Add S/λ² spectral sequence and B/Z/E classification API

ext/crates/sseq/src/coordinates/bze.rs

@@ -0,0 +1,50 @@
+use std::fmt::{self, Display, Formatter};
+
+use fp::matrix::Subquotient;
+
+/// Classification of a spectral sequence generator by its role relative to a differential on a
+/// given page. At page $r$, each generator in bidegree $b$ is exactly one of:
+///
+/// - **B** (boundary): in the image of $d_{r-1}$ from another bidegree.
+/// - **Z** (cycle): in the kernel of $d_r$, and not a boundary.
+/// - **E** (supports $d_r$): $d_r(x) \neq 0$.
+///
+/// Every spectral sequence admits this decomposition; the Adams spectral sequence for $S$ at $E_2$
+/// has everything as **Z** (degenerate), while $S/\lambda^2$ at $E_3$ has a nontrivial splitting.
+#[derive(Debug, Clone, Copy, PartialEq, Eq, Hash)]
+pub enum BZE {
+    /// Boundary: in the image of an incoming differential.
+    B,
+    /// Cycle mod boundary: survives to the next page.
+    Z,
+    /// Supports a differential: $d_r(x) \neq 0$.
+    E,
+}
+
+impl BZE {
+    /// Classify generator `idx` from a page's [`Subquotient`].
+    ///
+    /// The subquotient encodes $E_r = Z_r / B_r$ at a given bidegree:
+    /// - `page.zeros().pivots()[idx] >= 0` means `idx` is a boundary pivot (**B**).
+    /// - `page.complement_pivots()` yields generators not in the cycle subspace (**E**).
+    /// - Everything else is a cycle that is not a boundary (**Z**).
+    pub fn from_page_data(page: &Subquotient, idx: usize) -> Self {
+        if page.zeros().pivots()[idx] >= 0 {
+            return Self::B;
+        }
+        if page.complement_pivots().any(|p| p == idx) {
+            return Self::E;
+        }
+        Self::Z
+    }
+}
+
+impl Display for BZE {
+    fn fmt(&self, f: &mut Formatter) -> fmt::Result {
+        match self {
+            Self::B => write!(f, "B"),
+            Self::Z => write!(f, "Z"),
+            Self::E => write!(f, "E"),
+        }
+    }
+}

ext/crates/sseq/src/coordinates/mod.rs

@@ -1,10 +1,12 @@
+pub use bze::BZE;
 pub use degree::MultiDegree;
 pub use element::MultiDegreeElement;
 pub use generator::MultiDegreeGenerator;
 use maybe_rayon::prelude::*;
 use ordered::OrderedMultiDegree;
 pub use range::BidegreeRange;
 
+pub mod bze;
 pub mod degree;
 pub mod element;
 pub mod generator;

ext/crates/sseq/src/sseq.rs

@@ -9,7 +9,9 @@ use fp::{
 use once::MultiIndexed;
 
 use crate::{
-    coordinates::{Bidegree, BidegreeGenerator, degree::MultiDegree, element::MultiDegreeElement},
+    coordinates::{
+        BZE, Bidegree, BidegreeGenerator, degree::MultiDegree, element::MultiDegreeElement,
+    },
     differential::Differential,
 };
 
@@ -39,6 +41,28 @@ impl SseqProfile<2> for Adams {
     }
 }
 
+/// Trigraded Adams profile for S/λ². Coordinates are `(n, s, b)` = (stem, Adams filtration,
+/// Bockstein degree). The d₂ differential maps `(n, s, b) → (n − 1, s + 2, b + 1)`.
+pub struct AdamsLambda2;
+
+impl SseqProfile<3> for AdamsLambda2 {
+    const MIN_R: i32 = 2;
+
+    fn profile(r: i32, b: MultiDegree<3>) -> MultiDegree<3> {
+        let [n, s, bock] = b.coords();
+        MultiDegree::new([n - 1, s + r, bock + 1])
+    }
+
+    fn profile_inverse(r: i32, b: MultiDegree<3>) -> MultiDegree<3> {
+        let [n, s, bock] = b.coords();
+        MultiDegree::new([n + 1, s - r, bock - 1])
+    }
+
+    fn differential_length(offset: MultiDegree<3>) -> i32 {
+        offset.coords()[1]
+    }
+}
+
 pub struct Product<const N: usize> {
     pub b: MultiDegree<N>,
     /// Whether the product acts on the left or not. This affects the sign in the Leibniz rule.
@@ -368,6 +392,30 @@ impl<const N: usize, P: SseqProfile<N>> Sseq<N, P> {
         &self.data[b].page_data
     }
 
+    /// Classify generator `i` at degree `b` on page `r` as B (boundary), Z (cycle), or E
+    /// (supports a differential). Panics if the degree or page is not defined.
+    pub fn classify(&self, b: MultiDegree<N>, r: i32, i: usize) -> BZE {
+        let pd = &self.data[b].page_data;
+        let page = &pd[std::cmp::min(r, pd.len() - 1)];
+        BZE::from_page_data(page, i)
+    }
+
+    /// The B, Z, E generator indices at degree `b` on page `r`.
+    pub fn bze_indices(&self, b: MultiDegree<N>, r: i32) -> (Vec<usize>, Vec<usize>, Vec<usize>) {
+        let dim = self.data[b].dimension;
+        let mut bs = Vec::new();
+        let mut zs = Vec::new();
+        let mut es = Vec::new();
+        for i in 0..dim {
+            match self.classify(b, r, i) {
+                BZE::B => bs.push(i),
+                BZE::Z => zs.push(i),
+                BZE::E => es.push(i),
+            }
+        }
+        (bs, zs, es)
+    }
+
     /// Compute the product between `product` and the class `class`. Returns `None` if
     /// the product is not yet computed.
     pub fn multiply(

ext/examples/benchmarks/secondary-C2

@@ -1,12 +1,12 @@
 secondary -- C2 "" 30 7 ""
-d_2 x_(9, 2, 0) = [0]
-d_2 x_(10, 3, 0) = [0]
-d_2 x_(17, 3, 0) = [0]
-d_2 x_(17, 4, 0) = [1]
-d_2 x_(18, 2, 0) = [0]
-d_2 x_(18, 4, 0) = [0]
-d_2 x_(19, 5, 0) = [1]
-d_2 x_(20, 3, 0) = [0]
-d_2 x_(22, 3, 0) = [0]
-d_2 x_(22, 4, 0) = [0]
-d_2 x_(24, 5, 0) = [0]
+Z  x_(9, 2, 0)
+Z  x_(10, 3, 0)
+Z  x_(17, 3, 0)
+E  d_2 x_(17, 4, 0) = [1]
+Z  x_(18, 2, 0)
+Z  x_(18, 4, 0)
+E  d_2 x_(19, 5, 0) = [1]
+Z  x_(20, 3, 0)
+Z  x_(22, 3, 0)
+Z  x_(22, 4, 0)
+Z  x_(24, 5, 0)

ext/examples/benchmarks/secondary-C2-nassau

@@ -1,12 +1,12 @@
 secondary -- C2 /tmp/test_nassau_c2 30 7 ""
-d_2 x_(9, 2, 0) = [0]
-d_2 x_(10, 3, 0) = [0]
-d_2 x_(17, 3, 0) = [0]
-d_2 x_(17, 4, 0) = [1]
-d_2 x_(18, 2, 0) = [0]
-d_2 x_(18, 4, 0) = [0]
-d_2 x_(19, 5, 0) = [1]
-d_2 x_(20, 3, 0) = [0]
-d_2 x_(22, 3, 0) = [0]
-d_2 x_(22, 4, 0) = [0]
-d_2 x_(24, 5, 0) = [0]
+Z  x_(9, 2, 0)
+Z  x_(10, 3, 0)
+Z  x_(17, 3, 0)
+E  d_2 x_(17, 4, 0) = [1]
+Z  x_(18, 2, 0)
+Z  x_(18, 4, 0)
+E  d_2 x_(19, 5, 0) = [1]
+Z  x_(20, 3, 0)
+Z  x_(22, 3, 0)
+Z  x_(22, 4, 0)
+Z  x_(24, 5, 0)

ext/examples/benchmarks/secondary-S_2

@@ -1,19 +1,19 @@
 secondary -- S_2 "" 30 7 ""
-d_2 x_(1, 1, 0) = [0]
-d_2 x_(8, 2, 0) = [0]
-d_2 x_(15, 1, 0) = [1]
-d_2 x_(15, 2, 0) = [0]
-d_2 x_(15, 3, 0) = [0]
-d_2 x_(15, 4, 0) = [0]
-d_2 x_(16, 2, 0) = [0]
-d_2 x_(17, 4, 0) = [1]
-d_2 x_(17, 5, 0) = [0]
-d_2 x_(18, 2, 0) = [0]
-d_2 x_(18, 3, 0) = [0]
-d_2 x_(18, 4, 0) = [0]
-d_2 x_(18, 4, 1) = [1]
-d_2 x_(18, 5, 0) = [1]
-d_2 x_(19, 3, 0) = [0]
-d_2 x_(21, 3, 0) = [0]
-d_2 x_(24, 5, 0) = [0]
-d_2 x_(30, 5, 0) = [0]
+Z  x_(1, 1, 0)
+Z  x_(8, 2, 0)
+E  d_2 x_(15, 1, 0) = [1]
+Z  x_(15, 2, 0)
+Z  x_(15, 3, 0)
+Z  x_(15, 4, 0)
+Z  x_(16, 2, 0)
+E  d_2 x_(17, 4, 0) = [1]
+Z  x_(17, 5, 0)
+Z  x_(18, 2, 0)
+Z  x_(18, 3, 0)
+Z  x_(18, 4, 0)
+E  d_2 x_(18, 4, 1) = [1]
+E  d_2 x_(18, 5, 0) = [1]
+Z  x_(19, 3, 0)
+Z  x_(21, 3, 0)
+Z  x_(24, 5, 0)
+Z  x_(30, 5, 0)

ext/examples/benchmarks/secondary-tmf

@@ -1,13 +1,13 @@
 secondary -- A-mod-Sq1-Sq2-Sq4 "" 30 7 ""
-d_2 x_(1, 1, 0) = [0]
-d_2 x_(9, 4, 0) = [0]
-d_2 x_(9, 5, 0) = [0]
-d_2 x_(12, 3, 0) = [1]
-d_2 x_(12, 4, 0) = [1]
-d_2 x_(12, 5, 0) = [1]
-d_2 x_(15, 3, 0) = [1]
-d_2 x_(15, 4, 0) = [1]
-d_2 x_(17, 5, 0) = [0]
-d_2 x_(18, 4, 0) = [1]
-d_2 x_(21, 5, 0) = [0]
-d_2 x_(25, 5, 0) = [0]
+Z  x_(1, 1, 0)
+Z  x_(9, 4, 0)
+Z  x_(9, 5, 0)
+E  d_2 x_(12, 3, 0) = [1]
+E  d_2 x_(12, 4, 0) = [1]
+E  d_2 x_(12, 5, 0) = [1]
+E  d_2 x_(15, 3, 0) = [1]
+E  d_2 x_(15, 4, 0) = [1]
+Z  x_(17, 5, 0)
+E  d_2 x_(18, 4, 0) = [1]
+Z  x_(21, 5, 0)
+Z  x_(25, 5, 0)

ext/examples/benchmarks/secondary_product-h_0-S_2

@@ -1,38 +1,38 @@
 secondary_product -- S_2 "" 30 7 h_0 0 1 [1]
-[h_0] [x_(0, 0, 0)] = [1] + λ [0]
-[h_0] [x_(0, 1, 0)] = [1] + λ [1]
-[h_0] [x_(0, 2, 0)] = [1] + λ [0]
-[h_0] [x_(0, 3, 0)] = [1] + λ [1]
-[h_0] [x_(0, 4, 0)] = [1] + λ [0]
-[h_0] [x_(0, 5, 0)] = [1] + λ [1]
-[h_0] [x_(3, 1, 0)] = [1] + λ [1]
-[h_0] [x_(3, 2, 0)] = [1] + λ []
-[h_0] [x_(7, 1, 0)] = [1] + λ [0]
-[h_0] [x_(7, 2, 0)] = [1] + λ [1]
-[h_0] [x_(7, 3, 0)] = [1] + λ []
-[h_0] [x_(8, 2, 0)] = [0] + λ []
-[h_0] [x_(9, 3, 0)] = [0] + λ [0]
-[h_0] [x_(9, 4, 0)] = [0] + λ []
-[h_0] [x_(11, 5, 0)] = [1] + λ [1]
-[h_0] [x_(14, 2, 0)] = [1] + λ [1]
-[h_0] [x_(14, 3, 0)] = [0] + λ [0]
-[h_0] [x_(14, 4, 0)] = [1] + λ [1]
-[h_0] [x_(14, 5, 0)] = [1] + λ []
-[h_0] λ x_(15, 1, 0) = λ [1]
-[h_0] [x_(15, 2, 0)] = [1] + λ [1]
-[h_0] [x_(15, 3, 0)] = [1] + λ [0, 0]
-[h_0] [x_(15, 4, 0)] = [1, 0] + λ [1]
-[h_0] [x_(15, 5, 0)] = [1] + λ [0]
-[h_0] [x_(15, 5, 1)] = [0] + λ [0]
-[h_0] [x_(17, 3, 0)] = [0] + λ [0]
-[h_0] λ x_(17, 4, 0) = λ [1]
-[h_0] [x_(17, 5, 0)] = [1] + λ [0]
-[h_0] [x_(18, 2, 0)] = [1] + λ [1, 1]
-[h_0] [x_(18, 3, 0)] = [1, 0] + λ [1]
-[h_0] λ x_(18, 4, 1) = λ [1]
-[h_0] [x_(18, 4, 0)] = [0] + λ []
-[h_0] [x_(20, 4, 0)] = [1] + λ [0]
-[h_0] [x_(20, 5, 0)] = [1] + λ []
-[h_0] [x_(21, 3, 0)] = [] + λ [0]
-[h_0] [x_(23, 4, 0)] = [0] + λ [0]
-[h_0] [x_(23, 5, 0)] = [1] + λ [0]
+Z  [h_0] [x_(0, 0, 0)] = [x_(0, 1, 0)]
+Z  [h_0] [x_(0, 1, 0)] = [x_(0, 2, 0)] + λx_(0, 3, 0)
+Z  [h_0] [x_(0, 2, 0)] = [x_(0, 3, 0)]
+Z  [h_0] [x_(0, 3, 0)] = [x_(0, 4, 0)] + λx_(0, 5, 0)
+Z  [h_0] [x_(0, 4, 0)] = [x_(0, 5, 0)]
+Z  [h_0] [x_(0, 5, 0)] = [x_(0, 6, 0)] + λx_(0, 7, 0)
+Z  [h_0] [x_(3, 1, 0)] = [x_(3, 2, 0)] + λx_(3, 3, 0)
+Z  [h_0] [x_(3, 2, 0)] = [x_(3, 3, 0)]
+Z  [h_0] [x_(7, 1, 0)] = [x_(7, 2, 0)]
+Z  [h_0] [x_(7, 2, 0)] = [x_(7, 3, 0)] + λx_(7, 4, 0)
+Z  [h_0] [x_(7, 3, 0)] = [x_(7, 4, 0)]
+Z  [h_0] [x_(8, 2, 0)] = 0
+Z  [h_0] [x_(9, 3, 0)] = 0
+Z  [h_0] [x_(9, 4, 0)] = 0
+Z  [h_0] [x_(11, 5, 0)] = [x_(11, 6, 0)] + λx_(11, 7, 0)
+Z  [h_0] [x_(14, 2, 0)] = [x_(14, 3, 0)] + λx_(14, 4, 0)
+B  [h_0] [x_(14, 3, 0)] = 0
+Z  [h_0] [x_(14, 4, 0)] = [x_(14, 5, 0)] + λx_(14, 6, 0)
+Z  [h_0] [x_(14, 5, 0)] = [x_(14, 6, 0)]
+E  [h_0] λ x_(15, 1, 0) = λ [1]
+Z  [h_0] [x_(15, 2, 0)] = [x_(15, 3, 0)] + λx_(15, 4, 0)
+Z  [h_0] [x_(15, 3, 0)] = [x_(15, 4, 0)]
+Z  [h_0] [x_(15, 4, 0)] = [x_(15, 5, 0)] + λx_(15, 6, 0)
+Z  [h_0] [x_(15, 5, 0)] = [x_(15, 6, 0)]
+Z  [h_0] [x_(15, 5, 1)] = 0
+Z  [h_0] [x_(17, 3, 0)] = 0
+E  [h_0] λ x_(17, 4, 0) = λ [1]
+Z  [h_0] [x_(17, 5, 0)] = [x_(17, 6, 0)]
+Z  [h_0] [x_(18, 2, 0)] = [x_(18, 3, 0)] + λ(x_(18, 4, 0) + x_(18, 4, 1))
+Z  [h_0] [x_(18, 3, 0)] = [x_(18, 4, 0)] + λx_(18, 5, 0)
+E  [h_0] λ x_(18, 4, 1) = λ [1]
+Z  [h_0] [x_(18, 4, 0)] = 0
+Z  [h_0] [x_(20, 4, 0)] = [x_(20, 5, 0)]
+Z  [h_0] [x_(20, 5, 0)] = [x_(20, 6, 0)]
+Z  [h_0] [x_(21, 3, 0)] = 0
+Z  [h_0] [x_(23, 4, 0)] = 0
+Z  [h_0] [x_(23, 5, 0)] = [x_(23, 6, 0)]

ext/examples/benchmarks/secondary_product-h_0-S_2-nassau

@@ -1,38 +1,38 @@
 secondary_product -- S_2 /tmp/test_nassau_s_2 30 7 h_0 0 1 [1]
-[h_0] [x_(0, 0, 0)] = [1] + λ [0]
-[h_0] [x_(0, 1, 0)] = [1] + λ [1]
-[h_0] [x_(0, 2, 0)] = [1] + λ [0]
-[h_0] [x_(0, 3, 0)] = [1] + λ [1]
-[h_0] [x_(0, 4, 0)] = [1] + λ [0]
-[h_0] [x_(0, 5, 0)] = [1] + λ [1]
-[h_0] [x_(3, 1, 0)] = [1] + λ [1]
-[h_0] [x_(3, 2, 0)] = [1] + λ []
-[h_0] [x_(7, 1, 0)] = [1] + λ [0]
-[h_0] [x_(7, 2, 0)] = [1] + λ [1]
-[h_0] [x_(7, 3, 0)] = [1] + λ []
-[h_0] [x_(8, 2, 0)] = [0] + λ []
-[h_0] [x_(9, 3, 0)] = [0] + λ [0]
-[h_0] [x_(9, 4, 0)] = [0] + λ []
-[h_0] [x_(11, 5, 0)] = [1] + λ [1]
-[h_0] [x_(14, 2, 0)] = [1] + λ [0]
-[h_0] [x_(14, 3, 0)] = [0] + λ [1]
-[h_0] [x_(14, 4, 0)] = [1] + λ [0]
-[h_0] [x_(14, 5, 0)] = [1] + λ []
-[h_0] λ x_(15, 1, 0) = λ [1]
-[h_0] [x_(15, 2, 0)] = [1] + λ [1]
-[h_0] [x_(15, 3, 0)] = [1] + λ [0, 0]
-[h_0] [x_(15, 4, 0)] = [1, 0] + λ [1]
-[h_0] [x_(15, 5, 0)] = [1] + λ [0]
-[h_0] [x_(15, 5, 1)] = [0] + λ [0]
-[h_0] [x_(17, 3, 0)] = [0] + λ [0]
-[h_0] λ x_(17, 4, 0) = λ [1]
-[h_0] [x_(17, 5, 0)] = [1] + λ [0]
-[h_0] [x_(18, 2, 0)] = [1] + λ [1, 0]
-[h_0] [x_(18, 3, 0)] = [1, 0] + λ [0]
-[h_0] λ x_(18, 4, 1) = λ [1]
-[h_0] [x_(18, 4, 0)] = [0] + λ []
-[h_0] [x_(20, 4, 0)] = [1] + λ [1]
-[h_0] [x_(20, 5, 0)] = [1] + λ []
-[h_0] [x_(21, 3, 0)] = [] + λ [0]
-[h_0] [x_(23, 4, 0)] = [0] + λ [0]
-[h_0] [x_(23, 5, 0)] = [1] + λ [0]
+Z  [h_0] [x_(0, 0, 0)] = [x_(0, 1, 0)]
+Z  [h_0] [x_(0, 1, 0)] = [x_(0, 2, 0)] + λx_(0, 3, 0)
+Z  [h_0] [x_(0, 2, 0)] = [x_(0, 3, 0)]
+Z  [h_0] [x_(0, 3, 0)] = [x_(0, 4, 0)] + λx_(0, 5, 0)
+Z  [h_0] [x_(0, 4, 0)] = [x_(0, 5, 0)]
+Z  [h_0] [x_(0, 5, 0)] = [x_(0, 6, 0)] + λx_(0, 7, 0)
+Z  [h_0] [x_(3, 1, 0)] = [x_(3, 2, 0)] + λx_(3, 3, 0)
+Z  [h_0] [x_(3, 2, 0)] = [x_(3, 3, 0)]
+Z  [h_0] [x_(7, 1, 0)] = [x_(7, 2, 0)]
+Z  [h_0] [x_(7, 2, 0)] = [x_(7, 3, 0)] + λx_(7, 4, 0)
+Z  [h_0] [x_(7, 3, 0)] = [x_(7, 4, 0)]
+Z  [h_0] [x_(8, 2, 0)] = 0
+Z  [h_0] [x_(9, 3, 0)] = 0
+Z  [h_0] [x_(9, 4, 0)] = 0
+Z  [h_0] [x_(11, 5, 0)] = [x_(11, 6, 0)] + λx_(11, 7, 0)
+Z  [h_0] [x_(14, 2, 0)] = [x_(14, 3, 0)]
+B  [h_0] [x_(14, 3, 0)] = λx_(14, 5, 0)
+Z  [h_0] [x_(14, 4, 0)] = [x_(14, 5, 0)]
+Z  [h_0] [x_(14, 5, 0)] = [x_(14, 6, 0)]
+E  [h_0] λ x_(15, 1, 0) = λ [1]
+Z  [h_0] [x_(15, 2, 0)] = [x_(15, 3, 0)] + λx_(15, 4, 0)
+Z  [h_0] [x_(15, 3, 0)] = [x_(15, 4, 0)]
+Z  [h_0] [x_(15, 4, 0)] = [x_(15, 5, 0)] + λx_(15, 6, 0)
+Z  [h_0] [x_(15, 5, 0)] = [x_(15, 6, 0)]
+Z  [h_0] [x_(15, 5, 1)] = 0
+Z  [h_0] [x_(17, 3, 0)] = 0
+E  [h_0] λ x_(17, 4, 0) = λ [1]
+Z  [h_0] [x_(17, 5, 0)] = [x_(17, 6, 0)]
+Z  [h_0] [x_(18, 2, 0)] = [x_(18, 3, 0)] + λx_(18, 4, 0)
+Z  [h_0] [x_(18, 3, 0)] = [x_(18, 4, 0)]
+E  [h_0] λ x_(18, 4, 1) = λ [1]
+Z  [h_0] [x_(18, 4, 0)] = 0
+Z  [h_0] [x_(20, 4, 0)] = [x_(20, 5, 0)] + λx_(20, 6, 0)
+Z  [h_0] [x_(20, 5, 0)] = [x_(20, 6, 0)]
+Z  [h_0] [x_(21, 3, 0)] = 0
+Z  [h_0] [x_(23, 4, 0)] = 0
+Z  [h_0] [x_(23, 5, 0)] = [x_(23, 6, 0)]