Matching equations and finite bounds for normal coordinates

For an existing normal surface, the arcs on the two sides of every interior face are the same finite set, so their type counts agree. Distinct quadrilateral types in one tetrahedron separate different opposite vertex pairs and intersect, so a disjoint normal surface can use at most one of them. Conversely, for a vector satisfying these conditions, place the prescribed parallel copies of each triangle type and the single permitted quadrilateral type in every tetrahedron. In each triangular face, arcs of a fixed type are nested parallel copies; put the endpoint blocks in the same order on its three edges on both incident tetrahedra. The matching equations give equal counts for every type, so pair the $j$th arc on one side to the $j$th arc on the other for each $j$. These gluings identify disk sides in pairs and do not identify interior points, producing a properly embedded normal surface with precisely the given counts.

Put $m=7t$. A one-dimensional face of the pointed cone $C(T)$ is cut out by the matching equalities together with active coordinate equations $x_i=0$ whose combined row rank is $m-1$. Choose $m-1$ independent active rows and form the integer matrix $R$ with those rows. Each matching row has four entries in $\{+1,-1\}$ and Euclidean norm $2$; each coordinate row has norm $1$. The cofactor vector $w\in {\mathbb{Z}}^m$, whose $j$th coordinate is the signed determinant of the matrix obtained by deleting column $j$, is nonzero and satisfies $Rw=0$ by Laplace expansion. Hadamard's inequality bounds each such determinant by the product of the row norms, hence $\mathrm{\mid }{w}_j\mathrm{\mid }\le 2^{m-1}$. Dividing by the gcd of the coordinates gives the primitive integer generator on that ray and cannot increase any coordinate.

The cone $C(T)$ is pointed because it lies in the nonnegative orthant. Intersect it with ${\sum }_i{x}_i=1$; this is a bounded rational polytope. Triangulate that finite polytope by coning a triangulation of each face from its first vertex in a fixed finite vertex order. Coning the simplices back to the origin decomposes $C(T)$ into finitely many simplicial cones, each generated by at most $m$ extreme rays of $C(T)$.

Any nonzero vector satisfying the equations and quadrilateral condition is realized by step 1.1; counting its disk pieces returns the original vector. The zero vector has no disk pieces and represents the empty surface, which is admitted for coordinate addition. This proves the claimed characterization.

Let $v$ be a fundamental admissible integer vector. If $v=x+y$ with nonzero integer vectors $x,y\in C(T)$, then $0\le x_i,y_i\le v_i$ for every coordinate. Since the support of the quadrilateral coordinates of $v$ uses at most one quad type per tetrahedron, the coordinatewise sub-vectors $x,y$ satisfy the same quadrilateral restriction. Step 2.1 realizes both as nonempty normal surfaces, contradicting fundamentalness. Thus $v$ is indecomposable in the integer semigroup $C(T)\cap {\mathbb{Z}}^m$.

Choose a simplicial cone from step 1.3 containing $v$, with primitive ray generators $r_1,\dots ,r_d$, where $d\le m$. Write $v={\sum }_{j=1}^d{\lambda }_j{r}_j$ with ${\lambda }_j\ge 0$. If some ${\lambda }_j>1$, then $v-r_j$ is a nonnegative integer vector in the same cone and is nonzero; hence $v=(v-r_j)+r_j$ contradicts indecomposability. Therefore every ${\lambda }_j\le 1$, including the case $v=r_j$. For every coordinate $i$, step 1.2 gives $v_i\le {\sum }_j(r_j{)}_i\le d{2}^{m-1}\le 7t{2}^{7t-1}\le t{2}^{7t+2}$. This proves the fundamental-vector bound.