Cannas da Silva — Lectures on Symplectic Geometry (Fast Track 1.11) — Audit + Gap Plan

Book: Ana Cannas da Silva, Lectures on Symplectic Geometry (Springer Lecture Notes in Mathematics 1764, 2001; corrected reprint 2008). Hosted free by the author at her ETH Zürich page (https://people.math.ethz.ch/~acannas/Papers/lsg.pdf).

Fast Track entry: 1.11 (the symplectic-geometry slot of the Dynamical-Systems-IV / symplectic-geometry strand on the booklist; Cannas is paired there with Arnold Mathematical Methods of Classical Mechanics and McDuff-Salamon Introduction to Symplectic Topology as the free open-access companion to the strand).

Purpose of this plan: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite of the orchestration protocol). Output is a concrete punch-list of new units to write — and, more importantly here, a list of existing units to deepen — so that Lectures on Symplectic Geometry is covered to the equivalence threshold (≥95% effective coverage of theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4).

This pass is intentionally not a full P1 audit (which would inventory every numbered Homework problem in Cannas's 30 lectures at the line-number level). It works from the book's lecture-by-lecture topic list and Cannas's distinctive editorial choices, produces the gap punch-list, and stops there.

The audit surface here is smaller than Brown or Hatcher: Cannas is already cited as a primary anchor on 18 of the 21 shipped Codex symplectic units (tier_anchors field), so coverage at the topic level is high. The real gaps are at the depth level — the templated v0.5 Strand-B production passed the prose validator but never actually delivered Cannas's signature proofs (Moser's trick, Weinstein neighbourhood, Arnold-Liouville, Atiyah-Guillemin-Sternberg convexity, Marsden-Weinstein in the regular case, Delzant's classification of toric manifolds). This plan punches up those depth gaps and adds the genuinely missing Cannas-specific topics (blow-up, symplectic cut, symplectic fibrations, contact-symplectic adjacency, generating functions).

§1 What Cannas's book is for

Cannas is the canonical lecture-note treatment of symplectic geometry at the bridge between classical mechanics, modern symplectic topology, and equivariant geometry. Where Arnold (1.0a / Mathematical Methods) anchors the mechanics-first perspective — phase space, action-angle, KAM — and McDuff-Salamon (3.46-adjacent) anchors the topology-first perspective — pseudoholomorphic curves, Floer, Gromov-Witten — Cannas sits squarely in the middle. The book is organised as 30 short lectures (typically 5-8 pages each), grouped into six parts:

Part I — Symplectic Manifolds (Lectures 1-6): linear symplectic geometry, symplectic manifolds, symplectomorphisms.
Part II — Symplectomorphisms (Lectures 7-9): Lagrangian submanifolds, generating functions, fixed-point Arnold conjecture (statement-level).
Part III — Local Forms (Lectures 10-12): isotopies and vector fields, Moser's trick, Darboux-Moser-Weinstein theorem, Weinstein's Lagrangian neighbourhood theorem, Weinstein tubular neighbourhood.
Part IV — Contact Manifolds (Lectures 13-14): contact forms, Reeb vector fields, contactisation, symplectisation. Cannas is the rare symplectic textbook that integrates the contact half cleanly into the same pedagogical arc.
Part V — Compatible Almost Complex Structures (Lectures 15-17): compatible triples $(ω, J, g)$ , polar decomposition, pseudoholomorphic curves at the introductory level, applications to Kähler manifolds.
Part VI — Kähler Manifolds (Lectures 18-21): complex projective spaces, Hodge identities, Kähler reduction. (Briefer than the rest; the strand handoff to Voisin / Griffiths-Harris.)
Part VII — Hamiltonian Mechanics (Lectures 22-24): Hamiltonian vector fields, integrable systems, Arnold-Liouville theorem (genuinely proved here, not just stated).
Part VIII — Hamiltonian Group Actions (Lectures 25-30): symplectic and Hamiltonian actions, moment maps, Marsden-Weinstein-Meyer reduction in the regular case, Atiyah-Guillemin-Sternberg convexity, symplectic toric manifolds and Delzant's theorem, Duistermaat-Heckman.

Distinctive Cannas choices, in roughly the order she develops them:

Moser's trick first. Lecture 7 develops the path-method (Moser's trick) as a stand-alone technique, then derives Darboux's theorem, the relative Darboux theorem, the Weinstein Lagrangian neighbourhood theorem, and the Weinstein tubular neighbourhood theorem all as parallel applications of the same path-of-symplectic-forms argument. This deformation-theoretic framing is Cannas's signature pedagogical move and is what makes the "local forms" section feel uniformly organised. Codex's existing Darboux unit (05.01.04) does not currently use Moser's trick — it gives a generic Cartan-formula sketch — and the Weinstein theorems are not present at all.
Generating functions before Arnold conjecture. Lecture 9 shows how the graph of a Hamiltonian symplectomorphism on a cotangent bundle is generated by a function $S$ on the base, which gives the geometric setup for Arnold's conjecture (fixed points = critical points of $S$ ). Codex has the Arnold conjecture unit (05.08.01) but not the generating-function bridge.
Contact manifolds inside the symplectic narrative. Cannas treats contact geometry as the odd-dimensional cousin of symplectic geometry, with symplectisation $S (M) = M \times R$ and contactisation as paired functors. Codex has zero contact units today; this is the largest topic-level gap.
Toric symplectic geometry as the climax. Part VIII culminates in Delzant's theorem: every compact symplectic toric $2 n$ -manifold is determined up to equivariant symplectomorphism by its Delzant polytope. This is a complete classification result, the only such in symplectic geometry, and is the showcase example of how the moment-map framework buys a global theorem. Codex has the moment map and reduction units (05.04.01, 05.04.02) but no toric or Delzant unit.
Atiyah-Guillemin-Sternberg convexity proved. Lecture 27 walks through the convexity-of-image proof for moment maps of compact-torus actions. Codex's moment-map unit mentions convexity in passing but doesn't prove it.
Marsden-Weinstein-Meyer in the regular case. Lecture 23 / 26 of the Hamiltonian-actions part proves the regular reduction theorem carefully — symplectic structure descends to $M / / G = μ^{- 1} (0) / G$ when $0$ is a regular value of $μ$ and $G$ acts freely on $μ^{- 1} (0)$ . Codex's reduction unit (05.04.02) states the result but the proof is the templated generic Cartan-formula argument.
Duistermaat-Heckman. Lecture 30 ends with the Duistermaat-Heckman theorem on the variation of symplectic volume in reduced spaces, and the equivariant cohomology / localisation formula. This is the canonical bridge into equivariant symplectic geometry. Codex has nothing.
J-holomorphic curves at the introductory level. Lecture 17 states the Cauchy-Riemann equation for a smooth map $u : (Σ, j) \to (M, J)$ and gives the energy-area identity. Cannas does not develop the moduli space; she points to McDuff-Salamon for that. Codex's pseudoholomorphic-curve unit (05.06.02) is currently at Cannas's level (statement-only, no moduli) and that's appropriate; the deepening would be McDuff-Salamon territory, not Cannas.
Examples-first throughout. Each lecture leads with concrete examples ( $T^{*} X$ , $S^{2}$ as Kähler, $C P^{n}$ , coadjoint orbits) before the general theory. Codex's units mostly follow the reverse order; the "Worked example [Beginner]" sections are currently templated and could be replaced with Cannas's actual examples.

The book ends before serious symplectic topology (no Floer, no Gromov-Witten beyond the energy-area identity, no Hofer geometry). Cannas points the reader to McDuff-Salamon for those — so the Floer strand of Codex (05.08.*) is not a Cannas equivalence target; it's covered by the McDuff-Salamon entry whenever that book is audited.

§2 Coverage table (Codex vs Cannas)

Cross-referenced against the 21-unit Codex symplectic chapter and the 220-unit total corpus. ✓ = covered at Cannas-equivalent depth, △ = topic present but Codex unit shallower than Cannas's lecture (typically the templated v0.5 Strand-B prose), ✗ = not covered. The symplectic chapter has a uniform v0.5 Strand-B templated-prose problem: the "Key theorem with proof" in nearly every unit is the same generic Cartan-formula calculation regardless of the unit's stated theorem. Where the topic itself appears at the right scope but the templated proof doesn't actually prove the named theorem, the entry below is △.

Part I — Symplectic Manifolds (Lectures 1-6)

Cannas topic	Codex unit(s)	Status	Note
Symplectic linear algebra ( $V$ , $ω$ , isotropic / coisotropic / Lagrangian / symplectic subspaces, symplectic bases)	`05.01.01` symplectic-vector-space	△	Definitions present; Cannas's full subspace-classification table (the four types) and the standard-basis existence proof are not built out; Master section is templated.
Symplectic manifold (definition, $T^{*} X$ as canonical example, $ω = - d θ$ , Liouville form $θ = p d q$ )	`05.01.02` symplectic-manifold; `05.02.05` cotangent-bundle	△	Both units exist; cotangent-bundle unit covers $T^{} X$ but doesn't actually compute $ω = \sum d p_{i} \land d q_{i}$ as the canonical Darboux model; the Liouville one-form is mentioned but its naturality property ( $α^{} θ = α$ for any 1-form $α : X \to T^{*} X$ ) is missing.
Symplectomorphism group	`05.01.03` symplectic-group	△	Linear case treated; the diffeomorphism-group $Symp (M, ω)$ is mentioned but not developed.
Compatible triples $(ω, J, g)$ , polar decomposition	`05.06.01` almost-complex	△	Unit titled "almost-complex structure on a symplectic manifold"; existence of compatible $J$ is stated as theorem but the proof is the generic Cartan-formula template; polar decomposition argument missing.

Part II — Symplectomorphisms (Lectures 7-9)

Cannas topic	Codex unit(s)	Status	Note
Lagrangian submanifold (definition, half-dim isotropic)	`05.05.01` lagrangian-submanifold	△	Definition present; the canonical examples (zero section of $T^{*} X$ , graph of a closed 1-form, graph of a symplectomorphism in $M^{-} \times M$ ) are not all worked.
Graph criterion: $α : X \to T^{*} X$ is a section, $α$ is Lagrangian iff $d α = 0$	`05.05.01` (mention)	△	Stated as the unit's "Key theorem" but the templated proof doesn't carry it out.
Generating function for a symplectomorphism	—	✗	Gap. Cannas Lecture 9. The generating-function-on-the-base construction $ϕ_{S} (x, d S_{x}) = (x^{'}, ξ^{'})$ that gives the bridge to Arnold's conjecture is missing entirely.
Arnold conjecture (statement, fixed-point version on a cotangent bundle)	`05.08.01` arnold-conjecture	✓	Statement-level coverage matches Cannas's depth; deeper Floer-theoretic version is in `05.08.02`.

Part III — Local Forms (Lectures 10-12) — Cannas's signature section

Cannas topic	Codex unit(s)	Status	Note
Isotopies and time-dependent vector fields, $\frac{d}{d t} ρ_{t}^{} ω_{t} = ρ_{t}^{} (L_{V_{t}} ω_{t} + \frac{d ω _{t}}{d t})$	—	✗	Gap. Cannas Lecture 10. The base identity Moser's trick rests on. Used silently in `05.01.04` Darboux but never stated as its own lemma.
Moser's trick (path method): given a path $ω_{t}$ of symplectic forms with $\frac{d ω _{t}}{d t} = d β_{t}$ exact, find $V_{t}$ with $ι_{V_{t}} ω_{t} + β_{t} = 0$ , integrate to get $ρ_{t}^{*} ω_{t} = ω_{0}$	—	✗	Gap (high priority). Cannas Lecture 11. The most-load-bearing missing technique in the whole symplectic chapter. Used implicitly in (and mis-proved by) the Darboux unit.
Darboux theorem via Moser's trick	`05.01.04` darboux-theorem	△	The unit exists but the proof is the generic Cartan-formula template — it does not actually carry out Moser's path argument or land in the standard form $ω_{0} = \sum d q_{i} \land d p_{i}$ . Depth-gap, high priority.
Relative Darboux / Darboux-Moser-Weinstein	—	✗	Gap. Cannas Lecture 11. Two symplectic forms agreeing on a submanifold $X$ are equivalent on a neighbourhood.
Weinstein Lagrangian neighbourhood theorem	—	✗	Gap (high priority). Cannas Lecture 12. A neighbourhood of a Lagrangian $L \subset M$ is symplectomorphic to a neighbourhood of the zero section in $T^{} L$ . Load-bearing for Floer / Lagrangian Floer / generating-function constructions.*
Weinstein tubular neighbourhood theorem (general symplectic submanifold)	—	✗	Gap. Cannas Lecture 12. Generalisation of the Lagrangian case using the symplectic normal bundle.

Part IV — Contact Manifolds (Lectures 13-14)

Cannas topic	Codex unit(s)	Status	Note
Contact form, contact structure $ξ = ker α$ , $α \land (d α)^{n} \neq = 0$	—	✗	Gap (high priority — entire missing topic). No contact unit anywhere in Codex.
Reeb vector field $R_{α}$	—	✗	Gap.
Symplectisation $(M \times R, d (e^{t} α))$	—	✗	Gap.
Contactisation $M \times R$ for an exact symplectic $M$	—	✗	Gap.
Darboux for contact, Gray's theorem (contact analogue of Moser)	—	✗	Gap.

Part V — Compatible Almost Complex Structures (Lectures 15-17)

Cannas topic	Codex unit(s)	Status	Note
Compatible $J$ exists, contractibility of $J (ω)$	`05.06.01` almost-complex	△	Existence stated, contractibility not. Polar decomposition proof not carried out.
Integrable $J$ , Newlander-Nirenberg statement	—	✗	Gap. Cannas Lecture 16. Standard cross-strand topic; Codex's complex-geometry chapter (`04.*`) may cover Newlander-Nirenberg but it's not currently linked from the symplectic side.
Pseudoholomorphic curve, Cauchy-Riemann equation $d u \circ j = J \circ d u$ , energy-area identity	`05.06.02` pseudoholomorphic-curve	△	Definition present; energy-area identity stated as the "Key theorem" but proof is templated. Cannas's depth here is exactly statement-level + energy identity, so this unit could be at Cannas-equivalent depth with a non-templated rewrite.
Kähler manifold, Hodge identities, Kähler form	`04.05.*` (complex geometry strand, cross-link)	△	Cannas's Part VI is brief; the substantive Kähler exposition lives in Voisin / Griffiths-Harris and the Codex `04.*` strand. Cross-link rather than dedicated symplectic-Kähler unit suffices for Cannas equivalence.

Part VII — Hamiltonian Mechanics (Lectures 22-24)

Cannas topic	Codex unit(s)	Status	Note
Hamiltonian vector field, $ι_{X_{H}} ω = d H$ , conservation of energy	`05.02.01` hamiltonian-vector-field	△	Definition + identity present; the worked example is templated.
Poisson bracket, Jacobi identity, ${H, \cdot} = X_{H}$ derivation	`05.02.02` poisson-bracket	△	Definition present; Jacobi is stated, derivation property worked.
Integrable system (Liouville integrability: $n$ Poisson-commuting independent integrals on $M^{2 n}$ )	`05.02.03` integrable-system	△	Definition present; depth roughly Cannas's.
Arnold-Liouville theorem (compact connected level set is a torus, action-angle coordinates exist on a tubular neighbourhood)	`05.02.04` action-angle-coordinates	△	Title is "action-angle coordinates," theorem is named "Liouville-Arnold local normal form" — but the proof is the generic Cartan-formula template. The actual Arnold-Liouville statement (compact level set ⇒ torus, period lattice gives the action coordinates, Hamilton-Jacobi gives the angle coordinates) is not carried out. Depth-gap, high priority.

Part VIII — Hamiltonian Group Actions (Lectures 25-30) — Cannas's largest section

Cannas topic	Codex unit(s)	Status	Note
Symplectic and Hamiltonian $G$ -action, fundamental vector field	`05.04.01` moment-map (mention); `05.03.01` coadjoint-orbit (mention)	△	The setup-level facts (action by symplectomorphisms, fundamental vector field $X^{♯}$ ) appear in passing across these two units but are not developed cleanly.
Moment map (definition, three conditions: $d μ^{X} = ι_{X^{♯}} ω$ , equivariance, additivity)	`05.04.01` moment-map	△	Definition present; the three-condition characterisation is stated as the "Key theorem" but the templated proof does not actually establish the equivariance or the additivity-of-moments-of-Hamiltonians lemma.
Coadjoint orbit, Kirillov-Kostant-Souriau form	`05.03.01` coadjoint-orbit	△	Topic present; KKS form mentioned; the proof that KKS is symplectic and the moment map for the $G$ -action on $O_{ξ}$ is the inclusion is templated.
Marsden-Weinstein-Meyer reduction (regular case): $0$ regular value of $μ$ , $G$ acts freely on $μ^{- 1} (0)$ , then $M / / G = μ^{- 1} (0) / G$ is symplectic with the descended form	`05.04.02` symplectic-reduction	△	The unit exists; the regular case is what Cannas actually proves (vs. the singular case which needs stratifications). The current Codex unit's proof is templated. Depth-gap, high priority.
Reduction examples ( $S^{2}$ from $C^{2}$ , $C P^{n}$ from $C^{n + 1}$ , complex Grassmannians)	—	△	Cannas's worked examples in Lecture 26 are absent from the reduction unit. The Beginner "Worked example" slot in `05.04.02` is the templated phase-plane example.
Atiyah-Guillemin-Sternberg convexity theorem: image of moment map for compact-torus action on a connected compact symplectic $M$ is the convex hull of the images of the fixed points	—	✗	Gap (high priority). Cannas Lecture 27. One of the two named global theorems in Hamiltonian-actions (the other is Delzant).
Symplectic toric manifold: compact symplectic $2 n$ -manifold with effective Hamiltonian $T^{n}$ -action; the moment polytope	—	✗	Gap (high priority). Cannas Lectures 28-29.
Delzant's theorem: compact symplectic toric manifolds are classified by Delzant polytopes	—	✗	Gap (high priority). Cannas Lecture 29. Cannas's headline theorem; the showcase example of how the moment-map framework yields a complete classification.
Duistermaat-Heckman theorem: variation of symplectic volume in reduced spaces is polynomial of degree $n$ on regular values, with the equivariant-cohomology localisation formula as corollary	—	✗	Gap. Cannas Lecture 30.

Topics Cannas covers as remarks / pointers (no Codex equivalence-coverage required)

Cannas topic	Status	Note
KAM theorem (mentioned as motivation in §22)	—	Pointer-only in Cannas; Arnold is the primary reference. Defer to Arnold's per-book plan.
Floer homology (mentioned as the proof tool for Arnold conjecture, §9 / §17)	`05.08.02` floer-homology	△
Gromov non-squeezing (mentioned in §17 as the headline rigidity theorem)	`05.07.01` non-squeezing; `05.07.02` symplectic-capacity	△

Aggregate coverage estimate

Theorem layer: ~75% of Cannas's named theorems map to Codex units at the topic level; but only ~40% are at Cannas-equivalent proof depth. The remaining 25% (Moser's trick, Weinstein neighbourhoods, Arnold-Liouville proof, Atiyah-Guillemin-Sternberg, Delzant, Duistermaat-Heckman, contact geometry block) are absent at any depth. After the priority-1 punch-list below, the topic-level coverage rises to ~95% and the proof-depth coverage rises to ~80%; after priority-2 to ~90% proof-depth.

Exercise layer: not separately audited. Cannas's "Homework" sections (typically 4-6 problems per lecture, ~150 total) are standard symplectic exercises (compute moment map for an action, verify Jacobi identity, work out the integrable system on $T^{*} S^{n}$ ). Codex's symplectic exercise pack is the same templated 7-question block on every unit (the "what is the dimension if $n = 4$ " / "rewrite $Φ^{*} ω = ω$ " / "Cartan formula" set repeated 21 times); the actual Cannas-style exercises are not shipped. Defer to a dedicated symplectic-exercise-pack pass after the priority-1+2 batch lands.

Worked-example layer: ~25% covered. Cannas leads each lecture with a worked example; Codex's units share a single templated phase-plane example across all 21 units. The worked-example layer is the single largest exercise/example gap in the symplectic chapter.

Notation layer: ~80% aligned. Cannas writes $ω$ for the symplectic form, $θ$ for the Liouville one-form, $X^{♯}$ for the fundamental vector field of a $G$ -action, $μ$ for the moment map, $M / / G$ for symplectic reduction, $Δ$ for the moment polytope. Codex's units mostly match. A notation/cannas.md crosswalk is optional given the mostly-aligned conventions.

Sequencing layer: ~85%. The Codex DAG follows Cannas's prerequisite flow (linear → manifold → Hamiltonian VFs → moment map → reduction) faithfully. Missing edges: Moser's trick → Darboux, Weinstein neighbourhood → Floer setup, contact prerequisites for any future contact unit.

Intuition layer: ~50%. Cannas's lecture-style intuition is under-reproduced; the templated "phase space is paired" Beginner sections across all 21 units don't capture Cannas's lecture-by-lecture motivation.

Application layer: ~70%. Cannas's applications (geodesic flow as Hamiltonian on $T^{*} M$ , $C P^{n}$ as toric, the Hopf bundle as symplectic reduction, integrable systems on Lie groups) are partially covered via the Floer / Atiyah-Singer downstream units, but not as dense inline applications.

§3 Gap punch-list (P3-lite — units to write or deepen, priority-ordered)

The Codex symplectic chapter is mature in topic coverage. Most of the work below is deepening existing units to Cannas-proof-depth, plus a small number of genuinely new units (Moser, Weinstein neighbourhood, contact, Delzant, Duistermaat-Heckman). The recommended slot range is 05.09.* for new units and 05.10.* for the contact block, leaving the existing 05.01.*-05.08.* numbering stable. Where a deepening replaces the templated proof on an existing unit, no new ID is needed.

Priority 1 — load-bearing technique gaps and signature theorems

These items either provide infrastructure used silently elsewhere (Moser's trick, Weinstein neighbourhood, isotopies-and-vector-fields identity) or are Cannas's headline classification theorems (Delzant, Atiyah-Guillemin-Sternberg). Without them the symplectic chapter cannot honestly claim Cannas-equivalence.

05.01.05 Moser's trick (path method). Cannas Lecture 10-11 anchor; Moser 1965 On the volume elements on a manifold as originator-text. Three-tier; ~1800 words. The Master section gives the path-of-symplectic-forms statement, the cohomological primitivity hypothesis, and the integration of $V_{t}$ to get the isotopy. Used as a black box in items 2, 3, 7. Highest priority — every other item in this punch-list builds on it.
Deepen 05.01.04 Darboux's theorem. Replace the templated "Key theorem with proof" with the actual Moser-trick proof: take $ω_{0}$ to be $ω$ at $p$ in arbitrary coordinates, $ω_{1}$ to be the standard form $\sum d q_{i} \land d p_{i}$ in the chosen Darboux chart, the path $ω_{t} = (1 - t) ω_{0} + t ω_{1}$ is symplectic on a small neighbourhood, apply Moser. Cannas Lecture 8. No new unit ID; rewrite of the Intermediate "Key theorem" and Master "Full proof" sections.
05.05.02 Weinstein Lagrangian neighbourhood theorem. Cannas Lecture 12 anchor; Weinstein 1971 Symplectic manifolds and their Lagrangian submanifolds as originator-text. Three-tier; ~2000 words. Master section gives the proof: pick a metric, identify $T^{*} L$ with the symplectic-orthogonal complement of $T L$ in $T M ∣_{L}$ , apply the relative Moser theorem. Load-bearing for Floer setup (05.08.02), Lagrangian Floer (any future Lagrangian-intersection unit), and generating-function constructions.
Deepen 05.04.02 symplectic-reduction (Marsden-Weinstein-Meyer, regular case). Replace the templated proof with the actual regular-case argument: $0$ a regular value ⇒ $μ^{- 1} (0)$ is a smooth submanifold; $G$ -equivariance ⇒ $G$ acts on $μ^{- 1} (0)$ ; if the action is free, the quotient is smooth; the reduced form $ω_{red}$ on $M / / G$ is uniquely characterised by $π^{*} ω_{red} = ι^{*} ω$ . Cannas Lecture 23 anchor; Marsden-Weinstein 1974 Reduction of symplectic manifolds with symmetry and Meyer 1973 as joint originator-citations. No new unit ID; rewrite of the Intermediate and Master sections.
05.04.03 Atiyah-Guillemin-Sternberg convexity. Cannas Lecture 27 anchor; Atiyah 1982 Convexity and commuting Hamiltonians and Guillemin-Sternberg 1982 Convexity properties of the moment map as joint originator-citations. Three-tier; ~1800 words. Statement: for a Hamiltonian $T^{k}$ -action on a connected compact symplectic $M$ , the image $μ (M)$ is a convex polytope, equal to the convex hull of the images of the fixed points. Master section gives the proof via the Morse-theoretic argument on the components of $μ$ .
05.04.04 Symplectic toric manifold and Delzant's theorem. Cannas Lectures 28-29 anchor; Delzant 1988 Hamiltoniens périodiques et image convexe de l'application moment as originator-text. Three-tier; ~2200 words. Statement: compact symplectic toric $2 n$ -manifolds with effective Hamiltonian $T^{n}$ -action are classified up to equivariant symplectomorphism by Delzant polytopes. Master section sketches the symplectic-cut / reduction-from- $C^{d}$ construction. The signature global theorem of Cannas's book.
Deepen 05.02.04 action-angle-coordinates (Arnold-Liouville theorem). Replace the templated "Liouville-Arnold local normal form" with the actual statement and proof: a regular compact connected level set of an integrable system is a Lagrangian torus; on a tubular neighbourhood there exist coordinates $(I, θ)$ on $D \times T^{n}$ with $ω = \sum d I_{i} \land d θ_{i}$ and the Hamiltonian flows are linear on the torus. Cannas Lecture 24 anchor; Liouville 1855 + Arnold 1963 as originator-citations. No new unit ID; rewrite of the Intermediate and Master sections.

Priority 2 — Cannas-distinctive depth gaps on existing units, plus the contact block

These items either replace templated content with Cannas-anchored prose on existing units, or introduce the contact-symplectic adjacency block that Cannas treats as integral to the subject but Codex omits entirely.

05.05.03 Generating functions and Hamiltonian symplectomorphisms. Cannas Lecture 9 anchor. Three-tier; ~1500 words. The graph $gr (ϕ) \subset M^{-} \times M$ of a symplectomorphism is Lagrangian; for $ϕ$ Hamiltonian and isotopic to identity, the graph is generated by a function $S$ on the base. Bridge to 05.08.01 Arnold conjecture (fixed points = critical points of $S$ ). Master section gives the cotangent-bundle case in full.
05.10.01 Contact manifold. New chapter (05.10-contact/). Cannas Lectures 13-14 anchor; Cartan / Reeb originator-citations. Three-tier; ~1800 words. Definitions: contact form $α$ on $M^{2 n + 1}$ with $α \land (d α)^{n} \neq = 0$ , contact structure $ξ = ker α$ , Reeb vector field $R_{α}$ characterised by $ι_{R_{α}} d α = 0$ and $α (R_{α}) = 1$ . Worked example: $S^{2n+1} \subset \mathbb{C}^{n+1}$ with the standard contact structure as the kernel of $α = \frac{1}{2} \sum (x_{i} d y_{i} - y_{i} d x_{i})$ .
05.10.02 Symplectisation and contactisation. Cannas Lecture 14 anchor. Three-tier; ~1500 words. Symplectisation $S (M, α) = (M \times R, d (e^{t} α))$ as a functor from contact to symplectic. Contactisation $C (M, ω) = (M \times R, d t + θ)$ for $ω = d θ$ exact symplectic. Worked examples: $S (S^{2 n + 1}) = C^{n + 1} ∖ {0}$ , $C (T^{*} X) = T^{*} X \times R$ with the canonical contact structure for the jet bundle interpretation.
05.10.03 Gray's theorem (contact stability). Cannas Lecture 13 (in passing) + Gray 1959 originator-citation. Three-tier; ~1200 words. The contact analogue of Moser: a smooth path of contact structures on a compact $M$ is generated by a contact isotopy. Mirror Moser's-trick proof via the contact-form path method.
Deepen 05.06.01 almost-complex (compatible $J$ existence and contractibility). Replace the templated proof with the actual polar-decomposition argument: pick any Riemannian metric $g_{0}$ on $M$ ; $ω$ and $g_{0}$ together give an automorphism $A$ of $T M$ ; polar decompose $A = J ∣ A ∣$ with $J$ orthogonal and $J^{2} = - I$ ; show $J$ is compatible. Contractibility of $J (ω)$ via the Cayley-transform / convex-combination argument. Cannas Lecture 15. No new unit ID.
Deepen 05.04.01 moment-map. Replace the templated proof with the actual three-condition derivation: equivariance from differentiating the $G$ -action, additivity from the Hamiltonian bracket. Add Cannas's worked examples: rotation on $S^{2}$ , $T^{n}$ -action on $C^{n}$ , coadjoint-orbit case. Cannas Lectures 22, 25-26. No new unit ID.
Deepen 05.03.01 coadjoint-orbit. Replace the templated proof with the actual KKS-form construction: $\omega_\xi(X^\sharp, Y^\sharp) = \xi([X, Y])$ on the orbit, prove nondegeneracy from the kernel of $ad_{ξ}^{*}$ being the stabiliser, prove closedness from Jacobi. Cannas Lecture 14 (footnote in §V), Kirillov 1962, Kostant 1970, Souriau 1970. No new unit ID.

Priority 3 — Cannas-distinctive optional units (depth-completing)

These items round out Cannas's headline content but are not strictly required for FT equivalence: items 15-16 are showcase global theorems that round out the moment-map block; item 17 is a depth-completion on the cotangent-bundle unit; items 18-19 are Cannas-specific pedagogical units that don't have a single named theorem but matter for the exercise/example layer.

05.04.05 Duistermaat-Heckman theorem. Cannas Lecture 30 anchor; Duistermaat-Heckman 1982 On the variation in the cohomology of the symplectic form of the reduced phase space as originator-text. Three-tier; ~1800 words. Statement: for a Hamiltonian $T^{k}$ -action with proper moment map, the pushforward of the Liouville measure on a regular reduced level set is polynomial in the moment-map value. Master section sketches the equivariant-cohomology localisation derivation (Atiyah-Bott / Berline-Vergne).
05.04.06 Symplectic blow-up and symplectic cut. Cannas Lectures 28-29 (treated as construction tools for toric manifolds) + Lerman 1995 Symplectic cuts as originator-citation. Three-tier; ~1500 words. Symplectic blow-up replaces a point with a copy of $C P^{n - 1}$ ; symplectic cut is the reduction-by- $S^{1}$ -action construction that yields blow-up as a special case. Used as the building block in the Delzant proof (item 6 above), so produce alongside item 6.
05.02.06 Cotangent bundle (deepening of 05.02.05). Replace the templated proof with the actual computation $ω = - d θ$ , the naturality property $α^{*} θ = α$ , and the worked examples of the geodesic flow on $T^{*} M$ as a Hamiltonian system. Cannas Lecture 2 anchor. No new unit ID; rewrite of 05.02.05.
05.05.04 Lagrangian fibration. Cannas Lecture 29 (as backbone of toric manifolds) anchor. Three-tier; ~1500 words. A proper map $π : M \to B$ whose fibres are Lagrangian. Key examples: $T^{*} X \to X$ (cotangent projection), the moment-map of a toric manifold $μ : M \to Δ$ over the interior. Bridge to mirror symmetry literature (SYZ conjecture). Optional — the abstract concept matters but the headline theorems are in Gross / Hitchin downstream.

Priority 4 — Survey / pointer items, optional

05.10.04 Survey of contact topology and Reeb dynamics. Master-only, ~900 words. Pointer to Geiges An Introduction to Contact Topology, the Weinstein conjecture, recent Hofer-Wysocki-Zehnder. Deferred unless Codex expands into contact homology.
05.06.03 Newlander-Nirenberg integrability. Cannas Lecture 16 mentions; the actual treatment lives in the Codex complex geometry chapter. Optional cross-link unit; Master-only, ~700 words. Could be filed as 04.05.* instead; depends on where the cross-strand DAG settles.

§4 Implementation sketch (P3 → P4)

Minimum Cannas-equivalence batch = priority 1 only (items 1-7): 3 new units (05.01.05, 05.05.02, 05.04.03, 05.04.04) plus 4 deepenings (Darboux, reduction, action-angle, two more in priority 2 list). Realistic production estimate (mirroring earlier Lawson-Michelsohn / Bott-Tu / Brown-Hatcher batches):

~3 hours per new unit (research + draft + validate at 27/27 + Lean stub + Bridge / Synthesis prose).
~1.5 hours per deepening (replace templated section with Cannas-anchored prose; no new frontmatter / DAG edges in most cases).
Priority 1 totals: 4 new × 3 h + 4 deepenings × 1.5 h = ~18 hours.
Priority 1+2 totals: 4 new + 4 contact-block new + 1 generating-fn new + 4 priority-1 deepenings + 3 priority-2 deepenings = ~36 hours of focused production. At 4-6 production agents in parallel, this fits in a 2-3 day window with one integration agent stitching outputs.

Smaller surface than Brown / Hatcher. The Brown punch-list shipped ~7 priority-1 units and the Hatcher punch-list shipped 10 priority-1+2 units; this Cannas plan ships 3-4 new units at priority-1 and the balance of the work is depth deepening on already-shipped units. The production cost is comparable but the integration-agent load is lower because most deepenings are in-place rewrites of existing content/05-*/*.md files, not new files needing CONCEPT_CATALOG and deps.json edits.

Originator-prose targets. Cannas is herself a modern lecture-note synthesis in the Brown / Hatcher sense — the originator citations for the new priority-1 units are:

Moser's trick (item 1): Moser 1965, On the volume elements on a manifold, Trans. AMS 120, 286-294. Cannas Lecture 11 cites Moser directly.
Weinstein Lagrangian neighbourhood (item 3): Weinstein 1971, Symplectic manifolds and their Lagrangian submanifolds, Adv. Math. 6, 329-346.
Marsden-Weinstein-Meyer reduction (item 4 deepening): Marsden-Weinstein 1974 Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5, 121-130; Meyer 1973 Symmetries and integrals in mechanics (independently).
Atiyah-Guillemin-Sternberg convexity (item 5): Atiyah 1982 Convexity and commuting Hamiltonians, Bull. LMS 14; Guillemin-Sternberg 1982 Convexity properties of the moment mapping, Invent. Math. 67. Cite both.
Delzant's theorem (item 6): Delzant 1988 Hamiltoniens périodiques et image convexe de l'application moment, Bull. SMF 116, 315-339.
Arnold-Liouville theorem (item 7 deepening): Liouville 1855 (as the $L = {f_{i} = c_{i}}$ tori-foliation observation); Arnold 1963 (action-angle coordinate theorem in Mathematical Methods); cite both.
Duistermaat-Heckman (item 15): Duistermaat-Heckman 1982, Invent. Math. 69.
Symplectic cut (item 16): Lerman 1995, Math. Res. Lett. 2.
Contact-form (item 9): Sophus Lie 1872 (jet bundles); Cartan 1899-1909 contact transformations; Reeb 1952 Sur certaines propriétés topologiques des trajectoires des systèmes dynamiques.
Gray's theorem (item 11): Gray 1959 Some global properties of contact structures, Ann. Math. 69.

Each priority-1 unit's Master section should cite the originator paper in addition to Cannas.

Notation crosswalk. Cannas's notation is mostly aligned with Codex already (per §2 above). A notation/cannas.md file is not needed as a separate artifact; the existing tier_anchors field on each Cannas-anchored unit (citing Cannas da Silva §N) is sufficient. The one notation choice worth pinning down is the moment-map sign convention — Cannas uses $d μ^{X} = ι_{X^{♯}} ω$ (matching the Codex Hamiltonian-vector-field convention $ι_{X_{H}} ω = d H$ ). Record this in the rewritten 05.04.01 Master section.

DAG edges to add. New prerequisites arrows for the priority-1+2 batch:

05.01.05 (Moser) ← {05.01.04 Darboux, 05.05.02 Weinstein, 05.10.03 Gray}
05.01.04 (Darboux, after deepening) → still has its existing successors; no new arrows.
05.05.02 (Weinstein Lagrangian nbhd) ← {05.08.02 Floer homology, any future Lagrangian Floer unit}
05.05.03 (generating fns) ← {05.08.01 Arnold conjecture}
05.04.03 (AGS convexity) ← 05.04.04 Delzant
05.04.04 (Delzant) ← 05.04.06 symplectic cut (priority-3 item 16)
05.04.06 (sympl. cut) ← 05.04.04 Delzant (so this is a co-batch pair, see §6)
05.10.01 (contact mfd) → {05.10.02 symplectisation, 05.10.03 Gray}

Composite Cannas+Arnold batch recommendation. The Cannas Hamiltonian-mechanics block (Lectures 22-24, items 7 and 13 above) and the Arnold Mathematical Methods per-book plan (when produced) will share the action-angle / integrable-systems content. Defer the Arnold audit to the same campaign window so that the Arnold-Liouville deepening (item 7) covers both books with one rewrite.

§5 What this plan does NOT cover

A line-number-level inventory of every named theorem and Homework problem in Cannas's 30 lectures. (Would take the full P1 audit; deferred unless the priority-1+2 punch-list expands.)
The Cannas Homework / exercise pack (~150 problems across 30 lectures). Currently the Codex symplectic exercise pack is the templated 7-question block on every unit. A dedicated 05.E1/05.E2 Cannas-exercise-pack family is a P3-priority follow-up after the priority-1+2 theorem-layer batch closes.
McDuff-Salamon territory: the moduli space of $J$ -holomorphic curves, Gromov compactness with bubbling, Floer homology beyond the setup-level, Hofer geometry, symplectic topology beyond Gromov non-squeezing. Cannas points to McDuff-Salamon for these and so do we; that book gets its own per-book plan.
Arnold's Mathematical Methods of Classical Mechanics: the KAM-theorem part, the variational principles, the rigid-body examples. The Cannas/Arnold overlap in Hamiltonian mechanics (Lectures 22-24) is covered by the items 7, 13 deepenings; Arnold-only material (KAM, Hamilton-Jacobi PDE, rigid body, geodesic flow as a worked Hamiltonian system) is deferred to the Arnold per-book plan.
Voisin / Griffiths-Harris Kähler material. Cannas's Part VI Kähler block is brief; Codex's 04.05.* complex-geometry chapter is the primary equivalence target. Cross-link rather than duplicate.
Equivariant cohomology beyond the Duistermaat-Heckman corollary. The general theory (Atiyah-Bott / Berline-Vergne localisation, $K$ -theoretic versions) is Atiyah Yang-Mills (FT 3.20) territory, not Cannas's.
Floer homology depth gap. The shipped 05.08.01-04 Floer block is templated (same proof template across all four units); fixing this is not Cannas's job — it's McDuff-Salamon's. Defer to the McDuff-Salamon audit.
Worked-example densification across the existing 21 units. The templated phase-plane "Worked example [Beginner]" sections are a curriculum-wide v0.5-Strand-B pattern; replacing them with Cannas's actual lecture-opening examples ( $T^{*} X$ canonical 1-form, $S^{2} \subset R^{3}$ as area-form, $C P^{n}$ from reduction, etc.) is a large but optional pass. Only the priority-1 / priority-2 unit deepenings carry mandatory worked-example rewrites.

§6 Acceptance criteria for FT equivalence (Cannas)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4 and §9, the book is at equivalence-coverage when:

≥95% of Cannas's named theorems map to Codex units at Cannas-equivalent proof depth (currently ~40%; after priority-1 this rises to ~75%; after priority-1+2 to ~92%; after priority-3 deepenings to ~95%).
≥80% of Cannas's Homework problems have a Codex equivalent (currently ~5% — the templated 7-problem block; closing this requires the dedicated Cannas-exercise-pack pass per §5).
≥90% of Cannas's worked examples are reproduced in some Codex unit (currently ~25%; the priority-1+2 batch's required worked-example rewrites bring this to ~60%; the remainder requires the optional worked-example densification pass).
The notation alignment is recorded inline in the rewritten Master sections (no separate notation/cannas.md needed).
For every chapter dependency in Cannas (Part I → III → VII → VIII; V → III; II → III), there is a corresponding prerequisites arrow chain in Codex's DAG. The Moser-Weinstein-reduction-toric chain in particular must be unbroken after the priority-1 batch.
Pass-W weaving connects the new units (05.01.05, 05.05.02, 05.04.03, 05.04.04, 05.10.01-03) to the existing 05.02-hamiltonian/, 05.04-moment-reduction/, 05.05-lagrangian/, 05.06-almost-complex/, and 05.08-floer/ units via lateral connections.

The 7 priority-1 items (items 1-7) close the load-bearing-technique and signature-theorem gap. The 7 priority-2 items (items 8-14) close the contact block and the Cannas-distinctive depth gaps on existing units. The 6 priority-3+4 items (items 15-20) are depth-completion and survey pointers; they bring proof-depth coverage from ~92% to ~95%+ but are not strictly required for sign-off.

Composite Cannas + Arnold batch recommendation. Because Cannas Lectures 22-24 (Hamiltonian mechanics) and Arnold's Hamiltonian-mechanics Part III overlap on the Arnold-Liouville theorem and the Hamilton-Jacobi machinery, producing the Cannas priority-1 batch and the (yet-to-be- written) Arnold per-book plan's priority-1 batch together yields a Cannas+Arnold composite of ~10-12 units that closes both books' mechanics gaps simultaneously. This is the recommended execution path once Arnold is audited.

Honest scope. Cannas's Lectures on Symplectic Geometry is the smallest equivalence-gap book on the Tier-α-adjacent FT booklist: the symplectic chapter is already 21 units and the topic-level coverage is ~75%. The work in this plan is dominated by depth rewrites, not by new-unit production. The new units are concentrated in the contact block (05.10.* — entirely missing) and in the moment-map climax (05.04.03-06 — the Atiyah-Guillemin-Sternberg / Delzant / Duistermaat-Heckman global theorems). Everything else is in-place deepening of templated v0.5-Strand-B prose.

Cannas da Silva — *Lectures on Symplectic Geometry* (Fast Track 1.11) — Audit + Gap Plan