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Codex — v0.5 Lateral Expansion Plan

v0.x scope is complete: 66 units, fully closed DAG, spin-geometry strand (Wave 1) traversable from leaf to apex with three tiers. v0.5 is the first lateral expansion: producing additional apex strands beyond spin geometry to reach Fast Track parity.

Read docs/plans/PILOT_PLAN.md (v0.x retrospective), docs/plans/WAVE_3_PLAN.md and docs/plans/WAVE_4_PLAN.md (depth-completion plans), docs/pilot-lessons.md (consolidated production lessons), docs/specs/QUALITY_RUBRIC.md (the validator gate).

1. v0.5 scope

Five additional apex strands selected for breadth, canonisation, and tractability. Each strand:

  • Has 8–15 apex units culminating in a major synthesis theorem.
  • Pulls in 10–25 supporting units, most of whose foundations are already shipped in v0.x (smooth manifold, vector bundle, Lie group, de Rham cohomology, etc.).
  • Has well-defined Master-tier anchor texts from the Fast Track booklist.
  • Produces in roughly the same footprint as the original spin-geometry pilot (~30–40 units total per strand) but at substantially faster cadence because the foundational prereq chain is already in place.

The five strands:

# Strand Apex synthesis Anchors Est. units
A Algebraic geometry Riemann-Roch + Hodge decomposition + sheaf cohomology Hartshorne; Griffiths-Harris; Voisin; Vakil ~35
B Symplectic geometry Moment map + symplectic reduction + Floer homology Arnold; Audin; Cannas da Silva; Donaldson-Kronheimer ~30
C Riemann surfaces & complex analysis Riemann-Roch for curves + Abel-Jacobi + uniformisation Donaldson Riemann Surfaces; Forster; Farkas-Kra; Ahlfors-Sario ~25
D Representation theory Borel-Weil-Bott + Peter-Weyl + classification of compact simple Lie groups Fulton-Harris; Serre Linear Reps; Knapp; Bump ~30
E Statistical field theory & lattice models Onsager 2D Ising + RG flow + phase transitions Baxter; Itzykson-Drouffe; Kardar (MIT OCW 8.334) ~25

Estimated total v0.5 production: ~145 units across 5 strands. Combined with v0.x's 66 units, that's ~210 units total — roughly Fast Track parity for the full modern-geometry section.

2. Strand A — Algebraic geometry

Apex synthesis: Riemann-Roch, sheaf cohomology, Hodge theory.

Apex units (Wave 1 equivalent)

  1. 04.01.?? Sheaf — algebraic / topological foundation.
  2. 04.01.?? Coherent sheaf on a scheme.
  3. 04.02.?? Scheme — affine, gluing, examples.
  4. 04.02.?? Morphism of schemes.
  5. 04.03.?? Sheaf cohomology (Čech, derived functor).
  6. 04.03.?? Serre's vanishing theorem.
  7. 04.04.?? Divisor on a curve.
  8. 04.04.?? Riemann-Roch theorem (curves, then surfaces).
  9. 04.05.?? Hodge decomposition.
  10. 04.05.?? Kähler manifold.
  11. 04.06.?? Algebraic curve as Riemann surface (link to Strand C).

Wave 2 supporting units (unique to AG)

  • Commutative ring + ideal (extends our 03.01.02 / 03.01.03)
  • Localisation
  • Tensor product over a ring
  • Zariski topology
  • Spectrum of a ring Spec(A)
  • Quasi-coherent sheaf
  • Locally free sheaf / vector bundle on scheme
  • Étale topology (placeholder — full étale is v1+)
  • Algebraic variety (classical)
  • Projective space + projective scheme
  • Line bundle on projective scheme (Picard group)
  • Riemann's theorem on algebraic curves
  • Ext / Tor functors

Foundations already shipped (reuse)

  • Topological space, continuous map, metric space
  • De Rham cohomology, exterior derivative, Stokes' theorem
  • Differential forms, smooth manifold
  • Vector space, field, group, ring (function from gpt-codex's Wave 4 batch)
  • Tensor product, tensor algebra, quotient algebra, ideal
  • Homotopy, covering space

Estimated breakdown

  • Apex: 11 units
  • New supporting: ~13 units
  • Reused foundations: substantial (no new production needed)
  • Total new production: ~24 units

Owner: claude (one of three claude strands)

3. Strand B — Symplectic geometry

Apex synthesis: Moment map theory, symplectic reduction, applications to Hamiltonian dynamics and Floer homology.

Apex units (Wave 1 equivalent)

  1. 05.01.?? Symplectic vector space (linear algebra).
  2. 05.01.?? Symplectic manifold.
  3. 05.02.?? Hamiltonian vector field + symplectic vector field.
  4. 05.02.?? Poisson bracket + Poisson manifold.
  5. 05.03.?? Coadjoint orbit.
  6. 05.04.?? Moment map.
  7. 05.04.?? Marsden-Weinstein symplectic reduction.
  8. 05.05.?? Lagrangian submanifold.
  9. 05.06.?? Almost-complex structure on a symplectic manifold.
  10. 05.07.?? Gromov non-squeezing theorem.
  11. 05.08.?? Arnold conjecture (statement) → Floer homology setup.

Wave 2 supporting units (unique to SG)

  • Integrable system (Liouville-Arnold)
  • Action-angle coordinates
  • Cotangent bundle as canonical symplectic manifold
  • Darboux's theorem
  • Symplectic group (Lie group; reuses 03.03.01)
  • Maslov index
  • Pseudoholomorphic curve (J-holomorphic curve)
  • Symplectic capacity
  • Conley-Zehnder index
  • Floer homology (Hamiltonian / instanton flavors)

Foundations already shipped (reuse)

  • Smooth manifold, differential forms, exterior derivative
  • Lie group, Lie algebra, Lie group action
  • Bilinear form (skew-symmetric variant for symplectic forms)
  • De Rham cohomology
  • Hamilton's principle and variational calculus

Estimated breakdown

  • Apex: 11 units
  • New supporting: ~10 units
  • Total new production: ~21 units

Owner: gpt-codex (one of two gpt-codex strands)

4. Strand C — Riemann surfaces & complex analysis

Apex synthesis: Riemann-Roch for compact Riemann surfaces, Abel-Jacobi map, uniformisation theorem.

Apex units (Wave 1 equivalent)

  1. 06.01.?? Holomorphic function (single complex variable).
  2. 06.01.?? Cauchy-Riemann equations + complex differentiability.
  3. 06.02.?? Cauchy integral theorem + formula.
  4. 06.02.?? Residue theorem.
  5. 06.03.?? Riemann surface — definitions, examples.
  6. 06.03.?? Sheaf of holomorphic functions.
  7. 06.04.?? Divisor on a Riemann surface.
  8. 06.04.?? Riemann-Roch theorem (for compact Riemann surfaces).
  9. 06.05.?? Jacobian variety + Abel-Jacobi map.
  10. 06.05.?? Theta function on a Riemann surface.
  11. 06.06.?? Uniformisation theorem (Riemann mapping + uniformisation).
  12. 06.07.?? Algebraic curve ↔ Riemann surface correspondence (link to Strand A).

Wave 2 supporting units (unique to RS)

  • Power series in one complex variable
  • Holomorphic vector bundle on a Riemann surface
  • Meromorphic function
  • Branch point / branched covering
  • Genus of a Riemann surface
  • Hyperelliptic curve
  • Kähler form on a Riemann surface
  • Hodge theory for compact Riemann surfaces
  • Mittag-Leffler theorem
  • Weierstrass preparation theorem

Foundations already shipped (reuse)

  • Topological space, continuous map, metric space (for compactness)
  • Smooth manifold (Riemann surfaces are 2-real-dim with extra structure)
  • Differential forms, de Rham cohomology
  • Covering space, double cover
  • Lie group ( via Möbius transformations)

Estimated breakdown

  • Apex: 12 units
  • New supporting: ~10 units
  • Total new production: ~22 units

Owner: claude (one of three claude strands)

5. Strand D — Representation theory

Apex synthesis: Cartan-Weyl classification of finite-dimensional Lie algebra reps, Peter-Weyl theorem, Borel-Weil-Bott.

Apex units (Wave 1 equivalent)

  1. 07.01.?? Group representation (general).
  2. 07.01.?? Schur's lemma.
  3. 07.02.?? Character of a finite-group representation.
  4. 07.02.?? Character orthogonality + decomposition.
  5. 07.03.?? Highest weight representation of a complex semisimple Lie algebra.
  6. 07.03.?? Weight lattice + root system + Weyl group.
  7. 07.04.?? Cartan-Weyl classification of finite-dim irreducible reps.
  8. 07.05.?? Compact Lie group + maximal torus.
  9. 07.05.?? Peter-Weyl theorem.
  10. 07.06.?? Flag variety.
  11. 07.06.?? Borel-Weil-Bott theorem.
  12. 07.07.?? Connection to physics: , in particle physics.

Wave 2 supporting units (unique to RT)

  • Symmetric group representation
  • Young tableau + Young symmetriser
  • Cartan subalgebra + root space decomposition
  • Verma module + highest-weight construction
  • Universal enveloping algebra
  • Casimir element
  • Weyl character formula
  • Killing form (already in Lie algebra unit, refine here)
  • Schur-Weyl duality
  • Bruhat decomposition / parabolic subgroup

Foundations already shipped (reuse)

  • Group, group action, Lie group, Lie algebra
  • Vector space, bilinear form
  • Compact (topology), Hilbert space (for Peter-Weyl on )
  • Tensor product, tensor algebra
  • Spin group (already shipped as concrete Lie group example)

Estimated breakdown

  • Apex: 12 units
  • New supporting: ~10 units
  • Total new production: ~22 units

Owner: claude (one of three claude strands)

6. Strand E — Statistical field theory & lattice models

Apex synthesis: Onsager solution of 2D Ising, renormalisation group flow, universality classes for phase transitions.

Apex units (Wave 1 equivalent)

  1. 08.01.?? Partition function (statistical mechanics).
  2. 08.01.?? Ising model (definition + ferromagnetic / antiferromagnetic phases).
  3. 08.02.?? Mean-field theory + Curie-Weiss.
  4. 08.03.?? Onsager solution of 2D Ising (transfer matrix).
  5. 08.04.?? Renormalisation group (real-space block decimation).
  6. 08.04.?? Wilson-Fisher fixed point + universality.
  7. 08.05.?? Critical exponents + scaling laws.
  8. 08.06.?? Gaussian field theory + free boson.
  9. 08.06.?? Conformal symmetry at criticality (link to existing CFT unit).
  10. 08.07.?? Path integral formulation of statistical mechanics.
  11. 08.08.?? Wilson's lattice formulation of gauge theory.
  12. 08.09.?? Quantum-classical correspondence (Wick rotation).

Wave 2 supporting units (unique to SFT)

  • Boltzmann distribution + canonical ensemble
  • Free energy + Helmholtz / Gibbs
  • Correlation function (statistical mechanics)
  • Transfer matrix (general)
  • Spontaneous symmetry breaking
  • Mermin-Wagner theorem
  • Block-spin decimation
  • Beta function (RG)
  • Lattice gauge theory (Wilson action)
  • Effective field theory

Foundations already shipped (reuse)

  • Hilbert space (for path-integral Hilbert spaces)
  • Lie algebra (for gauge symmetries)
  • Differential forms (for action functionals)
  • CFT basics (already shipped as Wave 1 apex)
  • Tensor product

Estimated breakdown

  • Apex: 12 units
  • New supporting: ~10 units
  • Total new production: ~22 units

Owner: gpt-codex (one of two gpt-codex strands)

7. Production division of labor

claude (3 strands): A. Algebraic geometry, C. Riemann surfaces & complex analysis, D. Representation theory.

These three strands are mathematically adjacent (commutative algebra + complex/algebraic geometry + Lie/rep theory) and lean heavily on prose-density material. They share substantial supporting infrastructure: schemes appear in AG and Riemann surfaces, Lie reps appear in Borel-Weil and connect to compact-Lie classifications.

gpt-codex (2 strands): B. Symplectic geometry, E. Statistical field theory & lattice models.

These two strands are physics-flavored, computational, and somewhat self-contained. Symplectic geometry leans on Lie group + diffgeo foundations; SFT leans on CFT + Hilbert space. The gpt-codex session has shown strong production speed on physics-style technical content (CFT basics, Yang-Mills, Bott periodicity).

Why this split

  • Mathematical adjacency: claude's three strands form a connected sub-DAG (sheaf cohomology in AG ↔ holomorphic line bundles on Riemann surfaces ↔ characters of compact Lie groups → Borel-Weil). Producing them together lets cross-references resolve as production proceeds.
  • Prose density: AG, RS, RT have heavy prose components (definitions of schemes, divisors, weight lattices). claude's Master-tier prose pace fits this.
  • Computational density: SG and SFT have more action-functional / partition-function / commutator computation. gpt-codex has shown speed and accuracy on this style.
  • Independence: SG and SFT can run completely independently of claude's three strands; collisions on shared files (docs/catalogs/CONCEPT_CATALOG.md, manifests/deps.json, docs/pilot-lessons.md) are the only coordination point.

Estimated production time

Per strand: ~22 new units at the post-pilot scaffold cadence (~2–5 min per unit for the parallel session, ~15–30 min per unit for claude with full LM Master prose).

  • claude's 3 strands: ~66 units. At ~20 min/unit average, ~22 hours of focused work, distributable across multiple sessions.
  • gpt-codex's 2 strands: ~44 units. At ~5 min/unit, ~3.5 hours.

Realistic calendar: 2–3 weeks of intermittent work brings v0.5 to closure (with foundations already shipped, this is genuinely tractable).

8. Strand-batch files (paste-ready for gpt-codex)

The two gpt-codex strands need their own batch scaffolds. Written as separate files:

  • docs/batches/V05_GPT_BATCH_B_SYMPLECTIC.md — Strand B (Symplectic geometry) Wave 1 apex queue.
  • docs/batches/V05_GPT_BATCH_E_SFT.md — Strand E (Statistical field theory) Wave 1 apex queue.

Each file contains:

  • Read-first list (PILOT_PLAN, QUALITY_RUBRIC, etc.)
  • Production queue (~10 apex units in dependency order)
  • Per-unit production protocol
  • Hard conventions and prose standard
  • Stop conditions

The claude-side strands (A, C, D) are produced from docs/plans/CURRICULUM_V0_5_PLAN.md directly — claude reads the strand definition above and produces from it.

9. Numbering convention

Existing v0.x uses sections 00–03. v0.5 strands continue:

  • 04 — Algebraic geometry (Strand A)
  • 05 — Symplectic geometry (Strand B)
  • 06 — Riemann surfaces & complex analysis (Strand C)
  • 07 — Representation theory (Strand D)
  • 08 — Statistical field theory & lattice models (Strand E)

Each strand has chapters 01–10 (subject to refinement during production). Apex units are roughly in the higher chapter numbers per strand; supporting units in lower numbers.

10. Success criteria (v0.5)

v0.5 succeeds if:

  • All 5 strands ship at least 80% of their planned apex units (8 of ~10 each).
  • Each strand's apex units have pending_prereqs: false (every prereq resolves).
  • validate_all.py is green at unit count ≥ 200 (66 v0.x + ~135 v0.5).
  • pnpm build clean, all unit pages render.
  • Cross-strand cross-references work: a learner reading Strand A's Riemann-Roch can click through to Strand C's compact Riemann surface for the curve case.
  • LM editorial review pass on each strand's Master tier catches no flagrant violations.
  • No more than 2 units in the entire shipped DAG have pending_prereqs: true.

v0.5 fails if:

  • Any strand stalls below 50% apex completion.
  • Cross-strand references break (e.g., AG references a Strand C unit that was never produced).
  • The Master-prose drift reappears (suggesting the rubric needs further tightening).
  • The validator fails on more than 5 units after a strand reports.

11. Exit conditions

After v0.5 closes:

  • v1 launch readiness: rendering polish (KaTeX + theme + tier filter UX), exercise interactivity (interactive widgets — Neutron-side work), RAG layer over reference/, reviewer attestation pass.
  • v0.6 lateral expansion: 2–3 more strands beyond the v0.5 five (geometric quantization, number theory, advanced algebraic topology, etc.) — the same scaffold-driven pattern repeated.
  • Consolidation: rubric evolution (v3 of docs/specs/QUALITY_RUBRIC.md); LM-editorial-writer agent review on all v0.5 Master prose; sourcing-acquisition pass to resolve TODO_REF placeholders.

The v0.5 plan's purpose is twofold: (1) bring Codex to Fast Track parity in modern-geometry breadth, (2) prove the lateral-expansion model works at scale.

v0.5 Lateral Expansion Plan drafted 2026-04-28 immediately after Wave 5 closure. 5 strands, 2-session production split, ~145 new units, 2–3 week target window.