TENSOR SATURATION FOR EVEN SPIN GROUPS
A plain-language edition, written to be read front to back.

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READ ME FIRST
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This paper proves a theorem about when three quantum-mechanical
or geometric systems can be combined so that nothing changes under
a symmetry group. The group is the even spin group Spin(2n), the
double cover of the rotation group SO(2n). The theorem is a
saturation statement: if a certain combined system has an invariant
after every part is scaled by a positive integer N, then it already
had one before scaling.

Every technical word is defined the first time it appears. The
formal proof is long because it crosses representation theory,
geometry, and integer lattices. This edition follows the original
paper section by section, states the main theorem exactly, and
explains the job of each lemma. A glossary at the end collects the
symbols.

The whole paper in one paragraph:

For the group Spin(2n), take three irreducible representations with
highest weights lambda, mu, nu. A tensor invariant exists exactly
when there is a nonzero map from V(lambda) tensor V(mu) to V(nu*).
The obvious necessary condition is that lambda+mu+nu lies in the
root lattice Q(D_n). The saturation theorem says this condition is
enough: if an invariant exists for the dilated weights N lambda,
N mu, N nu for even one positive integer N, then an invariant exists
for the original weights. The proof uses real polygon geometry to
build paths, deforms those paths to make their integer switch
numbers integral (fixing a two-component parity obstruction), counts
comparisons across the two components, converts integral schedules
into tensor maps using fundamental columns, and removes a common
zero tail by a smaller-rank preprojective-module construction.

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PART 1. THE QUESTION
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1.1 Groups, representations, and invariants.

A group is a set with an associative multiplication, an identity,
and inverses. A representation of a group G on a complex vector
space V is a rule that assigns to each group element g a linear map
rho(g) on V, with rho(gh)=rho(g)rho(h) and rho(e)=identity.
A subspace W of V is invariant if every rho(g) sends W into W.
An irreducible representation has no invariant subspaces except 0
and V. We write V(lambda) for the irreducible representation with
highest weight lambda.

The tensor product V tensor W is the vector space spanned by
symbols v tensor w, with the group acting by
rho(g)(v tensor w)=(rho(g)v) tensor (rho(g)w). An invariant in
V tensor W tensor U is a nonzero vector fixed by every rho(g). It
is the same thing as a nonzero G-linear map from V tensor W to the
dual U*.

The problem in this paper is to decide when such an invariant
exists for three irreducible representations.

1.2 Highest weights.

For the groups in this paper, every irreducible representation is
classified by a highest weight. For Spin(2n), a weight is a vector

  x = (x_1, ..., x_n)

of real numbers. The Weyl group acts by permuting the coordinates
and by changing signs of an even number of coordinates.

The weight is dominant if

  x_1 >= x_2 >= ... >= x_{n-1} >= |x_n|.

The weight is integral if all coordinates are integers or all are
half-integers. Dominant integral weights are the highest weights of
irreducible representations. We write D_n for this root system.

The root lattice Q(D_n) is the set of integer vectors in the weight
lattice whose coordinate sum is even. The weight lattice consists
of all vectors whose coordinates are all integers or all
half-integers. Half-integer weights are the spin weights.

Small ranks: D_2 is A_1 x A_1, and D_3 is A_3. If n=1, Spin(2)
is a torus; then the statement is just the weight-sum equation.

1.3 The saturation question.

Let lambda, mu, nu be dominant integral weights. There is an
obvious necessary condition for an invariant in
V(lambda) tensor V(mu) tensor V(nu). The center of Spin(2n) acts
by a character on each irreducible representation. For the tensor
product to contain a vector on which the center acts trivially, the
sum lambda+mu+nu must lie in the root lattice Q(D_n). If it does
not, no invariant can exist.

Saturation asks whether this necessary condition is also
sufficient after rescaling. Suppose that for some positive integer N
the dilated triple N lambda, N mu, N nu has an invariant. Does the
original triple have one? Dilation means replacing each highest
weight by a positive integer multiple. The answer is not obvious:
a solution to a scaled integer problem need not come from a solution
at the original scale.

1.4 The main theorem.

Theorem (Type-D saturation). Let G=Spin(2n) with n>=2. Let lambda,
mu, nu be dominant integral weights such that lambda+mu+nu lies in
Q(D_n). Then for every positive integer N,

  (V(N lambda) tensor V(N mu) tensor V(N nu))^G is nonzero
  if and only if
  (V(lambda) tensor V(mu) tensor V(nu))^G is nonzero.

The nontrivial direction is: an invariant at some positive
dilation implies an invariant at scale one. The reverse direction
is automatic by taking the N-th tensor power of an invariant.

The theorem includes the half-integral spin weights. It also covers
D_2 and D_3 with the stated conventions. The stretching parameter N
is always a positive integer, not an arbitrary positive real.

Why this is hard: the root lattice condition is a parity condition,
but the invariant is an algebraic condition. One must show that the
parity condition can be realized by an actual tensor coefficient.
The proof does this by building a geometric path and then adjusting
its integer data without moving the endpoint.

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PART 2. THE TOOLS
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2.1 Roots, chambers, and Weyl reflections.

A root is a special vector that describes a mirror symmetry of the
weight space. For D_n the roots are

  +- e_i +- e_j    with i not equal to j,

where e_i is the i-th coordinate vector. Every root has squared
length two. The reflection in the root beta sends a weight x to

  s_beta(x) = x - (x dot beta) beta.

These reflections generate the Weyl group W.

The dominant chamber is the cone of weights satisfying
x_1 >= ... >= x_{n-1} >= |x_n|. Every Weyl orbit has exactly one
point in the closed dominant chamber. A weight is regular if it lies
in the interior of a chamber; regular dominant means all
inequalities are strict.

A shape is a polygonal path from 0 to mu whose every straight
increment lies in the dominant chamber. A schedule is a path that
starts at lambda, follows Weyl translates of the shape, and ends in
the Weyl orbit of xi = nu*. At a switch it changes from one Weyl
translate to another. The switch coefficient is an integer that
measures how much the path was reflected. A schedule is integral if
all its switch coefficients are integers.

The point of an integral schedule is that it is a discrete,
integer-compatible object. The theorem will build one, then turn it
into a tensor map.

2.2 Real moment solutions.

There is a second language for the same problem: matrices. Let K
be a compact form of the group, and identify weights with Hermitian
matrices in the Lie algebra of K. A real moment solution for
(lambda, mu, xi) is a triple of matrices A, B, C with

  A + B + C = 0,

where A lies in the compact orbit of lambda, B in the orbit of mu,
and C in the orbit of -xi. The zero-sum equation says that the three
orbits can be positioned so that their moment maps cancel.

A tensor invariant produces a real moment solution by maximizing the
invariant functional on unit vectors and differentiating. Conversely,
a real schedule produces a real moment solution by folding the path
into the dominant chamber and composing extremal intertwiners. These
two facts let the proof move freely between algebra and geometry.

2.3 The coefficient lemma.

The basic algebraic tool is a criterion for a highest-to-highest
tensor coefficient. Let h, h', theta be dominant integral weights.
Consider a linear map

  T : V(h) tensor V(theta) -> V(h').

Take the highest-weight vector v_h in V(h), and take the highest
output coordinate of T(v_h tensor -). The possible such linear forms
are exactly the forms supported at weight h'-h that annihilate the
subspace

  sum_i f_i^(h dot alpha_i + 1) V(theta),

where f_i are the negative simple root generators and alpha_i are
the simple roots. In particular, if h'=h+u theta is dominant, the
extremal coordinate at u theta gives a nonzero intertwiner.

This lemma is the bridge from root data to actual tensor maps. It is
used twice: once for a single path segment, and once for composing
many columns.

2.4 Flag composition.

Suppose we have a chain of intertwiners

  T_j : V(h_{j-1}) tensor V(theta_j) -> V(h_j).

If each T_j has a nonzero highest-to-highest coefficient at a flag
b_j, and if the flags occur in the right order under a one-parameter
flow, then the iterated composition is nonzero on the Cartan summand
of the tensor product of the V(theta_j). This is the flag composition
lemma. It turns a sequence of local coefficients into one global
tensor coefficient.

2.5 Fundamental columns.

A fundamental column is a straight path segment whose increment is
a fundamental weight. Every dominant integral weight is a
nonnegative integer sum of fundamental weights, so a shape can be
built by concatenating fundamental columns.

Minuscule columns are the easiest: all root pairings are 0 or +-1,
so there can be no effective interior switch. The column is straight,
and the coefficient lemma gives its intertwiner directly. The vector
column and the two spin columns are minuscule.

An ordinary exterior column has weight e_1+...+e_k with
2<=k<=n-2. Its only possible interior switch is at the midpoint.
The proof builds a common-wall invariant covector from coordinate
determinants and skew pairing forms, and chooses a flag at which its
coefficient is nonzero. A flag-composition argument then places the
product on the Cartan summand for mu.

2.6 Preprojective modules.

The last tool is algebraic and is used only for a common zero tail.
For a Dynkin diagram, a preprojective representation is a
representation of the doubled quiver whose two-arrow return maps
sum to zero. The semicanonical basis identifies the enveloping
algebra with functions on these representation varieties. The top
of a module is the quotient by incoming images; the socle is the
intersection of outgoing kernels. A tensor multiplicity is nonzero
exactly when some preprojective module satisfies top bounds from
lambda and socle bounds from mu, at every vertex.

This criterion is used to double a smaller-rank occurrence while
keeping both sets of bounds. The doubling is only for real
existence; the integral schedule is then built at the original
scale.

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PART 3. THE PROOF IN FIVE MOVES
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The proof has five moves. Each move solves one kind of problem:
real existence, integer obstruction, counting, tensor recovery, and
a degenerate zero tail.

3.1 Move 1: real schedules from geometry.

Start with a real moment solution for (lambda, mu, xi). We want a
schedule whose switch roots form a real basis of the weight space.
The construction has three steps.

First, approximate the moment data by strict regular data. A strict
triple of weighted flags has a positive margin in the inequality
sum_j m_theta_j(F_j,H) < 0 for every unit Hermitian H. Strict triples
are open, and every moment triple is a limit of strict ones. This is
the strict approximation lemma.

Second, build a triangle in the symmetric space G/K with prescribed
dominant side vectors R lambda, R mu, R xi*. The push of length R
theta from a point hK toward a flag F is the path
hk exp(s theta)K, 0<=s<=R, where h^{-1}F=kB. Composing three such
pushes gives a map whose large spheres point strictly inward, by the
radial bound limsup d/ds |P(s)| <= m_theta(F,H_0). Brouwer's
fixed-point theorem gives a fixed point, hence a triangle. This is
the symmetric-space triangle lemma.

Third, replace the middle side by the shape shrunken and scaled by R,
and let R tend to infinity. The radial Cartan parts converge to a
dominant path x(t) whose velocities are Weyl translates of the shape
velocities. The fixed flags on each open interval are torus-fixed,
and the Bruhat order of the frames is weakly decreasing. This is the
decreasing radial path lemma.

Now maximize the area A(x)=integral_0^1 rho dot x(t) dt over all such
decreasing paths with the same endpoints and shape. At a maximum,
the velocity differences at nonzero jumps away from shape breaks are
linearly independent. The proof is a variation argument: if they were
dependent, move the jump times in the null direction and get a
second-order improvement. This is the area variation lemma.

With generic endpoints, those independent differences form a basis
of the semisimple space. The endpoint balance has the form
y(1)-y(0)=b+sum_j tau_j(a_j-b_j), and a deficient span would confine
the free endpoints to a proper affine subspace.

Finally, expose each jump. Add a linear functional of the terminal
position to the area objective so that all basis-time derivatives
vanish. Insert an intermediate Bruhat state at the jump. Its endpoint
displacement is compensated exactly by shifting the basis times.
Maximality gives a strict inequality at every genuine insertion.
This is the strict exposure lemma.

The flag orbit polytope lemma then identifies the exposed jump with
an edge of a torus-orbit polytope. Such an edge is a root reflection,
and the two endpoint flags differ by a Bruhat cover. Therefore every
nonzero jump is a reflection and a cover. At a shape break there is
no nonzero jump, because the same functional would expose two
different unique vertices of two polytopes with the same labeled
normal fan. This is the exposed jumps are covers lemma.

The upshot is the basic real schedule proposition: every real moment
solution admits a schedule, for every shape ending at mu, whose
switch roots form a real basis. It is a limit of schedules for
nearby regular data whose switches are covers. Weak signs and
coincident times are allowed.

One more consequence is needed later. At a switch, suppose the two
exchanged signed labels have strict opposite position and moving-value
orders. Then no signed label on a third physical axis can lie
strictly between them in both orders. This is the no strict
third-axis obstruction corollary, and it follows because such a
third label would give two positive roots inverted by the incoming
direction, making the Bruhat length drop greater than one and
destroying the cover property.

3.2 Move 2: making the switches integral.

A real basis of roots need not be an integer basis of the root
lattice. So the basic schedule may have half-integer switch
coefficients. The second move fixes this.

Attach a signed graph to the switch roots. The vertices are the
coordinate axes. A root e_i - epsilon e_j is an edge that imposes the
signed identification x_i = epsilon x_j. A connected component is
balanced if these identifications are consistent around every cycle.
Changing signs of axes then makes the component a permutation
system. An unbalanced component has full coordinate span.

A connected unbalanced unicyclic component generates exactly the
even-sum integer lattice. Its coefficients are integral when the
right-hand side has even coordinate sum, and all half-odd-integral
on the cycle when the sum is odd. This is proved by removing leaves
and then solving the cycle equations. The determinant has absolute
value two.

If the switch roots split into commuting blocks, solve the blocks
separately and superpose the schedules. This block reduction lemma
covers a permutation block, at least three proper orthogonal blocks,
or two proper orthogonal blocks of even parity. Parity here means
the sum modulo two of the assigned difference lambda+mu-xi in that
block.

Suppose no integral schedule exists. Then the hard-case lemma says
any real schedule with at most l switches has exactly l independent
switch roots, and its graph has exactly two connected components,
both unbalanced and of odd parity. Each component is unicyclic. On
its cycle the switch coefficients are half-odd-integers, and off the
cycle they are integers.

The only obstruction is therefore a pair of odd unbalanced
components P and Q. To fix it, join them by one new edge. The
simplest picture is the half-relation deformation. The two components
are unbalanced, so each has a cycle with half-integral coefficients.
A joining root connects their even-sum lattices inside the full D_l
root lattice. For the displayed graph, half the primitive relation
has absolute value 1/2 on the cycles and 1 on the joining edge. More
generally the value 1 occurs on the whole connecting path, and
coefficients on attached trees vanish.

Moving along half of the primitive integer generator of the new
relation changes the switch coefficients by half-integers and reaches
integers without changing the endpoint. But reaching integer
coefficients arithmetically is not enough: the switch times and signs
must remain admissible throughout the motion. A local link is a
modification of the schedule at one time that replaces some switches
by one more switch, with the new roots generating the root lattice,
half-integral base coefficients, and a chosen orientation whose
candidate switches stay ordered and weakly feasible for small
positive parameter. Away from that time, the modification can be
forgotten by a common reflection of the suffix or by a change of
moving frame that fixes the whole shape.

The continuation proposition says that under the assumption of no
integral schedule, a local link cannot exist. The proof follows the
half-kernel direction through collisions. At a collision, the
candidate time for a root line depends only on that line and a common
potential, not on the route through the cluster. In the rank-two A_2
subsystem, every length-two Bruhat interval is a diamond, so the
other route through the diamond is also two downward covers. Sorting
by candidate time terminates because the candidate-time list decreases
lexicographically. At the integral endpoint, all coefficients are
integers, contradicting the assumption.

Thus it remains to show that the exceptional two-component schedule
must admit a link. That is the content of Move 3.

3.3 Move 3: the strict comparison count and knots.

Let P and Q be the two odd unbalanced components. Write P-hat and
Q-hat for their signed labels. For a signed label a, put e_{-a}=-e_a,
y_a = y dot e_a for positions, and v_a = (w mu) dot e_a for assigned
moving values. A cross-comparison is a pair (a,c) in P-hat x Q-hat.
Count each with weight 1/2, so a comparison and its signed mirror
together have weight one.

Set r=2|P|, h=2|Q|, and R=rh/2. Let p_lambda, p_mu, p_xi be the
weighted numbers of cross ties in the initial positions, assigned
moving values, and final positions. The middle number is independent
of the time at which the assigned values are read.

The goal is the strict inequality

  R > p_lambda + p_mu + p_xi.

This is the strict comparison count proposition. The moments on the
two components turn the tie counts into spectral multiplicity
counts. For a Hermitian triple on C^r, its profile is the collection
of multiplicities of each numerical eigenvalue of each of the three
matrices. The tie count is one half of the dot product of the two
profiles.

An orbit-dimension estimate gives the weak bound via Cauchy-Schwarz.
In one block, if z_i is the dimension of the kernel of the i-th
matrix, then twice the sum of the three stabilizer dimensions is
||m||^2 - sum_i z_i, and the orbit sum map is submersive onto the
skew-symmetric matrices. Comparing dimensions gives
||m||^2 - sum_i z_i <= r^2 - r. If the joint kernel is zero, the map
u -> (A_1 u, A_2 u, A_3 u) is injective into the kernel of addition
from the sum of images to the ambient space, so sum_i z_i <= r and
||m|| <= r. A joint kernel of dimension at least two would support a
commuting skew operator, so the only possible joint kernel is a
single line, which must belong to the block containing the sole
possible zero of the moving weight. Removing that line gives the
improved bound ||m||^2 <= r^2 + 2, and deleting the moving zero-level
entry makes Cauchy-Schwarz strict.

If there is no moving zero, equality in Cauchy-Schwarz forces
proportional profiles of full norms r and h. The profile rigidity
lemma says that two irreducible Hermitian zero-sum triples with
proportional profiles and norms r^2 and h^2 must have r=h and
identical profiles. Its proof uses flags with scalar endomorphisms,
direct sums filling an open set, and Krull-Schmidt uniqueness. Then
the two triples are isometric, and the skew operator rotating one
copy into the other commutes with the triple, giving a forbidden
permutation-type block. Equality is impossible. This proves the
strict count.

Now follow the cross-comparisons along the schedule. Strict
inversions alone do not record what happens on an interval where both
positions and moving values agree. Define the core phase p_a(t) for a
signed position label a. If its axis is off the unique cycle of its
component, the phase is zero until the switch on the first edge
toward that cycle and one afterward. If its axis lies on the cycle,
the phase is 0, 1/2, or 1, according as zero, one, or two of its cycle
incidences have occurred. The phase is unchanged on replacing a by
-a.

The algebraic description is: solve the component's root-basis
equation with right-hand side e_a. The cumulative signed a-coordinate
of the edges already used is p_a(t). For an off-cycle source,
pruning the attached trees leaves a unit contribution on its first
edge toward the cycle. At a cycle source, the two incident cycle
contributions are each one half.

A plateau lemma says: if cross labels a,c have equal position and
equal assigned moving value on an open interval between switch times,
then (p_a,p_c) is either (0,0) or (1,1). The proof reflects the whole
suffix by the reflection exchanging the two signed axes. If both
phases were half, each cut would open its cycle into a connected tree,
and the crosswise identification would give a connected l-edge graph
outside the exceptional configuration. If neither phase is half, one
cut piece contains the cycle and the other is a tree; the core piece
joins the two cores and leaves a separate tree component. Both are
excluded by the hard-case lemma.

Define J(t) as one half of the sum over cross pairs of:
+1 if the position and moving-value comparisons are opposite,
p_a+p_c-1 if both are equal,
0 otherwise.
Use one-sided values at isolated equalities. The direct links lemma
gives

  J(0+) >= R - p_lambda - p_mu,
  J(1-) <= p_xi.

Indeed strict position and velocity orders must begin discordant and
end concordant. An initial double tie has phases zero and contributes
-1, covered by subtracting both sorts of initial tie. At the final
endpoint a double tie has phases one and contributes +1; a position
tie with distinct velocities is an inversion just before the endpoint
and is also counted by p_xi. Away from the old switch times, any
changing cross equality with distinct velocities gives a direct
crossing link. Thus a decrease can occur only at an old switch time,
called a knot.

The monotonicity at knots proposition says J is nondecreasing at
every knot, and hence on the whole interval in the one-sided sense.
The proof makes the shape generic so that at any knot the nonzero
velocity differences of its switches are multiples of one form. It
groups signed labels by their assigned forms modulo that form. Each
class has either one value or two values H>L. Across different
classes, cross positions are unequal and cross velocities are
distinct; a swap changes the strict inversion count only if an
external spectator lies strictly between both swapped positions and
both swapped velocity values, and the no strict third-axis
obstruction corollary excludes exactly that.

Within a two-value class, a matching lemma decomposes the transfers
of the high value into disjoint matching pairs inside each component,
together with unused single labels. In an opposite class the matching
is disjoint on physical axes, with mirror transfers understood as the
same physical switches. The proof decomposes flow into directed paths
and shortcuts them; a cycle would sum strict inequalities to 0<0 and
cannot occur. In an opposite class, pair consecutive incidences and
organize edges into paths and cycles; a cycle again gives 0<0, and a
path is replaced by a direct flip of its two ends. Unless the old
switches were already disjoint pairs, this reduces their number,
contrary to the hard-case lemma.

Then check the three kinds of comparison: two unused labels, a
transfer and an unused label, and two transfers. For a transfer
a->b at positions x<z and a single label c at position h, the strict
count decreases by one precisely when x<=h<=z. Replace the transfer
by c->b and a->c. The graph subdivides the old edge through the other
component, is connected and unbalanced, and generates the full root
lattice. Its base coefficients are half-integral. The difference of
the two new derivatives is nonzero; choose its sign so the common
increasing coefficient function orders the two new times correctly.
If x<h<z, both gaps are positive and this is a forbidden feasible
link. At a tie h=x or h=z, the derivative of the zero gap is computed
from the gap formula. It is positive unless a plateau phase is zero;
in that case the plateau contribution -1 compensates for the strict
loss. After the knot the plateau has either two zero phases, giving a
link, or two unit phases, whose new contribution +1 compensates.
Thus every pair-single comparison is nondecreasing when links are
excluded. Two singles can decrease only by a crossing with distinct
velocities, which is a direct link.

For two transfers a->b at x<z and c->d at h<k, the loss is two
precisely when the closed intervals [x,z] and [h,k] overlap. A
touching endpoint permits a switched matching with a zero gap, which
deletes a switch and leaves fewer than l edges. If the overlap is
strict, the two old coefficients are equal. If they are half-odd,
both edges lie on cycles; switching the matching joins the two opened
cycles into a connected l-edge graph outside the exceptional
configuration. If they are integral, both old edges are bridges.
Use a star replacement at one endpoint: send c to one of b,d, then a
to c, then c to the other of b,d. The new graph joins the four pieces
of the two removed bridges, is connected, retains both unbalanced
cycles, and generates the full root lattice. Along the path between
its two cores, the primitive half-kernel coefficients have magnitude
one; off that path they vanish. The table of core endpoints shows
that one can order the wings so their derivatives surround the
middle derivative except when both cores are at donors. A receiver
star similarly works except when both cores are at receivers. If no
link exists, the donor-star and receiver-star obstructions cannot
both hold, so x=h or z=k (or both). These same-role coincidences
produce double plateaus on both sides of the knot, and the plateau
contribution rises from -1 to +1, compensating for the loss of two.
This exhausts all pair-pair comparisons. Mirrored transfers give the
same calculations, and the factor 1/2 applies uniformly.

Finally, the strict count and the boundary inequalities give
J(0+)>J(1-), contradicting the monotonicity at knots. Therefore the
exceptional configuration cannot exist, and an integral schedule must
exist. Uniform bounded-switch closedness passes from generic shapes
to an arbitrary shape with the same endpoint. This proves the
integral schedule theorem: for a permutation system, and for D_l
when mu has at most one zero coordinate.

3.4 Move 4: from integral schedules to tensor intertwiners.

The integral schedule has integer switch coefficients, but the
usual path-model condition also asks for a saturated chain in the
Bruhat order. The proof avoids strengthening the schedule. Instead it
uses fundamental columns and proves the needed representation
statement directly.

First fold the integral schedule into the dominant chamber. The
folded path has Weyl direction states in decreasing Bruhat order. Its
changes of state are an ordered sequence of downward reflections,
including coincident times. At each step from w to s_alpha w at time
t, the pairing alpha dot w eta(t) is an integer. For a concatenation
of fundamental columns, all column endpoint positions of the folded
path are dominant integral weights. This is the folding lemma. The
proof perturbs to separate sites and make ordinary chamber crossings
transverse, then checks integrality on the limiting chain. For an
original switch, changing the chamber frame changes both the root and
the direction by the same Weyl element, so the pairing is unchanged.
For a sorting crossing, the wall equation gives
alpha dot w eta = -alpha dot lambda + sum_{j<i} m_j alpha dot beta_j,
which is an integer by induction.

Now take one column of weight omega. Normalize its parameter to
0<=tau<=1, so the shape on this interval is chi + tau omega, with chi
a sum of preceding fundamental weights. Write x_0,x_1 for its dominant
endpoints and delta=x_1-x_0. Let u and v be the first and last Weyl
states in its open interior. Boundary-time steps are left between
columns, so the last state of one column dominates the first state of
the next. Since alpha dot w chi is integral, every interior step
satisfies tau alpha dot w omega is an integer.

For a minuscule weight, all root pairings are 0 or +-1. Therefore no
interior reflection can change w omega. The column is straight, with
delta=u omega. The extremal case of the coefficient lemma gives an
intertwiner V(x_0) tensor V(omega) -> V(x_1) whose
highest-to-highest coefficient is nonzero at the fixed flag u. The
vector column and both spin columns are minuscule. For D_2 and D_3
all fundamental columns are minuscule. Full Weyl states inside the
column may change while fixing omega, but that causes no problem:
the chosen fixed waypoint has future u, and u>=v dominates the next
column's initial state.

The only remaining case is an exterior column
omega=e_1+...+e_k with 2<=k<=n-2. Let E be the standard
2n-dimensional quadratic space with basis E_{+i},E_{-i} satisfying
(E_{+i},E_{-j})=delta_ij and all other pairings zero. The module
wedge^k E is the fundamental module V(omega); its flag vectors are
wedges of isotropic k-planes.

The coefficient lemma reduces the intertwiner problem to a functional
supported at weight delta. The common-walls lemma says: let J be the
set of simple roots alpha_i with x_0 dot alpha_i = x_1 dot alpha_i
= 0. A functional supported on the weight space of weight delta in
wedge^k E, invariant under the semisimple group generated by J, is an
allowed highest-to-highest coefficient for an intertwiner
V(x_0) tensor V(omega) -> V(x_1). The proof checks that a vector
contributing to its weight-delta component would have source weight
delta+(h_i+1)alpha_i, whose simple-root pairing is h_i+h'_i+2. All
weights of wedge^k E have coordinates in {0,1,-1}, so this pairing
cannot exceed two. Since h_i,h'_i are nonnegative integers, the source
is absent unless both vanish. In that case alpha_i is in J, and
invariance annihilates f_i V(omega).

The nonzero root pairings of omega have absolute value at most two.
By the interior integrality condition, an effective interior switch
occurs only at tau=1/2 and has absolute pairing two. It flips the
signs on two occupied axes. A root with pairing zero preserves both
the occupied support and its complement. Thus the support S of w omega
is fixed throughout the column, every interior reflection belongs to
D_S x D_{S^c}, and the endpoint directions a=u omega and a'=v omega
satisfy delta=(a+a')/2. The set R={i in S: a'_i=-a_i} has even
cardinality. On S\R, delta_i=a_i=a'_i is +-1; on R, delta_i=0.

Let H be the connected subgroup for D_S x D_{S^c}. The orbit H uB is
a flag variety of H. Each interior comparison is a downward Bruhat
reflection in this orbit. The resulting Bruhat inequality supplies a
waypoint b in U_H^+ u intersect U_H^- v, inside U^+ u intersect U^- v.
The wedge at b represents a maximal isotropic plane in the span of
E_{+i},E_{-i} for i in S. Its a and a' coordinates are nonzero.
Use the a coordinate as a graph chart: write its basis as
z_j = E_{a_j j} + sum_{i in S} T_{ij} E_{-a_i i}, for j in S.
Isotropy says T is skew-symmetric. The a' coordinate of the wedge of
all z_j is, up to sign, the determinant of T_{R,R}, so T_{R,R} is
nonsingular. When R is empty, the empty determinant is one.

There is also a sign restriction: if i<j and T_{ij} is nonzero, then
a_i=-1. A single replacement in the wedge gives a nonzero coordinate
of weight a-a_i e_i-a_j e_j. Since a is the past extremal limit, the
difference -a_i e_i-a_j e_j has positive pairing with the regular
dominant flow direction. This difference is a root, so it is positive.
In type D, a root supported on i<j is positive precisely when its
i-coefficient is positive.

Now build the common-wall invariant functional. The sign change on
the last coordinate preserves the positive root system, so apply it
if needed to arrange x_0>=0 coordinatewise. Let B be the collection
of blocks of axes having delta_i=0 and the same positive value of
x_{0i}. Assign a scalar c_B to each such block, put c_i=c_B for i in
B, and put c_i=0 on the zero-position, zero-delta tail. Define

  Theta = sum_B c_B sum_{i in B} E_{+i}^* wedge E_{-i}^*,
  Psi = (wedge_{i: delta_i nonzero} E_{delta_i i}^*) wedge
        Theta^(|R|/2).

Choose any fixed order for the coordinate wedge. Psi has degree k and
is supported on weight delta. It is invariant under the common-wall
semisimple group. On a block with nonzero delta, the coordinate wedge
is the determinant covector and is invariant under the special-linear
group. On a block with positive x_0 and zero delta, the two-form
sum E_{+i}^* wedge E_{-i}^* is the canonical invariant pairing form,
hence invariant under the whole general-linear group. The remaining
zero-position, zero-delta tail carries a special-orthogonal factor and
contributes no covectors.

It remains to choose the c_B so that Psi is nonzero at b. The
coordinate factors select exactly their own primal columns in the
graph chart. After these factors are removed, the pullback of Theta
to the remaining columns indexed by R has skew matrix
T-tilde_{ij}=(a_i c_i+a_j c_j)T_{ij}, up to one overall sign. The top
exterior power is nonzero precisely when T-tilde is nonsingular. This
is automatic for R empty. For nonempty R, partition R=N union P,
where N={i in R: x_{0i}>0, a_i=-1}. Thus P consists of positive signs
at positive positions, together with all indices in the zero tail.
Dominance gives the block form T_{R,R} = [[A,B_0],[-B_0^T,0]] in the
ordering N,P, and a nonzero entry from i in N to a positive-position
j in P must have i<j and x_{0i}>x_{0j}. For a zero-tail j in P, every
i in N already precedes j.

Order the blocks in B from larger to smaller position. Choose
positive exponents s_B strictly decreasing in this order, write B_i
for the block containing a positive-position index i in R, and set
c_B=1+t^{s_B} for t>0. Let t tend to 0+. For a permitted N,P entry
with both positions positive, the larger-position block has the larger
exponent, so its perturbation decays faster. This separates the
blockwise coefficients while leaving their common limit equal to one.
After scaling rows and columns by powers of t, the modified matrix
tends to [[-2A,B_0],[-B_0^T,0]]. This matrix is nonsingular. More
generally, if gamma is nonzero and (x,y) lies in the kernel after
replacing A by gamma A, then (x,y/gamma) lies in the kernel of the
original matrix. Nonsingularity therefore follows from the
nonsingularity of T_{R,R}. By continuity, T-tilde is nonsingular for
sufficiently small positive t. Thus Psi evaluates nontrivially on the
waypoint wedge, and the common-walls lemma supplies the required
single-column intertwiner.

Finally compose the column intertwiners. For each column j, write its
endpoints as h_{j-1},h_j, its weight as omega_j, and choose the
resulting intertwiner V(h_{j-1}) tensor V(omega_j) -> V(h_j). Its
highest-to-highest coefficient is nonzero on the chosen waypoint. For
an exterior column the past and future flags are its first and last
states u_j,v_j; for a minuscule column both may be taken to be u_j.
At a column boundary, v_j>=u_{j+1}. Thus the chosen future flag
dominates the next past flag: directly for an exterior column, and
through u_j>=v_j>=u_{j+1} for a minuscule column. The flag composition
lemma makes the composition nonzero on the Cartan summand
V(sum_j omega_j)=V(mu) inside the tensor product of the V(omega_j).
The first and last dominant positions are h_0=lambda and h_q=xi, so
this proves the column intertwiner proposition. For an empty list the
folded path is constant, so lambda=xi and the identity map suffices.

3.5 Move 5: removing a common zero tail.

The integral schedule theorem applies when the moving weight has at
most one zero coordinate. If all three labels have a longer zero tail,
reduce to a smaller even spin group. Suppose lambda, mu, xi are
dominant integral D_n-weights, all supported in coordinates 1,...,r,
where 1<=r<=n-2. If V(xi) occurs in V(lambda) tensor V(mu), then
V(2 xi-hat) occurs in V(2 lambda-hat) tensor V(2 mu-hat) for D_{r+1},
where lambda-hat=(L_1,...,L_r,0), and similarly for the other labels.
This is the zero-tail removal proposition.

The proof uses preprojective modules. Let D be a finite simply laced
Dynkin diagram, possibly disconnected. Choose an orientation and
adjoin a reverse arrow to every arrow. A representation of the
doubled quiver is preprojective if, at each vertex, the signed sum of
the two-arrow return maps is zero. Write Lambda(d) for the variety of
these representations on fixed spaces of dimension vector d=(d_i).
For M in Lambda(d), the top at vertex i is the dimension of M_i
modulo the sum of the images of incoming arrows. The socle at vertex
i is the dimension of the intersection of the kernels of outgoing
arrows.

The tensor criterion says: let lambda,mu,xi be dominant integral
weights for D, and suppose lambda+mu-xi=sum_i d_i alpha_i with d_i
nonnegative integers. Put lambda_i=lambda dot alpha_i and
mu_i=mu dot alpha_i. Then Hom(V(lambda) tensor V(mu),V(xi)) is
nonzero if and only if there exists M in Lambda(d) with
top_i M <= lambda_i and soc_i M <= mu_i for every i. The proof uses
the semicanonical basis and its left and right filtrations. The
generic evaluation map from the enveloping algebra in degree d to
functions on the irreducible components of Lambda(d) is an
isomorphism. The image of (f_i^p U)_d is supported on components whose
generic i-top is at least p; the image of (U f_i^p)_d is the analogous
subspace for the i-socle. The double quotient
U_d/(sum_i f_i^{lambda_i+1}U + sum_i U f_i^{mu_i+1}) is nonzero
exactly when a component satisfies all the top and socle bounds.

To shorten the diagram, keep an initial chain and replace its
continuation by two leaves. Let V_p=W=W_+ + W_- follow the trunk
V_1,...,V_{p-1}, with V_0=0. With arrows oriented to the right,
write u_i:V_{i-1}->V_i and v_i:V_i->V_{i-1} for the paired maps on
edge i. Here u_p combines the two leaf maps and v_p combines their
reverses. The relations are t_i=u_i v_i=v_{i+1}u_{i+1} for i<p,
t_p=u_p v_p, and the two diagonal blocks of t_p vanish. Since t_1=0,
all t_i are nilpotent. With Delta_j=L_j+M_j-X_j, the dimensions are
dim V_i=sum_{j<=i} Delta_j for 1<=i<=p. The top bounds on the trunk
are L_i-L_{i+1}, and the two leaf bounds are L_p-L_n and L_p+L_n;
socle bounds use M in place of L.

For a single edge, the nilpotent operator [[0,v_i],[u_i,0]] on
V_{i-1}+V_i has a Jordan basis whose rows alternate between the two
spaces. In one row, let h_L,h_R be the numbers of vectors on the left
and right, and let T,S in {0,1} record whether its top and bottom lie
on the right. Then h_R=h_L-1+T+S. For b>=1, define
q_i(b)=dim(ker t_i intersect im t_i^{b-1}), the number of t_i-rows of
length at least b. Let K_i^T(b) be the sum of T over edge-i rows with
h_R>=b, and K_i'^T(b) the same with h_L>=b. Define K_i^S,K_i'^S
similarly with S.

The threshold bounds lemma says: for every 1<=r<=p and b>=1,
K_r^T(b) <= L_r + q_r(b)/2 and K_r^S(b) <= M_r + q_r(b)/2. The proof
fixes b and works with the dual high-socle spaces
F_i^*(b)=ker t_i^* intersect im(t_i^*)^{b-1}. At a trunk vertex, the
two incoming maps give intersection dimensions K_i^T and
q_i-K_{i+1}'^T. Their common intersection is bounded by the vertex
top, giving K_i^T <= L_i-L_{i+1}+K_{i+1}'^T. At the leaves, the space
splits into two colors; adding the two leaf inequalities gives
K_p^T-q_p/2 <= L_p. A row comparison gives
K_i'^T-q_{i-1}/2 <= K_i^T-q_i/2, because a row contributes 1/2 when
it is gained or lost. Combining these inequalities downhill proves
the top assertion. The socle assertion is the same argument on the
primal high-socle space.

The filtered intersection lemma controls a moving subspace inside a
nilpotent filtration. Let F_1 superset F_2 superset ... superset
F_{s+1}=0 be a filtration of a finite-dimensional complex vector
space, and let P be its stabilizer in GL(H). For subspaces A,C write
a_b=dim(A intersect F_b), c_b=dim(C intersect F_b), and
f_b=dim F_b. For g in a nonempty Zariski-open subset of P,
dim(A intersect gC)=max(0,max_{1<=b<=s}(a_b+c_b-f_b)). The proof
chooses bases adapted to the filtration and uses Hall's
distinct-representatives criterion in the nested-support case. A
maximum matching leaves exactly the displayed deficiency unmatched,
and that deficiency is the intersection dimension.

The centralizer lemma says: let t be nilpotent on a complex vector
space V, and put F_b(t)=ker t intersect im t^{b-1} for b>=1. The
group C(t)={g in GL(V): gt=tg} induces the full stabilizer of this
filtration on ker t. It also induces the full stabilizer of the
corresponding filtration on ker t^* inside V^*, under the
contragredient action. The algebraic group C(t) is irreducible. The
proof writes V as a sum of Jordan row modules C[t]/(t^a). A module map
from a row of length a to a row of length b can send its bottom
nontrivially to the target bottom exactly when a<=b. These are
precisely the matrix entries permitted by the filtration F_b(t).
Every filtration-preserving bottom matrix lifts to a commuting
endomorphism, and invertibility on ker t makes the lift invertible.
The same argument applies to t^*, and C(t) is the determinant-nonzero
open subset of the vector space End_{C[t]}(V), so it is irreducible.

Now double. Occurrence implies that the simple-root coefficients of
lambda+mu-xi are nonnegative integers, so choose a module from the
tensor criterion. Keep the chain through V_r and take two copies of
it. The doubled space W-tilde=V_r+V_r is the new combined leaves.
Write t-tilde=t_r+t_r, u-tilde=u_r+u_r, and v-tilde=v_r+v_r. Color
every Jordan row of t_r alternately, and give its duplicate the
opposite colors. Thus t-tilde is off diagonal for the new splitting
W-tilde=W-tilde_+ + W-tilde_-. Each color has dimension dim V_r. A
row of length a and its duplicate contribute one bottom vector of
each color to every primal high-socle level with b<=a; the dual
vectors at their tops give the same contribution on the dual level.
Thus both colors have dimension q_r(b) on every primal and dual
high-socle level, inside a level of total dimension 2q_r(b).
Conjugating the coloring by g in C(t-tilde) preserves all these
properties.

Choose g to ensure the four leaf bounds. The subspace
A=Ann(im u-tilde) lies in ker t-tilde^*, because
im t-tilde is contained in im u-tilde. Its dual filtration profile is
2K_r^T(b). Each dual color has profile q_r(b). By the filtered
intersection and centralizer lemmas, a generic coloring has, in
either dual color, intersection with A of dimension
max(0,max_b{2K_r^T(b)-q_r(b)}) <= 2L_r, by the threshold bounds. This
is exactly the new leaf top. Similarly, ker v-tilde lies in ker
t-tilde and has profile 2K_r^S(b). A generic primal color meets it in
dimension at most 2M_r, giving the new leaf socle bound. Each of these
four requirements holds on a nonempty open subset of C(t-tilde), and
since this group is irreducible, all four hold simultaneously.

Let P_+ and P_- be the resulting color projections, and define the
new leaf arrows by u_+=P_+ u-tilde and v_+=v-tilde restricted to
W-tilde_+, and similarly for the minus sign. Their return maps are
zero because P_+ u-tilde v-tilde restricted to W-tilde_+ vanishes. At
the preceding trunk vertex, v_+u_+ + v_-u_- = v-tilde u-tilde. The
relation, incoming-image sum, and outgoing kernel at that vertex are
therefore unchanged from the doubled chain. All other trunk bounds
simply double. For i<r, the new trunk dimension is
2 sum_{j<=i} Delta_j. The two leaf coefficients for the D_{r+1}-weight
(2Delta_1,...,2Delta_r,0) are both sum_{j<=r} Delta_j = dim V_r,
precisely the two new color dimensions. The new leaf top bounds are
2L_r, and the socle bounds are 2M_r, as required. The tensor criterion
gives the asserted occurrence. When r=1, the retained edge is the
virtual edge from V_0=0; then t_1=u_1=v_1=0, and the construction
produces the two disconnected vertex spaces of D_2=A_1 x A_1. The same
bounds and dimension calculation apply.

3.6 Completing the main theorem.

Suppose first that the invariant at scale N is nonzero. Set xi=nu*,
so V(N xi) occurs in V(N lambda) tensor V(N mu), and
lambda+mu-xi lies in the root lattice. We may permute the three
invariant labels before designating the moving label mu.

If one of them has at most one zero coordinate, choose it as mu. The
invariant-to-moment lemma, followed by division by N, gives a real
moment solution for lambda,mu,xi. Choose the shape made of successive
fundamental columns summing to mu. The integral schedule theorem gives
an integral schedule for this shape, and the column intertwiner
proposition gives V(lambda) tensor V(mu) -> V(xi) nontrivially. This
is the desired invariant.

Otherwise all three labels have at least two zero coordinates. They
are then integer-coordinate weights, nonnegative and supported on
initial coordinate segments. Choose a label of largest support as mu,
and denote its support length by r<=n-2. If r=0, all three
representations are trivial. Assume r>=1. Apply the zero-tail removal
proposition to the occurrence at scale N. It gives the occurrence in
D_{r+1} for the truncated labels at scale 2N. The invariant-to-moment
lemma, divided by 2N, therefore gives a real moment solution for the
unscaled truncated labels.

Those labels remain integral, and their difference remains in
Q(D_{r+1}): deleting zero coordinates preserves the even coordinate
sum. The moving truncated label is positive in its first r coordinates
and has exactly one zero. Take the fundamental column shape for the
original D_n-weight mu. Since M_{r+1}=...=M_n=0, its fundamental
expansion uses only columns e_1+...+e_k with k<=r; every increment
therefore already vanishes beyond the retained coordinates. Restrict
each increment to the first r+1 coordinates. These increments are
dominant for D_{r+1}. They need not be its fundamental weights; the
integral schedule theorem applies to this arbitrary dominant-increment
shape and gives an unscaled integral schedule.

Pad this schedule with zero coordinates. Its roots and even signed
permutations belong to D_n, its endpoint lies in the required full
Weyl orbit, and every switch coefficient and sign is unchanged. It is
therefore an integral schedule for the original fundamental column
shape. The column intertwiner proposition again gives the unscaled
invariant. The use of 2N supplied only real existence in smaller rank
and has introduced no saturation factor.

Conversely, if an unscaled invariant functional is nonzero, the
compact translates of highest-weight vectors span each irreducible
representation, so the functional is nonzero on some triple of flag
vectors. Its N-th tensor power is nonzero on the N-th powers of those
same flag vectors. Those powers lie in the Cartan summands of highest
weights N lambda,N mu,N nu. Restriction gives the invariant at scale N.

The arguments include D_2=A_1 x A_1 and D_3=A_3 with the stated
conventions. If Spin(2) is included, it is a torus; its
representations are characters, and the assertion is the equivalence
of the weight-sum equation with its positive integral multiple. This
completes the proof.

================================================================
PART 4. A COMPLETE MAP OF THE LEMMAS
================================================================

This part is a reference. Each item says what the lemma guarantees
and where it is used. The names match the original paper's
statements so a reader can move between this edition and the formal
version.

4.1 Section 2: real moments and schedules.

Strict approximation lemma. Every real moment triple can be
approximated by triples of regular labels admitting strict flags.
Strict label triples form an open set, and one has open freedom to
vary the two endpoint labels of a nearby shape. Used to avoid ties
before the area argument.

Symmetric-space triangle lemma. If (lambda, mu, xi*) admits strict
flags, then for every R>0 there is a triangle in G/K with successive
dominant side vectors R lambda, R mu, R xi*. The proof uses the push
of length R theta toward a flag, the radial bound limsup d/ds |P(s)|
<= m_theta(F,H_0), compactness of unit directions, and Brouwer's
fixed-point theorem. Used to produce real paths.

Decreasing radial paths lemma. For generic strict labels and a
generic regular polygonal shape ending at mu, there is a dominant
path from lambda to xi whose velocities are Weyl translates of the
shape velocities, in weakly Bruhat-decreasing order. The proof takes
the triangle, replaces its middle side by the shape scaled by R, and
lets R tend to infinity. It uses a common flag z_R, operator
comparisons, a tropical formula for theta dot x(t), and a local
U^-U^+ factorization to prove the Bruhat comparison. Used to create
the path class for area maximization.

Area variation lemma. For an area-maximizing path with regular
shape, the velocity differences at its nonzero unfolded jumps away
from shape breaks are linearly independent. The proof moves jump
times in a null direction, checks admissibility after folding, and
shows a common two-sided first variation vanishes while one side has
a strictly positive second variation. Used to get a basis of jumps.

Strict exposure lemma. For a generic area maximum, a nonzero jump
has a linear functional whose maximum among the Bruhat interval is
attained exactly at the two endpoint velocities. At a shape break a
common functional has strict maxima at the two endpoint velocities.
The proof adds a terminal linear functional to the area objective so
all basis-time derivatives vanish, then inserts an intermediate
Bruhat state at the jump. Used to identify jumps with edges.

Flag orbit polytope lemma. For a flag b in U^+u intersect U^-v with
u>=v, the torus-fixed points of its orbit closure lie in [v,u] and
include u,v. For a regular dominant real weight d, the convex hull of
the vectors hd has a normal fan with vertices labeled by fixed flags,
independent of d. Every edge joins two vertices related by a root
reflection. The proof uses the Schubert closure, the weight polytope,
Minkowski sums of fundamental polytopes, and a one-dimensional torus
orbit. Used to identify exposed jumps.

Exposed jumps are covers lemma. The generic area maximum has no
nonzero jump at a shape break. Every other nonzero unfolded jump is a
reflection and is a Bruhat cover in its local chamber frame. The
proof combines strict exposure, the flag orbit polytope, and the
inversion count for beta=alpha+delta, where alpha and delta are
positive roots and both are inverted by the incoming direction. Used
to make switches cover reflections.

No strict third-axis obstruction corollary. At a switch of the
limiting schedule, if the two exchanged signed labels have strict
opposite position and moving-value orders, then no signed label on a
third physical axis lies strictly between them in both orders. Such
a label would give the two positive roots above, making the Bruhat
length drop greater than one. Used in the knot count.

Basic real schedule proposition. Let the root system be type D or a
permutation system, possibly with an orthogonal central space. If
(lambda, mu, xi) has a real moment solution, then every shape ending
at mu admits a schedule whose switch roots form a real basis of the
semisimple space. Weak signs, endpoint times, and coincident times
are allowed. It is a limit of schedules for nearby regular data whose
switches are covers in their local chamber frames.

Highest-to-highest coefficient lemma. Let h,h',theta be dominant
integral weights. The linear forms obtained by taking the
highest-output coordinate of T(v_h tensor -), for
T:V(h) tensor V(theta)->V(h'), are exactly the forms supported at
weight h'-h that annihilate
sum_i f_i^{h dot alpha_i+1} V(theta). In particular, if
h'=h+u theta is dominant, the extremal coordinate at u theta gives a
nonzero intertwiner. Used for every path segment and every column.

Flag composition lemma. Suppose h_0,...,h_s and theta_1,...,theta_s
are dominant integral weights and there are intertwiners
T_j:V(h_{j-1}) tensor V(theta_j)->V(h_j). Assume their
highest-to-highest coefficients are nonzero on flag vectors at flags
b_j. If the past and future limits of b_j under a(t) are u_j,v_j, and
v_j>=u_{j+1}, then the iterated composition is nonzero on
V(h_0) tensor V(theta_1+...+theta_s), where the second factor is the
Cartan summand in the tensor product of the V(theta_j). Used to
compose columns.

Invariants give moments lemma. A nonzero invariant in
V(lambda) tensor V(mu) tensor V(xi^*) gives a real moment solution
for (lambda,mu,xi). The same conclusion follows from an invariant for
simultaneous positive integer multiples of the labels. The proof
maximizes the invariant functional on unit flag vectors and
differentiates the three norms. Used at the start of the main
argument.

Schedules give moments lemma. A real schedule for (lambda,eta,xi)
gives a real moment solution for (lambda,mu,xi), where mu=eta(1). No
lattice condition is required. The proof perturbs to rational generic
data, projects to the dominant chamber, subdivides, dilates to make
positions integral, uses extremal intertwiners and flag composition,
then divides by the dilation and passes to the limit. Used in the
block reductions and in the main construction.

4.2 Section 3: lattices and continuation.

Integral schedule theorem. Let the three labels have integral root
pairings and satisfy lambda+mu-xi in Q. Let eta be any polygonal shape
with dominant increments and endpoint mu. A real moment solution gives
an integral schedule in either of two cases: (i) the root system is a
permutation system; (ii) the root system is D_l and mu has at most one
zero coordinate. An orthogonal central space is allowed in either
case. This is the main technical output of Moves 1-3.

Combining blocks lemma. Suppose a moment solution splits into
commuting coordinate blocks of permutation type and orthogonal type.
Assign the initial, moving, and final coordinates using this
splitting. Under the hypotheses of the integral schedule theorem (ii),
integral schedules can be combined from the blocks. In particular,
induction applies if there is a permutation block, if there are at
least three proper orthogonal blocks, or if there are two proper
orthogonal blocks of even parity. Here parity is the sum modulo two
of the assigned difference lambda+mu-xi in the block.

Exceptional case lemma. If no integral schedule exists, any real
schedule with at most l switches has exactly l independent switch
roots, and its graph has exactly two connected components, both
unbalanced and of odd parity. Each component is unicyclic. On its
cycle the switch coefficients are half-odd-integers, and off the
cycle they are integers. This isolates the only obstruction.

Closedness and generic shapes lemma. For fixed labels, the existence
of an integral schedule is closed under uniform convergence of shapes
with dominant increments. It suffices to prove the integral schedule
theorem for polygonal shapes whose slopes are relatively interior to
the smallest chamber face containing mu, with generic break values in
that face. The proof deletes removable switches, bounds their number
by the longest Weyl length, and passes to a subsequence with constant
integer levels.

Half-kernel bounds and potential lemma. Suppose the l+1 roots
generate Q(D_l), and their exact base coefficients m^0 are
half-integral. If m^0 is not already integral, either orientation d of
half the primitive integer relation among the roots satisfies
m^0+d in Z^{l+1}, |d_i|<=1, and |d_i^G|<=1, where d^G is its image
under the linear part of the gap formula. Moreover there is a vector L
such that d_i=L dot gamma_i. The proof uses signed-graph minors of
absolute value two or four, the primitive kernel vector, the
unimodular coefficient change from m to G, and the telescoping
identity sum_i (L dot gamma_i) beta_i=(1-p_k)L. Used to define the
half-relation interpolation.

Continuation proposition. Under the standing assumption of no
integral schedule, a local link cannot exist. The proof follows the
half-kernel direction through collisions. At a collision, candidate
times are determined by a common potential. In the rank-two A_2
subsystem, every length-two Bruhat interval is a diamond, so the
other route is also two downward covers. Sorting by candidate time
terminates lexicographically, and parameter one is attained with all
coefficients integral, a contradiction.

Direct links lemma. In the exceptional schedule with components P,Q,
each of the following produces a forbidden local link: (i) a signed
cross-component pair with strict concordant position and moving-value
orders at time zero; (ii) a signed cross-component pair with strict
discordant orders at time one; (iii) a position crossing of a
cross-component pair with distinct moving values between old switch
times; (iv) the crossing in (iii) at an old switch time, if both axes
are unused by its cluster. The proof inserts the cross swap or the
crossing swap, reflects the suffix, checks that the new roots
generate Q(D_l), and verifies the primitive half-kernel derivative on
the bridge has magnitude one. Used in the knot count.

4.3 Section 4: the strict count and simultaneous switches.

Profile rigidity lemma. Suppose two irreducible Hermitian zero-sum
triples have proportional profiles m,m', of sizes r,h, and
||m||^2=r^2, ||m'||^2=h^2. Then r=h and m=m'. The proof builds flags
with scalar endomorphisms, shows direct sums fill an open set, and
uses Krull-Schmidt uniqueness. Used to rule out equality in
Cauchy-Schwarz.

Strict comparison count proposition. In the standing exceptional
configuration, C=R-p_lambda-p_mu-p_xi is positive. The proof uses
the moment profiles of the two components, the orbit-dimension
estimate ||m||^2-sum_i z_i<=r^2-r, the joint-kernel reduction, and
profile rigidity. Used to get the strict numerical gap.

Plateau lemma. Suppose cross labels a,c have equal position and
equal assigned moving value on an open interval between switch times.
Then (p_a,p_c) is either (0,0) or (1,1). The proof reflects the whole
suffix by the reflection exchanging the two signed axes and uses the
hard-case lemma to exclude the other phase combinations. Used to
define the correction term in J.

Monotonicity at knots proposition. Under the standing
no-integral-schedule assumption, the function J is nondecreasing at
every knot, and hence on the whole interval in the one-sided sense.
The proof makes the shape generic, groups signed labels by forms
modulo the knot form, uses the no strict third-axis obstruction
corollary across classes, and reduces within a class to matching
pairs. It checks single-single, pair-single, and pair-pair
comparisons. A loss either gives a feasible extra-edge link,
forbidden by continuation, or is compensated by a double plateau.
Used to contradict the strict count.

4.4 Section 5: from integral schedules to tensor intertwiners.

Folding lemma. An integral schedule can be folded into a dominant
path with Weyl direction states in decreasing Bruhat order. Its
changes of state can be expressed as an ordered sequence of downward
reflections, including at coincident times. At each step from w to
s_alpha w at time t, alpha dot w eta(t) is an integer. For a
concatenation of fundamental columns, all column endpoint positions
of the folded path are dominant integral weights. Used to set up the
column argument.

Common-walls lemma. Let J be the set of simple roots alpha_i with
x_0 dot alpha_i=x_1 dot alpha_i=0. A functional supported on the
weight space of weight delta in wedge^k E, invariant under the
semisimple group generated by J, is an allowed highest-to-highest
coefficient for an intertwiner V(x_0) tensor V(omega)->V(x_1). The
proof checks that a vector contributing to its weight-delta component
would have source weight whose simple-root pairing is h_i+h'_i+2,
which exceeds the maximum coordinate pairing two in wedge^k E unless
both vanish. Used for exterior columns.

Column intertwiners proposition. Let lambda,mu,xi be dominant
integral weights of type D_n, n>=2. Choose a shape obtained by
concatenating straight segments whose increments are fundamental
weights and whose sum is mu. If this shape admits an integral schedule
from lambda to W xi, then Hom_G(V(lambda) tensor V(mu),V(xi)) is
nonzero. An empty list of columns is allowed when mu=0. The proof
treats a minuscule column directly and an exterior column at its
midpoint using the common-wall invariant, then composes with flag
composition.

4.5 Section 6: removing a zero tail.

Tensor criterion. Let lambda,mu,xi be dominant integral weights for
a finite simply laced Dynkin diagram D, and suppose
lambda+mu-xi=sum_i d_i alpha_i with d_i nonnegative integers. Put
lambda_i=lambda dot alpha_i and mu_i=mu dot alpha_i. Then
Hom(V(lambda) tensor V(mu),V(xi)) is nonzero if and only if there
exists M in Lambda(d) with top_i M<=lambda_i and soc_i M<=mu_i for
every i. The proof uses semicanonical generic evaluation and its left
and right filtrations.

Threshold bounds lemma. For every 1<=r<=p and b>=1,
K_r^T(b)<=L_r+q_r(b)/2 and K_r^S(b)<=M_r+q_r(b)/2. The proof uses
dual high-socle spaces, trunk inequalities, leaf inequalities, and a
row comparison that contributes 1/2 when a row is gained or lost.
Used to control every nilpotent filtration level before doubling.

Filtered intersection lemma. For a filtration F_1 superset ... superset
F_{s+1}=0, let P be its stabilizer in GL(H). For subspaces A,C write
a_b=dim(A intersect F_b), c_b=dim(C intersect F_b), and f_b=dim F_b.
For g in a nonempty Zariski-open subset of P,
dim(A intersect gC)=max(0,max_b(a_b+c_b-f_b)). The proof uses
adapted bases and Hall's distinct-representatives criterion. Used to
place a moving subspace inside a nilpotent filtration.

Centralizer on the socle lemma. Let t be nilpotent on V, and put
F_b(t)=ker t intersect im t^{b-1}. The group C(t)={g in GL(V):gt=tg}
induces the full stabilizer of this filtration on ker t, and the full
stabilizer of the corresponding filtration on ker t^* inside V^*. The
group C(t) is irreducible. The proof uses Jordan row modules and
inverse transpose. Used to choose a generic coloring.

Zero-tail removal proposition. Suppose lambda,mu,xi are dominant
integral D_n-weights, all supported in coordinates 1,...,r, where
1<=r<=n-2. If V(xi) occurs in V(lambda) tensor V(mu), then
V(2 xi-hat) occurs in V(2 lambda-hat) tensor V(2 mu-hat) for D_{r+1},
where lambda-hat=(L_1,...,L_r,0), and similarly for the other two
labels. The proof keeps a chain through V_r, takes two copies,
colors Jordan rows alternately, and uses the filtered intersection
and centralizer lemmas to choose a coloring satisfying both new leaf
top and socle bounds. Used to reduce a long zero tail.

Main theorem completion. If one label has at most one zero
coordinate, choose it as the moving weight, get a real moment
solution by dividing the dilated invariant, build an integral
schedule for a fundamental column shape, and convert it to a tensor
map. If all three labels have at least two zero coordinates, apply
zero-tail removal to get a doubled occurrence in smaller rank, divide
by 2N to get real existence at the unscaled truncated labels, build
an integral schedule there, pad it with zeros, and convert it to a
tensor map at the original scale. The reverse direction is the N-th
tensor power. This proves the main theorem.

================================================================
PART 5. EXAMPLES AND WHY THE DETAILS MATTER
================================================================

5.1 The root-lattice parity condition.

In D_n, the root lattice is the set of integer vectors with even
coordinate sum. Consider lambda=mu=nu=e_1. The sum is 3e_1, whose
coordinate sum is 3, odd. Therefore 3e_1 is not in the root lattice,
and there is no invariant in V(e_1) tensor V(e_1) tensor V(e_1). This
is the central-character obstruction.

If we dilate by N=2, the sum is 6e_1, whose coordinate sum is 6,
even. The root-lattice condition no longer obstructs. Saturation is a
statement about the opposite direction: an invariant at a dilation
forces an invariant at scale one, provided the root-lattice condition
holds at scale one. In the example above, the root-lattice condition
fails at scale one, so the theorem does not apply. This is a useful
check: saturation never claims an invariant when the central
character forbids it.

5.2 Spin weights.

Half-integer weights are allowed. For example, the spin
representation of Spin(2n) has a highest weight with coordinates
(1/2,...,1/2). The root lattice still consists of integer vectors
with even sum, while the weight lattice includes the half-integer
vectors. The theorem covers these spin weights, unlike the classical
Newell-Littlewood range for ordinary partitions.

5.3 Why positive integer stretching is essential.

Suppose one allowed arbitrary positive rational N. The original paper
gives a counterexample in D_n for n>=4. Take lambda=mu=nu=2e_1. The
triangle contraction (x,y)(y,z)(z,x) of the standard orthogonal form
defines a nonzero invariant on V(2e_1) tensor V(2e_1) tensor V(2e_1).
Indeed isotropic squares lie in V(2e_1), and in a hyperbolic basis
a_i,b_i with (a_i,b_j)=delta_ij, the isotropic vectors a_1, b_1, and
a_1+b_1+a_2-b_2 have nonzero pairwise products.

At N=1/2 the three labels are e_1, whose sum 3e_1 is outside Q(D_n),
so the triple has no invariant. Thus a rational dilation can have an
invariant while the original does not. The theorem avoids this by
requiring N to be a positive integer. This is why the statement says
"positive integral dilation" and not "positive real dilation".

5.4 Why an integral schedule is not a saturated chain.

In a highest-weight path model, the usual sufficient condition is an
integral saturated chain in the Bruhat order. The schedules in this
paper satisfy a weaker condition: only the switch coefficients are
integral. The column construction in Part 3.4 is designed to bridge
this gap. It proves the tensor implication directly from the weaker
condition, without strengthening the schedule. This is one of the
main innovations of the paper.

================================================================
PART 6. GLOSSARY
================================================================

A_2. The rank-two root system with three positive roots i, j, i+j.
Its Bruhat order has length-two diamonds, which are used in the
continuation proof.

Balanced component. A connected component of a signed root graph
whose signed identifications are consistent around every cycle. After
changing signs of axes, it becomes a permutation system.

Borel subgroup. A maximal connected solvable subgroup B of a
semisimple group. The flag variety is G/B. A highest-weight line is
fixed by B.

Bruhat cover. A strict Bruhat comparison u>v with no Weyl element
strictly between them.

Bruhat order. The partial order on Weyl group elements defined by
reflection-length drops, equivalently by containment of Schubert
cells. Downward steps are reflections that decrease length.

Cartan summand. The copy of V(theta_1+...+theta_s) inside the tensor
product of the V(theta_j) spanned by tensor products of highest-weight
vectors. Flag composition places the product of column coefficients
on this summand.

Column. A straight segment of a shape whose increment is a
fundamental weight. A fundamental column is the basic building block
of a shape.

Core phase. For a signed position label a, the number 0, 1/2, or 1
that records how many of the cycle incidences of its axis have
occurred. It is used to correct the comparison count on intervals
where positions and moving values both agree.

Cross-comparison. A pair of signed labels, one from each of the two
odd unbalanced components P,Q. Counted with weight 1/2 so a pair and
its signed mirror together have weight one.

D_n. The root system of Spin(2n). Weights are n-tuples. Roots are
+-e_i+-e_j for i not equal to j. The root lattice is the integer
vectors with even coordinate sum.

Dominant chamber. The cone x_1>=...>=x_{n-1}>=|x_n|. Every Weyl orbit
has a unique point in the closed dominant chamber.

Dominant weight. A weight in the dominant chamber. Dominant integral
weights classify irreducible representations.

Doubled quiver. The quiver obtained from a Dynkin diagram by adjoining
a reverse arrow to every arrow. Preprojective representations satisfy
the signed sum of the two-arrow return maps is zero.

Exterior column. A fundamental column with weight e_1+...+e_k for
2<=k<=n-2. Its only effective interior switch is at the midpoint.

Filtered intersection. The generic dimension of A intersect gC for a
filtration-preserving g is the maximum of 0 and a_b+c_b-f_b over the
filtration levels.

Flag. A full flag is a chain of subspaces 0=F_0 inside F_1 inside ...
inside F_N=V with dim F_i=i. A flag vector of weight h at a flag b is
a nonzero vector on the translate of the highest-weight line.

Flag composition. The lemma that a chain of intertwiners with
nonzero highest-to-highest coefficients at ordered flags composes to a
nonzero map on the Cartan summand.

Fundamental weight. One of the basic dominant weights omega_i whose
simple-root pairings are delta_ij. Every dominant integral weight is a
nonnegative integer sum of fundamental weights.

Half-relation. The primitive integer relation among the roots of a
connected unbalanced graph with one more edge than the rank. Half of
it moves half-integral switch coefficients to integers.

High-socle space. F_b(t)=ker t intersect im t^{b-1}. It counts the
number of Jordan rows of length at least b.

Integral schedule. A schedule all of whose switch coefficients are
integers.

Invariant. A nonzero vector fixed by every group element. In a tensor
product of three representations, an invariant is the same as a
nonzero G-linear map from the first two factors to the dual of the
third.

Knot. An old switch time at which the comparison function J can
decrease. The monotonicity at knots proposition says it cannot.

Local link. A modification of a schedule at one time that replaces
some switches by one more switch, with new roots generating the root
lattice, half-integral base coefficients, and a chosen orientation
whose candidate switches stay ordered and feasible for small positive
parameter.

Minuscule weight. A dominant weight whose root pairings are all 0 or
+-1. Minuscule columns have no effective interior switches.

Moment solution. A zero-sum triple of matrices in the compact orbits
of lambda, mu, -xi. It is the real-geometric form of the tensor
invariant problem.

Nonzero unfolded jump. A jump in the unfolded schedule whose two
states differ. Shape breaks do not count.

Parity of a block. The sum modulo two of the assigned difference
lambda+mu-xi in that block. It determines whether an orthogonal block
can be solved separately.

Permutation system. A root system whose roots are e_i-e_j. Its root
basis is a spanning tree, which is automatically an integral basis of
the root lattice.

Plateau. An open interval between switch times on which two cross
labels have equal position and equal assigned moving value. Its
phases are either (0,0) or (1,1).

Preprojective representation. A representation of the doubled quiver
whose signed two-arrow return maps sum to zero at every vertex. The
semicanonical basis identifies the enveloping algebra with functions
on these varieties.

Profile. For a Hermitian triple on C^r, the collection of
multiplicities of each numerical eigenvalue of each of the three
matrices. The tie count between two blocks is one half the dot
product of their profiles.

Root. A vector beta whose reflection s_beta(x)=x-(x dot beta)beta is
a symmetry of the root system. In D_n the roots are +-e_i+-e_j.

Root lattice Q(D_n). The integer vectors in the weight lattice with
even coordinate sum. The central-character obstruction lives here.

Schedule. A path that starts at lambda, follows Weyl translates of a
shape, and ends in the Weyl orbit of xi=nu*. At a switch it changes
from w to s_beta w.

Shape. A polygonal path from 0 to mu whose every increment is
dominant. A schedule follows Weyl translates of a shape.

Signed graph. A graph whose vertices are coordinate axes and whose
edges are signed identifications x_i=epsilon x_j coming from roots
e_i-epsilon e_j. Balanced components are permutation systems;
unbalanced components have full coordinate span.

Socle. For a module M over a quiver, the intersection of the kernels
of the outgoing arrows. It measures how many Jordan rows start at a
vertex.

Spin representation. An irreducible representation with half-integer
highest weight. It is minuscule in type D.

Strict triple. A triple of weighted flags with a positive margin in
the inequality sum_j m_theta_j(F_j,H)<0 for every unit Hermitian H.

Switch coefficient. The integer or half-integer m=-beta dot w eta(t)
at a switch from w to s_beta w. A schedule is integral when all these
numbers are integers.

Tensor product. The vector space spanned by symbols v tensor w, with
the group acting on both factors. An invariant in a tensor product is
a vector fixed by the group.

Top. For a module M over a quiver, the quotient of M_i by the sum of
the images of incoming arrows. It measures how many Jordan rows end at
a vertex.

Unbalanced component. A connected component of a signed root graph
whose signed identifications are inconsistent around some cycle. It
has full coordinate span and an even-sum integer lattice of index two.

Waypoint. A flag b in U^+u intersect U^-v used to evaluate a column
coefficient. Its past and future limits are the endpoint Weyl states.

Weyl group. The group generated by root reflections. For D_n it acts
by permuting coordinates and changing signs of an even number of
coordinates.

Weyl translate. A vector w eta obtained by applying a Weyl group
element w to a shape value eta. A schedule follows Weyl translates of
the shape.

Zero tail. A common terminal block of zero coordinates in all three
labels. The zero-tail removal proposition reduces a long zero tail to
a smaller-rank doubled occurrence.

================================================================
END OF EDITION
================================================================