@@ -63,6 +63,10 @@ required-features = ["std"]
 name = "ternlog_fold_probe"
 required-features = ["std"]
 
+[[example]]
+name = "perm_field_probe"
+required-features = ["std"]
+
 [[example]]
 name = "ternlog_popcnt_gap_probe"
 required-features = ["std"]
 
@@ -0,0 +1,327 @@
+//! One question over the whole 64×64 permutation field, answered by one fold.
+//!
+//! The expression is `f = ternlog(0x96)` (a XOR b XOR c) over a population of
+//! `N` 64-byte blocks. Each question is asked of the WHOLE field — all 4096
+//! cells of a `PermTable12` — and has one answer. Two executions:
+//!
+//! - **per-cell**: 4096 standalone folds. Each composes its cell's map,
+//!   shuffles the operands into that coordinate system, evaluates, and adds its
+//!   answer to the total.
+//! - **hot fold**: the data and the accumulator stay resident, and the field
+//!   streams through the one running fold as relations. Nothing is constructed
+//!   per cell. A shared basis disappears under an invariant terminal. A
+//!   relative basis or an output-coordinate mask stays in the fold through the
+//!   field's exposure (a 64×64 kernel built from the two 64-entry tables),
+//!   never as per-cell work.
+//!
+//! Ordered Keep is the one terminal that owes per-cell output. It is included
+//! to show where the per-cell cost genuinely remains.
+//!
+//! Every hot-fold answer is asserted equal to the per-cell answer. The
+//! counters count what this probe itself does: `evals` is the number of
+//! expression/predicate evaluations inside the fold, which is the incremental
+//! work that remains.
+//!
+//! ```sh
+//! cargo run --release --example perm_field_probe
+//! ```
+
+use ndarray::hpc::perm::{LaneOp, LaneTernlog, Perm64, PermBatch, PermTable12, Schedule};
+use std::time::Instant;
+
+const N: usize = 64;
+const CELLS: u16 = 4096;
+type Block = [u8; 64];
+
+#[derive(Default, Clone, Copy)]
+struct Counters {
+    compositions: u64,
+    shuffles: u64,
+    evals: u64,
+    bytes_written: u64,
+}
+
+struct Run<'a> {
+    t: &'a PermTable12,
+    c: Counters,
+}
+
+impl Run<'_> {
+    fn compose(&mut self, code: u16) -> Perm64 {
+        self.c.compositions += 1;
+        self.t.for_code(code)
+    }
+    fn shuffle(&mut self, p: Perm64, b: &Block) -> Block {
+        self.c.shuffles += 1;
+        self.c.bytes_written += 64;
+        p.materialize(b)
+    }
+    fn f(&mut self, a: &Block, b: &Block, c: &Block) -> Block {
+        self.c.evals += 64;
+        LaneTernlog(0x96).apply([a, b, c])
+    }
+}
+
+fn nonzero(v: &Block) -> u64 {
+    v.iter().filter(|&&x| x != 0).count() as u64
+}
+
+fn splitmix(s: &mut u64) -> u64 {
+    *s = s.wrapping_add(0x9E37_79B9_7F4A_7C15);
+    let mut z = *s;
+    z = (z ^ (z >> 30)).wrapping_mul(0xBF58_476D_1CE4_E5B9);
+    z = (z ^ (z >> 27)).wrapping_mul(0x94D0_49BB_1331_11EB);
+    z ^ (z >> 31)
+}
+
+fn shuffled(seed: u64) -> Perm64 {
+    let mut s = seed;
+    let mut idx: [u8; 64] = core::array::from_fn(|i| i as u8);
+    for i in (1..64).rev() {
+        idx.swap(i, (splitmix(&mut s) % (i as u64 + 1)) as usize);
+    }
+    Perm64::from_indices(idx).unwrap()
+}
+
+/// Sparse-ish bytes, so Count and Any are not saturated.
+fn population(seed: u64) -> Vec<Block> {
+    let mut s = seed;
+    (0..N)
+        .map(|_| {
+            core::array::from_fn(|_| {
+                if splitmix(&mut s).is_multiple_of(3) {
+                    (splitmix(&mut s) & 0xFF) as u8
+                } else {
+                    0
+                }
+            })
+        })
+        .collect()
+}
+
+struct Row {
+    name: &'static str,
+    c: Counters,
+    micros: f64,
+}
+
+fn run<T>(t: &PermTable12, name: &'static str, body: impl FnOnce(&mut Run) -> T) -> (T, Row) {
+    let mut r = Run {
+        t,
+        c: Counters::default(),
+    };
+    let start = Instant::now();
+    let out = body(&mut r);
+    let micros = start.elapsed().as_secs_f64() * 1e6;
+    (out, Row { name, c: r.c, micros })
+}
+
+fn report(case: &str, answer: &str, rows: &[Row]) {
+    println!("\n## {case}\nanswer: {answer} (all executions equal, asserted)");
+    println!(
+        "{:<10} {:>12} {:>10} {:>12} {:>13} {:>11}",
+        "execution", "compositions", "shuffles", "evals", "bytes_written", "µs"
+    );
+    for r in rows {
+        println!(
+            "{:<10} {:>12} {:>10} {:>12} {:>13} {:>11.1}",
+            r.name, r.c.compositions, r.c.shuffles, r.c.evals, r.c.bytes_written, r.micros
+        );
+    }
+}
+
+fn main() {
+    let table = PermTable12::new(core::array::from_fn(|i| shuffled(100 + i as u64)));
+    let (a, b, c) = (population(1), population(2), population(3));
+    println!("field: 4096 cells; population: {N} blocks per operand");
+
+    // ---- 1. Shared basis, Count over the field ---------------------------
+    // Σ over cells of Count(f(P·a, P·b, P·c)).
+    {
+        let (per_cell, r1) = run(&table, "per-cell", |r| {
+            let mut total = 0u64;
+            for code in 0..CELLS {
+                let p = r.compose(code);
+                for i in 0..N {
+                    let (pa, pb, pc) = (r.shuffle(p, &a[i]), r.shuffle(p, &b[i]), r.shuffle(p, &c[i]));
+                    total += nonzero(&r.f(&pa, &pb, &pc));
+                }
+            }
+            total
+        });
+        let (hot, r2) = run(&table, "hot fold", |r| {
+            // Equivariance moves P outside f, and Count removes it: every cell
+            // contributes the same amount. One pass, then scale.
+            let once: u64 = (0..N).map(|i| nonzero(&r.f(&a[i], &b[i], &c[i]))).sum();
+            once * CELLS as u64
+        });
+        assert_eq!(hot, per_cell);
+        assert_eq!(r2.c.compositions + r2.c.shuffles, 0);
+        report("shared basis, field-wide Count", &hot.to_string(), &[r1, r2]);
+    }
+
+    // ---- 2. Shared basis, Any over the field (stops at the first hit) -----
+    {
+        let (per_cell, r1) = run(&table, "per-cell", |r| {
+            for code in 0..CELLS {
+                let p = r.compose(code);
+                for i in 0..N {
+                    let (pa, pb, pc) = (r.shuffle(p, &a[i]), r.shuffle(p, &b[i]), r.shuffle(p, &c[i]));
+                    if nonzero(&r.f(&pa, &pb, &pc)) > 0 {
+                        return true;
+                    }
+                }
+            }
+            false
+        });
+        let (hot, r2) = run(&table, "hot fold", |r| (0..N).any(|i| nonzero(&r.f(&a[i], &b[i], &c[i])) > 0));
+        assert_eq!(hot, per_cell);
+        report("shared basis, field-wide Any (early exit)", &hot.to_string(), &[r1, r2]);
+    }
+
+    // ---- 3. Different bases, Count over the field -------------------------
+    // a is seen through P(cell); b and c through one fixed Q. The relation
+    // R = Q relative to P changes with the cell and survives Count, so it
+    // stays in the fold as the re-based field's exposure.
+    {
+        let q = table.for_code(0x5A5);
+        let (per_cell, r1) = run(&table, "per-cell", |r| {
+            let mut total = 0u64;
+            for code in 0..CELLS {
+                let p = r.compose(code);
+                for i in 0..N {
+                    let (pa, qb, qc) = (r.shuffle(p, &a[i]), r.shuffle(q, &b[i]), r.shuffle(q, &c[i]));
+                    total += nonzero(&r.f(&pa, &qb, &qc));
+                }
+            }
+            total
+        });
+        let (hot, r2) = run(&table, "hot fold", |r| {
+            let e = r.t.relative_field(q).exposure();
+            r.c.compositions += 128; // building the two re-based 64-entry tables
+            let mut total = 0u64;
+            for i in 0..N {
+                let (ab, bb, cb) = (&a[i], &b[i], &c[i]);
+                let mut evals = 0u64;
+                total += e.count_where(|lane, src| {
+                    evals += 1;
+                    ab[lane] ^ bb[src] ^ cb[src] != 0
+                });
+                r.c.evals += evals;
+            }
+            total
+        });
+        assert_eq!(hot, per_cell);
+        report("different bases (P per cell, fixed Q), field-wide Count", &hot.to_string(), &[r1, r2]);
+    }
+
+    // ---- 4. Mask fixed in OUTPUT coordinates, Count over the field ---------
+    // Σ over cells of Count(M ∧ P·f(a,b,c)). M does not move with P; it takes
+    // part in the fold through the exposure instead of per-cell transport.
+    {
+        let masks: Vec<u64> = {
+            let mut s = 77;
+            (0..N).map(|_| splitmix(&mut s)).collect()
+        };
+        let (per_cell, r1) = run(&table, "per-cell", |r| {
+            let mut total = 0u64;
+            for code in 0..CELLS {
+                let p = r.compose(code);
+                for i in 0..N {
+                    let (pa, pb, pc) = (r.shuffle(p, &a[i]), r.shuffle(p, &b[i]), r.shuffle(p, &c[i]));
+                    let v = r.f(&pa, &pb, &pc);
+                    total += (0..64)
+                        .filter(|&l| masks[i] >> l & 1 == 1 && v[l] != 0)
+                        .count() as u64;
+                }
+            }
+            total
+        });
+        let (hot, r2) = run(&table, "hot fold", |r| {
+            let e = r.t.exposure();
+            let mut total = 0u64;
+            for i in 0..N {
+                let v = r.f(&a[i], &b[i], &c[i]);
+                let m = masks[i];
+                let mut evals = 0u64;
+                total += e.count_where(|lane, src| {
+                    evals += 1;
+                    m >> lane & 1 == 1 && v[src] != 0
+                });
+                r.c.evals += evals;
+            }
+            total
+        });
+        assert_eq!(hot, per_cell);
+        report("mask fixed in output coordinates, field-wide Count", &hot.to_string(), &[r1, r2]);
+    }
+
+    // ---- 5. Ordered Keep: the one terminal that owes per-cell output ------
+    {
+        let (per_cell, r1) = run(&table, "per-cell", |r| {
+            let mut out = Vec::with_capacity(CELLS as usize * N);
+            for code in 0..CELLS {
+                let p = r.compose(code);
+                for i in 0..N {
+                    let (pa, pb, pc) = (r.shuffle(p, &a[i]), r.shuffle(p, &b[i]), r.shuffle(p, &c[i]));
+                    let v = r.f(&pa, &pb, &pc);
+                    out.push(v);
+                    r.c.bytes_written += 64;
+                }
+            }
+            out
+        });
+        let (hot, r2) = run(&table, "hot fold", |r| {
+            // Evaluate once in the shared basis; each cell is owed only its
+            // coordinates of the result.
+            let v: Vec<Block> = (0..N).map(|i| r.f(&a[i], &b[i], &c[i])).collect();
+            let mut out = Vec::with_capacity(CELLS as usize * N);
+            for code in 0..CELLS {
+                let p = r.compose(code);
+                for blk in &v {
+                    out.push(r.shuffle(p, blk));
+                }
+            }
+            out
+        });
+        assert_eq!(hot, per_cell);
+        report("shared basis, ordered Keep (per-cell output owed)", "4096 × N blocks", &[r1, r2]);
+    }
+
+    // ---- 6. Duplicate-heavy request stream, ordered Keep ------------------
+    {
+        let mut s = 5;
+        let stream: Vec<u16> = (0..20_000)
+            .map(|_| (splitmix(&mut s) % 300) as u16 * 13 % CELLS)
+            .collect();
+        let src = LaneTernlog(0x96).apply([&a[0], &b[0], &c[0]]);
+        let (per_request, r1) = run(&table, "per-cell", |r| {
+            stream
+                .iter()
+                .map(|&code| {
+                    let p = r.compose(code);
+                    r.shuffle(p, &src)
+                })
+                .collect::<Vec<Block>>()
+        });
+        let (dedup, r2) = run(&table, "dedup", |r| {
+            let composed = PermBatch::new(&stream, Schedule::Deduplicate).compose(r.t);
+            r.c.compositions += composed.compositions() as u64;
+            let mut out = vec![[0u8; 64]; stream.len()];
+            composed.materialize_into(&src, &mut out);
+            r.c.shuffles += out.len() as u64;
+            r.c.bytes_written += out.len() as u64 * 64;
+            out
+        });
+        assert_eq!(dedup, per_request, "multiplicity and order must survive deduplication");
+        let distinct = PermBatch::new(&stream, Schedule::Deduplicate)
+            .field()
+            .unwrap()
+            .len();
+        report(
+            &format!("{} requests over {distinct} distinct cells, ordered Keep", stream.len()),
+            "one block per request, in request order",
+            &[r1, r2],
+        );
+    }
+}