embedded3 A1-b: the tiled momentum volume as a host prototype (RTX_E3_MOMENTUM_VOLUME=tiled; pressure force, source and inertia on ½(V−+V+)hA; default off; the device refuses it) — REFUTED on the oblique instrument (level moves, the h-independent remainder and the near-wall pressure residual do not)

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

crates/specialized/rtx-cfd/src/solvers/incompressible/embedded3/step/cut_predictor.rs

@@ -344,8 +344,20 @@ impl Solver {
         let p_plus = lat.cell(cell_plus).map_or(0.0, |ci| field.p[ci]);
         let p_minus = lat.cell(cell_minus).map_or(0.0, |ci| field.p[ci]);
         let idx_f = lat.face(c, p).expect("the face");
-        let pressure = -(p_plus - p_minus) * cv.alpha * area[c] * mask.grad_weight(c, idx_f);
+        // A1-b: the control volume the pressure force, the source and the
-        let v_u = cv.alpha * h[c] * area[c];
+        // inertia act on — the aperture prism (default) or the tiled
+        // half-cells `½(V_− + V_+)` the convective / diffusive sides use.
+        let cell_mean = || {
+            let vol = |q: [i64; 3]| lat.cell(q).map_or(cv.alpha, |ci| mask.vol(ci));
+            0.5 * (vol(cell_minus) + vol(cell_plus))
+        };
+        let alpha_p = if self.params.momentum_volume_tiled {
+            cell_mean()
+        } else {
+            cv.alpha
+        };
+        let pressure = -(p_plus - p_minus) * alpha_p * area[c] * mask.grad_weight(c, idx_f);
+        let v_u = alpha_p * h[c] * area[c];
         let source = self.momentum_source.as_ref().map_or(0.0, |f| {
             let s = f(x[0], x[1], x[2], t_old);
             [s.0, s.1, s.2][c] * v_u
@@ -361,9 +373,9 @@ impl Solver {
         let shear = mu * a_w * c1;
         // The explicit part of the quadratic wall gradient (order 2).
         let shear_explicit = nb.map_or(0.0, |f| mu * a_w * c2 * (old[c][f] - ub));
-        let fraction = if self.params.momentum_volume_cell_mean {
+        let fraction = if self.params.momentum_volume_cell_mean || self.params.momentum_volume_tiled
-            let vol = |q: [i64; 3]| lat.cell(q).map_or(cv.alpha, |ci| mask.vol(ci));
+        {
-            0.5 * (vol(cell_minus) + vol(cell_plus))
+            cell_mean()
         } else {
             cv.alpha
         };

crates/specialized/rtx-cfd/src/solvers/incompressible/embedded3/step/device.rs

@@ -169,6 +169,10 @@ impl DeviceStep {
             !solver.params.pressure_centroid,
             "the centroid pressure gradient (S2-5) is a host prototype: the device kernels do not carry it"
         );
+        assert!(
+            !solver.params.momentum_volume_tiled,
+            "the tiled momentum volume (A1-b) is a host prototype: the device kernels do not carry it"
+        );
         let rt = runtime();
         let nu = (grid.nx + 1) * grid.ny * grid.nz;
         let nv = grid.nx * (grid.ny + 1) * grid.nz;

crates/specialized/rtx-cfd/src/solvers/incompressible/embedded3/step/mod.rs

@@ -139,6 +139,14 @@ pub struct Parameters {
     /// correction and the predictor's pressure force. The device path
     /// refuses it. `RTX_E3_PRESSURE_CENTROID=1`.
     pub pressure_centroid: bool,
+    /// HOST PROTOTYPE (A1-b, 2026-09-19): the pressure force, the momentum
+    /// source and the inertia of a face's control volume taken on the
+    /// TILED volume `½(V_− + V_+) h A` (the cell means the convective and
+    /// diffusive sides already use) instead of the aperture prism
+    /// `α h A`. On an oblique cut the two differ by O(1) in the cut
+    /// layer — the sharpened S2-7 hypothesis. The device path refuses it.
+    /// `RTX_E3_MOMENTUM_VOLUME=tiled`.
+    pub momentum_volume_tiled: bool,
     /// S2-5: the cross-direction diffusion between two faces over the
     /// distance between their OPEN-PART CENTROIDS (a cut face's velocity
     /// is its open part's mean, ½h(1 − α) off the face centre along the
@@ -181,6 +189,8 @@ impl Default for Parameters {
                 .is_ok_and(|v| v == "off"),
             wall_exchange_axis: std::env::var("RTX_E3_WALL_EXCHANGE").is_ok_and(|v| v == "axis"),
             pressure_centroid: std::env::var("RTX_E3_PRESSURE_CENTROID").is_ok_and(|v| v == "1"),
+            momentum_volume_tiled: std::env::var("RTX_E3_MOMENTUM_VOLUME")
+                .is_ok_and(|v| v == "tiled"),
             // ON by default since S2-5 (`=0` reproduces the records before it).
             diffusion_centroid: std::env::var("RTX_E3_DIFFUSION_CENTROID")
                 .map_or(true, |v| v != "0"),