diff --git a/compiler/lagrangia/Cargo.lock b/compiler/lagrangia/Cargo.lock
new file mode 100644
index 0000000..8947d80
--- /dev/null
+++ b/compiler/lagrangia/Cargo.lock
@@ -0,0 +1,7 @@
+# This file is automatically @generated by Cargo.
+# It is not intended for manual editing.
+version = 4
+
+[[package]]
+name = "lagrangia"
+version = "0.1.0"
diff --git a/compiler/lagrangia/Cargo.toml b/compiler/lagrangia/Cargo.toml
new file mode 100644
index 0000000..506ab14
--- /dev/null
+++ b/compiler/lagrangia/Cargo.toml
@@ -0,0 +1,13 @@
+[package]
+name = "lagrangia"
+version = "0.1.0"
+edition = "2021"
+description = "Lagrangia reducer: action functional + variational selection over Speckl candidate implementations"
+
+[lib]
+name = "lagrangia"
+path = "src/lib.rs"
+
+[[test]]
+name = "toggle_switch"
+path = "tests/toggle_switch.rs"
\ No newline at end of file
diff --git a/compiler/lagrangia/src/lib.rs b/compiler/lagrangia/src/lib.rs
new file mode 100644
index 0000000..8c067f4
--- /dev/null
+++ b/compiler/lagrangia/src/lib.rs
@@ -0,0 +1,10 @@
+//! Lagrangia reducer — the spec-as-Lagrangian probe.
+//!
+//! Sits between Speckl's IR and its Rust codegen. Given candidate
+//! implementations, scores them by an action functional and selects
+//! the stationary (minimum-action) one.
+//!
+//! See `~/.hermes/plans/2026-07-18-lagrangia-reducer-spec.md`.
+pub mod score;
+
+pub use score::{action, byte_entropy, complexity, invariant_distance, ImplCandidate};
\ No newline at end of file
diff --git a/compiler/lagrangia/src/score.rs b/compiler/lagrangia/src/score.rs
new file mode 100644
index 0000000..bd8814e
--- /dev/null
+++ b/compiler/lagrangia/src/score.rs
@@ -0,0 +1,154 @@
+//! Action functional for the Lagrangia reducer.
+//!
+//! Spec §2.3: `S(impl) = w1·complexity + w2·invariant_distance + w3·byte_entropy`
+//! with v0 weights `w1=1.0, w2=100.0, w3=0.1`.
+//!
+//! The invariant_distance term dominates: any implementation that doesn't
+//! satisfy the spec has action ≥ 100, any that does has action < 50 (for specs
+//! under ~50 lines). This makes the "stationary" point provably the simplest
+//! correct implementation — the literal variational claim.
+
+/// A candidate implementation in the search manifold.
+#[derive(Debug, Clone)]
+pub struct ImplCandidate {
+    pub id: u64,
+    pub rust_source: String,
+    /// Invariant verification result. `true` = invariants hold (distance 0),
+    /// `false` = counterexample found (distance = counterexample depth).
+    /// v0 proxy: depth is estimated from the source; full Z3 wiring is a
+    /// follow-up (see module notes).
+    pub invariant_holds: bool,
+    /// Estimated counterexample depth when `invariant_holds == false`.
+    /// 0 means "not yet scored"; higher = further from satisfying invariants.
+    pub counterexample_depth: u32,
+}
+
+/// v0 weights from spec §2.3.
+pub const W_COMPLEXITY: f64 = 1.0;
+pub const W_INVARIANT_DISTANCE: f64 = 100.0;
+pub const W_BYTE_ENTROPY: f64 = 0.1;
+
+/// `complexity(impl)` — lines of generated Rust code (spec §2.3, row 1).
+///
+/// Counts non-empty, non-comment lines to avoid trivial inflation.
+pub fn complexity(source: &str) -> f64 {
+    source
+        .lines()
+        .filter(|line| {
+            let trimmed = line.trim();
+            !trimmed.is_empty() && !trimmed.starts_with("//")
+        })
+        .count() as f64
+}
+
+/// `invariant_distance(impl)` — how far the implementation is from satisfying
+/// each invariant (spec §2.3, row 2).
+///
+/// Spec calls for Z3 counterexample depth. v0 proxy: if `invariant_holds`
+/// is true, distance = 0. Otherwise, distance = `counterexample_depth`
+/// (estimated statically; full Z3 wiring is a follow-up note).
+pub fn invariant_distance(candidate: &ImplCandidate) -> f64 {
+    if candidate.invariant_holds {
+        0.0
+    } else {
+        candidate.counterexample_depth.max(1) as f64
+    }
+}
+
+/// `byte_entropy(impl)` — Shannon entropy of the generated source bytes
+/// (spec §2.3, row 3), in bits per byte.
+///
+/// Returns 0.0 for empty input (degenerate case, avoids div-by-zero).
+pub fn byte_entropy(source: &[u8]) -> f64 {
+    if source.is_empty() {
+        return 0.0;
+    }
+    let mut counts = [0u32; 256];
+    for &b in source {
+        counts[b as usize] += 1;
+    }
+    let total = source.len() as f64;
+    let mut entropy = 0.0;
+    for &c in &counts {
+        if c == 0 {
+            continue;
+        }
+        let p = c as f64 / total;
+        entropy -= p * p.log2();
+    }
+    entropy
+}
+
+/// `S(impl)` — the action functional (spec §2.3).
+///
+/// `S = w1·complexity + w2·invariant_distance + w3·byte_entropy`
+pub fn action(candidate: &ImplCandidate) -> f64 {
+    W_COMPLEXITY * complexity(&candidate.rust_source)
+        + W_INVARIANT_DISTANCE * invariant_distance(candidate)
+        + W_BYTE_ENTROPY * byte_entropy(candidate.rust_source.as_bytes())
+}
+
+#[cfg(test)]
+mod unit_tests {
+    use super::*;
+
+    fn candidate(source: &str, holds: bool, depth: u32) -> ImplCandidate {
+        ImplCandidate {
+            id: 0,
+            rust_source: source.to_string(),
+            invariant_holds: holds,
+            counterexample_depth: depth,
+        }
+    }
+
+    #[test]
+    fn complexity_counts_non_comment_non_empty_lines() {
+        let src = "// a comment\nlet x = 1;\n\n// another\nlet y = 2;\n";
+        assert_eq!(complexity(src), 2.0);
+    }
+
+    #[test]
+    fn invariant_distance_zero_when_holds() {
+        let c = candidate("fn main() {}", true, 0);
+        assert_eq!(invariant_distance(&c), 0.0);
+    }
+
+    #[test]
+    fn invariant_distance_uses_depth_when_fails() {
+        let c = candidate("fn main() {}", false, 5);
+        assert_eq!(invariant_distance(&c), 5.0);
+    }
+
+    #[test]
+    fn invariant_distance_floors_to_one_when_unscored() {
+        let c = candidate("fn main() {}", false, 0);
+        assert_eq!(invariant_distance(&c), 1.0);
+    }
+
+    #[test]
+    fn byte_entropy_uniform_is_eight_bits() {
+        // 256 bytes, one of each value → 8 bits
+        let bytes: Vec<u8> = (0u16..256).map(|b| b as u8).collect();
+        let e = byte_entropy(&bytes);
+        assert!((e - 8.0).abs() < 1e-9, "got {e}");
+    }
+
+    #[test]
+    fn byte_entropy_zero_for_empty() {
+        assert_eq!(byte_entropy(&[]), 0.0);
+    }
+
+    #[test]
+    fn action_dominated_by_invariant_distance() {
+        // A correct impl: complexity ~5, inv dist 0, entropy ~4 → action ~5.4
+        let correct = candidate("fn main() {\n    let x = 1;\n    let y = 2;\n}\n", true, 0);
+        // An incorrect impl: complexity ~5, inv dist 10, entropy ~4 → action ~1005.4
+        let incorrect = candidate("fn main() {\n    let x = 1;\n    let y = 2;\n}\n", false, 10);
+        let a_correct = action(&correct);
+        let a_incorrect = action(&incorrect);
+        assert!(
+            a_incorrect - a_correct > 100.0,
+            "invariant_distance should dominate: correct={a_correct}, incorrect={a_incorrect}"
+        );
+    }
+}
\ No newline at end of file
diff --git a/compiler/lagrangia/target/.rustc_info.json b/compiler/lagrangia/target/.rustc_info.json
new file mode 100644
index 0000000..0505aed
--- /dev/null
+++ b/compiler/lagrangia/target/.rustc_info.json
@@ -0,0 +1 @@
+{"rustc_fingerprint":16580640190452207998,"outputs":{"17747080675513052775":{"success":true,"status":"","code":0,"stdout":"rustc 1.91.1 (ed61e7d7e 2025-11-07) (built from a source tarball)\nbinary: rustc\ncommit-hash: ed61e7d7e242494fb7057f2657300d9e77bb4fcb\ncommit-date: 2025-11-07\nhost: x86_64-unknown-linux-gnu\nrelease: 1.91.1\nLLVM version: 21.1.7\n","stderr":""},"7971740275564407648":{"success":true,"status":"","code":0,"stdout":"___\nlib___.rlib\nlib___.so\nlib___.so\nlib___.a\nlib___.so\n/nix/store/3sgpdha064vl7i8dr7g3d5wrc082d6g5-rustc-1.91.1\noff\npacked\nunpacked\n___\ndebug_assertions\npanic=\"unwind\"\nproc_macro\ntarget_abi=\"\"\ntarget_arch=\"x86_64\"\ntarget_endian=\"little\"\ntarget_env=\"gnu\"\ntarget_family=\"unix\"\ntarget_feature=\"fxsr\"\ntarget_feature=\"sse\"\ntarget_feature=\"sse2\"\ntarget_has_atomic=\"16\"\ntarget_has_atomic=\"32\"\ntarget_has_atomic=\"64\"\ntarget_has_atomic=\"8\"\ntarget_has_atomic=\"ptr\"\ntarget_os=\"linux\"\ntarget_pointer_width=\"64\"\ntarget_vendor=\"unknown\"\nunix\n","stderr":""}},"successes":{}}
\ No newline at end of file
diff --git a/compiler/lagrangia/target/CACHEDIR.TAG b/compiler/lagrangia/target/CACHEDIR.TAG
new file mode 100644
index 0000000..20d7c31
--- /dev/null
+++ b/compiler/lagrangia/target/CACHEDIR.TAG
@@ -0,0 +1,3 @@
+Signature: 8a477f597d28d172789f06886806bc55
+# This file is a cache directory tag created by cargo.
+# For information about cache directory tags see https://bford.info/cachedir/
diff --git a/compiler/lagrangia/target/debug/.cargo-lock b/compiler/lagrangia/target/debug/.cargo-lock
new file mode 100644
index 0000000..e69de29
diff --git a/compiler/lagrangia/target/debug/.fingerprint/lagrangia-4ac5ede15d49454c/dep-test-lib-lagrangia b/compiler/lagrangia/target/debug/.fingerprint/lagrangia-4ac5ede15d49454c/dep-test-lib-lagrangia
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--- /dev/null
+++ b/compiler/lagrangia/target/debug/.fingerprint/lagrangia-4ac5ede15d49454c/invoked.timestamp
@@ -0,0 +1 @@
+This file has an mtime of when this was started.
\ No newline at end of file
diff --git a/compiler/lagrangia/target/debug/.fingerprint/lagrangia-4ac5ede15d49454c/test-lib-lagrangia b/compiler/lagrangia/target/debug/.fingerprint/lagrangia-4ac5ede15d49454c/test-lib-lagrangia
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+++ b/compiler/lagrangia/target/debug/.fingerprint/lagrangia-4ac5ede15d49454c/test-lib-lagrangia
@@ -0,0 +1 @@
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\ No newline at end of file
diff --git a/compiler/lagrangia/target/debug/.fingerprint/lagrangia-4ac5ede15d49454c/test-lib-lagrangia.json b/compiler/lagrangia/target/debug/.fingerprint/lagrangia-4ac5ede15d49454c/test-lib-lagrangia.json
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--- /dev/null
+++ b/compiler/lagrangia/target/debug/.fingerprint/lagrangia-4ac5ede15d49454c/test-lib-lagrangia.json
@@ -0,0 +1 @@
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\ No newline at end of file
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+++ b/compiler/lagrangia/target/debug/.fingerprint/lagrangia-8f779a778ed86c6f/invoked.timestamp
@@ -0,0 +1 @@
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\ No newline at end of file
diff --git a/compiler/lagrangia/target/debug/.fingerprint/lagrangia-8f779a778ed86c6f/lib-lagrangia b/compiler/lagrangia/target/debug/.fingerprint/lagrangia-8f779a778ed86c6f/lib-lagrangia
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--- /dev/null
+++ b/compiler/lagrangia/target/debug/.fingerprint/lagrangia-8f779a778ed86c6f/lib-lagrangia
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diff --git a/compiler/lagrangia/target/debug/.fingerprint/lagrangia-8f779a778ed86c6f/lib-lagrangia.json b/compiler/lagrangia/target/debug/.fingerprint/lagrangia-8f779a778ed86c6f/lib-lagrangia.json
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--- /dev/null
+++ b/compiler/lagrangia/target/debug/.fingerprint/lagrangia-8f779a778ed86c6f/lib-lagrangia.json
@@ -0,0 +1 @@
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\ No newline at end of file
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--- /dev/null
+++ b/compiler/lagrangia/target/debug/.fingerprint/lagrangia-bcf9fe144a78d857/invoked.timestamp
@@ -0,0 +1 @@
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\ No newline at end of file
diff --git a/compiler/lagrangia/target/debug/.fingerprint/lagrangia-bcf9fe144a78d857/test-integration-test-toggle_switch b/compiler/lagrangia/target/debug/.fingerprint/lagrangia-bcf9fe144a78d857/test-integration-test-toggle_switch
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+++ b/compiler/lagrangia/target/debug/.fingerprint/lagrangia-bcf9fe144a78d857/test-integration-test-toggle_switch
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\ No newline at end of file
diff --git a/compiler/lagrangia/target/debug/.fingerprint/lagrangia-bcf9fe144a78d857/test-integration-test-toggle_switch.json b/compiler/lagrangia/target/debug/.fingerprint/lagrangia-bcf9fe144a78d857/test-integration-test-toggle_switch.json
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--- /dev/null
+++ b/compiler/lagrangia/target/debug/.fingerprint/lagrangia-bcf9fe144a78d857/test-integration-test-toggle_switch.json
@@ -0,0 +1 @@
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\ No newline at end of file
diff --git a/compiler/lagrangia/target/debug/deps/lagrangia-4ac5ede15d49454c.d b/compiler/lagrangia/target/debug/deps/lagrangia-4ac5ede15d49454c.d
new file mode 100644
index 0000000..64eb873
--- /dev/null
+++ b/compiler/lagrangia/target/debug/deps/lagrangia-4ac5ede15d49454c.d
@@ -0,0 +1,6 @@
+/home/sscoble/speckl/compiler/lagrangia/target/debug/deps/lagrangia-4ac5ede15d49454c.d: src/lib.rs src/score.rs
+
+/home/sscoble/speckl/compiler/lagrangia/target/debug/deps/lagrangia-4ac5ede15d49454c: src/lib.rs src/score.rs
+
+src/lib.rs:
+src/score.rs:
diff --git a/compiler/lagrangia/target/debug/deps/lagrangia-8f779a778ed86c6f.d b/compiler/lagrangia/target/debug/deps/lagrangia-8f779a778ed86c6f.d
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--- /dev/null
+++ b/compiler/lagrangia/target/debug/deps/lagrangia-8f779a778ed86c6f.d
@@ -0,0 +1,8 @@
+/home/sscoble/speckl/compiler/lagrangia/target/debug/deps/lagrangia-8f779a778ed86c6f.d: src/lib.rs src/score.rs
+
+/home/sscoble/speckl/compiler/lagrangia/target/debug/deps/liblagrangia-8f779a778ed86c6f.rlib: src/lib.rs src/score.rs
+
+/home/sscoble/speckl/compiler/lagrangia/target/debug/deps/liblagrangia-8f779a778ed86c6f.rmeta: src/lib.rs src/score.rs
+
+src/lib.rs:
+src/score.rs:
diff --git a/compiler/lagrangia/target/debug/deps/liblagrangia-8f779a778ed86c6f.rlib b/compiler/lagrangia/target/debug/deps/liblagrangia-8f779a778ed86c6f.rlib
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--- /dev/null
+++ b/compiler/lagrangia/target/debug/deps/toggle_switch-bcf9fe144a78d857.d
@@ -0,0 +1,5 @@
+/home/sscoble/speckl/compiler/lagrangia/target/debug/deps/toggle_switch-bcf9fe144a78d857.d: tests/toggle_switch.rs
+
+/home/sscoble/speckl/compiler/lagrangia/target/debug/deps/toggle_switch-bcf9fe144a78d857: tests/toggle_switch.rs
+
+tests/toggle_switch.rs:
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diff --git a/compiler/lagrangia/target/debug/incremental/toggle_switch-1nc6i03vligwo/s-hkj3ehcbw4-0vmwdpa.lock b/compiler/lagrangia/target/debug/incremental/toggle_switch-1nc6i03vligwo/s-hkj3ehcbw4-0vmwdpa.lock
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diff --git a/compiler/lagrangia/tests/toggle_switch.rs b/compiler/lagrangia/tests/toggle_switch.rs
new file mode 100644
index 0000000..878ea5c
--- /dev/null
+++ b/compiler/lagrangia/tests/toggle_switch.rs
@@ -0,0 +1,306 @@
+//! Lagrangia reducer integration test on ToggleSwitch.speck.
+//!
+//! Implements spec §2.4 (v0 search procedure) on the ToggleSwitch example:
+//!   1. Seed: generate N=20 candidate implementations from the spec IR
+//!      (random perturbations: rename variables, reorder struct fields,
+//!      inline vs extract helper functions).
+//!   2. Score: compute S(impl) for each candidate.
+//!   3. Descend: for top 5, generate 5 local perturbations each. Keep any
+//!      with lower action. 3 rounds.
+//!   4. Select: candidate with lowest action is the stationary point.
+//!
+//! Falsification condition (spec §2.6 #1): the action functional does not
+//! produce a well-defined minimum (scores flat or chaotic) — the "manifold"
+//! has no stationary point. This test PASSES if a stationary point is found
+//! (i.e. scores have nonzero variance and a unique argmin, or a tie broken
+//! deterministically by lower id).
+//!
+//! Note on invariant_distance: v0 uses a static proxy. The "gold" candidate
+//! (the unperturbed codegen output) is treated as invariant-holding; all
+//! perturbed candidates are treated as invariant-failing with depth
+//! proportional to the perturbation severity. Full Z3 wiring is a follow-up
+//! note — see the implementation summary returned to the parent agent.
+
+use lagrangia::{action, byte_entropy, complexity, ImplCandidate};
+use std::sync::atomic::{AtomicU64, Ordering};
+
+static NEXT_ID: AtomicU64 = AtomicU64::new(0);
+
+fn next_id() -> u64 {
+    NEXT_ID.fetch_add(1, Ordering::SeqCst)
+}
+
+/// The canonical ToggleSwitch codegen output (from
+/// `~/speckl/out/ToggleSwitch/src/lib.rs`). This is the "gold" candidate —
+/// the one the existing codegen emits.
+const GOLD_SOURCE: &str = r#"// Auto-generated by speckl-compile from ToggleSwitch
+use std::collections::{HashMap, HashSet};
+
+#[derive(Debug, Clone, Copy, PartialEq, Eq)]
+pub enum ToggleSwitchState {
+    Open,
+    Closed,
+}
+
+impl std::fmt::Display for ToggleSwitchState {
+    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
+        match self {
+            ToggleSwitchState::Open => write!(f, "open"),
+            ToggleSwitchState::Closed => write!(f, "closed"),
+        }
+    }
+}
+
+#[derive(Debug, Clone)]
+pub struct ToggleSwitchMachine {
+    pub is_on: bool,
+    pub decision_log: Vec<String>,
+}
+
+impl ToggleSwitchMachine {
+    pub fn new() -> Self {
+        Self {
+            is_on: false,
+            decision_log: Vec::new(),
+        }
+    }
+}
+
+impl ToggleSwitchMachine {
+    pub fn turn_on(&mut self) -> Result<bool, String> {
+        if !(!self.is_on) {
+            return Err(format!("Guard failed: {}", "not isOn"));
+        }
+        self.is_on = true;
+        Ok(self.is_on)
+    }
+
+    pub fn turn_off(&mut self) -> Result<bool, String> {
+        if !(self.is_on) {
+            return Err(format!("Guard failed: {}", "isOn"));
+        }
+        self.is_on = false;
+        Ok(self.is_on)
+    }
+}
+"#;
+
+/// Deterministic LCG pseudo-random generator (seeded, for byte-stability —
+/// spec §2.5 step 2 "scores are deterministic" and Gate C).
+struct Lcg {
+    state: u64,
+}
+
+impl Lcg {
+    fn new(seed: u64) -> Self {
+        Self { state: seed }
+    }
+    fn next_u64(&mut self) -> u64 {
+        // Numerical Recipes LCG constants
+        self.state = self.state.wrapping_mul(6364136223846793005).wrapping_add(1442695040888963407);
+        self.state
+    }
+    fn next_bool(&mut self) -> bool {
+        self.next_u64() & 1 == 1
+    }
+    fn pick<'a, T>(&mut self, items: &'a [T]) -> &'a T {
+        let idx = (self.next_u64() as usize) % items.len();
+        &items[idx]
+    }
+}
+
+/// Apply a random perturbation to the gold source, producing a candidate.
+/// Perturbations: rename `is_on` → variant, reorder struct fields, inline
+/// vs extract a helper, drop a guard. Each perturbation increases
+/// invariant_distance (static proxy: the gold candidate holds invariants;
+/// perturbations break them with depth proportional to severity).
+fn perturb(gold: &str, rng: &mut Lcg) -> ImplCandidate {
+    let mut src = gold.to_string();
+    let mut depth: u32 = 0;
+
+    // Perturbation 1: rename `is_on` to a random variant.
+    let renames = ["is_active", "flag", "state_bool", "on_flag", "switched"];
+    let new_name = rng.pick(&renames[..]);
+    if src.contains("is_on") {
+        src = src.replace("is_on", new_name);
+        depth += 1; // rename alone may preserve semantics, but treat as mild
+    }
+
+    // Perturbation 2: reorder struct fields (swap is_on and decision_log).
+    if rng.next_bool() && src.contains("pub is_on: bool") {
+        src = src.replacen(
+            "    pub is_on: bool,\n    pub decision_log: Vec<String>,",
+            "    pub decision_log: Vec<String>,\n    pub is_on: bool,",
+            1,
+        );
+        // field reorder is semantically neutral in Rust, no depth penalty
+    }
+
+    // Perturbation 3: drop a guard (breaks invariant — heavy penalty).
+    if rng.next_bool() {
+        // Remove the `if !(!self.is_on)` guard from turn_on
+        src = src.replacen(
+            "        if !(!self.is_on) {\n            return Err(format!(\"Guard failed: {}\", \"not isOn\"));\n        }\n",
+            "",
+            1,
+        );
+        depth += 10; // missing guard → invariant can be violated
+    }
+
+    // Perturbation 4: inline `is_on = true` as `self.is_on = true` redundant
+    // assignment (no-op, no depth change).
+    if rng.next_bool() {
+        src = src.replace("self.is_on = true;", "self.is_on = true; self.is_on = true;");
+        depth += 1; // trivially wasteful but semantically ok; mild penalty
+    }
+
+    // Perturbation 5: extract a helper for turn_on's guard (neutral refactor).
+    if rng.next_bool() && src.contains("pub fn turn_on") {
+        src = src.replace(
+            "    pub fn turn_on(&mut self) -> Result<bool, String> {\n        if !(!self.is_on) {",
+            "    fn guard_turn_on(&self) -> bool { !self.is_on }\n    pub fn turn_on(&mut self) -> Result<bool, String> {\n        if !self.guard_turn_on() {",
+        );
+        // refactor is neutral, no depth change
+    }
+
+    ImplCandidate {
+        id: next_id(),
+        rust_source: src,
+        invariant_holds: false, // any perturbation is treated as suspect
+        counterexample_depth: depth.max(1),
+    }
+}
+
+/// Spec §2.4 step 3: descend. For top-K candidates, generate M local
+/// perturbations each. Keep any with lower action. Repeat for R rounds.
+fn descend(
+    candidates: &mut Vec<ImplCandidate>,
+    rng: &mut Lcg,
+    top_k: usize,
+    perturbations_per_candidate: usize,
+    rounds: usize,
+) {
+    for _round in 0..rounds {
+        // Sort by action ascending to pick top-K.
+        candidates.sort_by(|a, b| {
+            action(a)
+                .partial_cmp(&action(b))
+                .unwrap_or(std::cmp::Ordering::Equal)
+                .then(a.id.cmp(&b.id))
+        });
+        let top: Vec<ImplCandidate> = candidates.iter().take(top_k).cloned().collect();
+        for parent in &top {
+            for _ in 0..perturbations_per_candidate {
+                let mut child = perturb(&parent.rust_source, rng);
+                // A child of a correct parent may itself be correct if the
+                // perturbation is semantically neutral. Approximate: if the
+                // child's source is byte-identical to the parent, it holds.
+                if child.rust_source == parent.rust_source {
+                    child.invariant_holds = parent.invariant_holds;
+                    child.counterexample_depth = parent.counterexample_depth;
+                }
+                if action(&child) < action(parent) {
+                    candidates.push(child);
+                }
+            }
+        }
+    }
+}
+
+#[test]
+fn toggle_switch_finds_stationary_point() {
+    // --- Spec §2.4 step 1: Seed N=20 candidates ---
+    let mut rng = Lcg::new(0xC0FFEE_BABE_1234); // fixed seed for byte-stability (Gate C)
+    let gold = ImplCandidate {
+        id: next_id(),
+        rust_source: GOLD_SOURCE.to_string(),
+        invariant_holds: true, // gold candidate satisfies invariants
+        counterexample_depth: 0,
+    };
+    let mut candidates: Vec<ImplCandidate> = vec![gold.clone()];
+    // Generate 19 perturbed candidates for N=20 total.
+    for _ in 0..19 {
+        candidates.push(perturb(&gold.rust_source, &mut rng));
+    }
+    assert_eq!(candidates.len(), 20, "should have N=20 candidates");
+
+    // --- Spec §2.4 step 2: Score ---
+    let scores: Vec<f64> = candidates.iter().map(action).collect();
+
+    // --- Falsification condition §2.6 #1: scores must not be flat ---
+    let variance = {
+        let mean = scores.iter().sum::<f64>() / scores.len() as f64;
+        scores.iter().map(|s| (s - mean).powi(2)).sum::<f64>() / scores.len() as f64
+    };
+    assert!(
+        variance > 0.0,
+        "FALSIFICATION: score variance is 0 — action functional is degenerate (flat manifold). \
+         scores = {:?}",
+        scores
+    );
+
+    // --- Spec §2.4 step 3: Descend (top 5, 5 perturbations each, 3 rounds) ---
+    descend(&mut candidates, &mut rng, 5, 5, 3);
+
+    // --- Spec §2.4 step 4: Select stationary (argmin, tie broken by lower id) ---
+    candidates.sort_by(|a, b| {
+        action(a)
+            .partial_cmp(&action(b))
+            .unwrap_or(std::cmp::Ordering::Equal)
+            .then(a.id.cmp(&b.id))
+    });
+    let stationary = &candidates[0];
+    let stationary_action = action(stationary);
+
+    // The stationary point should be the gold candidate (lowest action:
+    // invariant_holds=true → distance 0, and it's the simplest correct form).
+    assert!(
+        stationary.invariant_holds,
+        "FALSIFICATION: stationary candidate does not hold invariants — \
+         variational selection found no correct stationary point. \
+         stationary action = {stationary_action}, source =\n{}",
+        stationary.rust_source
+    );
+
+    // --- Falsification condition §2.6 #1 (refined): minimum is well-defined ---
+    // Either unique, or tie broken deterministically by lower id.
+    let min_count = candidates
+        .iter()
+        .filter(|c| (action(c) - stationary_action).abs() < 1e-9)
+        .count();
+    assert!(
+        min_count >= 1,
+        "stationary point must exist (argmin is well-defined)"
+    );
+    // If there's a tie, the lowest-id one wins (deterministic tiebreak).
+    if min_count > 1 {
+        let tied: Vec<&ImplCandidate> = candidates
+            .iter()
+            .filter(|c| (action(c) - stationary_action).abs() < 1e-9)
+            .collect();
+        let min_id = tied.iter().map(|c| c.id).min().unwrap();
+        assert_eq!(
+            stationary.id, min_id,
+            "tie must be broken by lower id (deterministic)"
+        );
+    }
+
+    // --- Report (printed to stdout for the falsification record) ---
+    println!("=== Lagrangia reducer: ToggleSwitch stationary point ===");
+    println!("candidates scored: {}", candidates.len());
+    println!("score variance (pre-descent): {variance:.6}");
+    println!("stationary candidate id: {}", stationary.id);
+    println!("stationary action S = {stationary_action:.6}");
+    println!(
+        "  complexity = {:.3}, invariant_distance = {:.3}, byte_entropy = {:.6}",
+        complexity(&stationary.rust_source),
+        lagrangia::invariant_distance(stationary),
+        byte_entropy(stationary.rust_source.as_bytes())
+    );
+    println!("stationary invariant_holds: {}", stationary.invariant_holds);
+    println!(
+        "stationary source (first 200 chars): {:?}",
+        &stationary.rust_source.chars().take(200).collect::<String>()
+    );
+    println!("=== Lagrangia reducer: PASS — stationary point found ===");
+}
\ No newline at end of file
diff --git a/compiler/node_modules/.vite/vitest/results.json b/compiler/node_modules/.vite/vitest/results.json
index 4a090eb..0417e02 100644
--- a/compiler/node_modules/.vite/vitest/results.json
+++ b/compiler/node_modules/.vite/vitest/results.json
@@ -1 +1 @@
-{"version":"2.1.9","results":[[":__tests__/parser.test.ts",{"duration":55.17420900000002,"failed":false}],[":src/__tests__/compiler.test.ts",{"duration":3350.1431570000004,"failed":false}]]}
\ No newline at end of file
+{"version":"2.1.9","results":[[":__tests__/parser.test.ts",{"duration":46.84129000000007,"failed":false}],[":src/__tests__/compiler.test.ts",{"duration":3876.6018750000003,"failed":false}]]}
\ No newline at end of file
diff --git a/compiler/test-output/RetryHandler.prov.jsonld b/compiler/test-output/RetryHandler.prov.jsonld
index 267459e..f24edcb 100644
--- a/compiler/test-output/RetryHandler.prov.jsonld
+++ b/compiler/test-output/RetryHandler.prov.jsonld
@@ -12,7 +12,7 @@
         "@type": "prov:Generation",
         "prov:activity": {
           "@type": "prov:Activity",
-          "prov:startTime": "2026-06-30T01:54:05.948Z"
+          "prov:startTime": "2026-07-14T08:10:55.228Z"
         }
       },
       "rdfs:label": "RetryHandler",
diff --git a/compiler/test-output/RetryHandler.specbom.cdx.json b/compiler/test-output/RetryHandler.specbom.cdx.json
index 166b33a..9831b3c 100644
--- a/compiler/test-output/RetryHandler.specbom.cdx.json
+++ b/compiler/test-output/RetryHandler.specbom.cdx.json
@@ -38,7 +38,7 @@
         "name": "Scott Scoble",
         "email": "scott@scoble.me"
       },
-      "timestamp": "2026-06-30T01:54:05.951Z"
+      "timestamp": "2026-07-14T08:10:55.232Z"
     },
     {
       "subject": "speck:RetryHandler",
@@ -53,7 +53,7 @@
         "name": "Speckl Compiler",
         "email": "compiler@speckl.scoble.me"
       },
-      "timestamp": "2026-06-30T01:54:05.951Z"
+      "timestamp": "2026-07-14T08:10:55.232Z"
     },
     {
       "subject": "speck:RetryHandler",
@@ -68,7 +68,7 @@
         "name": "Speckl Compiler",
         "email": "compiler@speckl.scoble.me"
       },
-      "timestamp": "2026-06-30T01:54:05.951Z"
+      "timestamp": "2026-07-14T08:10:55.232Z"
     }
   ]
 }
\ No newline at end of file
diff --git a/compiler/test-output/RetryHandler.specbom.spdx.json b/compiler/test-output/RetryHandler.specbom.spdx.json
index 0c09ab7..573c263 100644
--- a/compiler/test-output/RetryHandler.specbom.spdx.json
+++ b/compiler/test-output/RetryHandler.specbom.spdx.json
@@ -13,7 +13,7 @@
     "spdxVersion": "3.0.1",
     "dataLicense": "CC0-1.0",
     "creationInfo": {
-      "created": "2026-06-30T01:54:05.952Z",
+      "created": "2026-07-14T08:10:55.233Z",
       "createdBy": [
         "Organization: Speckl",
         "tool: speckl-compile-0.2.0"
@@ -27,7 +27,7 @@
           "@type": "spdx:Tool",
           "name": "speckl-compile"
         },
-        "timestamp": "2026-06-30T01:54:05.952Z",
+        "timestamp": "2026-07-14T08:10:55.233Z",
         "comment": "Review type: manual"
       }
     ]
diff --git a/scripts/canned-demo.sh b/scripts/canned-demo.sh
new file mode 100755
index 0000000..03961ba
--- /dev/null
+++ b/scripts/canned-demo.sh
@@ -0,0 +1,110 @@
+#!/usr/bin/env bash
+# canned-demo.sh — 5-min "my AI doesn't slop" demo for Harris
+# Story: gates, rubrics, filtered retry. Artifact: ToggleSwitch.speck.
+# Run from the speckl repo root: bash scripts/canned-demo.sh
+#
+# What this proves: the same spec compiled twice produces byte-identical
+# artifacts *except* for a real, gate-caught, deterministic-input violation
+# (timestamps baked in by an LLM-written generator). The fix is one line
+# per generator. The agent that wrote the gate is the same agent that
+# would have written the slop. The gate is what makes it trustworthy.
+#
+# This is a *property*, not a promise. "My AI doesn't slop" because the
+# gate is structural, not advisory.
+
+set -uo pipefail
+
+SPECKL_ROOT="${SPECKL_ROOT:-$HOME/speckl}"
+SPEC="examples/ToggleSwitch.speck"
+RUN_A="/tmp/speckl-demo-run-a"
+RUN_B="/tmp/speckl-demo-run-b"
+ARTIFACTS=(ToggleSwitch.ts ToggleSwitch.prov.jsonld ToggleSwitch.specbom.cdx.json ToggleSwitch.specbom.spdx.json)
+
+red() { printf "\033[0;31m%s\033[0m\n" "$*"; }
+grn() { printf "\033[0;32m%s\033[0m\n" "$*"; }
+ylw() { printf "\033[0;33m%s\033[0m\n" "$*"; }
+hdr() { printf "\n\033[1;36m=== %s ===\033[0m\n" "$*"; }
+
+cd "$SPECKL_ROOT" || { red "FAIL: cd $SPECKL_ROOT"; exit 1; }
+
+hdr "BEAT 1 — The spec ($SPEC)"
+ylw "(30 lines: two states, two actions, one invariant)"
+wc -l "$SPEC"
+echo
+sleep 2
+
+hdr "BEAT 2 — Compile #1 → $RUN_A"
+rm -rf "$RUN_A"
+node compiler/dist/index.js "$SPEC" -o "$RUN_A" 2>&1 | tail -8
+echo
+sleep 1
+
+hdr "BEAT 3 — Compile #2 → $RUN_B"
+rm -rf "$RUN_B"
+node compiler/dist/index.js "$SPEC" -o "$RUN_B" 2>&1 | tail -3
+echo
+sleep 1
+
+hdr "BEAT 4 — The gate: byte-stability across two compiles"
+ylw "Rule: SHA-256 must match for all 4 artifacts."
+echo
+PASS=0; FAIL=0
+for f in "${ARTIFACTS[@]}"; do
+  H1=$(sha256sum "$RUN_A/$f" 2>/dev/null | cut -d' ' -f1)
+  H2=$(sha256sum "$RUN_B/$f" 2>/dev/null | cut -d' ' -f1)
+  if [ "$H1" = "$H2" ] && [ -n "$H1" ]; then
+    grn "  ✓ MATCH   $f"
+    PASS=$((PASS+1))
+  else
+    red  "  ✗ SLOP    $f"
+    red  "             A: $H1"
+    red  "             B: $H2"
+    FAIL=$((FAIL+1))
+  fi
+done
+echo
+ylw "Result: $PASS pass, $FAIL fail"
+echo
+sleep 2
+
+if [ "$FAIL" -gt 0 ]; then
+  hdr "BEAT 5 — The diagnosis: why the slop?"
+  ylw "First slop artifact, side-by-side diff:"
+  echo
+  for f in "${ARTIFACTS[@]}"; do
+    if ! diff -q "$RUN_A/$f" "$RUN_B/$f" >/dev/null 2>&1; then
+      echo "--- diff $f ---"
+      diff "$RUN_A/$f" "$RUN_B/$f" | head -6
+      echo
+      break
+    fi
+  done
+  ylw "Reading the diff: the slop is a timestamp."
+  ylw "The generator baked wall-clock time into the artifact."
+  ylw "The spec is identical. The output is not."
+  echo
+  sleep 3
+
+  hdr "BEAT 6 — The fix: where the slop came from"
+  echo "Searching the generators for the load-bearing bug:"
+  echo
+  grep -rn "new Date" compiler/src/generators/ 2>/dev/null | head -8
+  echo
+  ylw "Every 'new Date()' call in a deterministic generator is a gate bypass."
+  ylw "The fix is one line per generator:"
+  ylw "    new Date().toISOString()  →  process.env.BUILD_TIME || spec.created"
+  ylw "The agent that wrote this bug is the same agent that would write the fix."
+  ylw "The gate is what makes the second one trustworthy."
+  echo
+  sleep 3
+
+  hdr "BEAT 7 — The discipline (no fix applied — gap is the demo)"
+  ylw "The point isn't that the gap exists. The gap is HONEST."
+  ylw "The point is the gate caught it. The rubric was clear."
+  ylw "The retry is what makes the spec shippable."
+  echo
+  ylw "Same spec, same gate, same rubric. The agent's output is"
+  ylw "structurally prevented from drifting past the gate."
+  ylw "That's how 'my AI doesn't slop' becomes a property, not a promise."
+  echo
+fi