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Codex — Wave 3 Plan

Wave 3. Production of the foundational prerequisites that Wave 2's pulled-prereq units declared as pending_prereqs: true. Where Wave 1 was the apex (10 units) and Wave 2 the first prereq layer (~30 units), Wave 3 reaches the foundational stratum: smooth manifold, Lie group, differential forms, topological space, normed/metric/inner-product spaces, basic algebra (group, ring, ideal), tensor product, covering space, etc.

Read OVERVIEW.md §5 (production strategy), docs/catalogs/DEPENDENCY_MAP.md §5–§6 (DAG layout and convergence), docs/plans/PILOT_PLAN.md (Wave 1 retrospective), docs/pilot-lessons.md (Wave 1 + Wave 2 production lessons), docs/specs/QUALITY_RUBRIC.md (the validator gate, including LM Master prose standard from §M).

1. Wave 3 scope

Wave 3 closes the remaining pending_prereqs: true flags across all currently-shipped units (Wave 1 + Wave 2). After Wave 3, every shipped unit's prerequisites should resolve to other shipped units — the DAG becomes traversable from leaf to apex.

Estimated unit count: ~18–22 foundational units, organized into 3–4 parallel branches that are each largely independent:

  • Topology branch (foundational): topological space, continuous map, compactness, Hausdorff, connectedness, covering space, homotopy
  • Smooth-manifold branch: smooth manifold, smooth map, tangent space, tangent bundle, differential forms, exterior derivative, integration prerequisites
  • Algebra branch: group, ring, ideal, associative algebra, tensor product, Lie group, group action, orthogonal group
  • Analysis branch: metric space, normed vector space, inner product space, completeness, differential operator (analytic side)

These branches are largely independent within Wave 3, so Wave 3 is the second real test of parallel agent production (after Wave 2). Wave 3 should ship in 3–4 batches over a similar window to Wave 2.

2. Pulled-prereq enumeration by source unit

What each Wave 2 unit declared as a still-pending prerequisite. Many entries are duplicated across units — those become high-leverage Wave 3 production targets.

From Phase 2.1 units

Wave 2 unit Pending prereqs (Wave 3 candidates)
01.01.03 Vector space Field, scalar, set theory primitives — if pre-vector-space units are wanted; alternatively, fold into vector space's own Beginner tier
03.05.01 Principal bundle Smooth manifold, Lie group, group action
03.05.02 Vector bundle Smooth manifold, vector space (shipped)
03.04.06 De Rham cohomology Smooth manifold, differential forms, exterior derivative

From Phase 2.2 units

Wave 2 unit Pending prereqs
01.01.15 Bilinear/quadratic form (clean — only depends on vector space, shipped)
03.09.03 Spin group Orthogonal group, Clifford (shipped)
02.11.04 Banach space Normed vector space, metric space, completeness
02.11.08 Hilbert space Inner product space, normed space (via Banach), completeness
03.05.04 Vector-bundle connection Vector bundle (shipped), smooth manifold
03.05.07 Principal-bundle connection Principal bundle (shipped), Lie algebra (shipped from Phase 2.3), smooth manifold

From Phase 2.3 units (assuming shipped)

Wave 2 unit Pending prereqs
03.04.01 Lie algebra Vector space (shipped), Lie group, smooth manifold
03.06.05 Invariant polynomial Lie algebra (shipped)
02.11.01 Bounded linear operators Banach (shipped)
02.11.05 Compact operators Bounded operators (shipped), Banach (shipped)
03.05.08 Complex vector bundle Vector bundle (shipped), complex structure on a vector space
03.05.09 Curvature of a connection Principal-bundle connection (shipped)

From Phase 2.4 units (planned)

Wave 2 unit Pending prereqs
02.11.03 Unbounded self-adjoint operators Hilbert (shipped), bounded (shipped), spectral theory rudiments
03.09.07 Symbol of a differential operator Differential operators, smooth manifold, cotangent bundle
03.09.09 Elliptic operators on a manifold Symbol (Phase 2.4), differential operators, smooth manifold
03.04.03 Integration on manifolds Differential forms, smooth manifold, orientation
03.04.08 Variational calculus on manifolds Differential forms, integration (Phase 2.4)
03.11.01 Central extension of a Lie algebra Lie algebra (shipped)

From Phase 2.5 units (planned)

Wave 2 unit Pending prereqs
03.05.03 Orthogonal frame bundle Principal bundle (shipped), orthogonal group
03.05.05 Double cover Covering space, principal bundle (shipped)
03.01.04 Tensor algebra Tensor product, vector space (shipped)
03.01.05 Quotient algebra Associative algebra, ideal
03.08.01 Topological K-theory Vector bundle (shipped), classifying space (Phase 2.5)
03.08.04 Classifying space Principal bundle (shipped), simplicial methods, homotopy
03.08.06 Stable homotopy Homotopy theory, homotopy groups, suspension
03.11.02 Infinite-dim Lie reps Lie algebra (shipped), central extension (Phase 2.4)
03.11.03 Virasoro algebra Central extension (Phase 2.4), Witt algebra

3. Consolidated Wave 3 production queue

Pulled prereqs that appear across multiple Wave 2 units (high leverage). Wave 3 should produce these first.

Tier A — high leverage (≥4 downstream uses)

Pulled ID (proposed) Title Appears in
03.02.?? (TBD) Smooth manifold Principal bundle, Vector bundle, De Rham, Vector-bundle connection, Principal-bundle connection, Lie algebra, Symbol of operator, Elliptic operators, Integration, Variational calculus — ≥10 units
03.04.?? (TBD) Differential forms De Rham, Integration, Variational calculus, exterior derivative — ≥3 units (and the foundation for several more)

Tier B — medium leverage (2–3 downstream uses)

Pulled ID (proposed) Title Appears in
03.??.?? Lie group Principal bundle, Lie algebra
03.??.?? Group action Principal bundle
02.??.?? Topological space Topology branch is foundational; needed by Banach, Hilbert, K-theory, classifying space, etc. — though most units don't explicitly cite it
02.??.?? Metric space Banach
02.??.?? Normed vector space Banach, Hilbert (via completeness)
02.??.?? Inner product space Hilbert
03.??.?? Tensor product Tensor algebra, Complex vector bundle
03.??.?? Orthogonal group Spin group, Orthogonal frame bundle
03.??.?? Covering space Double cover
03.??.?? Differential operator (general) Symbol, Elliptic operators
03.??.?? Homotopy / homotopy group Classifying space, Stable homotopy
03.??.?? Associative algebra Quotient algebra
03.??.?? Ideal (in an algebra) Quotient algebra
03.??.?? Witt algebra Virasoro (or fold into the Virasoro unit itself)

Tier C — convenience / completeness (singleton uses)

Pulled ID Title
03.??.?? Cotangent bundle (subsumes into smooth manifold or own unit)
03.??.?? Orientation
03.??.?? Suspension (topology)
03.??.?? Simplicial complex / simplicial methods

4. Suggested phase structure (3–4 batches)

Phase 3.1 — foundational (4–6 units, parallel-safe)

Units with no Wave 3 inter-deps. Produce in parallel; each unblocks several downstream Wave 3 / Wave 2 units.

  • ✅ Smooth manifold — 03.02.01, shipped 2026-04-28 (autonomous run)
  • ✅ Topological space — 02.01.01, shipped 2026-04-28 (autonomous run)
  • ✅ Metric space — 02.01.05, shipped 2026-04-28 (autonomous run)
  • ✅ Group (algebraic) — 01.02.01, shipped 2026-04-28 (Phase 3.2 batch via parallel session)
  • ✅ Lie group — 03.03.01, shipped 2026-04-28 (autonomous run)
  • ✅ Differential forms — 03.04.02, shipped 2026-04-28 (autonomous run)

Phase 3.1 complete (6 of 6).

Phase 3.2 — second layer (5–6 units)

Direct prereqs of Wave 2 / Phase 3.1 units. Underway via parallel session.

  • ✅ Continuous map — 02.01.02, shipped 2026-04-28
  • ✅ Group action — 03.03.02, shipped 2026-04-28
  • ✅ Orthogonal group — 03.03.03, shipped 2026-04-28
  • ⏳ Normed vector space — 02.11.06, in flight
  • ⏳ Inner product space — 02.11.07, in flight
  • ⏳ Spinor bundle — 03.09.05, shipped 2026-04-28 (closed Dirac apex)

Phase 3.2 progress: 4 of 6 shipped (or 5 of 6 counting spinor bundle, which I produced separately).

Phase 3.2 — second layer (5–7 units)

After Phase 3.1.

  • Continuous map (after topological space)
  • Normed vector space (after metric space)
  • Inner product space (after normed space)
  • Group action (after group)
  • Orthogonal group (after Lie group)
  • Tensor product (after vector space — already shipped — but conventions touch normed/inner-product)
  • Exterior derivative / differential forms refinement (after smooth manifold)

Phase 3.3 — third layer (5–7 units)

  • Covering space (after topological space + continuous map)
  • Homotopy / homotopy group (after topological space + continuous map)
  • Differential operator (after smooth manifold)
  • Cotangent bundle (after smooth manifold)
  • Associative algebra (algebra branch)
  • Ideal in an algebra (after associative algebra)

Phase 3.4 (optional cleanup) — specialized singletons

  • Witt algebra (or fold into Virasoro)
  • Suspension
  • Simplicial methods
  • Orientation

5. Numbering — section.chapter.ordinal assignments

The <section>.<chapter>.<ordinal> scheme expects each Wave 3 unit to have a permanent ID. Suggested chapters (subject to your reorganization preferences):

  • 01-foundations: foundational algebra (group, ring) goes here
  • 02-analysis/01-topology: topological space, continuous map, compactness, etc.
  • 02-analysis/02-metric-normed-inner: metric space, normed VS, inner product space
  • 03-modern-geometry/02-manifolds: smooth manifold, smooth maps, tangent / cotangent bundle
  • 03-modern-geometry/03-lie: Lie group, Lie group action, orthogonal group
  • 03-modern-geometry/04-differential-forms: differential forms, exterior derivative (chapter exists)
  • 03-modern-geometry/01-tensor-algebra (new): tensor product, associative algebra, ideal, tensor algebra (relocate 03.01.04 here; assign 03.01.?? for foundations)
  • 03-modern-geometry/12-homotopy (new): covering space, homotopy, suspension, simplicial methods

The numbering convention is non-dense (gaps reserved). Wave 3 will fill many of those gaps.

6. Success criteria (Wave 3)

Wave 3 succeeds if, by end:

  • All pending_prereqs: true flags across Wave 1 + Wave 2 units can be set to false (every declared prereq resolves to a shipped unit OR is explicitly deferred to Wave 4 with an integrator-approved waiver).
  • All prerequisites: IDs across the entire shipped DAG resolve.
  • validate_all.py is green at unit count ≥48 (10 Wave 1 + ~30 Wave 2 + ~18 Wave 3).
  • pnpm build clean.
  • Notation glossary at _meta/NOTATION.md updated to include all symbols introduced in Wave 3 units.
  • At least one lean_status: full unit ships in Wave 3 (foundational units like vector space, topological space, group should have substantial Mathlib coverage — Wave 3 is the natural place for Lean to start succeeding).

Wave 3 fails if:

  • Production diverges into curriculum-bloat territory (units that don't unblock anything Wave 2 declared).
  • The DAG can't be traversed from leaf to apex by end of wave.
  • Sign conventions or notation drift between Wave 3 foundational units and Wave 2 / Wave 1 units that depend on them.

7. Wave 3 is the first time foundational units exist

Wave 1 produced apex content. Wave 2 produced one layer below apex (still substantive graduate material). Wave 3 produces undergraduate-and-below material. This is a different audience-calibration problem:

  • Master tier at Wave 3 is still graduate-text voice (LM standard) but the content is much smaller per unit, since these are foundational definitions and basic theorems. Master sections will naturally be shorter than Wave 1's Master sections.
  • Intermediate tier at Wave 3 is the dominant tier for most learners — this is where Wave 3 units do the most pedagogical work.
  • Beginner tier at Wave 3 has the strongest claim to "approachability that exceeds Fast Track." A learner without prior exposure should be able to reach Master-tier comprehension by traversing Beginner → Intermediate → Master across many Wave 3 units.

This audience inversion means Wave 3 unit production should treat Beginner/Intermediate as the load-bearing tiers, with Master as the rigor backstop, not vice versa.

8. Exit to Wave 4 (or v1 launch?)

After Wave 3, options:

  • Wave 4 — even more foundational (high-school-algebra-prerequisite material). Per OVERVIEW.md §2, this is the "pre-calc ramp" that's been deliberately deferred. May or may not be in scope for v1.
  • v0.x → v1 launch decision — once Wave 3 ships, the DAG is traversable from foundational to apex. That's the minimum viable "high-school-to-graduate-mastery" promise of the project. v1 launch is gated on rendering, exercise interactivity, RAG, reviewer attestation — not on more units.
  • Lateral expansion — produce parallel apex strands beyond spin geometry (algebraic geometry, symplectic geometry, representation theory, statistical mechanics) following the same apex-first → pulled-prereq pattern.

Tyler's call which path Wave 4 takes. Wave 3 is the inflection point.

Plan drafted 2026-04-28 during Phase 2.3 / 2.4 production. Will be revised after Wave 2 closes and docs/pilot-lessons.md has accumulated foundational-tier observations.