Phase 2 — Math Layer

Status: complete. Epic E2 (#3, sub-issues #24–#32) merged.

Phase 2 delivered the header-only, type-generic math layer (math/include/branes/math/, namespace branes::math) that every estimator is built from. See the Math Layer Overview for the full surface; this page is the phase summary.

What landed

• Lie groups — SO3, SE3, Sim3 with exp/log, composition, adjoints, and left/right Jacobians (with inverses). See Lie Groups.
• Fixed-size primitives (lie/detail.hpp) — Vec<T,N>, Mat<T,R,C>, hat/vee, dot/cross/norm, and per-scalar math shims (sqrt_, sin_, atan2_, …).
• Non-linear least squares — Gauss-Newton and Levenberg-Marquardt with analytical, finite-difference-validated Jacobians and golden-value tests. See NLS.
• Camera models — pinhole+radtan, equidistant fisheye, unified omnidirectional, each with project/unproject/distort + analytical Jacobian. See Cameras.
• Linear algebra — tensor views, sparse matrices, Krylov inner solvers (MTL5 ITL), sparse direct solvers, dense Cholesky/SPD solve.

Design rules ratified

• Scalar-generic, never double-hardcoded — everything templates over branes::math::Scalar; the same code runs in double, float, and Universal posits.
• Call the shims, not std:: — algorithms call detail::sqrt_ etc. so they dispatch correctly per scalar type.
• Analytical + FD-validated Jacobians — hand-derived Jacobians are checked against finite differences, so higher layers can trust them at runtime.

Gotchas recorded

• Universal include order — include <number>/<type>.hpp before <number>/mathlib.hpp, guarded with // clang-format off so the include-regroup formatter doesn’t reorder them and break posit compilation.
• Braced float-narrowing — write parenthesized T(1e-12) rather than braced T{1e-12} for sub-epsilon constants, so they instantiate for float/posits.

This layer is the foundation for Phase 3 — VIO.