Snapshot WIP: solver HP epic progress, BPHX/HX physics, BMAD skill refresh.

Capture uncommitted solver robustness work (regularization, domain errors, linear solver lifecycle, tube DP/MSH), web workbench updates, and synced BMAD skills across IDE agent folders before starting BPHX pressure-drop.

Co-authored-by: Cursor <cursoragent@cursor.com>

crates/components/tests/test_msh_smoothness.rs

@@ -0,0 +1,66 @@
+//! Quick empirical check of MSH smoothness near x=1.
+//!
+//! Run with: cargo test --release -p entropyk-components --lib -- test_msh_smoothness_empirical real_check --nocapture
+use entropyk_components::heat_exchanger::two_phase_dp::{
+    friedel_gradient, msh_gradient, FriedelInput,
+};
+
+#[test]
+fn real_check() {
+    // R134a-like at 5 °C, mass flux typical of DX evaporator at opening=1.
+    let base = FriedelInput {
+        mass_flux: 300.0,
+        diameter: 0.0095,
+        quality: 0.5,
+        rho_liquid: 1290.0,
+        rho_vapor: 17.4,
+        mu_liquid: 250e-6,
+        mu_vapor: 11e-6,
+        sigma: 0.011,
+    };
+
+    let qualities = [
+        0.10, 0.50, 0.80, 0.90, 0.95, 0.97, 0.99, 0.995, 0.999, 0.9999, 1.0,
+    ];
+
+    println!(
+        "\n{:<10} {:<14} {:<14} {:<14}",
+        "x", "MSH", "dMSH/dx", "Friedel"
+    );
+    let mut prev_msh: Option<f64> = None;
+    let mut prev_x: Option<f64> = None;
+    for &x in &qualities {
+        let inp = FriedelInput { quality: x, ..base };
+        let msh = msh_gradient(&inp);
+        let fri = friedel_gradient(&inp);
+        let dmsdh = match (prev_msh, prev_x) {
+            (Some(pm), Some(px)) => (msh - pm) / (x - px),
+            _ => f64::NAN,
+        };
+        println!("{:<10.5} {:<14.3} {:<14.3} {:<14.3}", x, msh, dmsdh, fri);
+        prev_msh = Some(msh);
+        prev_x = Some(x);
+    }
+
+    // Analytic derivative check: d/dx[linear·(1-x)^(1/3)] near x=1
+    println!("\nAnalytic derivative of linear·(1-x)^(1/3) term:");
+    let a: f64 =
+        2.0 * 0.079 * (300.0_f64 * 0.0095 / 250e-6).powf(-0.25) * 300.0 * 300.0 / (0.0095 * 1290.0);
+    let b: f64 =
+        2.0 * 0.079 * (300.0_f64 * 0.0095 / 11e-6).powf(-0.25) * 300.0 * 300.0 / (0.0095 * 17.4);
+    println!("  A = liquid-only gradient = {:.3} Pa/m", a);
+    println!("  B = vapor-only gradient  = {:.3} Pa/m", b);
+    println!("  Ratio B/A = {:.1}", b / a);
+    for &x in &[0.9_f64, 0.95, 0.99, 0.999, 0.9999] {
+        let linear = a + 2.0 * (b - a) * x;
+        let term = linear * (1.0 - x).powf(1.0 / 3.0);
+        // d/dx[linear·(1-x)^(1/3)] = 2(b-a)·(1-x)^(1/3) - (1/3)·linear·(1-x)^(-2/3)
+        let d_term = 2.0 * (b - a) * (1.0 - x).powf(1.0 / 3.0)
+            - (1.0 / 3.0) * linear * (1.0 - x).powf(-2.0 / 3.0);
+        let d_total = d_term + 3.0 * b * x * x;
+        println!(
+            "  x={:.5}: term={:>10.2}  d(term)/dx={:>14.2}  d(MSH)/dx={:>14.2}",
+            x, term, d_term, d_total
+        );
+    }
+}