Math Layer Overview

The math layer (math/include/branes/math/, namespace branes::math) is header-only, C++20, and type-generic. Every primitive is templated over a scalar that satisfies the branes::math::Scalar concept, so the same code compiles in double, float, or a Universal posit for mixed-precision study. Nothing hardcodes double.

It depends only on Universal (custom arithmetic) and MTL5 (matrix template library) — and on nothing heavier. No I/O, no middleware, no hardware awareness.

What it provides

Area	Header(s)	Summary
Lie groups	lie/{so3,se3,sim3}.hpp	SO(3), SE(3), Sim(3) with exp/log, composition, adjoints, left/right Jacobians. See Lie Groups.
Lie detail	lie/detail.hpp	Fixed-size Vec<T,N> / Mat<T,R,C>, hat/vee, scalar shims (sqrt_, sin_, atan2_, …) that dispatch to the right overloads per scalar type.
Non-linear least squares	nls.hpp, nls/{dense_linalg,solver}.hpp	Gauss-Newton and Levenberg-Marquardt with analytical Jacobians. See NLS.
Camera models	cameras.hpp, cameras/*	Pinhole+radtan, equidistant fisheye, unified omnidirectional — project/unproject/distort + analytical Jacobians. See Cameras.
Linear algebra	tensor.hpp, sparse.hpp, krylov.hpp, sparse_direct.hpp, arithmetic.hpp	Tensor views, sparse matrices, Krylov inner solvers (via MTL5 ITL), sparse direct solvers.

Design choices that recur

• Scalar-generic, with shims. Free functions like branes::math::lie::detail::sqrt_ exist so a single algorithm body works whether T is double (calls std::sqrt) or a Universal type (calls the type’s sqrt). Algorithms call the shim, never std::sqrt directly.
• Analytical Jacobians, finite-difference-validated. Where a Jacobian is derived by hand, a test checks it against a numerical finite-difference of the function.
• Fixed-size where the dimension is known. Poses and small blocks use compile-time Vec/Mat; only runtime-sized state (e.g. the growing MSCKF covariance) uses a dynamic matrix.
• Clean-room. Everything is implemented from the mathematical definitions and published papers — never copied from GPL or other restrictively-licensed source. See Algorithms & Provenance.

Status

Epic E2 (Phase 2) is complete: the surface above is in place and unit-tested, including golden-value tests for the NLS solvers. It is the foundation the VIO layer is built on.