R-parity tolerance audit¶

tests/r_parity/compare.py::TOLERANCES is the single pre-registered tolerance budget for the StatsPAI ↔ R (and, where materialized, ↔ Stata) parity harness. Every number in that table is a claim: "on the committed golden fixtures, StatsPAI and the canonical reference agree to within this relative difference." A reviewer who sees a 500% entry with no justification is right to discount the whole table, so this document grades every loose entry, records the gap actually observed in the committed results/*.json artifacts, and lists — honestly — the entries we cannot yet justify.

Audited 2026-06-10 at 64 materialized modules; re-audited 2026-09-05 at 87 modules after the JSS parity-closure pass (Sun–Abraham, GLMM, PPML-HDFE, panel SFA, VAR, TWFE event study) and the addition of modules 86 (fect) and 87 (interflex). Re-audited 2026-09-22 at 89 modules (88 rdbwselect and 89 rdms joined the machine tier; 27 glmm_aghq moved to the iterative tier, see the loosening record below). The budget is enforced by tests/test_parity_harness_contract.py::test_headline_passes_are_inside_registered_r_tolerance (R side) and test_stata_headline_over_budget_modules_are_explicitly_registered (Stata side); the golden JSONs themselves are hash-locked by TIER_A_FIXTURE_LOCK.json, so neither side of any comparison can drift silently.

The three tolerance tiers¶

Every entry belongs to one of three regimes. Conflating them in a single column is what makes a tolerance table look arbitrary, so name the regime explicitly:

Machine precision (1e-6 and below). Same estimand, same convention, closed form or tightly converged optimizer. The residual is floating-point noise (typically 1e-15 to 1e-9; cross-BLAS reassociation in sandwich "meat" sums can reach ~1e-8, see verify_reproduce.py::REPRO_TOL_OVERRIDE). 79 of 89 modules register here on the point estimate.
Convention gap (1e-4 to 5e-2). Both implementations are correct, but they compute a documented different quantity: degrees-of-freedom divisors (T vs T−k), small-sample cluster corrections (ssc), expected vs observed information, analytic vs influence-function SEs. The budget bounds the size of the named convention difference, and the mechanism must be stated next to the entry.
Methodological (T3/T4, above 5e-2). The two sides cannot agree deterministically: independent forest RNG (combined Monte Carlo error), bootstrap vs delta-method inference, non-unique SCM donor weights. The budget bounds a documented methodological disagreement, the verdict is graded T3/T4 rather than treated as an ordinary deterministic pass, and the residual-noise source is recorded in the module's extra block.

Orthogonal to all three is the reproducibility tolerance (verify_reproduce.py, 1e-9): same code, same data, same packages must reproduce the committed golden values nearly bit-exactly. A parity tolerance never excuses a reproducibility drift.

Grading scheme¶

Each entry at or above 5e-2 (plus the formerly loose entries) gets a grade:

A — mechanistic. The two sides compute different, individually documented quantities by construction. The specific method on each side is named (from our source and the R function's documented method).
B — empirical. Like-for-like comparison with residual numerical or Monte Carlo noise; we report the observed gap and the margin (tolerance ÷ observed gap).
C — unjustified. Flagged for future work; no mechanism pinned and/or the budget does not actually bound the rows it appears to cover.

Audit rules. Never loosen a value without re-registering it here. Tighten when the observed gap (recomputed from the committed JSONs, worst across the R and Stata sides) is more than 5× smaller than the budget, to ≈3× the observed gap rounded to a clean number, floored at the harness-wide 1e-6 machine tier.

Scope of the 2026-06 audit round: the 5× rule was first applied to the legacy loose tier (budgets ≥ 1e-2). A follow-up pass then swept the mid-tier modules whose margin also exceeded 5×, recomputing the worst observed rel_se across both reference sides from the committed JSONs and tightening each to ≈3× that gap (floored at the 1e-6 machine tier):

module	old `rel_se`	obs worst (both sides)	new `rel_se`
`42_nbreg`	`1e-2`	`1.4e-3`	`5e-3`
`43_heckman`	`1e-3`	`8.6e-5`	`5e-4`
`28_frontier`	`1e-4`	`1.3e-5`	`5e-5`
`44_mlogit`	`1e-4`	`1.2e-5`	`5e-5`
`41_tobit`	`1e-3`	`2.0e-6`	`1e-5`
`14_ols_cluster`	`1e-3`	`6.1e-9`	`1e-6`
`24_coxph`	`1e-3`	`2.6e-15`	`1e-6`
`46_clogit`	`1e-3`	`2.7e-9`	`1e-6`
`49_oprobit`	`1e-3`	`3.0e-7`	`1e-6`

45_ologit (margin 5.1×, obs 2.0e-6) sits at the rule boundary and is left at 1e-5. No value was loosened; the harness contract test and the offline render both pass at the new budgets.

Reproducing the audit¶

Run from the repository root. This recomputes, for every module, the worst joined SE gap across both reference sides and flags sentinel (no-joined-SE-row) budgets and over-budget non-headline rows:

import importlib.util
import sys
from pathlib import Path

spec = importlib.util.spec_from_file_location(
    "compare", Path("tests/r_parity/compare.py")
)
compare = importlib.util.module_from_spec(spec)
sys.modules["compare"] = compare
spec.loader.exec_module(compare)

for module, tol in sorted(compare.TOLERANCES.items()):
    rows = compare.collect(module)
    worst_se = max(
        (v for d in rows for v in (d.rel_se, d.rel_se_st) if v is not None),
        default=None,
    )
    budget = tol.get("rel_se", tol.get("abs_se"))
    if worst_se is None:
        print(f"{module:22s} rel_se budget {budget:8.2g}  no joined SE row (sentinel)")
    elif worst_se > budget:
        print(f"{module:22s} rel_se budget {budget:8.2g}  OVER: observed {worst_se:.3g}")

The two enforcement commands (both must pass after any tolerance change):

# from the repository root
python3 tests/r_parity/compare.py          # re-render parity tables (idempotent)
python3 -m pytest tests/test_parity_harness_contract.py -q -o addopts=''

Entries at or above 5e-2 — full justification¶

Observed gaps are recomputed from the committed tests/r_parity/results/<module>_{py,R}.json and tests/stata_parity/results/<module>_Stata.json artifacts (worst row; R side / Stata side). "Margin" is tolerance ÷ worst observed gap.

Module	Quantity	Tolerance	Grade	Observed gap (R / Stata)	Justification
`05_sunab`	`rel_se`	3e-2	A	0.0093 / 8e-12	Sun–Abraham (`sun2021estimating`) event-time IW SEs. The default `att_rel_<e>` rows carry the Prop. 3 cohort-share term (Stata `eventstudyinteract` convention; py == Stata to 8e-12 on every row), while `fixest::sunab` treats the shares as fixed, so R differs by exactly that positive term at multi-cohort relative times (worst 0.93% at e=1) and by 8e-12 at single-cohort times and on the `agg="att"` ATT. The `att_rel_<e>_fixedshare` rows (`share_variance=False`) pin the fixest convention itself to 8e-12. Both sides use the fixest/reghdfe nested-K rule (12 observed cells + 5 year effects = 17 on mpdta); before 1.24.0 StatsPAI counted 9 unobserved cells and no time effects, which was the whole of the old "unpinned" 0.08% base gap. Margin 3.2×.
`06_rd`	`rel_se`	0.10	A	0.0671 / 0.0671	Default-bandwidth rows delegate to the official CCT port (`calonico2014robust`) and match R/Stata `rdrobust` at ~1e-12. The budget is bound only by the deliberately retained legacy internal SE rows at a forced common `h` (`forced_h*` diagnostics, observed 6.7%), kept to document the pre-delegation convention. Margin 1.5×.
`07_scm`	`rel_est`	1.0	A (T4)	headline 0.0239 vs R; donor weights up to 1.78 vs R, 0.0072 vs Stata	Classical SCM on the Basque fixture (`abadie2003economic`; method `abadie2010synthetic`). The donor-weight solution is not unique under the ADH nested-V specification: multi-start diagnostics find multiple near-best weight classes, and R `Synth::synth` and Stata `synth` land on measurably different local optima. StatsPAI's native solver tracks Stata. Verdict is GAP (T4 reference-disagreement disclosure), not PASS; module `52_scm_unique` certifies exact recovery on an identified DGP.
`13_causal_forest`	`rel_est`	0.01	B (T3)	0.0047 / —	AIPW doubly robust ATE/ATT on both sides (`sp.causal_forest` vs `grf::causal_forest` + `average_treatment_effect`; `wager2018estimation`, `athey2019generalized`). The two forests cannot share RNG, so the row is graded against combined Monte Carlo error (~0.05 combined SE on the clean-overlap DGP); a multi-seed truth-recovery pytest guard backs it. Widened from 0.005 when the ATT row moved onto grf's own plug-in + Hajek-correction estimator: both sides now run the same estimator, so the residual 0.47% is pure forest MC, and 0.005 left only 1.07× margin against machine-to-machine drift. The ATE headline sits at 0.19%. Margin 2.1×.
`13_causal_forest`	`rel_se`	0.25	B	0.077 / —	The AIPW operator is now pinned exactly — fed grf's own forest outputs, StatsPAI reproduces `grf::get_scores` elementwise to 2.3e-14 and grf's ATE/ATT estimate and `std.err` to 1e-15 (`tests/reference_parity/test_grf_aipw_operator_parity.py`) — so this budget covers forest RNG only, not an unresolved formula difference. Tightened from 0.50 after the ATT convention fix removed the historical 14.6% ATT row (now 0.087%); the remaining worst is the ATE row at 7.7%, whose SE depends on the forest's own `tau.hat` dispersion. Margin 3.2×.
`29_panel_sfa`	`rel_se`	5e-2	B	0.0184 / 2.9e-6	Half-normal Pitt–Lee panel SFA (`pitt1981measurement`). All three sides now fit the same likelihood: the Stata do-file constrains `xtfrontier, ti`'s truncated-normal `mu` to 0 (before 2026-09 the Stata rows were a different model, mislabelled as a "scale" difference; the intercept and `sigma_u` now agree to 1e-6). Stata's analytic-Hessian OIM (`ml` method `d2`) matches StatsPAI's central-difference OIM to 3e-6 on every SE row, which pins the Python Hessian as exact; `frontier::sfa`'s `mleCov` (Coelli's FRONTIER 4.1 routine) is 0.1%–1.8% off, worst on the intercept. Budget = 3× the worst R-side row; margin 2.7×.
`30_oaxaca`	`rel_se`	0.05	A	0.0125 / 0.0122	Blinder–Oaxaca (`blinder1973wage`, `oaxaca1973male`; cf. `jann2008blinder`). StatsPAI reports closed-form delta-method SEs (`src/statspai/decomposition/oaxaca.py`); `oaxaca::oaxaca` reports seeded bootstrap SEs with `R=100` replications, whose own Monte Carlo noise is ~`(2R)^{-1/2}` ≈ 7% of the SE. Tightened 2026-06-10 from 1.0 (4× margin); a future regeneration that changes the bootstrap RNG stream may legitimately require re-registration.
`36_mediation`	`rel_se`	0.10	A	0.0701 / 0.0321	Causal mediation (`imai2010general`). StatsPAI uses bootstrap inference (B=1000); `mediation::mediate` uses quasi-Bayesian Monte Carlo with `sims=200` (~5% MC noise by itself); the Stata bridge uses delta-method SEs. Different inference algorithms by construction; point effects match at 1e-15. Margin 1.4×. Frozen by the contract test.
`40_qreg`	`rel_se`	0.10	A	0.0734 / 0.0302	Median regression (`koenker2005quantile`). StatsPAI uses the Powell-type iid kernel sandwich (`src/statspai/regression/quantile.py`, kernel estimate of the residual density at zero); the R fixture deliberately reports `summary(rq, se="nid")` — the Hendricks–Koenker difference-quotient sandwich — chosen to match Stata `qreg`'s default. Different sparsity estimators by construction. Margin 1.4×.

All remaining entries are at 3e-2 or tighter and are graded inline in compare.py. Modules 26_glmm_logit, 27_glmm_aghq and 47_ppmlhdfe_3fe left this table on 2026-09-05: their 1.9% / 1.9% / 1.8% "convention" gaps were StatsPAI defects (fixed-effect covariance conditional on the variance components; a small-sample factor matching neither reference) and are now at 1e-2 / 2e-5 / 1e-6.

Tightenings applied 2026-06-10¶

Rule applied: observed gap (worst over R and Stata sides, all joined rows — stricter than the headline-only enforcement) more than 5× smaller than the budget → tighten to ≈3× observed, floored at the harness machine tier 1e-6. No value was loosened.

Module	Quantity	Old	New	Observed worst gap	Basis
`03_hdfe`	`rel_se`	1e-2	1e-6	8.4e-15	The "1-df convention gap" comment was stale: with `ssc='fixest'`, IID SEs match `fixest::feols`/`reghdfe` at machine level on both sides.
`15_hdfe_cluster`	`rel_se`	5e-2	1e-6	1.25e-11	Stale "ssc convention": CR1 nested-FE cluster SEs now match on both sides.
`30_oaxaca`	`rel_se`	1.0	0.05	1.25e-2	Grade-A delta-vs-bootstrap gap; 3× observed ≈ 0.0375, rounded to 0.05.
`11_psm`	`rel_se`	5.0	1e-6	sentinel (no joined `se` field)	`att_psm` carries `se=None` on all three sides by design; see "Sentinel entries". Since 2026-09-22 the point-valued `se_teffects_ai` row joins Stata under `rel_est` at 7e-8 (`se_method='abadie_imbens_2016'`, the Abadie-Imbens 2016 estimated-score variance `teffects psmatch` reports); the default `se_abadie_imbens` (psmatch2 `ai(1)`, 643.35 vs 621.79) is a documented convention -- sample- vs population-ATT target and no estimated-score term.
`12_sdid`	`rel_se`	5e-2	1e-6	sentinel	Point-only ATT row (`arkhangelsky2021synthetic`); placebo SEs are backend-native diagnostics under distinct names.
`16_bjs`	`rel_se`	0.25	1e-6	sentinel	BJS imputation (`borusyak2024revisiting`); SE rows are side-specific (`se_cluster_if` / `se_didimputation` / `se_stata_did_imputation`).
`07_scm`	`rel_se`	1.0	1e-6	sentinel	All SCM rows are point-only.
`18_augsynth`	`rel_se`	1.0	1e-6	sentinel	`augsynth` fixture (`benmichael2021augmented`) emits no joinable SE. The `rel_est` budget (2e-5) is now grade A: the 7.9e-6 R residual is `augsynth::synth_qp`'s OSQP stopping tolerance (eps 1e-8); with OSQP at 1e-13 augsynth agrees with StatsPAI's exact simplex QP and the audited Stata/Mata bridge (`18_augsynth.do`) to 1e-11.
`19_gsynth`	`rel_se`	1.0	1e-6	sentinel	`gsynth` fixture (`xu2017generalized`) emits no joinable SE. An audited Stata/Mata bridge (`19_gsynth.do`, gsynth's control-only factor convention with r fixed at the R `r.cv`) joins the ATT at 2.2e-15; `fect_stata method(ife)` is a different estimator (IFEct) and is kept as an unjoined diagnostic.
`20_bacon`	`rel_se`	1.0	1e-6	sentinel	The Goodman–Bacon decomposition (`goodmanbacon2021difference`, `goodmanbacon2019bacondecomp`) has no SEs on any side.
`31_dfl`	`rel_se`	1.0	1e-6	sentinel	Point-only decomposition rows.
`39_arima`	`rel_se`	1e-2	1e-6	sentinel	No SE row joins on this fixture.
`52_scm_unique`	`rel_se`	1.0	1e-6	sentinel	All rows point-only.

Verification: after these edits, python3 tests/r_parity/compare.py re-rendered all parity tables byte-identically (the budget does not enter any rendered artifact for rel_se-only changes, and no rel_est was touched, so the strictness tiers and the JSS Appendix B tables are unchanged), and python3 -m pytest tests/test_parity_harness_contract.py -q -o addopts='' passes every tolerance-related contract. A row-level checker asserting the new values against all joined SE rows on both reference sides (not just headline rows) also passes for every tightened entry.

Closed 2026-09-22 (no tolerance change)¶

73_did2s SE promoted to T2. sp.gardner_did's default vce='analytic' now computes the did2s corrected two-stage clustered variance (stage-1 estimation error propagated through γ = (X10'X10)^{-1} X1'X2), matching R did2s at 2.7e-10 and Stata did2s at 1.3e-14 on the static ATT and all 26 event-study horizons at 2.1e-14. rel_se registered at the 1e-6 machine tier; the pre-1.29 stage-2-only SE (25.9% low) is reachable as vce='stage2' and kept as the unjoined static_ATT_stage2_se row.
04_csdid Stata SE convention pinned as a row. group_overall_fixedshare = sp.aggte(share_variance=False) joins csdid's estat group GAverage at 6.2e-15; STATA_SE_GAP_NOTES keeps the 0.27% note for the default row only.
18_augsynth / 19_gsynth Stata bridges materialised (see the sentinel table above); STATA_SKIP_REASON now lists four modules (13, 77, 79, 80). Stata-side reproducibility: 85/85 modules, zero drift.

Loosening applied 2026-09-22: `27_glmm_aghq` point estimate 1e-6 → 5e-6 (grade B)¶

The only budget ever loosened under the audit rule, registered here with the measurement that justifies it.

The likelihood is shared. The logLik row joins R lme4::glmer (nAGQ = 8) at rel 1e-13, and lme4's deviance function (devFunOnly = TRUE) evaluated at StatsPAI's (θ, β) is within 4.7e-11 of its value at lme4's own optimum. Both sides compute the same 8-point adaptive Gauss–Hermite approximation.
The objective is flat. Along the (intercept, √var) direction a 5e-11 deviance step moves the intercept by 5e-7, i.e. the argmax is numerically undetermined at the 1e-6 level for a 1500-observation log-likelihood evaluated in double precision.
The reference's own optimisers disagree at that level. On the committed fixture lme4 with bobyqa (the script's setting, rhoend = 1e-12), nloptwrap (xtol_abs = ftol_abs = 1e-14), optimx/nlminb and Nelder_Mead return intercepts 1.1e-6, 1.5e-6 and 1.4e-5 apart from one another while all reproduce the deviance to 1e-10. Stata melogit, intpoints(8) (default mvaghermite, tight tolerance(1e-10) ltolerance(1e-13)) lands 2e-8 from StatsPAI on the intercept; its logLik differs by 5e-6 because Stata's mean–variance adaptive grid is a different 8-point approximation, and all three sides agree to 3e-8 at 30 points (intmethod(mcaghermite) gives the same 30-point numbers to 1e-15).
Budget. 5e-6 ≈ 3× the widest lme4-vs-lme4 spread that still reproduces the deviance to 1e-10. The module therefore reports in the iterative strictness tier (79 machine / 8 iterative / 1 moderate / 1 T4 as of this change), not the machine tier. The SE budget (2e-5) is unchanged; observed 4.8e-7 (R) / 4.2e-6 (Stata).
Guard. test_glmm_parity_uses_tight_optimizer_solution_with_se_convention_guard additionally asserts logLik rel < 1e-9, so a future drift in the quadrature itself cannot hide behind the optimiser-noise budget.

Sentinel entries¶

Several modules are point-only by design: their SE estimators differ by construction, so the harness stores them as side-specific diagnostic rows with distinct statistic names that never join, and the headline row carries se=None. Example — module 11_psm: StatsPAI reports the Abadie–Imbens (2006, eq. 14) sample-ATT SE (se_abadie_imbens = 643.35), the estimator Stata psmatch2, ai(J) reports; MatchIt::matchit documents no canonical analytic SE for nearest-neighbor matching with replacement, so the R fixture records a weighted-lm-on-matched-data diagnostic (se_matchit_lm); Stata teffects psmatch reports its own Abadie–Imbens variant (abadie2016matching, se_teffects_ai = 621.79), 3.4% below ours. That gap was localised on 2026-09-22 (previously an open item under the §5 rule): teffects psmatch reports the Abadie & Imbens (2016) variance for matching on the estimated propensity score, σ²_δ − c'V_γ c + d'V_γ d, whose two extra terms (−74 905 + 8 036 on NSW-DW) are negative here, and whose base term targets the population ATT (it includes the (τ_i − τ)² heterogeneity term) rather than the sample ATT of eq. 14. sp.match(se_method='abadie_imbens_2016') rebuilds the number from StatsPAI's own matched frame, within-arm conditional variances and logit information matrix to rel 6.4e-8 (the residual is Stata's ML stopping rule), so the point-valued se_teffects_ai row now joins Stata under the module's rel_est budget of 1e-6. The default se_abadie_imbens (psmatch2 ai(1)) is unchanged and documented as the sample-ATT, known-score convention.

For such modules the rel_se budget is vacuous — no value, loose or tight, is ever exercised. The previous loose values (up to 5.0 for 11_psm) were leftovers from the original 2026-05-04 harness commit, before the SE rows were split into side-specific diagnostics, and read as if we tolerated a 500% SE gap. They are now pinned at the 1e-6 machine floor as sentinels: if a future fixture regeneration ever makes an SE row join, the contract fails loudly and the maintainer must consciously register a justified budget instead of inheriting a stale loose one.

Demonstrating a convention gap: the VAR df divisor¶

Module 33_var's SE budget illustrates why "both correct, different convention" must be stated mechanistically. StatsPAI's default matches Stata var (conditional-MLE divisor T); vars::VAR runs per-equation lm() (divisor T−k). The entire R-side SE gap is the deterministic ratio sqrt(T/(T−k)):

import numpy as np
import pandas as pd
import statspai as sp

rng = np.random.default_rng(42)
n = 200
y1, y2 = np.zeros(n), np.zeros(n)
for t in range(2, n):
    y1[t] = 0.5 * y1[t - 1] - 0.2 * y2[t - 2] + rng.standard_normal()
    y2[t] = 0.3 * y2[t - 1] + 0.1 * y1[t - 1] + rng.standard_normal()
df = pd.DataFrame({"y1": y1, "y2": y2})

stata_side = sp.var(df, lags=2, se_df="stata")  # divisor T   (Stata var)
r_side = sp.var(df, lags=2, se_df="r")          # divisor T-k (vars::VAR)

T = stata_side.n_obs
k = 2 * 2 + 1  # 2 lags x 2 variables + constant per equation
ratio = np.asarray(r_side.se["y1"]) / np.asarray(stata_side.se["y1"])
print(f"observed SE ratio = {ratio.flat[0]:.6f} "
      f"(identical for every coefficient: {np.allclose(ratio, ratio.flat[0])})")
print(f"sqrt(T / (T - k)) = {np.sqrt(T / (T - k)):.6f}")

Output: observed SE ratio = 1.012871 … — the ratio−1 form (1.287%) of the max-normalised 0.0127 gap recorded for every 33_var SE row in parity_table.md (the committed fixture has T=198, k=5, sqrt(198/193) − 1 = 1.287%; the table normalises by the larger side).

Known weak spots¶

The honest list, re-audited 2026-09-05. Every entry that used to sit here as grade C or "budget bounds nothing" has been closed by locating the mechanism on the StatsPAI side and, where a like-for-like row exists, pinning both references at machine level. The closures are kept below (struck through) because the pattern is the lesson; the open items follow.

~~29_panel_sfa (rel_se 1e-3) — grade C.~~ Closed. The Stata reference was a different likelihood (truncated-normal Battese–Coelli ti with free mu); constrained to mu = 0 it agrees with StatsPAI to 3e-6 on every SE and 1e-6 on every point row, so the registered 5e-2 now bounds only frontier's approximate covariance (graded B above).
~~33_var (rel_se 1e-3) — budget mis-keyed.~~ Closed. The Python side emits eq_* (divisor T, Stata) and eq_*__Tk (divisor T-k, R vars::VAR) rows and each reference side emits only its own convention; every compared SE is at 1e-15 and the budget is 1e-6.
~~05_sunab — fixest-side mechanism unpinned.~~ Closed. The 17.1% figure came from the old R script re-aggregating cohort×period cells by hand without their covariance; the native fixest aggregation is now used. The residual 0.08% at single-cohort times was a StatsPAI degrees-of-freedom defect (nine all-zero cohort×relative-time columns counted in K, time effects not counted), fixed in 1.24.0. What remains is the documented Sun–Abraham Prop. 3 share term (mechanism A), demonstrated by the att_rel_<e>_fixedshare rows matching fixest to 8e-12 and the default rows matching Stata eventstudyinteract to 8e-12.
~~26_glmm_logit / 27_glmm_aghq — SE convention not derived.~~ Closed. Not a convention: the fixed-effect covariance omitted the variance-component uncertainty. The full observed-information Hessian of the marginal log-likelihood (Stata vce(oim)) puts AGHQ at 5e-7 (R) / 4e-6 (Stata) and Laplace at 4e-6 (Stata); the remaining 0.27% on the Laplace R side is lme4's own optimum sitting 1.5e-4 away in β (its logLik is 2.8e-7 lower).
~~47_ppmlhdfe_3fe — R-side residual unpinned.~~ Closed. StatsPAI applied (N-1)/(N-k) with k = slopes only, matching neither reference. ssc="stata" (N/(N-1)) matches ppmlhdfe to 1e-11 and ssc="fixest" (N/(N-K), K counting absorbed FE levels) matches fixest::fepois to 1e-8; modules 37 and 47 emit one row per convention.
13_causal_forest rel_se 0.25 — open by construction (T3). The residual is forest-RNG dispersion in tau.hat and is not tightenable; the binding evidence is the exact AIPW-operator pin plus the Track B coverage sweep, not the relative band.
~~Headline-only enforcement.~~ Closed. test_every_r_se_row_is_inside_budget gates the registered rel_se on every R-joined SE row of every PASS module (all 87 pass), and test_every_stata_se_row_is_inside_budget_or_registered requires any Stata SE row over budget to carry a mechanism in compare.py::STATA_SE_GAP_NOTES (two entries: 04_csdid group_overall 0.27%, fixed-share aggregation; 71_dml_family PLIV, ivreg's N/(N-K). The 83_lpdid lead-row entry, 3.9e-4 and attributed to reghdfe's K, was removed on 2026-09-07: the gap was a StatsPAI lead-sample rule one period stricter than lpdid's, and the rows now agree to 2e-16).
~~04_csdid group_overall Stata SE (0.27%) — open.~~ Closed (2026-09-05). csdid's estat group GAverage aggregates the per-cohort influence functions with the cohort shares held fixed; did::aggte(type = "group") and sp.aggte add the share-estimation term (did:::wif). Rebuilding the fixed-share aggregate from StatsPAI's joint cell influence functions reproduces csdid's SE to 1e-14 (the earlier independent-cell attempt overshot because it dropped the cross-cell covariance). Mechanism A, registered in STATA_SE_GAP_NOTES and pinned by test_csdid_group_overall_stata_se_is_the_fixed_share_aggregation.
Contract-frozen values. tests/test_parity_harness_contract.py asserts exact equality for several budgets (e.g. 26_glmm_logit, 27_glmm_aghq, 36_mediation, 38_drdid, 10_honest_did, 11_psm rel_est). Any future change must touch both files in the same commit, deliberately.

Last audited: 2026-09-22 (StatsPAI 1.29.0 source snapshot, 89 parity modules). Re-run the snippet above and refresh this document whenever a TOLERANCES entry changes.