rtx-cfd/rtx-fsi: overset A-P0 GATED + M1 precision probe — curvilinear collocated PISO: relative-reduction pressure stop (the absolute stop floored |du/dt| at 2e-4 on 64²), line-implicit-n sign fix, adjustPhi; gates: Cartesian reduction 1.37–1.40x the staggered error at orders 0.83/0.90; skewed stretched periodic annulus Stokes orders 2.30/2.06 (explicit and line-implicit), upwind 1.08/0.80; Poiseuille exact to 1e-9 on Cartesian and affine-sheared periodic channels (both diffusion variants), varying-skew channel order 2.02 (v 1.9), cell mass 1e-14; divergence ≤ 1e-11 relative every step; snapshot/restore bit-identical. M1: poisson.rs multigrid hierarchy generic over MgScalar (f32/f64), f64 CG keeps its own fine level; MgPrecision on MultigridParameters/EmbeddedParameters/PisoParameters, set_poisson_precision, harness RTX_FSI2_POISSON_F32 (march + noise probe, printed marker); f64 arm bit-identical in vivo (FSI2 default line-for-line with 08-31), f32 arm holds the noise floor and stall pins and the FSI2 band; poisson_equivalence f32 arm

Co-Authored-By: Claude Fable 5.1 <[email protected]>
Claude-Session: https://claude.ai/code/session_01X2GmJXeQ2njUecEKiJZ1G2

crates/specialized/rtx-cfd/src/mesh/patch_gen.rs

@@ -124,6 +124,7 @@ pub fn channel_sheared(
 /// A channel whose skew varies smoothly along x: `x' = x + alpha · y ·
 /// sin(2π x / lx)` (zero at both ends, so it can be periodic), with the
 /// across-channel spacing stretched by `stretch` toward the top wall.
+/// Folds when `2π alpha ly / lx > 1`; keep `alpha` around 0.1.
 pub fn channel_varying_skew(
     lx: f64,
     ly: f64,

crates/specialized/rtx-cfd/src/solvers/incompressible/curvilinear/mod.rs

@@ -95,7 +95,12 @@ pub struct CurvilinearParameters {
     /// Projections per step (the first removes the divergence; the rest
     /// mop up inner-solver truncation).
     pub corrector_steps: usize,
-    /// Pressure-correction stop, relative to the step's flux scale.
+    /// Pressure-correction stop: each projection reduces the L1 cell mass
+    /// imbalance by this factor (plus a rounding floor of 1e-15 × Σ|F|).
+    /// Relative to the incoming divergence, not to the flux scale, so the
+    /// stop tightens as the flow settles — an absolute stop leaves a
+    /// velocity-noise floor that grows with the grid (measured: |du/dt|
+    /// floored at 2e-4 on the 64² Cartesian MMS with `1e-10 × Σ|F|`).
     pub tolerance: f64,
     /// Convection scheme.
     pub convection: PatchConvection,
@@ -111,7 +116,7 @@ impl Default for CurvilinearParameters {
     fn default() -> Self {
         Self {
             corrector_steps: 2,
-            tolerance: 1e-8,
+            tolerance: 1e-4,
             convection: PatchConvection::Upwind,
             normal_diffusion: NormalDiffusion::Explicit,
             boundaries: PatchBoundaries::default(),
@@ -304,22 +309,24 @@ impl CurvilinearPisoSolver {
         }
         let (_, matrix, anchor) = self.matrix.as_ref().expect("assembled");
         let flux_scale: f64 = field.flux.iter().map(|f| f.abs()).sum::<f64>().max(1e-300);
-        let tolerance = self.params.tolerance * flux_scale;
+        let floor = 1e-15 * flux_scale;
 
         let mut iterations = 0;
         let mut converged = true;
         let mut performed = 0;
         let mut max_div = self.max_divergence(&field.flux);
         for _ in 0..self.params.corrector_steps {
+            let incoming = self.divergence_l1(&field.flux);
+            if incoming <= floor {
+                break;
+            }
+            let tolerance = self.params.tolerance * incoming + floor;
             let (pc, out) = self.solve_pressure_correction(matrix, *anchor, &field.flux, tolerance);
             iterations += out.iterations;
             converged &= out.converged;
             self.apply_correction(field, &pc, dt);
             performed += 1;
             max_div = self.max_divergence(&field.flux);
-            if max_div <= tolerance {
-                break;
-            }
         }
         self.time = t_new;
         Ok(CurvilinearResult {

crates/specialized/rtx-cfd/src/solvers/incompressible/curvilinear/predictor.rs

@@ -64,21 +64,22 @@ impl CurvilinearPisoSolver {
                 let mut lu = ops.face_gradient_flux(mesh, f, &field.u, &un, bu);
                 let mut lv = ops.face_gradient_flux(mesh, f, &field.v, &vn, bv);
                 if implicit_n && is_n && (side.is_none() || bval.is_some()) {
+                    // The orthogonal part, `alpha (u_other − u_c)` into the
+                    // cell whichever side owns the face, goes implicit: take
+                    // it out of the explicit flux here, put it back in
+                    // `solve_lines`.
                     let alpha = ops.face(f).alpha;
                     let (pu, pv) = match (face.owner, face.neigh) {
                         (Some(p), Some(q)) => {
                             let other = if p == c { q } else { p };
                             (
-                                sign * alpha * (field.u[other] - field.u[c]),
+                                alpha * (field.u[other] - field.u[c]),
-                                sign * alpha * (field.v[other] - field.v[c]),
+                                alpha * (field.v[other] - field.v[c]),
                             )
                         }
                         _ => {
                             let b = bval.expect("dirichlet");
-                            (
+                            (alpha * (b.0 - field.u[c]), alpha * (b.1 - field.v[c]))
-                                sign * alpha * (b.0 - field.u[c]),
-                                sign * alpha * (b.1 - field.v[c]),
-                            )
                         }
                     };
                     lu = sign * lu - pu;

crates/specialized/rtx-cfd/src/solvers/incompressible/curvilinear/projection.rs

@@ -207,6 +207,20 @@ impl CurvilinearPisoSolver {
         }
     }
 
+    /// Total cell mass imbalance `Σ_c |Σ_f sign F_f|`.
+    pub(super) fn divergence_l1(&self, flux: &[f64]) -> f64 {
+        let mesh = &self.mesh;
+        (0..mesh.cell_count())
+            .map(|c| {
+                mesh.cell_faces(c)
+                    .iter()
+                    .map(|&(f, sign)| sign * flux[f])
+                    .sum::<f64>()
+                    .abs()
+            })
+            .sum()
+    }
+
     /// Largest cell mass imbalance `|Σ sign F_f|`.
     pub(super) fn max_divergence(&self, flux: &[f64]) -> f64 {
         let mesh = &self.mesh;

crates/specialized/rtx-cfd/src/solvers/incompressible/embedded.rs

@@ -38,7 +38,9 @@
 
 use super::ale::{AleBoundaries, SideBoundary};
 use super::embedded_body::{EmbeddedBody, EmbeddedMask, FaceKind};
-use super::poisson::{MultigridParameters, PoissonProblem, PoissonSolverKind, solve_multigrid_pcg};
+use super::poisson::{
+    MgPrecision, MultigridParameters, PoissonProblem, PoissonSolverKind, solve_multigrid_pcg,
+};
 use super::simple::ConvectionScheme;
 use super::{FlowField, SolverResult};
 use crate::{CfdConfig, CfdError, CfdResult};
@@ -61,6 +63,12 @@ pub struct EmbeddedParameters {
     /// [`PoissonSolverKind::Sor`]). Both solve the same system to the same
     /// true-residual stop; multigrid's cost is mesh-independent.
     pub poisson_solver: PoissonSolverKind,
+    /// Precision of the multigrid V-cycle (default [`MgPrecision::F64`] =
+    /// bit-identical). `F32` is the M1 precision probe of
+    /// `overset_metal_campaign.md` §3.2 / §5.2: the CG, its true residual
+    /// and the stop stay f64; only the preconditioner runs in single
+    /// precision. No effect with [`PoissonSolverKind::Sor`].
+    pub poisson_precision: MgPrecision,
     /// Convective face values in the explicit predictor (default
     /// [`ConvectionScheme::Upwind`], which is bit-identical to the fixed-grid
     /// PISO). The TVD schemes add SIMPLE's limited correction to each
@@ -82,6 +90,7 @@ impl Default for EmbeddedParameters {
             tolerance: 1e-6,
             boundaries: AleBoundaries::default(),
             poisson_solver: PoissonSolverKind::Sor,
+            poisson_precision: MgPrecision::F64,
             convection_scheme: ConvectionScheme::Upwind,
         }
     }
@@ -174,6 +183,12 @@ impl EmbeddedPisoSolver {
         })
     }
 
+    /// Precision of the multigrid V-cycle (see
+    /// [`EmbeddedParameters::poisson_precision`]); the M1 probe knob.
+    pub fn set_poisson_precision(&mut self, precision: MgPrecision) {
+        self.parameters.poisson_precision = precision;
+    }
+
     /// Mask hysteresis for the moving-body rebuild, as a fraction of the
     /// min cell size (default 0, exactly the plain rebuild). With a band,
     /// a cell within `band * h_min` of the surface keeps the
@@ -902,7 +917,10 @@ impl EmbeddedPisoSolver {
             let solution = solve_multigrid_pcg(
                 &problem,
                 &mut p_prime,
-                &MultigridParameters::default(),
+                &MultigridParameters {
+                    precision: self.parameters.poisson_precision,
+                    ..MultigridParameters::default()
+                },
                 inner_stop,
                 anchor_cell,
             );

crates/specialized/rtx-cfd/src/solvers/incompressible/mod.rs

@@ -57,7 +57,9 @@ pub use flow_field::FlowField;
 pub use piso::{PisoParameters, PisoResult, PisoSolver};
 #[cfg(feature = "cuda")]
 pub use piso_gpu::PisoGpuSolver;
-pub use poisson::{MultigridParameters, PoissonProblem, PoissonSolution, PoissonSolverKind};
+pub use poisson::{
+    MgPrecision, MultigridParameters, PoissonProblem, PoissonSolution, PoissonSolverKind,
+};
 pub use polygon_sdf::PolygonSdf;
 pub use simple::{ConvectionScheme, SimpleParameters, SimpleResult, SimpleSolver};
 #[cfg(feature = "cuda")]

crates/specialized/rtx-cfd/src/solvers/incompressible/piso.rs

@@ -37,7 +37,9 @@
 //! - Convective face fluxes fell back to the centre value at the sweep edges
 //!   instead of using the prescribed boundary faces that exist there.
 
-use super::poisson::{MultigridParameters, PoissonProblem, PoissonSolverKind, solve_multigrid_pcg};
+use super::poisson::{
+    MgPrecision, MultigridParameters, PoissonProblem, PoissonSolverKind, solve_multigrid_pcg,
+};
 use super::{BoundaryConditions, FlowField, IncompressibleSolver, SolverResult};
 use crate::{CfdConfig, CfdResult};
 use async_trait::async_trait;
@@ -59,6 +61,9 @@ pub struct PisoParameters {
     /// [`PoissonSolverKind::Sor`]). Both solve the same system to the same
     /// true-residual stop; multigrid's cost is mesh-independent.
     pub poisson_solver: PoissonSolverKind,
+    /// Precision of the multigrid V-cycle (default [`MgPrecision::F64`] =
+    /// bit-identical; `F32` = the M1 precision probe, no effect with SOR).
+    pub poisson_precision: MgPrecision,
 }
 
 impl Default for PisoParameters {
@@ -68,6 +73,7 @@ impl Default for PisoParameters {
             time_step: 0.001,
             tolerance: 1e-6,
             poisson_solver: PoissonSolverKind::Sor,
+            poisson_precision: MgPrecision::F64,
         }
     }
 }
@@ -398,7 +404,10 @@ impl PisoSolver {
             let solution = solve_multigrid_pcg(
                 &problem,
                 &mut p_prime,
-                &MultigridParameters::default(),
+                &MultigridParameters {
+                    precision: self.parameters.poisson_precision,
+                    ..MultigridParameters::default()
+                },
                 inner_stop,
                 Some(nx + 1),
             );

crates/specialized/rtx-cfd/src/solvers/incompressible/poisson.rs

@@ -252,9 +252,27 @@ impl PoissonProblem {
     }
 }
 
+/// Arithmetic precision of the multigrid PRECONDITIONER (the conjugate
+/// gradient around it, its true residual, the mean projections and the
+/// stopping rule stay `f64`). `F32` is the registered mixed-precision
+/// design of `overset_metal_campaign.md` §3.2 M1 / §5.2: it decides
+/// whether an fp32-only device (Apple Metal) can carry the pressure solve
+/// of a coupled march. Default `F64` = bit-identical to the historical
+/// solver.
+#[derive(Debug, Clone, Copy, PartialEq, Eq, Default)]
+pub enum MgPrecision {
+    /// Double precision throughout (the default; bit-identical).
+    #[default]
+    F64,
+    /// Single-precision V-cycle inside the double-precision CG.
+    F32,
+}
+
 /// Multigrid preconditioner parameters.
 #[derive(Debug, Clone)]
 pub struct MultigridParameters {
+    /// Precision of the V-cycle (see [`MgPrecision`]).
+    pub precision: MgPrecision,
     /// Symmetric Gauss–Seidel sweeps before AND after the coarse correction
     /// (default 2; a value of 0 is treated as 1). One count for both on
     /// purpose: unequal pre/post counts make the V-cycle non-symmetric and
@@ -271,6 +289,7 @@ pub struct MultigridParameters {
 impl Default for MultigridParameters {
     fn default() -> Self {
         Self {
+            precision: MgPrecision::F64,
             smoother_sweeps: 2,
             coarsest_cells: 32,
             max_iterations: 500,
@@ -335,14 +354,77 @@ const COARSE_CORRECTION: f64 = 2.0;
 /// Hard cap on the number of levels (a 1×1 coarsest is reached long before).
 const MAX_LEVELS: usize = 64;
 
-/// One level of the hierarchy: the problem, its diagonal, the list of
+/// The scalar the V-cycle runs in. `f64` reproduces the historical solver
-/// active cells that carry an equation (row-major), the parent map into the
+/// bit for bit (same operations in the same order on the same values);
-/// next coarser level, and the V-cycle work vectors.
+/// `f32` is the mixed-precision probe.
-struct Level {
+pub trait MgScalar:
+    Copy
+    + PartialEq
+    + PartialOrd
+    + std::ops::Add<Output = Self>
+    + std::ops::Sub<Output = Self>
+    + std::ops::Mul<Output = Self>
+    + std::ops::Div<Output = Self>
+    + std::ops::AddAssign
+    + Send
+    + Sync
+    + 'static
+{
+    /// Zero.
+    const ZERO: Self;
+    /// From double.
+    fn from_f64(v: f64) -> Self;
+    /// To double.
+    fn to_f64(self) -> f64;
+    /// Absolute value.
+    fn abs(self) -> Self;
+}
+
+impl MgScalar for f64 {
+    const ZERO: Self = 0.0;
+    #[inline]
+    fn from_f64(v: f64) -> Self {
+        v
+    }
+    #[inline]
+    fn to_f64(self) -> f64 {
+        self
+    }
+    #[inline]
+    fn abs(self) -> Self {
+        f64::abs(self)
+    }
+}
+
+impl MgScalar for f32 {
+    const ZERO: Self = 0.0;
+    #[inline]
+    fn from_f64(v: f64) -> Self {
+        v as f32
+    }
+    #[inline]
+    fn to_f64(self) -> f64 {
+        f64::from(self)
+    }
+    #[inline]
+    fn abs(self) -> Self {
+        f32::abs(self)
+    }
+}
+
+/// One level of the hierarchy: the problem (kept in `f64` for coarsening
+/// and inspection), its coefficients and diagonal in the V-cycle scalar,
+/// the list of active cells that carry an equation (row-major), and the
+/// parent map into the next coarser level.
+struct Level<T: MgScalar> {
     problem: PoissonProblem,
     /// `problem.active && ap > 0`: cells with an equation.
     active: Vec<bool>,
-    ap: Vec<f64>,
+    ae: Vec<T>,
+    aw: Vec<T>,
+    an: Vec<T>,
+    as_: Vec<T>,
+    ap: Vec<T>,
     /// Row-major indices of the active cells.
     cells: Vec<usize>,
     /// Fine index → coarse index (empty on the coarsest level).
@@ -350,26 +432,26 @@ struct Level {
 }
 
 /// V-cycle work vectors of one level.
-struct Work {
+struct Work<T: MgScalar> {
     /// Right-hand side of the residual equation on this level.
-    b: Vec<f64>,
+    b: Vec<T>,
     /// Correction on this level.
-    x: Vec<f64>,
+    x: Vec<T>,
     /// Residual.
-    r: Vec<f64>,
+    r: Vec<T>,
 }
 
-impl Work {
+impl<T: MgScalar> Work<T> {
     fn new(n: usize) -> Self {
         Self {
-            b: vec![0.0; n],
+            b: vec![T::ZERO; n],
-            x: vec![0.0; n],
+            x: vec![T::ZERO; n],
-            r: vec![0.0; n],
+            r: vec![T::ZERO; n],
         }
     }
 }
 
-impl Level {
+impl<T: MgScalar> Level<T> {
     fn new(mut problem: PoissonProblem) -> Self {
         let (nx, ny) = (problem.nx, problem.ny);
         let n = nx * ny;
@@ -405,10 +487,15 @@ impl Level {
             }
         }
         let cells: Vec<usize> = (0..n).filter(|&idx| active[idx]).collect();
+        let cast = |v: &[f64]| v.iter().map(|&x| T::from_f64(x)).collect::<Vec<T>>();
         Self {
+            ae: cast(&problem.ae),
+            aw: cast(&problem.aw),
+            an: cast(&problem.an),
+            as_: cast(&problem.as_),
+            ap: cast(&ap),
             problem,
             active,
-            ap,
             cells,
             coarse_of: Vec::new(),
         }
@@ -418,38 +505,38 @@ impl Level {
     /// read (their coefficients are the only non-zero ones after
     /// sanitising).
     #[inline]
-    fn neighbour_sum(&self, x: &[f64], idx: usize) -> f64 {
+    fn neighbour_sum(&self, x: &[T], idx: usize) -> T {
         let nx = self.problem.nx;
-        let mut s = 0.0;
+        let mut s = T::ZERO;
-        let ae = self.problem.ae[idx];
+        let ae = self.ae[idx];
-        if ae != 0.0 {
+        if ae != T::ZERO {
             s += ae * x[idx + 1];
         }
-        let aw = self.problem.aw[idx];
+        let aw = self.aw[idx];
-        if aw != 0.0 {
+        if aw != T::ZERO {
             s += aw * x[idx - 1];
         }
-        let an = self.problem.an[idx];
+        let an = self.an[idx];
-        if an != 0.0 {
+        if an != T::ZERO {
             s += an * x[idx + nx];
         }
-        let as_ = self.problem.as_[idx];
+        let as_ = self.as_[idx];
-        if as_ != 0.0 {
+        if as_ != T::ZERO {
             s += as_ * x[idx - nx];
         }
         s
     }
 
     /// `y = A x` on the active cells.
-    fn apply(&self, x: &[f64], y: &mut [f64]) {
+    fn apply(&self, x: &[T], y: &mut [T]) {
         for &idx in &self.cells {
             y[idx] = self.ap[idx] * x[idx] - self.neighbour_sum(x, idx);
         }
     }
 
     /// `r = b - A x` on the active cells; returns its L1 norm.
-    fn residual(&self, b: &[f64], x: &[f64], r: &mut [f64]) -> f64 {
+    fn residual(&self, b: &[T], x: &[T], r: &mut [T]) -> T {
-        let mut l1 = 0.0;
+        let mut l1 = T::ZERO;
         for &idx in &self.cells {
             let v = b[idx] - (self.ap[idx] * x[idx] - self.neighbour_sum(x, idx));
             r[idx] = v;
@@ -459,7 +546,7 @@ impl Level {
     }
 
     /// One symmetric Gauss–Seidel sweep (forward then backward) on `A x = b`.
-    fn symmetric_gs(&self, b: &[f64], x: &mut [f64]) {
+    fn symmetric_gs(&self, b: &[T], x: &mut [T]) {
         for &idx in &self.cells {
             x[idx] = (b[idx] + self.neighbour_sum(x, idx)) / self.ap[idx];
         }
@@ -506,24 +593,24 @@ impl Level {
 }
 
 /// The multigrid hierarchy: level 0 is the fine problem.
-pub(crate) struct Hierarchy {
+pub(crate) struct Hierarchy<T: MgScalar = f64> {
-    levels: Vec<Level>,
+    levels: Vec<Level<T>>,
-    work: Vec<Work>,
+    work: Vec<Work<T>>,
     sweeps: usize,
 }
 
-impl Hierarchy {
+impl<T: MgScalar> Hierarchy<T> {
     /// Builds the hierarchy down to `coarsest_cells` active cells (or until
     /// coarsening stops reducing the count).
     pub(crate) fn build(problem: &PoissonProblem, params: &MultigridParameters) -> Self {
-        let mut levels = vec![Level::new(problem.clone())];
+        let mut levels = vec![Level::<T>::new(problem.clone())];
         while levels.len() < MAX_LEVELS {
             let fine = levels.last().expect("at least one level");
             if fine.cells.len() <= params.coarsest_cells.max(1) {
                 break;
             }
             let (coarse, coarse_of) = fine.coarsen();
-            let coarse = Level::new(coarse);
+            let coarse = Level::<T>::new(coarse);
             if coarse.cells.len() >= fine.cells.len() {
                 break;
             }
@@ -570,7 +657,7 @@ impl Hierarchy {
         let depth = self.levels.len();
         let levels = &self.levels;
         for &idx in &levels[0].cells {
-            self.work[0].b[idx] = r[idx];
+            self.work[0].b[idx] = T::from_f64(r[idx]);
         }
         // Down: smooth from zero, restrict the residual.
         for l in 0..depth - 1 {
@@ -578,14 +665,14 @@ impl Hierarchy {
             let (head, tail) = self.work.split_at_mut(l + 1);
             let (wf, wc) = (&mut head[l], &mut tail[0]);
             for &idx in &fine.cells {
-                wf.x[idx] = 0.0;
+                wf.x[idx] = T::ZERO;
             }
             for _ in 0..self.sweeps {
                 fine.symmetric_gs(&wf.b, &mut wf.x);
             }
             fine.residual(&wf.b, &wf.x, &mut wf.r);
             for &idx in &coarse.cells {
-                wc.b[idx] = 0.0;
+                wc.b[idx] = T::ZERO;
             }
             for &idx in &fine.cells {
                 let c = fine.coarse_of[idx];
@@ -601,7 +688,7 @@ impl Hierarchy {
             let bottom = &levels[depth - 1];
             let wb = &mut self.work[depth - 1];
             for &idx in &bottom.cells {
-                wb.x[idx] = 0.0;
+                wb.x[idx] = T::ZERO;
             }
             for _ in 0..COARSEST_SWEEPS {
                 bottom.symmetric_gs(&wb.b, &mut wb.x);
@@ -613,14 +700,14 @@ impl Hierarchy {
             let (head, tail) = self.work.split_at_mut(l + 1);
             let (wf, wc) = (&mut head[l], &tail[0]);
             for &idx in &fine.cells {
-                wf.x[idx] += COARSE_CORRECTION * wc.x[fine.coarse_of[idx]];
+                wf.x[idx] += T::from_f64(COARSE_CORRECTION) * wc.x[fine.coarse_of[idx]];
             }
             for _ in 0..self.sweeps {
                 fine.symmetric_gs(&wf.b, &mut wf.x);
             }
         }
         for &idx in &levels[0].cells {
-            z[idx] = self.work[0].x[idx];
+            z[idx] = self.work[0].x[idx].to_f64();
         }
     }
 }
@@ -649,6 +736,23 @@ pub fn solve_multigrid_pcg(
     params: &MultigridParameters,
     tolerance: f64,
     anchor: Option<usize>,
+) -> PoissonSolution {
+    match params.precision {
+        MgPrecision::F64 => solve_pcg_with::<f64>(problem, p, params, tolerance, anchor),
+        MgPrecision::F32 => solve_pcg_with::<f32>(problem, p, params, tolerance, anchor),
+    }
+}
+
+/// The CG driver: `f64` throughout, with the V-cycle in `T`. The fine
+/// level used for the CG's own products and true residual is a separate
+/// `f64` level, so the precision of the preconditioner never enters the
+/// stopping rule or the reported residual.
+fn solve_pcg_with<T: MgScalar>(
+    problem: &PoissonProblem,
+    p: &mut [f64],
+    params: &MultigridParameters,
+    tolerance: f64,
+    anchor: Option<usize>,
 ) -> PoissonSolution {
     let n = problem.nx * problem.ny;
     assert_eq!(p.len(), n, "p must have nx*ny entries");
@@ -658,8 +762,9 @@ pub fn solve_multigrid_pcg(
         problem.validate()
     );
 
-    let mut hier = Hierarchy::build(problem, params);
+    let mut hier = Hierarchy::<T>::build(problem, params);
-    let cells: Vec<usize> = hier.levels[0].cells.clone();
+    let fine = Level::<f64>::new(problem.clone());
+    let cells: Vec<usize> = fine.cells.clone();
     let active_n = cells.len();
     if active_n == 0 {
         return PoissonSolution {
@@ -710,9 +815,9 @@ pub fn solve_multigrid_pcg(
     let mut q = vec![0.0; n];
 
     let true_residual =
-        |p: &[f64], r: &mut [f64], hier: &Hierarchy| -> f64 { hier.levels[0].residual(&b, p, r) };
+        |p: &[f64], r: &mut [f64], fine: &Level<f64>| -> f64 { fine.residual(&b, p, r) };
 
-    let anchor = anchor.filter(|&a| a < n && hier.levels[0].active[a]);
+    let anchor = anchor.filter(|&a| a < n && fine.active[a]);
     let finish = |p: &mut [f64], iterations: usize, residual: f64| {
         // Level of each singular component: the anchor's component is
         // shifted so p[anchor] == 0, every other singular component to mean
@@ -737,7 +842,7 @@ pub fn solve_multigrid_pcg(
         }
     };
 
-    let mut res = true_residual(p, &mut r, &hier);
+    let mut res = true_residual(p, &mut r, &fine);
     if res < tolerance {
         return finish(p, 0, res);
     }
@@ -759,12 +864,12 @@ pub fn solve_multigrid_pcg(
     let mut last_true = res;
     while iterations < params.max_iterations {
         iterations += 1;
-        hier.levels[0].apply(&d, &mut q);
+        fine.apply(&d, &mut q);
         let dq = dot(&d, &q);
         if !dq.is_finite() || dq <= 0.0 || !rz.is_finite() || rz <= 0.0 {
             // Breakdown (r = 0 to rounding, or a non-positive curvature
             // from rounding in the null space): stop on the true residual.
-            res = true_residual(p, &mut r, &hier);
+            res = true_residual(p, &mut r, &fine);
             return finish(p, iterations, res);
         }
         let alpha = rz / dq;
@@ -775,7 +880,7 @@ pub fn solve_multigrid_pcg(
         if l1(&r) < tolerance {
             // The recurrence residual passed: confirm against the TRUE
             // residual, and resynchronise if rounding has let them drift.
-            res = true_residual(p, &mut r, &hier);
+            res = true_residual(p, &mut r, &fine);
             if res < tolerance || res > 0.9 * last_true {
                 return finish(p, iterations, res);
             }
@@ -792,7 +897,7 @@ pub fn solve_multigrid_pcg(
             d[idx] = z[idx] + beta * d[idx];
         }
     }
-    res = true_residual(p, &mut r, &hier);
+    res = true_residual(p, &mut r, &fine);
     finish(p, iterations, res)
 }
 

crates/specialized/rtx-cfd/src/solvers/incompressible/poisson/tests.rs

@@ -446,7 +446,7 @@ fn galerkin_coarse_operator_matches_r_a_p() {
         }
     }
     pr.validate().expect("weighted problem is valid");
-    let hier = Hierarchy::build(&pr, &MultigridParameters::default());
+    let hier = Hierarchy::<f64>::build(&pr, &MultigridParameters::default());
     assert!(hier.depth() >= 3, "depth {}", hier.depth());
     let mut g = Lcg(5);
     for l in 0..hier.depth() - 1 {
@@ -570,7 +570,7 @@ fn v_cycle_preconditioner_is_symmetric() {
         ),
         ("dirichlet", assemble(n, n, |_, _| true, [true; 4])),
     ] {
-        let mut hier = Hierarchy::build(&pr, &params);
+        let mut hier = Hierarchy::<f64>::build(&pr, &params);
         let mut g = Lcg(11);
         let cells = active_indices(&pr);
         let mut x = vec![0.0; n * n];

crates/specialized/rtx-cfd/tests/curvilinear_mms.rs

@@ -69,7 +69,7 @@ async fn march(
             .unwrap_or(d)
     };
     let dt = env("RTX_CURV_DTFRAC", 0.4) * (h * h / (4.0 * nu)).min(h);
-    let tol = env("RTX_CURV_TOL", 1e-10);
+    let tol = env("RTX_CURV_TOL", 1e-5);
     let config = CfdConfig::new()
         .with_density(RHO)
         .with_viscosity(MU)

crates/specialized/rtx-cfd/tests/curvilinear_poiseuille.rs

@@ -61,7 +61,7 @@ async fn steady(mesh: PatchMesh, diffusion: NormalDiffusion, dt_factor: f64) ->
     let dt = dt_factor * 0.4 * h * h / (4.0 * MU);
     let config = CfdConfig::new().with_density(1.0).with_viscosity(MU);
     let params = CurvilinearParameters {
-        tolerance: 1e-11,
+        tolerance: 1e-6,
         normal_diffusion: diffusion,
         ..CurvilinearParameters::default()
     };
@@ -72,9 +72,20 @@ async fn steady(mesh: PatchMesh, diffusion: NormalDiffusion, dt_factor: f64) ->
     solver.initialize(&mut field, |_, _| (0.0, 0.0));
     // Mass defect at the steady state, relative to the largest face flux
     // (during the transient it is the pressure solver's residual).
+    let env = |k: &str, d: f64| {
+        std::env::var(k)
+            .ok()
+            .and_then(|v| v.parse().ok())
+            .unwrap_or(d)
+    };
+    // 1e-9: the rounding floor of |du/dt| sits at ~2e-11 on the 32² channels
+    // (measured: 0 pressure iterations, divergence 1e-16) and the gates are
+    // at 1e-8.
+    let steady_tol = env("RTX_CURV_STEADY", 1e-9);
+    let trace = std::env::var("RTX_CURV_TRACE").is_ok();
     let mut worst_mass = 0.0_f64;
     let mut converged = false;
-    for _ in 0..2_000_000 {
+    for step in 0..2_000_000 {
         let before = field.u.clone();
         let r = solver.advance(&mut field, dt).await?;
         assert!(r.poisson_converged, "{r:?}");
@@ -86,7 +97,16 @@ async fn steady(mesh: PatchMesh, diffusion: NormalDiffusion, dt_factor: f64) ->
             .zip(&before)
             .map(|(a, b)| (a - b).abs())
             .fold(0.0, f64::max);
-        if change / dt < 1e-11 {
+        if trace && step % 50_000 == 0 {
+            println!(
+                "  step {step} t={:.2} |du/dt| {:.3e} iters {} div {:.2e}",
+                solver.time(),
+                change / dt,
+                r.poisson_iterations,
+                r.max_divergence
+            );
+        }
+        if change / dt < steady_tol {
             converged = true;
             break;
         }
@@ -155,9 +175,15 @@ async fn cartesian_and_sheared_channels_hit_the_discrete_profile_exactly() -> Cf
 async fn varying_skew_channel_converges_to_the_parabola_at_second_order() -> CfdResult<()> {
     let mut errs = Vec::new();
     let mut vs = Vec::new();
+    let only: Option<usize> = std::env::var("RTX_CURV_N")
+        .ok()
+        .and_then(|v| v.parse().ok());
     for n in [16usize, 32, 64] {
+        if only.is_some_and(|o| o != n) {
+            continue;
+        }
         let s = steady(
-            channel_varying_skew(1.0, 1.0, n, n, 0.3, 2.0, true)?,
+            channel_varying_skew(1.0, 1.0, n, n, 0.1, 2.0, true)?,
             NormalDiffusion::Explicit,
             1.0,
         )
@@ -179,6 +205,8 @@ async fn varying_skew_channel_converges_to_the_parabola_at_second_order() -> Cfd
     let o: Vec<f64> = errs.windows(2).map(|p| (p[0] / p[1]).log2()).collect();
     let ov: Vec<f64> = vs.windows(2).map(|p| (p[0] / p[1]).log2()).collect();
     println!("varying skew orders u {o:?}, v {ov:?}");
-    assert!(o.iter().all(|&x| x >= 1.8), "orders {o:?}");
+    if only.is_none() {
+        assert!(o.iter().all(|&x| x >= 1.8), "orders {o:?}");
+    }
     Ok(())
 }

crates/specialized/rtx-cfd/tests/embedded_fresh_cell_falsifier.rs

@@ -140,6 +140,7 @@ async fn run(moving: bool, dt: f64, t_end: f64) -> CfdResult<Vec<Record>> {
             corrector_steps: 2,
             tolerance: 1e-8,
             poisson_solver: PoissonSolverKind::Multigrid,
+            poisson_precision: rtx_cfd::solvers::incompressible::MgPrecision::F64,
             ..EmbeddedParameters::default()
         },
     )?;

crates/specialized/rtx-cfd/tests/embedded_moving.rs

@@ -80,6 +80,7 @@ fn solver(n: usize) -> CfdResult<EmbeddedPisoSolver> {
             corrector_steps: 2,
             tolerance: 1e-8,
             poisson_solver: PoissonSolverKind::Multigrid,
+            poisson_precision: rtx_cfd::solvers::incompressible::MgPrecision::F64,
             ..EmbeddedParameters::default()
         },
     )?;

crates/specialized/rtx-cfd/tests/poisson_equivalence.rs

@@ -30,8 +30,8 @@
 
 use rtx_cfd::solvers::incompressible::{
     AleBoundaries, BoundaryConditions, EmbeddedBody, EmbeddedParameters, EmbeddedPisoSolver,
-    FaceKind, FlowField, IncompressibleSolver, PisoParameters, PisoSolver, PoissonSolverKind,
+    FaceKind, FlowField, IncompressibleSolver, MgPrecision, PisoParameters, PisoSolver,
-    SideBoundary,
+    PoissonSolverKind, SideBoundary,
 };
 use rtx_cfd::{CfdConfig, CfdResult};
 use std::f64::consts::PI;
@@ -128,6 +128,7 @@ async fn piso_mms(n: usize, kind: PoissonSolverKind) -> CfdResult<SteadyMeasurem
             time_step: dt,
             tolerance: 1e-8,
             poisson_solver: kind,
+            poisson_precision: rtx_cfd::solvers::incompressible::MgPrecision::F64,
         },
     )?;
     solver.set_momentum_source(source);
@@ -291,6 +292,7 @@ async fn taylor_green(n: usize, kind: PoissonSolverKind) -> CfdResult<TaylorGree
             time_step: dt,
             tolerance: 1e-9,
             poisson_solver: kind,
+            poisson_precision: rtx_cfd::solvers::incompressible::MgPrecision::F64,
         },
     )?;
 
@@ -418,6 +420,16 @@ fn mms_initial_field(n: usize) -> CfdResult<FlowField> {
 /// `tests/embedded_mms.rs::measure`, velocity error and divergence only,
 /// with the inner solver selectable.
 async fn embedded_mms(n: usize, kind: PoissonSolverKind) -> CfdResult<SteadyMeasurement> {
+    embedded_mms_with(n, kind, MgPrecision::F64).await
+}
+
+/// `embedded_mms` with the multigrid V-cycle precision selectable (the M1
+/// precision probe of `overset_metal_campaign.md` §3.2).
+async fn embedded_mms_with(
+    n: usize,
+    kind: PoissonSolverKind,
+    precision: MgPrecision,
+) -> CfdResult<SteadyMeasurement> {
     let dx = 1.0 / n as f64;
     let dt = mms_time_step(n);
     let mut solver = EmbeddedPisoSolver::new(
@@ -426,6 +438,7 @@ async fn embedded_mms(n: usize, kind: PoissonSolverKind) -> CfdResult<SteadyMeas
             corrector_steps: 2,
             tolerance: 1e-8,
             poisson_solver: kind,
+            poisson_precision: precision,
             ..EmbeddedParameters::default()
         },
     )?;
@@ -526,6 +539,44 @@ async fn embedded_circle_steady_state_is_solver_independent() -> CfdResult<()> {
 /// solver assembles exactly the fixed-grid PISO's `PoissonProblem` (all
 /// cells active, anchor `(1, 1)`, no outlet), so the fields must still agree
 /// to the bit.
+/// M1 precision probe (`overset_metal_campaign.md` §3.2 M1, §5.2): with
+/// the V-cycle in single precision inside the f64 conjugate gradient, the
+/// embedded-circle steady state must be the same field to the projection
+/// stop — the f64 CG owns the true residual, so the preconditioner's
+/// precision may cost iterations, never accuracy. The measured gap is
+/// printed; the assert is at the tolerance scale, not at f32 eps.
+#[tokio::test]
+async fn embedded_circle_steady_state_survives_an_f32_vcycle() -> CfdResult<()> {
+    let n = 32;
+    let f64_arm = embedded_mms_with(n, PoissonSolverKind::Multigrid, MgPrecision::F64).await?;
+    let f32_arm = embedded_mms_with(n, PoissonSolverKind::Multigrid, MgPrecision::F32).await?;
+    println!(
+        "f32 V-cycle vs f64: L2 error {:.6e} vs {:.6e} (rel {:.2e}), max div {:.2e} vs {:.2e}, \
+         steps {} vs {}, wall {:.2} s vs {:.2} s",
+        f32_arm.l2_velocity,
+        f64_arm.l2_velocity,
+        rel(f32_arm.l2_velocity, f64_arm.l2_velocity),
+        f32_arm.max_div,
+        f64_arm.max_div,
+        f32_arm.steps,
+        f64_arm.steps,
+        f32_arm.seconds,
+        f64_arm.seconds
+    );
+    assert!(
+        rel(f32_arm.l2_velocity, f64_arm.l2_velocity) < 1e-5,
+        "f32 V-cycle changed the steady error: {:.6e} vs {:.6e}",
+        f32_arm.l2_velocity,
+        f64_arm.l2_velocity
+    );
+    assert!(
+        f32_arm.max_div < 1e-5,
+        "divergence with the f32 V-cycle {:.3e}",
+        f32_arm.max_div
+    );
+    Ok(())
+}
+
 #[tokio::test]
 async fn without_a_body_the_embedded_solver_is_piso_to_the_bit_with_multigrid() -> CfdResult<()> {
     let n = 16;
@@ -537,6 +588,7 @@ async fn without_a_body_the_embedded_solver_is_piso_to_the_bit_with_multigrid()
             time_step: dt,
             tolerance: 1e-8,
             poisson_solver: PoissonSolverKind::Multigrid,
+            poisson_precision: rtx_cfd::solvers::incompressible::MgPrecision::F64,
         },
     )?;
     piso.set_momentum_source(source);
@@ -547,6 +599,7 @@ async fn without_a_body_the_embedded_solver_is_piso_to_the_bit_with_multigrid()
             corrector_steps: 2,
             tolerance: 1e-8,
             poisson_solver: PoissonSolverKind::Multigrid,
+            poisson_precision: rtx_cfd::solvers::incompressible::MgPrecision::F64,
             ..EmbeddedParameters::default()
         },
     )?;
@@ -611,6 +664,7 @@ async fn channel_with_circle(kind: PoissonSolverKind) -> CfdResult<(FlowField, u
                 top: SideBoundary::Velocity,
             },
             poisson_solver: kind,
+            poisson_precision: rtx_cfd::solvers::incompressible::MgPrecision::F64,
             ..EmbeddedParameters::default()
         },
     )?;

crates/specialized/rtx-cfd/tests/turek_hron_cfd.rs

@@ -119,6 +119,7 @@ async fn run_cfd1(ny: usize) -> CfdResult<Cfd1> {
             top: SideBoundary::Velocity,
         },
         poisson_solver: PoissonSolverKind::Multigrid,
+        poisson_precision: rtx_cfd::solvers::incompressible::MgPrecision::F64,
         ..EmbeddedParameters::default()
     };
     let mut solver = EmbeddedPisoSolver::new(config, params)?;

crates/specialized/rtx-cfd/tests/turek_hron_cfd23.rs

@@ -126,6 +126,7 @@ impl Runner {
                 top: SideBoundary::Velocity,
             },
             poisson_solver: PoissonSolverKind::Multigrid,
+            poisson_precision: rtx_cfd::solvers::incompressible::MgPrecision::F64,
             // Upwind's numerical viscosity (|u| h / 2 ~ 10x the physical nu
             // on these grids) suppressed CFD3's vortex shedding entirely:
             // the ny = 41 upwind run produced ONE lift zero-crossing in

crates/specialized/rtx-fsi/tests/fsi2_harness/march.rs

@@ -122,6 +122,10 @@ pub struct MarchConfig {
     /// is repeated as 2, 4, 8, 16, 32 coupled substeps of dt/n — see
     /// `rescue.rs` and omni-cortex `docs/coupling_rescue_campaign.md`.
     pub coupling_rescue: bool,
+    /// M1 precision probe (`RTX_{prefix}_POISSON_F32`, default off =
+    /// bit-identical): the pressure multigrid's V-cycle in single
+    /// precision inside the f64 CG (`overset_metal_campaign.md` §3.2 M1).
+    pub poisson_f32: bool,
     /// Rung C (`RTX_{prefix}_CRESCUE_COARSE`, default 0 = off; needs
     /// `coupling_rescue`): on a trigger, instead of the substep ladder,
     /// reject the step and enter a coarse EPISODE of this many coupled
@@ -208,6 +212,7 @@ impl MarchConfig {
                 .and_then(|v| v.parse::<usize>().ok())
                 .unwrap_or(defaults.trace_from),
             coupling_rescue: num("CRESCUE", f64::from(u8::from(defaults.coupling_rescue))) != 0.0,
+            poisson_f32: num("POISSON_F32", f64::from(u8::from(defaults.poisson_f32))) != 0.0,
             coarse_episode: num("CRESCUE_COARSE", defaults.coarse_episode as f64) as usize,
             inc_trace: std::env::var(key("INCTRACE")).ok().or(defaults.inc_trace),
             increment_factor: num("CRESCUE_INC", defaults.increment_factor),
@@ -363,6 +368,7 @@ pub fn run_march(case: BenchmarkCase, config: &MarchConfig) -> MarchResult {
         trace_steps,
         trace_from,
         coupling_rescue,
+        poisson_f32,
         coarse_episode,
         ref inc_trace,
         increment_factor,
@@ -374,6 +380,10 @@ pub fn run_march(case: BenchmarkCase, config: &MarchConfig) -> MarchResult {
 
     let (harness, mut solver, mut field) = Fsi2Harness::build_case(case, ny, flag_nx, smooth_in_h);
     solver.set_mask_hysteresis(mask_hysteresis);
+    if poisson_f32 {
+        solver.set_poisson_precision(rtx_cfd::solvers::incompressible::MgPrecision::F32);
+        println!("  poisson V-cycle precision: F32 (M1 probe)");
+    }
     let dt_fluid = harness.dt_fluid;
     let dt = dt_fluid * subcycle as f64;
     let interface = &harness.interface;

crates/specialized/rtx-fsi/tests/fsi2_harness/mod.rs

@@ -438,6 +438,7 @@ impl Fsi2Harness {
                 top: SideBoundary::Velocity,
             },
             poisson_solver: PoissonSolverKind::Multigrid,
+            poisson_precision: rtx_cfd::solvers::incompressible::MgPrecision::F64,
             // TVD, deliberately: FSI2 marches in time and needs the
             // shedding physics upwind's numerical viscosity killed on
             // these grids (CFD3's finding). The limiter chatter that

crates/specialized/rtx-fsi/tests/fsi2_interface_noise.rs

@@ -99,6 +99,13 @@ fn fsi2_interface_noise_floor() {
         _ => fsi2_harness::FSI2,
     };
     let (mut harness, mut solver, mut field) = Fsi2Harness::build_case(case, ny, flag_nx, 0.0);
+    // M1 precision probe (`RTX_FSI2_POISSON_F32=1`): the pressure
+    // multigrid's V-cycle in f32 inside the f64 CG. Printed so the arm can
+    // never pass vacuously.
+    if env_or("RTX_FSI2_POISSON_F32", 0.0) != 0.0 {
+        solver.set_poisson_precision(rtx_cfd::solvers::incompressible::MgPrecision::F32);
+        println!("  poisson V-cycle precision: F32 (M1 probe)");
+    }
     let dt_fluid = harness.dt_fluid;
 
     // Rigid march to operating loads (the floor rides with the loads —

crates/specialized/rtx-fsi/tests/turek_hron_fsi1.rs

@@ -323,6 +323,7 @@ fn fsi1_coupled_cylinder_and_flag() {
             top: SideBoundary::Velocity,
         },
         poisson_solver: PoissonSolverKind::Multigrid,
+        poisson_precision: rtx_cfd::solvers::incompressible::MgPrecision::F64,
         // Upwind, deliberately: FSI1 is a steady FIXED-POINT problem, and
         // the TVD limiter's switching keeps the steady load chattering by
         // ~0.5% (a known property of limited schemes — they stall short of

crates/specialized/rtx-fsi/tests/turek_hron_fsi2.rs

@@ -162,6 +162,7 @@ fn fsi2_flapping_flag() {
             trace_steps: 0,
             trace_from: usize::MAX,
             coupling_rescue: false,
+            poisson_f32: false,
             coarse_episode: 0,
             inc_trace: None,
             increment_factor: 0.0,

crates/specialized/rtx-fsi/tests/turek_hron_fsi3.rs

@@ -134,6 +134,7 @@ fn fsi3_added_mass_flag() {
             trace_steps: 0,
             trace_from: usize::MAX,
             coupling_rescue: false,
+            poisson_f32: false,
             coarse_episode: 0,
             inc_trace: None,
             increment_factor: 0.0,