feat(math): add SE2(3) extended-pose Lie group — the R-IEKF state manifold (#347)

[Ravenwater]

Phase B foundation for the R-IEKF epic (#347). SE₂(3) — the extended-pose / double-direct-isometry group (R, v, p) — is the state manifold that makes the IMU dynamics group-affine, so the right-invariant error Jacobians are state-independent and the unobservable yaw/position subspace is preserved by construction. That's the cure Phase A validated by measurement (flat yaw leak #349, state-independent propagation Φ #351).

What's here

• math/lie/se23.hpp — exp/log (both translations share the SO(3) left Jacobian), compose, inverse, 5×5 matrix, 9×9 adjoint, in the rotation-first [φ; ν; ρ] ordering. Mirrors the se3.hpp/sim3.hpp idioms; clean-room (Solà 2018, Barrau-Bonnabel 2017).
• Tests (in lie_groups): exp/log round-trip, compose/inverse, and the adjoint identity X·exp(ξ)·X⁻¹ = exp(Ad_X·ξ) — 42 assertions, alongside the existing SO3/SE3/Sim3 suite (741 total, green).

Next in Phase B

The invariant IMU propagation and camera measurement update on this state, then the MsckfInvariantBackend wiring (parallel to the shipped EKF) + the EuRoC verdict.

Refs #347, #348, #212.

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Summary by CodeRabbit

• New Features

  • Added SE₂(3) extended-pose group: rotation plus velocity and position, with composition, inversion, exponential/logarithm maps, and homogeneous-like/adjoint matrix representations.

• Tests

  • Added tests for exp/log round-trips, composition and inversion identities, adjoint conjugation consistency, and robustness near half-turn rotations across numeric types.

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• tests/math/lie_groups.cpp

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Walkthrough

This PR introduces the SE₂(3) "extended pose" Lie group as a new C++20 header-only template class with rotation, velocity, and position components. It implements exp/log (using SO(3) Jacobians), group composition and inversion, 5×5 and 9×9 matrix/adjoint representations, and adds tests validating round-trips, group identities, adjoint action, multi-arithmetic coverage, and a near-half-turn regression.

Changes

SE₂(3) Extended Pose Lie Group Implementation

Layer / File(s)	Summary
SE23 class definition and public interface math/include/branes/math/lie/se23.hpp	Defines SE23<T> with SO3<T> rotation, Vector3 velocity, and Vector3 position; public constructors, component accessors, static exp(Tangent), and log() returning a 9-vector tangent using SO(3) left-Jacobian/inverse.
Group operations and matrix representations math/include/branes/math/lie/se23.hpp	Implements operator* (rotation compose, transform+sum velocity/position), inverse(), matrix() producing a 5×5 block representation, and adjoint() producing a 9×9 adjoint in [φ; ν; ρ] ordering using hat-maps and an internal place() helper.
SE23 test suite tests/math/lie_groups.cpp	Adds SE23 include, require_se23_close helper, tests for exp/log round-trip (including zero/identity), composition and inversion identities, adjoint conjugation identity X · exp(ξ) · X⁻¹ = exp(Ad_X · ξ), type coverage (float/posit), and a regression test near θ≈π.

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🎯 3 (Moderate) | ⏱️ ~25 minutes

🚥 Pre-merge checks | ✅ 3 | ❌ 2

❌ Failed checks (1 warning, 1 inconclusive)

Check name	Status	Explanation	Resolution
Docstring Coverage	⚠️ Warning	Docstring coverage is 38.46% which is insufficient. The required threshold is 80.00%.	Write docstrings for the functions missing them to satisfy the coverage threshold.
Description check	❓ Inconclusive	The description covers the key aspects (what, why, next steps) and references relevant design papers, but the PR template's Test Results table and several checklist items are incomplete or missing.	Complete the Test Results table with build/test outcomes for both gcc and clang, and explicitly check off (or address) the Conventional Commits and testing checklist items.

✅ Passed checks (3 passed)

Check name	Status	Explanation
Title check	✅ Passed	The title clearly and concisely describes the main change: adding the SE2(3) Lie group implementation as the R-IEKF state manifold, directly matching the primary purpose of this changeset.
Linked Issues check	✅ Passed	Check skipped because no linked issues were found for this pull request.
Out of Scope Changes check	✅ Passed	Check skipped because no linked issues were found for this pull request.

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Actionable comments posted: 1

🤖 Prompt for all review comments with AI agents

Verify each finding against current code. Fix only still-valid issues, skip the
rest with a brief reason, keep changes minimal, and validate.

Inline comments:
In `@tests/math/lie_groups.cpp`:
- Around line 412-450: SE23 tests only exercise double precision; add
multi-arithmetic round-trip and near-singular tests to match SO3/SE3/Sim3. Add a
TEST_CASE "SE23 round-trip across arithmetic types" that recreates the double
round-trip using lie::SE23<float> with det::Vec<float,9> and lie::SE23<posit32>
with det::Vec<posit32,9> (typedef posit32 = sw::universal::posit<32,2>),
checking X.log() ≈ xi with tolerance ~1e-4 via require_vec_close; also add
TEST_CASE "SE23 log is finite for a half-turn rotation" using lie::SE23<double>
with a Tan near θ≈π (use the suggested xi) and assert all entries of X.log() are
finite (std::isfinite) to exercise the SO3 inverse-left-Jacobian near its
removable singularity. Ensure test names match the pattern in the file and reuse
require_vec_close/REQUIRE as shown.

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• math/include/branes/math/lie/se23.hpp
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Actionable comments posted: 1

🤖 Prompt for all review comments with AI agents

Verify each finding against current code. Fix only still-valid issues, skip the
rest with a brief reason, keep changes minimal, and validate.

Inline comments:
In `@tests/math/lie_groups.cpp`:
- Around line 478-482: The test is asserting branch-cut-sensitive equality for
the half-turn case by checking X.log() == xi; instead, ensure the log is finite
and that the group invariant holds: replace the require_vec_close(X.log(), xi,
1e-9) call with two checks—verify the components of X.log() are finite (no
NaN/inf) and verify that lie::SE23<double>::exp(X.log()) is numerically close to
X (e.g., require_mat_close(lie::SE23<double>::exp(X.log()), X, 1e-9) or
similar). Keep the rest of the loop (xi, X, dxi) unchanged.

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