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  "engine_build": "2026-09-20",
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  "passed": 18,
  "failed": 0,
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      "passed": 7
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    "boundary": {
      "total": 3,
      "passed": 3
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    "property": {
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      "passed": 8
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  "expert_reviewed": false,
  "internally_reviewed": false,
  "review_kind": null,
  "source_checked": true,
  "reviewed_by": null,
  "reviewer_role": null,
  "last_reviewed": null,
  "expert_review": {
    "status": "optional_not_performed",
    "reviewer": null,
    "scope": [
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      "input semantics",
      "assumptions",
      "limitations",
      "sources",
      "tests"
    ],
    "note": "Domain expert review is optional under CVP (CVP-EXP-01). Absence does not block CVP Verified. See Review policy for supplemental domain assurance."
  },
  "internal_review": {
    "status": "not_applicable",
    "reviewer": null
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  "page": "https://www.calculatorx.com/calc/math/linear-equation",
  "api": "https://www.calculatorx.com/api/v1/calc/linear-equation",
  "schema_input": "https://www.calculatorx.com/schemas/math.algebra.linear_equation.input.json",
  "schema_output": "https://www.calculatorx.com/schemas/math.algebra.linear_equation.output.json",
  "assumptions": [
    "Degree 1 only",
    "Not quadratic"
  ],
  "limitations": [
    "Not ax² + bx + c = 0 — use /calc/math/quadratic",
    "Not a quadratic inequality",
    "Not a linear system",
    "Not CAS",
    "a = 0 and b = rhs → infinite (equation)",
    "a = 0 and b ≠ rhs → inconsistent (equation)",
    "a = 0 inequality → ℝ or ∅",
    "|ax+b|=rhs is mode=abs; rhs < 0 → empty",
    "|ax+b| ○ rhs is mode=abs_inequality; rhs < 0 → ℝ or ∅",
    "(ax+b)/(cx+d)=rhs is mode=rational; pole x=−d/c excluded; identically 0 denominator → INVALID_INPUT",
    "(ax+b)/(cx+d) ○ rhs is mode=rational_inequality; pole excluded; identically 0 denominator → INVALID_INPUT"
  ],
  "boundary_conditions": [
    "missing a or rhs → MISSING_REQUIRED_INPUT",
    "unknown mode → INVALID_MODE"
  ],
  "sources": [
    "ISO 80000-2:2019 Quantities and units — Mathematics"
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  "methods": [],
  "fingerprint": "sha256:377665020e0dc7814f58864ebd93118f1ea85ee2e7d982d8842b8ccfc1e4df2e",
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      "ok": true,
      "kind": "golden",
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        "b": 3,
        "rhs": 11
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        "status": "success"
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        "b": 3,
        "rhs": 11,
        "x": 4,
        "status": "success",
        "formula": "x = (rhs − b) / a",
        "mode": "solve"
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      "tol": null
    },
    {
      "id": "golden-linear-infinite",
      "ok": true,
      "kind": "golden",
      "inputs": {
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        "b": 5,
        "rhs": 5
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      "expect": {
        "status": "infinite"
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      "expectWarning": null,
      "actual": {
        "a": 0,
        "b": 5,
        "rhs": 5,
        "x": null,
        "status": "infinite",
        "formula": "0·x + b = rhs",
        "mode": "solve"
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      "tol": null
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    {
      "id": "golden-linear-empty",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "a": 0,
        "b": 5,
        "rhs": 6
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      "expect": {
        "status": "inconsistent"
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      "expectWarning": null,
      "actual": {
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        "b": 5,
        "rhs": 6,
        "x": null,
        "status": "inconsistent",
        "formula": "0·x + b = rhs",
        "mode": "solve"
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      "tol": null
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    {
      "id": "boundary-linear-missing",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "a": 2
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      "expectError": "MISSING_REQUIRED_INPUT",
      "code": "MISSING_REQUIRED_INPUT",
      "error": "Enter rhs."
    },
    {
      "id": "boundary-linear-mode",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "mode": "quadratic",
        "a": 2,
        "b": 3,
        "rhs": 11
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      "expectError": "INVALID_MODE",
      "code": "INVALID_MODE",
      "error": "mode must be solve, inequality, abs, abs_inequality, rational, or rational_inequality. Not quadratic, not a system, not CAS."
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    {
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      "ok": true,
      "kind": "golden",
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        "mode": "inequality",
        "a": 2,
        "b": 3,
        "rhs": 11,
        "op": "lt"
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      "expect": {
        "status": "interval",
        "bound": 4,
        "sense": "lt"
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      "expectWarning": null,
      "actual": {
        "a": 2,
        "b": 3,
        "rhs": 11,
        "mode": "inequality",
        "op": "lt",
        "op_symbol": "<",
        "sense": "lt",
        "sense_symbol": "<",
        "x": 4,
        "bound": 4,
        "status": "interval",
        "interval": "(-∞, 4)",
        "formula": "x < (rhs − b) / a"
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      "tol": null
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    {
      "id": "golden-ineq-flip",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "inequality",
        "a": -2,
        "b": 3,
        "rhs": 11,
        "op": "lt"
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      "expect": {
        "status": "interval",
        "bound": -4,
        "sense": "gt"
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      "expectWarning": null,
      "actual": {
        "a": -2,
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        "rhs": 11,
        "mode": "inequality",
        "op": "lt",
        "op_symbol": "<",
        "sense": "gt",
        "sense_symbol": ">",
        "x": -4,
        "bound": -4,
        "status": "interval",
        "interval": "(-4, ∞)",
        "formula": "a < 0 flips <"
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      "tol": null
    },
    {
      "id": "golden-ineq-all",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "inequality",
        "a": 0,
        "b": 5,
        "rhs": 6,
        "op": "lt"
      },
      "expect": {
        "status": "all"
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      "expectWarning": null,
      "actual": {
        "a": 0,
        "b": 5,
        "rhs": 6,
        "mode": "inequality",
        "op": "lt",
        "op_symbol": "<",
        "sense": "lt",
        "x": null,
        "bound": null,
        "status": "all",
        "interval": "ℝ",
        "formula": "0·x + b < rhs"
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      "tol": null
    },
    {
      "id": "golden-ineq-empty",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "mode": "inequality",
        "a": 0,
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        "rhs": 5,
        "op": "lt"
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      "expect": {
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      "expectWarning": null,
      "actual": {
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        "rhs": 5,
        "mode": "inequality",
        "op": "lt",
        "op_symbol": "<",
        "sense": "lt",
        "x": null,
        "bound": null,
        "status": "empty",
        "interval": "∅",
        "formula": "0·x + b < rhs"
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      "tol": null
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      "id": "boundary-ineq-op",
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      "kind": "boundary",
      "inputs": {
        "mode": "inequality",
        "a": 2,
        "b": 3,
        "rhs": 11,
        "op": "=="
      },
      "expectError": "INVALID_MODE",
      "code": "INVALID_MODE",
      "error": "op must be <, ≤, >, or ≥."
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    {
      "id": "property-linear-x",
      "ok": true,
      "kind": "property",
      "detail": "2x+3=11 → 4",
      "expect": 4,
      "actual": 4
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      "id": "property-linear-third",
      "ok": true,
      "kind": "property",
      "detail": "3x=1",
      "expect": 0.3333333333333333,
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      "id": "property-linear-id",
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      "kind": "property",
      "detail": "0x+2=2",
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      "id": "property-linear-neg",
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      "detail": "−x+4=1",
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      "detail": "2x+3<11 → x<4",
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      "id": "property-ineq-flip",
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      "kind": "property",
      "detail": "−2x+3<11 flips",
      "expect": -4,
      "actual": -4
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    {
      "id": "property-ineq-all",
      "ok": true,
      "kind": "property",
      "detail": "0x+1<2 → ℝ",
      "expect": "all",
      "actual": "all"
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  "tier": "engine_tested",
  "verification": {
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      "passed": 18,
      "total": 18,
      "golden": 7,
      "boundary": 3,
      "property": 8
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