Try using pinned vertices as the hard constraints

Implements regulators for the Euclidean coordinates of Point entities,
  automatically creating all three of them for each added point entity. When
  such a regulator is set, it freezes the corresponding representation
  coordinate to the set point. In addition, if all three coordinates of a
  given Point are set, the coradius coordinate (which holds the norm of the
  point) is frozen as well.

  Note that a PointCoordinateRegulator must be created with a Point as the
  subject. This commit modifies HalfCurvatureRegulator analogously, so that
  it can only be created with a Sphere.

  A couple of prospective issues that should be filed in association with
  this commit:
  * The new coordinate regulators create redundant display information with
    the raw representation coordinates of a point that are already shown in
    the outline view.
  * The optimization status of these regulators together with HalfCurvature
    regulators (i.e., the ones implemented by freezing coordinates) is different
    from InversiveDistance regulators when an Assembly is unrealizable: the
    frozen-coordinate constraints will be "hard" in that they will be forced
    to precisely equal their set point, whereas the distance regulators are
    "soft" in that they can be relaxed from their set points in an effort to
    minimize the loss function of the configuration as compared to the values
    of the constraints. Perhaps at some point we should/will have a mechanism
    to specify the softness/hardness of constraints, but in the meantime,
    there should not be two different categories of constraints. Suppose we
    decide that by default that all constraints are soft. Then the optimizer
    should be able to search changing, for example, the radius of a
    curvature-constrained sphere, so as to minimize the loss function (for a
    loss that would therefore presumably have a term akin to the square of the
    difference between the specified and actual half-curvature of the sphere).
    For example, suppose you specify that the half-curvature of a sphere is 1
    (so it has radius 1/2) but that its distance to a point is -1. These
    constraints cannot be satisfied, so the optimization fails, presumably
    with the point at the sphere center, and the sphere with radius 1/2.
    So all of the loss is concentrated in the difference between the actual
    point-sphere distance being -1/2, not -1. It would be more appropriate
    (in the all-soft constraint regime) to end up at something like a sphere of
    half-curvature 1/√2 with the point at the center, so that the loss is split
    between both the half-curvature and the distance to the sphere being off by
    1 - 1/√2. (At a guess, that would minimize the sum of the squares of the
    two differences.)

app-proto/Cargo.lock

@@ -255,6 +255,7 @@ dependencies = [
  "charming",
  "console_error_panic_hook",
  "dyna3",
+ "enum-iterator",
  "itertools",
  "js-sys",
  "lazy_static",
@@ -271,6 +272,26 @@ version = "1.13.0"
 source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "60b1af1c220855b6ceac025d3f6ecdd2b7c4894bfe9cd9bda4fbb4bc7c0d4cf0"
 
+[[package]]
+name = "enum-iterator"
+version = "2.3.0"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "a4549325971814bda7a44061bf3fe7e487d447cba01e4220a4b454d630d7a016"
+dependencies = [
+ "enum-iterator-derive",
+]
+
+[[package]]
+name = "enum-iterator-derive"
+version = "1.5.0"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "685adfa4d6f3d765a26bc5dbc936577de9abf756c1feeb3089b01dd395034842"
+dependencies = [
+ "proc-macro2",
+ "quote",
+ "syn",
+]
+
 [[package]]
 name = "equivalent"
 version = "1.0.1"

app-proto/Cargo.toml

@@ -10,6 +10,7 @@ default = ["console_error_panic_hook"]
 dev = []
 
 [dependencies]
+enum-iterator = "2.3.0"
 itertools = "0.13.0"
 js-sys = "0.3.70"
 lazy_static = "1.5.0"

app-proto/main.css

@@ -227,6 +227,16 @@ details[open]:has(li) .element-switch::after {
   border-radius: 8px;
 }
 
+#distortion-bar {
+  display: flex;
+  margin-top: 8px;
+  gap: 8px;
+}
+
+#distortion-gauge {
+  flex-grow: 1;
+}
+
 /* display */
 
 #display {

app-proto/src/assembly.rs

@@ -1,8 +1,11 @@
+use enum_iterator::{all, Sequence};
 use nalgebra::{DMatrix, DVector, DVectorView};
 use std::{
+    any::Any,
     cell::Cell,
     cmp::Ordering,
     collections::{BTreeMap, BTreeSet},
+    f64::consts::SQRT_2,
     fmt,
     fmt::{Debug, Formatter},
     hash::{Hash, Hasher},
@@ -26,6 +29,7 @@ use crate::{
         ConfigSubspace,
         ConstraintProblem,
         DescentHistory,
+        MatrixEntry,
         Realization,
     },
     specified::SpecifiedValue,
@@ -122,6 +126,11 @@ pub trait Element: Serial + ProblemPoser + DisplayItem {
     // be used carefully to preserve invariant (1), described in the comment on
     // the `tangent` field of the `Assembly` structure
     fn set_column_index(&self, index: usize);
+    
+    /* KLUDGE */
+    fn is_point(&self) -> bool {
+        false
+    }
 }
 
 impl Debug for dyn Element {
@@ -269,6 +278,7 @@ pub struct Point {
 
 impl Point {
     const WEIGHT_COMPONENT: usize = 3;
+    const NORM_COMPONENT: usize = 4;
     
     pub fn new(
         id: String,
@@ -303,6 +313,15 @@ impl Element for Point {
         )
     }
 
+    fn default_regulators(self: Rc<Self>) -> Vec<Rc<dyn Regulator>> {
+        all::<Axis>()
+            .map(|axis| {
+                Rc::new(PointCoordinateRegulator::new(self.clone(), axis))
+                    as Rc::<dyn Regulator>
+            })
+            .collect()
+    }
+    
     fn id(&self) -> &String {
         &self.id
     }
@@ -334,6 +353,10 @@ impl Element for Point {
     fn set_column_index(&self, index: usize) {
         self.column_index.set(Some(index));
     }
+    
+    fn is_point(&self) -> bool {
+        true
+    }
 }
 
 impl Serial for Point {
@@ -353,10 +376,17 @@ impl ProblemPoser for Point {
     }
 }
 
-pub trait Regulator: Serial + ProblemPoser + OutlineItem {
+pub trait Regulator: Any + Serial + ProblemPoser + OutlineItem {
     fn subjects(&self) -> Vec<Rc<dyn Element>>;
     fn measurement(&self) -> ReadSignal<f64>;
     fn set_point(&self) -> Signal<SpecifiedValue>;
+    fn soft(&self) -> Option<Signal<bool>> {
+        None
+    }
+    fn distortion(&self) -> Option<ReadSignal<f64>> { /* KLUDGE */
+        None
+    }
+    fn as_any(&self) -> &dyn Any;
 }
 
 impl Hash for dyn Regulator {
@@ -389,6 +419,8 @@ pub struct InversiveDistanceRegulator {
     pub subjects: [Rc<dyn Element>; 2],
     pub measurement: ReadSignal<f64>,
     pub set_point: Signal<SpecifiedValue>,
+    pub soft: Signal<bool>,
+    distortion: Option<ReadSignal<f64>>, /* KLUDGE */
     serial: u64,
 }
 
@@ -404,9 +436,24 @@ impl InversiveDistanceRegulator {
         });
         
         let set_point = create_signal(SpecifiedValue::from_empty_spec());
+        let distortion = if subjects.iter().all(|subj| subj.is_point()) {
+            Some(create_memo(move || {
+                let set_point_opt = set_point.with(|set_pt| set_pt.value);
+                let measurement_val = measurement.get();
+                match set_point_opt {
+                    None => 0.0,
+                    Some(set_point_val) => SQRT_2 * (
+                        (-measurement_val).sqrt() - (-set_point_val).sqrt()
+                    ),
+                }
+            }))
+        } else {
+            None
+        };
+        let soft = create_signal(false);
         let serial = Self::next_serial();
         
-        Self { subjects, measurement, set_point, serial }
+        Self { subjects, measurement, set_point, soft, distortion, serial }
     }
 }
 
@@ -422,6 +469,18 @@ impl Regulator for InversiveDistanceRegulator {
     fn set_point(&self) -> Signal<SpecifiedValue> {
         self.set_point
     }
+    
+    fn soft(&self) -> Option<Signal<bool>> {
+        Some(self.soft)
+    }
+    
+    fn distortion(&self) -> Option<ReadSignal<f64>> {
+        self.distortion
+    }
+    
+    fn as_any(&self) -> &dyn Any {
+        self
+    }
 }
 
 impl Serial for InversiveDistanceRegulator {
@@ -432,6 +491,7 @@ impl Serial for InversiveDistanceRegulator {
 
 impl ProblemPoser for InversiveDistanceRegulator {
     fn pose(&self, problem: &mut ConstraintProblem) {
+        let soft = self.soft.get_untracked();
         self.set_point.with_untracked(|set_pt| {
             if let Some(val) = set_pt.value {
                 let [row, col] = self.subjects.each_ref().map(
@@ -439,21 +499,25 @@ impl ProblemPoser for InversiveDistanceRegulator {
                         "Subjects should be indexed before inversive distance regulator writes problem data"
                     )
                 );
-                problem.gram.push_sym(row, col, val);
+                if soft {
+                    problem.soft.push_sym(row, col, val);
+                } else {
+                    problem.gram.push_sym(row, col, val);
+                }
             }
         });
     }
 }
 
 pub struct HalfCurvatureRegulator {
-    pub subject: Rc<dyn Element>,
+    pub subject: Rc<Sphere>,
     pub measurement: ReadSignal<f64>,
     pub set_point: Signal<SpecifiedValue>,
     serial: u64,
 }
 
 impl HalfCurvatureRegulator {
-    pub fn new(subject: Rc<dyn Element>) -> Self {
+    pub fn new(subject: Rc<Sphere>) -> Self {
         let measurement = subject.representation().map(
             |rep| rep[Sphere::CURVATURE_COMPONENT]
         );
@@ -477,6 +541,10 @@ impl Regulator for HalfCurvatureRegulator {
     fn set_point(&self) -> Signal<SpecifiedValue> {
         self.set_point
     }
+    
+    fn as_any(&self) -> &dyn Any {
+        self
+    }
 }
 
 impl Serial for HalfCurvatureRegulator {
@@ -498,6 +566,81 @@ impl ProblemPoser for HalfCurvatureRegulator {
     }
 }
 
+#[derive(Clone, Copy, Sequence)]
+pub enum Axis {X = 0, Y = 1, Z = 2}
+
+impl Axis {
+    pub const N_AXIS: usize = (Axis::Z as usize) + 1;
+    pub const NAME: [&str; Axis::N_AXIS] = ["X", "Y", "Z"];
+}
+
+pub struct PointCoordinateRegulator {
+    pub subject: Rc<Point>,
+    pub axis: Axis,
+    pub measurement: ReadSignal<f64>,
+    pub set_point: Signal<SpecifiedValue>,
+    serial: u64
+}
+
+impl PointCoordinateRegulator {
+    pub fn new(subject: Rc<Point>, axis: Axis) -> Self {
+        let measurement = subject.representation().map(
+            move |rep| rep[axis as usize]
+        );
+        let set_point = create_signal(SpecifiedValue::from_empty_spec());
+        Self { subject, axis, measurement, set_point, serial: Self::next_serial() }
+    }
+}
+
+impl Serial for PointCoordinateRegulator {
+    fn serial(&self) -> u64 { self.serial }
+}
+
+impl Regulator for PointCoordinateRegulator {
+    fn subjects(&self) -> Vec<Rc<dyn Element>> {
+        vec![self.subject.clone()]
+    }
+    
+    fn measurement(&self) -> ReadSignal<f64> {
+        self.measurement
+    }
+    
+    fn set_point(&self) -> Signal<SpecifiedValue> {
+        self.set_point
+    }
+    
+    fn as_any(&self) -> &dyn Any {
+        self
+    }
+}
+
+impl ProblemPoser for PointCoordinateRegulator {
+    fn pose(&self, problem: &mut ConstraintProblem) {
+        self.set_point.with_untracked(|set_pt| {
+            if let Some(val) = set_pt.value {
+                let col = self.subject.column_index().expect(
+                    "Subject must be indexed before point-coordinate regulator poses.");
+                problem.frozen.push(self.axis as usize, col, val);
+                // Check if all three coordinates have been frozen, and if so,
+                // freeze the coradius as well
+                let mut coords = [0.0; Axis::N_AXIS];
+                let mut nset: usize = 0;
+                for &MatrixEntry {index, value} in &(problem.frozen) {
+                    if index.1 == col && index.0 < Axis::N_AXIS {
+                        nset += 1;
+                        coords[index.0] = value
+                    }
+                }
+                if nset == Axis::N_AXIS {
+                    let [x, y, z] = coords;
+                    problem.frozen.push(
+                        Point::NORM_COMPONENT, col, point(x,y,z)[Point::NORM_COMPONENT]);
+                }
+            }
+        });
+    }
+}
+
 // the velocity is expressed in uniform coordinates
 pub struct ElementMotion<'a> {
     pub element: Rc<dyn Element>,
@@ -705,7 +848,7 @@ impl Assembly {
         
         // look for a configuration with the given Gram matrix
         let Realization { result, history } = realize_gram(
-            &problem, 1.0e-12, 0.5, 0.9, 1.1, 200, 110
+            &problem, 1.0e-20, 0.5, 0.9, 1.1, 400, 110
         );
         
         /* DEBUG */

app-proto/src/components/diagnostics.rs

@@ -111,6 +111,94 @@ fn StepInput() -> View {
     }
 }
 
+#[component]
+fn DistortionGauge() -> View {
+    let state = use_context::<AppState>();
+    let total_distortion = create_memo(move || {
+        state.assembly.regulators.with(|regs| {
+            let mut total = 0.0;
+            for reg in regs {
+                if let Some(distortion) = reg.distortion() {
+                    total += distortion.get().abs();
+                }
+            }
+            total
+        })
+    });
+    
+    view! {
+        div(id = "distortion-gauge") {
+            "Distortion: " (total_distortion.with(|distort| distort.to_string()))
+        }
+    }
+}
+
+#[component]
+fn PrintButton() -> View {
+    view! {
+        button(
+            on:click = |_| {
+                let state = use_context::<AppState>();
+                
+                // print the edge length distortions
+                let mut hard_distortion_table = String::new();
+                let mut soft_distortion_table = String::new();
+                let mut highest_distortion = f64::NEG_INFINITY;
+                let mut lowest_distortion = f64::INFINITY;
+                let mut largest_hard_distortion = f64::NEG_INFINITY;
+                state.assembly.regulators.with_untracked(|regs| {
+                    for reg in regs {
+                        if let Some(distortion) = reg.distortion() {
+                            let distortion_val = distortion.get();
+                            let subjects = reg.subjects();
+                            let distortion_line = format!(
+                                "{}, {}: {distortion_val}\n",
+                                subjects[0].id(),
+                                subjects[1].id(),
+                            );
+                            match reg.soft() {
+                                Some(soft) if soft.get() => {
+                                    soft_distortion_table += &distortion_line;
+                                    highest_distortion = highest_distortion.max(distortion_val);
+                                    lowest_distortion = lowest_distortion.min(distortion_val);
+                                },
+                                _ => {
+                                    hard_distortion_table += &distortion_line;
+                                    largest_hard_distortion = largest_hard_distortion.max(distortion_val.abs());
+                                }
+                            };
+                        }
+                    }
+                });
+                console_log!("\
+                    === Distortions of flexible edges (for labels) ===\n\n\
+                    --- Range ---\n\n\
+                    Highest: {highest_distortion}\n\
+                    Lowest: {lowest_distortion}\n\n\
+                    --- Table ---\n\n{soft_distortion_table}\n\
+                    === Distortions of rigid edges (for validation) ===\n\n\
+                    These values should be small relative to the ones for the flexible edges\n\n\
+                    --- Range ---\n\n\
+                    Largest absolute: {largest_hard_distortion}\n\n\
+                    --- Table ---\n\n{hard_distortion_table}\
+                ");
+                
+                // print the vertex coordinates
+                let mut coords_table = String::new();
+                state.assembly.elements.with_untracked(|elts| {
+                    for elt in elts.iter().filter(|elt| elt.is_point()) {
+                        let (x, y, z) = elt.representation().with(
+                            |rep| (rep[0], rep[1], rep[2])
+                        );
+                        coords_table += &format!("{}: {x}, {y}, {z}\n", elt.id());
+                    }
+                });
+                console_log!("=== Vertex coordinates ===\n\n{coords_table}");
+            },
+        ) { "Print" }
+    }
+}
+
 fn into_log10_time_point((step, value): (usize, f64)) -> Vec<Option<f64>> {
     vec![
         Some(step as f64),
@@ -315,6 +403,10 @@ pub fn Diagnostics() -> View {
             }
             DiagnosticsPanel(name = "loss") { LossHistory {} }
             DiagnosticsPanel(name = "spectrum") { SpectrumHistory {} }
+            div(id = "distortion-bar") {
+                DistortionGauge {}
+                PrintButton {}
+            }
         }
     }
 }

app-proto/src/components/display.rs

@@ -597,16 +597,16 @@ pub fn Display() -> View {
                 |status| status.is_ok()
             );
             let step_val = state.assembly.step.with_untracked(|step| step.value);
-            let on_init_step = step_val.is_some_and(|n| n == 0.0);
             let on_last_step = step_val.is_some_and(
                 |n| state.assembly.descent_history.with_untracked(
                     |history| n as usize + 1 == history.config.len().max(1)
                 )
             );
-            let on_manipulable_step =
-                !realization_successful && on_init_step
-                || realization_successful && on_last_step;
-            if on_manipulable_step && state.selection.with(|sel| sel.len() == 1) {
+            if
+                state.selection.with(|sel| sel.len() == 1)
+                && realization_successful
+                && on_last_step
+            {
                 let sel = state.selection.with(
                     |sel| sel.into_iter().next().unwrap().clone()
                 );

app-proto/src/components/outline.rs

@@ -6,9 +6,11 @@ use web_sys::{KeyboardEvent, MouseEvent, wasm_bindgen::JsCast};
 use crate::{
     AppState,
     assembly::{
+        Axis,
         Element,
         HalfCurvatureRegulator,
         InversiveDistanceRegulator,
+        PointCoordinateRegulator,
         Regulator,
     },
     specified::SpecifiedValue
@@ -119,6 +121,19 @@ impl OutlineItem for HalfCurvatureRegulator {
     }
 }
 
+impl OutlineItem for PointCoordinateRegulator {
+    fn outline_item(self: Rc<Self>, _element: &Rc<dyn Element>) -> View {
+        view! {
+            li(class = "regulator") {
+                div(class = "regulator-label") { (Axis::NAME[self.axis as usize]) }
+                div(class = "regulator-type") { "Coordinate" }
+                RegulatorInput(regulator = self)
+                div(class = "status")
+            }
+        }
+    }
+}
+
 // a list item that shows an element in an outline view of an assembly
 #[component(inline_props)]
 fn ElementOutlineItem(element: Rc<dyn Element>) -> View {

app-proto/src/engine.rs

@@ -1,6 +1,7 @@
 use lazy_static::lazy_static;
 use nalgebra::{Const, DMatrix, DVector, DVectorView, Dyn, SymmetricEigen};
 use std::fmt::{Display, Error, Formatter};
+use sycamore::prelude::console_log; /* DEBUG */
 
 // --- elements ---
 
@@ -46,14 +47,14 @@ pub fn project_sphere_to_normalized(rep: &mut DVector<f64>) {
 
 // normalize a point's representation vector by scaling
 pub fn project_point_to_normalized(rep: &mut DVector<f64>) {
-    rep.scale_mut(0.5 / rep[3]);
+    rep.scale_mut(0.5 / rep[3]); //FIXME: This 3 should be Point::WEIGHT_COMPONENT
 }
 
 // --- partial matrices ---
 
 pub struct MatrixEntry {
-    index: (usize, usize),
-    value: f64,
+    pub index: (usize, usize),
+    pub value: f64,
 }
 
 pub struct PartialMatrix(Vec<MatrixEntry>);
@@ -240,6 +241,7 @@ impl DescentHistory {
 
 pub struct ConstraintProblem {
     pub gram: PartialMatrix,
+    pub soft: PartialMatrix,
     pub frozen: PartialMatrix,
     pub guess: DMatrix<f64>,
 }
@@ -249,6 +251,7 @@ impl ConstraintProblem {
         const ELEMENT_DIM: usize = 5;
         Self {
             gram: PartialMatrix::new(),
+            soft: PartialMatrix::new(),
             frozen: PartialMatrix::new(),
             guess: DMatrix::<f64>::zeros(ELEMENT_DIM, element_count),
         }
@@ -258,6 +261,7 @@ impl ConstraintProblem {
     pub fn from_guess(guess_columns: &[DVector<f64>]) -> Self {
         Self {
             gram: PartialMatrix::new(),
+            soft: PartialMatrix::new(),
             frozen: PartialMatrix::new(),
             guess: DMatrix::from_columns(guess_columns),
         }
@@ -280,14 +284,18 @@ lazy_static! {
 struct SearchState {
     config: DMatrix<f64>,
     err_proj: DMatrix<f64>,
+    loss_hard: f64,
     loss: f64,
 }
 
 impl SearchState {
-    fn from_config(gram: &PartialMatrix, config: DMatrix<f64>) -> Self {
-        let err_proj = gram.sub_proj(&(config.tr_mul(&*Q) * &config));
+    fn from_config(gram: &PartialMatrix, soft: &PartialMatrix, softness: f64, config: DMatrix<f64>) -> Self {
+        let config_gram = &(config.tr_mul(&*Q) * &config);
+        let err_proj_hard = gram.sub_proj(config_gram);
+        let err_proj = &err_proj_hard + softness * soft.sub_proj(config_gram);
+        let loss_hard = err_proj_hard.norm_squared();
         let loss = err_proj.norm_squared();
-        Self { config, err_proj, loss }
+        Self { config, err_proj, loss_hard, loss }
     }
 }
 
@@ -331,6 +339,8 @@ pub fn local_unif_to_std(v: DVectorView<f64>) -> DMatrix<f64> {
 // use backtracking line search to find a better configuration
 fn seek_better_config(
     gram: &PartialMatrix,
+    soft: &PartialMatrix,
+    softness: f64,
     state: &SearchState,
     base_step: &DMatrix<f64>,
     base_target_improvement: f64,
@@ -341,7 +351,7 @@ fn seek_better_config(
     let mut rate = 1.0;
     for backoff_steps in 0..max_backoff_steps {
         let trial_config = &state.config + rate * base_step;
-        let trial_state = SearchState::from_config(gram, trial_config);
+        let trial_state = SearchState::from_config(gram, soft, softness, trial_config);
         let improvement = state.loss - trial_state.loss;
         if improvement >= min_efficiency * rate * base_target_improvement {
             return Some((trial_state, backoff_steps));
@@ -376,7 +386,7 @@ pub fn realize_gram(
     max_backoff_steps: i32,
 ) -> Realization {
     // destructure the problem data
-    let ConstraintProblem { gram, guess, frozen } = problem;
+    let ConstraintProblem { gram, soft, guess, frozen } = problem;
     
     // start the descent history
     let mut history = DescentHistory::new();
@@ -403,13 +413,18 @@ pub fn realize_gram(
     let scale_adjustment = (gram.0.len() as f64).sqrt();
     let tol = scale_adjustment * scaled_tol;
     
+    // set up constants and variables related to minimizing the soft loss
+    const GRAD_TOL: f64 = 1e-12;
+    let mut grad_size = f64::INFINITY;
+    let mut softness = 1.0;
+    
     // convert the frozen indices to stacked format
     let frozen_stacked: Vec<usize> = frozen.into_iter().map(
         |MatrixEntry { index: (row, col), .. }| col*element_dim + row
     ).collect();
     
     // use a regularized Newton's method with backtracking
-    let mut state = SearchState::from_config(gram, frozen.freeze(guess));
+    let mut state = SearchState::from_config(gram, soft, softness, frozen.freeze(guess));
     let mut hess = DMatrix::zeros(element_dim, assembly_dim);
     for _ in 0..max_descent_steps {
         // find the negative gradient of the loss function
@@ -426,7 +441,7 @@ pub fn realize_gram(
                 let neg_d_err =
                     basis_mat.tr_mul(&*Q) * &state.config
                     + state.config.tr_mul(&*Q) * &basis_mat;
-                let neg_d_err_proj = gram.proj(&neg_d_err);
+                let neg_d_err_proj = gram.proj(&neg_d_err) + softness * soft.proj(&neg_d_err);
                 let deriv_grad = 4.0 * &*Q * (
                     -&basis_mat * &state.err_proj
                     + &state.config * &neg_d_err_proj
@@ -455,10 +470,13 @@ pub fn realize_gram(
             hess[(k, k)] = 1.0;
         }
         
-        // stop if the loss is tolerably low
+        // stop if the hard loss is tolerably low and the total loss is close to
+        // stationary. we use `neg_grad_stacked` to measure the size of the
+        // gradient because it's been projected onto the frozen subspace
         history.config.push(state.config.clone());
-        history.scaled_loss.push(state.loss / scale_adjustment);
-        if state.loss < tol { break; }
+        history.scaled_loss.push(state.loss_hard / scale_adjustment);
+        grad_size = neg_grad_stacked.norm_squared();
+        if state.loss_hard < tol && grad_size < softness * GRAD_TOL { break; }
         
         // compute the Newton step
         /* TO DO */
@@ -482,7 +500,7 @@ pub fn realize_gram(
         
         // use backtracking line search to find a better configuration
         if let Some((better_state, backoff_steps)) = seek_better_config(
-            gram, &state, &base_step, neg_grad.dot(&base_step),
+            gram, soft, softness, &state, &base_step, neg_grad.dot(&base_step),
             min_efficiency, backoff, max_backoff_steps,
         ) {
             state = better_state;
@@ -493,8 +511,18 @@ pub fn realize_gram(
                 history,
             };
         }
+        
+        // if we're near a minimum of the total loss, but the hard loss still
+        // isn't tolerably low, make the soft constraints softer
+        const SOFTNESS_BACKOFF_THRESHOLD: f64 = 1e-6;
+        const SOFTNESS_BACKOFF: f64 = 0.95;
+        if state.loss_hard >= tol && grad_size < softness * SOFTNESS_BACKOFF_THRESHOLD {
+            softness *= SOFTNESS_BACKOFF;
+            state = SearchState::from_config(gram, soft, softness, state.config);
+            console_log!("Softness decreased to {softness}");
+        }
     }
-    let result = if state.loss < tol {
+    let result = if state.loss_hard < tol && grad_size < softness * GRAD_TOL {
         // express the uniform basis in the standard basis
         const UNIFORM_DIM: usize = 4;
         let total_dim_unif = UNIFORM_DIM * assembly_dim;