Implicit flex elasticity in the CG constraint solver via an effective metric

This CL replaces the post-hoc implicit flex correction (`flexInterp_cgsolve`) with a **linearly-implicit effective metric** `M̃ = M + (h² + h·damping)·K` carried by the CG constraint solver itself. Contact/friction forces and implicit flex elasticity are now computed against one consistent metric, instead of the solver seeing `M` and a post-solve correction changing `qacc` behind its back.

Gate (unchanged semantics): `solver="CG"` + implicit/implicitfast integrator + pyramidal cones + flex stiffness present. Newton and PGS are untouched. `solver="CG"` remains the user-facing contract — the factorization is an implementation detail of the preconditioner.

### What's in the metric

- **mjData `efm_*`** (arena, efc-like lifetime/skip semantics; built in `mj_fwdPosition`, value-refreshed in `mj_fwdVelocity`): the per-step stiffness CSR `efm_B_*`, its reverse-Cholesky factor `efm_dofid` + `efm_L_*` (nested-dissection ordered, separators-first for the reverse factorization), and the smooth-force shift `efm_c = h·K·qvel`.
- **`mjd_flexStiff_assemble`** now assembles stretch (Gauss–Newton), standard dim-2 bending, and — via the cached corotated stiffness `d->flexelem_krot` — interp stiffness (all node bodies on simple sliders: point Jacobian is I₃, `flex_centered` not required; fixed nodes drop like pins) into one dof-level CSR. `mjd_effMulAdd`/`mjd_effSolve` apply the metric, with matrix-free operator fallbacks where assembly does not apply.
- **mjModel `efm0_*`** (`nefm0dof`/`nefm0L`): the constant part of the metric factor — currently the dim-2 bending factor, computed once in `mj_setConst` — so bending-only models pay zero per-step factorization cost. Naming mirrors mjData's `efm_*` with the standard `0`-suffix (reference/constant) idiom, and is deliberately not bending-specific: future constant contributors extend it without renames.
- The solver consumes the metric through pre-shifted `qfrc_smooth` and the metric products `Ma`/`Mv`/`Mgrad`; `qacc_smooth` becomes the unconstrained minimizer of the implicit dynamics, which makes the no-constraint shortcut and the warmstart choice consistent by construction.
- **`mj_inverse` adds `B·qacc − c`**, making inverse dynamics discrete-consistent with the gated forward dynamics — exact, since the gated path has no qDeriv term (new test `ForwardTest.GatedFlexInverseConsistency`).

### Performance

All numbers: ms/step over the same 2000-step window, models as shipped on each side (old code with the old model settings vs this CL with the new ones).

The new solver path activates on exactly two shipped models — the ponchos, the only flex models that need an implicit integrator (poncho on Euler degenerates to >200 ms/step). For them, this CL trades speed for consistency: the implicit bending solve now runs inside every solver iteration, where the contact solve can see the stiffness, instead of once after the solve. Solver iterations drop because the curvature is visible, but each iteration pays for the implicit solve:

| model | before | after | solver iters/step |
|---|---|---|---|
| poncho | 2.47 | 3.30 (1.33×) | 16.8 → 11.8 |
| poncho_edgeequality | 1.96 | 2.72 (1.39×) | 13.2 → 10.0 |

What that price buys: contact forces consistent with the implicit elasticity (previously the post-hoc correction changed `qacc` after the constraint solve), discrete-consistent inverse dynamics, and the removal of the post-hoc special case from the integration path. Raising poncho's timestep from 2 to 5 ms leaves its per-step cost nearly flat, so the consistency price can be recovered by taking fewer steps where accuracy allows.

Every other flex model was measured stable on Euler at its shipped timestep and switches to it (these models predate the post-hoc integrator; implicit was never load-bearing for them). They end up equal or faster than before: bunny_multicell 0.47 → 0.40, trampoline 0.28 → 0.25, plate 1.02 → 0.99, pancake 0.34 → 0.33.

Finally, the per-step factorization makes configurations practical that the old code could only integrate explicitly: implicit stretch elasticity (`elastic2d="stretch"`/`"both"`, dim-3 solids) and factorized interp stiffness. No before/after exists for these — stock has no implicit treatment of stretch at all.

### Behavior changes

- With the post-hoc correction deleted, interp/bending models running `solver="Newton"` (or elliptic cones, or islands) now integrate flex elasticity **explicitly** (previously: post-hoc implicit). Affects e.g. `gripper_trilinear` (stable, and faster, but different semantics). Follow-up options: Newton-side metric support, or a documented fallback.
- With the gate on, `mj_forward` outputs are timestep-dependent for gated models (they answer the linearly-implicit discrete problem); `qacc_smooth` and `mj_inverse` change accordingly. Non-gated models are bit-identical (full suite green throughout).

### Validation

- 1737/1737 tests, including new: `FlexStretchDerivatives` (FD-validated GN operator), `FlexStiffAssemble`/`FlexStiffAssembleInterp` (CSR ≡ operators), `GatedFlexInverseConsistency` (fails pre-change), equivalence tests vs the old post-hoc treatment (bending matches to 2e-11).
- Fingerprint discipline throughout: bending-only models bit-exact across every refactor; permutation/kernel changes verified iteration-identical.

### Known follow-ups (not in this CL)

3×3-block sparse Cholesky kernel (the numeric factorization is index-bound; projected ~3× on the factor); mjModel persistence of the factor's symbolic pattern (rest-pose ND makes sizes compile-time); the general effective-metric mode (all solvers, all PSD-safe force classes, behind an enable flag).

PiperOrigin-RevId: 948561856
Change-Id: I8b8e32ebd0428042af71647d0470d10773bf6daf

src/engine/engine_derivative.h

@@ -65,6 +65,39 @@ MJAPI void mjd_flexInterp_cacheKrot(const mjModel* m, mjData* d, mjtNum* K_rot_o
 MJAPI void mjd_flexBend_mul(const mjModel* m, mjData* d, mjtNum* res, const mjtNum* vec,
                             mjtNum s1, mjtNum s2);
 
+// compute res += scale * K_stretch * vec for standard (non-interp) flex stretch,
+// K_stretch the Gauss-Newton Hessian of the passive stretch force at the current state
+//   scale = s1 + s2 * flex_damping[f]  per flex
+MJAPI void mjd_flexStretch_mul(const mjModel* m, mjData* d, mjtNum* res, const mjtNum* vec,
+                               mjtNum s1, mjtNum s2);
+
+// assemble the standard-flex implicit stiffness (s1 + s2*damping)*(K_bend + K_stretch) into
+// dof-level CSR; phase 1 (colind==NULL) fills rownnz/rowadr and returns total nnz, phase 2
+// fills colind/val. Interp flexes are assembled iff Krot (mjd_flexInterp_cacheKrot cache) is
+// non-NULL and the centered fast path applies (check mjd_flexInterpAssemblable first).
+MJAPI int mjd_flexStiff_assemble(const mjModel* m, mjData* d, int* rownnz, int* rowadr,
+                                 int* colind, mjtNum* val, mjtNum s1, mjtNum s2,
+                                 int flg_bend, int flg_stretch, const mjtNum* Krot);
+
+// can all interp flexes be assembled to dof-level CSR? (centered fast path everywhere)
+MJAPI mjtBool mjd_flexInterpAssemblable(const mjModel* m);
+
+// does any flex contribute assemblable implicit stiffness? (existence check)
+MJAPI mjtBool mjd_flexStiff_any(const mjModel* m, int flg_interp);
+
+// implicit effective metric Mtilde = M + (h^2+h*d)*K: per-step arena object (see mjdata.h efm_*)
+// build (or deactivate, active==0); the gate decision belongs to the caller
+MJAPI void mjd_effBuild(const mjModel* m, mjData* d, int active, int flg_factor);
+
+// refresh the metric's smooth-force shift c = h*K*qvel (values only, velocity stage)
+MJAPI void mjd_effShift(const mjModel* m, mjData* d);
+
+// res += B*vec (the stiffness part of the metric; caller supplies the M part)
+MJAPI void mjd_effMulAdd(const mjModel* m, mjData* d, mjtNum* res, const mjtNum* vec);
+
+// x = (M + B)^-1 b to 1e-10 relative; x = M^-1 b when the metric is inactive
+MJAPI void mjd_effSolve(const mjModel* m, mjData* d, mjtNum* x, const mjtNum* b);
+
 
 #ifdef __cplusplus
 }