Refactor(input): make relax_method a scalar option

[Stardust0831]

relax_method now selects each relaxation method with one scalar string. This removes the algorithm/variant pair from INPUT parsing and the relaxation interfaces, makes availability expressions refer to one well-defined value, and gives invalid inputs a complete list of accepted choices.

The relax_method values change as follows:

Previous value	New value
cg or cg 2	cg
cg 1	cg_nested
bfgs or bfgs 2	bfgs
bfgs 1	bfgs_traditional
cg_bfgs	cg_bfgs (unchanged)
lbfgs	lbfgs (unchanged)
sd	sd (unchanged)

The default remains simultaneous CG (cg). All two-token values are rejected; no compatibility aliases are retained.

Reminder

• I have read AGENTS.md and docs/developers_guide/agent_governance.md.
• I have linked an issue or explained why this PR does not need one.
• I have added adequate unit tests and/or case tests, or explained why not.
• I have listed the exact verification commands run and their results.
• I have described user-visible behavior changes, including INPUT parameter changes.
• I have explained core-module impact for ESolver, HSolver, ElecState, Hamilt, Operator, Psi, or other source/ changes.
• I have requested any needed governance exception below (none needed).

Linked Issue

Fixes #7822. Also addresses the availability-semantics discussion in #7783.

That discussion identified ambiguity between scalar membership (in) and vector containment (contains): when relax_method is a vector, it is unclear whether a condition refers to the whole vector, its first element, or one of its elements. A scalar parameter removes that ambiguity. The existing availability grammar remains unchanged.

Unit Tests and/or Case Tests for my changes

• Passed: PW/MPI build, 76 unit tests across seven CTest targets, and four-rank scalar broadcast.
• CLI checks accepted seven valid options and rejected nine invalid inputs with all choices listed. Help and generated documentation were verified.
• Whitespace and governance checks passed.

Not run: full trajectory comparisons, LCAO/GPU builds, or the full test suite; validation focused on PW/MPI. The broader MPI ParaRead fixture requires ELPA.

Verification commands and build details

GCC 13.4, PW/MPI, LCAO/ELPA disabled, assertions enabled, OMP_NUM_THREADS=1; executable version v3.11.0-beta10. $ABACUS_BUILD_DIR is the build directory; $PARAMETER_YAML and $INPUT_REFERENCE are generated scratch files. CLI input checks used copies of tests/01_PW/060_PW_RE_MG.

cmake --build "$ABACUS_BUILD_DIR" -j 8 --target MODULE_IO_read_input_serial MODULE_IO_read_item_serial MODULE_IO_availability MODULE_RELAX_ions_move_methods_test MODULE_RELAX_ions_move_cg_test MODULE_RELAX_relax_nsync_test MODULE_RELAX_relax_sync_test MODULE_IO_input_test_para abacus_pw_para
ctest --test-dir "$ABACUS_BUILD_DIR" --output-on-failure -R '^(MODULE_IO_(read_input_serial|read_item_serial|availability)|MODULE_RELAX_(ions_move_methods|ions_move_cg|relax_nsync|relax_sync)_test)$'
GTEST_FILTER=InputParaTest.TypedTDFieldLists ctest --test-dir "$ABACUS_BUILD_DIR" --timeout 60 --output-on-failure -R '^MODULE_IO_input_test_para_4$'
"$ABACUS_BUILD_DIR/abacus_pw_para" --version
"$ABACUS_BUILD_DIR/abacus_pw_para" -h relax_method
"$ABACUS_BUILD_DIR/abacus_pw_para" --check-input
"$ABACUS_BUILD_DIR/abacus_pw_para" --generate-parameters-yaml > "$PARAMETER_YAML"
python docs/generate_input_main.py "$PARAMETER_YAML" --output "$INPUT_REFERENCE"
git diff --check upstream/develop...HEAD
python3 tools/03_code_analysis/agent_governance_check.py --base upstream/develop --head HEAD --format text
git diff --name-only upstream/develop...HEAD | rg '\.(cpp|h)$' | xargs python3 tools/03_code_analysis/code_quality_score.py

Quality scoring: 15/17 files passed; both existing low scores are unchanged from the base.

What's changed?

• Store and pass relax_method as one std::string. Accept exactly one of cg (default), cg_nested, bfgs, bfgs_traditional, cg_bfgs, lbfgs, or sd; reject empty, unknown, and multi-token values without compatibility aliases.
• Simplify parsing, MPI synchronization, dispatch, and reporting to use scalar names. Preserve the existing numerical algorithms, CG-to-traditional-BFGS transition, and BFGS history resets after cell changes.
• Update Input_Item type/default/help and dependent availability expressions. Invalid empty, unknown, and multi-token values now share one diagnostic that lists every accepted scalar choice. Add assertions that scalar membership is valid and vector containment is rejected for relax_method.
• Update the geometry-optimization manual and availability-guide examples to use scalar option names. Remove obsolete spellings, numbered variants, and migration prose from the manual and CLI help.
• Correct example and regression-test INPUT files, test fixtures, and their comments to use current names. Retain invalid legacy input only in rejection tests.

The diff contains 40 files: 9 implementation files, 10 unit-test/fixture files, 2 documentation files, and 19 example/regression INPUT files. Each example/regression INPUT change replaces one relaxation selection.

Governance Notes

• INPUT/docs changes: relax_method is now scalar throughout. Input_Item metadata is the source of truth; generated YAML and Markdown references were checked but are not committed.
• Core module impact: changes are confined to IO representation and relaxation selection/reporting. No numerical kernels or new global dependencies are introduced.
• Historical quality debt: two existing large IO test files remain below the quality-score threshold, with unchanged scores relative to the base. No new governance findings; no exceptions requested.