{
  "capability_id": "math.exponential_growth",
  "tool_id": "exponential-growth",
  "engine": "exponential-growth",
  "calculation_version": "1.0.7",
  "generated_at": "2026-09-16T22:03:43.783Z",
  "engine_build": "2026-09-10",
  "test_count": 20,
  "passed": 20,
  "failed": 0,
  "tests_summary": {
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      "total": 14,
      "passed": 14
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    "boundary": {
      "total": 6,
      "passed": 6
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    "property": {
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  "expert_reviewed": false,
  "internally_reviewed": false,
  "review_kind": null,
  "source_checked": true,
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    "scope": [
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      "input semantics",
      "assumptions",
      "limitations",
      "sources",
      "tests"
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    "note": "Domain expert review is optional under CVP (CVP-EXP-01). Absence does not block CVP Verified. See Review policy for supplemental domain assurance."
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  "assumptions": [
    "Constant rate over the interval (real populations often need logistic models long-term).",
    "CalculatorX defines the periodic model domain as r > −100% so the growth factor stays positive and inverse/logarithmic solving remains well-defined."
  ],
  "limitations": [
    "Periodic model requires r > −100% → RATE_MUST_BE_GT_MINUS_100",
    "Non-finite inputs → INVALID_NUMBER"
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  "boundary_conditions": [
    "Need three known values to solve the fourth.",
    "All four values enter verify / consistency mode rather than being rejected.",
    "Grouped inputs such as 1,705.97 are canonicalized to 1705.97.",
    "Displayed x(t) may match while a small Difference remains — that residual is from the unrounded result.",
    "Periodic r ≤ −100% is rejected as outside the model domain (not because 0^t is undefined for every forward case).",
    "x₀ = 0 yields a trivial zero path."
  ],
  "sources": [
    {
      "title": "NIST DLMF §4.2(i) The Logarithm",
      "section": "Principal logarithm ln z",
      "url": "https://dlmf.nist.gov/4.2",
      "supports": "Inverse solving for t; doubling time t₂=ln 2 / ln(1+r/100); half-life via ln(1/2)",
      "kind": "reference",
      "accessed": "2026-08-25"
    },
    {
      "title": "NIST DLMF §4.2(iii) The Exponential Function",
      "section": "exp z and the real exponential",
      "url": "https://dlmf.nist.gov/4.2",
      "supports": "Continuous model x(t)=x₀ e^(kt)",
      "kind": "reference",
      "accessed": "2026-08-25"
    },
    {
      "title": "NIST DLMF §4.2(iv) Powers",
      "section": "Principal value z^a = exp(a ln z)",
      "url": "https://dlmf.nist.gov/4.2",
      "supports": "Periodic model x(t)=x₀(1+r/100)^t as a real power of the positive growth factor 1+r/100",
      "kind": "reference",
      "accessed": "2026-08-25"
    }
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  "fingerprint": "sha256:db1f1770a512fc80cc710f57cdb3bb6c1ff47cf809b3c9b304d0a686c2a0e6dc",
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      "tol": 0.01,
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    {
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      "inputs": {
        "x0": 10000,
        "ratePct": 5,
        "t": 11
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      "tol": 0,
      "engine": "exponential-growth-x0",
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          "comparison_basis": "display_quantization",
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          "half_ulp": 0.005,
          "rounding_rule": "consistent iff formatGrowthAmount(calculated) === formatGrowthAmount(provided) using the same Result Card quantization; half_ulp and relative_error are diagnostics (numeric_close), not acceptance shortcuts",
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      "code": "RATE_MUST_BE_GT_MINUS_100",
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