feat: add second order to SScalar

[oberbichler]

Completes the three-step plan for SScalar: characterize (#84), change the storage (#85), add second order (this).

SScalar carried a value and a gradient keyed by name. This adds the Hessian, so named derivatives reach the same order the indexed DDScalar already offered. The Python types are SScalar (order 1) and SSScalar (order 2), following the one-letter-per-order naming of DScalar/DDScalar; dd(a, b) is absent from the first-order class rather than raising.

x = hj.SSScalar.variable("x", 0.5)
y = hj.SSScalar.variable("y", 0.4)

f = (x * y).sqrt()
f.d("x")          # 0.4472135954999579
f.dd("x", "y")    # 0.5590169943749475
f.dd("x", "z")    # 0.0 -- an unknown name, as with d

Why dense over the names

The Hessian is dense over the names a value carries, laid out as the upper triangle in the same packing DDScalar uses. That was measured rather than assumed: any nonlinear function contributes the full outer product of its own gradient, so a simulation of the fill rate reaches 100% after two operations. What stays sparse is the set of names — which is the point of the type, since it does not have to be known in advance. A pair-keyed map would allocate per entry to store what is dense anyway (25 allocations against 0 for k=6).

Cost

Order 1 pays nothing. The Hessian storage sits behind [[no_unique_address]] and collapses to an empty member:

	sizeof	runtime, k=6	allocations/iter
SScalar<1>	32 B	1.93 µs	60
SScalar<2>	56 B	2.52 µs	73

Second order costs 1.31× the first — for six names and 21 Hessian entries.

Tests

34 new C++ cases (each checking the value, three gradient entries and all six Hessian entries, plus symmetry) and 8 new Python cases. The expected values were generated symbolically with sympy, so they share no formulas with the header.

Writing them first paid off. Of 413 assertions, 174 failed at the start; the last 8 to fall pointed at four operators — -u, s - u, u * s, u / s — that copied the storage and rewrote value and gradient in place without ever touching the Hessian. Routing them through the existing apply() helper fixes it and is shorter than what it replaced.

Verification

• C++ 118 cases / 1886 assertions, Debug and Release with -Werror
• clang ASan + UBSan clean
• Python 259 tests
• clang-format, ruff clean; just tidy 35 findings against 41 on main (all pre-existing, none in the new code)