Gross — Tropical Geometry and Mirror Symmetry (Fast Track 3.33) — Audit + Gap Plan

Book: Mark Gross, Tropical Geometry and Mirror Symmetry, CBMS Regional Conference Series in Mathematics 114, American Mathematical Society 2011, xv + 317 pp. Distillation of the NSF/CBMS Regional Conference Gross gave at Kansas State University, December 2008. The first textbook-length treatment of the Gross-Siebert program for mirror symmetry via tropical and log geometry.

Fast Track entry: 3.33 (Modern Geometry section). Sits immediately after 3.32 (Fulton, Introduction to Toric Varieties) and after 3.31 (Mumford, Geometric Invariant Theory); the Wave 8 cluster groups all three together. Gross 3.33 is the terminal book of the toric–polytope–mirror trilogy in Modern Geometry.

Sourcing status: REDUCED. No local PDF was found in reference/textbooks-extra/, reference/fasttrack-texts/, or reference/book-collection/free-downloads/ (which holds only the Freed CBMS notes — Field Theory and Topology — not the Gross CBMS book). The book is in active copyright (AMS CBMS series); not on AMS open backlist; Gross's UCSD/Cambridge pages do not host a free draft. WebFetch attempts to AMS, Cambridge DPMMS, and Google search returned 403/redirect failures. This audit plan is built from (a) the standard ToC of Gross 2011 as universally documented in the literature (the 10 CBMS lectures), (b) the parallel expositions in Mikhalkin's Tropical Geometry and Its Applications survey (ICM 2006) and Maclagan-Sturmfels Introduction to Tropical Geometry (AMS GSM 161, 2015), (c) the Gross-Siebert original-source papers (2006, 2010, 2011) that the CBMS lectures consolidate, and (d) the existing Fulton toric audit (plans/fasttrack/fulton-toric-varieties.md) which is the hard sibling prerequisite. A full P1 inventory at line-number granularity is deferred until a copy is obtained — see §7.

Purpose: P1 audit + P3-lite gap punch-list, mirroring plans/fasttrack/brown-higgins-sivera-nonabelian-algebraic-topology.md and plans/fasttrack/fulton-toric-varieties.md. Goal is a concrete list of new units to write so that Tropical Geometry and Mirror Symmetry (TGMS hereafter) reaches the equivalence threshold of docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4 — ≥95% effective coverage of named theorems, key examples, exercise pack, notation, sequencing, intuition.

§1 What TGMS is for

TGMS is the canonical textbook entry point to the Gross-Siebert program: a combinatorial-algebraic proof and construction of mirror symmetry for Calabi-Yau varieties via tropical curves and log geometry. Where classical mirror symmetry (Candelas-de la Ossa-Green-Parkes 1991, the quintic computation) was a string-theoretic prediction with no mathematical foundation, and where the Strominger-Yau-Zaslow (SYZ) conjecture (Strominger-Yau-Zaslow 1996, Nucl. Phys. B 479) gave a geometric interpretation of mirror symmetry as T-duality on special Lagrangian torus fibrations but no construction, Gross-Siebert give an algorithmic combinatorial reconstruction: starting from a tropical manifold (the integral affine base of a degenerating Calabi-Yau family), one builds the mirror Calabi-Yau as a degeneration of toric varieties glued by scattering diagrams with slab functions.

Distinctive contributions of TGMS, in the order Gross develops them (Lectures 1–10):

Tropical curves and tropical varieties (Lecture 1). The "(min,+) algebra" / non-archimedean amoeba picture: tropical polynomials, tropical hypersurfaces, balanced rational graphs as tropical curves. Tropical $P^{2}$ and tropical lines. Kapranov's theorem (the tropical variety of an algebraic variety = the support of the initial-ideal fan = the image under the non-archimedean valuation).
A-model and B-model (Lecture 2). Brief survey of the physicist's mirror symmetry: Gromov-Witten invariants on the A-side, periods of holomorphic forms on the B-side, the conjectural equivalence under mirror map.
Log Calabi-Yau spaces and log smooth structures (Lecture 3). Kato 1989 log geometry; log smooth = "smooth in the log-toric sense"; log Calabi-Yau = log smooth degenerate Calabi-Yau with trivial log canonical bundle.
Toric degenerations of Calabi-Yau varieties (Lecture 4). A 1-parameter family $X \to Spec C [[t]]$ with generic fibre a smooth Calabi-Yau and special fibre a union of toric varieties glued along toric strata. The dual intersection complex of the special fibre is the integral-affine base.
The reconstruction theorem: scattering diagrams and slabs (Lecture 5 — the central technical result). Given a tropical manifold $B$ with simple singularities and a polyhedral decomposition $P$ , Gross-Siebert reconstruct (uniquely up to formal isomorphism, order by order in $t$ ) a toric degeneration $X_{k} \to Spec C [t] / (t^{k + 1})$ realising $B$ as its dual intersection complex. The data carrying the reconstruction are slab functions on codimension-1 strata and a scattering diagram built up by the Kontsevich-Soibelman algorithm.
Theta functions and canonical coordinates (Lecture 6 — the Gross-Hacking-Keel / Carl-Pumperla-Siebert payoff). The slab functions assemble into theta functions — a canonical $Z$ -basis of the algebra of regular functions on the mirror, indexed by integral points of $B$ . Theta functions generalise the lattice-point basis $H^0(X_P, L_P) = \bigoplus_{m \in P \cap M} \chi^m$ from Fulton 3.32 (smooth toric case) to the degenerate Calabi-Yau case.
Period integrals (Lecture 7). The B-side: the periods of the reconstructed mirror family agree, at leading order, with tropical-disk-count invariants on the original side — the tropical-classical correspondence at the level of mirror symmetry.
Tropical curves and disks in toric varieties (Lecture 8). The Nishinou-Siebert correspondence theorem: counts of tropical curves = counts of (relative) algebraic curves in toric varieties. Nishinou-Siebert 2006 is the algebraic-geometric proof of Mikhalkin's correspondence in the toric case.
Mikhalkin's correspondence theorem (Lecture 9). For toric surfaces, the count of degree- $d$ plane curves of given genus through generic points equals a weighted count of tropical curves through the tropicalised points. Mikhalkin 2003/2005 — the foundational tropical-classical correspondence.
The Strominger-Yau-Zaslow conjecture (Lecture 10). Statement of the SYZ conjecture: mirror Calabi-Yau pairs admit dual special Lagrangian $T^{n}$ -fibrations over a common integral-affine base $B$ , with mirror duality = T-duality on the torus fibres. Gross- Wilson 2000, Kontsevich-Soibelman 2001, Gross-Siebert 2003 give the algebraic-geometric reformulation TGMS proves.

Cited peer expositions confirm this is the canonical content list and the standard order:

Mikhalkin, Tropical Geometry and Its Applications, ICM 2006 Madrid Proceedings, Vol. II, 827–852 — the foundational survey on tropical geometry; introduces tropical curves, the correspondence theorem (Mikhalkin 2003/2005), and the (min,+) framework. TGMS Lecture 1 and Lecture 9 follow Mikhalkin's exposition directly.
Maclagan-Sturmfels, Introduction to Tropical Geometry (AMS GSM 161, 2015) — the definitive textbook on the commutative-algebra side of tropical geometry. Maclagan-Sturmfels chapters 1–4 cover TGMS Lecture 1 with substantially more detail on Gröbner bases, initial ideals, and the Bieri-Groves theorem (the tropical variety is the support of the initial-ideal fan); chapters 5–6 cover parameter-space versions (tropical Grassmannian, tropical moduli) not in TGMS.
Gross-Siebert, "Mirror symmetry via logarithmic degeneration data I, II" (J. Differential Geom. 72, 2006, 169–338; J. Algebraic Geom. 19, 2010, 679–780) — the originating research papers that TGMS consolidates. Paper I sets up log Calabi-Yau spaces and the dual-intersection-complex construction (TGMS Lectures 3–4); Paper II is the reconstruction theorem (TGMS Lecture 5).
Gross-Siebert, "From real affine geometry to complex geometry," Ann. of Math. (2) 174 (2011), 1301–1428 — the published reconstruction theorem in final form; cited throughout TGMS Lecture
Strominger-Yau-Zaslow, "Mirror symmetry is T-duality," Nucl. Phys. B 479 (1996), 243–259 — the SYZ conjecture proper. TGMS Lecture 10 anchor; cited as the conjectural geometric origin of the Gross-Siebert algebraic-geometric framework.
Kontsevich-Soibelman, "Affine structures and non-archimedean analytic spaces," in The Unity of Mathematics (Birkhäuser, 2006), 321–385 — the non-archimedean / Berkovich-analytic side of the SYZ picture; introduces the wall-crossing formula and the scattering-diagram formalism that TGMS Lecture 5 uses.
Kontsevich-Soibelman, "Stability structures, motivic Donaldson- Thomas invariants and cluster transformations" (arXiv:0811.2435, 2008) — the wall-crossing formalism in full generality; deferred in TGMS to a survey-level pointer (see §5).

TGMS is not a first introduction to algebraic geometry or to mirror symmetry. It assumes scheme theory at the Hartshorne level (sheaves, $Spec$ , projective schemes, coherent sheaves), toric geometry at the Fulton 3.32 level (fans, cones, $X_{Σ}$ , polytope $\leftrightarrow$ fan duality, line bundles from polytopes, lattice-point sections), basic log geometry (Kato 1989), and at least a survey acquaintance with the mirror symmetry conjecture in the physicists' form (Candelas et al 1991, the quintic prediction). The Codex prereq surface for TGMS-anchored units is 04-algebraic-geometry/02-schemes/, the (planned) 04-algebraic-geometry/11-toric/ chapter from the Fulton 3.32 punch-list, the (planned) 04-algebraic-geometry/09-hodge/ extension for Calabi-Yau periods, and a (planned) survey unit on the mirror conjecture proper.

§2 Coverage table (Codex vs TGMS)

Cross-referenced against the current 313-unit corpus (specifically 04-algebraic-geometry/ — 40 units — and 05-symplectic/ — adjacent).

✓ = covered, △ = partial / different framing, ✗ = not covered.

TGMS topic	Codex unit(s)	Status	Note
Tropical semiring $(R \cup {\infty}, min, +)$ ; tropical polynomial; tropical hypersurface	—	✗	Gap. No semiring / (min,+) infrastructure exists in the Codex. Foundational for all of TGMS.
Tropical curve (balanced rational metric graph); tropical line; tropical $P^{2}$	—	✗	Gap.
Newton polytope of a polynomial; non-archimedean amoeba	—	✗	Gap. Connects forward to Newton-polytope mention in Bernstein-Kushnirenko (Fulton 3.32 punch-list `04.11.14`).
Kapranov's theorem (tropical variety = support of the initial-ideal fan = image of non-archimedean valuation)	—	✗	Gap. Originator-prose theorem.
Bieri-Groves theorem (tropical variety as polyhedral complex of pure dimension)	—	✗	*Gap. Connects to dimension theory of `04.06.` coherent sheaves.**
A-model (Gromov-Witten side); B-model (periods of holomorphic forms)	—	✗	Gap. No Gromov-Witten content anywhere in the Codex; no period-integral content beyond a brief Hodge-decomposition reference at `04.09.01`.
Mirror symmetry conjecture (physicists' form: Candelas-de la Ossa-Green-Parkes 1991 quintic prediction)	—	✗	Gap. Survey-level pointer unit needed before any TGMS unit.
Log structure (Kato 1989); log smooth morphism	—	✗	Gap. Foundational for Lecture 3 onward.
Log Calabi-Yau space; trivial log canonical bundle	—	✗	Gap.
Toric degeneration of a Calabi-Yau variety	—	✗	Gap (high priority — TGMS Lecture 4 central object).
Dual intersection complex of a degenerate Calabi-Yau; integral affine structure with singularities	—	✗	Gap (high priority — the combinatorial input data).
Polyhedral decomposition $P$ of an integral affine manifold $B$	—	✗	Gap.
Scattering diagram (Kontsevich-Soibelman); wall-crossing formula	—	✗	Gap. Master-tier pointer; deferred deep content — see §5.
Slab function; structure of a tropical manifold	—	✗	Gap.
Gross-Siebert reconstruction theorem	—	✗	Gap (high priority — TGMS's central theorem; statement-level unit at FT-equivalence).
Theta function of a polarised tropical manifold	—	✗	Gap. Bridges to Fulton 3.32 line-bundle-from-polytope construction (`04.11.10`); TGMS theta functions generalise lattice-point sections.
Period integral and the mirror map	—	✗	Gap. Master-tier pointer.
Tropical curve count in a toric surface	—	✗	Gap.
Mikhalkin's correspondence theorem	—	✗	Gap (high priority — TGMS Lecture 9; Mikhalkin 2003/2005 originator).
Nishinou-Siebert correspondence (toric, all dimensions)	—	✗	Gap.
Strominger-Yau-Zaslow (SYZ) conjecture; special Lagrangian torus fibration; T-duality	—	✗	Gap (high priority — TGMS Lecture 10). Connects to `05-symplectic/lagrangian/` (Lagrangian content) and to the Calabi-Yau-Kähler material.
Calabi-Yau manifold (algebraic-geometric definition: trivial canonical bundle)	—	✗	Gap. Foundational prereq for all mirror-symmetry material.
Calabi-Yau metric (Calabi conjecture; Yau's theorem)	—	✗	Gap. Differential-geometric companion; pointer to `05-symplectic/almost-complex/` Kähler material.
Reflexive polytope; Batyrev mirror duality	—	✗	Gap. Already noted in Fulton 3.32 punch-list as `04.11.16` (priority 4 pointer); TGMS picks up where Batyrev leaves off.

Aggregate coverage estimate: ~0% of TGMS has corresponding Codex units. The gap is total. This is unsurprising — TGMS is a research-level monograph on a programme (Gross-Siebert 2003–2011) that postdates the core algebraic-geometry foundations the Codex currently covers.

Distinctive Codex situation. TGMS sits at the apex of three prerequisite stacks:

Toric geometry (Fulton 3.32 audit, fully punch-listed in plans/fasttrack/fulton-toric-varieties.md — 16 units across four priority bands). All of TGMS's combinatorial infrastructure (cones, fans, polytopes, lattice-point sections) is the Fulton 3.32 toolkit. Hard prerequisite.
Hodge theory and Calabi-Yau geometry (partially covered at 04.09.01-hodge-decomposition.md and 04.09.02-kodaira-vanishing.md). TGMS Lecture 2 (A/B-model survey) and Lecture 7 (period integrals) need a Calabi-Yau definition unit, a Calabi-Yau period unit, and a mirror-conjecture survey unit. Soft prerequisite — can be written in parallel.
Log geometry (Kato 1989). Completely absent from the Codex and from any other Fast Track book. TGMS Lectures 3–6 are not readable without it. Hard prerequisite — must be written first as its own unit cluster.

The natural new chapter location is 04-algebraic-geometry/12-tropical- mirror/ (new chapter folder appended to the existing 10-chapter AG section, after the planned 11-toric/ Fulton chapter). The log-geometry prerequisite units should go in 04-algebraic-geometry/13-log/ (separate new chapter), or as a 12-tropical-mirror/00-log-preliminaries/ sub-section.

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

Priority 0 — blocked by Fulton 3.32 punch-list (hard prereq). Items 1–11 of plans/fasttrack/fulton-toric-varieties.md (algebraic torus, cones, fans, $X_{Σ}$ , orbit-cone correspondence, toric resolution, toric divisors, toric Picard, polytope-to-line-bundle, algebraic moment map) must ship before any TGMS unit can be written. TGMS inherits all Fulton 3.32 prereqs and additionally needs the priority-4 reflexive- polytope pointer (04.11.16).

Priority 0a — additional prereqs (independent of Fulton 3.32):

04.09.03 Calabi-Yau manifold (algebraic-geometric). Trivial canonical bundle; Kodaira-Spencer first-order deformation theory. Three-tier, ~1500 words. Anchors Lectures 2, 3, 4 of TGMS.
05.??.?? Special Lagrangian torus fibration. Located in 05-symplectic/lagrangian/. Pointer unit at FT-equivalence; references SYZ Lecture 10.
04.12.00 Log structure (Kato 1989) and log smooth morphism. Foundational. Three-tier, ~2500 words. Pre-stub for the 12-tropical- mirror/ chapter or its own 13-log/ chapter; the latter is cleaner long-term. Lectures 3–6 of TGMS are unreadable without this unit.

Priority 1 — high-leverage, captures TGMS's central content:

04.12.01 Tropical semiring and tropical polynomial. The $(R \cup {\infty}, min, +)$ semiring; tropical monomials and polynomials; tropical hypersurface as the non-differentiability locus. Worked examples: tropical line in $R^{2}$ , tropical conic, tropical $P^{2}$ . TGMS Lecture 1 anchor; Mikhalkin ICM 2006 §1 anchor; Maclagan-Sturmfels §1.1–1.3 anchor. Three-tier, ~2000 words.
04.12.02 Tropical curve (balanced rational metric graph). Vertices, edges with rational slope, balancing condition at each vertex; tropical degree; tropical genus. Worked examples: tropical degree- $d$ plane curves; tropical conic through 5 points. TGMS Lecture 1, 8, 9 anchor; Mikhalkin 2005 §2 anchor. Three-tier, ~2000 words.
04.12.03 Kapranov's theorem. Statement: for $V \subset (\mathbb{C}^*)^n $ana l g e b r ai c v a r i e t y w i t hi d e a l$ I$, the tropical variety $Trop (V) \subset R^{n}$ equals (i) the image of $V (C {{t}}))$ under the non-archimedean valuation $ν$ , and (ii) the support of the initial-ideal fan of $I$ . Kapranov 2000 (unpublished, but the result is universally attributed) and Speyer-Sturmfels 2004 give the proof now in the literature; cited via Maclagan-Sturmfels §3. Three-tier, ~2000 words. Originator-prose unit (Kapranov 2000).
04.12.04 Newton polytope and non-archimedean amoeba. Two constructions, one duality. Newton polytope $\mathrm{Newt}(f) \subset M_\mathbb{R} $o f$ f \in \mathbb{C}[x_1^{\pm}, \ldots, x_n^{\pm}] $; n o n - a r c him e d e anam oe ba o f$ V(f) $in$ \mathbb{R}^n$. Bridges to Fulton 3.32 04.11.14 (Bernstein-Kushnirenko / mixed volumes). Three-tier, ~1500 words.
04.12.05 Mikhalkin's correspondence theorem. Statement and sketch of proof: for a toric surface $X_{Δ}$ with polytope $Δ$ , the number of complex algebraic curves of bidegree $(d, g)$ in $X_{Δ}$ through generic points equals a weighted count of tropical curves through the tropicalised points; the weights are the multiplicities of Mikhalkin's "complex multiplicity formula". Mikhalkin 2003 arXiv / Mikhalkin 2005 JAMS 18, pp. 313–377. TGMS Lecture 9 anchor. Three-tier, ~2500 words. Originator-prose unit (Mikhalkin 2003/2005). The signature tropical-classical correspondence.
04.12.06 Nishinou-Siebert correspondence. The algebraic-geometric proof of Mikhalkin's correspondence via toric degenerations; extension to higher-dimensional toric varieties. Nishinou-Siebert 2006 Duke Math. J. 135, 1–51. Master-tier (the proof is technical); ~2000 words. Intermediate gives the statement only.
04.12.07 Toric degeneration of a Calabi-Yau variety. A 1-parameter flat family $\mathfrak{X} \to \mathrm{Spec}, \mathbb{C}[![t]!]$ with smooth generic fibre and special fibre a union of toric varieties glued along toric strata. Maximally unipotent monodromy / large-complex-structure limit. TGMS Lecture 4 anchor; Gross-Siebert 2006 J. Differential Geom. 72 §2 anchor. Three-tier, ~2500 words.
04.12.08 Dual intersection complex; tropical manifold $B$ . Construction of the dual intersection complex from a toric degeneration; integral affine structure on $B$ away from a codimension-2 singular locus $Δ$ ; the singular locus carries the "monodromy data" that distinguishes the Calabi-Yau case. Polyhedral decomposition $P$ of $B$ . TGMS Lecture 4 anchor. Three-tier, ~2500 words. Distinctive content; the combinatorial input for Lecture 5.
04.12.09 Gross-Siebert reconstruction theorem (statement). Given a tropical manifold $(B, P)$ with simple singularities and lifted gluing data, there is a unique (up to formal isomorphism) toric degeneration of Calabi-Yau varieties with dual intersection complex $(B, P)$ , reconstructed order by order in $t$ via the scattering / slab procedure. Gross-Siebert 2011 Ann. of Math. 174, pp. 1301–1428 (originating paper). TGMS Lecture 5 anchor (statement only at FT-equivalence; proof is a Master-only sketch). ~3000 words. TGMS's central theorem. Statement-level unit; proof deferred — see §5.
04.12.10 Strominger-Yau-Zaslow (SYZ) conjecture. Statement: mirror Calabi-Yau pairs $(X, X^{\lor})$ admit dual special Lagrangian $T^{n}$ -fibrations $X \to B$ and $X^{\lor} \to B$ over a common integral-affine base, with the mirror map = fibrewise T-duality. SYZ 1996 Nucl. Phys. B 479 anchor; TGMS Lecture 10 anchor. Three-tier, ~2000 words. Cross-pointer to special Lagrangian unit in 05-symplectic/lagrangian/ (P0a above). Originator-prose unit (Strominger-Yau-Zaslow 1996).

Priority 2 — theta functions and the polytope-to-mirror bridge:

04.12.11 Slab function and structure of a tropical manifold. Codimension-1 strata of $(B, P)$ carry slab functions — Laurent polynomials in the toric coordinates of the adjacent chambers, recording the gluing data. The collection of slab functions + scattering walls is the "structure" $S$ on $B$ . TGMS Lecture 5 anchor. Master-tier (the definition is technical); ~2500 words.
04.12.12 Theta function of a polarised tropical manifold. Theta functions $ϑ_{m}$ for $m$ an integral point of $B$ ; canonical $Z$ -basis of the algebra of regular functions on the mirror; broken-line construction. Generalises the lattice-point basis $H^{0} (X_{P}, L_{P}) = ⨁_{m \in P \cap M} χ^{m}$ from Fulton 3.32 04.11.10 (toric case) to the degenerate Calabi-Yau case. TGMS Lecture 6 anchor; Gross-Hacking-Keel 2015 Publ. IHÉS 122 + Carl-Pumperla-Siebert Forum Math. Sigma 8 (2020) anchors. Three-tier, ~2500 words. Distinctive content; the Calabi-Yau analogue of Fulton's polytope-to-sections theorem.
04.12.13 Period integral and the mirror map (pointer). For the reconstructed Calabi-Yau family $X_{t}$ , the periods $\int_{γ_{i}} Ω_{t}$ of the holomorphic volume form along a basis of cycles $γ_{i}$ satisfy a Picard-Fuchs equation whose leading-order behaviour is computed by tropical-disk counts on the A-side mirror. TGMS Lecture 7 anchor. Master-tier pointer unit; ~1500 words. Bridges to Hodge-theoretic Calabi-Yau periods in 04.09.*.

Priority 3 — log geometry deep content (prerequisite-completing):

04.12.14 (or 04.13.02) Log smooth morphism and log Calabi-Yau space. Builds on the P0a 04.12.00 log-structure unit. Log smooth = the log analogue of smooth; log Calabi-Yau = trivial log canonical. Kato 1989 Logarithmic structures of Fontaine-Illusie anchor; TGMS Lecture 3 anchor. Master-tier; ~2500 words.
04.12.15 Log Gromov-Witten invariants (pointer). Gross-Siebert 2013 J. Amer. Math. Soc. 26 / Abramovich-Chen 2014 Asian J. Math. 18. The A-side invariants the program ultimately computes. Master-only pointer; ~1500 words.

Priority 4 — scattering diagrams and wall-crossing (Master-tier, explicitly deferred deep content):

04.12.16 Scattering diagram (Kontsevich-Soibelman); wall-crossing formula (pointer). Algorithm constructing the scattering structure order by order in $t$ by adding walls to maintain consistency around each singular point of $B$ . Kontsevich-Soibelman 2006 Birkhäuser chapter + Kontsevich-Soibelman 2008 arXiv:0811.2435 originators. TGMS Lecture 5 anchor. Master-only; ~2000 words. Explicit pointer; deep content deferred — see §5.
04.12.17 Reflexive polytope and Batyrev mirror duality (full unit). Upgrade of Fulton 3.32 04.11.16 (pointer-only) to a full Intermediate+Master unit. Batyrev 1994 J. Algebraic Geom. 3, pp. 493–535. Concrete pre-Gross-Siebert mirror construction; one of the worked examples in TGMS. ~2000 words.

§4 Implementation sketch (P3 → P4)

For full TGMS coverage, items 1–10 are the minimum equivalence set (priority 1) plus the prerequisite log-structure unit P0a and the Fulton 3.32 priority-1+2 batch as hard prereqs. Realistic production estimate (mirroring earlier algebraic-geometry batches and accounting for the research-monograph density of TGMS):

~4–5 hours per unit. TGMS units skew higher than the AG corpus average because each unit needs to coordinate the (a) tropical / combinatorial side, (b) algebraic-geometric side via toric degeneration, and (c) mirror-symmetry interpretation. Worked examples are nontrivial (e.g. the quintic mirror, the K3 SYZ).
10 priority-1 units × ~4.5 hours = ~45 hours of focused production. Plus 3 prerequisite P0a units (~12 hours), 3 priority-2 units (~12 hours), 2 priority-3 units (~8 hours), and 2 priority-4 units (~8 hours) = ~85 hours total. Add the Fulton 3.32 priority-1+2 prereq batch (~38–40 hours) and the total is ~125 hours. Realistic 10–14 day window for a TGMS sprint after Fulton 3.32 ships.
10 priority-1 units alone (~45 hours) lift coverage from ~0% to ~55%; priority-2 (~12 hours) reaches ~75%; priority-3 (~8 hours) reaches ~85%; priority-4 (~8 hours) closes to ~95%.

Originator-prose target. TGMS's content has clear originator attributions per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10:

Kapranov 2000 (unpublished lectures, written up by Speyer- Sturmfels 2004 Adv. Geom.; canonical attribution in Maclagan- Sturmfels 2015) — originator of tropical curves and the tropical variety / non-archimedean valuation picture. Cite in 04.12.01– 04.12.03.
Mikhalkin 2003 (arXiv:math/0312530) and Mikhalkin 2005 (J. Amer. Math. Soc. 18, pp. 313–377) — originator of the tropical- classical correspondence theorem for toric surfaces. Cite in 04.12.05. Mikhalkin's ICM 2006 survey (Proceedings ICM Madrid, Vol. II, pp. 827–852) is the readable consolidated source.
Strominger-Yau-Zaslow 1996 (Nucl. Phys. B 479, pp. 243–259) — originator of the SYZ conjecture. Cite in 04.12.10. The paper is short (~17 pp.) and largely physical-intuition; the algebraic- geometric reformulation is Gross-Wilson 2000 and Kontsevich- Soibelman 2001.
Gross-Siebert 2006 (J. Differential Geom. 72, pp. 169–338) — originator of the log-geometric mirror programme. Cite in 04.12.07–04.12.08.
Gross-Siebert 2011 (Ann. of Math. 174, pp. 1301–1428) — originator of the reconstruction theorem. Cite in 04.12.09.
Kontsevich-Soibelman 2001 (Symplectic geometry and mirror symmetry, World Scientific, pp. 203–263) and Kontsevich-Soibelman 2006 (The Unity of Mathematics, Birkhäuser, pp. 321–385) — originators of scattering / wall-crossing in the SYZ-mirror context. Cite in 04.12.16.
Nishinou-Siebert 2006 (Duke Math. J. 135, pp. 1–51) — originator of the algebraic-geometric proof of the tropical- classical correspondence in arbitrary toric dimension. Cite in 04.12.06.
Gross 2011 (TGMS itself) — definitive textbook consolidation.

Each anchor unit's Master Historical section should carry at least one of these originator citations; 04.12.05 (Mikhalkin's theorem) and 04.12.09 (Gross-Siebert reconstruction) and 04.12.10 (SYZ) are the three units whose entire structure should be in the originator-prose mode per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10 — leading with the original statement and citation, supplying the historical context, and offering the modernised exposition only after the period-correct setup.

Notation crosswalk. TGMS uses $(B, P)$ for the tropical manifold with polyhedral decomposition, $S$ for the structure (slab functions + scattering walls), $Δ \subset B$ for the singular locus of the integral affine structure, and $X_{0}$ for the special fibre of a toric degeneration. The Codex notation decision (per docs/specs/UNIT_SPEC.md §11) should adopt these unchanged. The toric notation $(N, M, N_\mathbb{R}, M_\mathbb{R}, \sigma, \Sigma, X_\Sigma, U_\sigma)$ from Fulton 3.32 carries over to TGMS lectures 4–6; the tropical lectures 1–2 use $R^{n}$ and $R \cup {\infty}$ directly. Conflict to resolve: TGMS Lecture 10 (SYZ) writes $B$ for the SYZ base — same letter as the tropical-manifold base in Lectures 4–9. This is intentional: the SYZ base is the tropical manifold in the Gross-Siebert picture. The Codex units should make this identification explicit in 04.12.08 (dual intersection complex / tropical manifold) and again in 04.12.10 (SYZ statement).

§5 What this plan does NOT cover

A line-number-level inventory of every named theorem in TGMS. Deferred until a working copy of the book is sourced (see §7) — current sourcing status is reduced; the plan was built from the universally-documented 10-lecture structure of CBMS 114 plus the Mikhalkin / Maclagan-Sturmfels / Gross-Siebert original-paper cross-references.
Deep Gross-Siebert technical details. The proof of the reconstruction theorem (the scattering algorithm, the consistency arguments around each singular point of $B$ , the slab-function cocycle computations) is explicitly deferred. 04.12.09 is a statement-only unit with a Master-tier proof sketch. The full proof occupies Gross-Siebert 2011 Ann. of Math. 174 (128 pages) and is beyond FT-equivalence scope.
Kontsevich-Soibelman wall-crossing formula in full generality. 04.12.16 is a pointer unit. The motivic-DT / cluster-transformation generalisation (Kontsevich-Soibelman 2008 arXiv:0811.2435) is explicitly out of scope; future Fast Track entry candidate.
Gromov-Witten theory of Calabi-Yau threefolds. The A-side that mirror symmetry ultimately computes. TGMS Lecture 2 surveys it briefly; the Codex needs a Gromov-Witten chapter independently before that survey can land. Deferred to a separate Fast Track audit (Cox-Katz, Mirror Symmetry and Algebraic Geometry, FT 3.??).
Symplectic / Floer-theoretic mirror symmetry (Kontsevich's homological mirror conjecture, Fukaya category). Out of scope; FT 3.34+ candidate; pointer only in 04.12.10 Master.
The Donaldson-Thomas / Pandharipande-Thomas correspondence. Out of scope.
Tropical Grassmannian, tropical moduli spaces, tropical intersection theory beyond surfaces. Covered in Maclagan-Sturmfels Ch. 5–6 but not in TGMS; out of scope for this audit.
Exercise-pack production. TGMS exercises are mostly research-prompt level ("verify the following statement in the case ..."); the exercise pack is a P3-priority-3 follow-up after the priority-1 units ship.

§6 Acceptance criteria for FT equivalence (TGMS)

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

The Fulton 3.32 punch-list priority-1+2 units (04.11.01–04.11.11) have shipped (strict prereq).
The P0a prerequisite units have shipped: 04.09.03 (algebraic Calabi-Yau), special-Lagrangian-torus-fibration unit in 05-symplectic/lagrangian/, and 04.12.00/04.13.?? (log structure).
≥95% of TGMS's named theorems in Lectures 1–10 map to Codex units. Currently 0%. After priority-1 ships: ~55%. After priority-1+2: ~75%. After priority-1+2+3: ~85%. After priority-1+2+3+4: ≥95%.
≥90% of TGMS's worked computations have a direct unit or are referenced from a unit that covers them. The signature examples (the quintic mirror via toric degeneration, the K3 SYZ fibration, the focus-focus singularity of an integral affine $S^{2}$ , the tropical conic through 5 points realising Mikhalkin's correspondence in degree 2) are distributed across priority-1 and priority-2 units' worked-example sections.
Notation decisions are recorded (see §4 above).
Cross-pointers are explicit:
- From 04.12.07–04.12.09 back to Fulton 3.32 04.11.04 (fan and $X_{Σ}$ ) and 04.11.10 (polytope-to-line-bundle).
- From 04.12.10 (SYZ) laterally to the 05-symplectic/lagrangian/ special-Lagrangian unit and to the Calabi-Yau metric / Yau's theorem material.
- From 04.12.12 (theta functions) back to Fulton 3.32 04.11.10 (lattice-point basis) as the smooth-toric specialisation.
- From 04.12.17 (Batyrev) to the Fulton 3.32 04.11.16 pointer upgrade.
Pass-W weaving connects the new 04-algebraic-geometry/12-tropical- mirror/ chapter to 04-algebraic-geometry/11-toric/ (Fulton), to 04-algebraic-geometry/09-hodge/ (Calabi-Yau periods), to 04-algebraic-geometry/13-log/ (or in-chapter log preliminaries), and to 05-symplectic/lagrangian/ (SYZ side).

The 10 priority-1 units close the foundational gap. Priority-2 closes the theta-function and period-integral gap. Priority-3 closes the log-geometry deep content gap. Priority-4 is the scattering / Batyrev deepenings. Once all four priority bands are shipped, and the Fulton 3.32 and P0a prereqs are in place, TGMS is at the FT-equivalence threshold.

§7 Sourcing

Status: REDUCED. No local PDF was found in reference/textbooks-extra/, reference/fasttrack-texts/, or reference/book-collection/free-downloads/ (which holds the Freed CBMS notes — Field Theory and Topology — but not the Gross CBMS 114 book). The book is in active copyright (American Mathematical Society, CBMS Regional Conference Series in Mathematics 114, 2011) and is not on the AMS open backlist or any author-hosted page. WebFetch attempts to AMS bookstore, AMS books portal, Gross's Cambridge DPMMS page, and a Google search all returned 403 / certificate / non-content responses; the AMS retains the book behind the standard CBMS paywall.
Action required to upgrade to full P1 audit. Acquire a copy. Listed as BUY in docs/catalogs/FASTTRACK_BOOKLIST.md line for 3.33 ("Tropical Geometry and Mirror Symmetry — Mark Gross — Tropical methods, string theory — BUY") and similarly in docs/catalogs/NEED_TO_SOURCE.md ("3.33 | Tropical Geometry and Mirror Symmetry | Gross | AMS CBMS"). Estimated cost ~$54 paperback (AMS member ~$43). Library access via institutional ILL is the fastest path; the AMS sells CBMS books individually and AMS-member pricing is the standard route for a working copy. Once acquired, the lecture-level structure in §1 / §2 above can be upgraded to theorem-level inventory.
Local copy target. Add to reference/fasttrack-texts/03-modern- geometry/ as Gross-TropicalGeometryAndMirrorSymmetry.pdf to mirror the pattern of other paid FT texts.
Substitute expositions (used in this audit).
- Maclagan-Sturmfels, Introduction to Tropical Geometry (AMS GSM 161, 2015). The definitive textbook on the commutative-algebra and combinatorial side of tropical geometry. Open partial draft long available via Sturmfels' Berkeley page; final book is paywalled but the draft covers most of TGMS Lectures 1–2. M-S §1–4 anchors 04.12.01–04.12.04; M-S §5 (tropical varieties of higher dimension) supplements 04.12.05–04.12.06.
- Mikhalkin, Tropical Geometry and Its Applications, ICM 2006 Madrid Proceedings, Vol. II, pp. 827–852. Free at https://www.mathunion.org/icm/proceedings. Anchors 04.12.01–04.12.05 from the differential-geometric side; Mikhalkin is the originator of the correspondence theorem in 04.12.05.
- Gross-Siebert 2006 J. Differential Geom. 72 + 2010 J. Algebraic Geom. 19 + 2011 Ann. of Math. 174. The three originating research papers TGMS consolidates. Open via arXiv (math/0309070, math/0703822, math/0703822 sequel). The TGMS book is the readable reorganisation; for theorem-level statements the Ann. of Math. 2011 paper is the canonical citation.
- Strominger-Yau-Zaslow 1996 Nucl. Phys. B 479. Free via arXiv:hep-th/9606040. Short (~17 pp.); anchors 04.12.10 SYZ.
- Kontsevich-Soibelman 2001 + 2006. Free via arXiv:math/0011041 and arXiv:math/0406564. Anchors 04.12.16 scattering pointer.
Sibling-prereq audit. plans/fasttrack/fulton-toric-varieties.md is the hard prereq — also REDUCED, same paywall constraint (Princeton Annals of Math Studies 131). Both audits should be coordinated for a single acquisition pass: buying Fulton 3.32 and Gross 3.33 together is the natural batch. Cox-Little-Schenck Toric Varieties (used as Fulton substitute) also covers the toric-degeneration material at the level needed for 04.12.07–04.12.08, so the substitute corpus has substantial overlap across the two audits.

Unusual findings.

Total Codex gap. Unlike most Fast Track audits, where the target book has at least partial overlap with existing Codex content via a related differential-geometric, algebraic-topological, or analytic anchor, TGMS has zero overlap. No tropical content, no mirror-symmetry content, no log-geometry content, no Calabi-Yau-as-algebraic-variety content, no SYZ content. The 05-symplectic/lagrangian/ cluster mentions Lagrangians but no special Lagrangians or torus fibrations. The 04-algebraic- geometry/09-hodge/ cluster mentions Hodge decomposition but no Calabi-Yau periods. This is the deepest sole-coverage gap of any Fast Track audit completed to date.
Sibling-coordinated punch-list. Fulton 3.32 punch-list and TGMS punch-list together constitute the entire Wave 8 algebraic- geometry-and-mirror cluster: 16 (Fulton) + 17 (Gross) = 33 new units, plus 3 P0a prerequisite units = 36 total. The Gross 3.33 units depend strictly on the Fulton 3.32 priority-1+2 batch; the two punch-lists should be shipped in strict order (Fulton 3.32 priority-1+2 first, then Gross 3.33 priority-1, then both books' remaining priority bands in parallel). This is the largest single- wave punch-list in the Fast Track equivalence plan.
Log geometry is the chokepoint. Lectures 3–6 of TGMS (the heart of the Gross-Siebert programme) are inaccessible without log structures. Neither Fulton 3.32 nor any other shipped Codex chapter mentions Kato 1989. The P0a 04.12.00 (or its standalone 04.13.?? placement) is a hard prerequisite that does not appear on any other Fast Track punch-list. This unit must be carefully produced — log geometry has a steep terminology cost that ripples through 04.12.07–04.12.15.
Originator-prose density. TGMS is unusually rich in originator-prose anchor opportunities: Kapranov 2000, Mikhalkin 2003/2005, SYZ 1996, Gross-Siebert 2006/2010/2011, Kontsevich- Soibelman 2001/2006, Nishinou-Siebert 2006, Batyrev 1994, Kato 1989 — nine distinct originator citations across the 17-unit punch-list. This makes TGMS a high-leverage book for the FT- equivalence originator-prose target (FASTTRACK_EQUIVALENCE_PLAN.md §10).
The SYZ identification. TGMS's most striking conceptual move is the identification of the SYZ base (a real differential- geometric object: the quotient of a Calabi-Yau by a special Lagrangian $T^{n}$ -action) with the dual intersection complex of a toric degeneration (a purely combinatorial-algebraic object: an integral affine manifold with singularities). This identification is what makes the Gross-Siebert programme a constructive proof of mirror symmetry rather than just a reformulation. Units 04.12.08 and 04.12.10 should both call out this identification explicitly, with a lateral cross-pointer between them.

Gross — *Tropical Geometry and Mirror Symmetry* (Fast Track 3.33) — Audit + Gap Plan