# Prepared gravity subsystem and curvature correction

This branch introduces solver components; the complete equilibrium equations,
normalization, and model specification remain the source of truth.

## Block and lifetime contracts

`MakeGravityBlockPartition<Form>(layout)` selects the gravity-gradient and
gravity-potential blocks by their compiled value and residual types. Its
constraints reject forms lacking those blocks. The complement is retained in
root order, including any generated angular-momentum or central-density border.
There are no string identifiers or fixed root offsets in this selection.

`GravityBlockPartition` also has a checked explicit-index constructor for
algebraic tests and other compiled adapters. Local partitions may be empty;
gravity and retained local systems must be square. A globally empty gravity
system is rejected by the prepared linear backend. Gather/injection require
preallocated, nonoverlapping storage.

`PreparedGravitySubsystem` owns restricted operator/inverse workspaces and the
existing prepared linear backend. It borrows the full normalized operator,
inverse, and communicator at stable addresses. It is neither movable nor
copyable. Preparation records the root preparation generation. After candidate
preparation the caller must refresh its physical inverse and explicitly refresh
the subsystem; stale applications and solves fail. The subsystem does not
prepare a candidate, change an accepted state, or commit a step.

The exact gravity action is the principal restriction of the root Jacobian.
The preconditioner is the corresponding restriction of the supplied root
inverse. This reuses existing normalization, gravity boundary conditions, and
preconditioning recipes, but does not assert that the restricted inverse is an
exact gravity inverse. Every solve retains the backend's independently checked
true-defect gate. Both wrappers preallocate full-vector workspaces; applications
perform gather/scatter and the existing root actions without allocating their
own vectors. Full root actions may do unnecessary non-gravity work; a specialized
gravity action/inverse adapter remains a future optimization requiring equality
tests against this reference restriction.

Attempted solves, completed inner iterations, failed calls, completed solve time,
and the last completed solve report are available without history allocation.
Contract exceptions count as failed calls but do not trigger new collectives to
measure elapsed time during unwinding.

The existing FGMRES backend now admits ranks with zero local equations while
requiring a nonempty global system. `GetConfiguration()` returns its immutable
prescription so another prepared operator can reuse the same backend options.

## Curvature correction

`PreparedGravityCurvatureCorrection` captures `G(x)` and `J_G(x) p` before any
candidate preparation. For a prepared candidate `x + alpha p` it solves

```
G_z(x + alpha p) delta_z = G(x) + alpha J_G(x) p - G(x + alpha p).
```

Only the compact gravity correction is returned. The caller may inject it only
after a converged inner report, then must prepare and evaluate the complete
candidate again before applying its usual acceptance rules. In particular,
improving gravity does not certify the hydrostatic, material, or shape residual.

For smooth equations with uniformly bounded inverse gravity block, the right
side and correction are second order in `alpha`. This preserves the base and
Newton tangent even when the base does not lie on the gravity solution manifold.
It differs from projecting the candidate to `G=0`, which can introduce a finite
correction at zero step when the base gravity residual is nonzero. Inexact inner
tolerances control cancellation of the remainder; they cannot replace the outer
equilibrium and virial checks.

The owner captures a new base for each new Newton direction and invalidates the
cache when that direction is discarded. The stored base generation is exposed
for diagnostics; preparing trials intentionally does not invalidate the captured
base arrays. Reusing those arrays after replacing the base or direction violates
the caller's explicit lifecycle contract.

## Test coverage

`tests/solver/gravity_subsystem.cpp` covers typed forms with/without a central
border, noncontiguous and differently ordered row/value maps, invalid maps and
overlapping storage, exact restricted solves, true-defect failure, nonfinite
inputs, stale operator/inverse preparation, locally empty MPI partitions,
globally empty rejection, preserved backend options, zero affine curvature at
an off-manifold base, and the second-order correction at curved trial geometry.
The root integration campaign compiles these components and tests real stellar
states; this branch does not claim a new validated stellar equilibrium.