[M0-CORE-04] SimMath fpx16_16 backend (default)

[offdev]

What

Implements roadmap step M0-CORE-04 (ADR 0002, PRD §10.3, FR-3.3): the laige::fpx16_16 Q16.16 fixed-point type and its wiring in as the default SimMath backend, with the same op surface as M0-CORE-03.

• include/laige/fpx16_16.h — int32_t raw, value = raw/2^16, range [-32768, 32767.99998474], resolution 2^-16. All arithmetic in int64_t, defined for every input (CPP-004, no UB): saturating add/sub/mul/div/negate; round-to-nearest ties-to-even (the fixed-point analogue of IEEE round-to-nearest-even); divide-by-zero defined (x/0 → ±max with the sign of x, 0/0 → +0 — the saturation analogue of IEEE ±inf); sqrt(negative) → +0; documented fromInt32/toInt32/fromFloat/toFloat conversions (NaN → +0, saturation, single-rounding semantics). No NaN/Inf exist (total order); no implicit scalar constructors and no arithmetic operators — construction via fromInt32/fromFloat, computation via the named static ops / SimMath (PRD §10.3, CORE-008).
• sim_math.h — Fpx16_16 backend (delegates to the type ops) + SimMathFpx16 alias, marked as the default backend per ADR 0002 (the determinism.math config plumbing lands with the config step, FR-1.5). Additive template changes only: isNaN/isInf become backend-delegated (Fp32Pinned carries the IEEE implementation — semantics unchanged) and normalize compares against Scalar{} (identical for both backends).
• sim_math_fixed.cpp — explicit SimMath<Fpx16_16> instantiation (linkable symbol in both static and shared variants).
• tests/laige-core/math_fixed_tests.cpp — ctest -R math_fixed, 24 tests / 7 suites: exhaustive edge cases (min/max, wrap candidates), rounding ties for mul/div/toInt32/fromFloat (plus the no-ties-for-sqrt fact), the full saturation/divide-by-zero policy, conversion round trips (20k-raw LCG scan), the op surface, and the documented length accuracy bound (accurate while the sum of squares stays in the Q16.16 range, saturating beyond). Determinism property: a fixed 4096-tick op sequence run through the SimMath ops and through an independent raw-int64 reference agrees bit-for-bit, and its FNV-1a state hash equals the committed known-answer constant 0xF02728762777C581.
• Docs: docs/api/sim_math.md gains the fpx16_16 policy section (rounding, saturation, divide-by-zero, conversions, the length bound, the ARCH-010 determinism scope); quick start now leads with the default backend; building.md/README.md list the new entry; roadmap step marked done.

Verify (local, 2026-09-10)

• ctest -R math_fixed green; full ctest 10/10 on five trees: g++ 16.2.1 static, g++ ASan+UBSan (canonical LAIGE_ASAN=ON), g++ shared (LAIGE_BUILD_SHARED=ON), g++ TSan, and clang++ 22.1.8.
• The two-compiler property the step names: the known-answer determinism hash is identical on the two different compiler builds (g++ vs clang++). CI hookup for this property lands in M1-DET-04.
• Zero warnings under -Wall -Werror -fno-exceptions -fno-rtti (NFR-8.10).

Notes

• One additive change to the M0-CORE-03 template (backend-delegated classification) — fp32 behavior is bit-identical; ctest -R math_float stays green in every tree.
• Documented bound: length saturates beyond |v| ≈ 181.02 per component (the sum of squares must stay in the Q16.16 range). Local sim math stays well inside; world-span distances belong to M1-DET-02.