLMF §8.17 — Incomplete Beta Functions", "kind": "other", "reference": "§8.17", "url": "https://dlmf.nist.gov/8.17" } ] }, "controls": { "CVP-PROV-01": { "result": "PASS", "applicability": "required", "notes": "Sources present on evidence" }, "CVP-MODEL-01": { "result": "PASS", "applicability": "required", "notes": "Independent mpmath fixtures and exact identities across14 modes, including tiny tails, log-gamma range, branchpoints and incomplete-beta convergence" }, "CVP-SCOPE-01": { "result": "PASS", "applicability": "required", "notes": "Real erf/erfc, Gamma, lgamma, integer-order J/Y/I/K, Lambert W_0/W_{-1}, Beta, incomplete beta, sinc. Discovery pages lock mode. Not factorial. Not CAS." }, "CVP-UNIT-01": { "result": "PASS", "applicability": "required", "notes": "dimensionless" }, "CVP-REF-01": { "result": "PASS", "applicability": "required", "notes": "O1 model + O2 expected_values + O2 numerical_behavior" }, "CVP-GOLD-01": { "result": "PASS", "applicability": "required", "notes": "13/13 legacy golden; 1000/1000 oracle-backed numeric" }, "CVP-BND-01": { "result": "PASS", "applicability": "required", "notes": "7/7 numeric-edge Evidence + 1/1 near-singularity numeric; illegal enum/system is Invalid only" }, "CVP-INV-01": { "result": "PASS", "applicability": "required", "notes": "8 Evidence illegal-input + 16/16 oracle-backed error-code (exact_error_match, not ULP)" }, "CVP-NUM-01": { "result": "PASS", "applicability": "required", "notes": "max_error_observed_ulp=33692; policy declared a priori" }, "CVP-COV-01": { "result": "PASS", "applicability": "required", "notes": "15/15 declared partitions (erf, erfc, gamma, lgamma, bessel-j, bessel-y, bessel-i, bessel-k, lambert-w, lambert-wm1, beta, incomplete-beta, sinc, normalized-sinc, invalid-domain) · 13 legacy golden / 8 CVP boundary unique cases · 8 Evidence invalid-input (may overlap boundary)" }, "CVP-CONTRACT-01": { "result": "PASS", "applicability": "required", "notes": "6/6 CVP contract · release-artifacts-aligned=PASS · schema-output-required-declared=PASS · schema-valid-input-executes=PASS · engine-success-output-validates-schema=PASS · schema-rejects-incomplete-input=PASS · rest-success-payload-validates-schema=PASS" }, "CVP-DEP-01": { "result": "PASS", "applicability": "required", "notes": "engine special-functions; Node binary64" }, "CVP-ART-01": { "result": "PASS", "applicability": "required", "notes": "release_integrity.status=verified · digests: artifact_evidence_sha256=Evidence core (Trust freshness) · cvp_evidence_digest=artifacts.evidence_digest (fingerprint v2) · evidence_bundle_sha256=legacy Evidence JSON bytes · evidence_catalog_digest=attested_catalog_sha256 · production_sha256=bound production report · see artifacts.digest_registry" }, "CVP-PROD-01": { "result": "PASS", "applicability": "required", "notes": "this calculator CURRENT · bound snapshot pass @ 2026-10-19T07:58:46.038Z · live site-wide report stale @ 2026-10-19T07:58:59.631Z (4 other capabilities) · local checks pass · attestation_id=c64cc612dc6c3a945c8f29b84029807be91aeb07eee35eea41b94b1b8594e211 · This calculator CURRENT · This calculator is not affected · Bound attestation PASS @ 2026-10-19T07:58:46.038Z · Live production status STALE (4 capabilities; 372 remain CURRENT) @ 2026-10-19T07:58:59.631Z · Public schema 1.10.6 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓ · schema_const=1.10.6 · report output-const=pass · evidence-sha256=pass · semantic-contract=pass · capability-html-public-origin-agree=pass · cvp-manifest-public-origin-agree=pass · cvp-latest-alias-agrees=pass · cvp-immutable-manifest-fetch=pass · cvp-immutable-manifest-sha256=pass · cvp-latest-alias-matches-immutable=pass · calculator-html-cvp-projection=pass · evidence-html-cvp-projection=pass · attested_evidence_sha256=589b90803730d03fdb921029dbc9e570b3d8a9047e839174126db3b72f2cd1c6 · verified_at=2026-10-19T04:05:00.082Z · evidence_attested_at=2026-10-19T04:05:00.082Z · last_confirmed_current_at=2026-10-19T07:58:46.038Z · rendered_at=2026-10-19T07:59:00.247Z · binds=cvp_core_sha256", "check_result": "PASS", "production_state": "CURRENT", "binding_current": true }, "CVP-DIM-01": { "result": "PASS", "applicability": "applicable", "notes": "scalar special value" }, "CVP-PROP-01": { "result": "PASS", "applicability": "applicable", "notes": "1/1 property · enumerated fixed assertions (not generated; no seed)" }, "CVP-META-01": { "result": "PASS", "applicability": "applicable", "notes": "3/3 metamorphic" }, "CVP-ROUND-01": { "result": "SKIP", "applicability": "not_applicable", "notes": "No inverse / round-trip vectors in pack — control not in obligation set" }, "CVP-XCAL-01": { "result": "PASS", "applicability": "applicable", "notes": "1/1 cross-calculator · xcal-gamma-factorial PASS vs math.factorial" }, "CVP-IFACE-01": { "result": "PASS", "applicability": "applicable", "notes": "UI (SSR) · REST · MCP equivalent (MCP uses calc API handler). ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (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. Special response contract also checks all 14 mode formulas and unrounded numeric summaries, including erfc/K tails and a subnormal erf input; UI is shared presentation/SSR, not a live browser session." }, "CVP-ORACLE-01": { "result": "SKIP", "applicability": "higher_assurance", "notes": "O2 present; O3 not required this wave" }, "CVP-EXP-01": { "result": "SKIP", "applicability": "optional", "notes": "named expert review not performed" } }, "interfaces": { "compared": [ "ui-ssr", "rest", "mcp" ], "result": "PASS", "notes": "UI (SSR) · REST · MCP equivalent (MCP uses calc API handler). ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (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. Special response contract also checks all 14 mode formulas and unrounded numeric summaries, including erfc/K tails and a subnormal erf input; UI is shared presentation/SSR, not a live browser session." }, "artifacts": { "evidence_bundle_sha256": "af5699f5ab938af934ee120684176da8b9c7e4c238f117b39d673a84ab4a9d14", "evidence_digest": "a3f6d148d5414a4e99ff3d77f431d1b0818c94a7e122761a2239a86d2e1fdaae", "production_sha256": "589b90803730d03fdb921029dbc9e570b3d8a9047e839174126db3b72f2cd1c6", "attested_catalog_sha256": "93b711f8b5b8d602c9f2ecfb852b561e5cd33585603555116fdf45f0bd4f1957", "production_checks": [ { "id": "math.special:output-const", "status": "pass", "detail": "output const 1.10.6 vs capability 1.10.6", "scope": "last_attestation_snapshot", "current": true, "attestation_binding_current": true, "attested_at": "2026-10-19T04:05:00.082Z" }, { "id": "math.special:evidence-sha256", "status": "pass", "detail": "8cfeedba16d8", "scope": "last_attestation_snapshot", "current": true, "attestation_binding_current": true, "attested_at": "2026-10-19T04:05:00.082Z" }, { "id": "math.special:semantic-contract", "status": "pass", "detail": "57/58", "scope": "last_attestation_snapshot", "current": true, "attestation_binding_current": true, "attested_at": "2026-10-19T04:05:00.082Z" }, { "id": "math.special:capability-html-public-origin-agree", "status": "pass", "detail": "public 1.10.6 vs origin 1.10.6", "scope": "last_attestation_snapshot", "current": true, "attestation_binding_current": true, "attested_at": "2026-10-19T04:05:00.082Z" }, { "id": "math.special:cvp-manifest-public-origin-agree", "status": "pass", "detail": "math.special/1.10.6/2026-10-19.special-full-review-v8/core/a3f6d148d541", "scope": "last_attestation_snapshot", "current": true, "attestation_binding_current": true, "attested_at": "2026-10-19T04:05:00.082Z", "projection_compared": false }, { "id": "math.special:cvp-latest-alias-agrees", "status": "pass", "detail": "math.special/1.10.6/2026-10-19.special-full-review-v8/core/a3f6d148d541", "scope": "last_attestation_snapshot", "current": true, "attestation_binding_current": true, "attested_at": "2026-10-19T04:05:00.082Z", "projection_compared": false }, { "id": "math.special:cvp-immutable-manifest-fetch", "status": "pass", "detail": "public=200 origin=200", "scope": "last_attestation_snapshot", "current": true, "attestation_binding_current": true, "attested_at": "2026-10-19T04:05:00.082Z" }, { "id": "math.special:cvp-immutable-manifest-sha256", "status": "pass", "detail": "a3f6d148d541", "scope": "last_attestation_snapshot", "current": true, "attestation_binding_current": true, "attested_at": "2026-10-19T04:05:00.082Z" }, { "id": "math.special:cvp-latest-alias-matches-immutable", "status": "pass", "detail": "math.special/1.10.6/2026-10-19.special-full-review-v8/core/a3f6d148d541", "scope": "last_attestation_snapshot", "current": true, "attestation_binding_current": true, "attested_at": "2026-10-19T04:05:00.082Z", "projection_compared": false }, { "id": "math.special:calculator-html-cvp-projection", "status": "pass", "detail": "matches Trust display", "scope": "last_attestation_snapshot", "current": true, "attestation_binding_current": true, "attested_at": "2026-10-19T04:05:00.082Z" }, { "id": "math.special:evidence-html-cvp-projection", "status": "pass", "detail": "matches Trust display", "scope": "last_attestation_snapshot", "current": true, "attestation_binding_current": true, "attested_at": "2026-10-19T04:05:00.082Z" } ], "digest_registry": { "digest_algorithm": "sha256", "canonicalization": "json-sorted-keys-utf8", "entries": [ { "name": "cvp_core_sha256", "field": "artifacts.cvp_core_sha256", "digest_scope": "cvp_verification_fingerprint_v2", "value": "a3f6d148d5414a4e99ff3d77f431d1b0818c94a7e122761a2239a86d2e1fdaae", "note": "Immutable verification core; excludes attestation/live-status and Evidence contract counts" }, { "name": "cvp_evidence_digest", "field": "artifacts.evidence_digest", "digest_scope": "cvp_verification_fingerprint_v2", "value": "a3f6d148d5414a4e99ff3d77f431d1b0818c94a7e122761a2239a86d2e1fdaae", "note": "Alias of cvp_core_sha256 (historical field name)" }, { "name": "artifact_evidence_sha256", "field": "release_integrity.digests.evidence_core_sha256", "digest_scope": "evidence_artifact_core", "value": "8cfeedba16d810bfb48e8309870d38146725ccaa7ec049180ed7f67aea3f120d", "note": "Published Evidence JSON core — Trust freshness / production bind subject" }, { "name": "evidence_catalog_digest", "field": "artifacts.attested_catalog_sha256", "digest_scope": "capability_catalog", "value": "93b711f8b5b8d602c9f2ecfb852b561e5cd33585603555116fdf45f0bd4f1957", "note": "Capability catalog integrity; site-wide, not per-suite fingerprint" } ] }, "cvp_core_sha256": "a3f6d148d5414a4e99ff3d77f431d1b0818c94a7e122761a2239a86d2e1fdaae" }, "dependencies": [ { "name": "special-functions-engine", "version": "1.10.6" }, { "name": "special-functions-reference-oracle", "version": "1.0.0" }, { "name": "math.factorial", "version": "1.0.2", "role": "cross_calculator_peer", "capability_id": "math.factorial", "evidence_digest": "b7065c58307c3477dc76583b92ad80a0fffcbe5c4ed3ec567d0cd56fa6edf1a6", "evidence": "https://www.calculatorx.com/evidence/math.factorial/1.0.2.cvp.json" }, { "name": "node", "version": "22.20.0" } ], "attestation": { "status": "CURRENT", "href": "/.well-known/calculatorx-production-attestation.json", "id": "c64cc612dc6c3a945c8f29b84029807be91aeb07eee35eea41b94b1b8594e211", "attested_at": "2026-10-19T04:05:00.082Z", "last_confirmed_current_at": "2026-10-19T07:58:46.038Z", "site_attestation_current": false, "capability_attestation_current": true, "bound_snapshot_status": "pass", "live_report_status": "stale", "bound_attested_at": "2026-10-19T07:58:46.038Z", "live_updated_at": "2026-10-19T07:58:59.631Z", "bound_snapshot": { "status": "pass", "attested_at": "2026-10-19T07:58:46.038Z", "site_stale_count": 4 }, "live_site_status": { "href": "/.well-known/calculatorx-production-attestation.json" }, "bound_snapshot_site_stale_count": 4, "site_stale_capability_count": 4, "evidence_attested_at": "2026-10-19T04:05:00.082Z" }, "domain_review": { "status": "ABSENT", "spec_version": "1.10.6", "external_third_party": false, "note": "Named expert review not performed" }, "timestamps": { "verified_at": "2026-10-19T04:05:00.082Z", "generated_at": "2026-10-19T04:05:00.082Z", "rendered_at": "2026-10-19T07:59:00.247Z", "notes": "verified_at/generated_at mark the first publication of this cvp_core_sha256 (artifacts.evidence_digest alias) for suite verification. evidence_attested_at/attested_at is the first production bind of that core digest. last_confirmed_current_at is the latest site-wide report that still confirms CURRENT. rendered_at is this JSON serialization and is not part of cvp_core_sha256 / evidence_digest. Production attestation binds cvp_core_sha256 (binds evidence_digest). Causal order: verified_at ≤ evidence_attested_at ≤ last_confirmed_current_at. attestation.bound_snapshot.site_stale_count is the site-wide stale capability count at Manifest bind time — not a live value. Fetch attestation.live_site_status.href for the current site report." }, "tests": { "golden": { "passed": 1000, "total": 1000, "failed": 0 }, "boundary": { "passed": 8, "total": 8, "failed": 0 }, "invalid": { "passed": 24, "total": 24, "failed": 0, "comparison": "exact_error_match" }, "property": { "passed": 1, "total": 1, "failed": 0, "kind": "fixed" }, "round_trip": { "passed": 0, "total": 0, "failed": 0 }, "metamorphic": { "passed": 3, "total": 3, "failed": 0 }, "contract": { "passed": 6, "total": 6, "failed": 0 }, "cross_interface": { "passed": 24, "total": 24, "failed": 0 }, "cross_calculator": { "passed": 1, "total": 1, "failed": 0 } }, "numerical": { "representation": "IEEE-754 binary64; IUT publishes toPrecision(12) / roundSmart", "tolerance": { "type": "relative", "value": 1e-8, "rationale": "O2 is unrounded closed-form identities (antiderivative, quadratic formula, vertex, y0 e^{kΔt}, Lagrange/lerp). IUT is iterative/quadrature with 12-digit publication. Relative 1e-8 covers truncation on well-posed goldens plus roundSmart; |expected|<1e-8 uses absolute 1e-6 because golden/brent stop at default tol 1e-8. Declared a priori.", "declared_before_evaluation": true }, "max_error_observed": 33692 }, "coverage": { "partitions": [ { "id": "erf", "description": "erf", "exercised": true, "case_count": 82 }, { "id": "erfc", "description": "erfc", "exercised": true, "case_count": 44 }, { "id": "gamma", "description": "gamma", "exercised": true, "case_count": 28 }, { "id": "lgamma", "description": "lgamma", "exercised": true, "case_count": 32 }, { "id": "bessel-j", "description": "bessel_j", "exercised": true, "case_count": 212 }, { "id": "bessel-y", "description": "bessel_y", "exercised": true, "case_count": 155 }, { "id": "bessel-i", "description": "bessel_i", "exercised": true, "case_count": 209 }, { "id": "bessel-k", "description": "bessel_k", "exercised": true, "case_count": 155 }, { "id": "lambert-w", "description": "lambert_w", "exercised": true, "case_count": 22 }, { "id": "lambert-wm1", "description": "lambert_wm1", "exercised": true, "case_count": 10 }, { "id": "beta", "description": "beta", "exercised": true, "case_count": 15 }, { "id": "incomplete-beta", "description": "incomplete_beta", "exercised": true, "case_count": 65 }, { "id": "sinc", "description": "sinc", "exercised": true, "case_count": 14 }, { "id": "normalized-sinc", "description": "normalized_sinc", "exercised": true, "case_count": 13 }, { "id": "invalid-domain", "description": "Invalid inputs", "exercised": true, "case_count": 24 } ] }, "reference": { "items": [ { "class": "O1", "reference_scope": "model", "scope": "model", "label": "erf, Γ, Bessel, Lambert W, Beta, sinc as one capability" }, { "class": "O2", "reference_scope": "expected_values", "scope": "expected_values", "label": "Extended mpmath high-precision reference:100/160digits plus scale guards,526cases; independent Python implementation and exact binary64 input arguments", "engine": { "name": "mpmath1.4.1", "precision_bits": 332, "independent_of_iut": true, "same_runtime_as_iut": false } }, { "class": "O2", "reference_scope": "expected_values", "scope": "expected_values", "label": "erf/erfc and Bessel J/Y/I/K fixed references from mpmath 1.4.1, exact binary64 inputs; 90/140 decimal digits agree at 75 printed digits", "engine": { "name": "mpmath 1.4.1", "precision_bits": 299, "independent_of_iut": true, "same_runtime_as_iut": false } }, { "class": "O2", "reference_scope": "expected_values", "scope": "expected_values", "label": "Separate-module special-functions oracle — Γ(n)=(n−1)!, Γ(1/2)=√π, sinc identities, A&S erf, independent Bessel series, Lambert branch identities (not IUT Lanczos/Halley transcription)", "engine": { "name": "special-functions-reference", "precision_bits": 53, "independent_of_iut": true, "same_runtime_as_iut": true } }, { "class": "O2", "reference_scope": "numerical_behavior", "scope": "numerical_behavior", "label": "Same O2 module backs unrounded identities vs IUT roundSmart — Extended mpmath fixtures relative1e-10; sampled legacy identities retain relative1e-8/absolute1e-6; no full-domain guarantee", "engine": { "name": "special-functions-reference", "precision_bits": 53, "independent_of_iut": true, "same_runtime_as_iut": true } } ] }, "links": { "legacy_evidence": "https://www.calculatorx.com/evidence/math.special/1.10.6.json", "evidence_page": "https://www.calculatorx.com/evidence/math.special/1.10.6", "calculator": "https://www.calculatorx.com/calc/math/special-functions", "schema": "https://www.calculatorx.com/developers/cvp/schema/cvp-evidence-1.0.json", "protocol": "https://www.calculatorx.com/developers/cvp/spec/1.0", "manifest": "https://www.calculatorx.com/evidence/math.special/1.10.6.cvp.json", "manifest_immutable": "https://www.calculatorx.com/evidence/math.special/1.10.6/2026-10-19.special-full-review-v8.cvp.json", "manifest_core": "https://www.calculatorx.com/evidence/math.special/1.10.6/2026-10-19.special-full-review-v8.a3f6d148d541.cvp.json" }, "disclosure": { "policy": "calculatorx-public-evidence-v1", "full_validation": "private", "representation": "summary" } }