Proof. Take coinvariants for the diagonal negative nilpotent algebra on the tensor product. At weight \(h'\) this quotient records exactly the highest coordinate of each copy of \(V(h')\). Move negative generators from the first tensor factor to the second by the antipode. The highest-generator presentation \[V(h)=U(\mathfrak n^-) \Big/\sum_i U(\mathfrak n^-)f_i^{h\cdot\alpha_i+1}\] then gives precisely the quotient by (3): the antipode reverses products, so a relation \(Xf_i^{h\cdot\alpha_i+1}\) transfers into the image of the indicated power on an arbitrary second vector. This proves both necessity and sufficiency after dualizing the weight quotient.

The presentation itself follows by imposing the simple singular-vector relations on the Verma module. Each simple \(\mathfrak{sl}_2\) then acts integrably: local nilpotence extends from the generating vector using the adjoint action. The weights are therefore Weyl-invariant and bounded in the highest-weight order; together with PBW this makes the quotient finite-dimensional. Complete reducibility identifies it with the irreducible highest-weight module.

For an extremal weight \(u\theta\), the upward simple-string length is \(\max(0,-u\theta\cdot\alpha_i)\). It is at most \(h\cdot\alpha_i\) because \(h+u\theta\) is dominant. Thus its coordinate functional annihilates every image in (3). ◻

Proof. The empty composition is the identity; assume \(s\ge1\). We first explain why one positive-flow orbit can approximate the waypoints \(b_1,\ldots,b_s\) in their prescribed order, with successive time gaps tending to infinity. When \(v\ge u\), there is a bridge in \(U^+v\cap U^-u\). Indeed \(U^+v\) approaches \(u\), and a neighborhood of \(u\) is covered by small \(U^+U^-\) translates. Removing the small positive factor from a point of \(U^+v\) leaves a point in that intersection.

It remains to glue at a shared fixed flag \(p\). Suppose the preceding waypoint approximation lies at \(n^-p\in U^-p\) and the next desired point is \(n^+p\in U^+p\). Replace the former by \[ n^-a(-t)n^+a(t)p. \tag{4}\] As \(t\to\infty\) this tends to \(n^-p\), whereas after flowing for time \(t\) it tends to \(n^+p\). The remaining negative factor after that flow preserves the future cell containing \(n^+p\). Insert the bridges and repeat this construction, choosing each new time gap sufficiently large to preserve every earlier approximation.

For each such approximation, choose flag vectors \(z_j\in V(\theta_j)\) at its common starting flag \(z\), and times \(t_1<\cdots<t_s\). Let \(C\) denote the iterated composition of the \(T_j\), and set \[F_j(x)=T_j\bigl(x\otimes a(t_j)z_j\bigr),\qquad C_z(x)=C(x\otimes z_1\otimes\cdots\otimes z_s).\] Equivariance gives the exact identity \[F_s a(t_s-t_{s-1})F_{s-1}\cdots a(t_2-t_1)F_1 =a(t_s)C_z a(-t_1).\] Indeed, writing each \(F_j\) as \(a(t_j)T_j(a(-t_j){\cdot}\otimes z_j)\) cancels all the intermediate frame factors. For \(s=1\) the identity is just this equivariance formula.

Normalize each translated flag input so that it approaches the prescribed vector at \(b_j\). Normalize each intervening \(a(t_{j+1}-t_j)\) by its highest eigenvalue on \(V(h_j)\); as the time gaps tend to infinity, it approaches projection onto the highest line. The highest-input, highest-output coordinate of the left side therefore tends to the product of the nonzero waypoint coefficients. It is nonzero for some finite approximation. On the right side the outer frame factors act by nonzero scalars on those coordinates, so \(C_z\) is nonzero. The \(z_j\) lie at one common flag, and hence their tensor belongs to the Cartan summand. This proves the assertion. ◻

Proof. Take an invariant functional and maximize its absolute evaluation on triples of unit flag vectors, using compact-invariant Hermitian norms. The maximum is attained and is nonzero, since the flag-vector orbits span their representations. Vary all three vectors simultaneously by a Hermitian generator \(H\) and renormalize. Invariance makes the unnormalized numerator constant; differentiation of the three norms therefore gives zero sum of the expectations of \(H\). Since this holds for all \(H\), the three moments sum to zero. The moment of a compact translate of a unit highest vector is the corresponding compact translate of its weight. This is the required solution. For scaled labels divide the three Hermitian elements by the scale. ◻

Proof. Apply the three schedule simplifications. The remaining switch gaps, opposite moving-weight pairings, and distances from the time endpoints are strict. Perturb the initial position, shape, and times to rational generic data, allowing the shape endpoint and the final position to vary. Make the shape slopes regular, separate the switch times, and make all ordinary wall crossings simple, transversal, and disjoint from the switches and shape breaks. All strict signs persist.

Project the path to the dominant chamber. At an ordinary crossing, the incoming folded velocity has negative pairing with the crossed simple root, so the direction makes a downward simple Bruhat step. At an original switch, its positive gap makes its root positive in the local chamber frame, and the opposite pairing with the regular moving weight makes its inverse root negative. It too is a downward Bruhat reflection. Shape breaks alone do not change the Weyl frame.

Subdivide at every crossing, switch, and shape break. A common integer dilation makes the semisimple positions and shape increments integral. Each segment then has an extremal intertwiner from Lemma 4; its waypoint is the fixed flag of the segment’s direction. Their decreasing order permits Lemma 5. The resulting nonzero intertwiner gives a real moment solution by Lemma 6. Divide by the dilation and pass to the unperturbed labels by compact-orbit closedness. Central coordinates are omitted from the representation argument and restored by their unchanged addition equation; no integrality of a central coordinate is needed. ◻

Proof. A moment solution supplies semistable flags: each least supported level is at most its expectation, and the three expectations sum to zero.

For fixed labels, the strict and semistable loci of triples of flags are Zariski open. To see the finite incidence argument, write a test as \(H=\mathop{\mathrm{Ad}}(k)d\) with \(d\) dominant. A lower bound on a least level means that each relevant highest line lies in the \(k\)-translate of an upper level space \[\bigoplus_{\chi\cdot d\ge c} V_\chi.\] Such spaces are \(B\)-stable, and only finitely many choices occur in the finitely many fundamental representations. For each choice, incidence with a common translating flag is closed; its projection is closed because the flag variety is projective. Among these finitely many incidence types, retain those for which some dominant \(d\) violates the prescribed strict or weak inequality. Their union is exactly the failure locus.

Strict regular data exist. Take regular Hermitian elements \(X,Y\) with zero common semisimple Lie centralizer and with \(-X-Y\) regular. For example, one can start with \(X\) regular toral and choose \(Y\) generically with nonzero components on every simple root. Attach the three compact flags realizing these representatives. Equality in the sum of least-level inequalities forces each flag vector to be an eigenvector of the testing generator. For regular weights this makes the generator commute with all three representatives, so it must be zero. The resulting triple is strict.

Intersect the nonempty open flag locus for the original semistable labels with that for this auxiliary strict triple. The product of flag varieties is irreducible, so the intersection is nonempty. At a common triple of flags, a positive convex combination of the original and auxiliary labels is strict and regular. Such labels approach the original labels. Finally, the least supported level is upper semicontinuous in \(H\): disappearing components can only raise its value at the limit. Compactness of the unit sphere makes the strict margin uniform. Small changes of the finitely many weight coefficients preserve that margin. This proves openness and the stated freedom of endpoints. ◻

Proof. The push of length \(R\theta\) from \(hK\) toward a full flag \(F\) is the path \[hk\exp(s\theta)K,\qquad 0\le s\le R, \qquad h^{-1}F=kB,\quad k\in K.\] It is continuous and well-defined, and the same flag specifies its forward direction all along the push. Compose the three pushes. We prove that this composition has a fixed point.

In polar coordinates \(\exp(P)K\), radial distance is \(|P|\), and inverse polar coordinates are distance non-increasing. One direct verification uses the differential of the exponential: after translation back by \(\exp(-P)\), its Hermitian part is \[\frac{\sinh(\operatorname{ad}P)}{\operatorname{ad}P}\,\delta P.\] This expands lengths and is the identity in the radial direction. Polar decomposition gives global coordinates. In particular, points at a bounded mutual distance and tending to infinity have the same limiting radial unit direction.

At \(\exp(lH)K\), \(|H|=1\), the radial derivative of a push toward \(F\) is the pairing of \(H\) with the compact moment of \(\theta\) at \(\exp(-lH)F\). If \(l\to\infty\) and \(H\to H_0\), then \[ \limsup \frac{\,\mathrm d}{\,\mathrm ds}|P(s)| \le m_\theta(F,H_0). \tag{6}\] For a fundamental representation, take a narrow spectral window around the least supported level for \(H_0\). Before applying the exponential, the flag vector’s projection onto a slightly enlarged window has norm bounded below for all nearby \(H\). Multiplication by \(\exp(-lH)\) and normalization suppresses levels above that window exponentially. The resulting expectation has limsup at most the window’s upper endpoint. Shrinking the window proves (6); addition proves it for real dominant weights.

For fixed \(R\), all three pushes have bounded displacement. Along any sequence tending to infinity with limiting unit direction \(H_0\), the integrated radial derivatives therefore have total limsup at most \[R\sum_{j=1}^3 m_{\theta_j}(F_j,H_0)<0.\] The derivatives are bounded, so this estimate passes through the finite integrals. It follows, by compactness of unit directions, that every sufficiently large sphere maps strictly inward under the composition. Compose with radial retraction onto the corresponding closed ball and apply Brouwer’s fixed-point theorem (Brouwer 1911). A fixed point cannot lie on its boundary, by the strict inward property; an interior fixed point of the retracted map is a fixed point of the original composition. The three pushes from that point form the triangle. ◻

Proof. Move the first vertex of the triangle in Lemma 9 to \(K\). Replace its middle side by \[g_R\exp(R\eta(t))K.\] Here \(g_R\) includes the entering direction frame, so the replacement has the same endpoints. Its initial and final radial Cartan parts are \(R\lambda\) and \(R\xi\), the latter because the last side is reversed. Denote its radial Cartan part divided by \(R\) by \(x_R(t)\).

Let \(R\to\infty\). Use all fundamental representations and a regular integral highest-weight representation \(V(\sigma)\), with compact-invariant norms. We can choose a common flag \(z_R\) such that, in each of these representations and for every nonzero Cartan-weight projection, \[ \|\operatorname{pr}_\chi g_R^\dagger z_R\| \ge c\,\|\operatorname{pr}_\chi g_R^\dagger\|, \qquad c>0 \tag{7}\] with \(c\) independent of \(R\); use unit flag vectors for \(z_R\). Indeed normalize the finitely many nonzero projected operators to norm one. There are only finitely many possible nonzero-index patterns; for each pattern their tuples range in a compact space. A generic flag vector avoids the kernels of any such tuple, because each nonzero operator is nonzero on some flag vector and a finite union of proper closed subsets does not fill the flag variety. The maximum, over flags, of the minimum of the finitely many norms is positive and continuous in the tuple. Its compact minimum gives \(c\).

The adjoint operation is compatible across these representations. The highest-weight operator norm formula from compact-Cartan decomposition, applied to the adjoint product, and (7) show that, after a subsequence, \[ \theta\cdot x(t) =\max_\chi\bigl(c_{\theta,\chi}+\chi\cdot\eta(t)\bigr). \tag{8}\] Here \(c_{\theta,\chi}\) is the limit of \(R^{-1}\log\|\operatorname{pr}_\chi g_R^\dagger z_R\|\); values \(-\infty\) are allowed. To obtain the formula, the squared norm of \(\exp(R\eta(t))g_R^\dagger z_R\) is the sum of the squared norms of its weight projections. The logarithm of its norm therefore differs from the largest logarithm of a projection norm by a bounded amount. The logarithmic error in (7) is also bounded before division by \(R\). The coefficients are bounded above by the operator norm of \(g_R^\dagger\), whose Cartan part is \(R\lambda\). Thus subsequences exist simultaneously for all finitely many coefficients. In each fundamental representation, operator-norm submultiplicativity applied to \(\exp(R(\eta(t)-\eta(s)))\) and its inverse bounds \(|\theta\cdot(x_R(t)-x_R(s))|\) by \(\max_\chi |\chi\cdot(\eta(t)-\eta(s))|\). The fundamental weights span the Cartan space, so the paths \(x_R\) are uniformly Lipschitz. Their fixed endpoints make the limit a dominant path from \(\lambda\) to \(\xi\).

Choose the slopes of \(\eta\) to separate all distinct weight slopes of \(V(\sigma)\). Except at finitely many times, the maximum in its version of (8) has a unique weight. Every limit of the flags \(\exp(R\eta(t))g_R^\dagger z_R\) then lies in that weight space in the regular highest-weight embedding. Such a flag is torus-fixed. The fixed flags have distinct extremal weights, so it is a single fixed flag \(p\) on each open interval. The fundamental formulas give \(x'=p^{-1}\eta'\) there.

If a later interval gives \(q\), then \(p\ge q\). To see this, use a local factorization near \(p\) as small \(U^-U^+\) translates of \(p\). Multiplication by the exponential of the scaled regular dominant increment contracts the conjugated \(U^-\) factor to the identity. On the compact flag variety this factor acts uniformly close to the identity, even if the remaining \(U^+\) factor diverges. Consequently \(q\) belongs to the closure of \(U^+p\), which is exactly the asserted Bruhat comparison. Inversion preserves Bruhat order, so the velocity frames \(p^{-1}\) are weakly decreasing as well. ◻

Proof. An endpoint-preserving variation. Suppose those differences have a nontrivial linear dependence. Move the corresponding jump times linearly in a parameter \(s\) in the dependency direction, leaving the shape-break times fixed. For small positive and negative \(s\), the moved times stay in their original slope pieces. The unfolded endpoint change is exactly the sum of the velocity differences times the time changes, so it is zero.

Admissibility after folding. We check that folding these variations still gives admissible Bruhat-decreasing paths. Near an old jump position, only the wall subgroup \(W_I\) can be needed for sorting. Both states in (10) are left \(I\)-maximal. Consequently (9) preserves their comparison after any common local sorter \(g\in W_I\). Ordinary crossings introduced on either side are downward simple steps. If a moved jump and a wall crossing are simultaneous, sort on the incoming side, make the comparison, then perform the pure crossing on the outgoing direction. To verify the last step, the positive roots of the approached walls pull back under the incoming sorter to positive roots of the local \(I\)-system: the incoming slope has negative pairing with them. The other \(I\)-maximal state is also negative-facing on that approached subsystem. The outgoing pure sorting is therefore downward. The same reasoning works when a fixed shape break shares the old site. Away from the old sites and breaks there are no new wall contacts; a regular slope cannot run along a wall of a dominant path.

First and second variations of the folded area. We first show that \(\mathcal A(x_s)\) has a common two-sided derivative at zero. The issue is the change of chamber sorter near old wall sites. There are finitely many jump, break, and wall-contact times. On each adjacent open segment, regularity of the slope makes every vanishing wall coordinate grow at least linearly with distance from its contact time. The unfolded perturbation is uniformly \(O(|s|)\). Thus, outside neighborhoods of radius \(C|s|\) of those finitely many times, with \(C\) fixed sufficiently large, both signs of the perturbation have the same chamber sorter as the original path. They also have the same ordered prefix of moved jumps there. Consequently their unfolded positions have the form \(y_s(t)=y_0(t)+s z(t)\), with the same bounded \(z(t)\) for both parameter signs, and chamber projection applies the same linear Weyl frame. Chamber projection is globally Lipschitz (it is a continuous, finite piecewise-orthogonal map), so the omitted set, of length \(O(|s|)\), contributes only \(O(s^2)\) to the area difference. Omitting it from the bounded first-variation integrand also changes that integral by \(O(|s|)\). There is therefore a common two-sided first variation, which must vanish at an area maximum.

On either sufficiently small one-sided parameter interval, all resulting sites, including sorting crossings, have affine times \(\tau_j(s)\). Indeed positions are affine in \((t,s)\) in each fixed frame, and crossing times solve linear wall equations. Write \(v_j^-,v_j^+\) for the actual folded velocity vectors immediately before and after the \(j\)th site. After fixing the one-sided order of these finitely many sites, integration of the piecewise constant velocities with weight \(1-t\) gives \[ \frac{\,\mathrm d^2}{\,\mathrm ds^2}\mathcal A(x_s) =\sum_j \rho\cdot(v_j^+-v_j^-) \bigl(\tau_j'(s)\bigr)^2. \tag{11}\] A fixed shape break contributes zero. Every other term is nonnegative, because the corresponding comparison is downward and the local shape slope is regular dominant. A moved nonzero original jump gives a strictly positive term: its two unfolded states lie in distinct \(W_I\)-cosets, so sorting cannot erase their difference. The first variation vanishes and the one-sided second variation is strictly positive, contradicting maximality. ◻

Proof. The derivative of the unfolded endpoint with respect to the nonbreak jump times is an isomorphism: its columns are the basis of velocity differences. We can therefore add a linear functional of the unfolded terminal position to the area objective so that all unconstrained basis-time derivatives vanish. This functional is constant on the endpoint constraint, so the constrained maximization problem is unchanged.

At the chosen site, insert an intermediate state \(h\) from the Bruhat interval for a small duration, on the left or on the right. Its endpoint displacement is exactly compensated by linear shifts of the basis times. For sufficiently small duration, all these times remain in their fixed slope pieces, so compensation is exact, not merely to first order. If the chosen nonbreak site itself supplies a compensating variable, shift the inserted group as a whole. Chamber feasibility follows from the sorting argument in Lemma 11.

Use the fixed Weyl frame at the chosen off-wall site to identify its local coordinates with the common unfolded space. The first variation of the modified objective is then a linear functional \(f\) of the inserted local velocity difference. It is the same from either side of the site: it integrates the effect of that displacement on the later folded path, together with the terminal multiplier, and changing the lower time endpoint by a vanishing interval does not change the first variation. At a nonbreak site, shifting the original jump has vanishing first variation, so \(f(ud)=f(vd)\). Maximality gives \(f(hd)\le f(ud)\). At a break, a left insertion gives \(f(hd_-)\le f(ud_-)\) and a right insertion gives \(f(hd_+)\le f(vd_+)\). These include extension of the opposite endpoint state across the break.

For an inserted interval parameterized by its duration, the two boundary-time derivatives differ by one, so compensation cannot fix both. Thus each genuine intermediate insertion, or opposite-state extension at a break, moves at least one nonzero jump time. If its first-order inequality were equality, Equation (11) would give a strictly positive one-sided second variation and hence an improvement. Therefore every such inequality is strict. This proves both assertions. ◻

Proof. The orbit closure is contained in the Schubert closure through \(u\) and the opposite Schubert closure through \(v\), giving the interval assertion; the two flow limits supply its endpoints. For integral regular \(d\), this is the weight polytope of the torus orbit of the line of the flag vector in \(V(d)\). Vertices are obtained by generic one-parameter limits, and the fixed flags in this embedding have Weyl-extremal weights.

The flag vector for a sum of fundamental weights is the tensor of their flag vectors in the Cartan summand. Its torus weight polytope is therefore the corresponding Minkowski sum of the fundamental polytopes. For strictly positive coefficients its normal fan is their common refinement, independent of those coefficients. The labels are independent as well, since a generic torus limit is the same flag in all highest-line maps. Positive real coefficients give the same conclusion for real regular \(d\).

For integral \(d\), an edge is exposed by a one-parameter limit and comes from a one-dimensional torus orbit in the flag variety. Its codimension-one stabilizing subtorus annihilates a tangent root at a fixed endpoint, since the tangent weights of the flag variety there are roots. Thus the edge direction is a root direction. Its two Weyl vertices have equal norm, and two distinct equal-norm points on a line in root direction are exchanged by the corresponding root reflection. The common labeled fan carries this statement to every regular real \(d\). ◻

Proof. Choose the flag in Lemma 13. At a shape break, Lemma 12 would give two different unique maximizing vertex labels for the same functional on two polytopes with the same labeled normal fan. This is impossible. At a nonbreak site, that lemma exposes exactly the endpoint vertices, so they form an edge. Lemma 13 makes the jump a reflection.

Write it as \(u\to s_\beta u\), with \(\beta>0\) and \(u^{-1}\beta<0\). Suppose it is not a Bruhat cover. In these simply laced systems there are positive roots \(\alpha,\delta\) such that \[ \beta=\alpha+\delta,\qquad u^{-1}\alpha<0,\quad u^{-1}\delta<0. \tag{13}\] Here is the inversion count proving the assertion. Apart from \(\beta\), the positive roots taken negative by \(s_\beta\) are paired as \(\alpha,\beta-\alpha\). A pair with both roots inverted by \(u^{-1}\) contributes two to the length drop, whereas a mixed pair contributes zero. A pair with neither root inverted is impossible, since their sum has negative inverse image \(u^{-1}\beta\). Thus a drop greater than one forces (13).

In the rank-two subsystem on these three positive roots, both \(s_\alpha u\) and \(s_\delta u\) are proper intermediate states between \(u\) and \(s_\beta u\). Strict exposure therefore forces \(f\cdot\alpha<0\) and \(f\cdot\delta<0\): for example, \(s_\alpha ud-ud=-(ud\cdot\alpha)\alpha\) is a positive multiple of \(\alpha\). Equality of the two exposed endpoint values gives \(f\cdot\beta=0\), contrary to (13). The reflection is a cover. ◻

Proof of Proposition 3. First use generic strict labels and a generic regular shape. Lemma 10 supplies the nonempty compact path class. At an area maximum, Lemma 11 and the generic endpoint balance give a basis of nonbreak velocity differences; Lemma 14 removes all nonzero break jumps and makes every remaining nonzero jump a cover reflection. Its velocity difference is a nonzero scalar multiple of its root. Thus those roots form a basis. In the unfolded coordinates they obey (1): the local comparison root has nonnegative position pairing and negative pairing with the incoming moving weight. They therefore give a schedule.

For the prescribed moment data, use Lemma 8 and regular shapes converging to the prescribed shape. Such shapes can be obtained by small positive interpolation with shapes ending at the perturbing regular weights, followed by generic choices of their slopes and break values. At each stage the strict set gives the open endpoint freedom required in the area argument. The switch roots, frames, and endpoint Weyl elements take only finitely many values; pass to a subsequence on which these discrete data are fixed. The ordered switch times lie in a compact simplex. Uniform convergence of the shapes then gives convergence of the positions and switch coefficients. The roots remain the same independent basis, and the signs become weak. Central coordinates have been carried unchanged by addition throughout. This proves the stated limiting schedule. ◻

Proof. Such strict inequalities would persist in the regular approximating schedules. In their local sorted frame the third label would give the two positive roots in (13), both inverted by the incoming direction. The switch would then have Bruhat length drop greater than one, contrary to its cover property. ◻

Proof. We first explain the assignments and their arithmetic. In a schedule whose roots lie in the stated blocks, the initial moving frame assigns shape coordinates to each block; the endpoint assigns the final coordinates. Restricting the schedule gives a moment solution in the block by Lemma 7. Conversely, after sorting the assigned moving weight for the block, all restricted shape increments are dominant in the same frame. Indeed every root comparison has the same weak sign on every dominant increment as on their sum. The orthogonal central coordinates are simply retained. Schedules obtained in the blocks can thus be superposed. Their root inequalities are internal to the blocks, so no comparison with an outside coordinate is required.

The same assignments are available for a matrix splitting. Choose an oriented adapted orthonormal frame, and then choose representatives of each of the three matrices in its blocks. The assembled representatives belong to the original special-orthogonal orbits. For the negative end matrix, negate its representative to obtain the assigned end label for addition. Subgroup Weyl actions on these assignments are permitted in the full Weyl coset; a different initial representative can finally be returned to \(\lambda\) by a common Weyl action on the path.

Each type-\(D\) weight has a uniform coordinate fractional part, zero or one-half, invariant under signed permutations. The global root-lattice condition consequently makes every assigned coordinate difference an integer. On a permutation block, the central equation gives zero signed sum. Its difference lies in the permutation root lattice and has even ordinary sum as well. On an orthogonal block, changing allowed representatives changes the difference by a block root, so its parity is well-defined. The number of odd orthogonal blocks is even.

Even orthogonal blocks can be solved separately by induction; odd ones can be joined in pairs, using their direct-sum moment solution in the larger orthogonal block. If there are at least three orthogonal blocks, every such union is still proper. If there is a permutation block, all orthogonal blocks together occupy a proper subspace and have even total parity; solve that union and the permutation blocks separately. A single permutation block on all axes is covered by the first assertion of the theorem. A real two-dimensional orthogonal block has only its central equation and is even. Finally, a restriction of the moving coordinates retains the at-most-one-zero hypothesis. ◻

Proof. If the roots have deficient span, some connected component is balanced, possibly an isolated axis. Lemma 17 applies. Otherwise there are exactly \(l\) independent roots. Each component is then unbalanced and has as many edges as vertices, hence has a single cycle.

The roots of an unbalanced unicyclic component generate precisely its even-sum integer lattice. To see this, remove leaves successively. Their coefficients are determined integrally by the corresponding coordinate equations. The remaining cycle equations determine all cycle coefficients from one of them, with integral differences or sums; inconsistency of the signs closes the final equation with a factor two. The resulting determinant has absolute value two, so the image is the even-sum lattice. The coefficients are integral when the right-hand side has even coordinate sum and all half-odd-integral on the cycle when its coordinate sum is odd. This argument includes an unbalanced cycle with two edges.

A connected graph therefore already gives integral switch levels. Multiple components can be treated by Lemma 17, except for exactly two odd orthogonal components. These are the only possibility under the assumed failure of integral realizability. ◻

Proof. Delete zero-total, endpoint-time, and zero-gap switches by the simplifications in Definition 2. These operations preserve integrality. For the remaining strict switches, a small regular perturbation and chamber projection gives a strictly downward Bruhat step at each switch. Their number is therefore at most the length of the longest Weyl element. This bound is independent of the shape. There are finitely many ordered root lists and initial frames; after taking a subsequence they are constant and their times converge. The switch levels are bounded integers, hence are also constant on a further subsequence. Passing to the limit preserves the endpoint, weak signs, and these integer levels.

Let \(F\) be the smallest chamber face containing \(\mu\). Every increment of every allowed shape lies in \(F\): a simple-root pairing which vanishes on \(\mu\) is a sum of nonnegative increment pairings, each of which must vanish. Slight interpolation with the straight shape \(t\mu\) makes all slopes relatively interior to \(F\). Subdivision and perturbation of break values, keeping the endpoint, then impose any finite collection of open dense genericity conditions. The precise conditions used below are given in Section 4. ◻

Proof. A full-rank signed graph with \(c\) unbalanced components generates an integer lattice of index \(2^c\). Thus a graph generating \(Q(D_l)\) is connected and unbalanced. Deleting one edge leaves at most two components. Each nonzero maximal minor consequently has absolute value two or four: its components are unbalanced unicyclic graphs. Their gcd is two, the index of \(Q(D_l)\) in \(\mathbb Z^l\). Cofactors therefore give a primitive kernel vector with entries of magnitude at most two.

The velocity roots generate the same lattice as the position roots: \(\gamma_i\) equals \(\beta_i\) plus an integral combination of earlier roots, with a triangular inverse of the same form. Applying the preceding minor argument in that frame, and using the unimodular coefficient change (16), proves both derivative bounds. The endpoint right-hand side in (14) lies in \(Q(D_l)\), so it has an integer coefficient vector. Its difference from \(m^0\) is a multiple of the primitive relation. Since that relation has relatively prime entries and \(m^0\) is half-integral but not integral, the multiple is half-odd-integral. Adding either half relation makes all coefficients integral.

Finally, telescoping the reflections gives \[ \sum_i (L\cdot\gamma_i)\beta_i=(1-p_k)L. \tag{19}\] The determinant of \(p_k\) is \((-1)^{l+1}\). An orthogonal operator in dimension \(l\) without eigenvalue \(+1\) has determinant \((-1)^l\), by pairing its nonreal eigenvalues. Hence \(p_k\) has a nonzero fixed vector. Since the \(\gamma_i\) span, its pairings with them give a nonzero relation. The relation space is one-dimensional; scaling this fixed vector proves (18). ◻

Proof. Continuation may replace the ordered root list when times collide. We therefore follow coefficient lines in a finite set of root lists, not one fixed list. Each replacement preserves the endpoint and transforms the extrapolated coefficients at parameter one by an integral unimodular change. First we exclude removable switches at an interior parameter. We then use a common potential to select compatible times for alternative routes through a tied cluster. The local-link conditions start the motion at parameter zero, and compactness carries it to parameter one.

Proof. In the first case insert the cross swap at time zero, choosing the initial frame so that the old frame occurs just after the swap. Before the swap the velocity order is the strict inverse of its position order, so the gap is positive. Its coefficient is zero. The position graph has gained a bridge joining the two odd unbalanced components; its roots generate the full root lattice. In the primitive relation the bridge coefficient has magnitude two. Indeed, prune the off-cycle branches and follow the remaining connecting path: the cycle equations give equal coefficient magnitudes around each odd cycle, and their sign inconsistency makes the connecting flux twice that magnitude. Thus the primitive magnitudes are one on the cycles and two along the connecting path. The half-kernel derivative on the bridge has magnitude one. Choose its sign to make the new time move into \((0,1)\). All old gaps remain positive. The inserted edge can plainly be forgotten at the exact base time. The end case is the same construction with the old state before the new swap; choose the derivative to move its time leftward from one. Its base coefficient is the integral total root pairing.

For a crossing, insert the swap with its strict approaching inversion and reflect the entire old suffix. At the base point its gap is zero, its total is positive, and continuity holds because the crossing position is fixed by the reflection. In the velocity frame all old roots are retained. The two component root bases in that frame still generate their respective even-sum lattices, and the new root is a bridge between them. Hence the augmented roots generate \(Q(D_l)\), and the primitive half-kernel direction in the gap coordinates has nonzero, indeed unit, coordinate on the bridge. Choose its sign so that the new gap becomes positive. Its time remains interior. Formula (16), with zero base gap, shows that the inserted coefficient is half-integral; the other base coefficients remain half-integral. Forgetting is precisely reflection of the suffix back.

If the crossing lies at an old cluster unused by its two axes, its new root is orthogonal to all roots in that cluster. Commute these switches to put their nearby candidate times in order. This proves local feasibility also in the last case. Proposition 21 excludes each construction. ◻

Proof. Flags with scalar endomorphisms. For a profile \(m\), let \(\mathcal F(m)\) be the product of the three partial-flag varieties with the prescribed upper-level dimensions. Its complex dimension is \[ \dim_\mathbb C\mathcal F(m) =\frac{3r^2-\|m\|^2}{2}=r^2. \tag{24}\] The product of the corresponding Hermitian orbits has real dimension \(2r^2\). At an irreducible zero-sum triple, the sum map is submersive onto the traceless Hermitian matrices: the orthogonal cokernel consists of common commuting Hermitian matrices, which are scalar. Its local zero-sum fiber therefore has dimension \(r^2+1\), whereas simultaneous unitary conjugation has orbit dimension \(r^2-1\). Irreducibility is open. It follows that these fixed spectra admit arbitrarily many pairwise unitarily inequivalent irreducible zero-sum triples.

Their upper-level flags have scalar endomorphisms, and flags from two inequivalent triples have no nonzero morphism. Indeed, let \(L\) preserve all three upper-level filtrations, from a triple \((X_i)\) to a triple \((X'_i)\). In spectral coordinates its nonzero matrix entries go from a level \(a\) only to levels \(a'\ge a\). Consequently \[\mathop{\mathrm{tr}}(L^*X'_iL-L^*LX_i) =\sum_{a,a'}(a'-a)\|L_{a',a}\|^2\ge0.\] The sum of these three quantities is zero. Thus every nonzero entry has \(a'=a\), and \(X'_iL=LX_i\) for all \(i\). Irreducibility now gives the assertions. Here an invertible complex intertwiner between irreducible Hermitian triples is a scalar multiple of a unitary intertwiner, since \(L^*L\) commutes with the source triple.

Direct sums fill an open set. The Hermitian triples have supplied Hom-free spectral flags. We now work in the complex flag varieties: the configurations in the direct-sum families need not themselves arise from Hermitian zero-sum triples. For every positive integer \(b\), direct sums of \(b\) configurations in \(\mathcal F(m)\) having scalar endomorphisms, with arbitrary changes of frame, contain a nonempty Zariski-open subset of \(\mathcal F(bm)\). To prove this, choose \(b\) pairwise Hom-free configurations as above and examine the differential of the direct-sum map, including the action of \(\mathop{\mathrm{GL}}(br,\mathbb C)\). The flag tangent space splits into ordered blocks indexed by the summands. Its diagonal blocks are supplied by variation of the individual flags. Each off-diagonal ordered block has dimension \(r^2\): the contribution of its \(i\)-th flag is \[\sum_{a<b}m_i(a)m_i(b) =\frac{r^2-\sum_a m_i(a)^2}{2},\] and summing gives (24). Change of frame maps a space of \(r\times r\) matrices to that block; its kernel is exactly the space of filtration-preserving maps between the two summands, which is zero. Hence this differential is surjective. The direct-sum map is therefore dominant. Chevalley’s theorem makes its image constructible (The Stacks Project Authors 2026, Tag 054K); a dense constructible subset of an irreducible variety contains a nonempty Zariski-open subset. Requiring scalar endomorphisms on the summands is an open condition and includes our chosen point, so the same assertion holds with this requirement.

We finish with the scalar-endomorphism case of the Krull–Schmidt uniqueness argument; compare (Atiyah 1956, Theorem 1). Choose positive integers \(b,b'\) with \(bm=b'm'\). The two nonempty open subsets just obtained intersect in the irreducible variety \(\mathcal F(bm)\). One configuration in their intersection admits two direct-sum decompositions \[V=\bigoplus_i V_i=\bigoplus_j W_j,\] whose summands have scalar endomorphisms, with dimensions \(r\) and \(h\), respectively. Inclusions and projections preserve all three flags. On a fixed \(V_i\), the sum of the composites \(V_i\to W_j\to V_i\) is the identity. One composite is therefore a nonzero scalar. Write it as \(gf=c\,\mathrm{id}_{V_i}\), \(c\ne0\). The endomorphism \(fg\) of \(W_j\) is also scalar, and \((fg)f=cf\) shows that its scalar is \(c\). Thus \(f\) is an isomorphism, so \(r=h\). Proportional profiles of equal size are equal. ◻

Proof. Choose skew-symmetric zero-sum triples from the separate moment solutions in \(P\) and \(Q\), supplied by Lemma 7. Let \(m^P,m^Q\) be the profiles of \(i\) times these triples. Since the spectra of a real skew matrix occur in opposite pairs, \[ p_\lambda+p_\mu+p_\xi=\frac{m^P\cdot m^Q}{2}. \tag{26}\] A common nonzero skew operator commuting with either triple would give a forbidden splitting by Lemma 17. Thus both common skew stabilizers vanish.

The norm bound. Consider one block of real dimension \(r\), write its matrices as \(A_1,A_2,A_3\), and put \(z_i=\dim\ker A_i\). For one matrix, a kernel of dimension \(z_i\) contributes \(z_i(z_i-1)/2\) to its skew stabilizer; a positive rotation level of multiplicity \(k\) contributes \(k^2\). The squared profile norm instead has contributions \(z_i^2\) and \(2k^2\). Hence twice the sum of the three stabilizer dimensions is \(\|m\|^2-\sum_i z_i\). The orbit sum map is submersive onto \(\mathfrak{so}(r)\), because its orthogonal cokernel is the common skew stabilizer. Its kernel contains the simultaneous-conjugation tangent, of dimension \(\dim\mathop{\mathrm{SO}}(r)\). Comparing dimensions gives \[ \|m\|^2-\sum_i z_i\le r^2-r. \tag{27}\]

If the joint kernel is zero, the map \[u\longmapsto(A_1u,A_2u,A_3u)\] is injective into the kernel of addition from \(\bigoplus_i\mathop{\mathrm{im}}A_i\) to \(\mathbb R^r\). This addition map is onto: the orthogonal complement of the sum of the images is the joint kernel. Its kernel has dimension \(2r-\sum_i z_i\), so \(\sum_i z_i\le r\). Equation (27) yields \(\|m\|\le r\). The same proof works in odd dimension.

A joint kernel of dimension at least two would support a nonzero commuting skew operator. A possible joint kernel is therefore a single line, and it must belong to the block containing the sole possible zero of the moving weight. Remove this line. The reduced triple has neither a joint kernel nor a common skew stabilizer, so \[\|m\|^2 =\|m_{\mathrm{red}}\|^2+2\sum_i z_{i,\mathrm{red}}+3 \le(r-1)^2+2(r-1)+3=r^2+2.\] The moving zero-level entry in the original profile has multiplicity two and pairs with zero in the other block’s profile. Deleting that entry for the dot product improves the relevant squared bound to \(r^2-2\). Cauchy–Schwarz is then strict. If there is a moving zero but no joint kernel, the bounds \(\|m^P\|\le r\), \(\|m^Q\|\le h\) are already strict in Cauchy–Schwarz, since the moving zero entry prevents proportionality. Thus \(m^P\cdot m^Q<rh\) whenever the moving weight has a zero.

Excluding equality without a moving zero. The weak bound \(m^P\cdot m^Q\le rh\) holds by the same estimates. Equality forces proportional profiles, \(\|m^P\|=r\), \(\|m^Q\|=h\), and \(\sum_i z_i=r\) in the first block, with the analogous equality in the second. The moving matrices are invertible. Each of the other two matrices has a kernel: if one were invertible, the other would vanish, and the remaining pair \(A,-A\) would have \(A\ne0\) in its common skew stabilizer.

Both triples are absolutely irreducible. Indeed a proper common real invariant subspace has an invariant orthogonal complement. Each has even dimension, since the moving restriction is invertible and skew. Together with the other original block this gives at least three orthogonal blocks, contrary to Lemma 17. A proper complex invariant subspace for the Hermitian triple gives a nonscalar commuting Hermitian projection \(B+iD\), with \(B\) real symmetric and \(D\) real skew. Both commute with the real skew triple. A nonscalar \(B\) gives the forbidden real splitting, while nonzero \(D\) gives the forbidden skew stabilizer.

Lemma 23 now gives \(r=h\) and identical profiles, including their numerical levels. Replace the second skew triple by an orthogonal copy of the first. Choose the isometry’s orientation to match the moving special-orthogonal orbit. The other two matrices impose no additional orientation restriction: each has a kernel, and reflection in a kernel line is an orientation-reversing commuting isometry. Thus their orthogonal and special-orthogonal orbits coincide. On the sum of the two isometric copies, the skew operator rotating one copy into the other commutes with the triple. It gives a unitary, hence permutation-type, block in Lemma 17, again a contradiction. Equality is impossible. Equation (26) proves (25). ◻

Proof. The equality of moving values on distinct axes means that their assigned shape-coordinate forms agree on the whole minimal chamber face. Hence their shape values agree for every time. The initial coordinates have the same fractional part, and equality of the current positions implies that either both phases are half or neither is.

At any point of the plateau, reflect the entire suffix by the reflection exchanging the two signed axes, including every later position and switch root. This reflection fixes the position at the cut and fixes the current-frame shape \(w\eta(t)\) for every \(t\), since the two coordinate forms agree. Thus no new switch is needed at the cut, and the reflected endpoint remains in its original Weyl orbit. In the root graph, cut each involved vertex into its past and future copies, then identify opposite-time copies crosswise. If both phases are half, each cut opens its cycle into a connected tree. The crosswise identification gives 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, possibly an isolated vertex. The core piece uses the past copy at phase one and the future copy at phase zero. Opposite phases therefore join the two cores and leave a separate tree component. That too is excluded by Lemma 18. Only (28) remains. ◻

Proof. Every transfer sends \(H\) from a donor of lower position to a recipient of higher position. In an ordinary class, decompose this flow into directed paths from its net donors to its net recipients. Positions strictly increase, so no directed cycle occurs. Shortcut each path by its direct transfer. The resulting matching has the same final high and low assignments and legal positive gaps; see Equation (21). If the original transfers were not already disjoint pairs, this uses fewer switches, contrary to Lemma 18.

In an opposite class each physical swap flips two axes, producing two mirrored transfers. Mark each axis by its initial high sign. At any edge occurrence, the sum of the positions in the two currently high signs is strictly negative. At each vertex, pair consecutive incidences, leaving the first unpaired if their number is odd. The signed-position terms in each such pair cancel, because the high sign alternates. The paired incidences organize the edges into paths and cycles. A cycle would sum strict inequalities to \(0<0\), and cannot occur. On a path the sum leaves just the positions of its two ends in their initial high signs, with negative sum. A direct flip of these two axes is therefore feasible. Distinct paths have disjoint unpaired ends; their flips realize exactly the original odd-incidence axes, hence exactly the same final sign assignment. Unless the old switches were already physically disjoint, this again reduces their number. ◻

Proof of Proposition 26. Comparisons across distinct form classes are unchanged by the cover property. Within a class, Lemma 27 reduces all comparisons to the single–single, pair–single, and pair–pair cases just proved. Each possible loss either gives a feasible extra-edge link, forbidden by Proposition 21, or is compensated by its plateau contribution. The direct crossing links exclude decreases away from knots. Summing gives the assertion. ◻

Completion of the proof of Theorem 16. Proposition 24 and (30) imply \(J(0+)>J(1-)\), contradicting Proposition 26. This proves integral realizability for the generic shapes used above. The uniform bounded-switch closedness of Lemma 19 passes to an arbitrary shape with the same endpoint. Together with the permutation case and the rank reduction of Lemma 18, this completes that theorem. ◻

Proof. Put \(h_i=x_0\cdot\alpha_i\) and \(h'_i=x_1\cdot\alpha_i\). By Lemma 4, the functional must annihilate \(f_i^{h_i+1}V(\omega)\) for every \(i\). A vector contributing to its weight-\(\delta\) component would have source weight \(\delta+(h_i+1)\alpha_i\), whose simple-root pairing is \[\bigl(\delta+(h_i+1)\alpha_i\bigr)\cdot\alpha_i =h_i+h'_i+2.\] All weights of \(\bigwedge^k E\) 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 \(\alpha_i\in J\), and invariance annihilates \(f_iV(\omega)\). ◻

Proof of Proposition 28. Apply Lemma 29 and perform the preceding construction on each column. Write its endpoints as \(h_{j-1},h_j\), its weight as \(\omega_j\), and choose the resulting intertwiner \[V(h_{j-1})\otimes V(\omega_j)\longrightarrow V(h_j).\] 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, \(v_j\ge u_{j+1}\). Thus the chosen future flag dominates the next past flag: directly for an exterior column, and through \(u_j\ge v_j\ge u_{j+1}\) for a minuscule column. Lemma 5 therefore makes the composition nonzero on the Cartan summand \[V\Bigl(\sum_j\omega_j\Bigr)=V(\mu) \subset\bigotimes_jV(\omega_j).\] The first and last dominant positions are \(h_0=\lambda\) and \(h_q=\xi\), proving the assertion. For an empty list the folded path is constant, so \(\lambda=\xi\) and the identity map suffices. ◻

Proof. We recall precisely the semicanonical facts used here. On the preprojective varieties, convolution of invariant constructible functions is defined by integration with compactly supported Euler characteristic over submodules, with the submodule as the right factor. The constant function on the unique representation of dimension \(e_i\) represents \(f_i\). This identifies \(U=U(\mathfrak n^-)\) with the algebra generated by these functions; we identify the positive enveloping algebra in the cited conventions with the negative one by the Chevalley isomorphism. In each degree, generic evaluation on irreducible components is a vector-space isomorphism \[ U_d \longrightarrow \mathbb C^{\operatorname{Irr}\Lambda(d)} . \tag{47}\] Moreover, the image of \((f_i^pU)_d\) is the coordinate subspace supported on components whose generic \(i\)-top is at least \(p\). The image of \((Uf_i^p)_d\) is the analogous subspace for generic \(i\)-socle. Finite Dynkin type needs no additional nilpotence condition. For the constructible-function realization and generic component coordinates, see (Geiss et al. 2005, Proposition 3.1 and Sections 5.2–5.6); for their compatibility with the top and socle filtrations, see (Fang et al. 2023, Lemmas 5.1, 5.4, 5.6, and 5.9).

Here is the geometry behind the filtration statement, which also fixes the left/right convention. The preprojective equations describe the union of conormals to the finitely many ordinary-quiver orbits. Its components have dimension \[D(d)=\sum_{\{i,j\}\in\mathcal D}d_i d_j\] and are counted by the Kostant partitions of \(d\), hence by \(\dim U_d\). On the stratum with \(i\)-top exactly \(p\), remove the top by replacing \(M_i\) with its incoming image, of dimension \(a=d_i-p\). The lower representation has \(i\)-top zero. Conversely, put \(S=\sum_{j\sim i}d_j\). After choosing the retained \(a\)-plane, extensions add outgoing maps on the remaining \(p\) dimensions with values in the kernel of the signed incoming map, of dimension \(S-a\). Their parameter count, including the Grassmannian, is \[p(S-a)+pa=pS=D(d)-D(d-pe_i).\] The other relations are unchanged, because a path entering vertex \(i\) enters its retained image. This bundle construction gives the component correspondence for exact top \(p\).

At exact top \(p\), the submodule with quotient \(S_i^p\) is unique, so multiplication by \(f_i^{(p)}=f_i^p/p!\) transfers the generic value from the lower component. It vanishes when the top is smaller than \(p\). Descending induction on the top removes any contributions on larger-top components. Together with induction on dimension and the component count, this explains both (47) and its left filtration. Transposition of the arrows gives the right filtration and socle. Thus the same generic-coordinate basis is compatible with all the subspaces just described.

By Lemma 4 and the highest-generator presentation of \(V(\mu)\), the multiplicity in question is the dimension of \[ U_d\Big/ \left( \sum_i f_i^{\lambda_i+1}U+ \sum_i U f_i^{\mu_i+1} \right)_d . \tag{48}\] Under (47), this quotient is nonzero exactly when there is a component on which all the generic top and socle bounds in (46) hold. Those bounds are open rank conditions. Their intersection on such a component is nonempty; conversely, any point satisfying them lies on a component where they hold generically. This proves the criterion. ◻

Proof. Fix \(b\) and suppress it from the notation. For each vertex, write \[F_i^*(b)=\ker t_i^*\cap\mathop{\mathrm{im}}(t_i^*)^{b-1}\] for its dual high-socle space. At a trunk vertex \(i<p\), the two incoming maps give \[\begin{aligned} \dim\bigl(\operatorname{Ann}(\mathop{\mathrm{im}}u_i)\cap F_i^*(b)\bigr) &=K_i^T,\\ \dim\bigl(\operatorname{Ann}(\mathop{\mathrm{im}}v_{i+1})\cap F_i^*(b)\bigr) &=q_i-K_{i+1}^{\prime T}. \end{aligned}\] These identities are read directly from the alternating rows: the left image fails to contain the right-hand top exactly when \(T=1\), whereas the right image fails to contain a left-hand top exactly when \(T=0\). The intersection of the two annihilators is bounded by the full vertex top. The dimension inequality therefore gives \[ K_i^T\le L_i-L_{i+1}+K_{i+1}^{\prime T}\qquad(i<p). \tag{53}\]

At \(W\), the space \(F_p^*(b)\) splits into its two colors, since \(t_p\) is off diagonal. Let their dimensions be \(q_+,q_-\), so \(q_++q_-=q_p\). The left incoming-image annihilator has intersection dimension \(K_p^T\) with \(F_p^*(b)\). Its intersection with either color is bounded by the corresponding leaf top, whence \[K_p^T-q_-\le L_p+L_n,\qquad K_p^T-q_+\le L_p-L_n\] after naming the colors accordingly. Adding proves \[ K_p^T-\frac{q_p}{2}\le L_p. \tag{54}\] No equality of the old color dimensions is required.

For each edge, (51) also gives \[ K_i^{\prime T}-\frac{q_{i-1}}2 \le K_i^T-\frac{q_i}2 . \tag{55}\] Indeed, the contribution of one row to right minus left is \[\left(\mathbf1_{\{h_R\ge b\}}-\mathbf1_{\{h_L\ge b\}}\right) \left(T-\frac12\right).\] It is zero if the indicators agree. If a row is gained on the right, then \(T=S=1\); if it is lost, then \(T=S=0\). In either case its contribution is \(1/2\).

Combining (53) with (55) for edge \(i+1\) gives \[K_i^T-\frac{q_i}2 \le L_i-L_{i+1}+K_{i+1}^T-\frac{q_{i+1}}2 .\] Starting from (54) and descending proves the top assertion. At a trunk vertex, on the primal high-socle space \(\ker t_i\cap\mathop{\mathrm{im}}t_i^{b-1}\), the outgoing kernels \(\ker v_i\) and \(\ker u_{i+1}\) have intersection dimensions \(K_i^S\) and \(q_i-K_{i+1}^{\prime S}\), respectively. Their common intersection is bounded by the vertex socle. The same leaf estimate and row comparison, with \(T,L\) replaced by \(S,M\), therefore prove the socle assertion. ◻

Proof. Each term on the right is a lower bound by the dimension inequality inside \(F_b\). To attain all the bounds, choose coordinates adapted to the filtration. A basis for a subspace with profile \(a_b\) consists of \(a_b-a_{b+1}\) generic vectors with allowed coordinates in \(F_b\), for each \(b\); the analogous description applies to \(C\). Such bases have the required profiles on a nonempty open set.

The coordinate supports of the combined list of basis vectors are nested. This is the nested-support case of Hall’s distinct-representatives criterion (Hall 1935, Theorem 1). A maximum matching of these vectors to distinct allowed coordinates leaves exactly \[\max\left(0,\max_b(a_b+c_b-f_b)\right)\] unmatched vectors. Indeed any collection of vectors has, as its union of allowed coordinates, its largest allowed filtration level. For level \(F_b\), at most \(a_b+c_b\) vectors are constrained to that level or deeper. These are all the matching constraints, and they are attained by matching from the deepest level outward. A matching gives a nonzero minor, so the generic rank of the combined list has exactly this deficiency. Since each separate list is independent, the deficiency is its intersection dimension.

Finally, \(P\) is transitive on subspaces with a prescribed filtration profile: extend a filtration-adapted basis of the subspace to one of \(H\). We may therefore hold \(A\) fixed and vary only \(C\), which gives (56). Maximal rank is an open condition. ◻

Proof. Write \(V\) as a sum of Jordan row modules \(\mathbb C[t]/(t^a)\). 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 \(a\le b\). In that case, sending its top to a scalar multiple of \(t^{b-a}\) times the target top gives any desired bottom coefficient. These are precisely the matrix entries permitted by the filtration \(F_b(t)\). Every filtration-preserving bottom matrix therefore lifts to a commuting endomorphism. If its action on \(\ker t\) is invertible, the lift is invertible: a nonzero \(t\)-stable kernel would contain a nonzero vector in \(\ker t\). This proves the first assertion. Apply the same argument to \(t^*\) and use inverse transpose for the dual assertion.

Finally, \(C(t)\) is the determinant-nonzero open subset of the vector space \(\mathop{\mathrm{End}}_{\mathbb C[t]}(V)\), so it is irreducible. ◻

Proof of Proposition 31. Occurrence implies that the simple-root coefficients of \(\lambda+\mu-\xi\) are nonnegative integers, so choose a module from Proposition 32. Keep the chain through \(V_r\) and take two copies of it. The doubled space \(\widetilde W=V_r\oplus V_r\) will be the new combined leaves. Write \(\widetilde t=t_r\oplus t_r\), \(\widetilde u=u_r\oplus u_r\), and \(\widetilde v=v_r\oplus v_r\).

Color every Jordan row of \(t_r\) alternately, and give its duplicate the opposite colors. Thus \(\widetilde t\) is off diagonal for the new splitting \(\widetilde W=\widetilde W_+\oplus\widetilde W_-\). 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 \(b\le a\); 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 \(g\in C(\widetilde t)\) preserves all these properties.

We choose \(g\) to ensure the four leaf bounds. The subspace \[A=\operatorname{Ann}(\mathop{\mathrm{im}}\widetilde u)\] lies in \(\ker\widetilde t^*\), because \(\mathop{\mathrm{im}}\widetilde t\subseteq\mathop{\mathrm{im}}\widetilde u\). Its dual filtration profile is \(2K_r^T(b)\). Each dual color has profile \(q_r(b)\). By Lemmas 34 and 35, a generic coloring has, in either dual color, intersection with \(A\) of dimension \[\max\left(0,\max_b\{2K_r^T(b)-q_r(b)\}\right) \le 2L_r ,\] where the last inequality is Lemma 33. This is exactly the new leaf top: a functional on a color annihilates the projected incoming image if and only if its extension by zero to the other color annihilates \(\mathop{\mathrm{im}}\widetilde u\).

Similarly, \(\ker\widetilde v\subseteq\ker\widetilde t\) 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 \(C(\widetilde t)\). Since this group is irreducible, all four hold simultaneously. Surjectivity onto each of the primal and dual filtration stabilizers suffices here; the two actions need not be independently prescribable.

Let \(P_\pm\) be the resulting color projections, and define the new leaf arrows by \[u_\pm=P_\pm\widetilde u,\qquad v_\pm=\widetilde v|_{\widetilde W_\pm}.\] Their return maps are zero because \(P_\pm\widetilde u\widetilde v|_{\widetilde W_\pm}=0\). At the preceding trunk vertex, \[v_+u_++v_-u_-= \widetilde v(P_++P_-)\widetilde u =\widetilde v\widetilde u .\] The relation, incoming-image sum, and outgoing kernel at that vertex are therefore unchanged from the doubled chain. All other trunk bounds simply double.

It remains to check the dimension vector. For \(i<r\), the new trunk dimension is \(2\sum_{j\le i}\Delta_j\). The two leaf coefficients for the \(D_{r+1}\)-weight \[(2\Delta_1,\ldots,2\Delta_r,0)\] are both \(\sum_{j\le r}\Delta_j=\dim V_r\), precisely the two new color dimensions. The new leaf top bounds are \(2L_r\), and the socle bounds are \(2M_r\), as required. Applying Proposition 32 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 \(D_2=A_1\times A_1\). The same bounds and dimension calculation apply. ◻

Proof of Theorem 1. Suppose first that the invariant at scale \(N\in\mathbb Z_{>0}\) is nonzero. Set \(\xi=\nu^*\), so \(V(N\xi)\) occurs in \(V(N\lambda)\otimes V(N\mu)\), 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 \(\mu\). Lemma 6, followed by division by \(N\), gives a real moment solution for \(\lambda,\mu,\xi\). Choose the shape \(\eta\) made of successive fundamental columns summing to \(\mu\). Theorem 16 gives an integral schedule for this shape, and Proposition 28 gives \(V(\lambda)\otimes V(\mu)\longrightarrow V(\xi)\) 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 \(r\le n-2\). If \(r=0\), all three representations are trivial. Assume \(r\ge1\). Apply Proposition 31 to the occurrence at scale \(N\). It gives the occurrence in \(D_{r+1}\) for the truncated labels at scale \(2N\). Lemma 6, 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 \(M_{r+1}=\cdots=M_n=0\), its fundamental expansion uses only columns \(e_1+\cdots+e_k\) with \(k\le r\); 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; Theorem 16 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. Proposition 28 again gives the unscaled invariant. The use of \(2N\) supplied only real existence in smaller rank and has introduced no saturation factor.

Conversely, let an unscaled invariant functional be 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 \(D_2=A_1\times A_1\) and \(D_3=A_3\) with the stated conventions. If \(\mathop{\mathrm{Spin}}(2)\) 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. ◻