Artifact version: 1
Metrics¶
| Metric identity | Symbol | Value | Units | Status |
|---|---|---|---|---|
| exponential.absolute_residual | abs(f(x_star)) | 0.0 | function units | info |
| exponential.ad_derivative | `dx_star/dtheta | AD` | 0.5 | coordinate units per parameter unit |
| exponential.analytic_derivative | `dx_star/dtheta | analytic` | 0.5 | coordinate units per parameter unit |
| exponential.bracket_width | Delta_x | 0.0 | coordinate units | info |
| exponential.certificate | I_certified | 1 | dimensionless | info |
| exponential.fd_derivative | `dx_star/dtheta | FD` | 0.5000000000088267 | coordinate units per parameter unit |
| exponential.relative_ad_analytic_error | R_AD,analytic | 0.0 | dimensionless | info |
| exponential.relative_ad_fd_error | R_AD,FD | 1.7653434269648046e-11 | dimensionless | info |
| exponential.root | x_star | 0.6931471805599453 | coordinate units | info |
| exponential.slope_magnitude | abs(df/dx) | 2.0 | function units per coordinate unit | info |
| linear.absolute_residual | abs(f(x_star)) | 0.0 | function units | info |
| linear.ad_derivative | `dx_star/dtheta | AD` | 1.0 | coordinate units per parameter unit |
| linear.analytic_derivative | `dx_star/dtheta | analytic` | 1.0 | coordinate units per parameter unit |
| linear.bracket_width | Delta_x | 0.0 | coordinate units | info |
| linear.certificate | I_certified | 1 | dimensionless | info |
| linear.fd_derivative | `dx_star/dtheta | FD` | 1.0000000000065512 | coordinate units per parameter unit |
| linear.relative_ad_analytic_error | R_AD,analytic | 0.0 | dimensionless | info |
| linear.relative_ad_fd_error | R_AD,FD | 6.551204023665206e-12 | dimensionless | info |
| linear.root | x_star | 2.0 | coordinate units | info |
| linear.slope_magnitude | abs(df/dx) | 1.0 | function units per coordinate unit | info |
| quadratic.absolute_residual | abs(f(x_star)) | 4.440892098500626e-16 | function units | info |
| quadratic.ad_derivative | `dx_star/dtheta | AD` | 0.3535533905932738 | coordinate units per parameter unit |
| quadratic.analytic_derivative | `dx_star/dtheta | analytic` | 0.35355339059327373 | coordinate units per parameter unit |
| quadratic.bracket_width | Delta_x | 1.7763568394002505e-14 | coordinate units | info |
| quadratic.certificate | I_certified | 1 | dimensionless | info |
| quadratic.fd_derivative | `dx_star/dtheta | FD` | 0.35355339059739416 | coordinate units per parameter unit |
| quadratic.relative_ad_analytic_error | R_AD,analytic | 1.5700924586837752e-16 | dimensionless | info |
| quadratic.relative_ad_fd_error | R_AD,FD | 1.1654168283690369e-11 | dimensionless | info |
| quadratic.root | x_star | 1.414213562373095 | coordinate units | info |
| quadratic.slope_magnitude | abs(df/dx) | 2.82842712474619 | function units per coordinate unit | info |
Comparisons¶
| Comparison | Metric | Relation | Reference | Units | Absolute tolerance | Relative tolerance | Status | Note |
|---|---|---|---|---|---|---|---|---|
| exponential.absolute_residual.gate | exponential.absolute_residual | less_equal | 1e-12 | function units | 0.0 | 0.0 | pass | Existing implicit-root validation threshold. |
| exponential.bracket_width.gate | exponential.bracket_width | less_equal | 1e-12 | coordinate units | 0.0 | 0.0 | pass | Existing implicit-root validation threshold. |
| exponential.certificate.gate | exponential.certificate | equal | 1 | dimensionless | 0.0 | 0.0 | pass | The complete runtime certificate accepted the case. |
| exponential.relative_ad_analytic_error.gate | exponential.relative_ad_analytic_error | less_equal | 1e-09 | dimensionless | 0.0 | 0.0 | pass | Existing implicit-root validation threshold. |
| exponential.relative_ad_fd_error.gate | exponential.relative_ad_fd_error | less_equal | 1e-06 | dimensionless | 0.0 | 0.0 | pass | Existing implicit-root validation threshold. |
| exponential.slope_magnitude.gate | exponential.slope_magnitude | greater_equal | 1e-08 | function units per coordinate unit | 0.0 | 0.0 | pass | Implicit derivative requires adequate root-slope conditioning. |
| linear.absolute_residual.gate | linear.absolute_residual | less_equal | 1e-12 | function units | 0.0 | 0.0 | pass | Existing implicit-root validation threshold. |
| linear.bracket_width.gate | linear.bracket_width | less_equal | 1e-12 | coordinate units | 0.0 | 0.0 | pass | Existing implicit-root validation threshold. |
| linear.certificate.gate | linear.certificate | equal | 1 | dimensionless | 0.0 | 0.0 | pass | The complete runtime certificate accepted the case. |
| linear.relative_ad_analytic_error.gate | linear.relative_ad_analytic_error | less_equal | 1e-09 | dimensionless | 0.0 | 0.0 | pass | Existing implicit-root validation threshold. |
| linear.relative_ad_fd_error.gate | linear.relative_ad_fd_error | less_equal | 1e-06 | dimensionless | 0.0 | 0.0 | pass | Existing implicit-root validation threshold. |
| linear.slope_magnitude.gate | linear.slope_magnitude | greater_equal | 1e-08 | function units per coordinate unit | 0.0 | 0.0 | pass | Implicit derivative requires adequate root-slope conditioning. |
| quadratic.absolute_residual.gate | quadratic.absolute_residual | less_equal | 1e-12 | function units | 0.0 | 0.0 | pass | Existing implicit-root validation threshold. |
| quadratic.bracket_width.gate | quadratic.bracket_width | less_equal | 1e-12 | coordinate units | 0.0 | 0.0 | pass | Existing implicit-root validation threshold. |
| quadratic.certificate.gate | quadratic.certificate | equal | 1 | dimensionless | 0.0 | 0.0 | pass | The complete runtime certificate accepted the case. |
| quadratic.relative_ad_analytic_error.gate | quadratic.relative_ad_analytic_error | less_equal | 1e-09 | dimensionless | 0.0 | 0.0 | pass | Existing implicit-root validation threshold. |
| quadratic.relative_ad_fd_error.gate | quadratic.relative_ad_fd_error | less_equal | 1e-06 | dimensionless | 0.0 | 0.0 | pass | Existing implicit-root validation threshold. |
| quadratic.slope_magnitude.gate | quadratic.slope_magnitude | greater_equal | 1e-08 | function units per coordinate unit | 0.0 | 0.0 | pass | Implicit derivative requires adequate root-slope conditioning. |
Environment policy¶
environment is an emission snapshot; --check gates deterministic controls, schema, units, and algorithmic metrics, not current revision
Limitations¶
Certification relies on caller assertions of uniqueness and smoothness.
Flat-slope rejection is validated in executable tests, not represented as a certified case.
Method payload¶
{
"cases": [
{
"absolute_residual": {
"unit": "function units",
"value": 0.0
},
"ad_derivative": {
"unit": "coordinate units per parameter unit",
"value": 1.0
},
"analytic_derivative": {
"unit": "coordinate units per parameter unit",
"value": 1.0
},
"bracket_width": {
"unit": "coordinate units",
"value": 0.0
},
"certified": true,
"fd_derivative": {
"unit": "coordinate units per parameter unit",
"value": 1.0000000000065512
},
"name": "linear",
"relative_ad_analytic_error": {
"unit": "dimensionless",
"value": 0.0
},
"relative_ad_fd_error": {
"unit": "dimensionless",
"value": 6.551204023665206e-12
},
"root": {
"unit": "coordinate units",
"value": 2.0
},
"slope_magnitude": {
"unit": "function units per coordinate unit",
"value": 1.0
},
"status": 0
},
{
"absolute_residual": {
"unit": "function units",
"value": 4.440892098500626e-16
},
"ad_derivative": {
"unit": "coordinate units per parameter unit",
"value": 0.3535533905932738
},
"analytic_derivative": {
"unit": "coordinate units per parameter unit",
"value": 0.35355339059327373
},
"bracket_width": {
"unit": "coordinate units",
"value": 1.7763568394002505e-14
},
"certified": true,
"fd_derivative": {
"unit": "coordinate units per parameter unit",
"value": 0.35355339059739416
},
"name": "quadratic",
"relative_ad_analytic_error": {
"unit": "dimensionless",
"value": 1.5700924586837752e-16
},
"relative_ad_fd_error": {
"unit": "dimensionless",
"value": 1.1654168283690369e-11
},
"root": {
"unit": "coordinate units",
"value": 1.414213562373095
},
"slope_magnitude": {
"unit": "function units per coordinate unit",
"value": 2.82842712474619
},
"status": 0
},
{
"absolute_residual": {
"unit": "function units",
"value": 0.0
},
"ad_derivative": {
"unit": "coordinate units per parameter unit",
"value": 0.5
},
"analytic_derivative": {
"unit": "coordinate units per parameter unit",
"value": 0.5
},
"bracket_width": {
"unit": "coordinate units",
"value": 0.0
},
"certified": true,
"fd_derivative": {
"unit": "coordinate units per parameter unit",
"value": 0.5000000000088267
},
"name": "exponential",
"relative_ad_analytic_error": {
"unit": "dimensionless",
"value": 0.0
},
"relative_ad_fd_error": {
"unit": "dimensionless",
"value": 1.7653434269648046e-11
},
"root": {
"unit": "coordinate units",
"value": 0.6931471805599453
},
"slope_magnitude": {
"unit": "function units per coordinate unit",
"value": 2.0
},
"status": 0
}
],
"controls": {
"fd_step": {
"unit": "parameter units",
"value": 1e-05
},
"residual_limit": {
"unit": "function units",
"value": 1e-12
},
"slope_floor": {
"unit": "function units per coordinate unit",
"value": 1e-08
},
"width_limit": {
"unit": "coordinate units",
"value": 1e-12
}
},
"environment": {
"device": "cpu:0",
"git_revision": "a3f0d4bd563555e67a3ec0ca8e1ed0e8be671f60",
"jax_backend": "cpu",
"jax_version": "0.10.1",
"measured_at_utc": "2026-07-13T03:01:34.508984+00:00",
"platform": "macOS-26.1-arm64-arm-64bit-Mach-O",
"python_version": "3.13.7",
"working_tree_dirty": true
},
"precision": "float64",
"provenance_policy": "environment is an emission snapshot; --check gates deterministic controls, schema, units, and algorithmic metrics, not current revision",
"schema_version": 2
}