Add reusable SpectraREML batch AI-REML engine

2026-08-09 14:13:38 +08:00
parent 777eeb288c
commit da971a2ca7
22 changed files with 4736 additions and 0 deletions
--- a/.gitignore
+++ b/.gitignore
@@ -0,0 +1,10 @@
+/build*/
+/*.obj
+/*.lib
+/*.exp
+/out/
+/results/
+__pycache__/
+*.py[cod]
+*.tmp.*
+/examples/synthetic_work/
--- a/CMakeLists.txt
+++ b/CMakeLists.txt
@@ -0,0 +1,118 @@
+cmake_minimum_required(VERSION 3.20)
+
+project(
+  SpectraREML
+  VERSION 0.1.0
+  DESCRIPTION "Reusable batched spectral AI-REML engine"
+  LANGUAGES CXX
+)
+
+include(GNUInstallDirs)
+include(CTest)
+
+list(PREPEND CMAKE_MODULE_PATH "${CMAKE_CURRENT_SOURCE_DIR}/cmake")
+
+set(REML_BLAS "MKL" CACHE STRING "BLAS/LAPACKE backend: MKL or OPENBLAS")
+set_property(CACHE REML_BLAS PROPERTY STRINGS MKL OPENBLAS)
+string(TOUPPER "${REML_BLAS}" REML_BLAS)
+
+option(REML_ENABLE_OPENMP "Parallelize independent REML tasks with OpenMP" ON)
+option(REML_ENABLE_NATIVE_ARCH "Enable compiler tuning for the build host" OFF)
+
+add_library(reml_blas_lapacke INTERFACE)
+add_library(REML::BLAS_LAPACKE ALIAS reml_blas_lapacke)
+
+if(REML_BLAS STREQUAL "MKL")
+  # These variables are consumed by Intel's MKLConfig.cmake.  Sequential is
+  # deliberately the portable default because tasks are parallelized with
+  # OpenMP; a threaded BLAS can otherwise create nested thread pools.
+  set(MKL_INTERFACE "lp64" CACHE STRING "oneMKL integer interface")
+  set(MKL_LINK "dynamic" CACHE STRING "oneMKL link type")
+  set(MKL_THREADING "sequential" CACHE STRING "oneMKL threading layer")
+  set_property(CACHE MKL_INTERFACE PROPERTY STRINGS lp64 ilp64)
+  set_property(CACHE MKL_LINK PROPERTY STRINGS dynamic static)
+  set_property(CACHE MKL_THREADING PROPERTY STRINGS sequential intel_thread gnu_thread tbb_thread)
+
+  find_package(MKL CONFIG REQUIRED)
+  if(NOT TARGET MKL::MKL)
+    message(FATAL_ERROR "MKL was found, but its MKL::MKL CMake target is missing")
+  endif()
+  target_link_libraries(reml_blas_lapacke INTERFACE MKL::MKL)
+  target_compile_definitions(reml_blas_lapacke INTERFACE REML_BLAS_MKL=1)
+elseif(REML_BLAS STREQUAL "OPENBLAS")
+  find_package(REML_OpenBLAS REQUIRED)
+  target_link_libraries(reml_blas_lapacke INTERFACE REML::OpenBLASLAPACKE)
+  target_compile_definitions(reml_blas_lapacke INTERFACE REML_BLAS_OPENBLAS=1)
+else()
+  message(FATAL_ERROR "Unsupported REML_BLAS='${REML_BLAS}'. Choose MKL or OPENBLAS.")
+endif()
+
+if(REML_ENABLE_OPENMP)
+  find_package(OpenMP REQUIRED COMPONENTS CXX)
+endif()
+
+add_library(spectra_reml_core STATIC
+  src/linalg.cpp
+  src/grm.cpp
+  src/reml_core.cpp
+  src/batch_io.cpp
+)
+add_library(SpectraREML::core ALIAS spectra_reml_core)
+
+target_compile_features(spectra_reml_core PUBLIC cxx_std_17)
+target_include_directories(spectra_reml_core
+  PUBLIC
+    $<BUILD_INTERFACE:${CMAKE_CURRENT_SOURCE_DIR}/include>
+    $<INSTALL_INTERFACE:${CMAKE_INSTALL_INCLUDEDIR}>
+)
+target_link_libraries(spectra_reml_core PUBLIC REML::BLAS_LAPACKE)
+
+if(REML_ENABLE_OPENMP)
+  target_link_libraries(spectra_reml_core PUBLIC OpenMP::OpenMP_CXX)
+  target_compile_definitions(spectra_reml_core PUBLIC REML_USE_OPENMP=1)
+endif()
+
+if(MSVC)
+  target_compile_options(spectra_reml_core PRIVATE /W4 /permissive- /EHsc)
+else()
+  target_compile_options(spectra_reml_core PRIVATE -Wall -Wextra -Wpedantic)
+  target_compile_definitions(spectra_reml_core PUBLIC _FILE_OFFSET_BITS=64)
+  if(REML_ENABLE_NATIVE_ARCH)
+    target_compile_options(spectra_reml_core PRIVATE -march=native)
+  endif()
+endif()
+
+add_executable(spectra_reml src/main.cpp)
+target_link_libraries(spectra_reml PRIVATE SpectraREML::core)
+target_compile_features(spectra_reml PRIVATE cxx_std_17)
+
+if(MSVC)
+  target_compile_options(spectra_reml PRIVATE /W4 /permissive- /EHsc)
+else()
+  target_compile_options(spectra_reml PRIVATE -Wall -Wextra -Wpedantic)
+  if(REML_ENABLE_NATIVE_ARCH)
+    target_compile_options(spectra_reml PRIVATE -march=native)
+  endif()
+endif()
+
+if(BUILD_TESTING)
+  add_executable(test_reml_synthetic tests/cpp/test_reml_synthetic.cpp)
+  target_link_libraries(test_reml_synthetic PRIVATE SpectraREML::core)
+  add_test(NAME reml_synthetic COMMAND test_reml_synthetic)
+
+  add_executable(test_grm_io tests/cpp/test_grm_io.cpp)
+  target_link_libraries(test_grm_io PRIVATE SpectraREML::core)
+  add_test(NAME grm_io COMMAND test_grm_io)
+endif()
+
+install(TARGETS spectra_reml_core spectra_reml
+  RUNTIME DESTINATION ${CMAKE_INSTALL_BINDIR}
+  LIBRARY DESTINATION ${CMAKE_INSTALL_LIBDIR}
+  ARCHIVE DESTINATION ${CMAKE_INSTALL_LIBDIR}
+)
+install(DIRECTORY include/ DESTINATION ${CMAKE_INSTALL_INCLUDEDIR})
+
+message(STATUS "SpectraREML ${PROJECT_VERSION}")
+message(STATUS "  BLAS/LAPACKE backend : ${REML_BLAS}")
+message(STATUS "  OpenMP               : ${REML_ENABLE_OPENMP}")
+message(STATUS "  Native architecture  : ${REML_ENABLE_NATIVE_ARCH}")
--- a/README.md
+++ b/README.md
@@ -1,2 +1,155 @@
 # SpectraREML

+SpectraREML is a reusable batch AI-REML engine for many continuous traits that share one genomic relationship matrix (GRM).
+
+For each task, it fits
+
+\[
+y = X\beta + Zu + e, \qquad
+u \sim N(0,\sigma_g^2G), \qquad
+e \sim N(0,\sigma_e^2I).
+\]
+
+The engine diagonalizes the GRM once, rotates the common design and every unique task-specific covariate once, rotates phenotypes in blocks, and fits independent REML tasks in parallel.
+
+## Features
+
+- C++17 numerical core using oneMKL or OpenBLAS/LAPACKE.
+- GRM eigendecomposition once per task set.
+- AI-REML with a strong-Wolfe line search; no EM updates.
+- Signed standard-deviation parameterization with a separate one-sided KKT check for the \(\sigma_g^2=0\) boundary.
+- No explicit dense \(P\) matrix.
+- Variable numbers of extra fixed-effect covariates per task through a CSR index.
+- Atomic block output and safe resume/force semantics.
+- Generic Python CLI for manifest creation, validation, execution, provenance signatures, status, and result export.
+
+The numerical core has no knowledge of cohorts, molecular assay types, or domain-specific variable names.
+
+## Repository layout
+
+```text
+include/spectra_reml/  public C++ API
+src/                   numerical core, batch I/O, and CLI
+python/spectra_reml.py generic Python CLI
+examples/example.py    fully synthetic input example
+scripts/run_server.sh  generic Linux build/run wrapper
+tests/                 numerical and file-contract tests
+docs/FORMAT.md         binary and tabular file contract
+```
+
+## Build with Intel oneMKL
+
+Load the oneAPI environment first:
+
+```bash
+source /opt/intel/oneapi/setvars.sh
+
+cmake -S . -B build-mkl \
+  -DCMAKE_BUILD_TYPE=Release \
+  -DREML_BLAS=MKL \
+  -DMKL_INTERFACE=lp64 \
+  -DMKL_LINK=dynamic \
+  -DMKL_THREADING=sequential
+cmake --build build-mkl --parallel
+ctest --test-dir build-mkl --output-on-failure
+```
+
+For a nonstandard installation, locate `MKLConfig.cmake` and set its directory explicitly:
+
+```bash
+cmake -S . -B build-mkl \
+  -DREML_BLAS=MKL \
+  -DMKL_DIR=/path/to/mkl/latest/lib/cmake/mkl \
+  -DMKL_THREADING=sequential
+```
+
+Sequential BLAS is recommended because independent tasks are already parallelized by OpenMP.
+
+## Build with OpenBLAS
+
+```bash
+cmake -S . -B build-openblas \
+  -DCMAKE_BUILD_TYPE=Release \
+  -DREML_BLAS=OPENBLAS \
+  -DOpenBLAS_ROOT=/path/to/openblas \
+  -DLAPACKE_ROOT=/path/to/lapacke
+cmake --build build-openblas --parallel
+ctest --test-dir build-openblas --output-on-failure
+```
+
+The configuration performs a real CBLAS/LAPACKE link check. It supports LAPACKE either inside OpenBLAS or in a separate library.
+
+## Quick start
+
+Create a synthetic bundle:
+
+```bash
+python examples/example.py
+```
+
+Create a manifest for your own data:
+
+```bash
+python python/spectra_reml.py make-manifest \
+  --manifest work/manifest.json \
+  --grm-bin data/example.grm.bin \
+  --grm-id data/example.grm.id \
+  --base-x data/base_x.f64.bin \
+  --phenotypes data/phenotypes.f64.bin \
+  --extra-covariates data/extra_covariates.f32.bin \
+  --tasks data/tasks.tsv \
+  --extra-offsets data/extra_offsets.i64.bin \
+  --extra-indices data/extra_indices.i32.bin \
+  --output-dir work/blocks \
+  --n-samples 3523 \
+  --n-base-covariates 8 \
+  --n-phenotype-rows 10000 \
+  --n-extra-covariate-rows 2000
+```
+
+Validate, run, and export results:
+
+```bash
+python python/spectra_reml.py validate --manifest work/manifest.json
+
+python python/spectra_reml.py run \
+  --manifest work/manifest.json \
+  --engine build-mkl/spectra_reml \
+  --threads 28 \
+  --blas-threads 1 \
+  --block-size 256 \
+  --resume
+
+python python/spectra_reml.py finalize \
+  --manifest work/manifest.json \
+  --output work/results.tsv.gz
+```
+
+`results.tsv.gz` retains the complete per-task summary and stores the fixed-effect vector and row-wise packed lower covariance as JSON arrays.
+
+## Recovery and provenance
+
+Each block is written as four files, with `.complete` renamed last. The Python layer adds `run.signature.json`, which binds the canonical manifest, engine SHA-256, numerical options, thread settings, and block size.
+
+- `--resume` reuses complete blocks only when the signature matches exactly.
+- `--force` invalidates the old signature before deleting old blocks and starting a new generation.
+- `finalize` refuses blocks that are not bound to the current manifest.
+- `--dry-run` does not mutate output state.
+
+## Threading
+
+Use one BLAS thread with multiple outer task threads unless benchmarking shows otherwise:
+
+```bash
+export MKL_NUM_THREADS=1
+export OPENBLAS_NUM_THREADS=1
+export OMP_DYNAMIC=FALSE
+```
+
+Then set `--threads` to the physical cores allocated to the process.
+
+See [docs/FORMAT.md](docs/FORMAT.md) for the exact file contract.
+
+## License
+
+BSD 3-Clause. See [LICENSE](LICENSE).
--- a/cmake/FindREML_OpenBLAS.cmake
+++ b/cmake/FindREML_OpenBLAS.cmake
@@ -0,0 +1,135 @@
+# Locate an LP64 OpenBLAS implementation together with the CBLAS and LAPACKE
+# C interfaces.  LAPACKE may be provided by libopenblas itself or by a separate
+# liblapacke library.
+#
+# Result:
+#   REML_OpenBLAS_FOUND
+#   REML::OpenBLASLAPACKE
+#
+# Optional hints/overrides:
+#   OpenBLAS_ROOT
+#   LAPACKE_ROOT
+#   REML_OPENBLAS_LIBRARY
+#   REML_LAPACKE_LIBRARY
+#   REML_CBLAS_INCLUDE_DIR
+#   REML_LAPACKE_INCLUDE_DIR
+
+include(CheckCXXSourceCompiles)
+include(CMakePushCheckState)
+include(FindPackageHandleStandardArgs)
+
+set(_reml_openblas_hints)
+foreach(_root IN ITEMS "${OpenBLAS_ROOT}" "$ENV{OpenBLAS_ROOT}" "$ENV{OPENBLAS_ROOT}" "$ENV{CONDA_PREFIX}")
+  if(_root)
+    list(APPEND _reml_openblas_hints "${_root}")
+  endif()
+endforeach()
+
+set(_reml_lapacke_hints ${_reml_openblas_hints})
+foreach(_root IN ITEMS "${LAPACKE_ROOT}" "$ENV{LAPACKE_ROOT}")
+  if(_root)
+    list(PREPEND _reml_lapacke_hints "${_root}")
+  endif()
+endforeach()
+
+# Prefer package-provided imported targets because they carry static-library
+# dependencies (Fortran runtime, pthreads, and libm) correctly.
+find_package(OpenBLAS CONFIG QUIET)
+if(TARGET OpenBLAS::OpenBLAS)
+  set(_reml_openblas_link OpenBLAS::OpenBLAS)
+elseif(TARGET OpenBLAS)
+  set(_reml_openblas_link OpenBLAS)
+endif()
+
+if(NOT _reml_openblas_link)
+  find_library(REML_OPENBLAS_LIBRARY
+    NAMES openblas libopenblas
+    HINTS ${_reml_openblas_hints}
+    PATH_SUFFIXES lib lib64 Library/lib
+  )
+  if(REML_OPENBLAS_LIBRARY)
+    set(_reml_openblas_link "${REML_OPENBLAS_LIBRARY}")
+  endif()
+endif()
+
+find_path(REML_CBLAS_INCLUDE_DIR
+  NAMES cblas.h
+  HINTS ${_reml_openblas_hints}
+  PATH_SUFFIXES include include/openblas Library/include Library/include/openblas
+)
+find_path(REML_LAPACKE_INCLUDE_DIR
+  NAMES lapacke.h
+  HINTS ${_reml_lapacke_hints}
+  PATH_SUFFIXES include include/openblas include/lapacke Library/include Library/include/openblas
+)
+
+find_package(LAPACKE CONFIG QUIET)
+if(TARGET LAPACKE::LAPACKE)
+  set(_reml_lapacke_link LAPACKE::LAPACKE)
+elseif(TARGET LAPACKE)
+  set(_reml_lapacke_link LAPACKE)
+else()
+  find_library(REML_LAPACKE_LIBRARY
+    NAMES lapacke liblapacke
+    HINTS ${_reml_lapacke_hints}
+    PATH_SUFFIXES lib lib64 Library/lib
+  )
+  if(REML_LAPACKE_LIBRARY)
+    set(_reml_lapacke_link "${REML_LAPACKE_LIBRARY}")
+  endif()
+endif()
+
+if(_reml_lapacke_link)
+  # Keep the static-link order dependency first, provider second:
+  # liblapacke calls LAPACK symbols supplied by libopenblas.
+  set(_reml_link_items ${_reml_lapacke_link} ${_reml_openblas_link})
+else()
+  set(_reml_link_items ${_reml_openblas_link})
+endif()
+
+unset(REML_OPENBLAS_LINK_OK CACHE)
+if(REML_CBLAS_INCLUDE_DIR AND REML_LAPACKE_INCLUDE_DIR AND _reml_openblas_link)
+  cmake_push_check_state(RESET)
+  set(CMAKE_REQUIRED_INCLUDES "${REML_CBLAS_INCLUDE_DIR};${REML_LAPACKE_INCLUDE_DIR}")
+  set(CMAKE_REQUIRED_LIBRARIES ${_reml_link_items})
+  check_cxx_source_compiles([[
+    #include <cblas.h>
+    #include <lapacke.h>
+    int main() {
+      double a[1] = {1.0};
+      double b[1] = {1.0};
+      double c[1] = {0.0};
+      cblas_dgemm(CblasRowMajor, CblasNoTrans, CblasNoTrans,
+                  1, 1, 1, 1.0, a, 1, b, 1, 0.0, c, 1);
+      return LAPACKE_dpotrf(LAPACK_ROW_MAJOR, 'L', 1, a, 1);
+    }
+  ]] REML_OPENBLAS_LINK_OK)
+  cmake_pop_check_state()
+else()
+  set(REML_OPENBLAS_LINK_OK FALSE)
+endif()
+
+find_package_handle_standard_args(REML_OpenBLAS
+  REQUIRED_VARS
+    REML_CBLAS_INCLUDE_DIR
+    REML_LAPACKE_INCLUDE_DIR
+    _reml_openblas_link
+    REML_OPENBLAS_LINK_OK
+  FAIL_MESSAGE
+    "OpenBLAS with LP64 CBLAS and LAPACKE is required. Set OpenBLAS_ROOT/LAPACKE_ROOT, or the REML_* cache variables listed in cmake/FindREML_OpenBLAS.cmake."
+)
+
+if(REML_OpenBLAS_FOUND AND NOT TARGET REML::OpenBLASLAPACKE)
+  add_library(REML::OpenBLASLAPACKE INTERFACE IMPORTED)
+  set_property(TARGET REML::OpenBLASLAPACKE PROPERTY
+    INTERFACE_INCLUDE_DIRECTORIES "${REML_CBLAS_INCLUDE_DIR};${REML_LAPACKE_INCLUDE_DIR}")
+  set_property(TARGET REML::OpenBLASLAPACKE PROPERTY
+    INTERFACE_LINK_LIBRARIES "${_reml_link_items}")
+endif()
+
+mark_as_advanced(
+  REML_CBLAS_INCLUDE_DIR
+  REML_LAPACKE_INCLUDE_DIR
+  REML_OPENBLAS_LIBRARY
+  REML_LAPACKE_LIBRARY
+)
--- a/docs/FORMAT.md
+++ b/docs/FORMAT.md
@@ -0,0 +1,162 @@
+# SpectraREML file contract
+
+All raw binary files are little-endian, headerless, and contiguous. Integer indices and element offsets are zero based.
+
+## Shared sample order
+
+The following matrices must use exactly the same sample order:
+
+1. GRM;
+2. common design matrix;
+3. phenotype matrix;
+4. extra-covariate matrix.
+
+When supplied, `--grm-id` is checked for the expected number of nonempty rows. Domain adapters remain responsible for verifying the actual identifiers and order.
+
+## GRM
+
+`--grm-bin` uses the GCTA lower-triangle packed `float32` layout:
+
+```text
+G[0,0],
+G[1,0], G[1,1],
+G[2,0], G[2,1], G[2,2], ...
+```
+
+For `n` samples, the exact file size is `4 * n * (n + 1) / 2` bytes.
+
+## Common design
+
+`--base-x` is a row-major `float64` matrix with shape
+
+```text
+n_samples × n_base_covariates
+```
+
+It must already contain every common fixed effect, including an intercept if required. SpectraREML does not add or standardize columns.
+
+## Phenotypes
+
+`--phenotypes` is a row-major `float64` matrix with shape
+
+```text
+n_phenotype_rows × n_samples
+```
+
+Each task selects one row through `phenotype_row`.
+
+## Extra fixed-effect covariates
+
+`--extra-covariates` is an optional row-major `float32` matrix with shape
+
+```text
+n_extra_covariate_rows × n_samples
+```
+
+Only rows referenced by at least one task are read and rotated. The file may be omitted when the row count and all task-specific counts are zero.
+
+## Task table
+
+`--tasks` is a UTF-8 tab-separated file with exactly four columns:
+
+```text
+task_index  task_id  phenotype_row  n_extra_covariates
+0           trait_a  0              0
+1           trait_b  1              2
+```
+
+Requirements:
+
+- `task_index` is consecutive and zero based;
+- `task_id` is nonempty and unique;
+- `phenotype_row` is within the phenotype matrix;
+- `n_extra_covariates` agrees with the CSR offsets.
+
+## CSR task-to-covariate mapping
+
+`--extra-offsets` is an `int64` array of length `n_tasks + 1`. It begins with zero and is nondecreasing.
+
+`--extra-indices` is an `int32` array of length `offsets[-1]`. For task `i`, its extra-covariate row indices are
+
+```text
+indices[offsets[i]:offsets[i+1]]
+```
+
+An index must be in `[0, n_extra_covariate_rows)`, and a task cannot reference the same row twice.
+
+## Block output
+
+For block number `KKKKKK`:
+
+```text
+block_KKKKKK.summary.tsv
+block_KKKKKK.beta.f64.bin
+block_KKKKKK.cov.f64.bin
+block_KKKKKK.complete
+```
+
+The summary header is:
+
+```text
+task_index
+task_id
+status
+n_fixed
+n_extra_covariates
+beta_offset
+cov_offset
+sigma_g2
+sigma_e2
+h2
+logL
+iterations
+line_search_steps
+grad_inf
+error
+```
+
+`beta_offset` and `cov_offset` count `float64` elements, not bytes. A negative offset indicates that no estimates were emitted for that task.
+
+The covariance array uses the row-wise packed lower triangle:
+
+```text
+(0,0), (1,0), (1,1), (2,0), (2,1), (2,2), ...
+```
+
+The `.complete` marker is written last and contains tab-separated key/value rows:
+
+```text
+format  spectra-reml-block-v1
+block   0
+tasks   256
+beta_elements  4096
+cov_elements   34816
+```
+
+Consumers must ignore blocks without `.complete`.
+
+## Status values
+
+```text
+converged
+converged_boundary
+max_iterations
+line_search_failed
+rank_deficient
+invalid_input
+non_positive_covariance
+numerical_error
+```
+
+`converged_boundary` is a successful residual-only solution accepted after the one-sided variance-component score and likelihood checks.
+
+## Generic finalized output
+
+The Python CLI exports one TSV row per task. It includes the full summary plus:
+
+```text
+beta_json
+covariance_packed_lower_json
+```
+
+Project-specific software can attach coefficient names and derive contrasts without changing the numerical engine.
--- a/examples/example.py
+++ b/examples/example.py
@@ -0,0 +1,83 @@
+#!/usr/bin/env python3
+"""Create a small, fully synthetic SpectraREML input bundle."""
+
+from __future__ import annotations
+
+import sys
+from pathlib import Path
+
+import numpy as np
+
+PROJECT = Path(__file__).resolve().parents[1]
+sys.path.insert(0, str(PROJECT / "python"))
+import spectra_reml  # noqa: E402
+
+
+def main() -> None:
+    output = Path(__file__).resolve().parent / "synthetic_work"
+    output.mkdir(parents=True, exist_ok=True)
+    rng = np.random.default_rng(20260809)
+    n = 12
+
+    z = rng.normal(size=(n, 5))
+    grm = z @ z.T / z.shape[1]
+    diagonal = np.sqrt(np.diag(grm))
+    grm /= np.outer(diagonal, diagonal)
+    packed = np.asarray(
+        [grm[row, col] for row in range(n) for col in range(row + 1)],
+        dtype="<f4",
+    )
+    packed.tofile(output / "example.grm.bin")
+    (output / "example.grm.id").write_text(
+        "".join("sample{0}\tsample{0}\n".format(i + 1) for i in range(n)),
+        encoding="utf-8",
+    )
+
+    base_x = np.column_stack((np.ones(n), np.linspace(-1.0, 1.0, n)))
+    extra = rng.normal(size=(2, n)).astype("<f4")
+    phenotypes = np.vstack(
+        (
+            0.3 + 0.5 * base_x[:, 1] + rng.normal(scale=0.2, size=n),
+            -0.2 + 0.8 * extra[0] + rng.normal(scale=0.2, size=n),
+            0.1 + 0.4 * extra[0] - 0.3 * extra[1] + rng.normal(scale=0.2, size=n),
+        )
+    )
+    np.asarray(base_x, dtype="<f8").tofile(output / "base_x.f64.bin")
+    np.asarray(phenotypes, dtype="<f8").tofile(output / "phenotypes.f64.bin")
+    extra.tofile(output / "extra_covariates.f32.bin")
+    (output / "tasks.tsv").write_text(
+        "task_index\ttask_id\tphenotype_row\tn_extra_covariates\n"
+        "0\ttrait_0\t0\t0\n"
+        "1\ttrait_1\t1\t1\n"
+        "2\ttrait_2\t2\t2\n",
+        encoding="utf-8",
+    )
+    np.asarray([0, 0, 1, 3], dtype="<i8").tofile(output / "extra_offsets.i64.bin")
+    np.asarray([0, 0, 1], dtype="<i4").tofile(output / "extra_indices.i32.bin")
+
+    manifest = output / "manifest.json"
+    spectra_reml.main(
+        [
+            "make-manifest",
+            "--manifest", str(manifest),
+            "--grm-bin", str(output / "example.grm.bin"),
+            "--grm-id", str(output / "example.grm.id"),
+            "--base-x", str(output / "base_x.f64.bin"),
+            "--phenotypes", str(output / "phenotypes.f64.bin"),
+            "--extra-covariates", str(output / "extra_covariates.f32.bin"),
+            "--tasks", str(output / "tasks.tsv"),
+            "--extra-offsets", str(output / "extra_offsets.i64.bin"),
+            "--extra-indices", str(output / "extra_indices.i32.bin"),
+            "--output-dir", str(output / "blocks"),
+            "--n-samples", str(n),
+            "--n-base-covariates", str(base_x.shape[1]),
+            "--n-phenotype-rows", str(phenotypes.shape[0]),
+            "--n-extra-covariate-rows", str(extra.shape[0]),
+        ]
+    )
+    print("\nNext:")
+    print("python python/spectra_reml.py run --manifest {} --engine /path/to/spectra_reml --threads 4".format(manifest))
+
+
+if __name__ == "__main__":
+    main()
--- a/include/spectra_reml/batch_io.hpp
+++ b/include/spectra_reml/batch_io.hpp
@@ -0,0 +1,62 @@
+#pragma once
+
+#include "spectra_reml/grm.hpp"
+#include "spectra_reml/types.hpp"
+
+#include <cstddef>
+#include <cstdint>
+#include <filesystem>
+#include <string>
+#include <vector>
+
+namespace spectra::reml {
+
+struct BatchInputPaths {
+    std::filesystem::path grm_bin;
+    std::filesystem::path grm_id;
+    std::filesystem::path base_x_f64;
+    std::filesystem::path phenotype_f64;
+    std::filesystem::path extra_covariate_f32;
+    std::filesystem::path tasks_tsv;
+    std::filesystem::path extra_offsets_i64;
+    std::filesystem::path extra_indices_i32;
+    std::filesystem::path output_directory;
+};
+
+struct BatchDimensions {
+    std::size_t sample_count = 0;
+    std::size_t base_covariate_count = 0;
+    std::size_t phenotype_row_count = 0;
+    std::size_t extra_covariate_row_count = 0;
+};
+
+struct BatchOptions {
+    std::size_t block_size = 256;
+    std::size_t outer_threads = 1;
+    bool resume = false;
+    bool overwrite = false;
+    RemlOptions reml;
+};
+
+[[nodiscard]] ColMajorMatrix read_row_major_f64_matrix(
+    const std::filesystem::path& path, std::size_t rows, std::size_t cols);
+
+[[nodiscard]] std::vector<RemlTask> read_tasks(
+    const std::filesystem::path& tasks_tsv,
+    const std::filesystem::path& extra_offsets_i64,
+    const std::filesystem::path& extra_indices_i32,
+    const BatchDimensions& dimensions);
+
+// Runs one complete task set.  The GRM and common variables are transformed
+// once; phenotypes are transformed in blocks and tasks are fitted in parallel.
+// Each completed block contains:
+//   block_NNNNNN.summary.tsv
+//   block_NNNNNN.beta.f64.bin
+//   block_NNNNNN.cov.f64.bin
+//   block_NNNNNN.complete
+// The .complete marker is renamed last and is the sole resume criterion.
+void run_task_batch(const BatchInputPaths& paths,
+                    const BatchDimensions& dimensions,
+                    const BatchOptions& options);
+
+}  // namespace spectra::reml
--- a/include/spectra_reml/grm.hpp
+++ b/include/spectra_reml/grm.hpp
@@ -0,0 +1,36 @@
+#pragma once
+
+#include "spectra_reml/types.hpp"
+
+#include <cstddef>
+#include <filesystem>
+#include <vector>
+
+namespace spectra::reml {
+
+struct SpectralGrm {
+    std::vector<double> eigenvalues;
+    ColMajorMatrix eigenvectors;
+    double minimum_eigenvalue = 0.0;
+    double maximum_eigenvalue = 0.0;
+};
+
+// GCTA stores the lower triangle, row by row, as IEEE-754 float32 values:
+// (0,0), (1,0), (1,1), (2,0), ... .  The returned matrix is full symmetric,
+// double precision and column-major.
+[[nodiscard]] ColMajorMatrix read_gcta_grm_lower_triangle(
+    const std::filesystem::path& grm_bin, std::size_t sample_count);
+
+// Reads and diagonalizes a GRM once.  Eigenvalues are deliberately not
+// clipped: preserving the supplied GRM is required for numerical equivalence.
+// The REML evaluator rejects parameter trials for which sigma_g^2 lambda_i +
+// sigma_e^2 is not positive.
+[[nodiscard]] SpectralGrm read_and_diagonalize_gcta_grm(
+    const std::filesystem::path& grm_bin, std::size_t sample_count);
+
+// Optional structural check for a GCTA .grm.id file.  The Python driver owns
+// sample-ID alignment; this guard catches wrong dimensions at the CLI boundary.
+void validate_grm_id_count(const std::filesystem::path& grm_id,
+                           std::size_t expected_sample_count);
+
+}  // namespace spectra::reml
--- a/include/spectra_reml/linalg.hpp
+++ b/include/spectra_reml/linalg.hpp
@@ -0,0 +1,41 @@
+#pragma once
+
+#include "spectra_reml/types.hpp"
+
+#include <cstddef>
+#include <string>
+#include <vector>
+
+namespace spectra::reml {
+
+// C = A' B.  A is n x k and B is n x m; C is k x m.
+[[nodiscard]] ColMajorMatrix cross_product(const ColMajorMatrix& a,
+                                           const ColMajorMatrix& b);
+
+// C = A' B, specialized for an orthogonal n x n matrix A.  This is kept as a
+// distinct API because it is the dominant level-3 BLAS operation in a batch.
+[[nodiscard]] ColMajorMatrix rotate_to_eigenspace(
+    const ColMajorMatrix& eigenvectors, const ColMajorMatrix& variables);
+
+// Overwrites a symmetric matrix with its eigenvectors (columns) and returns
+// eigenvalues in ascending order.  Only the lower triangle is inspected.
+[[nodiscard]] std::vector<double> symmetric_eigen_decomposition(
+    ColMajorMatrix& symmetric_matrix);
+
+// Cholesky helpers.  The factor is lower triangular and stored in the lower
+// triangle of a full column-major matrix.
+[[nodiscard]] bool cholesky_factor_in_place(std::vector<double>& matrix,
+                                            std::size_t order,
+                                            std::string* error = nullptr);
+[[nodiscard]] bool cholesky_solve_in_place(
+    const std::vector<double>& factor, std::size_t order, double* rhs,
+    std::size_t rhs_columns, std::string* error = nullptr);
+[[nodiscard]] bool cholesky_inverse(
+    const std::vector<double>& factor, std::size_t order,
+    std::vector<double>& inverse, std::string* error = nullptr);
+
+[[nodiscard]] double dot(const std::vector<double>& a,
+                         const std::vector<double>& b);
+[[nodiscard]] double infinity_norm(const std::vector<double>& x);
+
+}  // namespace spectra::reml
--- a/include/spectra_reml/reml.hpp
+++ b/include/spectra_reml/reml.hpp
@@ -0,0 +1,49 @@
+#pragma once
+
+#include "spectra_reml/types.hpp"
+
+#include <limits>
+#include <string>
+#include <vector>
+
+namespace spectra::reml {
+
+// Fits y = X beta + g + e in the GRM eigenspace, where
+// Var(y) = sigma_g^2 diag(lambda) + sigma_e^2 I.
+//
+// The parameters are the signed standard deviations (sigma_e, sigma_g), as in
+// thesis Equations 4.24-4.25.  The covariance depends on their squares, so the
+// optimization is unconstrained.  It uses AI-REML and a strong-Wolfe line
+// search; no EM update is performed and P is never materialized.
+[[nodiscard]] RemlResult fit_ai_reml_spectral(
+    const std::vector<double>& y_star, const ColMajorMatrix& x_star,
+    const std::vector<double>& eigenvalues,
+    const RemlOptions& options = {});
+
+// Exposed for finite-difference and independent-oracle tests.  The ordering is
+// theta=(sigma_e,sigma_g); ai is [ee,eg;eg,gg] in column-major order.
+struct RemlEvaluation {
+    bool valid = false;
+    double log_likelihood = -std::numeric_limits<double>::infinity();
+    double gradient_e = std::numeric_limits<double>::quiet_NaN();
+    double gradient_g = std::numeric_limits<double>::quiet_NaN();
+    // Scores with respect to the variance parameters v_e=sigma_e^2 and
+    // v_g=sigma_g^2.  These remain informative when sigma_g=0, unlike the
+    // signed-standard-deviation gradient gradient_g=2*sigma_g*score_v_g.
+    double variance_score_e = std::numeric_limits<double>::quiet_NaN();
+    double variance_score_g = std::numeric_limits<double>::quiet_NaN();
+    double ai_ee = std::numeric_limits<double>::quiet_NaN();
+    double ai_eg = std::numeric_limits<double>::quiet_NaN();
+    double ai_gg = std::numeric_limits<double>::quiet_NaN();
+    std::vector<double> beta;
+    std::vector<double> beta_covariance;
+    std::string error;
+};
+
+[[nodiscard]] RemlEvaluation evaluate_reml_spectral(
+    const std::vector<double>& y_star, const ColMajorMatrix& x_star,
+    const std::vector<double>& eigenvalues, double sigma_e, double sigma_g,
+    bool compute_ai = true, bool compute_beta_covariance = false,
+    double covariance_floor_relative = 1e-12);
+
+}  // namespace spectra::reml
--- a/include/spectra_reml/types.hpp
+++ b/include/spectra_reml/types.hpp
@@ -0,0 +1,133 @@
+#pragma once
+
+#include <cmath>
+#include <cstddef>
+#include <cstdint>
+#include <limits>
+#include <stdexcept>
+#include <string>
+#include <utility>
+#include <vector>
+
+namespace spectra::reml {
+
+// Dense matrices use Fortran/BLAS column-major storage throughout the
+// numerical core.  Keeping this convention at the API boundary prevents
+// hidden transposes before LAPACK calls.
+class ColMajorMatrix {
+public:
+    ColMajorMatrix() = default;
+    ColMajorMatrix(std::size_t rows, std::size_t cols)
+        : rows_(rows), cols_(cols), values_(rows * cols, 0.0) {}
+    ColMajorMatrix(std::size_t rows, std::size_t cols,
+                   std::vector<double> values)
+        : rows_(rows), cols_(cols), values_(std::move(values)) {
+        if (values_.size() != rows_ * cols_) {
+            throw std::invalid_argument("ColMajorMatrix data has the wrong size");
+        }
+    }
+
+    [[nodiscard]] std::size_t rows() const noexcept { return rows_; }
+    [[nodiscard]] std::size_t cols() const noexcept { return cols_; }
+    [[nodiscard]] std::size_t size() const noexcept { return values_.size(); }
+    [[nodiscard]] bool empty() const noexcept { return values_.empty(); }
+    [[nodiscard]] std::size_t leading_dimension() const noexcept { return rows_; }
+
+    double* data() noexcept { return values_.data(); }
+    const double* data() const noexcept { return values_.data(); }
+    std::vector<double>& values() noexcept { return values_; }
+    const std::vector<double>& values() const noexcept { return values_; }
+
+    double& operator()(std::size_t row, std::size_t col) noexcept {
+        return values_[row + col * rows_];
+    }
+    double operator()(std::size_t row, std::size_t col) const noexcept {
+        return values_[row + col * rows_];
+    }
+
+private:
+    std::size_t rows_ = 0;
+    std::size_t cols_ = 0;
+    std::vector<double> values_;
+};
+
+enum class FitStatus {
+    converged,
+    converged_boundary,
+    max_iterations,
+    line_search_failed,
+    rank_deficient,
+    invalid_input,
+    non_positive_covariance,
+    numerical_error
+};
+
+[[nodiscard]] const char* to_string(FitStatus status) noexcept;
+
+struct RemlOptions {
+    std::size_t max_iterations = 100;
+    std::size_t line_search_max_evaluations = 48;
+    std::size_t line_search_max_zoom_iterations = 48;
+    double gradient_absolute_tolerance = 1e-7;
+    double gradient_relative_tolerance = 1e-8;
+    double step_relative_tolerance = 1e-9;
+    double likelihood_relative_tolerance = 1e-11;
+    double wolfe_c1 = 1e-4;
+    double wolfe_c2 = 0.9;
+    double initial_line_search_step = 1.0;
+    double maximum_line_search_step = 64.0;
+    double covariance_floor_relative = 1e-12;
+    double rank_tolerance_relative = 1e-10;
+    double ai_ridge_relative = 1e-10;
+    std::size_t ai_ridge_attempts = 8;
+
+    // KKT check for the residual-only boundary sigma_g^2=0.  The boundary is
+    // accepted only when the one-sided variance-component score is no larger
+    // than boundary_score_tolerance and its likelihood is not inferior to the
+    // current interior iterate (within the stated relative roundoff tolerance).
+    double boundary_score_tolerance = 1e-8;
+    double boundary_likelihood_relative_tolerance = 1e-12;
+    double boundary_h2_trigger = 1e-3;
+
+    // NaN means use the paper's initial value: both standard deviations are
+    // sqrt(OLS residual mean square / 2).
+    double initial_sigma_e = std::numeric_limits<double>::quiet_NaN();
+    double initial_sigma_g = std::numeric_limits<double>::quiet_NaN();
+};
+
+struct RemlResult {
+    FitStatus status = FitStatus::invalid_input;
+    double sigma_e = std::numeric_limits<double>::quiet_NaN();
+    double sigma_g = std::numeric_limits<double>::quiet_NaN();
+    double sigma_e2 = std::numeric_limits<double>::quiet_NaN();
+    double sigma_g2 = std::numeric_limits<double>::quiet_NaN();
+    double h2 = std::numeric_limits<double>::quiet_NaN();
+    // Equation B.5 of the thesis, omitting additive constants.
+    double log_likelihood = -std::numeric_limits<double>::infinity();
+    std::size_t iterations = 0;
+    std::size_t line_search_evaluations = 0;
+    double gradient_inf_norm = std::numeric_limits<double>::infinity();
+    std::vector<double> beta;
+    // Row-wise packed lower triangle:
+    // (0,0), (1,0), (1,1), (2,0), (2,1), (2,2), ...
+    std::vector<double> beta_covariance_packed_lower;
+    std::string error;
+
+    [[nodiscard]] bool has_estimates() const noexcept {
+        return !beta.empty() && std::isfinite(log_likelihood);
+    }
+};
+
+struct RemlTask {
+    std::uint64_t task_index = 0;
+    std::string task_id;
+    std::uint64_t phenotype_row = 0;
+    std::vector<std::int32_t> extra_covariate_rows;
+};
+
+struct TaskResult {
+    RemlTask task;
+    RemlResult fit;
+};
+
+}  // namespace spectra::reml
--- a/python/spectra_reml.py
+++ b/python/spectra_reml.py
@@ -0,0 +1,766 @@
+#!/usr/bin/env python3
+"""Generic manifest, execution, recovery, and result CLI for SpectraREML."""
+
+from __future__ import annotations
+
+import argparse
+import csv
+import gzip
+import hashlib
+import json
+import os
+import shutil
+import subprocess
+import sys
+import time
+from datetime import datetime, timezone
+from pathlib import Path
+from typing import Any, Mapping, NoReturn, Sequence
+
+for _name in (
+    "OMP_NUM_THREADS",
+    "OPENBLAS_NUM_THREADS",
+    "MKL_NUM_THREADS",
+    "NUMEXPR_NUM_THREADS",
+):
+    os.environ.setdefault(_name, "1")
+
+try:
+    import numpy as np
+except ImportError as exc:  # pragma: no cover
+    raise SystemExit("NumPy is required: {}".format(exc))
+
+
+MANIFEST_FORMAT = "spectra-reml-manifest-v1"
+BLOCK_FORMAT = "spectra-reml-block-v1"
+RUN_SIGNATURE_FORMAT = "spectra-reml-run-signature-v1"
+FINALIZE_FORMAT = "spectra-reml-finalize-v1"
+TASK_HEADER = (
+    "task_index",
+    "task_id",
+    "phenotype_row",
+    "n_extra_covariates",
+)
+SUMMARY_HEADER = (
+    "task_index",
+    "task_id",
+    "status",
+    "n_fixed",
+    "n_extra_covariates",
+    "beta_offset",
+    "cov_offset",
+    "sigma_g2",
+    "sigma_e2",
+    "h2",
+    "logL",
+    "iterations",
+    "line_search_steps",
+    "grad_inf",
+    "error",
+)
+SUCCESS_STATUSES = frozenset(("converged", "converged_boundary"))
+BLOCK_PATTERNS = (
+    "block_*.summary.tsv",
+    "block_*.beta.f64.bin",
+    "block_*.cov.f64.bin",
+    "block_*.complete",
+)
+
+
+def fail(message: str) -> NoReturn:
+    raise RuntimeError(message)
+
+
+def log(message: str) -> None:
+    print(time.strftime("%Y-%m-%d %H:%M:%S"), "|", message, flush=True)
+
+
+def utc_now() -> str:
+    return datetime.now(timezone.utc).isoformat(timespec="seconds")
+
+
+def resolved(path: Path) -> Path:
+    return path.expanduser().resolve()
+
+
+def file_identity(path: Path) -> dict[str, Any]:
+    path = resolved(path)
+    stat = path.stat()
+    return {
+        "path": str(path),
+        "size_bytes": int(stat.st_size),
+        "mtime_ns": int(stat.st_mtime_ns),
+    }
+
+
+def sha256_file(path: Path) -> str:
+    digest = hashlib.sha256()
+    with path.open("rb") as handle:
+        for block in iter(lambda: handle.read(1024 * 1024), b""):
+            digest.update(block)
+    return digest.hexdigest()
+
+
+def canonical_sha256(value: Any) -> str:
+    encoded = json.dumps(
+        value,
+        ensure_ascii=False,
+        allow_nan=False,
+        sort_keys=True,
+        separators=(",", ":"),
+    ).encode("utf-8")
+    return hashlib.sha256(encoded).hexdigest()
+
+
+def atomic_json(path: Path, value: Mapping[str, Any]) -> None:
+    path.parent.mkdir(parents=True, exist_ok=True)
+    temporary = path.with_name(path.name + ".tmp.{}".format(os.getpid()))
+    try:
+        with temporary.open("w", encoding="utf-8", newline="") as handle:
+            json.dump(value, handle, ensure_ascii=False, allow_nan=False, indent=2)
+            handle.write("\n")
+            handle.flush()
+            os.fsync(handle.fileno())
+        os.replace(temporary, path)
+    finally:
+        try:
+            temporary.unlink()
+        except FileNotFoundError:
+            pass
+
+
+def read_json(path: Path) -> dict[str, Any]:
+    try:
+        with path.open("r", encoding="utf-8-sig") as handle:
+            value = json.load(handle)
+    except (OSError, json.JSONDecodeError) as exc:
+        fail("Cannot read JSON {}: {}".format(path, exc))
+    if not isinstance(value, dict):
+        fail("JSON root is not an object: {}".format(path))
+    return value
+
+
+def require_file(path: Path, label: str, allow_empty: bool = False) -> None:
+    if not path.is_file():
+        fail("{} is missing: {}".format(label, path))
+    if not allow_empty and path.stat().st_size == 0:
+        fail("{} is empty: {}".format(label, path))
+
+
+def require_size(path: Path, expected: int, label: str) -> None:
+    require_file(path, label, allow_empty=(expected == 0))
+    observed = path.stat().st_size
+    if observed != expected:
+        fail("{} size mismatch: observed {}; expected {}: {}".format(
+            label, observed, expected, path
+        ))
+
+
+def manifest_paths(manifest: Mapping[str, Any]) -> dict[str, Path | None]:
+    raw = manifest.get("paths")
+    if not isinstance(raw, dict):
+        fail("Manifest has no paths object.")
+    required = (
+        "grm_bin",
+        "base_x_f64",
+        "phenotype_f64",
+        "tasks_tsv",
+        "extra_offsets_i64",
+        "extra_indices_i32",
+        "output_directory",
+    )
+    missing = [name for name in required if name not in raw]
+    if missing:
+        fail("Manifest paths are missing: {}".format(", ".join(missing)))
+    result: dict[str, Path | None] = {}
+    for name in required + ("grm_id", "extra_covariate_f32"):
+        value = raw.get(name)
+        result[name] = Path(str(value)) if value not in (None, "") else None
+    return result
+
+
+def read_tasks(path: Path) -> list[dict[str, str]]:
+    with path.open("r", encoding="utf-8-sig", newline="") as handle:
+        reader = csv.DictReader(handle, delimiter="\t")
+        if tuple(reader.fieldnames or ()) != TASK_HEADER:
+            fail("Task header must be exactly: {}".format("\t".join(TASK_HEADER)))
+        return list(reader)
+
+
+def validate_manifest(path: Path) -> dict[str, Any]:
+    path = resolved(path)
+    manifest = read_json(path)
+    if manifest.get("format") != MANIFEST_FORMAT:
+        fail("Unsupported manifest format: {!r}".format(manifest.get("format")))
+    dimensions = manifest.get("dimensions")
+    if not isinstance(dimensions, dict):
+        fail("Manifest has no dimensions object.")
+    names = (
+        "sample_count",
+        "base_covariate_count",
+        "phenotype_row_count",
+        "extra_covariate_row_count",
+        "task_count",
+        "extra_index_count",
+    )
+    parsed: dict[str, int] = {}
+    for name in names:
+        value = dimensions.get(name)
+        if isinstance(value, bool) or not isinstance(value, int) or value < 0:
+            fail("Invalid dimension {}={!r}".format(name, value))
+        parsed[name] = value
+    if min(parsed["sample_count"], parsed["base_covariate_count"], parsed["task_count"]) <= 0:
+        fail("Sample, base-covariate, and task counts must be positive.")
+
+    paths = manifest_paths(manifest)
+    n = parsed["sample_count"]
+    require_size(paths["grm_bin"], n * (n + 1) // 2 * 4, "GRM")  # type: ignore[arg-type]
+    require_size(paths["base_x_f64"], n * parsed["base_covariate_count"] * 8, "base design")  # type: ignore[arg-type]
+    require_size(paths["phenotype_f64"], n * parsed["phenotype_row_count"] * 8, "phenotypes")  # type: ignore[arg-type]
+    require_size(paths["extra_offsets_i64"], (parsed["task_count"] + 1) * 8, "extra offsets")  # type: ignore[arg-type]
+    require_size(paths["extra_indices_i32"], parsed["extra_index_count"] * 4, "extra indices")  # type: ignore[arg-type]
+    if parsed["extra_covariate_row_count"]:
+        if paths["extra_covariate_f32"] is None:
+            fail("extra_covariate_f32 is required when its row count is nonzero.")
+        require_size(paths["extra_covariate_f32"], n * parsed["extra_covariate_row_count"] * 4, "extra covariates")
+    if paths["grm_id"] is not None:
+        require_file(paths["grm_id"], "GRM IDs")
+        with paths["grm_id"].open("r", encoding="utf-8-sig") as handle:
+            if sum(bool(line.strip()) for line in handle) != n:
+                fail("GRM ID row count differs from sample_count.")
+
+    sources = manifest.get("source_files")
+    if not isinstance(sources, dict) or not sources:
+        fail("Manifest has no source identities.")
+    for label, expected in sources.items():
+        if not isinstance(expected, dict):
+            fail("Malformed source identity: {}".format(label))
+        source = Path(str(expected.get("path", "")))
+        if not source.is_file():
+            fail("Source is missing: {}: {}".format(label, source))
+        observed = source.stat()
+        if (
+            observed.st_size != int(expected.get("size_bytes", -1))
+            or observed.st_mtime_ns != int(expected.get("mtime_ns", -1))
+        ):
+            fail("Source changed since manifest creation: {}: {}".format(label, source))
+
+    controls = {
+        "base_x_f64": paths["base_x_f64"],
+        "tasks_tsv": paths["tasks_tsv"],
+        "extra_offsets_i64": paths["extra_offsets_i64"],
+        "extra_indices_i32": paths["extra_indices_i32"],
+    }
+    hashes = manifest.get("control_sha256")
+    if not isinstance(hashes, dict) or set(hashes) != set(controls):
+        fail("control_sha256 does not cover all control files.")
+    for name, control in controls.items():
+        assert control is not None
+        if sha256_file(control) != hashes[name]:
+            fail("Control checksum mismatch: {}".format(control))
+
+    tasks = read_tasks(paths["tasks_tsv"])  # type: ignore[arg-type]
+    if len(tasks) != parsed["task_count"]:
+        fail("Task count differs from manifest.")
+    offsets = np.fromfile(paths["extra_offsets_i64"], dtype="<i8")
+    indices = np.fromfile(paths["extra_indices_i32"], dtype="<i4")
+    if offsets[0] != 0 or np.any(offsets[1:] < offsets[:-1]) or int(offsets[-1]) != indices.size:
+        fail("Extra-covariate CSR offsets are invalid.")
+    if indices.size and (
+        int(indices.min()) < 0
+        or int(indices.max()) >= parsed["extra_covariate_row_count"]
+    ):
+        fail("An extra-covariate index is outside its matrix.")
+    seen_ids: set[str] = set()
+    for expected, row in enumerate(tasks):
+        try:
+            task_index = int(row["task_index"])
+            phenotype_row = int(row["phenotype_row"])
+            extra_count = int(row["n_extra_covariates"])
+        except ValueError as exc:
+            fail("Invalid task integer at row {}: {}".format(expected, exc))
+        if task_index != expected:
+            fail("Task indices must be consecutive and zero based.")
+        if not row["task_id"] or row["task_id"] in seen_ids:
+            fail("Task IDs must be nonempty and unique.")
+        seen_ids.add(row["task_id"])
+        if phenotype_row < 0 or phenotype_row >= parsed["phenotype_row_count"]:
+            fail("Task phenotype row is outside its matrix.")
+        if extra_count != int(offsets[expected + 1] - offsets[expected]):
+            fail("Task extra count disagrees with CSR offsets.")
+    return manifest
+
+
+def make_manifest(args: argparse.Namespace) -> Path:
+    target = resolved(args.manifest)
+    paths: dict[str, Path | None] = {
+        "grm_bin": resolved(args.grm_bin),
+        "grm_id": resolved(args.grm_id) if args.grm_id else None,
+        "base_x_f64": resolved(args.base_x),
+        "phenotype_f64": resolved(args.phenotypes),
+        "extra_covariate_f32": resolved(args.extra_covariates) if args.extra_covariates else None,
+        "tasks_tsv": resolved(args.tasks),
+        "extra_offsets_i64": resolved(args.extra_offsets),
+        "extra_indices_i32": resolved(args.extra_indices),
+        "output_directory": resolved(args.output_dir),
+    }
+    tasks = read_tasks(paths["tasks_tsv"])  # type: ignore[arg-type]
+    extra_index_count = paths["extra_indices_i32"].stat().st_size // 4  # type: ignore[union-attr]
+    source_names = ("grm_bin", "phenotype_f64")
+    sources = {name: file_identity(paths[name]) for name in source_names}  # type: ignore[arg-type]
+    for name in ("grm_id", "extra_covariate_f32"):
+        if paths[name] is not None:
+            sources[name] = file_identity(paths[name])  # type: ignore[arg-type]
+    controls = ("base_x_f64", "tasks_tsv", "extra_offsets_i64", "extra_indices_i32")
+    manifest = {
+        "format": MANIFEST_FORMAT,
+        "created_utc": utc_now(),
+        "dimensions": {
+            "sample_count": args.n_samples,
+            "base_covariate_count": args.n_base_covariates,
+            "phenotype_row_count": args.n_phenotype_rows,
+            "extra_covariate_row_count": args.n_extra_covariate_rows,
+            "task_count": len(tasks),
+            "extra_index_count": extra_index_count,
+        },
+        "paths": {name: str(value) if value is not None else None for name, value in paths.items()},
+        "source_files": sources,
+        "control_sha256": {name: sha256_file(paths[name]) for name in controls},  # type: ignore[arg-type]
+    }
+    atomic_json(target, manifest)
+    validate_manifest(target)
+    log("manifest created: {} tasks -> {}".format(len(tasks), target))
+    return target
+
+
+def resolve_engine(value: str) -> Path:
+    candidate = Path(value).expanduser()
+    if candidate.is_file():
+        return candidate.resolve()
+    found = shutil.which(value)
+    if found:
+        return Path(found).resolve()
+    fail("SpectraREML engine was not found: {}".format(value))
+
+
+def build_engine_command(engine: Path, manifest: Mapping[str, Any], args: argparse.Namespace) -> list[str]:
+    paths = manifest_paths(manifest)
+    dims = manifest["dimensions"]
+    command = [
+        str(engine),
+        "--grm-bin", str(paths["grm_bin"]),
+        "--base-x", str(paths["base_x_f64"]),
+        "--phenotypes", str(paths["phenotype_f64"]),
+        "--tasks", str(paths["tasks_tsv"]),
+        "--extra-offsets", str(paths["extra_offsets_i64"]),
+        "--extra-indices", str(paths["extra_indices_i32"]),
+        "--out-dir", str(paths["output_directory"]),
+        "--n-samples", str(dims["sample_count"]),
+        "--n-base-covariates", str(dims["base_covariate_count"]),
+        "--n-phenotype-rows", str(dims["phenotype_row_count"]),
+        "--n-extra-covariate-rows", str(dims["extra_covariate_row_count"]),
+        "--block-size", str(args.block_size),
+        "--threads", str(args.threads),
+    ]
+    if paths["grm_id"] is not None:
+        command.extend(("--grm-id", str(paths["grm_id"])))
+    if int(dims["extra_index_count"]):
+        command.extend(("--extra-covariates", str(paths["extra_covariate_f32"])))
+    option_map = (
+        ("max_iterations", "--max-iterations"),
+        ("line_search_max_evals", "--line-search-max-evals"),
+        ("line_search_max_zoom", "--line-search-max-zoom"),
+        ("gradient_abs_tol", "--gradient-abs-tol"),
+        ("gradient_rel_tol", "--gradient-rel-tol"),
+        ("step_rel_tol", "--step-rel-tol"),
+        ("likelihood_rel_tol", "--likelihood-rel-tol"),
+        ("wolfe_c1", "--wolfe-c1"),
+        ("wolfe_c2", "--wolfe-c2"),
+        ("initial_step", "--initial-step"),
+        ("maximum_step", "--maximum-step"),
+        ("rank_tol", "--rank-tol"),
+        ("covariance_floor", "--covariance-floor"),
+        ("boundary_h2_trigger", "--boundary-h2-trigger"),
+        ("boundary_score_tol", "--boundary-score-tol"),
+        ("boundary_logl_rel_tol", "--boundary-logl-rel-tol"),
+    )
+    for attribute, option in option_map:
+        command.extend((option, repr(getattr(args, attribute))))
+    for attribute, option in (("initial_sigma_e", "--initial-sigma-e"), ("initial_sigma_g", "--initial-sigma-g")):
+        value = getattr(args, attribute)
+        if value is not None:
+            command.extend((option, repr(value)))
+    if args.resume:
+        command.append("--resume")
+    elif args.force:
+        command.append("--overwrite")
+    return command
+
+
+def engine_identity(engine: Path) -> dict[str, Any]:
+    value = file_identity(engine)
+    value["sha256"] = sha256_file(engine)
+    return value
+
+
+def signature_payload(manifest: Mapping[str, Any], engine: Path, args: argparse.Namespace) -> dict[str, Any]:
+    excluded = {"action", "manifest", "engine", "dry_run", "output"}
+    options = {
+        key: value for key, value in vars(args).items()
+        if key not in excluded and key not in {"resume", "force"}
+    }
+    return {
+        "manifest_sha256": canonical_sha256(manifest),
+        "engine": engine_identity(engine),
+        "options": options,
+    }
+
+
+def make_signature(manifest: Mapping[str, Any], engine: Path, args: argparse.Namespace) -> dict[str, Any]:
+    payload = signature_payload(manifest, engine, args)
+    return {
+        "format": RUN_SIGNATURE_FORMAT,
+        "created_utc": utc_now(),
+        "signature_sha256": canonical_sha256(payload),
+        "payload": payload,
+    }
+
+
+def read_signature(path: Path) -> dict[str, Any]:
+    value = read_json(path)
+    payload = value.get("payload")
+    if value.get("format") != RUN_SIGNATURE_FORMAT or not isinstance(payload, dict):
+        fail("Invalid run signature: {}".format(path))
+    if value.get("signature_sha256") != canonical_sha256(payload):
+        fail("Run signature integrity check failed: {}".format(path))
+    return value
+
+
+def block_files(output: Path) -> list[Path]:
+    values: set[Path] = set()
+    for pattern in BLOCK_PATTERNS:
+        values.update(path for path in output.glob(pattern) if path.is_file())
+    return sorted(values)
+
+
+def validate_resume_layout(output: Path, task_count: int, block_size: int) -> None:
+    for marker in output.glob("block_*.complete"):
+        digits = marker.name[len("block_"):-len(".complete")]
+        if len(digits) != 6 or not digits.isdigit():
+            fail("Invalid block marker: {}".format(marker))
+        index = int(digits)
+        expected = min(block_size, max(0, task_count - index * block_size))
+        values: dict[str, str] = {}
+        with marker.open("r", encoding="utf-8") as handle:
+            for line in handle:
+                fields = line.rstrip("\r\n").split("\t", 1)
+                if len(fields) != 2:
+                    fail("Malformed marker: {}".format(marker))
+                values[fields[0]] = fields[1]
+        if expected <= 0 or values.get("format") != BLOCK_FORMAT or int(values.get("tasks", -1)) != expected:
+            fail("Block marker is incompatible with this manifest/block size: {}".format(marker))
+
+
+def read_marker(path: Path) -> dict[str, str]:
+    values: dict[str, str] = {}
+    with path.open("r", encoding="utf-8") as handle:
+        for line in handle:
+            fields = line.rstrip("\r\n").split("\t", 1)
+            if len(fields) != 2 or not fields[0] or fields[0] in values:
+                fail("Malformed completion marker: {}".format(path))
+            values[fields[0]] = fields[1]
+    return values
+
+
+def run_engine(manifest: Mapping[str, Any], args: argparse.Namespace) -> None:
+    if args.dry_run:
+        print(json.dumps(build_engine_command(Path(args.engine), manifest, args), ensure_ascii=False))
+        return
+    engine = resolve_engine(args.engine)
+    expected_signature = make_signature(manifest, engine, args)
+    paths = manifest_paths(manifest)
+    output = paths["output_directory"]
+    assert output is not None
+    output.mkdir(parents=True, exist_ok=True)
+    signature_path = output / "run.signature.json"
+    markers = list(output.glob("block_*.complete"))
+    if args.force:
+        try:
+            signature_path.unlink()
+        except FileNotFoundError:
+            pass
+        for path in block_files(output):
+            path.unlink()
+        atomic_json(signature_path, expected_signature)
+    elif args.resume:
+        if markers:
+            if not signature_path.is_file():
+                fail("Completed blocks have no run signature; use --force.")
+            observed = read_signature(signature_path)
+            if observed["signature_sha256"] != expected_signature["signature_sha256"]:
+                fail("Run signature differs from engine, manifest, or options; use --force.")
+            validate_resume_layout(output, int(manifest["dimensions"]["task_count"]), args.block_size)
+        else:
+            atomic_json(signature_path, expected_signature)
+    else:
+        if block_files(output):
+            fail("Block outputs already exist; use --resume or --force.")
+        atomic_json(signature_path, expected_signature)
+
+    command = build_engine_command(engine, manifest, args)
+    environment = os.environ.copy()
+    environment.update({
+        "OMP_NUM_THREADS": str(args.threads),
+        "MKL_NUM_THREADS": str(args.blas_threads),
+        "OPENBLAS_NUM_THREADS": str(args.blas_threads),
+        "OMP_DYNAMIC": "FALSE",
+    })
+    log("starting SpectraREML engine")
+    with (output / "engine.log").open("a" if args.resume else "w", encoding="utf-8") as handle:
+        handle.write("# {}\n# command={}\n".format(utc_now(), json.dumps(command, ensure_ascii=False)))
+        handle.flush()
+        completed = subprocess.run(command, stdout=handle, stderr=subprocess.STDOUT, env=environment, check=False)
+    atomic_json(output / "run.json", {
+        "finished_utc": utc_now(),
+        "return_code": completed.returncode,
+        "command": command,
+        "run_signature_sha256": expected_signature["signature_sha256"],
+    })
+    if completed.returncode:
+        fail("SpectraREML failed with exit code {}; see engine.log".format(completed.returncode))
+
+
+def finalizable_signature(manifest: Mapping[str, Any]) -> dict[str, Any]:
+    output = manifest_paths(manifest)["output_directory"]
+    assert output is not None
+    path = output / "run.signature.json"
+    if not path.is_file():
+        fail("Cannot finalize without a run signature.")
+    value = read_signature(path)
+    if value["payload"].get("manifest_sha256") != canonical_sha256(manifest):
+        fail("Run signature does not belong to the current manifest.")
+    return value
+
+
+def read_completed(manifest: Mapping[str, Any]) -> list[dict[str, Any]]:
+    output = manifest_paths(manifest)["output_directory"]
+    assert output is not None
+    task_count = int(manifest["dimensions"]["task_count"])
+    results: dict[int, dict[str, Any]] = {}
+    for marker in sorted(output.glob("block_*.complete")):
+        stem = marker.name[:-len(".complete")]
+        summary_path = output / (stem + ".summary.tsv")
+        beta_path = output / (stem + ".beta.f64.bin")
+        cov_path = output / (stem + ".cov.f64.bin")
+        marker_values = read_marker(marker)
+        try:
+            declared_tasks = int(marker_values["tasks"])
+            beta_elements = int(marker_values["beta_elements"])
+            cov_elements = int(marker_values["cov_elements"])
+        except (KeyError, ValueError) as exc:
+            fail("Malformed completion counts in {}: {}".format(marker, exc))
+        if marker_values.get("format") != BLOCK_FORMAT or min(
+            declared_tasks, beta_elements, cov_elements
+        ) < 0:
+            fail("Invalid completion marker: {}".format(marker))
+        require_size(beta_path, beta_elements * 8, "block beta")
+        require_size(cov_path, cov_elements * 8, "block covariance")
+        with summary_path.open("r", encoding="utf-8-sig", newline="") as handle:
+            reader = csv.DictReader(handle, delimiter="\t")
+            if tuple(reader.fieldnames or ()) != SUMMARY_HEADER:
+                fail("Unexpected summary header: {}".format(summary_path))
+            rows = list(reader)
+        if len(rows) != declared_tasks:
+            fail("Summary row count disagrees with marker: {}".format(marker))
+        beta = np.fromfile(beta_path, dtype="<f8")
+        cov = np.fromfile(cov_path, dtype="<f8")
+        next_beta_offset = 0
+        next_cov_offset = 0
+        for row in rows:
+            index = int(row["task_index"])
+            if index in results or index < 0 or index >= task_count:
+                fail("Duplicate/out-of-range task index: {}".format(index))
+            p = int(row["n_fixed"])
+            beta_offset, cov_offset = int(row["beta_offset"]), int(row["cov_offset"])
+            if beta_offset < 0 or cov_offset < 0:
+                if beta_offset != cov_offset:
+                    fail("Only one output offset is negative for task {}".format(index))
+                beta_values: list[float] = []
+                cov_values: list[float] = []
+            else:
+                packed = p * (p + 1) // 2
+                if beta_offset != next_beta_offset or cov_offset != next_cov_offset:
+                    fail("Non-contiguous output offsets for task {}".format(index))
+                if beta_offset + p > beta.size or cov_offset + packed > cov.size:
+                    fail("Output offsets exceed binary arrays for task {}".format(index))
+                beta_values = beta[beta_offset:beta_offset + p].tolist()
+                cov_values = cov[cov_offset:cov_offset + packed].tolist()
+                next_beta_offset += p
+                next_cov_offset += packed
+            result: dict[str, Any] = dict(row)
+            result["beta_json"] = json.dumps(beta_values, separators=(",", ":"))
+            result["covariance_packed_lower_json"] = json.dumps(cov_values, separators=(",", ":"))
+            results[index] = result
+        if next_beta_offset != beta_elements or next_cov_offset != cov_elements:
+            fail("Block binary arrays contain unused elements: {}".format(stem))
+    missing = sorted(set(range(task_count)).difference(results))
+    if missing:
+        fail("Completed blocks are missing {} tasks.".format(len(missing)))
+    return [results[index] for index in range(task_count)]
+
+
+def finalize(manifest: Mapping[str, Any], output: Path) -> Path:
+    signature = finalizable_signature(manifest)
+    rows = read_completed(manifest)
+    output = resolved(output)
+    output.parent.mkdir(parents=True, exist_ok=True)
+    temporary = output.with_name(output.name + ".tmp.{}".format(os.getpid()))
+    opener = gzip.open if output.suffix == ".gz" else open
+    fields = list(SUMMARY_HEADER) + ["beta_json", "covariance_packed_lower_json"]
+    try:
+        with opener(temporary, "wt", encoding="utf-8", newline="") as handle:
+            writer = csv.DictWriter(handle, fieldnames=fields, delimiter="\t", lineterminator="\n")
+            writer.writeheader()
+            writer.writerows(rows)
+        os.replace(temporary, output)
+    finally:
+        try:
+            temporary.unlink()
+        except FileNotFoundError:
+            pass
+    counts: dict[str, int] = {}
+    for row in rows:
+        counts[row["status"]] = counts.get(row["status"], 0) + 1
+    atomic_json(output.with_suffix(output.suffix + ".json"), {
+        "format": FINALIZE_FORMAT,
+        "finished_utc": utc_now(),
+        "manifest_sha256": canonical_sha256(manifest),
+        "run_signature_sha256": signature["signature_sha256"],
+        "tasks": len(rows),
+        "status_counts": counts,
+    })
+    log("finalized {} tasks -> {}".format(len(rows), output))
+    return output
+
+
+def status(manifest: Mapping[str, Any]) -> None:
+    output = manifest_paths(manifest)["output_directory"]
+    assert output is not None
+    counts: dict[str, int] = {}
+    completed = 0
+    for marker in output.glob("block_*.complete"):
+        summary = output / (marker.name[:-len(".complete")] + ".summary.tsv")
+        with summary.open("r", encoding="utf-8-sig", newline="") as handle:
+            for row in csv.DictReader(handle, delimiter="\t"):
+                completed += 1
+                counts[row.get("status", "unknown")] = counts.get(row.get("status", "unknown"), 0) + 1
+    print(json.dumps({
+        "tasks": manifest["dimensions"]["task_count"],
+        "completed": completed,
+        "status_counts": counts,
+        "output_directory": str(output),
+    }, ensure_ascii=False, indent=2))
+
+
+def positive_integer(value: str) -> int:
+    parsed = int(value)
+    if parsed <= 0:
+        raise argparse.ArgumentTypeError("must be positive")
+    return parsed
+
+
+def add_run_options(parser: argparse.ArgumentParser) -> None:
+    parser.add_argument("--engine", default="spectra_reml")
+    parser.add_argument("--threads", type=positive_integer, default=1)
+    parser.add_argument("--blas-threads", type=positive_integer, default=1)
+    parser.add_argument("--block-size", type=positive_integer, default=256)
+    parser.add_argument("--max-iterations", type=positive_integer, default=100)
+    parser.add_argument("--line-search-max-evals", type=positive_integer, default=48)
+    parser.add_argument("--line-search-max-zoom", type=positive_integer, default=48)
+    parser.add_argument("--gradient-abs-tol", type=float, default=1e-7)
+    parser.add_argument("--gradient-rel-tol", type=float, default=1e-8)
+    parser.add_argument("--step-rel-tol", type=float, default=1e-9)
+    parser.add_argument("--likelihood-rel-tol", type=float, default=1e-11)
+    parser.add_argument("--wolfe-c1", type=float, default=1e-4)
+    parser.add_argument("--wolfe-c2", type=float, default=0.9)
+    parser.add_argument("--initial-step", type=float, default=1.0)
+    parser.add_argument("--maximum-step", type=float, default=64.0)
+    parser.add_argument("--rank-tol", type=float, default=1e-10)
+    parser.add_argument("--covariance-floor", type=float, default=1e-12)
+    parser.add_argument("--boundary-h2-trigger", type=float, default=1e-3)
+    parser.add_argument("--boundary-score-tol", type=float, default=1e-8)
+    parser.add_argument("--boundary-logl-rel-tol", type=float, default=1e-12)
+    parser.add_argument("--initial-sigma-e", type=float)
+    parser.add_argument("--initial-sigma-g", type=float)
+    mode = parser.add_mutually_exclusive_group()
+    mode.add_argument("--resume", action="store_true")
+    mode.add_argument("--force", action="store_true")
+    parser.add_argument("--dry-run", action="store_true")
+
+
+def build_parser() -> argparse.ArgumentParser:
+    parser = argparse.ArgumentParser(description=__doc__)
+    subparsers = parser.add_subparsers(dest="action", required=True)
+    make = subparsers.add_parser("make-manifest")
+    make.add_argument("--manifest", type=Path, required=True)
+    make.add_argument("--grm-bin", type=Path, required=True)
+    make.add_argument("--grm-id", type=Path)
+    make.add_argument("--base-x", type=Path, required=True)
+    make.add_argument("--phenotypes", type=Path, required=True)
+    make.add_argument("--extra-covariates", type=Path)
+    make.add_argument("--tasks", type=Path, required=True)
+    make.add_argument("--extra-offsets", type=Path, required=True)
+    make.add_argument("--extra-indices", type=Path, required=True)
+    make.add_argument("--output-dir", type=Path, required=True)
+    make.add_argument("--n-samples", type=positive_integer, required=True)
+    make.add_argument("--n-base-covariates", type=positive_integer, required=True)
+    make.add_argument("--n-phenotype-rows", type=positive_integer, required=True)
+    make.add_argument("--n-extra-covariate-rows", type=int, default=0)
+    for action in ("validate", "status"):
+        command = subparsers.add_parser(action)
+        command.add_argument("--manifest", type=Path, required=True)
+    finalize_parser = subparsers.add_parser("finalize")
+    finalize_parser.add_argument("--manifest", type=Path, required=True)
+    finalize_parser.add_argument("--output", type=Path, required=True)
+    for action in ("run", "all"):
+        command = subparsers.add_parser(action)
+        command.add_argument("--manifest", type=Path, required=True)
+        if action == "all":
+            command.add_argument("--output", type=Path, required=True)
+        add_run_options(command)
+    return parser
+
+
+def main(argv: Sequence[str] | None = None) -> int:
+    args = build_parser().parse_args(argv)
+    if args.action == "make-manifest":
+        make_manifest(args)
+        return 0
+    manifest = validate_manifest(args.manifest)
+    if args.action == "validate":
+        print(json.dumps({
+            "manifest": str(resolved(args.manifest)),
+            "manifest_sha256": canonical_sha256(manifest),
+            "dimensions": manifest["dimensions"],
+        }, ensure_ascii=False, indent=2))
+    elif args.action == "status":
+        status(manifest)
+    elif args.action == "finalize":
+        finalize(manifest, args.output)
+    elif args.action == "run":
+        run_engine(manifest, args)
+    elif args.action == "all":
+        run_engine(manifest, args)
+        if not args.dry_run:
+            finalize(manifest, args.output)
+    return 0
+
+
+if __name__ == "__main__":
+    try:
+        raise SystemExit(main())
+    except KeyboardInterrupt:
+        raise SystemExit(130)
+    except Exception as exc:
+        print("ERROR: {}".format(exc), file=sys.stderr)
+        raise SystemExit(1)
--- a/scripts/run_server.sh
+++ b/scripts/run_server.sh
@@ -0,0 +1,112 @@
+#!/usr/bin/env bash
+
+set -euo pipefail
+
+ACTION="${1:-}"
+if [[ ! "$ACTION" =~ ^(build|test|validate|run|dry-run|finalize|status|all)$ ]]; then
+  echo "Usage: bash $0 {build|test|validate|run|dry-run|finalize|status|all}" >&2
+  exit 2
+fi
+
+SCRIPT_DIR="$(cd -- "$(dirname -- "${BASH_SOURCE[0]}")" && pwd)"
+PROJECT_DIR="${PROJECT_DIR:-$(cd -- "$SCRIPT_DIR/.." && pwd)}"
+PYTHON="${PYTHON:-python3}"
+BACKEND="${BACKEND:-MKL}"
+BACKEND="${BACKEND^^}"
+BUILD_DIR="${BUILD_DIR:-$PROJECT_DIR/build-${BACKEND,,}}"
+ENGINE="${ENGINE:-$BUILD_DIR/spectra_reml}"
+MANIFEST="${MANIFEST:-}"
+OUTPUT="${OUTPUT:-}"
+THREADS="${THREADS:-1}"
+BLAS_THREADS="${BLAS_THREADS:-1}"
+BLOCK_SIZE="${BLOCK_SIZE:-256}"
+RESUME="${RESUME:-1}"
+FORCE="${FORCE:-0}"
+CLI="$PROJECT_DIR/python/spectra_reml.py"
+
+die() {
+  echo "ERROR: $*" >&2
+  exit 1
+}
+
+[[ "$BACKEND" == "MKL" || "$BACKEND" == "OPENBLAS" ]] || die "BACKEND must be MKL or OPENBLAS"
+[[ "$THREADS" =~ ^[1-9][0-9]*$ ]] || die "THREADS must be positive"
+[[ "$BLAS_THREADS" =~ ^[1-9][0-9]*$ ]] || die "BLAS_THREADS must be positive"
+[[ "$BLOCK_SIZE" =~ ^[1-9][0-9]*$ ]] || die "BLOCK_SIZE must be positive"
+[[ "$RESUME" == "0" || "$RESUME" == "1" ]] || die "RESUME must be 0 or 1"
+[[ "$FORCE" == "0" || "$FORCE" == "1" ]] || die "FORCE must be 0 or 1"
+[[ "$RESUME" != "1" || "$FORCE" != "1" ]] || die "RESUME and FORCE are mutually exclusive"
+
+build_project() {
+  local args=(
+    -S "$PROJECT_DIR"
+    -B "$BUILD_DIR"
+    -DCMAKE_BUILD_TYPE=Release
+    -DREML_BLAS="$BACKEND"
+    -DREML_ENABLE_OPENMP=ON
+  )
+  if [[ "$BACKEND" == "MKL" ]]; then
+    args+=(
+      -DMKL_INTERFACE=lp64
+      -DMKL_LINK=dynamic
+      -DMKL_THREADING=sequential
+    )
+  fi
+  cmake "${args[@]}"
+  cmake --build "$BUILD_DIR" --parallel
+}
+
+if [[ "$ACTION" == "build" ]]; then
+  build_project
+  exit 0
+fi
+
+command -v "$PYTHON" >/dev/null 2>&1 || die "Python was not found: $PYTHON"
+[[ -s "$CLI" ]] || die "Generic CLI is missing: $CLI"
+
+if [[ "$ACTION" == "test" ]]; then
+  [[ -d "$BUILD_DIR" ]] || die "Build first: bash $0 build"
+  ctest --test-dir "$BUILD_DIR" --output-on-failure
+  "$PYTHON" "$PROJECT_DIR/tests/test_math_reference.py"
+  "$PYTHON" "$PROJECT_DIR/tests/python/test_spectra_reml_cli.py"
+  exit 0
+fi
+
+[[ -n "$MANIFEST" ]] || die "Set MANIFEST=/absolute/path/to/manifest.json"
+
+case "$ACTION" in
+  validate|status)
+    "$PYTHON" "$CLI" "$ACTION" --manifest "$MANIFEST"
+    ;;
+  finalize)
+    [[ -n "$OUTPUT" ]] || die "Set OUTPUT=/absolute/path/to/results.tsv.gz"
+    "$PYTHON" "$CLI" finalize --manifest "$MANIFEST" --output "$OUTPUT"
+    ;;
+  run|dry-run|all)
+    if [[ "$ACTION" != "dry-run" ]]; then
+      [[ -x "$ENGINE" ]] || die "Engine is missing/not executable: $ENGINE"
+    fi
+    mode=()
+    if [[ "$FORCE" == "1" ]]; then
+      mode+=(--force)
+    elif [[ "$RESUME" == "1" ]]; then
+      mode+=(--resume)
+    fi
+    common=(
+      --manifest "$MANIFEST"
+      --engine "$ENGINE"
+      --threads "$THREADS"
+      --blas-threads "$BLAS_THREADS"
+      --block-size "$BLOCK_SIZE"
+      "${mode[@]}"
+    )
+    if [[ "$ACTION" == "dry-run" ]]; then
+      "$PYTHON" "$CLI" run "${common[@]}" --dry-run
+    elif [[ "$ACTION" == "all" ]]; then
+      [[ -n "$OUTPUT" ]] || die "Set OUTPUT=/absolute/path/to/results.tsv.gz"
+      "$PYTHON" "$CLI" all "${common[@]}" --output "$OUTPUT"
+    else
+      "$PYTHON" "$CLI" run "${common[@]}"
+    fi
+    ;;
+esac
--- a/src/batch_io.cpp
+++ b/src/batch_io.cpp
@@ -0,0 +1,711 @@
+#include "spectra_reml/batch_io.hpp"
+
+#include "spectra_reml/linalg.hpp"
+#include "spectra_reml/reml.hpp"
+
+#include <algorithm>
+#include <array>
+#include <chrono>
+#include <cmath>
+#include <cstddef>
+#include <cstdint>
+#include <cstring>
+#include <filesystem>
+#include <fstream>
+#include <iomanip>
+#include <iostream>
+#include <limits>
+#include <set>
+#include <sstream>
+#include <stdexcept>
+#include <string>
+#include <thread>
+#include <unordered_map>
+#include <unordered_set>
+#include <utility>
+#include <vector>
+
+#ifdef _OPENMP
+#include <omp.h>
+#endif
+
+namespace spectra::reml {
+namespace {
+
+constexpr const char* kOutputFormatVersion = "spectra-reml-block-v1";
+
+std::size_t checked_product(std::size_t left, std::size_t right,
+                            const char* description) {
+    if (left != 0 && right > std::numeric_limits<std::size_t>::max() / left) {
+        throw std::overflow_error(std::string(description) + " size overflow");
+    }
+    return left * right;
+}
+
+void require_little_endian_ieee754() {
+    static_assert(sizeof(float) == 4, "batch format requires 32-bit float");
+    static_assert(sizeof(double) == 8, "batch format requires 64-bit double");
+    const std::uint32_t marker = 0x01020304U;
+    const auto* bytes = reinterpret_cast<const unsigned char*>(&marker);
+    if (bytes[0] != 0x04U || !std::numeric_limits<float>::is_iec559 ||
+        !std::numeric_limits<double>::is_iec559) {
+        throw std::runtime_error(
+            "binary batch format requires little-endian IEEE-754 hardware");
+    }
+}
+
+void validate_file_size(const std::filesystem::path& path,
+                        std::uintmax_t expected_bytes,
+                        const char* description) {
+    std::error_code error;
+    const auto actual_bytes = std::filesystem::file_size(path, error);
+    if (error) {
+        throw std::runtime_error("cannot stat " + std::string(description) +
+                                 " file " + path.string() + ": " +
+                                 error.message());
+    }
+    if (actual_bytes != expected_bytes) {
+        std::ostringstream message;
+        message << description << " file size mismatch for " << path.string()
+                << ": expected " << expected_bytes << " bytes, found "
+                << actual_bytes;
+        throw std::runtime_error(message.str());
+    }
+}
+
+template <typename T>
+std::vector<T> read_exact_binary_vector(const std::filesystem::path& path,
+                                        std::size_t count,
+                                        const char* description) {
+    const std::size_t bytes = checked_product(count, sizeof(T), description);
+    validate_file_size(path, static_cast<std::uintmax_t>(bytes), description);
+    std::vector<T> values(count);
+    std::ifstream input(path, std::ios::binary);
+    if (!input) {
+        throw std::runtime_error("cannot open " + std::string(description) +
+                                 " file " + path.string());
+    }
+    if (bytes != 0) {
+        input.read(reinterpret_cast<char*>(values.data()),
+                   static_cast<std::streamsize>(bytes));
+    }
+    if (!input || input.gcount() != static_cast<std::streamsize>(bytes)) {
+        throw std::runtime_error("short read from " + std::string(description) +
+                                 " file " + path.string());
+    }
+    return values;
+}
+
+std::vector<std::string> split_tab(const std::string& line) {
+    std::vector<std::string> fields;
+    std::size_t begin = 0;
+    while (true) {
+        const std::size_t end = line.find('\t', begin);
+        fields.push_back(line.substr(begin, end - begin));
+        if (end == std::string::npos) {
+            break;
+        }
+        begin = end + 1;
+    }
+    return fields;
+}
+
+std::size_t find_required_column(const std::vector<std::string>& header,
+                                 const std::string& name) {
+    const auto found = std::find(header.begin(), header.end(), name);
+    if (found == header.end()) {
+        throw std::runtime_error("tasks TSV is missing required column " + name);
+    }
+    return static_cast<std::size_t>(std::distance(header.begin(), found));
+}
+
+std::uint64_t parse_u64(const std::string& text, const char* field,
+                        std::size_t line_number) {
+    if (text.empty() || text.front() == '-') {
+        std::ostringstream message;
+        message << "invalid " << field << " at tasks TSV line " << line_number;
+        throw std::runtime_error(message.str());
+    }
+    std::size_t consumed = 0;
+    unsigned long long value = 0;
+    try {
+        value = std::stoull(text, &consumed, 10);
+    } catch (const std::exception&) {
+        std::ostringstream message;
+        message << "invalid " << field << " at tasks TSV line " << line_number;
+        throw std::runtime_error(message.str());
+    }
+    if (consumed != text.size()) {
+        std::ostringstream message;
+        message << "invalid " << field << " at tasks TSV line " << line_number;
+        throw std::runtime_error(message.str());
+    }
+    return static_cast<std::uint64_t>(value);
+}
+
+std::string sanitize_tsv(std::string text) {
+    for (char& character : text) {
+        if (character == '\t' || character == '\n' || character == '\r') {
+            character = ' ';
+        }
+    }
+    return text;
+}
+
+ColMajorMatrix read_selected_f32_rows(
+    const std::filesystem::path& path, std::size_t total_rows,
+    std::size_t sample_count, const std::vector<std::int32_t>& selected_rows) {
+    const std::size_t element_count =
+        checked_product(total_rows, sample_count, "extra_covariates matrix");
+    const std::size_t byte_count =
+        checked_product(element_count, sizeof(float), "extra_covariates matrix");
+    validate_file_size(path, byte_count, "extra_covariates matrix");
+    ColMajorMatrix selected(sample_count, selected_rows.size());
+    if (selected_rows.empty()) {
+        return selected;
+    }
+    std::ifstream input(path, std::ios::binary);
+    if (!input) {
+        throw std::runtime_error("cannot open extra_covariates matrix " + path.string());
+    }
+    std::vector<float> buffer(sample_count);
+    const std::size_t row_bytes =
+        checked_product(sample_count, sizeof(float), "extra_covariates row");
+    for (std::size_t column = 0; column < selected_rows.size(); ++column) {
+        const auto row = selected_rows[column];
+        if (row < 0 || static_cast<std::size_t>(row) >= total_rows) {
+            throw std::runtime_error("extra_covariates row index is outside the matrix");
+        }
+        const std::uintmax_t offset =
+            static_cast<std::uintmax_t>(row) * row_bytes;
+        input.seekg(static_cast<std::streamoff>(offset), std::ios::beg);
+        input.read(reinterpret_cast<char*>(buffer.data()),
+                   static_cast<std::streamsize>(row_bytes));
+        if (!input || input.gcount() != static_cast<std::streamsize>(row_bytes)) {
+            throw std::runtime_error("short read from extra_covariates row " +
+                                     std::to_string(row));
+        }
+        for (std::size_t sample = 0; sample < sample_count; ++sample) {
+            const double value = static_cast<double>(buffer[sample]);
+            if (!std::isfinite(value)) {
+                throw std::runtime_error("non-finite extra_covariates at row " +
+                                         std::to_string(row) + ", sample " +
+                                         std::to_string(sample));
+            }
+            selected(sample, column) = value;
+        }
+        input.clear();
+    }
+    return selected;
+}
+
+ColMajorMatrix read_selected_f64_rows(
+    std::ifstream& input, const std::filesystem::path& path,
+    std::size_t total_rows, std::size_t sample_count,
+    const std::vector<RemlTask>& tasks, std::size_t begin, std::size_t end) {
+    if (end < begin || end > tasks.size()) {
+        throw std::invalid_argument("invalid phenotypes block bounds");
+    }
+    ColMajorMatrix selected(sample_count, end - begin);
+    const std::size_t row_bytes =
+        checked_product(sample_count, sizeof(double), "phenotypes row");
+    for (std::size_t column = 0; column < end - begin; ++column) {
+        const auto row = tasks[begin + column].phenotype_row;
+        if (row >= total_rows) {
+            throw std::runtime_error("phenotypes row index is outside the matrix");
+        }
+        const std::uintmax_t offset = row * row_bytes;
+        input.seekg(static_cast<std::streamoff>(offset), std::ios::beg);
+        input.read(reinterpret_cast<char*>(selected.data() + column * sample_count),
+                   static_cast<std::streamsize>(row_bytes));
+        if (!input || input.gcount() != static_cast<std::streamsize>(row_bytes)) {
+            throw std::runtime_error("short read from phenotypes file " +
+                                     path.string() + " at row " +
+                                     std::to_string(row));
+        }
+        input.clear();
+    }
+    return selected;
+}
+
+std::string block_stem(std::size_t block_index) {
+    std::ostringstream name;
+    name << "block_" << std::setw(6) << std::setfill('0') << block_index;
+    return name.str();
+}
+
+std::filesystem::path temporary_path(const std::filesystem::path& final_path) {
+    const auto clock_value =
+        std::chrono::high_resolution_clock::now().time_since_epoch().count();
+    const auto thread_hash =
+        std::hash<std::thread::id>{}(std::this_thread::get_id());
+    return final_path.string() + ".tmp." + std::to_string(clock_value) + "." +
+           std::to_string(thread_hash);
+}
+
+void rename_checked(const std::filesystem::path& source,
+                    const std::filesystem::path& destination) {
+    std::error_code error;
+    std::filesystem::rename(source, destination, error);
+    if (error) {
+        throw std::runtime_error("cannot atomically rename " + source.string() +
+                                 " to " + destination.string() + ": " +
+                                 error.message());
+    }
+}
+
+void remove_if_exists(const std::filesystem::path& path) {
+    std::error_code error;
+    std::filesystem::remove(path, error);
+    if (error) {
+        throw std::runtime_error("cannot remove existing output " + path.string() +
+                                 ": " + error.message());
+    }
+}
+
+void write_binary_doubles(const std::filesystem::path& path,
+                          const std::vector<double>& values) {
+    std::ofstream output(path, std::ios::binary | std::ios::trunc);
+    if (!output) {
+        throw std::runtime_error("cannot create output " + path.string());
+    }
+    if (!values.empty()) {
+        output.write(reinterpret_cast<const char*>(values.data()),
+                     static_cast<std::streamsize>(values.size() * sizeof(double)));
+    }
+    output.flush();
+    if (!output) {
+        throw std::runtime_error("failed writing output " + path.string());
+    }
+}
+
+void write_block_atomic(const std::filesystem::path& output_directory,
+                        std::size_t block_index,
+                        const std::vector<TaskResult>& results,
+                        std::size_t base_covariate_count,
+                        bool overwrite) {
+    const std::string stem = block_stem(block_index);
+    const auto summary_path = output_directory / (stem + ".summary.tsv");
+    const auto beta_path = output_directory / (stem + ".beta.f64.bin");
+    const auto covariance_path = output_directory / (stem + ".cov.f64.bin");
+    const auto complete_path = output_directory / (stem + ".complete");
+    const std::array<std::filesystem::path, 4> final_paths = {
+        summary_path, beta_path, covariance_path, complete_path};
+    for (const auto& path : final_paths) {
+        if (std::filesystem::exists(path)) {
+            if (!overwrite) {
+                throw std::runtime_error("output already exists: " + path.string());
+            }
+            remove_if_exists(path);
+        }
+    }
+
+    std::vector<double> beta_values;
+    std::vector<double> covariance_values;
+    std::vector<std::int64_t> beta_offsets(results.size(), -1);
+    std::vector<std::int64_t> covariance_offsets(results.size(), -1);
+    for (std::size_t index = 0; index < results.size(); ++index) {
+        const auto& fit = results[index].fit;
+        if (!fit.has_estimates()) {
+            continue;
+        }
+        beta_offsets[index] = static_cast<std::int64_t>(beta_values.size());
+        covariance_offsets[index] =
+            static_cast<std::int64_t>(covariance_values.size());
+        beta_values.insert(beta_values.end(), fit.beta.begin(), fit.beta.end());
+        covariance_values.insert(covariance_values.end(),
+                                 fit.beta_covariance_packed_lower.begin(),
+                                 fit.beta_covariance_packed_lower.end());
+    }
+
+    const auto summary_temp = temporary_path(summary_path);
+    const auto beta_temp = temporary_path(beta_path);
+    const auto covariance_temp = temporary_path(covariance_path);
+    const auto complete_temp = temporary_path(complete_path);
+    const std::array<std::filesystem::path, 4> temporary_paths = {
+        summary_temp, beta_temp, covariance_temp, complete_temp};
+    try {
+        write_binary_doubles(beta_temp, beta_values);
+        write_binary_doubles(covariance_temp, covariance_values);
+        {
+            std::ofstream summary(summary_temp, std::ios::trunc);
+            if (!summary) {
+                throw std::runtime_error("cannot create output " +
+                                         summary_temp.string());
+            }
+            summary
+                << "task_index\ttask_id\tstatus\tn_fixed\tn_extra_covariates\tbeta_offset"
+                   "\tcov_offset\tsigma_g2\tsigma_e2\th2\tlogL\titerations"
+                   "\tline_search_steps\tgrad_inf\terror\n";
+            summary << std::setprecision(17);
+            for (std::size_t index = 0; index < results.size(); ++index) {
+                const auto& item = results[index];
+                const auto& fit = item.fit;
+                const std::size_t expected_fixed =
+                    fit.beta.empty() ? base_covariate_count +
+                                           item.task.extra_covariate_rows.size()
+                                     : fit.beta.size();
+                summary << item.task.task_index << '\t'
+                        << sanitize_tsv(item.task.task_id) << '\t'
+                        << to_string(fit.status) << '\t' << expected_fixed << '\t'
+                        << item.task.extra_covariate_rows.size() << '\t'
+                        << beta_offsets[index] << '\t' << covariance_offsets[index]
+                        << '\t' << fit.sigma_g2 << '\t' << fit.sigma_e2 << '\t'
+                        << fit.h2 << '\t' << fit.log_likelihood << '\t'
+                        << fit.iterations << '\t' << fit.line_search_evaluations
+                        << '\t' << fit.gradient_inf_norm << '\t'
+                        << sanitize_tsv(fit.error) << '\n';
+            }
+            summary.flush();
+            if (!summary) {
+                throw std::runtime_error("failed writing output " +
+                                         summary_temp.string());
+            }
+        }
+        {
+            std::ofstream complete(complete_temp, std::ios::trunc);
+            if (!complete) {
+                throw std::runtime_error("cannot create completion marker " +
+                                         complete_temp.string());
+            }
+            complete << "format\t" << kOutputFormatVersion << '\n'
+                     << "block\t" << block_index << '\n'
+                     << "tasks\t" << results.size() << '\n'
+                     << "beta_elements\t" << beta_values.size() << '\n'
+                     << "cov_elements\t" << covariance_values.size() << '\n';
+            complete.flush();
+            if (!complete) {
+                throw std::runtime_error("failed writing completion marker " +
+                                         complete_temp.string());
+            }
+        }
+        rename_checked(beta_temp, beta_path);
+        rename_checked(covariance_temp, covariance_path);
+        rename_checked(summary_temp, summary_path);
+        rename_checked(complete_temp, complete_path);
+    } catch (...) {
+        for (const auto& path : temporary_paths) {
+            std::error_code ignored;
+            std::filesystem::remove(path, ignored);
+        }
+        throw;
+    }
+}
+
+bool block_is_complete(const std::filesystem::path& output_directory,
+                       std::size_t block_index) {
+    return std::filesystem::exists(
+        output_directory / (block_stem(block_index) + ".complete"));
+}
+
+}  // namespace
+
+ColMajorMatrix read_row_major_f64_matrix(const std::filesystem::path& path,
+                                         std::size_t rows,
+                                         std::size_t cols) {
+    require_little_endian_ieee754();
+    const std::size_t element_count =
+        checked_product(rows, cols, "float64 matrix");
+    const auto row_major =
+        read_exact_binary_vector<double>(path, element_count, "float64 matrix");
+    ColMajorMatrix result(rows, cols);
+    for (std::size_t row = 0; row < rows; ++row) {
+        for (std::size_t col = 0; col < cols; ++col) {
+            const double value = row_major[row * cols + col];
+            if (!std::isfinite(value)) {
+                throw std::runtime_error("non-finite value in float64 matrix " +
+                                         path.string());
+            }
+            result(row, col) = value;
+        }
+    }
+    return result;
+}
+
+std::vector<RemlTask> read_tasks(
+    const std::filesystem::path& tasks_tsv,
+    const std::filesystem::path& extra_offsets_i64,
+    const std::filesystem::path& extra_indices_i32,
+    const BatchDimensions& dimensions) {
+    require_little_endian_ieee754();
+    std::ifstream input(tasks_tsv);
+    if (!input) {
+        throw std::runtime_error("cannot open tasks TSV " + tasks_tsv.string());
+    }
+    std::string line;
+    if (!std::getline(input, line)) {
+        throw std::runtime_error("tasks TSV is empty");
+    }
+    if (!line.empty() && line.back() == '\r') {
+        line.pop_back();
+    }
+    const auto header = split_tab(line);
+    const auto task_column = find_required_column(header, "task_index");
+    const auto task_id_column = find_required_column(header, "task_id");
+    const auto phenotype_column =
+        find_required_column(header, "phenotype_row");
+    const auto extra_count_column = find_required_column(header, "n_extra_covariates");
+    const std::size_t required_field_count =
+        std::max({task_column, task_id_column, phenotype_column, extra_count_column}) +
+        1;
+
+    struct TaskRow {
+        RemlTask task;
+        std::size_t declared_extra_count = 0;
+    };
+    std::vector<TaskRow> task_rows;
+    std::unordered_set<std::uint64_t> task_indices;
+    std::size_t line_number = 1;
+    while (std::getline(input, line)) {
+        ++line_number;
+        if (!line.empty() && line.back() == '\r') {
+            line.pop_back();
+        }
+        if (line.empty()) {
+            continue;
+        }
+        const auto fields = split_tab(line);
+        if (fields.size() < required_field_count) {
+            throw std::runtime_error("too few fields at tasks TSV line " +
+                                     std::to_string(line_number));
+        }
+        TaskRow row;
+        row.task.task_index =
+            parse_u64(fields[task_column], "task_index", line_number);
+        row.task.task_id = fields[task_id_column];
+        row.task.phenotype_row =
+            parse_u64(fields[phenotype_column], "phenotype_row", line_number);
+        row.declared_extra_count = static_cast<std::size_t>(
+            parse_u64(fields[extra_count_column], "n_extra_covariates", line_number));
+        if (row.task.task_id.empty()) {
+            throw std::runtime_error("empty task_id at tasks TSV line " +
+                                     std::to_string(line_number));
+        }
+        if (row.task.phenotype_row >= dimensions.phenotype_row_count) {
+            throw std::runtime_error("phenotype_row outside matrix at tasks TSV line " +
+                                     std::to_string(line_number));
+        }
+        if (!task_indices.insert(row.task.task_index).second) {
+            throw std::runtime_error("duplicate task_index in tasks TSV: " +
+                                     std::to_string(row.task.task_index));
+        }
+        task_rows.push_back(std::move(row));
+    }
+    if (!input.eof()) {
+        throw std::runtime_error("failed while reading tasks TSV " +
+                                 tasks_tsv.string());
+    }
+
+    const auto offsets = read_exact_binary_vector<std::int64_t>(
+        extra_offsets_i64, task_rows.size() + 1, "extra-covariate offsets");
+    if (offsets.empty() || offsets.front() != 0) {
+        throw std::runtime_error("extra-covariate offsets must start at zero");
+    }
+    for (std::size_t i = 1; i < offsets.size(); ++i) {
+        if (offsets[i] < offsets[i - 1]) {
+            throw std::runtime_error("extra-covariate offsets must be non-decreasing");
+        }
+    }
+    if (offsets.back() < 0) {
+        throw std::runtime_error("final extra-covariate offset is negative");
+    }
+    const auto extra_indices = read_exact_binary_vector<std::int32_t>(
+        extra_indices_i32, static_cast<std::size_t>(offsets.back()), "extra-covariate indices");
+
+    std::vector<RemlTask> tasks;
+    tasks.reserve(task_rows.size());
+    for (std::size_t i = 0; i < task_rows.size(); ++i) {
+        const auto begin = static_cast<std::size_t>(offsets[i]);
+        const auto end = static_cast<std::size_t>(offsets[i + 1]);
+        if (end - begin != task_rows[i].declared_extra_count) {
+            throw std::runtime_error("n_extra_covariates disagrees with offsets for task " +
+                                     std::to_string(task_rows[i].task.task_index));
+        }
+        std::unordered_set<std::int32_t> within_task;
+        for (std::size_t offset = begin; offset < end; ++offset) {
+            const auto row = extra_indices[offset];
+            if (row < 0 ||
+                static_cast<std::size_t>(row) >= dimensions.extra_covariate_row_count) {
+                throw std::runtime_error("extra-covariate row outside extra_covariates matrix for task " +
+                                         std::to_string(
+                                             task_rows[i].task.task_index));
+            }
+            if (!within_task.insert(row).second) {
+                throw std::runtime_error("duplicate extra covariate within task " +
+                                         std::to_string(
+                                             task_rows[i].task.task_index));
+            }
+            task_rows[i].task.extra_covariate_rows.push_back(row);
+        }
+        tasks.push_back(std::move(task_rows[i].task));
+    }
+    return tasks;
+}
+
+void run_task_batch(const BatchInputPaths& paths,
+                    const BatchDimensions& dimensions,
+                    const BatchOptions& options) {
+    require_little_endian_ieee754();
+    if (dimensions.sample_count == 0 ||
+        dimensions.base_covariate_count == 0 || options.block_size == 0 ||
+        options.outer_threads == 0) {
+        throw std::invalid_argument(
+            "sample, covariate, block and thread counts must be positive");
+    }
+    if (options.resume && options.overwrite) {
+        throw std::invalid_argument("--resume and --overwrite are mutually exclusive");
+    }
+    std::error_code directory_error;
+    std::filesystem::create_directories(paths.output_directory, directory_error);
+    if (directory_error) {
+        throw std::runtime_error("cannot create output directory " +
+                                 paths.output_directory.string() + ": " +
+                                 directory_error.message());
+    }
+    if (!paths.grm_id.empty()) {
+        validate_grm_id_count(paths.grm_id, dimensions.sample_count);
+    }
+
+    const auto tasks = read_tasks(paths.tasks_tsv, paths.extra_offsets_i64,
+                                  paths.extra_indices_i32, dimensions);
+    if (tasks.empty()) {
+        throw std::runtime_error("tasks TSV contains no tasks");
+    }
+    std::cerr << "Reading and diagonalizing the GRM for "
+              << dimensions.sample_count << " samples...\n";
+    const SpectralGrm spectral = read_and_diagonalize_gcta_grm(
+        paths.grm_bin, dimensions.sample_count);
+    std::cerr << "GRM eigenvalue range: [" << std::setprecision(17)
+              << spectral.minimum_eigenvalue << ", "
+              << spectral.maximum_eigenvalue << "]\n";
+
+    const ColMajorMatrix base_x = read_row_major_f64_matrix(
+        paths.base_x_f64, dimensions.sample_count,
+        dimensions.base_covariate_count);
+    const ColMajorMatrix base_x_star =
+        rotate_to_eigenspace(spectral.eigenvectors, base_x);
+
+    std::set<std::int32_t> unique_extra_set;
+    for (const auto& task : tasks) {
+        unique_extra_set.insert(task.extra_covariate_rows.begin(), task.extra_covariate_rows.end());
+    }
+    const std::vector<std::int32_t> unique_extra_rows(unique_extra_set.begin(),
+                                                    unique_extra_set.end());
+    std::unordered_map<std::int32_t, std::size_t> extra_to_column;
+    extra_to_column.reserve(unique_extra_rows.size());
+    for (std::size_t column = 0; column < unique_extra_rows.size(); ++column) {
+        extra_to_column.emplace(unique_extra_rows[column], column);
+    }
+    ColMajorMatrix extra_covariate_star(dimensions.sample_count, 0);
+    if (!unique_extra_rows.empty()) {
+        if (paths.extra_covariate_f32.empty()) {
+            throw std::runtime_error(
+                "extra_covariates file is required because tasks contain extra covariates");
+        }
+        std::cerr << "Reading and rotating " << unique_extra_rows.size()
+                  << " unique extra covariate rows...\n";
+        const ColMajorMatrix extra_covariates = read_selected_f32_rows(
+            paths.extra_covariate_f32, dimensions.extra_covariate_row_count,
+            dimensions.sample_count, unique_extra_rows);
+        extra_covariate_star =
+            rotate_to_eigenspace(spectral.eigenvectors, extra_covariates);
+    }
+
+    const std::size_t phenotype_elements = checked_product(
+        dimensions.phenotype_row_count, dimensions.sample_count,
+        "phenotypes matrix");
+    const std::size_t phenotype_bytes = checked_product(
+        phenotype_elements, sizeof(double), "phenotypes matrix");
+    validate_file_size(paths.phenotype_f64, phenotype_bytes,
+                       "phenotypes matrix");
+    std::ifstream phenotype_input(paths.phenotype_f64, std::ios::binary);
+    if (!phenotype_input) {
+        throw std::runtime_error("cannot open phenotypes matrix " +
+                                 paths.phenotype_f64.string());
+    }
+
+#ifdef _OPENMP
+    if (options.outer_threads >
+        static_cast<std::size_t>(std::numeric_limits<int>::max())) {
+        throw std::invalid_argument("thread count exceeds OpenMP integer range");
+    }
+    omp_set_dynamic(0);
+    omp_set_num_threads(static_cast<int>(options.outer_threads));
+#else
+    if (options.outer_threads != 1) {
+        std::cerr << "Warning: binary was built without OpenMP; using one outer "
+                     "thread.\n";
+    }
+#endif
+
+    const std::size_t block_count =
+        (tasks.size() + options.block_size - 1) / options.block_size;
+    for (std::size_t block = 0; block < block_count; ++block) {
+        if (options.resume && block_is_complete(paths.output_directory, block)) {
+            std::cerr << "Skipping completed " << block_stem(block) << "\n";
+            continue;
+        }
+        const std::size_t begin = block * options.block_size;
+        const std::size_t end =
+            std::min(tasks.size(), begin + options.block_size);
+        std::cerr << "Processing " << block_stem(block) << " (tasks " << begin
+                  << ".." << (end - 1) << ")...\n";
+        const ColMajorMatrix phenotypes = read_selected_f64_rows(
+            phenotype_input, paths.phenotype_f64,
+            dimensions.phenotype_row_count, dimensions.sample_count, tasks,
+            begin, end);
+        const ColMajorMatrix phenotype_star =
+            rotate_to_eigenspace(spectral.eigenvectors, phenotypes);
+        std::vector<TaskResult> results(end - begin);
+
+#ifdef _OPENMP
+#pragma omp parallel for schedule(dynamic)
+#endif
+        for (std::int64_t local_signed = 0;
+             local_signed < static_cast<std::int64_t>(end - begin);
+             ++local_signed) {
+            const auto local = static_cast<std::size_t>(local_signed);
+            const RemlTask& task = tasks[begin + local];
+            TaskResult item;
+            item.task = task;
+            try {
+                const std::size_t fixed_count =
+                    dimensions.base_covariate_count + task.extra_covariate_rows.size();
+                ColMajorMatrix design(dimensions.sample_count, fixed_count);
+                std::copy(base_x_star.values().begin(),
+                          base_x_star.values().end(), design.values().begin());
+                for (std::size_t extra = 0; extra < task.extra_covariate_rows.size(); ++extra) {
+                    const auto found = extra_to_column.find(task.extra_covariate_rows[extra]);
+                    if (found == extra_to_column.end()) {
+                        throw std::runtime_error(
+                            "internal error: rotated extra covariate is missing");
+                    }
+                    const double* source =
+                        extra_covariate_star.data() +
+                        found->second * dimensions.sample_count;
+                    double* destination =
+                        design.data() +
+                        (dimensions.base_covariate_count + extra) *
+                            dimensions.sample_count;
+                    std::copy(source, source + dimensions.sample_count,
+                              destination);
+                }
+                std::vector<double> phenotype(
+                    phenotype_star.data() + local * dimensions.sample_count,
+                    phenotype_star.data() +
+                        (local + 1) * dimensions.sample_count);
+                item.fit = fit_ai_reml_spectral(
+                    phenotype, design, spectral.eigenvalues, options.reml);
+            } catch (const std::exception& exception) {
+                item.fit.status = FitStatus::numerical_error;
+                item.fit.error = exception.what();
+            }
+            results[local] = std::move(item);
+        }
+        write_block_atomic(paths.output_directory, block, results,
+                           dimensions.base_covariate_count,
+                           options.overwrite || options.resume);
+    }
+}
+
+}  // namespace spectra::reml
--- a/src/grm.cpp
+++ b/src/grm.cpp
@@ -0,0 +1,121 @@
+#include "spectra_reml/grm.hpp"
+
+#include "spectra_reml/linalg.hpp"
+
+#include <algorithm>
+#include <cmath>
+#include <cstdint>
+#include <fstream>
+#include <limits>
+#include <sstream>
+#include <stdexcept>
+#include <string>
+#include <vector>
+
+namespace spectra::reml {
+namespace {
+
+std::uintmax_t expected_grm_bytes(std::size_t n) {
+    if (n == 0) {
+        throw std::invalid_argument("sample count must be positive");
+    }
+    if (n > (std::numeric_limits<std::size_t>::max() / (n + 1))) {
+        throw std::overflow_error("sample count overflows GRM element count");
+    }
+    const std::size_t elements = n * (n + 1) / 2;
+    if (elements > std::numeric_limits<std::uintmax_t>::max() / sizeof(float)) {
+        throw std::overflow_error("GRM byte count overflow");
+    }
+    return static_cast<std::uintmax_t>(elements) * sizeof(float);
+}
+
+}  // namespace
+
+ColMajorMatrix read_gcta_grm_lower_triangle(
+    const std::filesystem::path& grm_bin, std::size_t sample_count) {
+    const auto expected_bytes = expected_grm_bytes(sample_count);
+    std::error_code file_error;
+    const auto actual_bytes = std::filesystem::file_size(grm_bin, file_error);
+    if (file_error) {
+        throw std::runtime_error("cannot stat GRM file " + grm_bin.string() +
+                                 ": " + file_error.message());
+    }
+    if (actual_bytes != expected_bytes) {
+        std::ostringstream message;
+        message << "GRM file size mismatch for " << grm_bin.string()
+                << ": expected " << expected_bytes << " bytes for "
+                << sample_count << " samples, found " << actual_bytes;
+        throw std::runtime_error(message.str());
+    }
+
+    const std::size_t element_count = sample_count * (sample_count + 1) / 2;
+    std::vector<float> lower_triangle(element_count);
+    std::ifstream input(grm_bin, std::ios::binary);
+    if (!input) {
+        throw std::runtime_error("cannot open GRM file " + grm_bin.string());
+    }
+    input.read(reinterpret_cast<char*>(lower_triangle.data()),
+               static_cast<std::streamsize>(expected_bytes));
+    if (!input || input.gcount() != static_cast<std::streamsize>(expected_bytes)) {
+        throw std::runtime_error("short read from GRM file " + grm_bin.string());
+    }
+
+    ColMajorMatrix grm(sample_count, sample_count);
+    std::size_t source = 0;
+    for (std::size_t row = 0; row < sample_count; ++row) {
+        for (std::size_t col = 0; col <= row; ++col, ++source) {
+            const double value = static_cast<double>(lower_triangle[source]);
+            if (!std::isfinite(value)) {
+                std::ostringstream message;
+                message << "non-finite GRM value at (" << row << ',' << col
+                        << ')';
+                throw std::runtime_error(message.str());
+            }
+            grm(row, col) = value;
+            grm(col, row) = value;
+        }
+    }
+    return grm;
+}
+
+SpectralGrm read_and_diagonalize_gcta_grm(
+    const std::filesystem::path& grm_bin, std::size_t sample_count) {
+    SpectralGrm result;
+    result.eigenvectors =
+        read_gcta_grm_lower_triangle(grm_bin, sample_count);
+    result.eigenvalues =
+        symmetric_eigen_decomposition(result.eigenvectors);
+    result.minimum_eigenvalue = result.eigenvalues.front();
+    result.maximum_eigenvalue = result.eigenvalues.back();
+    return result;
+}
+
+void validate_grm_id_count(const std::filesystem::path& grm_id,
+                           std::size_t expected_sample_count) {
+    std::ifstream input(grm_id);
+    if (!input) {
+        throw std::runtime_error("cannot open GRM ID file " + grm_id.string());
+    }
+    std::size_t count = 0;
+    std::string line;
+    while (std::getline(input, line)) {
+        if (!line.empty() && line.back() == '\r') {
+            line.pop_back();
+        }
+        if (line.find_first_not_of(" \t") != std::string::npos) {
+            ++count;
+        }
+    }
+    if (!input.eof()) {
+        throw std::runtime_error("failed while reading GRM ID file " +
+                                 grm_id.string());
+    }
+    if (count != expected_sample_count) {
+        std::ostringstream message;
+        message << "GRM ID count mismatch: expected " << expected_sample_count
+                << ", found " << count;
+        throw std::runtime_error(message.str());
+    }
+}
+
+}  // namespace spectra::reml
--- a/src/linalg.cpp
+++ b/src/linalg.cpp
@@ -0,0 +1,201 @@
+#include "spectra_reml/linalg.hpp"
+
+#if defined(REML_BLAS_MKL)
+#include <mkl_cblas.h>
+#include <mkl_lapacke.h>
+#else
+#include <cblas.h>
+#include <lapacke.h>
+#endif
+
+#include <algorithm>
+#include <cmath>
+#include <limits>
+#include <sstream>
+#include <stdexcept>
+
+namespace spectra::reml {
+namespace {
+
+lapack_int checked_lapack_int(std::size_t value, const char* name) {
+    if (value > static_cast<std::size_t>(std::numeric_limits<lapack_int>::max())) {
+        throw std::overflow_error(std::string(name) + " exceeds LAPACK integer range");
+    }
+    return static_cast<lapack_int>(value);
+}
+
+int checked_blas_int(std::size_t value, const char* name) {
+    if (value > static_cast<std::size_t>(std::numeric_limits<int>::max())) {
+        throw std::overflow_error(std::string(name) + " exceeds CBLAS integer range");
+    }
+    return static_cast<int>(value);
+}
+
+}  // namespace
+
+ColMajorMatrix cross_product(const ColMajorMatrix& a,
+                             const ColMajorMatrix& b) {
+    if (a.rows() != b.rows()) {
+        throw std::invalid_argument("cross_product requires equal row counts");
+    }
+    ColMajorMatrix result(a.cols(), b.cols());
+    cblas_dgemm(CblasColMajor, CblasTrans, CblasNoTrans,
+                checked_blas_int(a.cols(), "A columns"),
+                checked_blas_int(b.cols(), "B columns"),
+                checked_blas_int(a.rows(), "row count"), 1.0, a.data(),
+                checked_blas_int(a.leading_dimension(), "A leading dimension"),
+                b.data(),
+                checked_blas_int(b.leading_dimension(), "B leading dimension"),
+                0.0, result.data(),
+                checked_blas_int(result.leading_dimension(),
+                                 "result leading dimension"));
+    return result;
+}
+
+ColMajorMatrix rotate_to_eigenspace(const ColMajorMatrix& eigenvectors,
+                                    const ColMajorMatrix& variables) {
+    if (eigenvectors.rows() != eigenvectors.cols()) {
+        throw std::invalid_argument("eigenvector matrix must be square");
+    }
+    if (eigenvectors.rows() != variables.rows()) {
+        throw std::invalid_argument(
+            "eigenvectors and variables have inconsistent sample counts");
+    }
+    ColMajorMatrix rotated(variables.rows(), variables.cols());
+    cblas_dgemm(
+        CblasColMajor, CblasTrans, CblasNoTrans,
+        checked_blas_int(eigenvectors.cols(), "eigenvector column count"),
+        checked_blas_int(variables.cols(), "variable column count"),
+        checked_blas_int(eigenvectors.rows(), "sample count"), 1.0,
+        eigenvectors.data(),
+        checked_blas_int(eigenvectors.leading_dimension(),
+                         "eigenvector leading dimension"),
+        variables.data(),
+        checked_blas_int(variables.leading_dimension(),
+                         "variable leading dimension"),
+        0.0, rotated.data(),
+        checked_blas_int(rotated.leading_dimension(),
+                         "rotated leading dimension"));
+    return rotated;
+}
+
+std::vector<double> symmetric_eigen_decomposition(
+    ColMajorMatrix& symmetric_matrix) {
+    if (symmetric_matrix.rows() != symmetric_matrix.cols()) {
+        throw std::invalid_argument("symmetric eigen decomposition requires a square matrix");
+    }
+    const auto n = checked_lapack_int(symmetric_matrix.rows(), "matrix order");
+    std::vector<double> eigenvalues(symmetric_matrix.rows());
+    const lapack_int info = LAPACKE_dsyevd(
+        LAPACK_COL_MAJOR, 'V', 'L', n, symmetric_matrix.data(), n,
+        eigenvalues.data());
+    if (info < 0) {
+        std::ostringstream message;
+        message << "LAPACKE_dsyevd rejected argument " << -info;
+        throw std::runtime_error(message.str());
+    }
+    if (info > 0) {
+        std::ostringstream message;
+        message << "LAPACKE_dsyevd failed to converge (info=" << info << ')';
+        throw std::runtime_error(message.str());
+    }
+    return eigenvalues;
+}
+
+bool cholesky_factor_in_place(std::vector<double>& matrix, std::size_t order,
+                              std::string* error) {
+    if (matrix.size() != order * order) {
+        if (error != nullptr) {
+            *error = "Cholesky matrix has the wrong size";
+        }
+        return false;
+    }
+    const auto n = checked_lapack_int(order, "Cholesky order");
+    const lapack_int info =
+        LAPACKE_dpotrf(LAPACK_COL_MAJOR, 'L', n, matrix.data(), n);
+    if (info == 0) {
+        return true;
+    }
+    if (error != nullptr) {
+        std::ostringstream message;
+        if (info < 0) {
+            message << "LAPACKE_dpotrf rejected argument " << -info;
+        } else {
+            message << "matrix is not positive definite at leading minor "
+                    << info;
+        }
+        *error = message.str();
+    }
+    return false;
+}
+
+bool cholesky_solve_in_place(const std::vector<double>& factor,
+                             std::size_t order, double* rhs,
+                             std::size_t rhs_columns, std::string* error) {
+    if (factor.size() != order * order || rhs == nullptr) {
+        if (error != nullptr) {
+            *error = "invalid Cholesky solve dimensions";
+        }
+        return false;
+    }
+    const auto n = checked_lapack_int(order, "Cholesky order");
+    const auto nrhs = checked_lapack_int(rhs_columns, "right-hand-side count");
+    const lapack_int info = LAPACKE_dpotrs(
+        LAPACK_COL_MAJOR, 'L', n, nrhs, factor.data(), n, rhs, n);
+    if (info == 0) {
+        return true;
+    }
+    if (error != nullptr) {
+        std::ostringstream message;
+        message << "LAPACKE_dpotrs failed (info=" << info << ')';
+        *error = message.str();
+    }
+    return false;
+}
+
+bool cholesky_inverse(const std::vector<double>& factor, std::size_t order,
+                      std::vector<double>& inverse, std::string* error) {
+    if (factor.size() != order * order) {
+        if (error != nullptr) {
+            *error = "invalid Cholesky inverse dimensions";
+        }
+        return false;
+    }
+    inverse = factor;
+    const auto n = checked_lapack_int(order, "Cholesky order");
+    const lapack_int info =
+        LAPACKE_dpotri(LAPACK_COL_MAJOR, 'L', n, inverse.data(), n);
+    if (info != 0) {
+        if (error != nullptr) {
+            std::ostringstream message;
+            message << "LAPACKE_dpotri failed (info=" << info << ')';
+            *error = message.str();
+        }
+        inverse.clear();
+        return false;
+    }
+    for (std::size_t col = 0; col < order; ++col) {
+        for (std::size_t row = 0; row < col; ++row) {
+            inverse[row + col * order] = inverse[col + row * order];
+        }
+    }
+    return true;
+}
+
+double dot(const std::vector<double>& a, const std::vector<double>& b) {
+    if (a.size() != b.size()) {
+        throw std::invalid_argument("dot product requires equal vector lengths");
+    }
+    return cblas_ddot(checked_blas_int(a.size(), "vector length"), a.data(), 1,
+                      b.data(), 1);
+}
+
+double infinity_norm(const std::vector<double>& x) {
+    double result = 0.0;
+    for (const double value : x) {
+        result = std::max(result, std::abs(value));
+    }
+    return result;
+}
+
+}  // namespace spectra::reml
--- a/src/main.cpp
+++ b/src/main.cpp
@@ -0,0 +1,354 @@
+#include "spectra_reml/batch_io.hpp"
+
+#include <cmath>
+#include <cstddef>
+#include <cstdlib>
+#include <exception>
+#include <filesystem>
+#include <iostream>
+#include <limits>
+#include <stdexcept>
+#include <string>
+#include <unordered_map>
+#include <unordered_set>
+
+namespace {
+
+using spectra::reml::BatchDimensions;
+using spectra::reml::BatchInputPaths;
+using spectra::reml::BatchOptions;
+
+void print_usage(std::ostream& output) {
+    output
+        << "spectra_reml - reusable spectral AI-REML batch engine\n\n"
+        << "Required input paths:\n"
+        << "  --grm-bin PATH              GCTA lower-triangle .grm.bin\n"
+        << "  --base-x PATH               row-major n x p0 float64 matrix\n"
+        << "  --phenotypes PATH           row-major rows x n float64 matrix\n"
+        << "  --tasks PATH                TSV with task_index,task_id,phenotype_row,n_extra_covariates\n"
+        << "  --extra-offsets PATH         int64 CSR offsets, m+1 entries\n"
+        << "  --extra-indices PATH         int32 zero-based extra-covariate row indices\n"
+        << "  --out-dir PATH              task-set output directory\n\n"
+        << "Required dimensions:\n"
+        << "  --n-samples N\n"
+        << "  --n-base-covariates P\n"
+        << "  --n-phenotype-rows R\n"
+        << "  --n-extra-covariate-rows R\n\n"
+        << "Conditionally required:\n"
+        << "  --extra-covariates PATH      row-major rows x n float32 matrix; required\n"
+        << "                               when any task uses extra covariates\n\n"
+        << "Optional validation and scheduling:\n"
+        << "  --grm-id PATH               verify .grm.id line count\n"
+        << "  --block-size N              tasks per atomic block (default 256)\n"
+        << "  --threads N                 OpenMP task threads (default 1)\n"
+        << "  --resume                    skip blocks with .complete marker\n"
+        << "  --overwrite                 replace existing block files\n\n"
+        << "Optional AI-REML controls:\n"
+        << "  --max-iterations N          default 100\n"
+        << "  --line-search-max-evals N   default 48\n"
+        << "  --line-search-max-zoom N    default 48\n"
+        << "  --gradient-abs-tol X        default 1e-7\n"
+        << "  --gradient-rel-tol X        default 1e-8\n"
+        << "  --step-rel-tol X            default 1e-9\n"
+        << "  --likelihood-rel-tol X      default 1e-11\n"
+        << "  --wolfe-c1 X                default 1e-4\n"
+        << "  --wolfe-c2 X                default 0.9\n"
+        << "  --initial-step X            default 1\n"
+        << "  --maximum-step X            default 64\n"
+        << "  --initial-sigma-e X         override OLS-based initial value\n"
+        << "  --initial-sigma-g X         override OLS-based initial value\n"
+        << "  --rank-tol X                default 1e-10\n"
+        << "  --covariance-floor X        relative floor, default 1e-12\n"
+        << "  --boundary-h2-trigger X     test v_g=0 once h2<=X (default 1e-3)\n"
+        << "  --boundary-score-tol X      one-sided score tolerance (default 1e-8)\n"
+        << "  --boundary-logl-rel-tol X   likelihood comparison tolerance (default 1e-12)\n"
+        << "  --help                      show this message\n\n"
+        << "All binary inputs and outputs are little-endian. Offsets are counted in\n"
+        << "elements, not bytes. BLAS thread counts are configured outside this CLI.\n";
+}
+
+struct Arguments {
+    std::unordered_map<std::string, std::string> values;
+    std::unordered_set<std::string> flags;
+};
+
+Arguments parse_arguments(int argc, char** argv) {
+    const std::unordered_set<std::string> flag_names = {
+        "--help", "--resume", "--overwrite"};
+    Arguments result;
+    for (int i = 1; i < argc; ++i) {
+        const std::string name = argv[i];
+        if (name.rfind("--", 0) != 0) {
+            throw std::invalid_argument("unexpected positional argument: " + name);
+        }
+        if (flag_names.count(name) != 0U) {
+            if (!result.flags.insert(name).second) {
+                throw std::invalid_argument("duplicate flag: " + name);
+            }
+            continue;
+        }
+        if (i + 1 >= argc) {
+            throw std::invalid_argument("missing value for " + name);
+        }
+        if (!result.values.emplace(name, argv[++i]).second) {
+            throw std::invalid_argument("duplicate option: " + name);
+        }
+    }
+    return result;
+}
+
+const std::string& require_value(const Arguments& arguments,
+                                 const std::string& name) {
+    const auto found = arguments.values.find(name);
+    if (found == arguments.values.end() || found->second.empty()) {
+        throw std::invalid_argument("missing required option " + name);
+    }
+    return found->second;
+}
+
+std::string optional_value(const Arguments& arguments, const std::string& name,
+                           const std::string& fallback = {}) {
+    const auto found = arguments.values.find(name);
+    return found == arguments.values.end() ? fallback : found->second;
+}
+
+std::size_t parse_size(const std::string& text, const std::string& name,
+                       bool allow_zero = false) {
+    if (text.empty() || text.front() == '-') {
+        throw std::invalid_argument("invalid integer for " + name + ": " + text);
+    }
+    std::size_t consumed = 0;
+    unsigned long long parsed = 0;
+    try {
+        parsed = std::stoull(text, &consumed, 10);
+    } catch (const std::exception&) {
+        throw std::invalid_argument("invalid integer for " + name + ": " + text);
+    }
+    if (consumed != text.size() ||
+        parsed > static_cast<unsigned long long>(
+                     std::numeric_limits<std::size_t>::max()) ||
+        (!allow_zero && parsed == 0)) {
+        throw std::invalid_argument("invalid integer for " + name + ": " + text);
+    }
+    return static_cast<std::size_t>(parsed);
+}
+
+double parse_double(const std::string& text, const std::string& name,
+                    bool require_positive = false,
+                    bool allow_zero = false) {
+    std::size_t consumed = 0;
+    double parsed = 0.0;
+    try {
+        parsed = std::stod(text, &consumed);
+    } catch (const std::exception&) {
+        throw std::invalid_argument("invalid number for " + name + ": " + text);
+    }
+    if (consumed != text.size() || !std::isfinite(parsed) ||
+        (require_positive &&
+         (allow_zero ? parsed < 0.0 : parsed <= 0.0))) {
+        throw std::invalid_argument("invalid number for " + name + ": " + text);
+    }
+    return parsed;
+}
+
+template <typename Setter>
+void set_if_present(const Arguments& arguments, const std::string& name,
+                    Setter setter) {
+    const auto found = arguments.values.find(name);
+    if (found != arguments.values.end()) {
+        setter(found->second);
+    }
+}
+
+void reject_unknown_options(const Arguments& arguments) {
+    const std::unordered_set<std::string> known = {
+        "--grm-bin",
+        "--grm-id",
+        "--base-x",
+        "--phenotypes",
+        "--extra-covariates",
+        "--tasks",
+        "--extra-offsets",
+        "--extra-indices",
+        "--out-dir",
+        "--n-samples",
+        "--n-base-covariates",
+        "--n-phenotype-rows",
+        "--n-extra-covariate-rows",
+        "--block-size",
+        "--threads",
+        "--max-iterations",
+        "--line-search-max-evals",
+        "--line-search-max-zoom",
+        "--gradient-abs-tol",
+        "--gradient-rel-tol",
+        "--step-rel-tol",
+        "--likelihood-rel-tol",
+        "--wolfe-c1",
+        "--wolfe-c2",
+        "--initial-step",
+        "--maximum-step",
+        "--initial-sigma-e",
+        "--initial-sigma-g",
+        "--rank-tol",
+        "--covariance-floor",
+        "--boundary-h2-trigger",
+        "--boundary-score-tol",
+        "--boundary-logl-rel-tol"};
+    for (const auto& [name, value] : arguments.values) {
+        (void)value;
+        if (known.count(name) == 0U) {
+            throw std::invalid_argument("unknown option: " + name);
+        }
+    }
+}
+
+}  // namespace
+
+int main(int argc, char** argv) {
+    try {
+        const Arguments arguments = parse_arguments(argc, argv);
+        if (arguments.flags.count("--help") != 0U) {
+            print_usage(std::cout);
+            return EXIT_SUCCESS;
+        }
+        reject_unknown_options(arguments);
+
+        BatchInputPaths paths;
+        paths.grm_bin = require_value(arguments, "--grm-bin");
+        paths.grm_id = optional_value(arguments, "--grm-id");
+        paths.base_x_f64 = require_value(arguments, "--base-x");
+        paths.phenotype_f64 = require_value(arguments, "--phenotypes");
+        paths.extra_covariate_f32 = optional_value(arguments, "--extra-covariates");
+        paths.tasks_tsv = require_value(arguments, "--tasks");
+        paths.extra_offsets_i64 = require_value(arguments, "--extra-offsets");
+        paths.extra_indices_i32 = require_value(arguments, "--extra-indices");
+        paths.output_directory = require_value(arguments, "--out-dir");
+
+        BatchDimensions dimensions;
+        dimensions.sample_count = parse_size(
+            require_value(arguments, "--n-samples"), "--n-samples");
+        dimensions.base_covariate_count = parse_size(
+            require_value(arguments, "--n-base-covariates"),
+            "--n-base-covariates");
+        dimensions.phenotype_row_count = parse_size(
+            require_value(arguments, "--n-phenotype-rows"),
+            "--n-phenotype-rows");
+        dimensions.extra_covariate_row_count = parse_size(
+            require_value(arguments, "--n-extra-covariate-rows"),
+            "--n-extra-covariate-rows", true);
+
+        BatchOptions options;
+        options.resume = arguments.flags.count("--resume") != 0U;
+        options.overwrite = arguments.flags.count("--overwrite") != 0U;
+        set_if_present(arguments, "--block-size", [&](const std::string& value) {
+            options.block_size = parse_size(value, "--block-size");
+        });
+        set_if_present(arguments, "--threads", [&](const std::string& value) {
+            options.outer_threads = parse_size(value, "--threads");
+        });
+        set_if_present(arguments, "--max-iterations",
+                       [&](const std::string& value) {
+                           options.reml.max_iterations =
+                               parse_size(value, "--max-iterations");
+                       });
+        set_if_present(arguments, "--line-search-max-evals",
+                       [&](const std::string& value) {
+                           options.reml.line_search_max_evaluations =
+                               parse_size(value, "--line-search-max-evals");
+                       });
+        set_if_present(arguments, "--line-search-max-zoom",
+                       [&](const std::string& value) {
+                           options.reml.line_search_max_zoom_iterations =
+                               parse_size(value, "--line-search-max-zoom");
+                       });
+        set_if_present(arguments, "--gradient-abs-tol",
+                       [&](const std::string& value) {
+                           options.reml.gradient_absolute_tolerance =
+                               parse_double(value, "--gradient-abs-tol", true,
+                                            true);
+                       });
+        set_if_present(arguments, "--gradient-rel-tol",
+                       [&](const std::string& value) {
+                           options.reml.gradient_relative_tolerance =
+                               parse_double(value, "--gradient-rel-tol", true,
+                                            true);
+                       });
+        set_if_present(arguments, "--step-rel-tol",
+                       [&](const std::string& value) {
+                           options.reml.step_relative_tolerance =
+                               parse_double(value, "--step-rel-tol", true, true);
+                       });
+        set_if_present(arguments, "--likelihood-rel-tol",
+                       [&](const std::string& value) {
+                           options.reml.likelihood_relative_tolerance =
+                               parse_double(value, "--likelihood-rel-tol", true,
+                                            true);
+                       });
+        set_if_present(arguments, "--wolfe-c1", [&](const std::string& value) {
+            options.reml.wolfe_c1 = parse_double(value, "--wolfe-c1", true);
+        });
+        set_if_present(arguments, "--wolfe-c2", [&](const std::string& value) {
+            options.reml.wolfe_c2 = parse_double(value, "--wolfe-c2", true);
+        });
+        set_if_present(arguments, "--initial-step",
+                       [&](const std::string& value) {
+                           options.reml.initial_line_search_step =
+                               parse_double(value, "--initial-step", true);
+                       });
+        set_if_present(arguments, "--maximum-step",
+                       [&](const std::string& value) {
+                           options.reml.maximum_line_search_step =
+                               parse_double(value, "--maximum-step", true);
+                       });
+        set_if_present(arguments, "--initial-sigma-e",
+                       [&](const std::string& value) {
+                           options.reml.initial_sigma_e =
+                               parse_double(value, "--initial-sigma-e");
+                       });
+        set_if_present(arguments, "--initial-sigma-g",
+                       [&](const std::string& value) {
+                           options.reml.initial_sigma_g =
+                               parse_double(value, "--initial-sigma-g");
+                       });
+        set_if_present(arguments, "--rank-tol", [&](const std::string& value) {
+            options.reml.rank_tolerance_relative =
+                parse_double(value, "--rank-tol", true, true);
+        });
+        set_if_present(arguments, "--covariance-floor",
+                       [&](const std::string& value) {
+                           options.reml.covariance_floor_relative = parse_double(
+                               value, "--covariance-floor", true, true);
+                       });
+        set_if_present(arguments, "--boundary-h2-trigger",
+                       [&](const std::string& value) {
+                           options.reml.boundary_h2_trigger = parse_double(
+                               value, "--boundary-h2-trigger", true, true);
+                           if (options.reml.boundary_h2_trigger > 1.0) {
+                               throw std::invalid_argument(
+                                   "--boundary-h2-trigger must be <= 1");
+                           }
+                       });
+        set_if_present(arguments, "--boundary-score-tol",
+                       [&](const std::string& value) {
+                           options.reml.boundary_score_tolerance = parse_double(
+                               value, "--boundary-score-tol", true, true);
+                       });
+        set_if_present(arguments, "--boundary-logl-rel-tol",
+                       [&](const std::string& value) {
+                           options.reml.boundary_likelihood_relative_tolerance =
+                               parse_double(value, "--boundary-logl-rel-tol",
+                                            true, true);
+                       });
+
+        spectra::reml::run_task_batch(paths, dimensions, options);
+        return EXIT_SUCCESS;
+    } catch (const std::invalid_argument& exception) {
+        std::cerr << "Argument error: " << exception.what() << "\n\n";
+        print_usage(std::cerr);
+        return 2;
+    } catch (const std::exception& exception) {
+        std::cerr << "Fatal error: " << exception.what() << '\n';
+        return EXIT_FAILURE;
+    }
+}
--- a/src/reml_core.cpp
+++ b/src/reml_core.cpp
@@ -0,0 +1,842 @@
+#include "spectra_reml/reml.hpp"
+
+#include "spectra_reml/linalg.hpp"
+
+#include <algorithm>
+#include <array>
+#include <cmath>
+#include <cstddef>
+#include <limits>
+#include <sstream>
+#include <stdexcept>
+#include <string>
+#include <utility>
+#include <vector>
+
+namespace spectra::reml {
+namespace {
+
+constexpr double kTiny = 64.0 * std::numeric_limits<double>::epsilon();
+
+struct ValidatedProblem {
+    const std::vector<double>& y;
+    const ColMajorMatrix& x;
+    const std::vector<double>& lambda;
+    std::size_t n;
+    std::size_t p;
+};
+
+ValidatedProblem validate_problem(const std::vector<double>& y,
+                                  const ColMajorMatrix& x,
+                                  const std::vector<double>& lambda) {
+    if (y.empty()) {
+        throw std::invalid_argument("phenotype is empty");
+    }
+    if (x.rows() != y.size()) {
+        throw std::invalid_argument("phenotype and design row counts differ");
+    }
+    if (lambda.size() != y.size()) {
+        throw std::invalid_argument("phenotype and eigenvalue counts differ");
+    }
+    if (x.cols() == 0) {
+        throw std::invalid_argument("design matrix has no columns");
+    }
+    if (x.cols() >= x.rows()) {
+        throw std::invalid_argument("REML requires more observations than fixed effects");
+    }
+    for (std::size_t i = 0; i < y.size(); ++i) {
+        if (!std::isfinite(y[i])) {
+            throw std::invalid_argument("phenotype contains a non-finite value");
+        }
+        if (!std::isfinite(lambda[i])) {
+            throw std::invalid_argument("GRM eigenvalues contain a non-finite value");
+        }
+    }
+    for (const double value : x.values()) {
+        if (!std::isfinite(value)) {
+            throw std::invalid_argument("design matrix contains a non-finite value");
+        }
+    }
+    return {y, x, lambda, y.size(), x.cols()};
+}
+
+void fill_symmetric_upper(std::vector<double>& matrix, std::size_t p) {
+    for (std::size_t col = 0; col < p; ++col) {
+        for (std::size_t row = 0; row < col; ++row) {
+            matrix[row + col * p] = matrix[col + row * p];
+        }
+    }
+}
+
+bool factor_is_numerically_full_rank(const std::vector<double>& factor,
+                                     std::size_t p,
+                                     double relative_tolerance) {
+    double min_diagonal = std::numeric_limits<double>::infinity();
+    double max_diagonal = 0.0;
+    for (std::size_t i = 0; i < p; ++i) {
+        const double diagonal = std::abs(factor[i + i * p]);
+        min_diagonal = std::min(min_diagonal, diagonal);
+        max_diagonal = std::max(max_diagonal, diagonal);
+    }
+    if (!(max_diagonal > 0.0) || !std::isfinite(min_diagonal)) {
+        return false;
+    }
+    const double ratio = min_diagonal / max_diagonal;
+    return ratio * ratio > relative_tolerance;
+}
+
+bool apply_p(const ValidatedProblem& problem,
+             const std::vector<double>& inverse_h,
+             const std::vector<double>& cholesky,
+             const std::vector<double>& input,
+             std::vector<double>& output, std::string& error) {
+    const auto n = problem.n;
+    const auto p = problem.p;
+    std::vector<double> rhs(p, 0.0);
+    for (std::size_t col = 0; col < p; ++col) {
+        const double* x_col = problem.x.data() + col * n;
+        double value = 0.0;
+        for (std::size_t i = 0; i < n; ++i) {
+            value += x_col[i] * inverse_h[i] * input[i];
+        }
+        rhs[col] = value;
+    }
+    if (!cholesky_solve_in_place(cholesky, p, rhs.data(), 1, &error)) {
+        return false;
+    }
+    output.resize(n);
+    for (std::size_t i = 0; i < n; ++i) {
+        double fitted = 0.0;
+        for (std::size_t col = 0; col < p; ++col) {
+            fitted += problem.x(i, col) * rhs[col];
+        }
+        output[i] = inverse_h[i] * (input[i] - fitted);
+    }
+    return true;
+}
+
+RemlEvaluation evaluate_impl(const ValidatedProblem& problem, double sigma_e,
+                             double sigma_g, bool compute_ai,
+                             bool compute_beta_covariance,
+                             double covariance_floor_relative,
+                             double rank_tolerance_relative) {
+    RemlEvaluation result;
+    if (!std::isfinite(sigma_e) || !std::isfinite(sigma_g)) {
+        result.error = "variance parameters are not finite";
+        return result;
+    }
+    const double sigma_e2 = sigma_e * sigma_e;
+    const double sigma_g2 = sigma_g * sigma_g;
+    if (!std::isfinite(sigma_e2) || !std::isfinite(sigma_g2) ||
+        !(sigma_e2 > 0.0 || sigma_g2 > 0.0)) {
+        result.error = "both variance components are zero or non-finite";
+        return result;
+    }
+
+    const auto n = problem.n;
+    const auto p = problem.p;
+    double covariance_scale = sigma_e2;
+    for (const double lambda : problem.lambda) {
+        covariance_scale =
+            std::max(covariance_scale, std::abs(sigma_g2 * lambda));
+    }
+    const double covariance_floor = std::max(
+        std::numeric_limits<double>::min(),
+        covariance_floor_relative * std::max(covariance_scale,
+                                             std::numeric_limits<double>::min()));
+
+    std::vector<double> inverse_h(n);
+    double log_determinant_h = 0.0;
+    double sum_inverse_h = 0.0;
+    double sum_lambda_inverse_h = 0.0;
+    for (std::size_t i = 0; i < n; ++i) {
+        const double h = sigma_e2 + sigma_g2 * problem.lambda[i];
+        if (!std::isfinite(h) || h <= covariance_floor) {
+            std::ostringstream message;
+            message << "non-positive spectral covariance at index " << i
+                    << " (value=" << h << ')';
+            result.error = message.str();
+            return result;
+        }
+        inverse_h[i] = 1.0 / h;
+        log_determinant_h += std::log(h);
+        sum_inverse_h += inverse_h[i];
+        sum_lambda_inverse_h += problem.lambda[i] * inverse_h[i];
+    }
+
+    std::vector<double> normal_matrix(p * p, 0.0);
+    std::vector<double> normal_rhs(p, 0.0);
+    for (std::size_t i = 0; i < n; ++i) {
+        const double weight = inverse_h[i];
+        for (std::size_t row = 0; row < p; ++row) {
+            const double x_row = problem.x(i, row);
+            normal_rhs[row] += weight * x_row * problem.y[i];
+            for (std::size_t col = 0; col <= row; ++col) {
+                normal_matrix[row + col * p] +=
+                    weight * x_row * problem.x(i, col);
+            }
+        }
+    }
+    fill_symmetric_upper(normal_matrix, p);
+    std::vector<double> cholesky = normal_matrix;
+    std::string linear_algebra_error;
+    if (!cholesky_factor_in_place(cholesky, p, &linear_algebra_error)) {
+        result.error = "fixed-effect normal matrix is singular: " +
+                       linear_algebra_error;
+        return result;
+    }
+    if (!factor_is_numerically_full_rank(cholesky, p,
+                                         rank_tolerance_relative)) {
+        result.error = "fixed-effect normal matrix is numerically rank deficient";
+        return result;
+    }
+
+    double log_determinant_normal = 0.0;
+    for (std::size_t i = 0; i < p; ++i) {
+        log_determinant_normal += 2.0 * std::log(cholesky[i + i * p]);
+    }
+    result.beta = normal_rhs;
+    if (!cholesky_solve_in_place(cholesky, p, result.beta.data(), 1,
+                                 &linear_algebra_error)) {
+        result.error = linear_algebra_error;
+        result.beta.clear();
+        return result;
+    }
+
+    std::vector<double> py(n);
+    double quadratic = 0.0;
+    for (std::size_t i = 0; i < n; ++i) {
+        double fitted = 0.0;
+        for (std::size_t col = 0; col < p; ++col) {
+            fitted += problem.x(i, col) * result.beta[col];
+        }
+        py[i] = inverse_h[i] * (problem.y[i] - fitted);
+        quadratic += problem.y[i] * py[i];
+    }
+    result.log_likelihood =
+        -0.5 * (log_determinant_normal + log_determinant_h + quadratic);
+    if (!std::isfinite(result.log_likelihood)) {
+        result.error = "restricted log-likelihood is not finite";
+        result.beta.clear();
+        return result;
+    }
+
+    std::vector<double> correction_e(p * p, 0.0);
+    std::vector<double> correction_g(p * p, 0.0);
+    for (std::size_t i = 0; i < n; ++i) {
+        const double weight2 = inverse_h[i] * inverse_h[i];
+        const double lambda_weight2 = problem.lambda[i] * weight2;
+        for (std::size_t row = 0; row < p; ++row) {
+            const double x_row = problem.x(i, row);
+            for (std::size_t col = 0; col <= row; ++col) {
+                const double product = x_row * problem.x(i, col);
+                correction_e[row + col * p] += weight2 * product;
+                correction_g[row + col * p] += lambda_weight2 * product;
+            }
+        }
+    }
+    fill_symmetric_upper(correction_e, p);
+    fill_symmetric_upper(correction_g, p);
+    if (!cholesky_solve_in_place(cholesky, p, correction_e.data(), p,
+                                 &linear_algebra_error) ||
+        !cholesky_solve_in_place(cholesky, p, correction_g.data(), p,
+                                 &linear_algebra_error)) {
+        result.error = linear_algebra_error;
+        result.beta.clear();
+        return result;
+    }
+    double trace_correction_e = 0.0;
+    double trace_correction_g = 0.0;
+    for (std::size_t i = 0; i < p; ++i) {
+        trace_correction_e += correction_e[i + i * p];
+        trace_correction_g += correction_g[i + i * p];
+    }
+    const double trace_p = sum_inverse_h - trace_correction_e;
+    const double trace_p_lambda =
+        sum_lambda_inverse_h - trace_correction_g;
+
+    double py_squared = 0.0;
+    double py_lambda_py = 0.0;
+    for (std::size_t i = 0; i < n; ++i) {
+        const double square = py[i] * py[i];
+        py_squared += square;
+        py_lambda_py += problem.lambda[i] * square;
+    }
+    result.gradient_e = -sigma_e * (trace_p - py_squared);
+    result.gradient_g = -sigma_g * (trace_p_lambda - py_lambda_py);
+    result.variance_score_e = 0.5 * (py_squared - trace_p);
+    result.variance_score_g = 0.5 * (py_lambda_py - trace_p_lambda);
+
+    if (compute_ai) {
+        std::vector<double> p_py;
+        if (!apply_p(problem, inverse_h, cholesky, py, p_py,
+                     linear_algebra_error)) {
+            result.error = linear_algebra_error;
+            result.beta.clear();
+            return result;
+        }
+        std::vector<double> lambda_py(n);
+        for (std::size_t i = 0; i < n; ++i) {
+            lambda_py[i] = problem.lambda[i] * py[i];
+        }
+        std::vector<double> p_lambda_py;
+        if (!apply_p(problem, inverse_h, cholesky, lambda_py, p_lambda_py,
+                     linear_algebra_error)) {
+            result.error = linear_algebra_error;
+            result.beta.clear();
+            return result;
+        }
+        const double y_p3_y = dot(py, p_py);
+        const double y_pp_lambda_p_y = dot(py, p_lambda_py);
+        const double y_p_lambda_p_lambda_p_y =
+            dot(lambda_py, p_lambda_py);
+        result.ai_ee = 2.0 * sigma_e2 * y_p3_y;
+        result.ai_eg =
+            2.0 * sigma_e * sigma_g * y_pp_lambda_p_y;
+        result.ai_gg = 2.0 * sigma_g2 * y_p_lambda_p_lambda_p_y;
+    }
+
+    if (compute_beta_covariance) {
+        if (!cholesky_inverse(cholesky, p, result.beta_covariance,
+                              &linear_algebra_error)) {
+            result.error = linear_algebra_error;
+            result.beta.clear();
+            return result;
+        }
+    }
+
+    if (!std::isfinite(result.gradient_e) ||
+        !std::isfinite(result.gradient_g) ||
+        !std::isfinite(result.variance_score_e) ||
+        !std::isfinite(result.variance_score_g) ||
+        (compute_ai &&
+         (!std::isfinite(result.ai_ee) || !std::isfinite(result.ai_eg) ||
+          !std::isfinite(result.ai_gg)))) {
+        result.error = "gradient or average-information matrix is not finite";
+        result.beta.clear();
+        result.beta_covariance.clear();
+        return result;
+    }
+    result.valid = true;
+    return result;
+}
+
+struct InitialGuess {
+    bool valid = false;
+    double sigma_e = 0.0;
+    double sigma_g = 0.0;
+    double residual_mean_square = 0.0;
+    std::string error;
+};
+
+InitialGuess initial_guess(const ValidatedProblem& problem,
+                           const RemlOptions& options) {
+    InitialGuess result;
+    const auto n = problem.n;
+    const auto p = problem.p;
+    std::vector<double> normal(p * p, 0.0);
+    std::vector<double> rhs(p, 0.0);
+    for (std::size_t i = 0; i < n; ++i) {
+        for (std::size_t row = 0; row < p; ++row) {
+            const double x_row = problem.x(i, row);
+            rhs[row] += x_row * problem.y[i];
+            for (std::size_t col = 0; col <= row; ++col) {
+                normal[row + col * p] += x_row * problem.x(i, col);
+            }
+        }
+    }
+    fill_symmetric_upper(normal, p);
+    std::vector<double> factor = normal;
+    if (!cholesky_factor_in_place(factor, p, &result.error)) {
+        result.error = "ordinary fixed-effect design is rank deficient: " +
+                       result.error;
+        return result;
+    }
+    if (!factor_is_numerically_full_rank(factor, p,
+                                         options.rank_tolerance_relative)) {
+        result.error = "ordinary fixed-effect design is numerically rank deficient";
+        return result;
+    }
+    if (!cholesky_solve_in_place(factor, p, rhs.data(), 1, &result.error)) {
+        return result;
+    }
+    double residual_sum_squares = 0.0;
+    double phenotype_sum_squares = 0.0;
+    for (std::size_t i = 0; i < n; ++i) {
+        double fitted = 0.0;
+        for (std::size_t col = 0; col < p; ++col) {
+            fitted += problem.x(i, col) * rhs[col];
+        }
+        const double residual = problem.y[i] - fitted;
+        residual_sum_squares += residual * residual;
+        phenotype_sum_squares += problem.y[i] * problem.y[i];
+    }
+    const double residual_mean_square =
+        residual_sum_squares / static_cast<double>(n - p);
+    result.residual_mean_square = residual_mean_square;
+    const double phenotype_rms =
+        std::sqrt(phenotype_sum_squares / static_cast<double>(n));
+    const double sigma_floor =
+        std::max(1e-12, std::sqrt(std::numeric_limits<double>::epsilon()) *
+                            std::max(1.0, phenotype_rms));
+    const double paper_initial =
+        std::max(sigma_floor, std::sqrt(0.5 * residual_mean_square));
+    result.sigma_e = std::isfinite(options.initial_sigma_e)
+                         ? options.initial_sigma_e
+                         : paper_initial;
+    result.sigma_g = std::isfinite(options.initial_sigma_g)
+                         ? options.initial_sigma_g
+                         : paper_initial;
+    if (!std::isfinite(result.sigma_e) || !std::isfinite(result.sigma_g) ||
+        !(result.sigma_e != 0.0 || result.sigma_g != 0.0)) {
+        result.error = "initial variance parameters are invalid";
+        return result;
+    }
+    result.valid = true;
+    return result;
+}
+
+bool ai_direction(const RemlEvaluation& evaluation,
+                  const RemlOptions& options, std::array<double, 2>& direction,
+                  std::string& error) {
+    const double scale = std::max(
+        {1.0, std::abs(evaluation.ai_ee), std::abs(evaluation.ai_eg),
+         std::abs(evaluation.ai_gg)});
+    double ridge = 0.0;
+    for (std::size_t attempt = 0; attempt <= options.ai_ridge_attempts;
+         ++attempt) {
+        const double a = evaluation.ai_ee + ridge;
+        const double b = evaluation.ai_eg;
+        const double c = evaluation.ai_gg + ridge;
+        const double determinant = a * c - b * b;
+        if (a > 0.0 && c > 0.0 && determinant > kTiny * scale * scale) {
+            direction[0] =
+                (c * evaluation.gradient_e - b * evaluation.gradient_g) /
+                determinant;
+            direction[1] =
+                (a * evaluation.gradient_g - b * evaluation.gradient_e) /
+                determinant;
+            const double directional_derivative =
+                evaluation.gradient_e * direction[0] +
+                evaluation.gradient_g * direction[1];
+            if (std::isfinite(direction[0]) && std::isfinite(direction[1]) &&
+                directional_derivative > 0.0) {
+                return true;
+            }
+        }
+        ridge = attempt == 0 ? options.ai_ridge_relative * scale : ridge * 10.0;
+    }
+    error = "average-information matrix cannot produce an ascent direction";
+    return false;
+}
+
+struct LineSearchResult {
+    bool success = false;
+    double alpha = 0.0;
+    RemlEvaluation evaluation;
+    std::size_t evaluations = 0;
+    std::string error;
+};
+
+double directional_derivative(const RemlEvaluation& evaluation,
+                              const std::array<double, 2>& direction) {
+    return evaluation.gradient_e * direction[0] +
+           evaluation.gradient_g * direction[1];
+}
+
+LineSearchResult strong_wolfe_line_search(
+    const ValidatedProblem& problem, double sigma_e, double sigma_g,
+    const RemlEvaluation& initial, const std::array<double, 2>& direction,
+    const RemlOptions& options) {
+    LineSearchResult result;
+    const double derivative_zero = directional_derivative(initial, direction);
+    if (!(derivative_zero > 0.0) || !std::isfinite(derivative_zero)) {
+        result.error = "line-search direction is not an ascent direction";
+        return result;
+    }
+
+    auto evaluate_alpha = [&](double alpha) {
+        ++result.evaluations;
+        return evaluate_impl(problem, sigma_e + alpha * direction[0],
+                             sigma_g + alpha * direction[1], false, false,
+                             options.covariance_floor_relative,
+                             options.rank_tolerance_relative);
+    };
+    auto armijo_bound = [&](double alpha) {
+        return initial.log_likelihood +
+               options.wolfe_c1 * alpha * derivative_zero;
+    };
+
+    auto zoom = [&](double alpha_lo, double alpha_hi,
+                    RemlEvaluation evaluation_lo) -> LineSearchResult {
+        LineSearchResult zoom_result;
+        // Keep the global evaluation counter in result; copy it at return.
+        for (std::size_t iteration = 0;
+             iteration < options.line_search_max_zoom_iterations &&
+             result.evaluations < options.line_search_max_evaluations;
+             ++iteration) {
+            const double alpha = 0.5 * (alpha_lo + alpha_hi);
+            const double interval_scale =
+                std::max({1.0, std::abs(alpha_lo), std::abs(alpha_hi)});
+            if (std::abs(alpha_hi - alpha_lo) <=
+                8.0 * std::numeric_limits<double>::epsilon() * interval_scale) {
+                zoom_result.error = "strong-Wolfe zoom interval collapsed";
+                break;
+            }
+            RemlEvaluation trial = evaluate_alpha(alpha);
+            if (!trial.valid || trial.log_likelihood < armijo_bound(alpha) ||
+                trial.log_likelihood <= evaluation_lo.log_likelihood) {
+                alpha_hi = alpha;
+                continue;
+            }
+            const double derivative =
+                directional_derivative(trial, direction);
+            if (std::abs(derivative) <=
+                options.wolfe_c2 * std::abs(derivative_zero)) {
+                zoom_result.success = true;
+                zoom_result.alpha = alpha;
+                zoom_result.evaluation = std::move(trial);
+                zoom_result.evaluations = result.evaluations;
+                return zoom_result;
+            }
+            if (derivative * (alpha_hi - alpha_lo) <= 0.0) {
+                alpha_hi = alpha_lo;
+            }
+            alpha_lo = alpha;
+            evaluation_lo = std::move(trial);
+        }
+        zoom_result.evaluations = result.evaluations;
+        if (zoom_result.error.empty()) {
+            zoom_result.error = "strong-Wolfe zoom exceeded its evaluation limit";
+        }
+        return zoom_result;
+    };
+
+    double alpha_previous = 0.0;
+    RemlEvaluation evaluation_previous = initial;
+    double alpha = std::min(options.initial_line_search_step,
+                            options.maximum_line_search_step);
+    for (std::size_t iteration = 0;
+         result.evaluations < options.line_search_max_evaluations;
+         ++iteration) {
+        RemlEvaluation trial = evaluate_alpha(alpha);
+        if (!trial.valid || trial.log_likelihood < armijo_bound(alpha) ||
+            (iteration > 0 &&
+             trial.log_likelihood <= evaluation_previous.log_likelihood)) {
+            return zoom(alpha_previous, alpha, std::move(evaluation_previous));
+        }
+        const double derivative = directional_derivative(trial, direction);
+        if (std::abs(derivative) <=
+            options.wolfe_c2 * std::abs(derivative_zero)) {
+            result.success = true;
+            result.alpha = alpha;
+            result.evaluation = std::move(trial);
+            return result;
+        }
+        if (derivative <= 0.0) {
+            return zoom(alpha, alpha_previous, std::move(trial));
+        }
+        if (alpha >= options.maximum_line_search_step) {
+            result.error =
+                "strong-Wolfe curvature condition not reached at maximum step";
+            return result;
+        }
+        alpha_previous = alpha;
+        evaluation_previous = std::move(trial);
+        alpha = std::min(2.0 * alpha, options.maximum_line_search_step);
+    }
+    result.error = "strong-Wolfe line search exceeded its evaluation limit";
+    return result;
+}
+
+void finalize_result(RemlResult& result, const ValidatedProblem& problem,
+                     double sigma_e, double sigma_g,
+                     const RemlOptions& options) {
+    RemlEvaluation final =
+        evaluate_impl(problem, sigma_e, sigma_g, false, true,
+                      options.covariance_floor_relative,
+                      options.rank_tolerance_relative);
+    if (!final.valid) {
+        result.status = FitStatus::numerical_error;
+        result.error = "final REML evaluation failed: " + final.error;
+        return;
+    }
+    result.sigma_e = sigma_e;
+    result.sigma_g = sigma_g;
+    result.sigma_e2 = sigma_e * sigma_e;
+    result.sigma_g2 = sigma_g * sigma_g;
+    const double total_variance = result.sigma_e2 + result.sigma_g2;
+    result.h2 = total_variance > 0.0 ? result.sigma_g2 / total_variance
+                                    : std::numeric_limits<double>::quiet_NaN();
+    result.log_likelihood = final.log_likelihood;
+    result.gradient_inf_norm =
+        std::max(std::abs(final.gradient_e), std::abs(final.gradient_g));
+    result.beta = std::move(final.beta);
+    result.beta_covariance_packed_lower.clear();
+    result.beta_covariance_packed_lower.reserve(
+        problem.p * (problem.p + 1) / 2);
+    for (std::size_t row = 0; row < problem.p; ++row) {
+        for (std::size_t col = 0; col <= row; ++col) {
+            result.beta_covariance_packed_lower.push_back(
+                final.beta_covariance[row + col * problem.p]);
+        }
+    }
+}
+
+void validate_options(const RemlOptions& options) {
+    if (options.max_iterations == 0) {
+        throw std::invalid_argument("max_iterations must be positive");
+    }
+    if (!(options.wolfe_c1 > 0.0 && options.wolfe_c1 < options.wolfe_c2 &&
+          options.wolfe_c2 < 1.0)) {
+        throw std::invalid_argument("strong-Wolfe constants must satisfy 0<c1<c2<1");
+    }
+    if (!(options.initial_line_search_step > 0.0) ||
+        !(options.maximum_line_search_step >=
+          options.initial_line_search_step)) {
+        throw std::invalid_argument("invalid line-search step bounds");
+    }
+    if (!(options.covariance_floor_relative >= 0.0) ||
+        !(options.rank_tolerance_relative >= 0.0) ||
+        !(options.boundary_score_tolerance >= 0.0) ||
+        !(options.boundary_likelihood_relative_tolerance >= 0.0) ||
+        !(options.boundary_h2_trigger >= 0.0 &&
+          options.boundary_h2_trigger <= 1.0)) {
+        throw std::invalid_argument("numerical tolerances must be non-negative");
+    }
+}
+
+}  // namespace
+
+const char* to_string(FitStatus status) noexcept {
+    switch (status) {
+        case FitStatus::converged:
+            return "converged";
+        case FitStatus::converged_boundary:
+            return "converged_boundary";
+        case FitStatus::max_iterations:
+            return "max_iterations";
+        case FitStatus::line_search_failed:
+            return "line_search_failed";
+        case FitStatus::rank_deficient:
+            return "rank_deficient";
+        case FitStatus::invalid_input:
+            return "invalid_input";
+        case FitStatus::non_positive_covariance:
+            return "non_positive_covariance";
+        case FitStatus::numerical_error:
+            return "numerical_error";
+    }
+    return "unknown";
+}
+
+RemlEvaluation evaluate_reml_spectral(
+    const std::vector<double>& y_star, const ColMajorMatrix& x_star,
+    const std::vector<double>& eigenvalues, double sigma_e, double sigma_g,
+    bool compute_ai, bool compute_beta_covariance,
+    double covariance_floor_relative) {
+    try {
+        const auto problem = validate_problem(y_star, x_star, eigenvalues);
+        return evaluate_impl(problem, sigma_e, sigma_g, compute_ai,
+                             compute_beta_covariance,
+                             covariance_floor_relative, 1e-12);
+    } catch (const std::exception& exception) {
+        RemlEvaluation result;
+        result.error = exception.what();
+        return result;
+    }
+}
+
+RemlResult fit_ai_reml_spectral(const std::vector<double>& y_star,
+                                const ColMajorMatrix& x_star,
+                                const std::vector<double>& eigenvalues,
+                                const RemlOptions& options) {
+    RemlResult result;
+    try {
+        validate_options(options);
+        const auto problem = validate_problem(y_star, x_star, eigenvalues);
+        const InitialGuess guess = initial_guess(problem, options);
+        if (!guess.valid) {
+            result.status = FitStatus::rank_deficient;
+            result.error = guess.error;
+            return result;
+        }
+        double sigma_e = guess.sigma_e;
+        double sigma_g = guess.sigma_g;
+        RemlEvaluation current =
+            evaluate_impl(problem, sigma_e, sigma_g, true, false,
+                          options.covariance_floor_relative,
+                          options.rank_tolerance_relative);
+        if (!current.valid) {
+            result.status = current.error.find("rank") != std::string::npos
+                                ? FitStatus::rank_deficient
+                                : FitStatus::non_positive_covariance;
+            result.error = "initial REML evaluation failed: " + current.error;
+            return result;
+        }
+
+        // At v_g=sigma_g^2=0 the signed-sigma gradient is identically zero,
+        // even when the variance-component KKT condition is not satisfied.
+        // Construct the analytically optimized residual-only candidate once:
+        // v_e = RSS_OLS/(n-p).  It is accepted below only if the one-sided
+        // score dL/dv_g is non-positive and its likelihood is not inferior to
+        // the current interior iterate.  This is a boundary comparison, not an
+        // EM update and not an unconditional truncation of small estimates.
+        const double boundary_sigma_e =
+            guess.residual_mean_square > 0.0
+                ? std::sqrt(guess.residual_mean_square)
+                : 0.0;
+        RemlEvaluation genetic_boundary;
+        if (boundary_sigma_e > 0.0 && std::isfinite(boundary_sigma_e)) {
+            genetic_boundary =
+                evaluate_impl(problem, boundary_sigma_e, 0.0, false, false,
+                              options.covariance_floor_relative,
+                              options.rank_tolerance_relative);
+        } else {
+            genetic_boundary.error =
+                "residual-only boundary has zero residual variance";
+        }
+        auto near_genetic_boundary = [&]() {
+            const double sigma_e2 = sigma_e * sigma_e;
+            const double sigma_g2 = sigma_g * sigma_g;
+            const double total = sigma_e2 + sigma_g2;
+            return total > 0.0 &&
+                   sigma_g2 <= options.boundary_h2_trigger * total;
+        };
+        auto accept_genetic_boundary =
+            [&](const RemlEvaluation& reference) -> bool {
+            if (!genetic_boundary.valid || !reference.valid ||
+                genetic_boundary.variance_score_g >
+                    options.boundary_score_tolerance) {
+                return false;
+            }
+            const double likelihood_scale =
+                std::max({1.0, std::abs(genetic_boundary.log_likelihood),
+                          std::abs(reference.log_likelihood)});
+            const double likelihood_tolerance =
+                options.boundary_likelihood_relative_tolerance *
+                likelihood_scale;
+            if (genetic_boundary.log_likelihood + likelihood_tolerance <
+                reference.log_likelihood) {
+                return false;
+            }
+            sigma_e = boundary_sigma_e;
+            sigma_g = 0.0;
+            result.status = FitStatus::converged_boundary;
+            result.error.clear();
+            return true;
+        };
+
+        result.status = FitStatus::max_iterations;
+        for (std::size_t iteration = 0; iteration < options.max_iterations;
+             ++iteration) {
+            // Once the interior path is sufficiently close to v_g=0, test the
+            // already-computed KKT boundary immediately.  Acceptance still
+            // requires both the one-sided score and likelihood conditions, so
+            // this is an early exact boundary decision rather than truncation.
+            if (near_genetic_boundary() &&
+                accept_genetic_boundary(current)) {
+                break;
+            }
+            const double gradient_norm =
+                std::max(std::abs(current.gradient_e),
+                         std::abs(current.gradient_g));
+            const double gradient_threshold =
+                options.gradient_absolute_tolerance +
+                options.gradient_relative_tolerance *
+                    std::max(1.0, std::abs(current.log_likelihood));
+            if (gradient_norm <= gradient_threshold) {
+                // Do not mistake the automatic factor 2*sigma_g in the
+                // signed-parameter gradient for convergence when the
+                // one-sided variance score still points into the interior.
+                if (near_genetic_boundary() &&
+                    current.variance_score_g >
+                        options.boundary_score_tolerance) {
+                    // Continue with the AI direction below.
+                } else {
+                    result.status = FitStatus::converged;
+                    break;
+                }
+            }
+
+            std::array<double, 2> direction{};
+            std::string direction_error;
+            if (!ai_direction(current, options, direction, direction_error)) {
+                if (!accept_genetic_boundary(current)) {
+                    result.status = FitStatus::numerical_error;
+                    result.error = direction_error;
+                }
+                break;
+            }
+            LineSearchResult line_search = strong_wolfe_line_search(
+                problem, sigma_e, sigma_g, current, direction, options);
+            result.line_search_evaluations += line_search.evaluations;
+            if (!line_search.success) {
+                if (!accept_genetic_boundary(current)) {
+                    result.status = FitStatus::line_search_failed;
+                    result.error = line_search.error;
+                }
+                break;
+            }
+
+            const double previous_likelihood = current.log_likelihood;
+            const double previous_sigma_e = sigma_e;
+            const double previous_sigma_g = sigma_g;
+            sigma_e += line_search.alpha * direction[0];
+            sigma_g += line_search.alpha * direction[1];
+            current = std::move(line_search.evaluation);
+            ++result.iterations;
+
+            // The line search only needs score, not AI.  Re-evaluate once at
+            // the accepted point to construct the next AI direction.
+            current = evaluate_impl(problem, sigma_e, sigma_g, true, false,
+                                    options.covariance_floor_relative,
+                                    options.rank_tolerance_relative);
+            if (!current.valid) {
+                result.status = FitStatus::numerical_error;
+                result.error = "accepted REML point could not be re-evaluated: " +
+                               current.error;
+                break;
+            }
+
+            const double step_scale =
+                std::max({1.0, std::abs(previous_sigma_e),
+                          std::abs(previous_sigma_g)});
+            const double step_norm =
+                std::max(std::abs(sigma_e - previous_sigma_e),
+                         std::abs(sigma_g - previous_sigma_g));
+            const double likelihood_change =
+                std::abs(current.log_likelihood - previous_likelihood);
+            if (step_norm <= options.step_relative_tolerance * step_scale &&
+                likelihood_change <=
+                    options.likelihood_relative_tolerance *
+                        std::max(1.0, std::abs(previous_likelihood))) {
+                if (near_genetic_boundary() &&
+                    accept_genetic_boundary(current)) {
+                    break;
+                }
+                if (!(near_genetic_boundary() &&
+                      current.variance_score_g >
+                          options.boundary_score_tolerance)) {
+                    result.status = FitStatus::converged;
+                    break;
+                }
+            }
+        }
+        if (result.status == FitStatus::max_iterations) {
+            (void)accept_genetic_boundary(current);
+        }
+        finalize_result(result, problem, sigma_e, sigma_g, options);
+        return result;
+    } catch (const std::invalid_argument& exception) {
+        result.status = FitStatus::invalid_input;
+        result.error = exception.what();
+        return result;
+    } catch (const std::exception& exception) {
+        result.status = FitStatus::numerical_error;
+        result.error = exception.what();
+        return result;
+    }
+}
+
+}  // namespace spectra::reml
--- a/tests/cpp/test_grm_io.cpp
+++ b/tests/cpp/test_grm_io.cpp
@@ -0,0 +1,130 @@
+#include "spectra_reml/grm.hpp"
+
+#include <algorithm>
+#include <chrono>
+#include <cmath>
+#include <cstdlib>
+#include <filesystem>
+#include <fstream>
+#include <iostream>
+#include <stdexcept>
+#include <string>
+#include <vector>
+
+namespace {
+
+void require(bool condition, const std::string& message) {
+    if (!condition) {
+        throw std::runtime_error(message);
+    }
+}
+
+void require_near(double actual, double expected, double tolerance,
+                  const std::string& message) {
+    if (std::abs(actual - expected) > tolerance) {
+        throw std::runtime_error(message + ": actual=" +
+                                 std::to_string(actual) +
+                                 ", expected=" + std::to_string(expected));
+    }
+}
+
+class TemporaryDirectory {
+public:
+    TemporaryDirectory() {
+        const auto seed =
+            std::chrono::high_resolution_clock::now().time_since_epoch().count();
+        path_ = std::filesystem::temp_directory_path() /
+                ("spectra_reml_grm_test_" + std::to_string(seed));
+        std::filesystem::create_directories(path_);
+    }
+    ~TemporaryDirectory() {
+        std::error_code ignored;
+        std::filesystem::remove_all(path_, ignored);
+    }
+    const std::filesystem::path& path() const noexcept { return path_; }
+
+private:
+    std::filesystem::path path_;
+};
+
+void write_floats(const std::filesystem::path& path,
+                  const std::vector<float>& values) {
+    std::ofstream output(path, std::ios::binary | std::ios::trunc);
+    output.write(reinterpret_cast<const char*>(values.data()),
+                 static_cast<std::streamsize>(values.size() * sizeof(float)));
+    if (!output) {
+        throw std::runtime_error("failed to create GRM fixture");
+    }
+}
+
+void test_read_and_decompose() {
+    TemporaryDirectory temporary;
+    const auto grm_path = temporary.path() / "fixture.grm.bin";
+    const auto id_path = temporary.path() / "fixture.grm.id";
+    // Symmetric positive-definite matrix, packed as GCTA lower triangle.
+    const std::vector<float> packed = {1.2F, 0.2F, 1.1F,
+                                       -0.1F, 0.15F, 0.9F};
+    write_floats(grm_path, packed);
+    {
+        std::ofstream ids(id_path);
+        ids << "F1 I1\nF2 I2\nF3 I3\n";
+    }
+    spectra::reml::validate_grm_id_count(id_path, 3);
+
+    const auto matrix = spectra::reml::read_gcta_grm_lower_triangle(grm_path, 3);
+    require_near(matrix(0, 0), 1.2, 1e-7, "wrong GRM diagonal");
+    require_near(matrix(1, 0), 0.2, 1e-7, "wrong GRM lower triangle");
+    require_near(matrix(0, 1), 0.2, 1e-7, "GRM was not symmetrized");
+    require_near(matrix(2, 0), -0.1, 1e-7, "wrong GRM packed order");
+    require_near(matrix(1, 2), 0.15, 1e-7, "wrong GRM upper triangle");
+
+    const auto spectral =
+        spectra::reml::read_and_diagonalize_gcta_grm(grm_path, 3);
+    require(spectral.eigenvalues.size() == 3, "wrong eigenvalue count");
+    require(std::is_sorted(spectral.eigenvalues.begin(),
+                           spectral.eigenvalues.end()),
+            "LAPACK eigenvalues are not ascending");
+    require(spectral.minimum_eigenvalue > 0.0,
+            "positive-definite fixture has non-positive eigenvalue");
+
+    // Reconstruct U diag(lambda) U' and compare to the original matrix.
+    for (std::size_t row = 0; row < 3; ++row) {
+        for (std::size_t col = 0; col < 3; ++col) {
+            double reconstructed = 0.0;
+            for (std::size_t axis = 0; axis < 3; ++axis) {
+                reconstructed += spectral.eigenvectors(row, axis) *
+                                 spectral.eigenvalues[axis] *
+                                 spectral.eigenvectors(col, axis);
+            }
+            require_near(reconstructed, matrix(row, col), 2e-12,
+                         "eigendecomposition does not reconstruct GRM");
+        }
+    }
+}
+
+void test_wrong_file_size_is_rejected() {
+    TemporaryDirectory temporary;
+    const auto grm_path = temporary.path() / "short.grm.bin";
+    write_floats(grm_path, {1.0F, 0.0F});
+    bool rejected = false;
+    try {
+        (void)spectra::reml::read_gcta_grm_lower_triangle(grm_path, 2);
+    } catch (const std::runtime_error&) {
+        rejected = true;
+    }
+    require(rejected, "short GRM file was not rejected");
+}
+
+}  // namespace
+
+int main() {
+    try {
+        test_read_and_decompose();
+        test_wrong_file_size_is_rejected();
+        std::cout << "test_grm_io: PASS\n";
+        return EXIT_SUCCESS;
+    } catch (const std::exception& exception) {
+        std::cerr << "test_grm_io: FAIL: " << exception.what() << '\n';
+        return EXIT_FAILURE;
+    }
+}
--- a/tests/cpp/test_reml_synthetic.cpp
+++ b/tests/cpp/test_reml_synthetic.cpp
@@ -0,0 +1,226 @@
+#include "spectra_reml/reml.hpp"
+
+#include <algorithm>
+#include <cmath>
+#include <cstdlib>
+#include <iostream>
+#include <stdexcept>
+#include <string>
+#include <vector>
+
+namespace {
+
+void require(bool condition, const std::string& message) {
+    if (!condition) {
+        throw std::runtime_error(message);
+    }
+}
+
+void require_near(double actual, double expected, double relative_tolerance,
+                  const std::string& message) {
+    const double scale = std::max({1.0, std::abs(actual), std::abs(expected)});
+    if (!std::isfinite(actual) || !std::isfinite(expected) ||
+        std::abs(actual - expected) > relative_tolerance * scale) {
+        throw std::runtime_error(message + ": actual=" +
+                                 std::to_string(actual) +
+                                 ", expected=" + std::to_string(expected));
+    }
+}
+
+struct Fixture {
+    std::vector<double> y;
+    spectra::reml::ColMajorMatrix x;
+    std::vector<double> lambda;
+
+    Fixture() : x(18, 3) {
+        lambda = {0.12, 0.18, 0.25, 0.31, 0.39, 0.48, 0.58, 0.69, 0.81,
+                  0.95, 1.10, 1.28, 1.47, 1.69, 1.94, 2.22, 2.55, 2.91};
+        y.resize(lambda.size());
+        for (std::size_t i = 0; i < lambda.size(); ++i) {
+            const double z = (static_cast<double>(i) - 8.5) / 5.0;
+            x(i, 0) = 1.0;
+            x(i, 1) = z;
+            x(i, 2) = (i % 3 == 0 ? -0.8 : (i % 3 == 1 ? 0.1 : 0.7));
+            const double deterministic_noise =
+                std::sin(1.7 * static_cast<double>(i) + 0.3) *
+                    std::sqrt(0.35 + 0.55 * lambda[i]) +
+                0.22 * std::cos(0.37 * static_cast<double>(i));
+            y[i] = 0.4 - 0.25 * x(i, 1) + 0.15 * x(i, 2) +
+                   deterministic_noise;
+        }
+    }
+};
+
+void test_score_matches_finite_difference() {
+    Fixture fixture;
+    constexpr double sigma_e = 0.73;
+    constexpr double sigma_g = 0.46;
+    const auto evaluated = spectra::reml::evaluate_reml_spectral(
+        fixture.y, fixture.x, fixture.lambda, sigma_e, sigma_g, true, true);
+    require(evaluated.valid, "reference REML evaluation failed: " +
+                                 evaluated.error);
+    require(evaluated.beta.size() == fixture.x.cols(),
+            "beta has wrong length");
+    require(evaluated.beta_covariance.size() ==
+                fixture.x.cols() * fixture.x.cols(),
+            "beta covariance has wrong size");
+
+    const double epsilon_e = 2e-6 * std::max(1.0, std::abs(sigma_e));
+    const double epsilon_g = 2e-6 * std::max(1.0, std::abs(sigma_g));
+    const auto e_plus = spectra::reml::evaluate_reml_spectral(
+        fixture.y, fixture.x, fixture.lambda, sigma_e + epsilon_e, sigma_g,
+        false, false);
+    const auto e_minus = spectra::reml::evaluate_reml_spectral(
+        fixture.y, fixture.x, fixture.lambda, sigma_e - epsilon_e, sigma_g,
+        false, false);
+    const auto g_plus = spectra::reml::evaluate_reml_spectral(
+        fixture.y, fixture.x, fixture.lambda, sigma_e, sigma_g + epsilon_g,
+        false, false);
+    const auto g_minus = spectra::reml::evaluate_reml_spectral(
+        fixture.y, fixture.x, fixture.lambda, sigma_e, sigma_g - epsilon_g,
+        false, false);
+    require(e_plus.valid && e_minus.valid && g_plus.valid && g_minus.valid,
+            "finite-difference REML evaluation failed");
+    const double finite_difference_e =
+        (e_plus.log_likelihood - e_minus.log_likelihood) / (2.0 * epsilon_e);
+    const double finite_difference_g =
+        (g_plus.log_likelihood - g_minus.log_likelihood) / (2.0 * epsilon_g);
+    require_near(evaluated.gradient_e, finite_difference_e, 2e-6,
+                 "sigma_e score disagrees with finite difference");
+    require_near(evaluated.gradient_g, finite_difference_g, 2e-6,
+                 "sigma_g score disagrees with finite difference");
+    require(evaluated.ai_ee > 0.0 && evaluated.ai_gg > 0.0,
+            "AI diagonal must be positive");
+    require(evaluated.ai_ee * evaluated.ai_gg -
+                    evaluated.ai_eg * evaluated.ai_eg >=
+                -1e-10,
+            "AI matrix must be positive semidefinite");
+}
+
+void test_signed_parameterization_and_row_permutation() {
+    Fixture fixture;
+    const auto positive = spectra::reml::evaluate_reml_spectral(
+        fixture.y, fixture.x, fixture.lambda, 0.61, 0.52, true, false);
+    const auto negative = spectra::reml::evaluate_reml_spectral(
+        fixture.y, fixture.x, fixture.lambda, -0.61, -0.52, true, false);
+    require(positive.valid && negative.valid, "signed evaluations failed");
+    require_near(positive.log_likelihood, negative.log_likelihood, 1e-13,
+                 "likelihood must depend on squared standard deviations");
+    require_near(positive.gradient_e, -negative.gradient_e, 1e-12,
+                 "sigma_e gradient must change sign");
+    require_near(positive.gradient_g, -negative.gradient_g, 1e-12,
+                 "sigma_g gradient must change sign");
+
+    std::vector<double> permuted_y(fixture.y.rbegin(), fixture.y.rend());
+    std::vector<double> permuted_lambda(fixture.lambda.rbegin(),
+                                        fixture.lambda.rend());
+    spectra::reml::ColMajorMatrix permuted_x(fixture.x.rows(), fixture.x.cols());
+    for (std::size_t i = 0; i < fixture.x.rows(); ++i) {
+        for (std::size_t j = 0; j < fixture.x.cols(); ++j) {
+            permuted_x(i, j) = fixture.x(fixture.x.rows() - 1 - i, j);
+        }
+    }
+    const auto permuted = spectra::reml::evaluate_reml_spectral(
+        permuted_y, permuted_x, permuted_lambda, 0.61, 0.52, true, false);
+    require(permuted.valid, "permuted evaluation failed");
+    require_near(positive.log_likelihood, permuted.log_likelihood, 2e-13,
+                 "orthogonal-axis permutation changed REML likelihood");
+    require_near(positive.gradient_e, permuted.gradient_e, 2e-12,
+                 "orthogonal-axis permutation changed sigma_e score");
+    require_near(positive.gradient_g, permuted.gradient_g, 2e-12,
+                 "orthogonal-axis permutation changed sigma_g score");
+}
+
+void test_ai_reml_fit_improves_likelihood() {
+    Fixture fixture;
+    spectra::reml::RemlOptions options;
+    options.initial_sigma_e = 0.9;
+    options.initial_sigma_g = 0.9;
+    options.max_iterations = 150;
+    options.gradient_absolute_tolerance = 1e-8;
+    options.gradient_relative_tolerance = 1e-9;
+    const auto initial = spectra::reml::evaluate_reml_spectral(
+        fixture.y, fixture.x, fixture.lambda, options.initial_sigma_e,
+        options.initial_sigma_g, true, false);
+    require(initial.valid, "initial fit evaluation failed");
+    const auto fitted = spectra::reml::fit_ai_reml_spectral(
+        fixture.y, fixture.x, fixture.lambda, options);
+    require(fitted.has_estimates(),
+            "AI-REML returned no estimates: " + fitted.error);
+    require(fitted.status == spectra::reml::FitStatus::converged,
+            std::string("AI-REML did not converge: ") +
+                spectra::reml::to_string(fitted.status) + " " + fitted.error);
+    require(fitted.log_likelihood >= initial.log_likelihood - 1e-10,
+            "AI-REML decreased the likelihood");
+    require(fitted.gradient_inf_norm < 2e-5,
+            "AI-REML final score is too large");
+    require(fitted.beta.size() == fixture.x.cols(),
+            "fitted beta has wrong length");
+    require(fitted.beta_covariance_packed_lower.size() ==
+                fixture.x.cols() * (fixture.x.cols() + 1) / 2,
+            "packed beta covariance has wrong length");
+    require_near(fitted.sigma_e2, fitted.sigma_e * fitted.sigma_e, 1e-14,
+                 "reported sigma_e2 is inconsistent");
+    require_near(fitted.sigma_g2, fitted.sigma_g * fitted.sigma_g, 1e-14,
+                 "reported sigma_g2 is inconsistent");
+}
+
+void test_residual_only_kkt_boundary() {
+    constexpr std::size_t n = 14;
+    spectra::reml::ColMajorMatrix x(n, 1);
+    std::vector<double> lambda(n);
+    std::vector<double> y(n, 0.0);
+    for (std::size_t i = 0; i < n; ++i) {
+        x(i, 0) = 1.0;
+        lambda[i] = 0.1 * static_cast<double>(i + 1);
+    }
+    // The residual energy is confined to the two smallest GRM eigenvalues,
+    // making the one-sided score for v_g at zero strictly negative.
+    y[0] = 1.0;
+    y[1] = -1.0;
+    const double expected_sigma_e2 = 2.0 / static_cast<double>(n - 1);
+    const auto boundary = spectra::reml::evaluate_reml_spectral(
+        y, x, lambda, std::sqrt(expected_sigma_e2), 0.0, false, true);
+    require(boundary.valid, "residual-only boundary evaluation failed: " +
+                                boundary.error);
+    require(boundary.variance_score_g < -1e-3,
+            "fixture does not have a negative one-sided genetic score");
+
+    spectra::reml::RemlOptions options;
+    options.max_iterations = 100;
+    options.gradient_absolute_tolerance = 1e-10;
+    options.gradient_relative_tolerance = 1e-11;
+    const auto fitted =
+        spectra::reml::fit_ai_reml_spectral(y, x, lambda, options);
+    require(fitted.has_estimates(),
+            "boundary AI-REML returned no estimates: " + fitted.error);
+    require(fitted.status == spectra::reml::FitStatus::converged_boundary,
+            std::string("residual-only optimum was not accepted as a KKT boundary: ") +
+                spectra::reml::to_string(fitted.status) + " " + fitted.error);
+    require(fitted.iterations <= 10,
+            "near-boundary KKT check did not terminate promptly");
+    require_near(fitted.sigma_g2, 0.0, 1e-15,
+                 "genetic boundary must report exactly zero variance");
+    require_near(fitted.h2, 0.0, 1e-15,
+                 "genetic boundary must report exactly zero h2");
+    require_near(fitted.sigma_e2, expected_sigma_e2, 2e-12,
+                 "residual-only variance is not RSS/(n-p)");
+    require_near(fitted.log_likelihood, boundary.log_likelihood, 2e-12,
+                 "reported boundary likelihood is inconsistent");
+}
+
+}  // namespace
+
+int main() {
+    try {
+        test_score_matches_finite_difference();
+        test_signed_parameterization_and_row_permutation();
+        test_ai_reml_fit_improves_likelihood();
+        test_residual_only_kkt_boundary();
+        std::cout << "test_reml_synthetic: PASS\n";
+        return EXIT_SUCCESS;
+    } catch (const std::exception& exception) {
+        std::cerr << "test_reml_synthetic: FAIL: " << exception.what() << '\n';
+        return EXIT_FAILURE;
+    }
+}
--- a/tests/python/test_spectra_reml_cli.py
+++ b/tests/python/test_spectra_reml_cli.py
@@ -0,0 +1,163 @@
+from __future__ import annotations
+
+import csv
+import gzip
+import importlib.util
+import json
+import tempfile
+import unittest
+from pathlib import Path
+from types import SimpleNamespace
+from unittest import mock
+
+import numpy as np
+
+
+SCRIPT = Path(__file__).resolve().parents[2] / "python" / "spectra_reml.py"
+SPEC = importlib.util.spec_from_file_location("spectra_reml", SCRIPT)
+assert SPEC is not None and SPEC.loader is not None
+cli = importlib.util.module_from_spec(SPEC)
+SPEC.loader.exec_module(cli)
+
+
+class SpectraRemlCliTests(unittest.TestCase):
+    def fixture(self, root: Path) -> tuple[Path, dict]:
+        n, p0 = 3, 2
+        np.zeros(n * (n + 1) // 2, dtype="<f4").tofile(root / "grm.bin")
+        (root / "grm.id").write_text("F1 I1\nF2 I2\nF3 I3\n", encoding="utf-8")
+        np.asarray([[1, -1], [1, 0], [1, 1]], dtype="<f8").tofile(root / "base.bin")
+        np.zeros((2, n), dtype="<f8").tofile(root / "phenotypes.bin")
+        np.asarray([[0, 1, 2]], dtype="<f4").tofile(root / "extra.bin")
+        (root / "tasks.tsv").write_text(
+            "task_index\ttask_id\tphenotype_row\tn_extra_covariates\n"
+            "0\ttrait_a\t0\t0\n"
+            "1\ttrait_b\t1\t1\n",
+            encoding="utf-8",
+        )
+        np.asarray([0, 0, 1], dtype="<i8").tofile(root / "offsets.bin")
+        np.asarray([0], dtype="<i4").tofile(root / "indices.bin")
+        manifest_path = root / "manifest.json"
+        args = cli.build_parser().parse_args(
+            [
+                "make-manifest",
+                "--manifest", str(manifest_path),
+                "--grm-bin", str(root / "grm.bin"),
+                "--grm-id", str(root / "grm.id"),
+                "--base-x", str(root / "base.bin"),
+                "--phenotypes", str(root / "phenotypes.bin"),
+                "--extra-covariates", str(root / "extra.bin"),
+                "--tasks", str(root / "tasks.tsv"),
+                "--extra-offsets", str(root / "offsets.bin"),
+                "--extra-indices", str(root / "indices.bin"),
+                "--output-dir", str(root / "blocks"),
+                "--n-samples", "3",
+                "--n-base-covariates", "2",
+                "--n-phenotype-rows", "2",
+                "--n-extra-covariate-rows", "1",
+            ]
+        )
+        cli.make_manifest(args)
+        return manifest_path, cli.validate_manifest(manifest_path)
+
+    @staticmethod
+    def write_blocks(root: Path) -> None:
+        root.mkdir(parents=True, exist_ok=True)
+        (root / "block_000000.summary.tsv").write_text(
+            "\t".join(cli.SUMMARY_HEADER) + "\n"
+            "0\ttrait_a\tconverged\t2\t0\t0\t0\t1\t1\t0.5\t-1\t4\t8\t1e-8\t\n"
+            "1\ttrait_b\tconverged_boundary\t3\t1\t2\t3\t0\t1\t0\t-2\t3\t6\t1e-9\t\n",
+            encoding="utf-8",
+        )
+        np.asarray([1, 2, 3, 4, 5], dtype="<f8").tofile(root / "block_000000.beta.f64.bin")
+        np.arange(1, 10, dtype="<f8").tofile(root / "block_000000.cov.f64.bin")
+        (root / "block_000000.complete").write_text(
+            "format\t{}\nblock\t0\ntasks\t2\nbeta_elements\t5\ncov_elements\t9\n".format(cli.BLOCK_FORMAT),
+            encoding="utf-8",
+        )
+
+    def test_manifest_and_generic_command(self) -> None:
+        with tempfile.TemporaryDirectory() as temporary:
+            root = Path(temporary)
+            _, manifest = self.fixture(root)
+            args = cli.build_parser().parse_args(
+                ["run", "--manifest", str(root / "manifest.json"), "--engine", "spectra_reml"]
+            )
+            command = cli.build_engine_command(Path("spectra_reml"), manifest, args)
+            for option in (
+                "--phenotypes",
+                "--extra-covariates",
+                "--extra-offsets",
+                "--extra-indices",
+                "--n-phenotype-rows",
+                "--n-extra-covariate-rows",
+            ):
+                self.assertEqual(command.count(option), 1)
+
+    def test_signature_recovery_and_finalize(self) -> None:
+        with tempfile.TemporaryDirectory() as temporary:
+            root = Path(temporary)
+            manifest_path, manifest = self.fixture(root)
+            engine = root / "spectra_reml"
+            engine.write_bytes(b"synthetic engine")
+            args = cli.build_parser().parse_args(
+                ["run", "--manifest", str(manifest_path), "--engine", str(engine), "--resume"]
+            )
+            with mock.patch.object(cli.subprocess, "run", return_value=SimpleNamespace(returncode=0)):
+                cli.run_engine(manifest, args)
+            output = Path(manifest["paths"]["output_directory"])
+            signature = cli.read_signature(output / "run.signature.json")
+            self.assertEqual(signature["format"], cli.RUN_SIGNATURE_FORMAT)
+
+            self.write_blocks(output)
+            changed = cli.build_parser().parse_args(
+                [
+                    "run", "--manifest", str(manifest_path), "--engine", str(engine),
+                    "--resume", "--max-iterations", "101",
+                ]
+            )
+            with self.assertRaisesRegex(RuntimeError, "signature differs"):
+                cli.run_engine(manifest, changed)
+
+            result = root / "results.tsv.gz"
+            cli.finalize(manifest, result)
+            with gzip.open(result, "rt", encoding="utf-8", newline="") as handle:
+                rows = list(csv.DictReader(handle, delimiter="\t"))
+            self.assertEqual([row["task_id"] for row in rows], ["trait_a", "trait_b"])
+            self.assertEqual(json.loads(rows[0]["beta_json"]), [1.0, 2.0])
+            self.assertEqual(len(json.loads(rows[1]["covariance_packed_lower_json"])), 6)
+
+            altered = dict(manifest)
+            altered["created_utc"] = "changed"
+            with self.assertRaisesRegex(RuntimeError, "does not belong"):
+                cli.finalize(altered, root / "unsafe.tsv")
+
+    def test_force_invalidates_before_cleanup_and_dry_run_is_read_only(self) -> None:
+        with tempfile.TemporaryDirectory() as temporary:
+            root = Path(temporary)
+            manifest_path, manifest = self.fixture(root)
+            engine = root / "spectra_reml"
+            engine.write_bytes(b"engine")
+            output = Path(manifest["paths"]["output_directory"])
+            self.write_blocks(output)
+            resume_args = cli.build_parser().parse_args(
+                ["run", "--manifest", str(manifest_path), "--engine", str(engine), "--resume"]
+            )
+            cli.atomic_json(output / "run.signature.json", cli.make_signature(manifest, engine, resume_args))
+            before = sorted(path.name for path in output.iterdir())
+            dry = cli.build_parser().parse_args(
+                ["run", "--manifest", str(manifest_path), "--engine", "missing", "--force", "--dry-run"]
+            )
+            cli.run_engine(manifest, dry)
+            self.assertEqual(before, sorted(path.name for path in output.iterdir()))
+
+            force = cli.build_parser().parse_args(
+                ["run", "--manifest", str(manifest_path), "--engine", str(engine), "--force"]
+            )
+            with mock.patch.object(cli, "block_files", side_effect=RuntimeError("cleanup interrupted")):
+                with self.assertRaisesRegex(RuntimeError, "cleanup interrupted"):
+                    cli.run_engine(manifest, force)
+            self.assertFalse((output / "run.signature.json").exists())
+
+
+if __name__ == "__main__":
+    unittest.main()
--- a/tests/test_math_reference.py
+++ b/tests/test_math_reference.py
@@ -0,0 +1,128 @@
+#!/usr/bin/env python3
+"""Independent NumPy checks for the spectral-space AI-REML equations.
+
+This file deliberately does not import the production C++ implementation.  It
+serves as a small, readable oracle for gradient and transformation tests.
+"""
+
+from __future__ import annotations
+
+import math
+
+import numpy as np
+
+
+def restricted_loglik_and_derivatives(
+    theta: np.ndarray,
+    eigenvalues: np.ndarray,
+    x_star: np.ndarray,
+    y_star: np.ndarray,
+) -> tuple[float, np.ndarray, np.ndarray]:
+    """Return paper-scale REML log likelihood, score, and AI matrix.
+
+    Parameters are the standard-deviation coordinates ``(sigma_e, sigma_g)``.
+    The function forms P explicitly because it is only used on tiny test data.
+    """
+
+    sigma_e, sigma_g = (float(theta[0]), float(theta[1]))
+    diagonal = sigma_e**2 + sigma_g**2 * eigenvalues
+    if np.any(diagonal <= 0.0):
+        raise ValueError("H is not positive definite")
+    w = 1.0 / diagonal
+    h_inv = np.diag(w)
+    c = x_star.T @ h_inv @ x_star
+    c_inv = np.linalg.inv(c)
+    p = h_inv - h_inv @ x_star @ c_inv @ x_star.T @ h_inv
+    py = p @ y_star
+
+    loglik = -0.5 * (
+        np.linalg.slogdet(c)[1]
+        + np.log(diagonal).sum()
+        + float(y_star @ py)
+    )
+
+    score_e = -sigma_e * (np.trace(p) - float(py @ py))
+    score_g = -sigma_g * (
+        np.trace(p @ np.diag(eigenvalues))
+        - float(py @ (eigenvalues * py))
+    )
+
+    p_py = p @ py
+    lambda_py = eigenvalues * py
+    p_lambda_py = p @ lambda_py
+    ai_ee = 2.0 * sigma_e**2 * float(py @ p_py)
+    ai_eg = 2.0 * sigma_e * sigma_g * float(py @ p_lambda_py)
+    ai_gg = 2.0 * sigma_g**2 * float(lambda_py @ p_lambda_py)
+    ai = np.asarray([[ai_ee, ai_eg], [ai_eg, ai_gg]], dtype=np.float64)
+    return loglik, np.asarray([score_e, score_g]), ai
+
+
+def finite_difference_gradient(
+    theta: np.ndarray,
+    eigenvalues: np.ndarray,
+    x_star: np.ndarray,
+    y_star: np.ndarray,
+    step: float = 1e-6,
+) -> np.ndarray:
+    result = np.empty(2, dtype=np.float64)
+    for index in range(2):
+        delta = np.zeros(2, dtype=np.float64)
+        delta[index] = step
+        high = restricted_loglik_and_derivatives(
+            theta + delta, eigenvalues, x_star, y_star
+        )[0]
+        low = restricted_loglik_and_derivatives(
+            theta - delta, eigenvalues, x_star, y_star
+        )[0]
+        result[index] = (high - low) / (2.0 * step)
+    return result
+
+
+def test_score_matches_finite_difference() -> None:
+    rng = np.random.default_rng(70123)
+    n = 17
+    p = 4
+    eigenvalues = np.linspace(0.05, 2.1, n)
+    x_star = np.column_stack((np.ones(n), rng.normal(size=(n, p - 1))))
+    y_star = rng.normal(size=n)
+    theta = np.asarray([0.83, 0.57])
+    _, score, ai = restricted_loglik_and_derivatives(
+        theta, eigenvalues, x_star, y_star
+    )
+    numerical = finite_difference_gradient(theta, eigenvalues, x_star, y_star)
+    np.testing.assert_allclose(score, numerical, rtol=2e-6, atol=2e-6)
+    np.testing.assert_allclose(ai, ai.T, rtol=0.0, atol=1e-12)
+    assert np.linalg.eigvalsh(ai).min() > 0.0
+
+
+def test_orthogonal_transformation_preserves_reml() -> None:
+    rng = np.random.default_rng(70124)
+    n = 15
+    q, _ = np.linalg.qr(rng.normal(size=(n, n)))
+    eigenvalues = np.linspace(0.1, 1.8, n)
+    grm = q @ np.diag(eigenvalues) @ q.T
+    x = np.column_stack((np.ones(n), rng.normal(size=(n, 3))))
+    y = rng.normal(size=n)
+    theta = np.asarray([0.76, 0.62])
+    sigma_e, sigma_g = theta
+
+    h = sigma_g**2 * grm + sigma_e**2 * np.eye(n)
+    h_inv = np.linalg.inv(h)
+    c = x.T @ h_inv @ x
+    p = h_inv - h_inv @ x @ np.linalg.inv(c) @ x.T @ h_inv
+    original = -0.5 * (
+        np.linalg.slogdet(c)[1]
+        + np.linalg.slogdet(h)[1]
+        + float(y @ p @ y)
+    )
+
+    transformed = restricted_loglik_and_derivatives(
+        theta, eigenvalues, q.T @ x, q.T @ y
+    )[0]
+    assert math.isclose(original, transformed, rel_tol=2e-12, abs_tol=2e-12)
+
+
+if __name__ == "__main__":
+    test_score_matches_finite_difference()
+    test_orthogonal_transformation_preserves_reml()
+    print("PASS: spectral AI-REML reference checks")