feat: UI quality-of-life batch (#293–#304) + solution group projection — 3.1.0.dev1

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UI quality-of-life batch (issues #293–#304) + solution group projection

These rough edges surfaced while implementing the numerical real cellular decomposition algorithm in
Python. The through-line goal: get from a polynomial system to the projection values of critical
points as real_mp, straight through numpy, without a code maze.

Per the "put reusable primitives in core" doctrine, anything with a C++ home landed in core +
bindings (behind C++ tests); only presentation/interop sugar stayed pure-Python.

C++ core + bindings

• random real slice isn't real; hard to make random high-precision numbers #294 — Slice::RandomReal's orthogonal path was silently using the complex conjugate-orthonormal
  matrix, so a "real" slice came out complex; now genuinely real. New random_real / random_complex /
  random_vector factories (continuous, seed-reproducible — the generic-direction tool, unlike the
  quantized orthonormal random_matrix).
• add is_distinct_up_to #304 — reusable IsDistinct / IsSamePoint infinity-norm util → bertini.is_distinct_up_to
  (accepts real_mp / complex_mp / real_dbl / complex_dbl points).
• i want for solutions() etc to NOT duplicate the >1 multiplicity solutions #299 — merge_multiplicities (default True) on the solver's own SolutionsWhere + finite
  accessors — dedup where the multiplicity clustering lives.
• solution_metadata should have a way to get the metadata, given just a point #302 — metadata_for(point): returns the cluster representative by default (one record,
  carrying .multiplicity), or coincident=True for the list of every copy — return type never
  depends on the point's multiplicity.
• sys.functions() should return a list of functions #297 — System.functions() / copy_functions().
• function.eval() needs to accept a 1d numpy array #300 — node.eval accepts a point array or a {Variable-or-name: value} dict (not just kwargs),
  still returning complex_mp natively.
• Group projection — System.coordinates_of(point, group) + a group= keyword on every solution
  getter and to_dataframe: project solutions onto one variable group (projective groups returned
  as-is). Reusable for the eventual C++ decomposition.

Python layer

• TypeError when getting the variable group of a system. #293 — variables(['x','y','z']); add_variable_group accepts loose variables / a list / a single
  Variable / a VariableGroup; Slice.random_* gives a clear message (and unwraps a one-group sequence)
  instead of an opaque converter error.
• i need to be able to copy a system to make a new system more quickly #296 — sys.clone() method.
• i want to be able to sympy directly from bertini2 symbolic objects #295 — bertini.sympy_bridge exposed lazily + nodes auto-sympify (_sympy_), so
  sympy.sympify(node) / sympy.Matrix(array) / sympy.det(J) just work.
• random real slice isn't real; hard to make random high-precision numbers #294 sugar — VariableGroup @ coeffs → the linear-combination node.

#298 / #301 (numpy interop) — note the overlap

Adds mp-native elementwise helpers bertini.real/imag/abs/conj/round/sum/norm/is_real (the
numpy↔eigenpy dtype boundary is unpatchable for identity-seeded reductions; documented in
known_gotchas). These likely overlap with the native ufunc coverage on feature/numpy_compat —
left for the maintainer to reconcile (the helpers still stand alone for is_real / dedup / norm).

Deferred

• do i need a HomogeneousVariableGroup type? #303 (HomogeneousVariableGroup) — out of scope by decision.

Verification

• C++: test_classes + test_nag_algorithms green (new regression tests for every C++ change).
• Python: 660 passed, 4 skipped (the one initial failure was the intended i want for solutions() etc to NOT duplicate the >1 multiplicity solutions #299 default change; that
  test was updated).
• Acceptance artifact: a polished marimo notebook (in the sibling curve_decomposition_in_python repo,
  original left untouched) runs end-to-end — projection values arrive as real_mp, plus a 3D plot of
  the real critical points.

Heads-up for review

• Version bump 3.0.0rc1 → 3.1.0.dev1 is included (this branch is the 3.1.0.dev1 work). Drop the
  first commit if you'd rather bump separately.
• The projective-nullvector reformulation does not track — mathematically cleaner (only critical
  points survive) but the mhom homotopy fails every path for this bilinear structure. The notebook uses
  the working affine+patch formulation and documents the projective attempt as an open question for the
  start-system / endgame work.
• int(real_mp) segfaults (pre-existing real_mp binding bug; float/str fine) — hit
  incidentally, out of scope here, worth a follow-up (real_mp.__int__).
• doclint not run locally (no doxygen in this env); every new public C++ entity carries a matching
  Doxygen comment, so the CI gate should pass.

Closes #293, #294, #295, #296, #297, #299, #300, #302, #304.
(#298 and #301 addressed via helpers — see the overlap note above before closing.)

🤖 Generated with Claude Code

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[ofloveandhate]

there's some competing work in here, with #306 . will deal with in a minute