diff --git a/src/case_fold.rs b/src/case_fold.rs
index 3c41879..df9f787 100644
--- a/src/case_fold.rs
+++ b/src/case_fold.rs
@@ -386,6 +386,47 @@ mod tests {
         }
     }
 
+    /// Tier-3 exhaustive gate for the case-fold invariants over every code point.
+    ///
+    /// `fold_case` is a *per-code-point* transform (no composition, no cross-char
+    /// state), so a string's fold is exactly the concatenation of its chars' folds.
+    /// That makes single-code-point enumeration a **complete proof** for all inputs —
+    /// not the sampling the `\PC*` proptests below do. Cheap (~1.1M single chars), and
+    /// it catches any table entry whose folded form is not itself fully folded
+    /// (idempotency), reintroduces an ASCII uppercase, or maps to empty. `#[ignore]`
+    /// (Tier 3); run via the `--lib -- --ignored` step.
+    #[test]
+    #[ignore = "exhaustive: every code point through fold_case; run in Tier 3 / pre-release"]
+    fn exhaustive_fold_case_invariants() {
+        for cp in 0u32..=0x0010_FFFF {
+            let Some(ch) = char::from_u32(cp) else {
+                continue; // surrogates
+            };
+            let s = ch.to_string();
+            let once = fold_case_impl(&s);
+            // idempotent
+            assert_eq!(
+                once,
+                fold_case_impl(&once),
+                "fold_case not idempotent on U+{cp:04X}"
+            );
+            // never drops the char
+            assert!(!once.is_empty(), "fold_case emptied U+{cp:04X}");
+            // no residual ASCII uppercase
+            assert!(
+                !once.chars().any(|c| c.is_ascii_uppercase()),
+                "fold_case left ASCII uppercase for U+{cp:04X}: {once:?}"
+            );
+            // ASCII in ⇒ ASCII out
+            if ch.is_ascii() {
+                assert!(
+                    once.is_ascii(),
+                    "fold_case of ASCII U+{cp:04X} is non-ASCII"
+                );
+            }
+        }
+    }
+
     // ── Property-based tests ─────────────────────────────────────────
 
     mod proptest_properties {
diff --git a/src/filename.rs b/src/filename.rs
index 2c77e6b..a9b3857 100644
--- a/src/filename.rs
+++ b/src/filename.rs
@@ -492,6 +492,52 @@ mod tests {
         // Must be valid UTF-8 (implicit — Rust String guarantees this)
     }
 
+    /// Tier-3 exhaustive gate for `collapse_dot_sequences`.
+    ///
+    /// The collapse state machine turns purely on `.`-vs-not, so two exhaustive sweeps
+    /// prove it completely where the `\PC*` proptests sample: (1) every pattern over
+    /// {`.`, `a`} up to length 12, proving no `".."`, idempotency, and single-dot
+    /// preservation over every dot arrangement; and (2) every non-`.` code point is
+    /// preserved verbatim (per-char property). `#[ignore]` (Tier 3); run via
+    /// `--lib -- --ignored`.
+    #[test]
+    #[ignore = "exhaustive: collapse_dot_sequences over every dot pattern + code point; Tier 3"]
+    fn exhaustive_collapse_dot_sequences() {
+        // (1) every dot arrangement up to length 12.
+        let alphabet = ['.', 'a'];
+        let mut stack = vec![String::new()];
+        while let Some(s) = stack.pop() {
+            let once = collapse_dot_sequences(&s);
+            assert!(!once.contains(".."), "double dots from {s:?} → {once:?}");
+            assert_eq!(
+                once,
+                collapse_dot_sequences(&once),
+                "not idempotent on {s:?}"
+            );
+            if !s.contains("..") {
+                assert_eq!(once, s, "single-dot input altered: {s:?}");
+            }
+            if s.len() < 12 {
+                for &a in &alphabet {
+                    let mut n = s.clone();
+                    n.push(a);
+                    stack.push(n);
+                }
+            }
+        }
+        // (2) every non-dot code point passes through unchanged.
+        for cp in 0u32..=0x0010_FFFF {
+            let Some(c) = char::from_u32(cp) else {
+                continue;
+            };
+            if c == '.' {
+                continue;
+            }
+            let s = c.to_string();
+            assert_eq!(collapse_dot_sequences(&s), s, "dropped non-dot U+{cp:04X}");
+        }
+    }
+
     mod proptest_properties {
         use super::*;
         use proptest::prelude::*;
diff --git a/src/presets.rs b/src/presets.rs
index 023dec8..a79b7b7 100644
--- a/src/presets.rs
+++ b/src/presets.rs
@@ -2460,6 +2460,64 @@ mod tests {
             })
         }
 
+        /// Tier-3 exhaustive gate for preset idempotency (#416/#467/#498/#523 class).
+        ///
+        /// The key presets are fixed points: `canonicalize(canonicalize(x)) ==
+        /// canonicalize(x)`, and likewise for `sort_key`/`search_key`/`catalog_key`/
+        /// `ml_normalize`. The `adversarial()` proptests sample `any::<char>()`; this
+        /// enumerates the two domains where non-idempotency actually lives. (1) Every
+        /// single code point — the #498 class (a base exposed by NFKD/strip that only
+        /// resolves on a second pass). (2) Every BMP base × every combining diacritical
+        /// (U+0300–036F) — the #523 class (a fold/transliterate output that composes with
+        /// a following mark, or a composition that exposes a new fold); BMP covers every
+        /// composable Latin/Greek/Cyrillic base, and the astral planes are covered by (1).
+        /// `#[ignore]` (Tier 3): ~1.1M scalars across 5 presets plus ~7.1M BMP base×mark
+        /// pairs across 2 presets, each checked twice — on the order of 40M preset calls,
+        /// a few seconds in release.
+        #[test]
+        #[ignore = "exhaustive: preset idempotency over code points + base×mark; Tier 3"]
+        fn exhaustive_preset_idempotency() {
+            // Generic (monomorphized) so the tens-of-millions of calls in the inner loops
+            // pay no vtable dispatch — Tier-3 runtime stays predictable.
+            fn idem<F: Fn(&str) -> String>(label: &str, f: F, s: &str) {
+                let once = f(s);
+                assert_eq!(once, f(&once), "{label} not idempotent on {s:?}");
+            }
+            let cat = |s: &str| catalog_key(s, None, false).unwrap().into_owned();
+            let ml = |s: &str| ml_normalize(s, None, "cldr").unwrap().into_owned();
+
+            // (1) every single code point, across the key presets.
+            for cp in 0u32..=0x0010_FFFF {
+                let Some(c) = char::from_u32(cp) else {
+                    continue;
+                };
+                let s = c.to_string();
+                idem(
+                    "canonicalize",
+                    |x| canonicalize(x).unwrap().into_owned(),
+                    &s,
+                );
+                idem("sort_key", |x| sort_key(x, None).unwrap().into_owned(), &s);
+                idem(
+                    "search_key",
+                    |x| search_key(x, None).unwrap().into_owned(),
+                    &s,
+                );
+                idem("catalog_key", cat, &s);
+                idem("ml_normalize", ml, &s);
+            }
+
+            // (2) every BMP base × every combining diacritical — the compose/fold class.
+            let marks: Vec<char> = (0x0300u32..=0x036F).filter_map(char::from_u32).collect();
+            for base in (0u32..=0xFFFF).filter_map(char::from_u32) {
+                for &m in &marks {
+                    let s: String = [base, m].iter().collect();
+                    idem("catalog_key", cat, &s);
+                    idem("ml_normalize", ml, &s);
+                }
+            }
+        }
+
         proptest! {
             #![proptest_config(ProptestConfig::with_cases(1000))]
 
diff --git a/src/slugify.rs b/src/slugify.rs
index 94bee11..f76846c 100644
--- a/src/slugify.rs
+++ b/src/slugify.rs
@@ -1295,6 +1295,37 @@ mod tests {
         assert!(err.contains("[unclosed"), "pattern not echoed: {err}");
     }
 
+    /// Tier-3 exhaustive gate for the slug *codomain* over every code point.
+    ///
+    /// A slug's character set is a per-code-point property: every output char comes from
+    /// transliterating one input char to ASCII and slug-filtering it, so if every single
+    /// code point slugs to `[a-z0-9-]` (ASCII), so does every string. That makes this a
+    /// **complete proof** of `slugify_output_is_ascii` and the charset-membership half of
+    /// `slugify_output_charset`, where the `\PC*` proptests only sample. (The separator-
+    /// *position* rules — no leading/trailing/`--` — are cross-char and stay with the
+    /// proptests.) `#[ignore]` (Tier 3); run via `--lib -- --ignored`.
+    #[test]
+    #[ignore = "exhaustive: every code point through slugify; run in Tier 3 / pre-release"]
+    fn exhaustive_slug_codomain() {
+        let config = default_config();
+        for cp in 0u32..=0x0010_FFFF {
+            let Some(ch) = char::from_u32(cp) else {
+                continue; // surrogates
+            };
+            let result = slugify_impl(&ch.to_string(), &config);
+            assert!(
+                result.is_ascii(),
+                "non-ASCII slug for U+{cp:04X}: {result:?}"
+            );
+            for c in result.chars() {
+                assert!(
+                    c.is_ascii_lowercase() || c.is_ascii_digit() || c == '-',
+                    "slug of U+{cp:04X} has out-of-charset {c:?}: {result:?}"
+                );
+            }
+        }
+    }
+
     mod proptest_properties {
         use super::*;
         use proptest::prelude::*;
diff --git a/src/whitespace.rs b/src/whitespace.rs
index e3403ad..98ca5ba 100644
--- a/src/whitespace.rs
+++ b/src/whitespace.rs
@@ -257,6 +257,56 @@ mod tests {
         }
     }
 
+    /// Tier-3 exhaustive gate for the whitespace-collapse invariants (#433).
+    ///
+    /// `collapse_whitespace` treats every whitespace code point *uniformly* (any run of
+    /// fold-whitespace / blank-render collapses to one space, the ends are trimmed), so
+    /// its state machine turns on the whitespace-vs-not *pattern*. Two exhaustive sweeps
+    /// together pin it where the `\PC*` proptests only sample: (1) every code point in a
+    /// run context (`x␟c␟c␟y c z`), proving each is classified so runs collapse
+    /// idempotently with no `"  "` and trimmed ends; and (2) every pattern over {two
+    /// distinct whitespace chars, a non-ws letter} up to length 7, proving the
+    /// collapse/trim state machine on mixed runs and boundaries. `#[ignore]` (Tier 3);
+    /// run via `--lib -- --ignored`.
+    #[test]
+    #[ignore = "exhaustive: whitespace-collapse over every code point + patterns; Tier 3"]
+    fn exhaustive_collapse_whitespace() {
+        let check = |s: &str| {
+            let once = collapse_whitespace(s);
+            assert_eq!(once, collapse_whitespace(&once), "not idempotent on {s:?}");
+            assert!(!once.contains("  "), "double space from {s:?} → {once:?}");
+            if !once.is_empty() {
+                assert_ne!(once.as_bytes()[0], b' ', "leading space from {s:?}");
+                assert_ne!(
+                    *once.as_bytes().last().unwrap(),
+                    b' ',
+                    "trailing space {s:?}"
+                );
+            }
+        };
+        // (1) every code point in a collapsing run context.
+        for cp in 0u32..=0x0010_FFFF {
+            let Some(c) = char::from_u32(cp) else {
+                continue;
+            };
+            check(&format!("x{c}{c}y{c}z"));
+        }
+        // (2) every pattern over {SPACE, NBSP, 'x'} up to length 7 (uniform ws handling
+        //     means two distinct ws chars suffice to exercise mixed runs).
+        let alphabet = [' ', '\u{00A0}', 'x'];
+        let mut stack = vec![String::new()];
+        while let Some(s) = stack.pop() {
+            check(&s);
+            if s.chars().count() < 7 {
+                for &a in &alphabet {
+                    let mut n = s.clone();
+                    n.push(a);
+                    stack.push(n);
+                }
+            }
+        }
+    }
+
     mod proptest_properties {
         use super::*;
         use proptest::prelude::*;