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

[JoeyBF]

Summary

Implements the computational machinery for S/λ² (the cofiber of the Adams d₂ differential) from "The Cofiber of the Adams d₂ Differential", integrated into SecondaryExtAlgebra.

• Trigraded spectral sequence (Sseq<3, AdamsLambda2>): E2 has two copies of Ext at each (n, s) — one at Bockstein degree 0 and one at 1. The d₂ maps (n, s, 0) → (n−1, s+2, 1). E3 = E∞ gives π(S/λ²).
• B/Z/E classification API: BZE enum with SecondaryExtAlgebra::classify() and BZE::from_page_data() to decompose Ext = B ⊕ Z ⊕ E at each bidegree, where d₂: E_{(n,s)} ≅ B_{(n−1,s+2)}.
• lambda2 example binary: Generates the four tables from the paper (B/Z/E decomposition, Ext products, conical basis for π(S/λ²), secondary products in Mod_{Cλ²}).
• Edge case fix in SecondaryLift::compute_homotopies: guard against empty homotopy range (min.s >= max.s) which panicked for multipliers at high filtration near the resolution boundary.

Changes

File	Description
ext/crates/sseq/src/sseq.rs	Add AdamsLambda2 profile implementing SseqProfile<3>
ext/src/ext_algebra/secondary.rs	Add BZE enum, classify(), build_lambda2_sseq(), lambda2_page_data(), lambda2_e3_dimension()
ext/src/ext_algebra/mod.rs	Re-export BZE
ext/src/secondary.rs	Fix compute_homotopies empty-range panic
ext/examples/lambda2.rs	New example binary for the four tables

Test plan

• test_lambda2_sseq — verifies dim(E) = dim(B) across bidegrees, h₄ killed, h₀ survives
• test_sphere_d2 — verifies d₂(h₄) = h₀h₃² and permanent cycles
• test_sphere_products — verifies h₀² ≠ 0 and h₀h₁ = 0
• All four tables run without panic on S_2 at (n=16, s=6)
• Full test suite passes (only pre-existing test_tempdir_lock failure)

🤖 Generated with Claude Code

https://claude.ai/code/session_017kezHiWcYod7hYovRPczz9

---

Generated by Claude Code

Summary by CodeRabbit

• New Features
  • Added end-user support for the (S/\lambda^2) context with a dedicated spectral sequence, plus APIs to inspect its pages/dimensions.
  • Introduced a new B/Z/E classification for (d_2)-related behavior and surfaced it in related computations and examples.
  • Added ( \pi(S/\lambda^2) ) helpers (Pi basis and projection) and a new example that prints multiple (S/\lambda^2) tables.
• Bug Fixes
  • Prevented homotopy computations from iterating over empty/inverted degree ranges.
• Tests
  • Added/expanded tests for (S/\lambda^2), indexing, and ( \pi )-type behavior.

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ℹ️ Review info

⚙️ Run configuration

Configuration used: Organization UI

Review profile: ASSERTIVE

Plan: Pro

Run ID: d2e35793-20fe-4348-a025-e7f4d08a80ee

📥 Commits

Reviewing files that changed from the base of the PR and between fa37ad9 and ef09a7a.

📒 Files selected for processing (4)

• ext/examples/lambda2.rs
• ext/examples/secondary_product.rs
• ext/src/ext_algebra/mod.rs
• ext/src/ext_algebra/secondary.rs

📝 Walkthrough

Walkthrough

Adds S/λ² classification and spectral-sequence support, stores and queries a lambda2 spectral sequence in the secondary algebra, and updates examples and benchmark outputs to use the new B/Z/E labels and secondary element formatting.

Changes

S/λ² spectral sequence and secondary classifications

Layer / File(s)	Summary
Secondary degree and element model ext/src/secondary.rs	Defines the secondary weight, degree, generator, and element types and adds the compute_homotopies guard for empty s-ranges.
AdamsLambda2 profile and BZE classification ext/crates/sseq/src/coordinates/bze.rs, ext/crates/sseq/src/coordinates/mod.rs, ext/crates/sseq/src/sseq.rs, ext/src/ext_algebra/mod.rs	Introduces BZE, re-exports it from coordinates, adds AdamsLambda2 as an SseqProfile<3>, and adds Sseq classification helpers and ExtAlgebra spectral-sequence wiring.
Secondary lambda2 cache and Pi projection ext/src/ext_algebra/secondary.rs	Adds the lambda2 spectral-sequence cache, builds it during extension, classifies generators, projects products to Pi data, updates secondary multiplication output, and extends tests.
Example output updates ext/examples/lambda2.rs, ext/examples/secondary.rs, ext/examples/secondary_product.rs, ext/examples/secondary_massey.rs	Adds the lambda2 example tables and rewrites the secondary, secondary_product, and secondary_massey examples to use BZE and the new secondary element formatting.
Benchmark trace rewrites ext/examples/benchmarks/secondary-C2, ext/examples/benchmarks/secondary-C2-nassau, ext/examples/benchmarks/secondary-S_2, ext/examples/benchmarks/secondary-tmf, ext/examples/benchmarks/secondary_product-h_0-S_2, ext/examples/benchmarks/secondary_product-h_0-S_2-nassau, ext/examples/benchmarks/secondary_product-v_1-C2	Rewrites recorded secondary benchmark outputs to use Z and E trace lines and updated equation forms.

Estimated code review effort

🎯 4 (Complex) | ⏱️ ~50 minutes

Possibly related PRs

• SpectralSequences/sseq#218: Adds the underlying Sseq refactor that the new AdamsLambda2 and page-data helpers extend.
• SpectralSequences/sseq#225: Introduces the const-generic multi-degree machinery used by the new trigraded spectral-sequence code.
• SpectralSequences/sseq#241: Extends the secondary algebra layer that this PR builds on with new classification, Pi, and lambda2_sseq APIs.

Suggested reviewers

• hoodmane

🐇 I found a λ² trail to hop,
With B, Z, and E that never stop.
The tables now gleam,
In one spectral stream,
And secondary rabbits say “pop!”

🚥 Pre-merge checks | ✅ 5

✅ Passed checks (5 passed)

Check name	Status	Explanation
Description Check	✅ Passed	Check skipped - CodeRabbit’s high-level summary is enabled.
Title check	✅ Passed	The title accurately summarizes the main additions: the S/λ² spectral sequence and the new B/Z/E classification API.
Docstring Coverage	✅ Passed	Docstring coverage is 100.00% which is sufficient. The required threshold is 80.00%.
Linked Issues check	✅ Passed	Check skipped because no linked issues were found for this pull request.
Out of Scope Changes check	✅ Passed	Check skipped because no linked issues were found for this pull request.

✨ Finishing Touches

🧪 Generate unit tests (beta)

• Create PR with unit tests

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[coderabbitai]

Actionable comments posted: 2

🤖 Prompt for all review comments with AI agents

Verify each finding against current code. Fix only still-valid issues, skip the
rest with a brief reason, keep changes minimal, and validate.

Inline comments:
In `@ext/examples/lambda2.rs`:
- Around line 139-147: print_table_iii is currently printing raw
lambda2_page_data() entries instead of the advertised B/Z/E-annotated conical
basis. Update print_table_iii to use the new BZE-related API when iterating the
E3/E∞ generators so each row includes the correct B/Z/E type alongside the
tridegree, and keep the output aligned with the table description in the
function’s doc comment.
- Around line 177-183: The Table IV output in the lambda2 example is missing the
commutator column even though the doc comment and table description say it
should be included. Update the Table IV generation logic in the function that
prints the secondary products so it also computes and prints the commutator from
the existing stem parities (using the x·y and y·x relation). Keep the secondary
product output, but extend the same loop/row formatting to include the
commutator information for each pair.

🪄 Autofix (Beta)

Fix all unresolved CodeRabbit comments on this PR:

• Push a commit to this branch (recommended)
• Create a new PR with the fixes

---

ℹ️ Review info

⚙️ Run configuration

Configuration used: Organization UI

Review profile: ASSERTIVE

Plan: Pro

Run ID: 96149d7c-f3a8-4050-b59d-348c29f958e1

📥 Commits

Reviewing files that changed from the base of the PR and between faff1d6 and 1c4e6f3.

📒 Files selected for processing (5)

• ext/crates/sseq/src/sseq.rs
• ext/examples/lambda2.rs
• ext/src/ext_algebra/mod.rs
• ext/src/ext_algebra/secondary.rs
• ext/src/secondary.rs

[coderabbitai]

Caution

Some comments are outside the diff and can’t be posted inline due to platform limitations.

⚠️ Outside diff range comments (1)

ext/src/ext_algebra/secondary.rs (1)

228-244: 🎯 Functional Correctness | 🟠 Major | 🏗️ Heavy lift

Don’t expose λ² E3 dimensions when the underlying d₂ is uncomputed.

Line 240 skips unknown d₂ maps, but the λ² copies are already defined and the accessors return definite dimensions. Near the computation boundary this treats “d₂ target/source not computed” as “d₂ is zero”, unlike d2() which returns None. Gate λ² degree definition/accessors on the same computability range, and add a boundary regression test.

Possible direction

+        let d2_shift = Bidegree::n_s(-1, 2);
+
         for b in res.iter_stem() {
             let dim = res.number_of_gens_in_bidegree(b);
             let [n, s] = b.coords();
-            sseq.set_dimension(MultiDegree::new([n, s, 0]), dim);
-            sseq.set_dimension(MultiDegree::new([n, s, 1]), dim);
+
+            if b.t() <= 0 || res.has_computed_bidegree(b + d2_shift) {
+                sseq.set_dimension(MultiDegree::new([n, s, 0]), dim);
+            }
+
+            let incoming_source = b - d2_shift;
+            if incoming_source.s() < 0
+                || incoming_source.t() <= 0
+                || res.has_computed_bidegree(incoming_source)
+            {
+                sseq.set_dimension(MultiDegree::new([n, s, 1]), dim);
+            }
         }

Also consider returning Option<usize> from lambda2_e3_dimension if None can mean “not computed” rather than “zero”.

Also applies to: 272-299, 439-503

🤖 Prompt for AI Agents

Verify each finding against current code. Fix only still-valid issues, skip the
rest with a brief reason, keep changes minimal, and validate.

In `@ext/src/ext_algebra/secondary.rs` around lines 228 - 244, The λ² E3 dimension
accessors in secondary.rs are exposing definite dimensions even when the
corresponding d₂ data is still uncomputed, unlike d2() which reports None.
Update the λ² degree setup and the related accessors (including
lambda2_e3_dimension and the code around res_lift/homotopy lookups) to gate on
the same computability boundary as d₂, so unknown source/target bidegrees are
not treated as zero. If appropriate, make the accessor return Option<usize> for
not-yet-computed degrees, and add a regression test covering the boundary case.

🤖 Prompt for all review comments with AI agents

Verify each finding against current code. Fix only still-valid issues, skip the
rest with a brief reason, keep changes minimal, and validate.

Outside diff comments:
In `@ext/src/ext_algebra/secondary.rs`:
- Around line 228-244: The λ² E3 dimension accessors in secondary.rs are
exposing definite dimensions even when the corresponding d₂ data is still
uncomputed, unlike d2() which reports None. Update the λ² degree setup and the
related accessors (including lambda2_e3_dimension and the code around
res_lift/homotopy lookups) to gate on the same computability boundary as d₂, so
unknown source/target bidegrees are not treated as zero. If appropriate, make
the accessor return Option<usize> for not-yet-computed degrees, and add a
regression test covering the boundary case.

---

ℹ️ Review info

⚙️ Run configuration

Configuration used: Organization UI

Review profile: ASSERTIVE

Plan: Pro

Run ID: 3feb8d09-2af3-42c4-918d-a3fd3fbf3fc1

📥 Commits

Reviewing files that changed from the base of the PR and between 1c4e6f3 and 2c19f37.

📒 Files selected for processing (8)

• ext/examples/benchmarks/secondary-C2
• ext/examples/benchmarks/secondary-C2-nassau
• ext/examples/benchmarks/secondary-S_2
• ext/examples/benchmarks/secondary-tmf
• ext/examples/lambda2.rs
• ext/examples/secondary.rs
• ext/examples/secondary_product.rs
• ext/src/ext_algebra/secondary.rs

[coderabbitai]

Actionable comments posted: 5

🤖 Prompt for all review comments with AI agents

Verify each finding against current code. Fix only still-valid issues, skip the
rest with a brief reason, keep changes minimal, and validate.

Inline comments:
In `@ext/examples/lambda2.rs`:
- Around line 35-46: Reject non-unit modules before building the tables in
lambda2’s main example flow: the current use of query_module(...),
ExtAlgebra::from_resolution, and later e2.resolution()/e2.unit_generator(...)
assumes unit-generator indexing that is only valid for a unit input. Update the
lambda2 example to fail fast unless e2.is_unit() is true, and keep the
resolution-dependent table generation behind that check so Table II cannot read
generators from the wrong basis.

In `@ext/examples/secondary_product.rs`:
- Around line 116-123: The classification in `secondary_multiply_into` is
currently inferred from `prod.source.vec().iter_nonzero().next()`, which
mislabels mixed sources by basis order instead of class. Update the
`secondary_product.rs` printing logic to use `SecondaryExtAlgebra::classify()`
for `prod.source` (or explicitly split mixed sources before formatting) so the
`B|Z|E` label reflects the full class rather than the first nonzero basis
element.

In `@ext/src/ext_algebra/mod.rs`:
- Around line 78-84: The cached sseq in ExtAlgebra becomes stale after
compute_through_bidegree() extends the underlying resolution, so sseq() can
return an outdated snapshot that no longer matches dimension() and basis() for
newly computed bidegrees. Update ExtAlgebra::new and the sseq()/build_sseq()
flow so the spectral sequence is rebuilt or invalidated whenever the resolution
grows, or make sseq() compute from the current resolution state instead of
storing a one-time cached value.

In `@ext/src/ext_algebra/secondary.rs`:
- Around line 496-528: `project_to_pi` is building `PiElement` with coordinates
from `sq.reduce(...)` but `basis_indices` from ambient generator filtering, so
the two bases can mismatch. Fix this in `secondary.rs` by deriving the basis
indices from the actual `Subquotient` generator basis used by
`lambda2_page_data(td)`/`Sq::reduce`, or by projecting into the same ambient
basis before constructing `PiElement`. Ensure `PiElement::coords`,
`basis_indices`, and `Display` all refer to the same basis.
- Around line 420-432: The current `classify()` implementation only delegates to
`lambda2_sseq().classify(...)` on the `(n, s, 0)` copy, so it misses the
weight-1 boundary case and can mislabel true `B` classes as `Z`. Update
`SecondaryExt::classify` to use the same boundary-aware logic as
`adams_classify()` (or otherwise incorporate the weight-1 incoming `d₂` check)
while preserving the existing `BZE` decomposition behavior for
`BidegreeGenerator`.

🪄 Autofix (Beta)

Fix all unresolved CodeRabbit comments on this PR:

• Push a commit to this branch (recommended)
• Create a new PR with the fixes

---

ℹ️ Review info

⚙️ Run configuration

Configuration used: Organization UI

Review profile: ASSERTIVE

Plan: Pro

Run ID: 42d44cc6-5fbc-4ffd-83ae-46ef7ed3bc60

📥 Commits

Reviewing files that changed from the base of the PR and between 2c19f37 and fa37ad9.

📒 Files selected for processing (17)

• ext/crates/sseq/src/coordinates/bze.rs
• ext/crates/sseq/src/coordinates/mod.rs
• ext/crates/sseq/src/sseq.rs
• ext/examples/benchmarks/secondary-C2
• ext/examples/benchmarks/secondary-C2-nassau
• ext/examples/benchmarks/secondary-S_2
• ext/examples/benchmarks/secondary-tmf
• ext/examples/benchmarks/secondary_product-h_0-S_2
• ext/examples/benchmarks/secondary_product-h_0-S_2-nassau
• ext/examples/benchmarks/secondary_product-v_1-C2
• ext/examples/lambda2.rs
• ext/examples/secondary.rs
• ext/examples/secondary_massey.rs
• ext/examples/secondary_product.rs
• ext/src/ext_algebra/mod.rs
• ext/src/ext_algebra/secondary.rs
• ext/src/secondary.rs