Chapter 10 — Tree Space, or: the Grassmannian Is a Line · Learn You a Tropical Geometry for Great Good!

Parts I–IV built objects: curves, counts, matroids. Part V asks where those objects live. A moduli space is a space whose points are themselves geometric objects — here, trees. The classical object is the Grassmannian Gr(2,4): all 2-planes through the origin in 4-space. A 2-plane is recorded by its six 2×2 minors, the Plücker coordinates p_ij, which obey one relation, the Plücker quadric

p12·p34 − p13·p24 + p14·p23 = 0.

Tropicalise it: replace each p_ij by a weight w_ij ∈ ℝ (think: a valuation), and ask which weight vectors can occur. Speyer–Sturmfels’ theorem (arXiv:math/0304218, Thm 3.4) says the answer for Gr(2,n) is phylogenetic tree space: the weights are exactly the (negatives of) tree metrics. This chapter does the first nontrivial case, n = 4, end to end — and finds that Chapter 1’s object was a moduli space all along.

10.1 The four-point condition

For a tree on leaves {1,2,3,4} with leaf-to-leaf distances d_ij, the three ways of pairing the leaves give three sums

s1 = d12 + d34,   s2 = d13 + d24,   s3 = d14 + d23,

and a classical fact about trees says: the two largest of s1, s2, s3 are equal — the pairing that respects the tree’s one internal edge wins twice, the cross pairing loses. With the book’s min-plus sign convention (w = −d), that becomes the paper’s four-point condition (6):

min(s1, s2, s3) is attained at least twice, where now si are the sums of the w’s.

Predict first. Three tree topologies exist for four leaves — the cherries can be {1,2}|{3,4}, {1,3}|{2,4}, or {1,4}|{2,3} — plus the degenerate star tree (no internal edge, all three sums equal). So the tropical Grassmannian should have three maximal cones glued along a common ridge. The program checks exactly that (§10.2).

10.2 Three cones, and the quotient is a tropical line

Two kinds of freedom move w without changing the tree’s essential shape. Scaling column i of the 2-plane by a constant shifts w_ij ↦ w_ij + a_i + a_j; the map φ(a)_ij = a_i + a_j sweeps out a 4-dimensional subspace image(φ) ⊂ ℝ⁶ (rank exactly 4, asserted; note the scalar shift (1,…,1) already lies in it, as φ(½,½,½,½) — the lineality is 4-dimensional, not 5). Dimension bookkeeping: a maximal cone is image(φ) plus one ray, so 5-dimensional in ℝ⁶ — that is the affine cone over the 4-dimensional Gr(2,4) ⊂ ℙ⁵; mod out ℝ·(1,…,1) the fan is 4-dimensional, and mod out all of image(φ) it is the 2-dimensional quotient below. Tropically, image(φ) is pendant-edge length: change it and the internal structure does not move. The internal edge itself contributes the cut vector

E_{A,B} = Σ_{i∈A, j∈B} e_ij ,

one for each split A|B. The chapter program verifies, on 4000 random integer points with exact fractions:

w satisfies the four-point condition iff w lies in one of the three closed cones image(φ) − ℝ≥0·E_topo, one per topology;
the decomposition is constructive: w = φ(a) − t·E_topo with internal edge length t = (s_strict − s_tied)/2 ≥ 0, recovered uniquely, and t is φ-invariant (500 trials) — pendant edges are exactly what the quotient mods out.

Now quotient: ℝ⁶ / image(φ) ≅ ℝ² via (s1−s2, s1−s3). The three cone generators land on the rays

(1,1),   (−1,0),   (0,−1)

— primitive in ℤ², and (1,1)+(−1,0)+(0,−1) = 0. Balanced. The quotient fan is a tropical line (the paper’s Ex. 3.2 says so in words; the program checks the balancing sum). Chapter 1’s tripod returns, one level up: it is the moduli space of 4-leaf trees, i.e. of 2-planes in 4-space, with the star tree at the vertex and the three resolved topologies on the rays.

Tree space for four leaves: the quotient fan is a tropical line with the three tree topologies on its rays and the star tree at the origin; a worked example runs tree → weights → four-point tie → lift matrix

Figure 10.1 — A: the quotient fan (tropical line) with the three topologies; B: a worked example (tree → w → four-point tie → the lift matrix of §10.3). Every number drawn was asserted in the chapter program or the figure script before rendering.

10.3 Initial forms, and lifting back to an actual plane

A weight is in the tropical variety iff the initial form of the Plücker quadric — the terms attaining the minimum weight — is not a monomial (Thm 2.1(c)). On cone interiors the program finds the binomial for that topology; off the fan, a monomial; 2000 random points, no exceptions.

topology	strict max	initial form `in_w`	quotient ray
`12	34`	`s1`	`−p13·p24 + p14·p23`
`13	24`	`s2`	`p12·p34 + p14·p23`
`14	23`	`s3`	`p12·p34 − p13·p24`

The three cones of the tropical Grassmannian Gr(2,4): one per tree topology, each with its strict-maximum pair sum, its initial binomial, and its quotient ray

Figure 10.2 — The table above, drawn. Every cell — the strict maximum, the tie, the binomial, the ray — was read off the chapter program by the figure script and asserted before rendering.

More: every fan point lifts. Corollary 2.2 made constructive: given w = φ(a) − t·E_{12,34} with t > 0, the 2×4 matrix over ℂ((π)) with columns

(1,0),  (0,1),  (π^{−t}, π^{−t}),  (π^{−t}, π^{−t} − π^{t}),

column i then scaled by π^{a_i}, has all six 2×2 minors with valuations exactly w_ij — verified by exact π-adic series arithmetic on 8 test points, including the ridge and the origin, and the six minors of each lift satisfy the Plücker relation exactly, as series. The tropical point is not an approximation of a plane; it is the valuation shadow of one, and the shadow can be reversed.

10.4 The combinatorics of T₄

Tree space T₄ itself: 2³ − 4 − 1 = 3 vertices (the three splits, as rays), 3!! = 3 facets (the three trivalent trees), pure of dimension n − 3 = 1 — a tripod, again. The next chapter takes n = 5, where this graph becomes a famous one.

Honest Box — a sign typo in the paper, caught by computation. Equation (8) of Speyer–Sturmfels as printed gives the cone generators as +E_{A,B}. With the paper’s own convention “−w_ij is the distance” and its four-point condition (6), a tree metric d = t·E_{12,34} gives w = −t·E, which satisfies (6) but is not in image(φ) + ℝ≥0·E (writing −tE = φ(a) + sE, s ≥ 0, forces φ(a) = cE with c < 0, and a1+a2 = 0, a1+a3 = c, a3+a4 = 0 give c = 0 — contradiction). The correct generators are −E_{A,B}. This was confirmed against the arXiv TeX source, not just a rendering: the sign in (8) is a typo in the paper. The program asserts both directions: −E lies in the fan’s interior for each topology; +E violates the four-point condition outright.

10.5 The whole chapter, as a program

Everything in this chapter is one runnable program: ch10_grassmannian.py — exact Fractions and exact π-adic series throughout, no dependencies beyond the standard library. Its assertions are the chapter’s claims, in ten groups: the sign check on (8) (−E in the fan’s interior, +E out); four-point ⇔ union of the three cones image(φ) − t·E_topo on 4000 random integer points, with the decomposition round-tripped; the quotient fan’s three rays primitive and summing to zero; initial forms binomial on the fan and monomial off it (2000 points); 8 fan points lifted to 2×4 matrices over ℂ((π)) with valuations exact; topology and internal edge t recovered and φ-invariant (500 trials); the T₄ counts (3 vertices, 3 facets); and — added when the draft was re-verified for the book — the Try It answer key below, rank(φ) = 4 with (1,…,1) = φ(½,½,½,½), and the lifted minors satisfying the Plücker relation exactly for all three topologies. Run it, and if it prints its final line, the claims are what we said they are. (This section used to print the listing inline; a screenful of code you can download is just a download with extra scrolling.)

10.6 Try It

Let w have pair sums s1 = 5, s2 = 3, s3 = 3. Does w satisfy the four-point condition? If so, which topology, and what is the internal edge length t?
Why does adding φ(a) to w never change t — in tree language, what does φ(a) move?
The quotient rays are (1,1), (−1,0), (0,−1). Verify the balancing condition, and say which topology each ray carries.

Answers

1. Yes — the minimum 3 is attained twice. The unique strict maximum is s1, so the cherries are {1,2} and {3,4} (topology 12|34), and t = (5 − 3)/2 = 1.
2. φ(a) shifts every pair sum by the same Σa (each pair sum adds two disjoint pair-terms covering all four indices once), so the differences s_strict − s_tied — and hence t — are unchanged. In tree language φ(a) moves pendant (leaf) edge lengths only; the internal edge is the tree’s internal structure, which column scaling cannot touch.
3. (1,1) + (−1,0) + (0,−1) = (0,0) — balanced. In the program’s coordinates: (1,1) carries 12|34, (−1,0) carries 13|24, (0,−1) carries 14|23.

What we learned

Chapter 1’s tripod is a moduli space. The tropical line is Gr(2,4) modulo its lineality: its three rays are the three ways to resolve a 4-leaf tree, its vertex is the star tree, and its balancing is the balancing of §10.2 — checked, not quoted.
The tropical point remembers a plane. Every weight in the fan is the valuation vector of an actual 2-plane, by an explicit matrix whose minors satisfy the one Plücker relation exactly — tropicalisation here loses nothing that the valuations were keeping.
Trust the check, not the printed formula. The paper’s equation (8) has the wrong sign for its own conventions; the four-point condition itself, run on +E, is what convicts it — confirmed against the TeX source before we said so in print.

One rung up, n = 5: tree space T₅ is no longer a tripod but a graph with 10 vertices and 15 edges — and it is a famous graph. Chapter 11 proves the Petersen graph is a moduli space: the tropical Grassmannian Gr(2,5), built from splits.

Source & provenance: Day 10 (2026-10-20) of the tropical geometry study — code hidden_files/code/day10_grassmannian.py, figure files/figures/day10-tree-space.png (generator hidden_files/code/day10_figures.py), study log Day 10 entry, run log hidden_files/day10-run.log. Source: D. Speyer & B. Sturmfels, “The tropical Grassmannian”, arXiv:math/0304218 — Thm 3.4, four-point condition (6), cone generators (8) with the sign correction of the Honest Box, Ex. 3.2, Cor. 2.2, Thm 2.1(c) — verified against the arXiv TeX source on Day 10. The chapter program was merged into one self-contained file (Day 10 program + the book cycle’s Phase-2 verification script) and re-run end-to-end on 2026-11-10, all ten assertion groups passing (run log hidden_files/ch10-run.log). Figures regenerated as SVG the same day: code hidden_files/code/ch10_extra_figures.py, which imports the chapter program, asserts every value it draws — including every cell of the §10.3 table — and only then renders. Referee pass 2026-11-09 (Nyx, #minplus-book): the dimension bookkeeping of §10.2 (lineality 4, maximal cones 5 in ℝ⁶, the 4-dimensional figure being the projective one) is hers as much as ours — her four sanity anchors were checked one by one, the one bookkeeping disagreement resolved in her favour on the substance and stated in §10.2 as above.

Chapter code

The full program behind this chapter, as a download: ch10_grassmannian.py — the same file §10.5 points at. Its assertions are the chapter’s claims; run it.