Add SecondaryExtAlgebra: the d2 differential layer

[JoeyBF]

Introduce ext::ext_algebra::secondary::SecondaryExtAlgebra<CC>, the secondary layer over ExtAlgebra. It composes the primary algebra with the secondary resolutions of M and the unit k, and exposes the secondary differential d2 (with a survives/permanent-cycle check), the E3-page page_data, and the Mod_Cλ² secondary product secondary_multiply_into.

These wrap SecondaryResolution and SecondaryResolutionHomomorphism; no new linear algebra is added. The layer is split from ExtAlgebra because the secondary machinery requires CC::Algebra: PairAlgebra, a bound the primary layer does not impose. To support this, ext_algebra.rs becomes the module directory ext_algebra/{mod,secondary}.rs. A compute_partial passthrough and a public secondary_product_lift preserve the per-s sharding workflow, and a unit-free ExtAlgebra::without_unit constructor supports the d2-only path.

Refactors the secondary (d2 lister) and secondary_product examples onto the layer; both produce byte-identical output to before (verified on S_2). Adds a unit test asserting the first Adams differential d2(h4) = h0 h3^2 is nonzero and that h0, h1, h2 are permanent.

Claude-Session: https://claude.ai/code/session_01XYWcvWZPm3YJkVpCmeGYsP

Summary by CodeRabbit

• New Features
  • Added a new secondary ($d_2$) layer for computing secondary differentials and secondary products via the Ext algebra API.
  • Introduced a new constructor to build Ext computations without requiring a separate unit.
  • Updated the secondary example programs to use the new secondary workflow and to produce clearer, structured output.
• Bug Fixes
  • Improved secondary-product handling by switching to the dedicated secondary algebra implementation with built-in permanence checks.
• Tests
  • Added/extended a test to validate $d_2$ behavior on a standard resolution.

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ℹ️ Review info

⚙️ Run configuration

Configuration used: Organization UI

Review profile: ASSERTIVE

Plan: Pro

Run ID: 75c52520-ab71-4407-9f18-3fa7b096cd55

📥 Commits

Reviewing files that changed from the base of the PR and between a16676e and 9d47357.

📒 Files selected for processing (4)

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

📝 Walkthrough

Walkthrough

Adds a new secondary Ext algebra module, exposes ExtAlgebra::without_unit, and rewrites the secondary examples to use d2, survival checks, and secondary product computation through the new APIs.

Changes

Secondary Ext algebra and examples

Layer / File(s)	Summary
Secondary module surface ext/src/ext_algebra/mod.rs, ext/src/ext_algebra/secondary.rs	Exports secondary, adds ExtAlgebra::without_unit, and defines SecondaryExtAlgebra and SecondaryProduct.
Secondary d2 and product engine ext/src/ext_algebra/secondary.rs	Implements new, extend_all, compute_partial, d2, survives, page accessors, secondary product lifts, secondary product computation, and the sphere d2 test.
Secondary differential example ext/examples/secondary.rs	Builds SecondaryExtAlgebra from a unitless resolution, supports partial computation, and prints d2 values from sec_e2.d2(...).
Secondary product example ext/examples/secondary_product.rs	Builds ExtAlgebra and SecondaryExtAlgebra, constructs the input class, checks survival, and prints secondary products from secondary_multiply_into.

Estimated code review effort

🎯 4 (Complex) | ⏱️ ~60 minutes

Possibly related PRs

• SpectralSequences/sseq#240: Introduces the ExtAlgebra API that this PR extends with without_unit and the new secondary algebra layer.

Suggested reviewers

• hoodmane

Poem

A rabbit hopped where spectra glow,
and found new d2 winds to know.
With carrots of ext and lambda twine,
the secondary bunnies leap just fine. 🐇

🚥 Pre-merge checks | ✅ 4 | ❌ 1

❌ Failed checks (1 warning)

Check name	Status	Explanation	Resolution
Docstring Coverage	⚠️ Warning	Docstring coverage is 0.00% which is insufficient. The required threshold is 80.00%.	Write docstrings for the functions missing them to satisfy the coverage threshold.

✅ Passed checks (4 passed)

Check name	Status	Explanation
Description Check	✅ Passed	Check skipped - CodeRabbit’s high-level summary is enabled.
Title check	✅ Passed	The title clearly summarizes the main addition: a new SecondaryExtAlgebra layer for d2 computations.
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.

_{✏️ Tip: You can configure your own custom pre-merge checks in the settings.}

✨ 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/secondary_product.rs`:
- Around line 62-73: The partial-job path in secondary_product.rs is doing too
much work because sec_e2.extend_all() and the full secondary-layer setup happen
before checking ext::utils::secondary_job(). Reorder the logic so the
secondary_job() branch is handled first, and only build the full secondary layer
with SecondaryExtAlgebra::new, extend_all(), and the survival assertion on the
non-partial path. Keep secondary_product_lift and lift.compute_partial(s) inside
the partial-job branch so it skips the full E3 computation.

In `@ext/src/ext_algebra/secondary.rs`:
- Around line 150-153: The survivors check in Secondary algebra should not treat
an unknown d2 differential as a surviving class. Update the survives method on
Secondary (and any related d2 lookup handling) so that self.d2(x) returning None
is treated as unknown/false rather than true, while still returning true only
when the computed d2 vector is zero. Keep the existing zero-vector case in
survives, but distinguish it from the uncomputed target case documented in d2.

🪄 Autofix (Beta)

Fix all unresolved CodeRabbit comments on this PR:

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

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ℹ️ Review info

⚙️ Run configuration

Configuration used: Organization UI

Review profile: ASSERTIVE

Plan: Pro

Run ID: 0c0db09f-7e7c-4aaa-a278-03fb6ed053f6

📥 Commits

Reviewing files that changed from the base of the PR and between cb256a6 and d1a7927.

📒 Files selected for processing (4)

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