> ## Documentation Index
> Fetch the complete documentation index at: https://offthepace.mintlify.site/llms.txt
> Use this file to discover all available pages before exploring further.

# Identity-closure tests: the additive identities, enforced

> The fourteen hand-written assertions that prove the transform layer's additive identities and shrinkage bounds hold exactly, not approximately, on every row of every build.

This is where "the identity closes to 0.1 ms" stops being a claim about the SQL and becomes a property CI checks on every row, every build. Fourteen singular tests: thirteen run real logic today; one is a placeholder, documented honestly below rather than silently passing as if it were full coverage.

## The shared closure macro

Most of these tests share one shape: a total column must equal the sum of named components plus a residual, within a tolerance. Rather than writing that comparison fourteen times, [`assert_additive_identity`](/reference/macros/assert_additive_identity) takes the model, the total column, the component list, and the residual column, and returns whichever rows violate the sum:

```sql theme={null}
{{ assert_additive_identity(
     ref('int_lap_residual_decomposed'),
     'pace_delta_s',
     ['fuel_component_s', 'compound_component_s', 'rubber_component_s',
      'ambient_component_s', 'constructor_component_s', 'dirty_air_tax_s'],
     'driver_skill_residual_s',
     tolerance=0.0001
) }}
```

That's the literal body of `assert_lap_7term_identity`, the "Lap 7-term residual identity" test below that enforces the [seven-term identity](/decomposition/seven-term-identity) itself. Three more of the fourteen tests below restate the same kind of sum-equals-total check by hand, at sector and qualifying grain, where the column lists differ enough that a direct `SELECT` reads more clearly than a macro call.

<AccordionGroup>
  <Accordion title="Residual decomposition identity">
    `driver_skill_residual_s` must equal `pace_delta_s − total_explained_s`, within 0.0001 s. This is the identity's original two-term form total explained vs. residual that the seven-term test below further decomposes into six named physics components. `track_unexplained_s` is explicitly excluded from both sides: it's the track-evolution model's own internal residual, informational, not part of the closure. Guards [`int_lap_residual_decomposed`](/reference/models/int/int_lap_residual_decomposed). **Active.**
  </Accordion>

  <Accordion title="Lap 7-term residual identity">
    The headline test. After `dirty_air_tax_s` is extracted as its own term, the closure is:

    $\text{pace\_delta\_s} = \text{fuel} + \text{compound} + \text{rubber} + \text{ambient} + \text{constructor} + \text{dirty\_air\_tax} + \text{driver\_skill\_residual}$

    enforced via [`assert_additive_identity`](/reference/macros/assert_additive_identity) at 0.0001 s tolerance. Full derivation and the meaning of each term: [the Seven-Term Identity](/decomposition/seven-term-identity). The model this closes: [`families/residual`](/transform/families/residual). Guards [`int_lap_residual_decomposed`](/reference/models/int/int_lap_residual_decomposed). **Active.**
  </Accordion>

  <Accordion title="Qualifying 7-term identity">
    The same identity, re-fit at qualifying grain rather than mirrored from the race-grain coefficients qualifying laps have their own fuel, compound, rubber, ambient, constructor, and dirty-air terms because a single push lap has a fundamentally different fuel and tyre-temperature profile than a race stint:

    $\text{quali\_pace\_delta\_s} = \sum(\text{6 quali-fit components}) + \text{quali\_skill\_residual\_s}$

    Same 0.0001 s tolerance as the race-grain test proof that the qualifying-specific coefficients are mathematically sound, not just plausible. Guards [`int_qualifying_decomposed`](/reference/models/int/int_qualifying_decomposed). **Active.**
  </Accordion>

  <Accordion title="Sector residual identity">
    Each of a lap's three sectors closes its own copy of the identity `sector_pace_delta_s` equals the sum of six sector-grain components plus `sector_driver_skill_residual_s`, at the same 0.0001 s tolerance the lap grain uses (the linear time-share allocation that produces sector components from lap components introduces no meaningful error). Guards [`int_sector_residual_decomposed`](/reference/models/int/int_sector_residual_decomposed). **Active.**
  </Accordion>

  <Accordion title="Sector-to-lap re-aggregation">
    Checks whether `SUM(sector_driver_skill_residual_s)` over a lap's three sectors equals that lap's own `driver_skill_residual_s` from the lap-grain model. It doesn't, today, by more than the 0.001 s tolerance and the gap is structural, not a bug: the sector-grain baseline (`field_sector_pace_smoothed_s`, a per-sector median) and the lap-grain baseline (`base_track_pace_s`, a trimmed-mean smoothed value) are fit independently, and the median of a sum isn't the sum of medians. Closing this exactly would mean fitting a joint sector-grain baseline via panel regression instead of three independent sector medians a larger change than this test's scope. Tagged `placeholder`; returns no rows by construction until that joint baseline exists. **Placeholder.**
  </Accordion>

  <Accordion title="Corner closure">
    `braking_loss_s + mid_corner_residual_s + exit_residual_s` must equal `corner_residual_total_s` within 0.001 s, checked only where all three phase residuals are non-`NULL`. Unlike the sector split, this isn't a proportional time-share allocation of a known total each phase residual is measured independently against a field median so the sum equaling the total is a real, checked property rather than true by construction. Guards [`int_corner_skill_residuals`](/reference/models/int/int_corner_skill_residuals). **Active.**
  </Accordion>

  <Accordion title="Ghost-car self-consistency">
    A degenerate-case identity rather than an additive one: when the ghost car's ego driver and host constructor are the same (i.e. the counterfactual recombination is asking "what if this driver drove their own car"), `predicted_lap_time_s` must equal the driver's own `actual_lap_time_s` exactly, within 0.0001 s. If recombination is mathematically correct, the self-case has to be a no-op; any drift here means the recombination formula itself is wrong, not just imprecise for genuine swaps. Guards [`fct_ghost_car_pace`](/reference/models/fct/fct_ghost_car_pace). **Active.**
  </Accordion>

  <Accordion title="Example identity closure">
    The same macro call as the lap 7-term test above, byte-for-byte kept as a standalone, readable usage example of `assert_additive_identity` rather than removed once the canonical test existed. It doesn't add coverage beyond what the lap 7-term test already enforces; its job is documentation-by-example for anyone adding a new closure test, not an independent guarantee. Guards [`int_lap_residual_decomposed`](/reference/models/int/int_lap_residual_decomposed). **Active.**
  </Accordion>

  <Accordion title="Deg-slope centering">
    An internal-consistency check on the field-mean itself, not on any one constructor's slope: the precision-weighted mean of `(deg_slope_raw_s_per_lap − field_mean_slope_s_per_lap)` must be exactly 0 within every `(race_year, compound)` cell, at 1e-9 s/lap tolerance (low-sample cells excluded they don't enter the field mean). If this drifts, "deviation from the field-average compound curve" stops meaning what every downstream consumer of the centred slope assumes it means. Guards [`int_constructor_deg_sensitivity`](/reference/models/int/int_constructor_deg_sensitivity). **Active.**
  </Accordion>

  <Accordion title="Cliff-hinge centering">
    The same centring property, applied to the post-cliff-onset hinge coefficient instead of the pre-cliff slope: the precision-weighted mean of `(cliff_hinge_coef_s_per_lap − cliff_hinge_field_mean_s_per_lap)` must be 0 within every `(race_year, compound)` cell, same 1e-9 s/lap tolerance, low-sample cliff cells excluded. Guards [`int_constructor_deg_sensitivity`](/reference/models/int/int_constructor_deg_sensitivity). **Active.**
  </Accordion>

  <Accordion title="Affinity shrinkage bounds">
    Three tests below share one statistical property: Bayesian shrinkage toward a proper prior is convex, so the posterior (shrunk) mean is mathematically guaranteed to land between the raw observed value and the prior mean never outside either:

    $\text{shrunk} \in \big[\min(\text{raw}, \text{prior}),\ \max(\text{raw}, \text{prior})\big]$

    This one checks it for per-circuit driver affinity, at floating-point epsilon (1e-9 s). A violation isn't a borderline statistical judgment it means the shrinkage formula itself is wrong, or the prior and observed values were mismatched (e.g. computed over different row sets). Guards [`int_driver_circuit_affinity`](/reference/models/int/int_driver_circuit_affinity). **Active.**
  </Accordion>

  <Accordion title="Era-affinity shrinkage bounds">
    The same convexity bound, checked within each era cell instead of pooled across eras. Guards [`int_driver_circuit_era_affinity`](/reference/models/int/int_driver_circuit_era_affinity). **Active.**
  </Accordion>

  <Accordion title="Era-rating shrinkage bounds">
    The same convexity bound again, applied to era-normalized driver ratings: `shrunk_residual_s` must land between the driver's raw residual mean and the season mean. Guards [`int_driver_season_ratings`](/reference/models/int/int_driver_season_ratings). **Active.**
  </Accordion>

  <Accordion title="Affinity CI brackets mean">
    A 95% credible interval built symmetrically around a posterior mean (`mean ± 1.96·se`) has to bracket that mean by construction `ci_low ≤ mean ≤ ci_high`. This test checks that property across both circuit-affinity models at once (era and non-era), wherever the standard error is estimable; degenerate single-observation cells carry `NULL` bounds by design and are excluded rather than flagged. A violation means the mean and the CI bounds were computed from mismatched intermediates a stale standard error, or low/high swapped. Guards [`int_driver_circuit_affinity`](/reference/models/int/int_driver_circuit_affinity) and [`int_driver_circuit_era_affinity`](/reference/models/int/int_driver_circuit_era_affinity). **Active.**
  </Accordion>
</AccordionGroup>