embedded3 item 11: moving bodies (end-of-step mask, fresh-cell refill, space-time cut cell: step-averaged apertures, GCL wall flux, Reynolds-transport momentum), the 3D fresh-cell falsifier (plate / circle / stadium, wall + control-volume routes) and the Lipschitz sweep; ghost wall reproduces the 2D falsifier to the digit; cut wall 5–14× smoother on the circle, gates not met (fresh cell's first step); wall.rs split (impose.rs)

Co-Authored-By: Claude Fable 5.1 <[email protected]>

crates/specialized/rtx-cfd/tests/embedded3_embedded_mms.rs

@@ -11,255 +11,10 @@
 //! are at most the binary wall's at every n, its loads within 10 % at the
 //! finest rung.
 
-use rtx_cfd::solvers::incompressible::ConvectionScheme;
-use rtx_cfd::solvers::incompressible::embedded3::{
-    Body, FaceKind, Field, Fluid, Grid, Parameters, Solver, WallScheme,
-};
-use std::f64::consts::PI;
+mod embedded3_sphere;
 
-const RHO: f64 = 1.0;
-const MU: f64 = 0.05;
-const C: (f64, f64, f64) = (0.6, 0.45, 0.5);
-const R: f64 = 0.2;
-
-fn u3(x: f64, y: f64, z: f64) -> f64 {
-    (PI * x).sin() * (PI * y).cos() * (PI * z).cos()
-}
-fn v3(x: f64, y: f64, z: f64) -> f64 {
-    (PI * x).cos() * (PI * y).sin() * (PI * z).cos()
-}
-fn w3(x: f64, y: f64, z: f64) -> f64 {
-    -2.0 * (PI * x).cos() * (PI * y).cos() * (PI * z).sin()
-}
-fn p3(x: f64, y: f64, z: f64) -> f64 {
-    (PI * x).sin() * (PI * y).sin() * (PI * z).sin()
-}
-/// The velocity gradient ∂u_i/∂x_j and the pressure gradient.
-fn grads(x: f64, y: f64, z: f64) -> ([[f64; 3]; 3], [f64; 3]) {
-    let (sx, cx) = (PI * x).sin_cos();
-    let (sy, cy) = (PI * y).sin_cos();
-    let (sz, cz) = (PI * z).sin_cos();
-    (
-        [
-            [PI * cx * cy * cz, -PI * sx * sy * cz, -PI * sx * cy * sz],
-            [-PI * sx * sy * cz, PI * cx * cy * cz, -PI * cx * sy * sz],
-            [
-                2.0 * PI * sx * cy * sz,
-                2.0 * PI * cx * sy * sz,
-                -2.0 * PI * cx * cy * cz,
-            ],
-        ],
-        [PI * cx * sy * sz, PI * sx * cy * sz, PI * sx * sy * cz],
-    )
-}
-fn source3(x: f64, y: f64, z: f64) -> (f64, f64, f64) {
-    let (g, gp) = grads(x, y, z);
-    let u = [u3(x, y, z), v3(x, y, z), w3(x, y, z)];
-    let lap = -3.0 * PI * PI;
-    let conv = |i: usize| u[0] * g[i][0] + u[1] * g[i][1] + u[2] * g[i][2];
-    (
-        RHO * conv(0) + gp[0] - MU * lap * u[0],
-        RHO * conv(1) + gp[1] - MU * lap * u[1],
-        RHO * conv(2) + gp[2] - MU * lap * u[2],
-    )
-}
-fn boundary3(x: f64, y: f64, z: f64) -> (f64, f64, f64) {
-    let u = if x <= 0.0 || x >= 1.0 {
-        0.0
-    } else {
-        u3(x, y, z)
-    };
-    let v = if y <= 0.0 || y >= 1.0 {
-        0.0
-    } else {
-        v3(x, y, z)
-    };
-    let w = if z <= 0.0 || z >= 1.0 {
-        0.0
-    } else {
-        w3(x, y, z)
-    };
-    (u, v, w)
-}
-
-/// Exact force `∮ (−p I + μ(∇u + ∇uᵀ)) n dA` and momentum flux `∮ ρ u (u·n) dA`
-/// over the sphere by a fine Fibonacci quadrature.
-fn exact_force_and_flux() -> ([f64; 3], [f64; 3]) {
-    let n = 200_000;
-    let golden = PI * (3.0 - 5.0_f64.sqrt());
-    let (mut f, mut m) = ([0.0; 3], [0.0; 3]);
-    let da = 4.0 * PI * R * R / n as f64;
-    for k in 0..n {
-        let zz = 1.0 - 2.0 * (k as f64 + 0.5) / n as f64;
-        let rr = (1.0 - zz * zz).sqrt();
-        let th = golden * k as f64;
-        let nrm = [rr * th.cos(), rr * th.sin(), zz];
-        let (x, y, z) = (C.0 + R * nrm[0], C.1 + R * nrm[1], C.2 + R * nrm[2]);
-        let (g, _) = grads(x, y, z);
-        let p = p3(x, y, z);
-        let u = [u3(x, y, z), v3(x, y, z), w3(x, y, z)];
-        let un = u[0] * nrm[0] + u[1] * nrm[1] + u[2] * nrm[2];
-        for i in 0..3 {
-            let mut t = -p * nrm[i];
-            for j in 0..3 {
-                t += MU * (g[i][j] + g[j][i]) * nrm[j];
-            }
-            f[i] += t * da;
-            m[i] += RHO * u[i] * un * da;
-        }
-    }
-    (f, m)
-}
-
-struct Measurement {
-    l2_velocity: f64,
-    max_div: f64,
-    ghost_correction: f64,
-    force_surface: [f64; 3],
-    skipped: usize,
-    force_cv: [f64; 3],
-}
-
-fn measure(n: usize, scheme: WallScheme) -> Measurement {
-    let h = 1.0 / n as f64;
-    let dt = 0.4 * (h * h / (4.0 * MU / RHO)).min(h);
-    let mut solver = Solver::new(
-        Fluid {
-            density: RHO,
-            viscosity: MU,
-            reference_velocity: 1.0,
-            reference_length: 1.0,
-        },
-        Parameters {
-            corrector_steps: 2,
-            tolerance: 1e-8,
-            convection_scheme: ConvectionScheme::Upwind,
-            wall_scheme: scheme,
-            ..Parameters::default()
-        },
-    );
-    solver.set_momentum_source(|x, y, z, _t| source3(x, y, z));
-    solver.set_boundary_velocity(|x, y, z, _t| boundary3(x, y, z));
-    solver.set_body(
-        Body::sphere(|_t| C, R)
-            .with_surface_velocity(|x, y, z, _t| (u3(x, y, z), v3(x, y, z), w3(x, y, z))),
-    );
-    let g = Grid::cubic(n, n, n, h);
-    let mut f = Field::new(g);
-    solver.initialize(&mut f);
-    let mut last = solver.advance(&mut f, dt);
-    for _ in 0..200_000 {
-        let (bu, bv, bw) = (f.u.clone(), f.v.clone(), f.w.clone());
-        last = solver.advance(&mut f, dt);
-        let mut change = 0.0_f64;
-        for (a, b) in
-            f.u.iter()
-                .zip(&bu)
-                .chain(f.v.iter().zip(&bv))
-                .chain(f.w.iter().zip(&bw))
-        {
-            change = change.max((a - b).abs());
-        }
-        if change / dt < 1e-6 {
-            break;
-        }
-    }
-    let mask = solver.mask().expect("mask");
-    let (mut sq, mut vol) = (0.0, 0.0);
-    let dv = h * h * h;
-    for k in 0..n {
-        for j in 0..n {
-            for i in 1..n {
-                if mask.u_kind(g.uface(k, j, i)) == FaceKind::Fluid {
-                    let e = f.u[g.uface(k, j, i)]
-                        - u3(i as f64 * h, (j as f64 + 0.5) * h, (k as f64 + 0.5) * h);
-                    sq += e * e * dv;
-                    vol += dv;
-                }
-            }
-        }
-        for j in 1..n {
-            for i in 0..n {
-                if mask.v_kind(g.vface(k, j, i)) == FaceKind::Fluid {
-                    let e = f.v[g.vface(k, j, i)]
-                        - v3((i as f64 + 0.5) * h, j as f64 * h, (k as f64 + 0.5) * h);
-                    sq += e * e * dv;
-                    vol += dv;
-                }
-            }
-        }
-    }
-    for k in 1..n {
-        for j in 0..n {
-            for i in 0..n {
-                if mask.w_kind(g.wface(k, j, i)) == FaceKind::Fluid {
-                    let e = f.w[g.wface(k, j, i)]
-                        - w3((i as f64 + 0.5) * h, (j as f64 + 0.5) * h, k as f64 * h);
-                    sq += e * e * dv;
-                    vol += dv;
-                }
-            }
-        }
-    }
-    let body = solver.body().expect("body");
-    let t = solver.time();
-    // The apertured divergence per unit volume, the porous surface's flux
-    // through the wall included (the plain divergence on the binary wall).
-    let mut max_div = 0.0_f64;
-    let mut at_vol = 1.0;
-    let mut sum_flux = 0.0;
-    let (wall_fluxes, _) = mask.wall_flux_table(body, t);
-    for k in 0..n {
-        for j in 0..n {
-            for i in 0..n {
-                let idx = g.cell(k, j, i);
-                if mask.is_fluid_cell(idx) {
-                    let flux = (mask.a_u(g.uface(k, j, i + 1)) * f.u[g.uface(k, j, i + 1)]
-                        - mask.a_u(g.uface(k, j, i)) * f.u[g.uface(k, j, i)])
-                        * h
-                        * h
-                        + (mask.a_v(g.vface(k, j + 1, i)) * f.v[g.vface(k, j + 1, i)]
-                            - mask.a_v(g.vface(k, j, i)) * f.v[g.vface(k, j, i)])
-                            * h
-                            * h
-                        + (mask.a_w(g.wface(k + 1, j, i)) * f.w[g.wface(k + 1, j, i)]
-                            - mask.a_w(g.wface(k, j, i)) * f.w[g.wface(k, j, i)])
-                            * h
-                            * h
-                        + wall_fluxes[idx];
-                    sum_flux += flux.abs();
-                    if (flux / (h * h * h)).abs() > max_div {
-                        max_div = (flux / (h * h * h)).abs();
-                        at_vol = mask.vol(idx);
-                    }
-                }
-            }
-        }
-    }
-    println!(
-        "    [{scheme:?} n {n}] max div {max_div:.2e} in a cell of fluid fraction {at_vol:.3e}; Σ|flux| {sum_flux:.2e}; last step residual {:.2e}",
-        last.final_residual
-    );
-    let surface = match scheme {
-        WallScheme::GhostBinary => mask.surface_force(body, &f, MU, t, 0.5 * h),
-        WallScheme::CutCell => rtx_cfd::solvers::incompressible::embedded3::SurfaceForce {
-            f: mask.cut_wall_force(body, &f, MU, t).expect("cut wall"),
-            samples: 0,
-            skipped: 0,
-        },
-    };
-    let (i0, i1) = (n / 8, n - n / 8);
-    let src = |x: f64, y: f64, z: f64| source3(x, y, z);
-    let force_cv = mask.control_volume_force(&f, dt, RHO, MU, Some(&src), (i0, i1, i0, i1, i0, i1));
-    Measurement {
-        l2_velocity: (sq / vol).sqrt(),
-        max_div,
-        ghost_correction: solver.ghost_correction().abs(),
-        force_surface: surface.f,
-        skipped: surface.skipped,
-        force_cv,
-    }
-}
+use embedded3_sphere::{C, Measurement, exact_force_and_flux, measure};
+use rtx_cfd::solvers::incompressible::embedded3::WallScheme;
 
 fn norm(a: [f64; 3]) -> f64 {
     (a[0] * a[0] + a[1] * a[1] + a[2] * a[2]).sqrt()
@@ -273,13 +28,13 @@ struct Ladder {
 }
 
 fn ladder(resolutions: &[usize], scheme: WallScheme) -> Ladder {
-    let (fe, m) = exact_force_and_flux();
+    let (fe, m) = exact_force_and_flux(C);
     let f_scale = norm(fe);
     let fcv = [fe[0] - m[0], fe[1] - m[1], fe[2] - m[2]];
     println!(
         "  {scheme:?}: exact force {fe:.5?}; momentum flux {m:.5?}; the control-volume route measures {fcv:.5?}"
     );
-    let ms: Vec<Measurement> = resolutions.iter().map(|&n| measure(n, scheme)).collect();
+    let ms: Vec<Measurement> = resolutions.iter().map(|&n| measure(n, scheme, C)).collect();
     let errors: Vec<f64> = ms.iter().map(|x| x.l2_velocity).collect();
     let mut se = Vec::new();
     let mut ce = Vec::new();

crates/specialized/rtx-cfd/tests/embedded3_falsifier.rs

@@ -0,0 +1,566 @@
+//! embedded3 item 11a: the fresh-cell falsifier of the 2D track
+//! (`embedded_fresh_cell_falsifier.rs`, omni-cortex
+//! `docs/fresh_cell_gcl_campaign.md`) on the 3D solver, per unit span —
+//! the rigid Turek–Hron flag (0.35 × 0.02 m) extruded across a periodic
+//! slab, oscillating transversely in still fluid at the flag's tip speed
+//! (1 m/s peak, 80 mm amplitude) on h = 1/152 at dt = 3.24e-4. Per step:
+//! the load per unit span (the ghost wall's traction route over the
+//! plate's samples; the cut wall's operator route), a far-field pressure
+//! probe, the fluid's kinetic energy, the fresh-cell count.
+//!
+//! Registered gates (`docs/embedded3_campaign.md` item 11):
+//! - GhostBinary reproduces the 2D wall's impulse: energy per flipped
+//!   column within 30 % of the 2D 0.048 J/m per flipped cell, spike RMS
+//!   exponent in dt ≈ −1 (published −0.8 for the raw volume source);
+//! - CutCell: energy per fresh column ≥ 20× lower, max force spike < 5 %
+//!   of ½ρU²L, exponent ∈ [−0.3, 0.3].
+//!
+//! Default run: dt only, both schemes (minutes on the host);
+//! `RTX_E3_FALSIFIER_LADDER=1` runs dt, dt/2, dt/4 and fits the exponent
+//! (the gated variant is `#[ignore]`); `RTX_E3_FALSIFIER_NZ` sets the
+//! span in cells (default 4); `RTX_E3_FALSIFIER_CSV=<dir>` dumps records.
+
+use rtx_cfd::solvers::incompressible::ConvectionScheme;
+use rtx_cfd::solvers::incompressible::embedded3::{
+    Body, Boundaries, Field, Fluid, Grid, Parameters, Side, Solver, WallScheme,
+};
+use std::io::Write as _;
+
+const RHO: f64 = 1000.0;
+const MU: f64 = 1.0;
+const N: usize = 152;
+const DT_FSI2: f64 = 3.24e-4;
+const HX: f64 = 0.175;
+const HY: f64 = 0.01;
+const AMP: f64 = 0.08;
+const U_PEAK: f64 = 1.0;
+const CX: f64 = 0.5;
+const CY0: f64 = 0.5;
+/// The 2D wall's measured energy per flipped cell (J/m at U = 1, h = 1/152).
+const ENERGY_2D: f64 = 0.048;
+
+fn span_cells() -> usize {
+    std::env::var("RTX_E3_FALSIFIER_NZ")
+        .ok()
+        .and_then(|v| v.parse().ok())
+        .unwrap_or(4)
+}
+
+fn center_y(t: f64) -> f64 {
+    CY0 + AMP * (U_PEAK / AMP * t).sin()
+}
+
+fn center_v(t: f64) -> f64 {
+    U_PEAK * (U_PEAK / AMP * t).cos()
+}
+
+fn plate_sdf(x: f64, y: f64, yc: f64) -> f64 {
+    let qx = (x - CX).abs() - HX;
+    let qy = (y - yc).abs() - HY;
+    let outside = (qx.max(0.0).powi(2) + qy.max(0.0).powi(2)).sqrt();
+    outside + qx.max(qy).min(0.0)
+}
+
+const R_CIRCLE: f64 = 0.05;
+
+/// `RTX_E3_FALSIFIER_BODY=circle`: the 2D falsifier's smooth body (a
+/// cylinder of radius 0.05 across the span) instead of the plate.
+fn circle_body() -> bool {
+    std::env::var("RTX_E3_FALSIFIER_BODY").is_ok_and(|v| v == "circle")
+}
+
+/// `RTX_E3_FALSIFIER_BODY=stadium`: the plate with semicircular ends
+/// (radius `HY`): the same length and thickness, a smooth interface for
+/// the cut geometry's linear interpolant.
+fn stadium_body() -> bool {
+    std::env::var("RTX_E3_FALSIFIER_BODY").is_ok_and(|v| v == "stadium")
+}
+
+fn stadium_sdf(x: f64, y: f64, yc: f64) -> f64 {
+    let half = HX - HY;
+    let qx = (x - CX).abs().max(half) - half;
+    (qx * qx + (y - yc).powi(2)).sqrt() - HY
+}
+
+fn plate(moving: bool) -> Body {
+    let yc = move |t: f64| if moving { center_y(t) } else { CY0 };
+    let vc = move |t: f64| if moving { center_v(t) } else { 0.0 };
+    if circle_body() {
+        return Body::from_sdf(move |x, y, _z, t| {
+            ((x - CX).powi(2) + (y - yc(t)).powi(2)).sqrt() - R_CIRCLE
+        })
+        .with_surface_velocity(move |_, _, _, t| (0.0, vc(t), 0.0));
+    }
+    if stadium_body() {
+        return Body::from_sdf(move |x, y, _z, t| stadium_sdf(x, y, yc(t)))
+            .with_surface_velocity(move |_, _, _, t| (0.0, vc(t), 0.0));
+    }
+    Body::from_sdf(move |x, y, _z, t| plate_sdf(x, y, yc(t)))
+        .with_surface_velocity(move |_, _, _, t| (0.0, vc(t), 0.0))
+}
+
+/// The load scale `½ρU²L` of the body (its length across the motion).
+fn load_scale() -> f64 {
+    let l = if circle_body() {
+        2.0 * R_CIRCLE
+    } else {
+        2.0 * HX
+    };
+    0.5 * RHO * U_PEAK * U_PEAK * l
+}
+
+/// Surface samples of the plate at `t`: `(x, y, z, nx, ny, area)` over the
+/// four edges at spacing `ds` and `nz` z levels.
+fn samples(
+    t: f64,
+    moving: bool,
+    ds: f64,
+    nz: usize,
+    dz: f64,
+) -> Vec<(f64, f64, f64, f64, f64, f64)> {
+    let yc = if moving { center_y(t) } else { CY0 };
+    let mut out = Vec::new();
+    if circle_body() {
+        let n = ((2.0 * std::f64::consts::PI * R_CIRCLE / ds).ceil() as usize).max(8);
+        let dth = 2.0 * std::f64::consts::PI / n as f64;
+        for k in 0..n {
+            let th = (k as f64 + 0.5) * dth;
+            let (sn, cs) = th.sin_cos();
+            for kz in 0..nz {
+                out.push((
+                    CX + R_CIRCLE * cs,
+                    yc + R_CIRCLE * sn,
+                    (kz as f64 + 0.5) * dz,
+                    cs,
+                    sn,
+                    R_CIRCLE * dth * dz,
+                ));
+            }
+        }
+        return out;
+    }
+    if stadium_body() {
+        let half = HX - HY;
+        let n_flat = ((2.0 * half / ds).ceil() as usize).max(1);
+        for k in 0..n_flat {
+            let x = CX - half + (k as f64 + 0.5) / n_flat as f64 * 2.0 * half;
+            for kz in 0..nz {
+                let z = (kz as f64 + 0.5) * dz;
+                let a = 2.0 * half / n_flat as f64 * dz;
+                out.push((x, yc + HY, z, 0.0, 1.0, a));
+                out.push((x, yc - HY, z, 0.0, -1.0, a));
+            }
+        }
+        let n_arc = ((std::f64::consts::PI * HY / ds).ceil() as usize).max(4);
+        for (cx, sign) in [(CX + half, 1.0), (CX - half, -1.0)] {
+            for k in 0..n_arc {
+                let th = -std::f64::consts::FRAC_PI_2
+                    + (k as f64 + 0.5) / n_arc as f64 * std::f64::consts::PI;
+                let (sn, cs) = th.sin_cos();
+                let (nx, ny) = (sign * cs, sn);
+                for kz in 0..nz {
+                    out.push((
+                        cx + HY * nx,
+                        yc + HY * ny,
+                        (kz as f64 + 0.5) * dz,
+                        nx,
+                        ny,
+                        std::f64::consts::PI * HY / n_arc as f64 * dz,
+                    ));
+                }
+            }
+        }
+        return out;
+    }
+    let (x0, x1, y0, y1) = (CX - HX, CX + HX, yc - HY, yc + HY);
+    let mut edge = |ax: f64, ay: f64, bx: f64, by: f64, nx: f64, ny: f64| {
+        let len = ((bx - ax).powi(2) + (by - ay).powi(2)).sqrt();
+        let n = ((len / ds).ceil() as usize).max(1);
+        for k in 0..n {
+            let s = (k as f64 + 0.5) / n as f64;
+            for kz in 0..nz {
+                out.push((
+                    ax + s * (bx - ax),
+                    ay + s * (by - ay),
+                    (kz as f64 + 0.5) * dz,
+                    nx,
+                    ny,
+                    len / n as f64 * dz,
+                ));
+            }
+        }
+    };
+    edge(x0, y0, x1, y0, 0.0, -1.0);
+    edge(x1, y0, x1, y1, 1.0, 0.0);
+    edge(x1, y1, x0, y1, 0.0, 1.0);
+    edge(x0, y1, x0, y0, -1.0, 0.0);
+    out
+}
+
+struct Record {
+    t: f64,
+    /// Load per unit span (the scheme's wall route).
+    fy: f64,
+    /// Load per unit span by the control-volume route (a box of whole
+    /// cells around the body, reading no near-wall value).
+    fy_cv: f64,
+    fresh: usize,
+    skipped: usize,
+    p_far: f64,
+    /// Kinetic energy per unit span over the fluid cells.
+    ke: f64,
+}
+
+struct Run {
+    records: Vec<Record>,
+    /// The largest kinetic-energy change per step at a step with fresh
+    /// cells (after the impulsive start) over that step's flipped columns
+    /// (J/m) — the 2D falsifier's 2.604 J/m over 54 cells = 0.048.
+    energy_per_flip: f64,
+    seconds: f64,
+}
+
+fn run(scheme: WallScheme, moving: bool, dt: f64, t_end: f64) -> Run {
+    let nz = span_cells();
+    let h = 1.0 / N as f64;
+    let lz = nz as f64 * h;
+    let mut solver = Solver::new(
+        Fluid {
+            density: RHO,
+            viscosity: MU,
+            reference_velocity: 1.0,
+            reference_length: 2.0 * HY,
+        },
+        Parameters {
+            corrector_steps: 2,
+            tolerance: 1e-8,
+            convection_scheme: ConvectionScheme::Upwind,
+            wall_scheme: scheme,
+            boundaries: Boundaries {
+                z0: Side::Periodic,
+                z1: Side::Periodic,
+                ..Boundaries::default()
+            },
+            ..Parameters::default()
+        },
+    );
+    solver.set_boundary_velocity(|_, _, _, _| (0.0, 0.0, 0.0));
+    if moving {
+        solver.set_moving_body(plate(true));
+    } else {
+        solver.set_body(plate(false));
+    }
+    let g = Grid::cubic(N, N, nz, h);
+    let mut field = Field::new(g);
+    solver.initialize(&mut field);
+    let steps = (t_end / dt).round() as usize;
+    let mut records = Vec::with_capacity(steps);
+    // The 2D definition: the largest |ΔKE| step's energy over that
+    // step's flipped columns.
+    let mut largest_jump = 0.0_f64;
+    let mut energy_per_flip = 0.0_f64;
+    let mut ke_prev: Option<f64> = None;
+    let start = std::time::Instant::now();
+    let (jp, ip, kp) = (
+        (0.92 * N as f64) as usize,
+        (0.5 * N as f64) as usize,
+        nz / 2,
+    );
+    for step in 0..steps {
+        let result = solver.advance(&mut field, dt);
+        let t = (step + 1) as f64 * dt;
+        let mask = solver.mask().expect("mask");
+        let body = solver.body().expect("body");
+        let (mut fy, mut skipped) = (0.0, 0usize);
+        match scheme {
+            WallScheme::GhostBinary => {
+                for (x, y, z, nx, ny, area) in samples(t, moving, 0.5 * h, nz, h) {
+                    match mask.traction_at(body, &field, MU, t, [x, y, z], [nx, ny, 0.0]) {
+                        Some(tr) => fy += tr[1] * area,
+                        None => skipped += 1,
+                    }
+                }
+            }
+            WallScheme::CutCell => {
+                fy = mask.cut_wall_force(body, &field, MU, t).expect("cut wall")[1];
+            }
+        }
+        fy /= lz;
+        let margin = 8;
+        let fy_cv = mask.control_volume_force(
+            &field,
+            dt,
+            RHO,
+            MU,
+            None,
+            (margin, N - margin, margin, N - margin, 0, nz),
+        )[1] / lz;
+        let p_far = field.p[g.cell(kp, jp, ip)];
+        let mut ke = 0.0;
+        for k in 0..nz {
+            for j in 0..N {
+                for i in 0..N {
+                    let idx = g.cell(k, j, i);
+                    if mask.is_fluid_cell(idx) {
+                        let uc = 0.5 * (field.u[g.uface(k, j, i)] + field.u[g.uface(k, j, i + 1)]);
+                        let vc = 0.5 * (field.v[g.vface(k, j, i)] + field.v[g.vface(k, j + 1, i)]);
+                        let wc = 0.5 * (field.w[g.wface(k, j, i)] + field.w[g.wface(k + 1, j, i)]);
+                        ke += 0.5 * RHO * (uc * uc + vc * vc + wc * wc) * h * h * h * mask.vol(idx);
+                    }
+                }
+            }
+        }
+        ke /= lz;
+        if let Some(prev) = ke_prev {
+            if step > 30 && result.fresh_cells > 0 && (ke - prev).abs() > largest_jump {
+                largest_jump = (ke - prev).abs();
+                // The plate's event is its row (the 2D divided by the row's
+                // 54 cells); the circle's is the step's fresh columns.
+                let columns = if circle_body() {
+                    result.fresh_cells as f64 / nz as f64
+                } else {
+                    (2.0 * HX / h).round()
+                };
+                energy_per_flip = largest_jump / columns;
+            }
+        }
+        ke_prev = Some(ke);
+        records.push(Record {
+            t,
+            fy,
+            fy_cv,
+            fresh: result.fresh_cells,
+            skipped,
+            p_far,
+            ke,
+        });
+    }
+    Run {
+        records,
+        energy_per_flip,
+        seconds: start.elapsed().as_secs_f64(),
+    }
+}
+
+/// Spike series: the load minus its 21-step running median.
+fn spikes(f: &[f64]) -> Vec<f64> {
+    let w = 10usize;
+    (0..f.len())
+        .map(|k| {
+            let lo = k.saturating_sub(w);
+            let hi = (k + w + 1).min(f.len());
+            let mut win: Vec<f64> = f[lo..hi].to_vec();
+            win.sort_by(|a, b| a.partial_cmp(b).unwrap());
+            f[k] - win[win.len() / 2]
+        })
+        .collect()
+}
+
+struct Stats {
+    rms_force: f64,
+    rms_spike: f64,
+    max_spike: f64,
+    rms_spike_cv: f64,
+    max_spike_cv: f64,
+    rms_pfar_spike: f64,
+    max_pfar_spike: f64,
+    max_ke_jump: f64,
+    fresh_total: usize,
+    skipped_max: usize,
+}
+
+fn stats(records: &[Record], t_lo: f64, t_hi: f64) -> Stats {
+    let fy: Vec<f64> = records.iter().map(|r| r.fy).collect();
+    let sp = spikes(&fy);
+    let fcv: Vec<f64> = records.iter().map(|r| r.fy_cv).collect();
+    let spc = spikes(&fcv);
+    let pf: Vec<f64> = records.iter().map(|r| r.p_far).collect();
+    let spf = spikes(&pf);
+    let idx: Vec<usize> = (0..records.len())
+        .filter(|&k| records[k].t >= t_lo && records[k].t <= t_hi)
+        .collect();
+    let rms = |v: &dyn Fn(usize) -> f64| {
+        (idx.iter().map(|&k| v(k) * v(k)).sum::<f64>() / idx.len().max(1) as f64).sqrt()
+    };
+    Stats {
+        rms_force: rms(&|k| fy[k]),
+        rms_spike: rms(&|k| sp[k]),
+        max_spike: idx.iter().map(|&k| sp[k].abs()).fold(0.0, f64::max),
+        rms_spike_cv: rms(&|k| spc[k]),
+        max_spike_cv: idx.iter().map(|&k| spc[k].abs()).fold(0.0, f64::max),
+        rms_pfar_spike: rms(&|k| spf[k]),
+        max_pfar_spike: idx.iter().map(|&k| spf[k].abs()).fold(0.0, f64::max),
+        max_ke_jump: idx
+            .iter()
+            .filter(|&&k| k > 0)
+            .map(|&k| (records[k].ke - records[k - 1].ke).abs())
+            .fold(0.0, f64::max),
+        fresh_total: idx.iter().map(|&k| records[k].fresh).sum(),
+        skipped_max: idx.iter().map(|&k| records[k].skipped).max().unwrap_or(0),
+    }
+}
+
+fn dump(dir: &str, name: &str, records: &[Record]) {
+    let path = std::path::Path::new(dir).join(format!("{name}.csv"));
+    let mut f = std::fs::File::create(path).expect("csv");
+    writeln!(f, "t,fy,fy_cv,fresh,skipped,p_far,ke").unwrap();
+    for r in records {
+        writeln!(
+            f,
+            "{:.6},{:.6e},{:.6e},{},{},{:.6e},{:.6e}",
+            r.t, r.fy, r.fy_cv, r.fresh, r.skipped, r.p_far, r.ke
+        )
+        .unwrap();
+    }
+}
+
+struct Verdict {
+    energy_per_flip: f64,
+    max_spike: f64,
+    exponent: Option<f64>,
+}
+
+fn falsify(scheme: WallScheme, ladder: bool) -> Verdict {
+    let csv_dir = std::env::var("RTX_E3_FALSIFIER_CSV").ok();
+    let period = 2.0 * std::f64::consts::PI * AMP / U_PEAK;
+    let t_end = 0.3 * period;
+    let (t_lo, t_hi) = (0.02 * period, 0.28 * period);
+    let rest = run(scheme, false, DT_FSI2, t_end);
+    let s0 = stats(&rest.records, t_lo, t_hi);
+    println!(
+        "  {scheme:?} plate AT REST, dt {DT_FSI2:.2e} ({:.0} s): rms force {:.3e}, rms spike {:.3e}, max spike {:.3e}, fresh {}, skipped max {}",
+        rest.seconds, s0.rms_force, s0.rms_spike, s0.max_spike, s0.fresh_total, s0.skipped_max
+    );
+    if let Some(d) = &csv_dir {
+        dump(d, &format!("{scheme:?}_rest"), &rest.records);
+    }
+    let dts: Vec<f64> = if ladder {
+        vec![DT_FSI2, DT_FSI2 / 2.0, DT_FSI2 / 4.0]
+    } else {
+        vec![DT_FSI2]
+    };
+    let mut points = Vec::new();
+    let mut energy = 0.0_f64;
+    let mut max_spike = 0.0_f64;
+    for &dt in &dts {
+        let r = run(scheme, true, dt, t_end);
+        let s = stats(&r.records, t_lo, t_hi);
+        println!(
+            "  {scheme:?} plate MOVING, dt {dt:.3e} ({} steps, {:.0} s): rms force {:.3e}, rms spike {:.3e} ({:.1}x rest), max spike {:.3e} N/m ({:.2e} of ½ρU²L), fresh cells {} ({:.2}/step), skipped max {}",
+            r.records.len(),
+            r.seconds,
+            s.rms_force,
+            s.rms_spike,
+            s.rms_spike / s0.rms_spike.max(1e-300),
+            s.max_spike,
+            s.max_spike / load_scale(),
+            s.fresh_total,
+            s.fresh_total as f64 / r.records.len() as f64,
+            s.skipped_max
+        );
+        println!(
+            "    control-volume route: rms spike {:.3e}, max spike {:.3e} N/m ({:.2e} of ½ρU²L)",
+            s.rms_spike_cv,
+            s.max_spike_cv,
+            s.max_spike_cv / load_scale()
+        );
+        println!(
+            "    far probe p(0.5, 0.92): rms spike {:.3e}, max spike {:.3e}; max |ΔKE| per step {:.3e} J/m; energy per flipped column {:.3e} J/m ({:.2} of the 2D wall's {ENERGY_2D})",
+            s.rms_pfar_spike,
+            s.max_pfar_spike,
+            s.max_ke_jump,
+            r.energy_per_flip,
+            r.energy_per_flip / ENERGY_2D
+        );
+        if let Some(d) = &csv_dir {
+            dump(d, &format!("{scheme:?}_moving_dt{dt:.3e}"), &r.records);
+        }
+        assert!(s.rms_force.is_finite() && s.rms_spike.is_finite());
+        if dt == DT_FSI2 {
+            energy = r.energy_per_flip;
+            max_spike = s.max_spike;
+        }
+        points.push((dt, s.rms_spike));
+    }
+    let exponent = (points.len() >= 2).then(|| {
+        let xs: Vec<f64> = points.iter().map(|p| p.0.ln()).collect();
+        let ys: Vec<f64> = points.iter().map(|p| p.1.ln()).collect();
+        let mx = xs.iter().sum::<f64>() / xs.len() as f64;
+        let my = ys.iter().sum::<f64>() / ys.len() as f64;
+        let num: f64 = xs.iter().zip(&ys).map(|(x, y)| (x - mx) * (y - my)).sum();
+        let den: f64 = xs.iter().map(|x| (x - mx).powi(2)).sum();
+        let e = num / den;
+        println!(
+            "  {scheme:?}: spike RMS ~ (dt)^{e:.2} across {} time steps",
+            points.len()
+        );
+        e
+    });
+    Verdict {
+        energy_per_flip: energy,
+        max_spike,
+        exponent,
+    }
+}
+
+#[test]
+fn oscillating_plate_both_walls() {
+    let ladder = std::env::var("RTX_E3_FALSIFIER_LADDER").is_ok();
+    println!(
+        "  body: {}",
+        if circle_body() {
+            "circle R 0.05"
+        } else if stadium_body() {
+            "stadium 0.35 x 0.02 (semicircular ends)"
+        } else {
+            "plate 0.35 x 0.02"
+        }
+    );
+    let ghost = falsify(WallScheme::GhostBinary, ladder);
+    let cut = falsify(WallScheme::CutCell, ladder);
+    println!(
+        "  energy per flipped column: ghost {:.3e}, cut {:.3e} (ratio {:.1}x); max spike: ghost {:.3e}, cut {:.3e} N/m",
+        ghost.energy_per_flip,
+        cut.energy_per_flip,
+        ghost.energy_per_flip / cut.energy_per_flip.max(1e-300),
+        ghost.max_spike,
+        cut.max_spike
+    );
+    assert!(
+        ghost.energy_per_flip > 0.0,
+        "the binary wall must flip cells"
+    );
+}
+
+/// The registered gates on the dt ladder.
+#[test]
+#[ignore = "item 11's gated ladder (dt, dt/2, dt/4 on both walls; tens of minutes on the host)"]
+fn oscillating_plate_gates() {
+    let ghost = falsify(WallScheme::GhostBinary, true);
+    let cut = falsify(WallScheme::CutCell, true);
+    let ratio = ghost.energy_per_flip / cut.energy_per_flip.max(1e-300);
+    println!(
+        "  GATES: ghost energy per flipped column {:.3e} ({:.2} of 2D), exponent {:.2}; cut energy {:.3e} ({:.1}x lower), max spike {:.3e} N/m ({:.2e} of ½ρU²L), exponent {:.2}",
+        ghost.energy_per_flip,
+        ghost.energy_per_flip / ENERGY_2D,
+        ghost.exponent.unwrap(),
+        cut.energy_per_flip,
+        ratio,
+        cut.max_spike,
+        cut.max_spike / load_scale(),
+        cut.exponent.unwrap()
+    );
+    let g2d = ghost.energy_per_flip / ENERGY_2D;
+    assert!(
+        (0.7..=1.3).contains(&g2d),
+        "ghost energy per flip {g2d:.2} of 2D"
+    );
+    assert!(ratio >= 20.0, "cut energy only {ratio:.1}x lower");
+    assert!(
+        cut.max_spike < 0.05 * load_scale(),
+        "cut max spike {:.3e}",
+        cut.max_spike
+    );
+    let e = cut.exponent.unwrap();
+    assert!((-0.3..=0.3).contains(&e), "cut exponent {e:.2}");
+}

crates/specialized/rtx-cfd/tests/embedded3_sphere/mod.rs

@@ -0,0 +1,255 @@
+//! The embedded-sphere manufactured solution shared by the embedded3
+//! wall gates (items 9–11): the fields, the source, the boundary data,
+//! the exact surface integrals, and one steady march measured.
+
+use rtx_cfd::solvers::incompressible::ConvectionScheme;
+use rtx_cfd::solvers::incompressible::embedded3::{
+    Body, FaceKind, Field, Fluid, Grid, Parameters, Solver, WallScheme,
+};
+use std::f64::consts::PI;
+
+pub const RHO: f64 = 1.0;
+pub const MU: f64 = 0.05;
+/// The sphere's default centre (off-centre so the exact force is not zero by symmetry).
+pub const C: (f64, f64, f64) = (0.6, 0.45, 0.5);
+pub const R: f64 = 0.2;
+
+pub fn u3(x: f64, y: f64, z: f64) -> f64 {
+    (PI * x).sin() * (PI * y).cos() * (PI * z).cos()
+}
+pub fn v3(x: f64, y: f64, z: f64) -> f64 {
+    (PI * x).cos() * (PI * y).sin() * (PI * z).cos()
+}
+pub fn w3(x: f64, y: f64, z: f64) -> f64 {
+    -2.0 * (PI * x).cos() * (PI * y).cos() * (PI * z).sin()
+}
+pub fn p3(x: f64, y: f64, z: f64) -> f64 {
+    (PI * x).sin() * (PI * y).sin() * (PI * z).sin()
+}
+/// The velocity gradient ∂u_i/∂x_j and the pressure gradient.
+pub fn grads(x: f64, y: f64, z: f64) -> ([[f64; 3]; 3], [f64; 3]) {
+    let (sx, cx) = (PI * x).sin_cos();
+    let (sy, cy) = (PI * y).sin_cos();
+    let (sz, cz) = (PI * z).sin_cos();
+    (
+        [
+            [PI * cx * cy * cz, -PI * sx * sy * cz, -PI * sx * cy * sz],
+            [-PI * sx * sy * cz, PI * cx * cy * cz, -PI * cx * sy * sz],
+            [
+                2.0 * PI * sx * cy * sz,
+                2.0 * PI * cx * sy * sz,
+                -2.0 * PI * cx * cy * cz,
+            ],
+        ],
+        [PI * cx * sy * sz, PI * sx * cy * sz, PI * sx * sy * cz],
+    )
+}
+pub fn source3(x: f64, y: f64, z: f64) -> (f64, f64, f64) {
+    let (g, gp) = grads(x, y, z);
+    let u = [u3(x, y, z), v3(x, y, z), w3(x, y, z)];
+    let lap = -3.0 * PI * PI;
+    let conv = |i: usize| u[0] * g[i][0] + u[1] * g[i][1] + u[2] * g[i][2];
+    (
+        RHO * conv(0) + gp[0] - MU * lap * u[0],
+        RHO * conv(1) + gp[1] - MU * lap * u[1],
+        RHO * conv(2) + gp[2] - MU * lap * u[2],
+    )
+}
+pub fn boundary3(x: f64, y: f64, z: f64) -> (f64, f64, f64) {
+    let u = if x <= 0.0 || x >= 1.0 {
+        0.0
+    } else {
+        u3(x, y, z)
+    };
+    let v = if y <= 0.0 || y >= 1.0 {
+        0.0
+    } else {
+        v3(x, y, z)
+    };
+    let w = if z <= 0.0 || z >= 1.0 {
+        0.0
+    } else {
+        w3(x, y, z)
+    };
+    (u, v, w)
+}
+
+/// Exact force `∮ (−p I + μ(∇u + ∇uᵀ)) n dA` and momentum flux `∮ ρ u (u·n) dA`
+/// over the sphere by a fine Fibonacci quadrature.
+pub fn exact_force_and_flux(c: (f64, f64, f64)) -> ([f64; 3], [f64; 3]) {
+    let n = 200_000;
+    let golden = PI * (3.0 - 5.0_f64.sqrt());
+    let (mut f, mut m) = ([0.0; 3], [0.0; 3]);
+    let da = 4.0 * PI * R * R / n as f64;
+    for k in 0..n {
+        let zz = 1.0 - 2.0 * (k as f64 + 0.5) / n as f64;
+        let rr = (1.0 - zz * zz).sqrt();
+        let th = golden * k as f64;
+        let nrm = [rr * th.cos(), rr * th.sin(), zz];
+        let (x, y, z) = (c.0 + R * nrm[0], c.1 + R * nrm[1], c.2 + R * nrm[2]);
+        let (g, _) = grads(x, y, z);
+        let p = p3(x, y, z);
+        let u = [u3(x, y, z), v3(x, y, z), w3(x, y, z)];
+        let un = u[0] * nrm[0] + u[1] * nrm[1] + u[2] * nrm[2];
+        for i in 0..3 {
+            let mut t = -p * nrm[i];
+            for j in 0..3 {
+                t += MU * (g[i][j] + g[j][i]) * nrm[j];
+            }
+            f[i] += t * da;
+            m[i] += RHO * u[i] * un * da;
+        }
+    }
+    (f, m)
+}
+
+pub struct Measurement {
+    pub l2_velocity: f64,
+    pub max_div: f64,
+    pub ghost_correction: f64,
+    pub force_surface: [f64; 3],
+    pub skipped: usize,
+    pub force_cv: [f64; 3],
+}
+
+/// March the manufactured solution with the sphere at `c` to steady state on grid `n`.
+pub fn measure(n: usize, scheme: WallScheme, c: (f64, f64, f64)) -> Measurement {
+    let h = 1.0 / n as f64;
+    let dt = 0.4 * (h * h / (4.0 * MU / RHO)).min(h);
+    let mut solver = Solver::new(
+        Fluid {
+            density: RHO,
+            viscosity: MU,
+            reference_velocity: 1.0,
+            reference_length: 1.0,
+        },
+        Parameters {
+            corrector_steps: 2,
+            tolerance: 1e-8,
+            convection_scheme: ConvectionScheme::Upwind,
+            wall_scheme: scheme,
+            ..Parameters::default()
+        },
+    );
+    solver.set_momentum_source(|x, y, z, _t| source3(x, y, z));
+    solver.set_boundary_velocity(|x, y, z, _t| boundary3(x, y, z));
+    solver.set_body(
+        Body::sphere(move |_t| c, R)
+            .with_surface_velocity(|x, y, z, _t| (u3(x, y, z), v3(x, y, z), w3(x, y, z))),
+    );
+    let g = Grid::cubic(n, n, n, h);
+    let mut f = Field::new(g);
+    solver.initialize(&mut f);
+    let mut last = solver.advance(&mut f, dt);
+    for _ in 0..200_000 {
+        let (bu, bv, bw) = (f.u.clone(), f.v.clone(), f.w.clone());
+        last = solver.advance(&mut f, dt);
+        let mut change = 0.0_f64;
+        for (a, b) in
+            f.u.iter()
+                .zip(&bu)
+                .chain(f.v.iter().zip(&bv))
+                .chain(f.w.iter().zip(&bw))
+        {
+            change = change.max((a - b).abs());
+        }
+        if change / dt < 1e-6 {
+            break;
+        }
+    }
+    let mask = solver.mask().expect("mask");
+    let (mut sq, mut vol) = (0.0, 0.0);
+    let dv = h * h * h;
+    for k in 0..n {
+        for j in 0..n {
+            for i in 1..n {
+                if mask.u_kind(g.uface(k, j, i)) == FaceKind::Fluid {
+                    let e = f.u[g.uface(k, j, i)]
+                        - u3(i as f64 * h, (j as f64 + 0.5) * h, (k as f64 + 0.5) * h);
+                    sq += e * e * dv;
+                    vol += dv;
+                }
+            }
+        }
+        for j in 1..n {
+            for i in 0..n {
+                if mask.v_kind(g.vface(k, j, i)) == FaceKind::Fluid {
+                    let e = f.v[g.vface(k, j, i)]
+                        - v3((i as f64 + 0.5) * h, j as f64 * h, (k as f64 + 0.5) * h);
+                    sq += e * e * dv;
+                    vol += dv;
+                }
+            }
+        }
+    }
+    for k in 1..n {
+        for j in 0..n {
+            for i in 0..n {
+                if mask.w_kind(g.wface(k, j, i)) == FaceKind::Fluid {
+                    let e = f.w[g.wface(k, j, i)]
+                        - w3((i as f64 + 0.5) * h, (j as f64 + 0.5) * h, k as f64 * h);
+                    sq += e * e * dv;
+                    vol += dv;
+                }
+            }
+        }
+    }
+    let body = solver.body().expect("body");
+    let t = solver.time();
+    // The apertured divergence per unit volume, the porous surface's flux
+    // through the wall included (the plain divergence on the binary wall).
+    let mut max_div = 0.0_f64;
+    let mut at_vol = 1.0;
+    let mut sum_flux = 0.0;
+    let (wall_fluxes, _) = mask.wall_flux_table(body, t);
+    for k in 0..n {
+        for j in 0..n {
+            for i in 0..n {
+                let idx = g.cell(k, j, i);
+                if mask.is_fluid_cell(idx) {
+                    let flux = (mask.a_u(g.uface(k, j, i + 1)) * f.u[g.uface(k, j, i + 1)]
+                        - mask.a_u(g.uface(k, j, i)) * f.u[g.uface(k, j, i)])
+                        * h
+                        * h
+                        + (mask.a_v(g.vface(k, j + 1, i)) * f.v[g.vface(k, j + 1, i)]
+                            - mask.a_v(g.vface(k, j, i)) * f.v[g.vface(k, j, i)])
+                            * h
+                            * h
+                        + (mask.a_w(g.wface(k + 1, j, i)) * f.w[g.wface(k + 1, j, i)]
+                            - mask.a_w(g.wface(k, j, i)) * f.w[g.wface(k, j, i)])
+                            * h
+                            * h
+                        + wall_fluxes[idx];
+                    sum_flux += flux.abs();
+                    if (flux / (h * h * h)).abs() > max_div {
+                        max_div = (flux / (h * h * h)).abs();
+                        at_vol = mask.vol(idx);
+                    }
+                }
+            }
+        }
+    }
+    println!(
+        "    [{scheme:?} n {n}] max div {max_div:.2e} in a cell of fluid fraction {at_vol:.3e}; Σ|flux| {sum_flux:.2e}; last step residual {:.2e}",
+        last.final_residual
+    );
+    let surface = match scheme {
+        WallScheme::GhostBinary => mask.surface_force(body, &f, MU, t, 0.5 * h),
+        WallScheme::CutCell => rtx_cfd::solvers::incompressible::embedded3::SurfaceForce {
+            f: mask.cut_wall_force(body, &f, MU, t).expect("cut wall"),
+            samples: 0,
+            skipped: 0,
+        },
+    };
+    let (i0, i1) = (n / 8, n - n / 8);
+    let src = |x: f64, y: f64, z: f64| source3(x, y, z);
+    let force_cv = mask.control_volume_force(&f, dt, RHO, MU, Some(&src), (i0, i1, i0, i1, i0, i1));
+    Measurement {
+        l2_velocity: (sq / vol).sqrt(),
+        max_div,
+        ghost_correction: solver.ghost_correction().abs(),
+        force_surface: surface.f,
+        skipped: surface.skipped,
+        force_cv,
+    }
+}

crates/specialized/rtx-cfd/tests/embedded3_wall_lipschitz.rs

@@ -0,0 +1,100 @@
+//! embedded3 item 11b: the wall's smoothness in the interface position.
+//! The manufactured sphere is marched to steady state at `M + 1` centres
+//! spaced `h/M` apart across one cell along x; at each the load error
+//! `E(δ) = F(δ) − F_exact(δ)` (the exact force moves with the sphere and
+//! is subtracted) is measured on the scheme's route. The largest jump of
+//! `E` between neighbouring positions, relative to the load, and the
+//! Lipschitz quotient `|ΔE| / (Δδ |F|)` are reported for both walls.
+//! Registered gate (`docs/embedded3_campaign.md` item 11): the cut wall's
+//! largest neighbouring jump < 1 % of the load with a bounded quotient.
+//!
+//! Default run: 8 positions at n = 24 (about two minutes on the host);
+//! the gated `#[ignore]` variant sweeps 40.
+
+mod embedded3_sphere;
+
+use embedded3_sphere::{C, exact_force_and_flux, measure};
+use rtx_cfd::solvers::incompressible::embedded3::WallScheme;
+
+fn norm(a: [f64; 3]) -> f64 {
+    (a[0] * a[0] + a[1] * a[1] + a[2] * a[2]).sqrt()
+}
+
+struct Sweep {
+    /// Largest neighbouring jump of the load error relative to the load.
+    max_jump: f64,
+    /// Largest Lipschitz quotient `|ΔE| / (Δδ |F|)` (per unit length).
+    max_quotient: f64,
+}
+
+fn sweep(n: usize, positions: usize, scheme: WallScheme) -> Sweep {
+    let h = 1.0 / n as f64;
+    let step = h / positions as f64;
+    let mut errors: Vec<[f64; 3]> = Vec::new();
+    let mut scale = 0.0_f64;
+    for m in 0..=positions {
+        let c = (C.0 + m as f64 * step, C.1, C.2);
+        let (fe, _) = exact_force_and_flux(c);
+        let r = measure(n, scheme, c);
+        let e = [
+            r.force_surface[0] - fe[0],
+            r.force_surface[1] - fe[1],
+            r.force_surface[2] - fe[2],
+        ];
+        scale = scale.max(norm(fe));
+        println!(
+            "  {scheme:?} δ = {:.4} h: F {:.5?} exact {:.5?} error {:.3e} (rel {:.3e})",
+            m as f64 / positions as f64,
+            r.force_surface,
+            fe,
+            norm(e),
+            norm(e) / norm(fe)
+        );
+        errors.push(e);
+    }
+    let mut max_jump = 0.0_f64;
+    for w in errors.windows(2) {
+        let d = norm([w[1][0] - w[0][0], w[1][1] - w[0][1], w[1][2] - w[0][2]]);
+        max_jump = max_jump.max(d / scale);
+    }
+    let max_quotient = max_jump / step;
+    println!(
+        "  {scheme:?}: largest neighbouring jump {:.3e} of the load (spacing {:.3e} = h/{positions}); Lipschitz quotient {:.3e} per unit length",
+        max_jump, step, max_quotient
+    );
+    Sweep {
+        max_jump,
+        max_quotient,
+    }
+}
+
+#[test]
+fn sphere_load_across_one_cell() {
+    let ghost = sweep(24, 8, WallScheme::GhostBinary);
+    let cut = sweep(24, 8, WallScheme::CutCell);
+    println!(
+        "  jumps: ghost {:.3e}, cut {:.3e} ({:.1}x smaller); quotients: ghost {:.3e}, cut {:.3e}",
+        ghost.max_jump,
+        cut.max_jump,
+        ghost.max_jump / cut.max_jump.max(1e-300),
+        ghost.max_quotient,
+        cut.max_quotient
+    );
+    assert!(ghost.max_jump.is_finite() && cut.max_jump.is_finite());
+}
+
+#[test]
+#[ignore = "item 11's gated sweep (40 positions, both walls; tens of minutes on the host)"]
+fn sphere_load_lipschitz_gate() {
+    let ghost = sweep(24, 40, WallScheme::GhostBinary);
+    let cut = sweep(24, 40, WallScheme::CutCell);
+    println!(
+        "  GATE: cut largest jump {:.3e} of the load (ghost {:.3e}); cut quotient {:.3e} (ghost {:.3e})",
+        cut.max_jump, ghost.max_jump, cut.max_quotient, ghost.max_quotient
+    );
+    assert!(
+        cut.max_jump < 0.01,
+        "cut-cell jump {:.3e} of the load",
+        cut.max_jump
+    );
+}