Add Satterthwaite and Kenward-Roger fixed-effect tests

docs/FORMAT.md

@@ -92,6 +92,10 @@ For block number `KKKKKK`:
 block_KKKKKK.summary.tsv
 block_KKKKKK.beta.f64.bin
 block_KKKKKK.cov.f64.bin
+block_KKKKKK.fixed_se.f64.bin
+block_KKKKKK.fixed_stat.f64.bin
+block_KKKKKK.fixed_ddf.f64.bin
+block_KKKKKK.fixed_p.f64.bin
 block_KKKKKK.complete
 ```
 
@@ -112,10 +116,22 @@ logL
 iterations
 line_search_steps
 grad_inf
+fixed_test_method
+fixed_test_status
+fixed_test_offset
+extra_joint_num_df
+extra_joint_den_df
+extra_joint_f
+extra_joint_p
+fixed_test_error
 error
 ```
 
-`beta_offset` and `cov_offset` count `float64` elements, not bytes. A negative offset indicates that no estimates were emitted for that task.
+`beta_offset`, `cov_offset`, and `fixed_test_offset` count `float64`
+elements, not bytes. A negative beta/covariance offset means that no estimates
+were emitted. A negative fixed-test offset means that no valid coefficient-wise
+fixed-effect tests were emitted; successful REML estimates are retained even
+when inference fails.
 
 The covariance array uses the row-wise packed lower triangle:
 
@@ -126,11 +142,12 @@ The covariance array uses the row-wise packed lower triangle:
 The `.complete` marker is written last and contains tab-separated key/value rows:
 
 ```text
-format  spectra-reml-block-v1
+format  spectra-reml-block-v2
 block   0
 tasks   256
 beta_elements  4096
 cov_elements   34816
+fixed_test_elements  4096
 ```
 
 Consumers must ignore blocks without `.complete`.
@@ -150,6 +167,35 @@ numerical_error
 
 `converged_boundary` is a successful residual-only solution accepted after the one-sided variance-component score and likelihood checks.
 
+## Fixed-effect inference
+
+`--fixed-effect-test` selects `satterthwaite` (the default),
+`kenward-roger`, or `none`.
+
+The four `fixed_*` arrays have the same offsets and coefficient order as
+`beta`. They contain standard errors, statistics, denominator degrees of
+freedom, and p-values. With Satterthwaite inference, a coefficient statistic is
+a signed t statistic. With Kenward-Roger inference, it is an F statistic with
+one numerator degree of freedom. For every task with extra covariates, the
+summary also contains an F test of the joint null that all task-specific fixed
+effects are zero.
+
+`fixed_test_status` is one of:
+
+```text
+not_requested
+ok
+boundary_conditional
+fit_not_converged
+invalid_contrast
+information_singular
+numerical_error
+```
+
+At `converged_boundary`, inference conditions on the accepted active set
+`sigma_g2=0`; `fixed_test_status` is `boundary_conditional` and only residual
+variance uncertainty contributes to the small-sample adjustment.
+
 Phenotypes are scaled internally by their task-specific OLS residual RMS before
 optimization. Reported fixed effects, fixed-effect covariance, variance
 components, and restricted likelihood are transformed back to the original
@@ -168,6 +214,13 @@ The Python CLI exports one TSV row per task. It includes the full summary plus:
 ```text
 beta_json
 covariance_packed_lower_json
+fixed_effect_se_json
+fixed_effect_statistic_json
+fixed_effect_denominator_df_json
+fixed_effect_p_value_json
 ```
 
-Project-specific software can attach coefficient names and derive contrasts without changing the numerical engine.
+Project-specific software can attach coefficient names. The public C++
+inference API also accepts general linear hypotheses `L beta = rhs`; the batch
+format currently emits coefficient-wise tests and the joint extra-covariate
+test.