A negative solution to the finite lattice representation problem

Proof. For D\le U\le V\le X, the normality of R and the subgroup modular identity give

If D<DR<X, apply the fence to DR and its comparable complements U<V. Equation (2.1) would give V=U. Thus DR is an endpoint. In a modular lattice the same identity holds with any element in place of DR, so the existence of a fence also excludes modularity. Finally, taking inverse images identifies [D/F,X/F] with [D,X].□

Proof. Factor out F_X, so that the action of X on the cosets of D is faithful. Lemma 2.2 says that every nontrivial normal subgroup supplements D, or equivalently is transitive in this action. Let M be a minimal normal subgroup. If M were abelian, (2.2) would identify [D,X] with an interval of D-invariant subgroups of an abelian group. The latter lattice is modular: intersection is its meet and subgroup product is its join. This contradicts Lemma 2.2.

If M' were another minimal normal subgroup, then [M,M']\le M\cap M'=1. Every D-invariant subgroup of M would therefore be normalized by DM'=X, and so would be either 1 or M. The correspondence (2.2) would leave at most two elements in [D,X], again a contradiction.

A finite minimal normal subgroup is characteristically simple, hence a direct product of isomorphic simple groups. Since M is nonabelian, these simple groups are nonabelian. Conjugation by X is transitive on the direct factors: otherwise the product of one orbit of factors would be a proper nontrivial normal subgroup of X contained in M. This proves (2.3). Every nontrivial normal subgroup contains a minimal normal subgroup and therefore contains M. Its centralizer is normal and intersects M in Z(M)=1, so that centralizer is trivial. The assertion about soluble normal subgroups follows as well.

Finally X=DM. Inner automorphisms of M fix each of its simple direct factors. Thus D and X have the same image on their factor set. Restoring F_X proves the statements as written.□

Proof. The subgroup R\cap B is normal in B. Suppose first that R\cap B\le D. If B\le DR, then

a contradiction. Comparability therefore gives DR\le B, whence R=R\cap B\le D. Its contrapositive, followed by Lemma 2.2 in B, gives the asserted supplementation. The image of R\cap B modulo F_B then contains the nonabelian socle S_B, by Lemma 2.3. This is impossible if R is soluble. For the final assertion, F\le F_B because F is normal in B and contained in D. If both R and Q escaped D, their intersections with B would both map over S_B, while their images would commute. This would make S_B abelian.□

Proof. Induct on |I|, separating the last coordinate. The projection to the other coordinates is, by induction, a product T^r of diagonal strips. We have a subdirect subgroup of T^r\times T. The kernels of its two projections are normal in the respective factors. The kernel on the last T is either T, in which case this is the full product, or 1, in which case the subgroup identifies a quotient of T^r with T. Normal subgroups of T^r are products of its simple direct factors, so that quotient is one of the factors. The last coordinate is therefore joined to exactly one existing strip.

Two coordinates belong to the same strip precisely when the projection of E to their product is the graph of an isomorphism rather than the full product. This characterizes the partition and its links, and also proves the assertion about inclusion.□

Proof. The normal subgroup F_Y of X is contained in D, so F_Y\le F_X. Since Y=DN_Y and D\le X,

The image of E_X in X/F_X is a nontrivial normal subgroup and therefore contains S_X. Every simple composition factor of a subgroup of a direct product is a section of one of the factors: projection to one coordinate gives an exact sequence whose kernel lies in the product of the remaining coordinates, and induction applies. Thus T_X is a section of T_Y.

If T_X\cong T_Y, some coordinate projection of E_X is all of T_Y; otherwise none of those smaller groups could have a composition factor of order |T_Y|. The transitive action of D on I_Y makes all the projections surjective. Conversely, suppose E_X is subdirect. Lemma 2.5 expresses it as a product of diagonal copies of T_Y. The action of D on those copies is transitive. Since X/F_Y=(D/F_Y)E_X, the subgroup E_X is minimal normal in X/F_Y. Its intersection with F_X/F_Y is either trivial or all of E_X. The latter would imply X=D, which is excluded. Its image is therefore the unique minimal normal subgroup S_X, proving the equivalence and (2.4).

There are m_X strips on m_Y coordinates. If m_X=m_Y, every strip is a singleton, so E_X=S_Y and X=Y. Thus m_X<m_Y. The quotient identification with S_X assigns to every strip its actual simple factor in I_X and defines (2.5). The subgroup E_X is normal in X/F_Y, and its quotient identification with S_X commutes with conjugation by X; hence this support map is X-equivariant.

For composition, factor out F_Z and let E_Y=Y\cap N_Z. We have N_Y=F_YE_Y and F_Y\cap E_Y=1. Since F_Y\le X,

Thus the subdirect subgroup used for X<Y is the image of the one used for X<Z under the already specified identification E_Y\cong S_Y. Refinement of its strip partition is exactly composition of the support maps.□

Proof. Factor out F_Y, and let C be the product of the coordinate projections of B. It is X-invariant. The correspondence (2.2), with bottom X and normal subgroup S_Y, identifies [X,Y] with the X-invariant subgroups of S_Y containing B.

Suppose B<C<S_Y. If U<V are comparable complements of XC, put E_U=U\cap S_Y and E_V=V\cap S_Y. Their projections already contain the projections of C. Hence projecting \langle E_U,C\rangle=S_Y shows that E_U is subdirect; the same holds for E_V.

Every link within a strip of E_U carries the projection of B in one coordinate onto its projection in the other, because B\le E_U. Consequently E_U\cap C is a product of independent diagonal copies of those nontrivial projections, one for each strip of E_U. The proper inclusion E_U<E_V splits at least one strip, by Lemma 2.5. Intersecting with C then permits an additional independent nonidentity coordinate value, and gives E_U\cap C<E_V\cap C. This contradicts the required equality of both intersections with B. A fence therefore forces B=C. If X is a coatom, B<C<S_Y directly contradicts maximality.

A proper subdirect subgroup has a link between two distinct coordinates. The full product P^{I_Y} contains an element which is identity in the first of these coordinates and nonidentity in the second, contradicting that link. Now let Z be a coatom above X. If its projections were full, it would be such a proper subdirect subgroup. Otherwise its product closure is proper, and maximality forces it to equal that closure. Intersections of products are products.

Finally, for different labelled simple types, the projections are nontrivial because their image contains S_X, and transitivity makes all of them nontrivial. If one were full, all would be full, contradicting Proposition 2.6.□

Proof. For a coatom M above X, Proposition 2.7 writes its intersection with S_Y as P_M^{I_Y}, with 1<P_M<T_Y. Let J=H_*\cap T_Y. It normalizes P_M, so

The last inequality follows from simplicity and 1<P_M<T_Y. Both P_M and J are H_*-invariant, and their product therefore transports to a saturated overgroup of M which is still proper in Y. Maximality gives J\le P_M. Intersecting over the coatom representation of X gives J\le P_X; the reverse inclusion is immediate. This proves (2.6).

Products above P_X^{I_Y} correspond precisely to H_*-invariant subgroups P_Z\le T_Y containing P_X. By (2.6), these in turn correspond precisely to the subgroups K_Z of H_*T_Y containing H_*: the two maps are P_Z\mapsto H_*P_Z and K_Z\mapsto K_Z\cap T_Y. Since Z=X P_Z^{I_Y} and the product fixes the coordinate set, Z_i=X_iP_Z^{I_Y}, proving (2.7). Intersections and generated subgroups of coordinate products are again coordinate products, proving the sublattice assertion. Any intermediate projected subgroup would lift to an original intermediate saturated subgroup, which proves the assertion about covers. At a strict cover Proposition 2.7 supplies the product hypothesis, and the bottom is itself a coatom. The ambient group H_*T_Y is a subgroup of \operatorname{Aut}(T_Y) containing T_Y, hence is almost simple.□

Proof. Factor out F_Y throughout the proof. Put

Let Q_i be the projection of W to the ith coordinate, and let \widehat Q=\prod_iQ_i. These projections are transported into one another by D, and each is invariant under its coordinate stabilizer. Thus \widehat Q is normal in Z.

We claim that W=\widehat Q. For q,q'\in Q_i, choose w\in W whose ith coordinate is q and let e_i(q')\in E be supported only on coordinate i. Since W\trianglelefteq E,

It follows that [\widehat Q,\widehat Q]\le W. The image of \widehat Q in Z/F is therefore the abelian normal subgroup \widehat Q/W. Lemma 2.3 forces that image to be trivial. Hence W=\widehat Q, proving (2.8). This argument also covers Z=X at a covering comparison; it does not use a projected core with bottom equal to the whole projected group.

The normal subgroup EF/F of Z/F is nontrivial, since Z=DE and Z>D. It contains S_Z. Therefore, if L=N_Z\cap E, then

In particular L/W is a perfect normal subgroup of E/W=\prod_i(P_i/Q_i).

Every perfect normal subgroup V of a direct product \prod_iB_i is the product of its intersections with the factors. To see this, for v,w\in V the element [v,e_i(w_i)] lies in V\cap B_i and is the ith coordinate of [v,w]. Hence V'\le\prod_i(V\cap B_i)\le V, and V=V' gives equality. Apply this observation to L/W. It splits as a product of its coordinate intersections, each a direct product of copies of T_Z. Taking full inverse images gives L=R_Z^{I_Y} and (2.9)–(2.10). Conjugation by Z permutes these coordinate intersections and commutes with L/W\cong S_Z, so the resulting support map is Z-equivariant. Transitivity of D on I_Y makes the multiplicity a_Z the same in every coordinate.

Any K_Z-normal subgroup strictly between Q_Z and R_Z would transport to a Z-normal subgroup strictly between W and L. Its image would be a proper nontrivial normal subgroup of Z/F inside S_Z. This proves the chief-factor assertion and also the transitivity of K_Z on the a_Z simple factors of R_Z/Q_Z.

We have Z=DN_Z=DL. Since L\le N_Y fixes I_Y, for a chosen coordinate i this gives

The group R_Z acts by inner automorphisms on R_Z/Q_Z and fixes its simple factors. Thus \rho(D_i) and K_Z have the same image on that factor set. The kernel of \rho centralizes the ambient coordinate and so acts trivially on this section. This identifies the action with the original fiber action and proves (2.11) and its assertions about stabilizers.

For the additional projected-core assertion, let C_Z=\operatorname{core}_{K_Z}(H_*). The subgroup Q_Z is normal in K_Z and lies in P_X, because F_Z\le D\le X. By (2.6) it lies in H_*, and therefore Q_Z\le C_Z\cap P_Z. Conversely, transport C_Z\cap P_Z over all coordinates. Its product is normal in Z and is contained in P_X^{I_Y}\le X<Z. Lemma 2.2 for the original interval [D,Z] forces this product into D, hence into F_Z. Equation (2.8) now gives the reverse inclusion.

Since \rho(D_i)\le H_*\le K_Z and K_Z=\rho(D_i)R_Z, we obtain K_Z=H_*R_Z. Also R_Z\cap C_Z=Q_Z. The resulting image of R_Z modulo C_Z is minimal normal by the chief-factor assertion and is transitive by supplementation. Its factor action is the same for H_* and K_Z, because it is already induced by \rho(D_i).

Finally suppose a_Z=1. The support map (2.11) is then a bijection. The subgroup E fixes I_Y and consequently fixes I_Z. Conjugation embeds EF/F faithfully in \operatorname{Aut}(S_Z), since C_{Z/F}(S_Z)=1 by Lemma 2.3. Its image therefore lies in \operatorname{Aut}(T_Z)^{I_Y} and contains the inner subgroup S_Z. The quotient E/L\cong(P_Z/R_Z)^{I_Y} embeds in \operatorname{Out}(T_Z)^{I_Y}. Schreier’s Theorem, a consequence of the classification of finite simple groups, says that \operatorname{Out}(T_Z) is soluble; see also the explicit use in [28, Section 1]. Hence P_Z/R_Z is soluble.□

Proof. We spell out the factor actions. A subspace or paired-flag stabilizer has at most a bounded number of Levi components. A tensor product of two spaces has two component groups. Extension-field, subfield, form, and almost simple cases have one, except for the bounded low-dimensional coincidences. A direct-sum decomposition or a tensor-induced decomposition has the full symmetric action on its blocks, with a possible alternating restriction from the determinant or spinor condition. A block contributes at most a bounded number of simple factors; for instance an orthogonal plus-type four-space contributes two. Taking a point inside a block and then the block itself accounts for this extra bounded level. In the alternating case, the intransitive and affine alternatives have bounded component actions, while the imprimitive, diagonal, and product-action alternatives have this same full block action. Exceptional types have bounded rank and hence bounded numbers of components. Central, determinant, and form quotients do not split a simple component into more factors. This proves (C1); the explicit \mathcal C_2 and \mathcal C_7 component quotients can also be read in [10, proofs of Lemmas 5.4 and 5.9].□

Proof. We induct on r. The assertion for one step follows from primitivity of a natural giant, or from the bounded index in the other case. For the induction put N=\operatorname{core}_H(H_1). For an arbitrary U\in[A,H], the group A(U\cap N) lies in [A,H_1], and its intersection with N is U\cap N. The induction hypothesis therefore controls the intersection chain and its types. A strict increase in U must increase U\cap N or UN/N, since equality of both gives equality of orders. Furthermore

and the possible second factors are a fixed multiple of the possible indices [H_1:A(U\cap N)]. Thus it remains to control the images.

If H/N has bounded degree, its order is at most b! and there is nothing more to prove. Otherwise it is A_d or S_d, and H_1/N is its natural point stabilizer. Project the remaining route into H_1/N. A natural-giant step projects either to the same action or to a trivial action, because it is primitive; a bounded-degree step retains bounded degree. Starting with the natural A_{d-1} in H_1/N, retain an alternating group normal in the current projected group and supported on all but at most r points. A natural-giant quotient either kills this alternating group, which is then retained in the next subgroup, or maps it onto the natural alternating socle of its image. In the latter case equality of orders gives the same degree, and its point stabilizer removes one supported point. A step of bounded degree retains the alternating group once its order exceeds b!. After a step, take the union of all conjugates of the retained support in the new subgroup. The overlapping-support argument in the next paragraph supplies a normal alternating group on that union, without further loss of points. Apart from a possible sign on the large support, the residual quotient acts on the omitted points and has order at most 2r!. For d>2r+b+10, each step therefore loses at most one supported point, and the terminal image contains the alternating group supported on at least d-r points. Here the identification of the next point stabilizer uses \operatorname{Aut}(A_e)=S_e for e\geq7; the degree-six exception and all smaller first degrees are absorbed in the fixed finite universe. Thus we may take c=r in the large-degree argument.

An overgroup of an alternating group supported on d-c points has bounded height and boundedly many possible indices. To see this directly, assume d>2c+4. The supports of any two conjugates overlap in at least three points, so their alternating groups generate the alternating group on the union of their supports. The union of all conjugate supports is an orbit of the overgroup; its conjugate alternating groups generate the alternating group on that orbit. The orbit omits at most c points. Modulo the alternating group on that orbit, the overgroup has order at most 2c!. Its large simple type is therefore A_{d-j} with 0\leq j\leq c, and its possible indices belong to the finite list of falling factorials (d)_j divided or multiplied by integers bounded in terms of c. The excluded small d add only bounded groups. Combining this with the intersection induction proves all three conclusions. Invariant quotient blocks correspond exactly to the overgroups U already considered.□

Proof. Put d_0=2R+b+10 and M=d_0!, and define the positive rational atoms

We claim that every index on a route of length r is a product of at most L_r=r(r+3)/2 atoms or their inverses. Induct on r, putting N=\operatorname{core}_H(H_1) and B=A(U\cap N). Since U\cap N\trianglelefteq U, we have A\leq B\leq H_1 and B\cap N=U\cap N. Consequently

The image factor is bounded by M when the first action is small. Otherwise the supported alternating-group calculation in Lemma 3.5 gives

It therefore uses at most two atoms. The numerator [H_1:B] uses at most L_{r-1} atoms by induction.

The denominator is controlled across the whole collection, not just held fixed within one route. Project the suffix route into H_1/N. Its degree is exactly

Under any homomorphism with kernel C, a natural step L_0>L_1 projects to the same full natural action or to a trivial action: maximality gives L_1C=L_1 or L_0. A bounded step keeps bounded degree. Thus every factor on the right of (3.2) is an allowed natural degree, a bounded integer, or one, and is a single atom. In particular no arbitrary scalar-kernel divisor enters the denominator. Equation (3.1) gives L_r\leq L_{r-1}+2+(r-1), proving the claim.

There are at most M^2+K(R+1) atoms, so the union of all index palettes has cardinality at most

Only integral values of these rational recipes occur as actual indices. Invariant quotient blocks correspond to the overgroups U already allowed. If H' is any such block stabilizer, apply the argument anew to the next full route inside H'; it is not necessary that H' itself admit a natural-step route from H. Indices multiply through a bounded number of these operations, giving the final assertion.□

Proof. By Propositions 2.7 and 2.8, the starting group is a nontrivial proper coordinate product, and all designated intersections of coatoms are saturated. The shortcut a\prec v_{0N}\prec b projects to a two-step maximal chain in an almost simple group. Thus Proposition 3.3 gives the five-chief-factor bound for the actual projected bottom H_*. The image of the original D-stabilizer need not equal H_* as a group: in general H_* is that image multiplied by the starting coordinate product. What is required is equality of the induced active permutation actions, as follows.

For every label z, the equality z=DN_z implies that D and z have the same image on I_z, because N_z fixes its individual simple factors. If I_z\to I_w is a product-step factor map and g\in z fixes i\in I_w, write g=dn with d\in D and n\in N_z. Then d fixes i and has the same action as g on the fiber over i. Thus the bottom stabilizer induces the full local factor action at each level, not an arbitrary subgroup of the natural giant. This property survives the initial projection by Proposition 2.9.

We now account for all the shortcuts, including middles below a proper endpoint. An endpoint x_j is reached from b by the original shortcut b>v_{jN}>x_j; take an empty route for j=N. The first downward step is strict, because a proper subdirect group cannot contain the nontrivial full coordinate product at the bottom. Two strict covers give a concatenated route of length at most 2r_0. A strict cover followed by an equal-type merge instead makes the endpoint factor stabilizer an overgroup of the first route’s terminal stabilizer.

Starting at x_j, the middle v_{ij} and the lower endpoint x_i are reached by at most two further original covers. An equal-type operation takes an invariant quotient block. A strict operation supplies a fresh full local route inside the actual current projected factor stabilizer U\leq H_*. Indeed the image of the original D-stabilizer lies in U and already has the full local permutation image, whereas U acts through the labelled-node stabilizer because a lies below that node. The kernel of the original-coordinate projection centralizes that coordinate and acts trivially on its sections. Thus these are equal local permutation images; no equality of the stabilizer groups themselves is needed.

The factor maps and equal-type strips respect the original b coordinates. Here is the intersection identity behind this assertion. For labels x\leq z used in a route, work modulo F_b and write

These product descriptions follow from saturation and Proposition 2.9. Since F_z\leq D\leq x and N_z=F_z(N_z\cap N_b), intersection and the quotient map give

In an equal-type comparison this is the subdirect subgroup of S_z whose strips describe the merge. Because it is a full product over I_b, no strip can link factors in different original coordinates: such a link would prohibit independent elements in the two factors. For a strict split followed by a merge inside [x_i,x_j], repeat this identity with x_j as upper vertex. The fenced endpoint comparison makes x_i saturated in those coordinates, and the strict shortcut cover makes its middle saturated there. Thus the merge cannot cross those local starting fibers either. Altogether, every required factor, fiber, and quotient-block stabilizer uses at most four original covers, interpreted as full local routes or quotient operations. Four successive operations, each with route length at most 2r_0, give a conservative common bound, including all intermediate block stabilizers.

We establish the type pool before the degree and index pools. For fixed route parameters let

where c and the finite exceptional set \mathcal E are supplied by Lemma 3.5. If the whole current projected group has nonabelian composition types in \mathcal P, that lemma puts the types of every route prefix and every quotient-block overgroup in \Phi(\mathcal P). This includes any kernel of the corresponding permutation action. Retain the original types at every iteration. Starting with \operatorname{cf}_{\mathrm{na}}(H_*), of cardinality at most five, a bounded number of iterations gives one pool of absolute bounded cardinality for all these projected groups. Their induced images have no additional composition types. This proves (ii); it makes no claim about entire unprojected stabilizers in the original group.

For an equal-type stretch, fix its highest vertex. Equal-type transport identifies all earlier vertices with strip subgroups of that socle, whose support partitions strictly change along the stretch. Equal support partitions and subgroup inclusion would force equality of orders and hence equality of subgroups, including the diagonal twists. The preceding coordinate compatibility restricts these to invariant partitions of a single original b-coordinate fiber. Transitivity ensures that a strict change appears on that chosen fiber. All their block stabilizers are overgroups of a single route terminal supplied by the shortcut from b to the highest vertex. The height assertion of Lemma 3.5 proves (iii).

Each full giant of large degree d contributes the composition type A_d to its actual starting projected group. Every such group is a route prefix already covered by the preceding pool. Hence the set \mathcal D of giant degree values occurring anywhere in these bounded operations has absolute bounded cardinality. The values may be large. Apply Lemma 3.6 with this entire set \mathcal D, and multiply indices through the bounded number of operations. This yields a common index palette across all the actions, including arbitrary quotient-block choices followed by fresh full routes. In particular it is a union bound, not merely a separate cardinality bound for each action.

More explicitly, put q_i=m_{x_i}/m_b. The original shortcut from b to x_i realizes q_i as the degree on its active-factor fiber: a strict/strict path gives a route degree, and a strict/merge path gives a quotient-block degree. The common recipe palette bounds the cardinality of \{q_i\}. Since m_{x_i}/m_{x_j}=q_i/q_j, its quotient palette also has bounded cardinality. This proves (iv).□

Proof. The exact product/core identity in Proposition 2.9 gives, before projection,

Since F_j\leq D\leq x_i, we have Q_j\leq P_i. Intersecting with the lower group and then with its core gives the displayed formulas for J_{ij} and B_{ij}. Also C_j\leq C_i and R_j\leq P_j, whence

Thus uQ_j\mapsto uC_j embeds B_{ij} in C_i/C_j.

The C_i form a decreasing chain of normal subgroups of the one group H_*. Each nonsoluble successive quotient consumes at least one nonabelian chief factor of H_*. There are at most five, by Lemma 3.7. Cutting at these transitions leaves at most six stretches; one is arbitrarily long. The core of a shortcut middle lies between the cores of its two endpoints, so its killed layers are controlled by the same soluble quotient. No separate bound on a collection of unrelated shortcut cores is being assumed.

For the final assertion work with the original labelled vertices x_i<x_j and put

These subgroups are normal in x_i, with F_j\leq W\leq L. The kernel of the conjugation action of E on S_i=N_i/F_i is exactly W, because C_{x_i/F_i}(S_i)=1. When m_i=m_j, the product support map I_i\to I_j is a bijection. Since E\leq N_j fixes I_j, it fixes every factor of S_i. Its socle preimage is L, and therefore

which is soluble by Schreier’s Theorem. In the x_j coordinates write

Proposition 2.9 gives R_{ij}/Q_{ij}\cong T_i, and the preceding embedding makes P_{ij}/R_{ij} soluble.

The killed kernel is exactly the one controlled in the initial b projection. Indeed N_j=F_j(N_j\cap N_b) and F_j\leq F_i give

It is soluble on the retained stretch, so Q_{ij} is soluble as well. Thus P_{ij} has exactly its one active simple factor as a nonabelian composition factor. The same intersection calculation identifies the killed kernels for either half of a shortcut. This argument takes place in the original labelled x_i; it uses neither projected quasiprimitivity nor a supplementation dichotomy at an unlabelled bottleneck. A shortcut middle which still has several active factors can retain a nonabelian factor-permutation quotient; this is not asserted to be soluble. Its degrees are controlled by the common type pool, and it occurs only in the split–merge case considered separately below.□

Proof. Project the strict comparison from a to u using T4. The bottom is a full product and an intersection of coatoms. The coatoms and their intersection c_u are saturated, so Proposition 2.8 carries the entire interval [c_u,u] to an M_{16} interval ending at an almost simple group with socle T_u. For an alternating socle of degree at least seven that group is alternating or symmetric, contradicting Proposition 3.3. The exceptional automorphisms in degree six concern only bounded-order types.□

Proof. Apply [24, Theorem 0.5]. Outside (i), P lies in a proper algebraic subgroup B<X from a fixed constructible family. Apply [24, Theorem 12.3] to that family. It supplies an envelope H\leq B, with P\leq H, whose connected reductive quotient has simple adjoint factors X_1,\ldots,X_t such that the image L_i of P\cap H^\circ satisfies

with E_i simple. Each \dim X_i\leq\dim H^\circ<\dim X. The kernel of the map from H^\circ to the product of its simple adjoint quotients is soluble: it is an extension of the unipotent radical by the full center of the connected reductive quotient. That center can be disconnected, but is abelian. If L is the image of P\cap H^\circ in \prod_i L_i, then L' is a subdirect subgroup of \prod_iE_i, because each L_i/E_i is abelian. The subdirect product Lemma makes L' a product of full diagonals. Its simple factors are therefore among the E_i, counted at most once per position. There are at most \dim X/3 positions.

The component bound in [24, Proposition 1.4] bounds |H/H^\circ| throughout this constructible family. Its contribution has bounded order and bounded composition length. This proves (ii). Applying the same argument to every subgroup of a finite group of rank at most R proves the last assertion: the ambient algebraic dimensions range over a finite set, and the finitely many small orders and characteristics in the exceptional alternative are absorbed in a function of R.□

Proof. First, the pointwise centralizer of E=(X^{F_0})' in \operatorname{Aut}_{\rm alg}(X) is trivial when the field parameter is sufficiently large. Indeed the fixed groups X^a, for nonidentity algebraic automorphisms a, form a constructible family of proper closed subgroups: take the equalizer of the action and projection maps over the finite-type algebraic automorphism group, remove the identity section, and stratify. Now use [24, Proposition 3.5].

Here is the isogeny calculation needed to include field and exceptional graph operations. Let \mathcal I(X) be the group of bijections of X(\overline{\mathbb F}_p) generated by self-isogenies and their inverses. The isogeny Theorem [15, Theorem 6.1.16(1) and Proposition 6.1.13(2)] gives a homomorphism

For completeness, after fixing a Borel pair, a self-isogeny permutes the simple roots by \pi and raises their parameters to powers p^{e_i}. Use the row-coroot convention C_{ij}=\langle\alpha_j,\alpha_i^\vee\rangle for the Cartan matrix. The compatibility condition is

The labeled undirected Coxeter diagram is preserved. Unless a multiple edge is reversed, connectedness forces all e_i equal. An edge reversal is possible only for B_2,F_4 in characteristic two, or G_2 in characteristic three. The exponents on the two root lengths then differ by one. Thus every self-isogeny is a\theta^m, where a is algebraic, m\geq0, and \theta=F_p except in those three cases, when \theta^2=F_p; see also [15, Remark 6.2.10]. Root-group constants are absorbed by the algebraic automorphism, using the pinned uniqueness in the isogeny Theorem. Conjugation by \theta preserves algebraic automorphisms, so the same normal form holds with m\in\mathbb Z in \mathcal I(X). Inseparability degree proves uniqueness of m, and \nu(a\theta^m)=m is the asserted homomorphism. Positive \nu gives a Frobenius map: Lang–Steinberg removes its inner factor, leaving a positive diagram–field map [25, Proposition 2.7(i), p. 435]. Its field parameter is p^{\nu}, or p^{\nu/2} in the three exceptional cases.

Let C_E\leq\mathcal I(X) fix E pointwise. The preceding centralizer observation makes \nu|_{C_E} injective. Its image is a nonzero subgroup of \mathbb Z, since it contains \nu(F_0). Thus C_E=\langle F_1\rangle for a Frobenius map F_1, and F_0=F_1^b for some b\geq1. Put d=\dim X and let q_0 be the parameter of F_0. The fixed-point order bounds and bounded diagonal index in [24, Theorem 3.4(b),(d)] give

Enlarging c_\Phi includes the finitely many possibilities for the parameter of F_1 below four. For sufficiently large q_0, this forces b=1. Consequently C_E=\langle F_0\rangle. Since F fixes E, it follows that F=F_0^r with r>0.

The quotient P/E is abelian and E is perfect, so P'=E. Every m\in M therefore normalizes E. The finite automorphism Theorem [35, Theorems 30 and 36] lifts m to an element \widetilde m\in\mathcal I(X). The commutator \widetilde mF_0\widetilde m^{-1}F_0^{-1} has \nu=0 and fixes E pointwise. It is therefore the identity. Thus m normalizes E_e=(X^{F_0^e})' for every divisor e of r.

We have r>1, since r=1 would give T\leq M. If r were composite, choose 1<e<r dividing r. Fixed-point order estimates, after increasing the original field threshold, give E_e\not\leq P and 1<E_e<T. Hence

The last inequality follows because T is simple and cannot normalize a nontrivial proper subgroup of itself. This contradicts maximality and proves that r is prime.□

Proof. Consider a subsequence of distinct endpoint types of ranks at most R. A lower simple type is a section of each later one by Proposition 2.6; in particular their orders increase strictly. Remove the boundedly many types below the order thresholds of Lemma 3.10, for the finitely many relevant algebraic types. Every two-edge shortcut has intermediate types of rank bounded in terms of R, by the last assertion of that lemma. Increase the threshold accordingly. On each strict cover read downwards, the characteristic is then unchanged and the algebraic dimension does not increase. The only case of equal algebraic dimension is the sufficiently general alternative. An equal-type cover changes neither characteristic nor dimension.

There are finitely many possible dimensions and algebraic families. If the subsequence were arbitrarily long, it would have an arbitrarily long further subsequence with one dimension and family. On every prescribed two-edge shortcut between its endpoints the dimension cannot drop. Each strict cover is therefore a normalized prime subfield step by Lemma 3.11; a possible equal-type cover contributes no field change. With one fixed family at the endpoints, the conversion between its usual exponent f and the Frobenius parameter exponent cancels. Thus, for every i<j,

where \Omega counts prime factors with multiplicity. Four successive endpoints would give

a contradiction. All discarded sets and pigeonhole choices have bounds depending only on R, proving the first assertion.

For any desired rank threshold, remove the boundedly many endpoints below a sufficiently larger threshold. If a remaining shortcut middle had bounded rank, its lower endpoint, being a simple section of it, would have rank bounded by Lemma 3.10. The larger threshold was chosen to exclude this. Arbitrarily long chains remain, proving the second assertion.□

Proof. Take counterexamples of lengths tending to infinity, retaining the original endpoints initially. A lower simple type is a section of an upper one, so equality of the endpoint types of a segment forces equality throughout that segment. Lemma 3.7 bounds the lengths of these equal-type stretches. Choose one vertex from each stretch and discard the original initial stretch. There are arbitrarily many remaining vertices, and none has the type of the original a. Their shortcut middles are above a retained lower vertex, so also cannot have type T_a. Lemma 3.9 now excludes all large alternating types at both endpoints and middles. Sporadic groups and all other bounded-order coincidences form a finite list. Bounded-rank Lie types can be removed using Lemma 3.12. This gives (i), with arbitrarily large minimum rank.

All later coordinate projections still use the original tests. For a retained upper endpoint or middle u, condition T4 applies to a<u, with T_a\not\cong T_u. Proposition 2.8, with original lower vertex a, gives H_a\cap T_u=P_a. A retained vertex z saturated in these coordinates then has

because P_a\leq P_z. Saturation for a distinct-type endpoint comparison comes from its fence, and saturation at a strict shortcut cover comes from its coatom status. Thus the required inner-intersection identity never relies on inheriting a local coatom-meet representation from the global one below b, or on a new test at a retained bottom.

Use the common cardinality bound on ratios in Lemma 3.7 to color pairs by m_i/m_j. Finite Ramsey gives an arbitrarily long homogeneous subchain. On a homogeneous triple the common ratio t satisfies t^2=t; since t>0, it is one. This proves (ii). Next apply Lemma 3.8 in the original fixed projection: deleting the at most five nonsoluble transitions leaves an arbitrarily long stretch. All shortcut cores between its endpoints lie in the corresponding soluble gap. Passing to any later active coordinate takes subgroups and images of these soluble killed kernels, and hence preserves solubility. Lemma 3.8 and Proposition 2.9 give (iii).

The common type pool and index bounds are precisely Lemmas 3.7 and 3.6; none of these bounds counts abelian factors. They remain valid under all the preceding thinning, giving (iv).

Read a shortcut downwards from its upper endpoint. Its first cover cannot be a proper subdirect merge, because the lower endpoint is a nontrivial full product. If both covers are strict, their positive integral splitting multiplicities multiply to the endpoint ratio one. Both are therefore one. Their killed kernels are soluble by (iii), and their tails are soluble by Proposition 2.9. If the second cover is an equal-type merge, the first must split; the only unbounded-rank maximal-subgroup cases with such an active split are the geometric block and tensor cases described after Proposition 3.4. The merge stays inside each original upper coordinate. Its possible indices are already covered by (iv).

The inactive nonabelian labels in a shortcut belong to the pool in (iv). To see this, write its local inner group as P\trianglelefteq K, with soluble killed kernel Q and active chief factor R/Q\cong T^a. The faithful conjugation argument in Proposition 2.9 gives P/Q\hookrightarrow\operatorname{Aut}(T^a). The kernel of the factor-permutation map on P/R lies in \operatorname{Out}(T)^a and is soluble. Since P is normal in K, its permutation image is normal in the full local permutation image. Lemma 3.7 identifies that full image with an image of one of the projected stabilizers counted in (iv). Its normal subgroup therefore has its nonabelian composition types in the same pool. Thus all nonabelian composition types outside the active chief factor R/Q lie in this pool. The active type T can still vary along the chain.

Apply now the structural observations following Proposition 3.4. There are boundedly many choices of inactive simple labels by (iv), and the repetition indices also have boundedly many values. The resulting omitted dimensions therefore have boundedly many possible values. When all inactive components are soluble, their individual natural dimensions are absolutely bounded. This proves (v) and all the assertions passed to the next section.□

Proof. For the dimension and field statements, take the full stabilizers of the indicated structures. On a nondegenerate direct summand the active group has its natural module; on a totally singular summand the dual summand has the contragredient module. A tensor stabilizer acts as the tensor product of the component natural modules. Restriction of scalars multiplies dimension by its degree b and gives \epsilon_s f_s=b\epsilon_t f_t for the natural matrix fields. For example, the mixed \operatorname{GU}_m(q) row in \operatorname{Sp}_{2m}(q) has b=2 and equal usual exponents, since \epsilon_s=2 and \epsilon_t=1. It is a restriction of scalars, with a direct sum after scalar extension, and is not a twisted-tensor row. A subfield inclusion changes only the field. If an ordinary field degree were composite, its proper intermediate field structure would be preserved by the normalizer, contradicting maximality. The classical stabilizer descriptions with prime degree are given explicitly in [10, proofs of Lemmas 5.5 and 5.7]. Unitary conjugation and the diagram maps have bounded order, which accounts for the permitted bounded ratios in the remaining field operations. The twisted-tensor assertion is proved separately in Lemma 4.3 below.

The only soluble classical component groups have bounded natural dimension: a linear group in growing dimension contains a growing special linear simple section, and the corresponding assertions for the form groups follow by their natural hyperbolic subspaces. Thus the omitted dimensions in a one-active-component subspace or tensor stabilizer are bounded. The explicit block description of a parabolic gives a radical of order q^e, with e\leq Cn_t^2 and e\geq c n_t whenever that radical is nontrivial in the retained large-rank cases. For example, the linear radical attached to a k-space has exponent k(n_t-k); the symplectic and orthogonal exponents count the off-diagonal blocks subject to one symmetric or alternating relation. The smallest nonzero retained exponent is linear in n_t. The other factors are the displayed classical groups and scalar groups. Consequently their orders are determined by the dimensions, field, form signs, and the bounded choices of interchanged flags. Intersecting the full stabilizer with the ambient socle adds only its prescribed determinant or spinor condition; it does not introduce a freely chosen subgroup of a scalar group. This proves (C3).

For a symplectic-type normalizer put N=n_t=r^d, and let W be the projective image of its normal r-group, of order r^{2d}. The full inner-normalizer rows, including their minimal-field conditions, are given in [9, Section 2.2.6, pp. 70–71, Table 2.9]. Their small exceptions disappear on retaining d\geq4. Ordinary linear conjugation fixes scalars pointwise, so it preserves the nondegenerate scalar commutator pairing W\times W\to\mu_r. Moreover the projective linear centralizer of W is W: a centralizing operator twists the representing matrices by a character of W, that character is implemented by an element of W through the pairing, and the remaining centralizer is scalar by irreducibility.

In the linear and unitary rows, the normal group is an exponent-r extraspecial group for odd r, and a central product C_4\circ2^{1+2d} for r=2. In both cases the ordinary projective normalizer has quotient \operatorname{Sp}_{2d}(r) on W. Indeed every symplectic change of generators lifts to a center-fixing automorphism and is implemented by the irreducible representation with its specified central character. For r=2, multiplying a generator by the central element of order four restores its required square; thus a square form attached to a pure extraspecial subgroup is not an invariant of this scalar-extended normalizer. In the unitary case an implementing operator preserves the Hermitian form up to a multiplier, which is removed by a scalar using the surjective finite-field norm. The standard tensor generators of W have determinant one: for odd r this holds for the cyclic shift and diagonal root-of-unity generators, and for r=2 their determinants are (-1)^{2^{d-1}}=1. The map from this normalizer to the abelian quotient of the projective linear or unitary group by its simple socle therefore factors through \operatorname{Sp}_{2d}(r). That group is perfect in the retained range, so the map is trivial. Hence the full ordinary projective normalizer is already the inner intersection P. Quotienting by the entire ambient scalar center leaves kernel exactly W, and

This uses the quotient, without requiring the extension to split.

The pure symplectic and orthogonal form rows have r=2. Here the extraspecial square form is invariant. The cited inner rows give P/W=O_{2d}^{\varepsilon}(2) or \Omega_{2d}^{\varepsilon}(2), with active simple group T_s=\Omega_{2d}^{\varepsilon}(2) and index at most two. These are the inner rows after the determinant and spinor conditions; the scalar kernel of the natural cover is removed in W. Consequently N^2\leq |P|/|T_s|\leq2N^2 in every retained row. There is no extra factor from r-1 or the ambient field exponent.

The target-field bound uses the minimal-field condition of the maximal row. For odd r, the degree e=\operatorname{ord}_r(p) divides r-1; the usual target exponent is e in the linear row and e/2 in the unitary row. The binary and pure form rows have usual exponent one. Thus f_t\leq\max(1,r-1)\leq |T_s|. The representation characteristic is different from r, and the active source is over the prime field of order r. This proves (C4).

For an absolutely irreducible representation of a central cover \widetilde T_s, the dimension is at most |\widetilde T_s|^{1/2}: the square of each simple-module dimension is bounded by the dimension of the group algebra. The field-of-definition bound can be obtained directly, including in modular characteristic. For a finite group K in characteristic p, there are at most |K| absolute simple module types, by the semisimple quotient of its group algebra. Frobenius permutes these types, so an absolute simple representation has a Frobenius orbit of length a\leq |K|. An isomorphism with its p^a-Frobenius twist descends the representation to \mathbb F_{p^a}: choose a finite field containing its matrices and an intertwiner, scale the intertwiner to have norm one, and apply matrix Hilbert’s Theorem 90. The scalar norm adjustment is possible because finite-field norms are surjective. Apply this with K=\widetilde T_s. The Schur multipliers of the large classical groups have order at most a polynomial in |T_s|; the finitely many exceptional multipliers do not change the bound. Forms cost an extension of degree at most two. Maximality outside the subfield collection makes these bounds applicable to the target field itself. For the cross-characteristic lower bound, the classical tables in [20, Section 3, Tables 3–6] give degree at least q_s^r/3 once the source rank r\geq6, including the smaller Weil degrees. Their excluded families have bounded rank. As n_s\leq2r+1, this is at least q_s^{n_s/4} after increasing the rank threshold. This proves the numerical assertions of (C6).

For the defining-characteristic quadratic bound, use [26, Theorem 5.1 and Table 2] in rank r>11. Modules outside that table have dimension greater than r^3/8 in type A, and greater than r^3 in types B,C,D. Every listed nontrivial, nonnatural, nondual degree is at least r^2/2. Consequently the retained restricted module has dimension at least r^2/2\geq n_s^2/10. Type B in characteristic two is read through its type-C natural simple quotient, as in our parameter convention; the natural-size special-isogeny cases are already assigned to geometry.

A single occupied ordinary Frobenius digit has full source period: any shift strictly between zero and f_s moves it to a different position in the unique finite restricted label. Returning through f_s applies at most the bounded diagram operation. An internal tensor decomposition of that digit changes neither its occupied position nor this uniqueness argument. Thus the minimal scalar field degree is f_s times one of the bounded diagram periods. The matrix degree is separately \epsilon_t f_t. The form-compatible minimal-field argument in Lemma 4.2 bounds the ratio of this matrix degree to the minimal degree outside the geometric collections. This gives the fixed finite rational set of usual exponent ratios and proves (C5). Lemma 4.2 below supplies the final normalizer assertion.□

Proof. Lift the irreducible projective representation to the perfect central cover of Y. A projective linear element centralizing its image commutes with that representation up to a linear character of the perfect cover, hence commutes exactly. Schur’s Lemma makes it scalar, so it disappears projectively.

If a semilinear element in this kernel has scalar-field action \sigma, the representations \rho^\sigma and \rho are isomorphic. If \sigma has order a on the scalar field E, its intertwiner A satisfies

Surjectivity of the scalar norm adjusts A so that the product is one; matrix Hilbert’s Theorem 90 then gives a model over E^\sigma. An invariant bilinear form is a line defined by linear equations over the minimal field, and invariant quadratic-form coefficients satisfy linear equations there as well. In characteristic two, two invariant quadratic forms with the same polarization differ by the square of an invariant linear form. Absolute irreducibility makes that difference zero, so the prescribed quadratic form itself descends up to scalar. Matching finite-field form classes costs at most a quadratic extension. For a unitary group the involution on the minimal field is either nontrivial, giving Hermitian descent and surjective norms, or trivial, giving a preserved bilinear structure and hence a geometric form case. A larger field period would therefore provide a proper subfield structure normalized by the whole irreducible normalizer, contrary to this step being outside the subfield collection.

To justify the assertion about the whole normalizer, put an absolutely irreducible model over its minimal field E_0. A normalizing matrix intertwines this model with the model obtained by a source automorphism and a field automorphism; both have entries in E_0. The intertwining equations therefore have coefficients in E_0. Their solution space over the algebraic closure is one-dimensional by absolute irreducibility. It consequently has a nonzero E_0-rational solution, necessarily invertible. Up to a scalar, the normalizer thus preserves the projective E_0-structure. The same argument applies after dualizing in a graph component, and the uniqueness of the invariant form gives the already allowed quadratic extension. A graph operation contributes at most another factor of two in unbounded rank. This bounds the kernel independently of the ambient field degree.

Finally the diagonal, field, and graph description of automorphisms gives respectively factors at most n_Y, f_Y, and an absolute constant in the outer group.□

Proof. Let S be the simple source, M the almost simple maximal subgroup, and T the ambient simple classical group. Work over an algebraic closure on the cover supplied by Lemma 5.3, and write the irreducible module as

where V_i are the nonzero ordinary restricted Frobenius digits, with their indicated twists. The dimensions of these nontrivial digits tend to infinity with the source rank.

We first verify preservation of the actual tensor structure. A source field automorphism translates ordinary digit positions, applying the diagram operation when a position passes the source period; inner, diagonal, and diagram automorphisms act on the restricted modules at those positions. Ambient semilinear operations give the same shifts, and a graph operation dualizes the modules. Uniqueness of the finite restricted highest-weight parametrization matches the resulting whole digit modules. Tensor their individual intertwiners, with the matching permutation, to obtain an intertwiner of V. The discrepancy between a projective and linear intertwining identity is a linear character of the perfect cover and is trivial. Absolute irreducibility makes the space of whole-module intertwiners one-dimensional. Every actual normalizing operator therefore differs from this tensor intertwiner by a scalar and permutes the embedded algebras \mathcal A_i. For a graph operation the corresponding map on endomorphisms is an anti-isomorphism, which has the same conclusion. This argument preserves a whole ordinary digit even if that restricted module has an internal tensor decomposition in a very special characteristic.

Replace each digit by its full connected special linear group, or by its full connected bilinear-form group when required. Bilinear self-duality makes each ordinary digit self-dual separately. Unitary descent acts on the special linear factors, permitting duals. In characteristic two every nontrivial irreducible self-dual digit has an alternating form: a nonalternating invariant form would give an invariant linear functional by taking square roots of its diagonal. The tensor product of at least two nondegenerate alternating spaces has a quadratic form with the product polarization and value zero on pure tensors [25, Proposition 2.2(iv)]. It agrees with the prescribed finite quadratic form, after a scalar normalization of the polarization: their difference is the square of an invariant linear form, which vanishes by nontrivial absolute irreducibility. These constructions give a connected semisimple tensor subgroup K in the ambient form group.

The groups are intrinsic to the algebras \mathcal A_i and their restricted adjoint involutions. Hence the actual ambient Frobenius F and the full finite normalizer preserve K. In growing classical rank, finite ambient automorphisms have lifts commuting with F; diagonal lifts satisfy this projectively because F(x)x^{-1} is central. Graph and field lifts also commute, with only the bounded-rank triality and exceptional graph cases removed. This is the setup of [25, Theorem 2 and Section 4]. Henceforth view K, and later K', in the ambient adjoint group. Thus M normalizes the terminal derived subgroup H=(K^F)^{(\infty)}. Since H is perfect it lies in T, and S\leq H because S is a perfect subgroup of K^F.

For each orbit of F on the factors, fixed points are obtained by choosing one coordinate fixed by the corresponding power of F; see [25, Proposition 2.8(ii)]. Consequently H is a central product of classical components, one per orbit. It is proper in T also as a finite group. To check this uniformly, put q=p^{f_t}, let the individual digit dimensions be d_1,\ldots,d_k, put d_{\min}=\min_i d_i, and write n_t=\prod_i d_i. The earlier bound for nontrivial digits gives d_{\min}\to\infty with the source rank. Elementary classical order bounds give

For k\geq2, \sum_i d_i^2/n_t^2\leq k/d_{\min}^{2(k-1)}\leq2/d_{\min}^2. Thus H<T for all sufficiently large source ranks. If MH were the entire almost simple ambient group, its normal subgroup H\leq T would equal T. Hence MH is proper, and maximality gives H\leq M. But the perfect group H/S is a subgroup of the soluble group M/S, so H=S. There is therefore one Frobenius orbit, and its full natural classical component is the source. If its length is b, the digit dimension is n_s and n_t=n_s^b exactly.

The field relation is read from this actual natural-factor cycle. Put t=f_t. For a linear target, F^b on one factor is the standard field map with exponent bt, up to an inner conjugacy. For a unitary target, write F=F_{p^t}\tau with \tau the inverse-transpose operation. On one factor the return map is F_{p^{bt}}\tau^b, again up to an inner conjugacy. Its source is unitary when b is odd and linear when b is even, in both cases with usual exponent bt. An orthogonal graph choice changes the form sign, not this exponent. The characteristic-two quadratic construction likewise changes no field exponent. Thus

exactly, in the convention of this paper. The natural factors have positions 0,t,\ldots,(b-1)t modulo bt, which are distinct. For the odd unitary convention using positions 2jt, multiplication by two permutes the same residues modulo b; the prime two-cycle in a unitary target has a linear source. These are precisely the usual-field alternatives in [25, Section 4(5)].

Finally suppose b=uv, with u,v>1. Number the factors by \mathbb Z/b\mathbb Z, and group the cosets of its unique subgroup of order u. The normalizer’s permutation normalizes the regular cyclic descent action and preserves this block system. The algebra for a block B is the algebra generated by its \mathcal A_i, namely \operatorname{End}(\bigotimes_{i\in B}V_i) \otimes1. Replace each block by its full special linear or bilinear-form group. For a characteristic-two quadratic target use the full symplectic group on each grouped alternating space. The resulting quadratic tensor form is unchanged: it has the same polarization and vanishes on every fully pure original tensor, and such tensors span V. Hence this grouped subgroup K' still preserves the prescribed ambient form and is invariant under F and M.

Its terminal derived fixed-point subgroup H' has one classical component on dimension n_s^u over the field with usual exponent v f_t, with the unitary parity or orthogonal sign prescribed by the return map. For n_s\geq16, the elementary bounds Q^{d^2/8}\leq|\operatorname{Cl}_d(Q)|\leq Q^{2d^2} give

The strict inequalities follow from n_s^{2u-2}>16u and n_s^{2u(v-1)}>16v. As before M normalizes H' and S\leq H'. Its properness in T forces H'\leq M, and perfectness then forces H'=S, contradicting the first inequality. Therefore b is prime.□

Proof. Write the actual shortcut as U=z_i\prec V\prec W=z_j, and work modulo F_W. In one W-coordinate the last cover has Y\trianglelefteq P_A\le\operatorname{Aut}(Y). Its killed kernel from the middle is soluble by Proposition 3.13 and normal in P_A, hence is trivial. The active subgroup in that coordinate is therefore the actual simple group Y. Proposition 2.9 gives a V-equivariant bijection \beta:I_V\longrightarrow I_W because the multiplicities agree. For \alpha\in I_V and t=\beta(\alpha), the actual stabilizers in U coincide: U_\alpha=U_t. Let \rho_V,\rho_W be their middle and upper coordinate actions, and let \phi:N_{\operatorname{Aut}(T_j)}(Y)\longrightarrow\operatorname{Aut}(Y) be conjugation. The quotient-induced actions satisfy

where M_B is the actual first-cover coordinate stabilizer image.

The inner-intersection assertions use the original lower endpoint a from Definition 3.1. All retained comparisons start strictly above this a, so none uses an excluded middle v_{0j}. Condition T4 makes a an intersection of coatoms in [a,V] and [a,W] for every retained middle and upper endpoint. Proposition 2.8 in these original-a projections identifies the inner part of a saturated image with its transported inner projection. For U\prec V the coatom case of that Proposition also applies directly. In particular,

There is no assumption that a meet of coatoms in the original global interval is a meet of coatoms in every smaller interval. The bounded kernel from Lemma 4.2 now gives

which implies (4.5). This includes diagonal, field, and graph operations. All higher-core losses are already kernels of the displayed actual coordinate actions; no uncontrolled factor from the field of the final target is inserted.□

Proof. Let S contain the primes occurring in the numerators or denominators of members of \mathcal C. For \ell\in S, the integer v_\ell(x_j/x_i) lies in a fixed bounded interval: the contribution of c_{ij} is bounded, and that of u_{ij} has absolute value at most h. Outside S all valuations have sign \varepsilon, and the sum of their absolute values is at most h. Color (i,j) by the valuation vector on S and this last sum. This is a finite coloring with the asserted uniformity.

For i<j<k we have

On a homogeneous triangle each coordinate of the common valuation vector therefore equals twice itself, so it is zero. Outside S there is no cancellation because all valuations have the same sign. Their total absolute value likewise equals twice itself, hence is zero. Every valuation of every pair ratio is zero, so all ratios equal one. Finite Ramsey gives the last assertion.□

Proof. Color first by the order and kind of the two steps. By G1, their combined dimension formula is either

where b_{ij}>1 is prime or belongs to a fixed bounded set of integers. In the first case put x_i=n_i, and in the second put x_i=\log n_i. In either case x_j/x_i=b_{ij} has uniformly bounded prime complexity. Lemma 4.5, with \mathcal C=\{1\}, gives a homogeneous triangle with b_{ij}=1, a contradiction. Notice that field ratios alone would not suffice: ratios r_j/r_i of distinct primes are compatible on arbitrarily many vertices. It is the exact dimension formula that excludes this possibility.□

Proof. Color by the two cover categories, the endpoint family, and whether a split and merge occur. There are finitely many colors. The endpoint characteristic is common because every step is in defining characteristic. If both an increasing and a decreasing exceptional field move occur, Lemma 4.6 applies. Otherwise all exceptional primes in a pair ratio have the same sign. The product of the two bounded rational factors belongs to a fixed finite set, and at most two exceptional degrees occur. A bounded exceptional degree has uniformly bounded prime complexity as well. Apply Lemma 4.5 to x_i=f_i. We may therefore assume that all endpoint fields and families agree.

The exceptional degrees at the individual covers are now bounded integers. Indeed, with no opposite field moves their product, or its reciprocal, is a member of the fixed finite set of bounded twisting ratios, because f_j/f_i=1. Each positive degree divides the numerator or denominator of one of these finitely many rationals.

Fix a lower endpoint i. We show that its upper natural dimension has only boundedly many possible values. For block and parabolic shapes, the ambient dimension is the active dimension, or twice that dimension when its dual is present, plus the dimensions of the omitted blocks. The dimensions of nonabelian omitted blocks are determined, with bounded ambiguity, by the bounded-cardinality pool of inactive types. Soluble omitted blocks have bounded dimensions. A tensor shape multiplies the active dimension by a product of the omitted dimensions. A replicated base multiplies it by a replication degree from its bounded-cardinality pool. A form change has the same natural dimension, with the bounded natural-module exceptions included in the family color. Field restriction multiplies dimension by one of the bounded degrees just obtained, and a twisted-tensor step raises it to one of those degrees. Thus one geometric cover gives a bounded list of possible outputs from a fixed input. Two covers give a list whose size is at most the product of the two bounds. For a split followed by a merge the second step has the same simple label, so gives no new natural dimension at all.

All labels now have the same family and field. Their strict growth in order makes their natural dimensions distinct. There cannot be more upper endpoints than the bounded number of possible outputs from the fixed lower endpoint. This contradicts arbitrary length.□

Proof. The active factor T_i is a composition factor of P_{ij}, so its order divides |P_{ij}|. Saturation and supplementation give

Multiplying group indices through z_j and taking the positive mth root gives

Rearranging and dividing by |T_i| proves the formula. The same calculation applies at a shortcut because its three multiplicities agree. No identification of its intermediate label with a chain label is required.□

Proof. For U>z_i, the original test T5 provides a fenced interval or a labelled bottleneck at U. If U=z_i, the endpoint itself is labelled. Since F_j\trianglelefteq z_j and F_j\le D, apply Lemma 2.3 or Corollary 2.4 in the quotient by F_j, with bottom D/F_j. The soluble-normal conclusion gives (4.6). For (N2), the full inverse image of \overline K is normal in this same U and contained in D, so the commuting-normal conclusion kills one product. Since D\le z_i, a killed product has its coordinate subgroup in P_{ij}.□

Proof. Use the natural-space field E=\mathbb F_{p^e}; thus e=f, or e=2f in the unitary case. Apart from the bounded-rank graph exceptions already excluded, automorphisms in the form cases are represented by projective semilinear similarities. In the linear case there may also be correlations, which interchange the natural space and its dual. The ordinary orthogonal graph operation is a similarity in this description; characteristic-two type B uses its symplectic natural quotient.

First suppose that M acts on the natural-space side. Choose one semilinear representative s_g for each projective element, with s_1=1. The products satisfy s_gs_h=c(g,h)s_{gh} for nonzero scalars c(g,h)\in E. For an unknown vector v\in E^n impose

in a bilinear or Hermitian form case, and impose also Q(s_gv)=0 in a quadratic form case. In a plain linear space no equations are necessary. Field automorphisms and semilinear maps are \mathbb F_p-linear. These are therefore at most e(m^2+m) scalar polynomial equations of degree at most two in en prime-field variables. Their sum of degrees is less than en.

For completeness, the elementary polynomial count giving a nonzero solution is as follows. If F_1,\ldots,F_t have total degree sum less than the number N of prime-field variables, the number of their common zeros, reduced modulo p, is

Every expanded monomial has degree less than N(p-1), so some variable has exponent less than p-1. The sum over that variable is zero in \mathbb F_p, including exponent zero. The displayed sum is zero. All our equations vanish at zero, so their common zero set has another element. This is the needed Chevalley–Warning argument [13, 38].

Choose such a nonzero v and put

Then 1\le\dim U\le m. Scalar discrepancies between representatives do not change this span, so U is M-stable. The imposed equations make U totally singular in the form cases. In particular, in characteristic two we used the quadratic equations as well as the polar equations. Thus the stabilizer of U is a proper parabolic normalized by M.

Now let T be linear and suppose M contains correlations. Put H=M\cap\operatorname{P\Gamma L}(V) and choose d\in M\setminus H. On subspaces the correlation has the form

for an invertible semilinear map F:V\longrightarrow V^*. Choose representatives s_h, h\in H, and impose

Again there are at most e|H|^2 quadratic scalar equations. The same count gives v\ne0. Let U=\operatorname{span}_E\{s_hv:h\in H\} and W=d(U). The equations say U\le W. Since H is normal in M, both subspaces are H-stable. Moreover d(W)=d^2(U)=U, because d^2\in H. Thus the flag U<W is preserved by all of M, as an unordered pair under correlations. Its dimensions are r,n-r, where 1\le r\le |H|<n/2, so it is a proper flag. Its parabolic stabilizer is normalized by M. No involution assumption on d has been used.

In these constructions the small subspace has dimension 1\le r\le m<n/4 once n is sufficiently large. The upper triangular block matrices in the radical supply at least n/4 independent prime-parameter root coordinates, each of order at least q. For example the linear case has r(n-r) coordinates, and the flag case has at least r(n-2r); in the polar cases the block from the nondegenerate middle space to the singular r-space supplies at least r(n-2r-2) coordinates. These lower bounds are at least n/4 after enlarging the fixed dimension threshold. We may take a smaller absolute c_0 to cover all the families. Projectivization does not remove unipotent coordinates: the scalar center has order prime to p.

Let J be this proper parabolic and R its radical. The group R is soluble and normalized by M. In the coordinate application, take the actual enlargement (4.8). Its inner part is

The last inequality holds because J is a nontrivial proper subgroup of the simple group T_j. Thus U<z_j: properness only in \operatorname{Aut}(T_j) would not suffice. The soluble coordinate normal test gives \widehat R\le D/F_j, hence R\le P_{ij}\le M_{ij}. Its order contradicts q_j^{c_0n_j}>|M_{ij}|.□

Proof. Every ascending same-characteristic geometric step preserves both natural dimension and n\log q up to fixed positive lower factors, as established in G3. The same holds for a same-characteristic nonlinear step by its quadratic degree bound and fixed positive field ratio. If the last cover crosses, a preceding same-characteristic cover leaves n'\log q' at least a fixed multiple of n_i\log q_i. A preceding crossing has the same consequence from n'\ge\exp(c n_i\log q_i) and \log q'\ge\log2 at sufficiently large rank. The last crossing then gives the endpoint lower bound. If only the first cover crosses, it gives n'\ge\exp(c n_i\log q_i), and the last cover preserves natural dimension up to a fixed factor. Decreasing c and increasing the fixed minimum-rank threshold, we obtain in every case

Write u=n_i\log q_i. Classical order formulas give

After the first step every additional comparison gives at least the map x\mapsto F(x)=\exp(c\log(2)x). For fixed C_3,

If L(u)/R(u)\longrightarrow\infty and R(u)\longrightarrow\infty, then F(L(u))/\exp(R(u))\longrightarrow\infty as well. Induction therefore shows that h+2 comparisons dominate \mathcal E_h(\exp(C_3u^2)), and hence \mathcal E_h(a_i^C) when C_3\ge CC_2. This also proves the last assertion. The target prime characteristic need not be bounded in this argument: its only use in the iteration was \log q\ge\log2.□

Proof. Suppose otherwise, and color by the two step categories. Every pair in the resulting arbitrarily long chain has a crossing. There are three possibilities for its last step A.

The last step is geometric in the same characteristic. The first step B crosses. Hence n',f'\le a_i^C by (4.4). An ordinary geometric last step changes dimension by bounded factors, except for an extension degree. By (4.1) that degree is b=\epsilon'f'/(\epsilon_jf_j)\le2f'. A last subfield degree can be arbitrarily large but leaves dimension unchanged. A twisted-tensor last step has exponent b=f'/f_j\le f' by (4.2). Therefore

for a fixed C'. Lemma 4.11 contradicts this bound on a sufficiently distant pair.

The last step is symplectic type. Its source Y is a symplectic or orthogonal group over a prime field, say of order r, and its target dimension is r^{d} with 2d+O(1)=n'. First suppose B is in the same characteristic. Then r=p_i and f'=1. In a subfield first step the formula 1=\rho f_i b, with \rho in a fixed finite set of positive rationals, bounds both f_i and b; in the untwisted formula such a proper step is impossible. An ordinary extension degree in B satisfies b=\epsilon_i f_i\le2f_i, because the target here has f'=1 and \epsilon'=1; a twisted-tensor exponent satisfies b=f_i. The ordinary geometric formulas give a fixed-height tower bound on n' in terms of a_i. If B is nonlinear, its irreducible degree is at most a_i^C, giving the same conclusion. Since p_i\le a_i, the formula n_j=p_i^{d} gives n_j\le\mathcal E_h(a_i^{C'}) for fixed h,C'. This contradicts Lemma 4.11.

It remains in this case that B also crosses. Let b_{ij},s_{ij} be the inner multipliers of B,A. By G4,

The bound for b_{ij} is the outer-order bound when B is almost simple; in the symplectic-type case it follows from b_{ij}\le C(n')^2\le a_i^{C'}. For i<j<k, Lemma 4.8 now gives

The factors involving the common upper dimension n_k have canceled. This contradicts Lemma 4.11 on a sufficiently distant pair i<j.

The last step is almost simple irreducible. Let its source be Y. If B crosses, then n',f'\le a_i^C and |P_B|\le a_i^C: for an almost simple B this uses |P_B|\le a_i|\operatorname{Out}(T_i)|, and for a symplectic-type B it uses the preceding bound on its multiplier. Equation (4.5) gives

It is irrelevant here whether A itself is a crossing.

Suppose next that B is in the same characteristic, so that A crosses. If B is not a subfield step, its geometric dimension formulas, or its irreducible degree bound when nonlinear, bound n' by a fixed-height tower in a_i. Its field satisfies \log q'\le C\log q_i: restriction of scalars contributes the bounded factor \epsilon_i/\epsilon', and the other non-subfield rows have their fixed field factors. Thus |Y| is bounded by a fixed-height tower in a_i, and n_j\le |Y|^C contradicts Lemma 4.11.

Finally suppose B is subfield. Its rank equals the lower rank and its characteristic equals p_i. Its inner projection has order at most a_i^C, but f' can be unbounded. The last crossing gives

Equation (4.5), with n' comparable to n_i, therefore gives

The same upper bound includes (4.9). By Lemma 4.11, after a fixed separation we have a_i^{C'}\le n_j^{1/16}; and, for sufficiently large rank, (1+\log n_j)^{C'}\le n_j^{1/16}. Hence |M_{ij}|\le n_j^{1/8}. This satisfies the dimension inequality of Lemma 4.10, and q_j^{c_0n_j}>n_j^{1/8}\ge |M_{ij}|. That Lemma gives the final contradiction.

The three possibilities exhaust the last cover in a crossing path.□

Proof. Propositions 4.7 and 4.12 leave a defining-characteristic nonlinear cover on every comparison, with the common characteristic of the endpoints. Color by endpoint family and cover categories. A nonlinear cover has a bounded rational field ratio; only the other cover can carry an exceptional prime degree. Lemma 4.5 applied to f_i therefore gives equal endpoint exponents. The exceptional degree at the remaining cover must cancel a member of a fixed finite set of rational twisting ratios, so it is bounded. The intermediate ratios consequently belong to a fixed finite set.

The nonlinear cover increases natural dimension at least quadratically. All remaining geometric steps increase it by at least a fixed positive factor. Thus its position as first or last cover makes no difference to (4.11), after decreasing c.□

Proof. Use Lemma 4.13, and color by the position of the parabolic cover. The other cover is nonlinear.

If the parabolic cover is first, its active dimension is n_i and its ambient dimension is comparable to n_i. Its inner quotient is at most q^{C n_i^2}. The multiplier of the last nonlinear cover is at most |\operatorname{Out}(Y)|, which is absorbed by this bound, since n' is comparable to n_i and f'/f is bounded. Hence

For j<k the radical in its first cover also gives e_{jk}\ge q^{c_1 n_j}, with the bounded field ratio absorbed in c_1. By Lemma 4.8,

Iterating (4.11) through a fixed number of intermediate chain vertices makes n_j/n_i^2 tend to infinity as the minimum rank grows. The displayed inequalities are incompatible.

If the parabolic cover is last, write e_{ij}=b_{ij}s_{ij}, where b_{ij}\le|\operatorname{Out}(T_i)| is the nonlinear multiplier and s_{ij} is the parabolic multiplier. For a fixed upper endpoint k, the values s_{ik} have a uniformly bounded number of possibilities by G4. List those values in increasing order and color (i,k) by its position in that list. This gives a uniform finite coloring even though the lists for different k may be different. On a homogeneous triple, s_{ik}=s_{jk}, and therefore

But the parabolic radical on the pair (i,j) gives e_{ij}\ge q^{c_2n_j}. The classical outer-order bound is

so even (4.11) contradicts these inequalities at sufficiently large rank.□

Proof. There is at least one nonlinear cover. For a remaining geometric cover with active label T and natural dimension n, the other classical factors have bounded natural dimension and are soluble. Their orders, scalar factors, and bounded component-permutation factors are bounded by

Field and twisted-tensor degrees are bounded by Lemma 4.13; their semilinear factors satisfy the same bound. These are precisely the block, form, tensor, and field structures left after Lemma 4.14; no unipotent radical or extra nonabelian factor is being omitted from the estimate.

If the geometric cover is first, its intermediate natural dimension is at most C_1(1+n_i)^{b_1} for fixed constants. Its multiplier has the displayed bound in the lower parameters, and the last nonlinear multiplier is at most |\operatorname{Out}(Y)|\le Cn'f'. Since |T_i|\le q^{C n_i^2} and f\le q, their product satisfies (4.13).

If the nonlinear cover is first, its intermediate degree is at most a_i^{C_2} and f'/f is bounded. The geometric multiplier is bounded by q^{C_3}(1+n')^{C_3}(1+f')^{C_3}, which is at most q^{C_4 n_i^2}. Multiplying by a_i and the first nonlinear outer multiplier proves the same bound. With two nonlinear covers only the two outer multipliers occur, and the latter argument applies without the geometric factor.□

Proof. Parts (i)–(iv) collect Propositions 4.7 and 4.12 and Lemmas 4.13, 4.14, and 4.15. For the spacing assertion put x_i=c n_i. Then (4.11) says x_j\ge x_i^2 on every consecutive comparison. After s comparisons, n_j\ge c^{-1}(c n_i)^{2^s}. Choose 2^s>d and then increase the minimum rank to obtain n_j\ge n_i^d.

The coordinate normal test is inherited from the original tested intervals and is not altered by deleting chain vertices. Every proper parabolic of a large-rank classical group has unipotent radical of order at least q^{c_3 n_j} for an absolute c_3>0. One can see the uniform bound by choosing a maximal parabolic containing it: the omitted simple root has at least the relative rank many positive-root coordinates outside the Levi subsystem. Their product contributes at least q^{c_3 n_j}; unitary root orbits can only increase the corresponding root-group orders. The radical of the smaller parabolic contains that of the maximal one. Fixed-spacing thinning makes c_3n_j>Cn_i^2 uniformly. If M_{ij} normalized such a parabolic J with radical R, use (4.8). Since P_{ij}R\le N_{T_j}(J)<T_j, its full inverse image is proper in z_j. The soluble product \widehat R\trianglelefteq\overline U then satisfies (4.6). In particular R\le P_{ij}, contrary to |R|\ge q^{c_3 n_j}>q^{C n_i^2}\ge|P_{ij}|. This proves (v).□

Proof. Set Q=P_0\cap E_1 and A=\pi(Q). The group \overline P_0/A is a quotient of P_0/Q, which embeds in P_1/E_1, so it is soluble. A perfect group has trivial image in a soluble group. Consequently \overline E_0\leq A. Let B be its full inverse image in Q. Then B\lhd P_0 and

Because B/Z_0 is perfect, B=B'Z_0, and therefore B'=[B'Z_0,B'Z_0]=B''. Thus B' is perfect. The inverse image in B of Z(\overline E_0) has central commutators, and its intersection with B' is central in B': for an element of this inverse image, commutation with B' is a homomorphism from the perfect group B' to the abelian group Z_0. It is trivial. It follows that B' is a perfect central extension of the nonabelian simple group \overline E_0/Z(\overline E_0), hence is quasisimple.

The quotients P_0/Q, Q/B\cong A/\overline E_0, and B/B' are soluble. Hence P_0/B' is soluble. Since B' is perfect, it equals P_0^{(\infty)}. Finally B'\leq Q\leq E_1, and the terminal derived subgroup is characteristic.□

Proof. Steinberg’s multiplier Theorem has only a finite list of exceptions for the split groups, and the accompanying twisted statement has another finite list [36, Theorem 1.1 and the opening paragraph of p. 528]. Their ranks are bounded, independently of the field size. We supply perfectness separately: the root-presentation statement [35, Lemma 32', original pp. 51–53] applies to the universal split groups, and [35, Theorem 34 and its proof, original pp. 187–188] gives the twisted case outside its finite list of exceptions. The root-generated groups used there are the full fixed-point groups for simply connected groups over finite fields: see [35, Corollary 3 to Theorem 7, original p. 65, and the Corollary after Lemma 64, original p. 186]. Increase the discarded rank bound beyond these finite exception lists. Thus H_1(U,\mathbb Z)=H_2(U,\mathbb Z)=0 and u is the usual central simple quotient. This argument retains every field size, including q=2,3,4.

Let A=\ker a\leq Z(L). Pull back a along u to obtain the central extension

The action on A is trivial. The universal coefficient sequence

gives H^2(U,A)=0. The extension splits, and projection of a splitting to L supplies \pi. Two such maps differ by a homomorphism U\to A, which is zero because U is perfect. If H=\pi(U), then L=HA and A is central. Therefore

so \pi is onto. Its kernel is contained in \ker u\leq Z(U).

For a projective representation, compose with \pi and pull back \operatorname{GL}(V)\to\operatorname{PGL}(V). This is a central extension of U by k^\times. The same cohomology argument splits it; perfectness gives uniqueness of the resulting linear lift.□

Proof. Put I=\operatorname{ind}_{\mathbf G^\sigma}^{\mathbf G}k. The induction filtration of [6, arXiv version, Proposition 3.1.2], obtained from the coordinate-ring filtration and the Lang map, gives I a filtration with sections

each once. Write such a section as \nabla(\lambda)\otimes\nabla(\mu)^{[f]}, where \mu is obtained from \lambda^* by the diagram automorphism. Choose the regular height

It is invariant under diagram automorphisms and duality, is positive on every nonzero dominant weight and every positive root, and satisfies h(\mu)=h(\lambda).

We first justify removing the actual constants from this filtration. A nonconstant section cannot contain a trivial submodule. Indeed a map from k into it would transpose to a nonzero map \Delta(\lambda^*)\to\nabla(\mu)^{[f]}. The image contains the simple socle L(q\mu), so its highest weight must occur in the source. This gives qh(\lambda)\leq h(\lambda), impossible when \lambda\neq0. Let I_j be the first filtration stage containing the actual one-dimensional space of constants. That space injects into I_j/I_{j-1}, so this section is the unique section for \lambda=0. The constants split it and I_j=I_{j-1}\oplus k. The images of the earlier stages in I/k, followed by I_i/k for i\geq j, give a filtration of I/k; omitting the repeated stage removes exactly the constant section. Thus its sections are precisely the nonconstant ones.

We prove that a nonzero q-restricted simple L(\nu) cannot map to a nonconstant section. Suppose otherwise. Transposing the second tensor factor gives a nonzero homomorphism

Its image contains the simple socle L(\lambda). Filter the Weyl module by simple composition factors and apply Steinberg’s tensor product Theorem [34, Theorems 5.1 and 6.1]. Since \nu is q-restricted, each simple composition factor of the source has highest weight \nu+q\eta for some dominant \eta. Consequently

Transposing the first tensor factor instead gives a nonzero homomorphism

Its image contains L(q\mu), so the weight q\mu occurs in its source. The maximum height of a weight of that source is h(\lambda^*)+h(\nu)=h(\lambda)+h(\nu). Thus

Equations (5.1) and (5.3) are inconsistent when q>2.

When q=2, both inequalities are equalities. Hence \eta=0 and \lambda=\nu. The only weight of maximum height in the source of (5.2) is \lambda^*+\nu, and its weight space is the one-dimensional span of the tensor a\otimes b of highest-weight vectors. If 2\mu\neq\lambda^*+\nu, the required weight does not occur. Otherwise the homomorphism must be nonzero on a\otimes b. Choose a simple root \alpha with \langle\nu,\alpha^\vee\rangle=1, which is possible because \nu is nonzero and 2-restricted. For the Chevalley raising and lowering operators,

The target of (5.2) is Frobenius twisted, so its Lie algebra action is zero. Every module homomorphism to it kills the displayed vector, a contradiction.

A homomorphism from a finite-dimensional simple module to I/k has image in a finite stage of the filtration. Taking the first stage with nonzero intersection with that simple image gives a nonzero map into one of its sections. We have proved

Apply this also to the dual of the coefficient module. Tensoring 0\to k\to I\to I/k\to0 with L(\nu) shows that the map from H^1(\mathbf G,L(\nu)) to H^1(\mathbf G,L(\nu)\otimes I) is injective. The tensor identity for induction and derived Frobenius reciprocity identify this with the asserted restriction map. For split groups this is also the special case U=k of [14, Theorem 7.4]; the argument above includes the diagram twist and the equality case q=2.□

Proof. The polar form B of Q is nonzero: otherwise Q is the square of a linear form, and that linear form would be invariant under the nontrivial irreducible finite action. Its radical is invariant, so B is nondegenerate. Thus the finite representation is self-dual. The injective parametrization by q-restricted highest weights in Steinberg’s finite-group restriction Theorem [35, Theorem 43] implies that V\cong V^* rationally. Schur’s Lemma, first for \mathbf G and then for \mathbf G^\sigma, shows that their spaces of invariant bilinear forms are both one-dimensional and agree. In particular B itself is \mathbf G-invariant.

Polarization gives an exact sequence of rational modules

The first map sends a twisted linear form to its square. The obstruction to refining B by a rationally invariant quadratic form is its connecting class

This class restricts to zero on \mathbf G^\sigma, because Q is a refinement there. This description is also [18, Proposition 3.6 and Corollary 3.9].

Let \mathbf G_1 be the first Frobenius kernel. Its action on (V^*)^{[1]} is trivial. The inflation–restriction sequence therefore begins

The required first-kernel calculation is

It is reproduced in the proof of [18, Proposition 7.1, p. 411], citing [7, Theorem 3.1(C)(a),(f)]. Here the symplectic case includes A_1=C_1 and B_2=C_2. The exceptional module is L(\omega_1) in C_n numbering, including n=1, and L(\omega_2) in B_2 numbering. It is simple and self-dual. After untwisting, the last invariant space is therefore zero under our hypothesis. Inflation, followed by untwisting the quotient by \mathbf G_1, gives an isomorphism

Write c_B=F^*c, where F is ordinary Frobenius. Put \Gamma=\mathbf G^\sigma and M=V^*. Since F commutes with \sigma, its restriction F_\Gamma is an automorphism of \Gamma. Naturality gives the commutative square

The lower arrow is an isomorphism, with inverse obtained by pulling back along F_\Gamma^{-1} with the transported coefficients. This does not identify M with its Frobenius twist. Hence the vanishing of \operatorname{res}(c_B) implies that of \operatorname{res}(c). This and Lemma 5.4, applied to V^*, imply c=0 and c_B=0. Sequence (5.4) supplies a \mathbf G-invariant quadratic refinement Q_0 of B.

Finally Q-Q_0 has zero polarization, hence is the square of a linear form \ell. Its finite invariance implies that \ell is invariant under \mathbf G^\sigma. Such a linear form is zero on this nontrivial irreducible module. Therefore Q_0=Q.□

Proof. A line stabilizer fixes v, because a scalar preserving its nonzero quadratic value has square one, and the field has characteristic two. Choose a complement U to \langle v\rangle in v^\perp and choose t with B(v,t)=1 and B(t,U)=0. The space U is symplectic. An isometry inducing the identity on \overline W sends u\in U to u+\ell(u)v. Preservation of Q forces \ell(u)^2Q(v)=0, so it fixes U pointwise. Preservation of B then forces its image of t to be t+av. Preservation of Q says a^2Q(v)+a=0, hence a=0 or a=Q(v)^{-1}. These are exactly the identity and r_v.

For completeness, let s\in\operatorname{Sp}(U). The difference Q(u)+Q(su) has zero polarization, so, because the field is perfect, there is a unique linear form \ell satisfying

The map v\mapsto v, u\mapsto su+\ell(u)v is an isometry of v^\perp. Witt’s extension theorem [37, Theorem 7.4, pp. 57–58] extends this isometry to W: the ambient polar radical is zero, so its radical-intersection criterion is automatic even though v^\perp has degenerate polar form. This proves surjectivity. The formula for r_v also shows that it commutes with every isometry fixing v.

The Dickson invariant of an isometry in even characteristic is \operatorname{rank}(g-1) modulo two; in particular it is one on r_v. Its zero kernel is \Omega(W,Q) over a finite field of even order [37, Theorems 11.43 and 11.51, pp. 160, 164–165]; the exceptional group \Omega^+(4,2) is outside the dimension range here. Each fibre of the displayed surjection has two elements differing by r_v, so exactly one lies in this kernel. The last finite assertion follows.

We also justify the algebraic assertion, including its scheme convention. Put W_0=v^\perp. Restriction defines

The kernel of restriction from the full orthogonal vector stabilizer is the constant group of order two computed above: over an arbitrary k-algebra its remaining parameter satisfies a+a^2Q(v)=0, an étale equation. Its nonidentity element has Dickson invariant one, so the displayed map has trivial scheme-theoretic kernel. We give its inverse over an arbitrary k-algebra A. Put c=Q(v) and retain the decomposition W=kv\oplus U\oplus kt used above. An element h\in\operatorname{SO}(W_0)(A) has symplectic quotient action, of determinant one; the odd special orthogonal determinant condition therefore makes it fix v. Write h(u)=su+\ell(u)v for u\in U_A. There is a unique w\in U_A such that B(su,w)=\ell(u) for all u. The extension fixing v and restricting to h is obtained by setting

The bilinear condition is B(su,w)+\ell(u)=0, and the quadratic condition is exactly the displayed equation. Its derivative is one and its leading coefficient is a unit, so adjoining a solution gives an étale double cover. The two extensions differ by r_v and have opposite Dickson invariants. The unique Dickson-zero extension on this cover agrees on overlaps and descends to A. This is a functorial inverse, proving the asserted scheme isomorphism. The odd special orthogonal group is smooth, connected, and adjoint of type B_r by [15, author version, Theorem C.2.11 and Proposition C.3.10]. This restriction calculation is also the one in the proof of [15, author version, Proposition C.3.1]. Its action on W_0/\langle v\rangle is the purely inseparable isogeny to the symplectic group described in [15, author version, Remark C.3.6]; consequently this quotient is the simple module N_{B_r}. Both outer factors of the flag are trivial. We used the vector stabilizer: the line-stabilizer scheme \operatorname{SO}(W,Q)_{\langle v\rangle} is H\times\mu_2. Indeed its scalar on v has square one, and multiplying by the inverse central scalar makes it fix v; the two factors have trivial intersection. Its reduced subgroup and its finite-field points are the ones just described.□

Proof. Finite components and their normalization. We first specify the components in the structures of Proposition 4.1. A nondegenerate block or form subgroup contains the derived classical group on each active block. A tensor subgroup contains the central product of the derived groups on its tensor factors. An extension-field or subfield subgroup contains the derived classical group over its indicated field. In the retained twisted tensor construction, the single Frobenius orbit of the tensor factors has fixed-point group equal to the classical group over the corresponding extension field; its derived group is the active quasisimple group. The almost-simple case contains its quasisimple socle preimage.

The reductions leave one nonabelian active factor. Hence every permitted normalizer preserves it. All other base factors commute with it modulo scalars, and their surviving images are soluble. The retained diagonal, field, graph, and factor-permutation quotients are soluble as well; the exclusion of other nonabelian factors here includes factors in the permutation quotient, not just base factors. Thus the explicitly displayed active derived group is normal and the quotient is soluble. Scalar preimages introduce only central kernels: if C/Z is the displayed perfect active factor, with Z central, then C=C'Z and C'=C'', giving its quasisimple derived preimage. For the characteristic-two nonsingular-line case, Lemma 5.7 gives the asserted component directly. For a two-edge projection, we verify the precise hypothesis of Lemma 5.1. Use the common upper-coordinate notation P_0\leq P_1 and Q_1\leq R_1\leq P_1 from Lemma 3.8. Here Q_1 is soluble, R_1/Q_1 is the middle simple active factor, and P_1/R_1 is soluble. The upper retained edge supplies the quasisimple component E_1=P_1^{(\infty)}. Perfectness gives E_1\leq R_1. Its image in R_1/Q_1 is nontrivial, since Q_1 is soluble, and is normal, hence is the full simple group. Therefore

Both E_1 and Q_1 are normal in P_1. Thus, for x\in Q_1, commutation with x maps E_1 into E_1\cap Q_1=Z(E_1). It is a homomorphism to an abelian group, so perfectness makes it trivial. Consequently Q_1 centralizes E_1. Conjugation on E_1 agrees with conjugation on E_1/Z(E_1)\cong R_1/Q_1: an automorphism of the perfect group E_1 acting identically modulo its centre differs from the identity by a homomorphism E_1\to Z(E_1) and is therefore the identity.

The original-core product identity in Proposition 2.9 gives Q_1\leq P_0. Indeed its transported product comes from the middle original core, which is contained in D and hence in the lower labelled group. It follows that

is the actual first-edge inner projection of Lemma 3.8. Its retained structure supplies the normal quasisimple component J^{(\infty)} with soluble quotient. This is a structural assertion about this particular J; the excluded parabolic or module semidirect extensions have not been inferred to split.

Let \pi:P_1\to\operatorname{Aut}(E_1) be conjugation. By the centralization and simple-quotient identification just proved, \pi(P_0\cap R_1) is the inner image of J, isomorphic to J. It is normal in \pi(P_0), and \pi(P_0)/\pi(P_0\cap R_1) is soluble because it is a quotient of P_0/(P_0\cap R_1)\leq P_1/R_1. Hence the image of J^{(\infty)} is normal quasisimple in \pi(P_0), with soluble quotient. These are exactly the hypotheses of Lemma 5.1, which produces the actual lower component inside E_1.

Finally, if P\leq P' and L' is the component of P', the image of the perfect group L in the soluble group P'/L' is trivial. Hence L\leq L'. Characteristicity gives normalization by the common bottom. Each intermediate inner projection is normal in its own coordinate stabilizer, which contains the lower coordinate stabilizer; its characteristic component is therefore normalized by that lower stabilizer too.

To check that the active component is not killed, use the notation Q\leq R\leq P of Proposition 2.9. In our multiplicity-one coordinate, R/Q is the nonabelian simple active factor and P/R is soluble. Hence L\leq R. Since P/L is soluble, the simple group R/Q has trivial image modulo LQ, and LQ=R. Thus L maps onto R/Q. Transporting L over all upper coordinates gives a normal subgroup of the lower labelled group, mapping onto its socle modulo its original bottom core. If this product were contained in the common bottom, normality would put it inside that core, a contradiction.

For each actual finite component L_i, fix its standard cover U_i=\mathbf G_i^{\sigma_i} and the surjection \pi_i:U_i\twoheadrightarrow L_i of Lemma 5.3, over the specified simple quotient.

Rational maps for the geometric edges. We have obtained the finite components, their nesting, and their survival under the original bottom-core quotient. We now lift their inclusions. For the geometric cases we describe the maps on natural modules over k, which also proves the assertion about their composition factors. Write N for the source natural simple module.

1. (i)

   A block embedding acts on its active block by g, on a dual block, when present, by g^{-\mathsf T}, and trivially on omitted spaces. Its constituents are N, N^*, and trivial modules. A form embedding is the natural action with its prescribed invariant bilinear or quadratic form. These are rational matrix maps.

2. (ii)

   For a tensor factor with an omitted space U, the map is g\mapsto g\otimes1_U. On the active factor its natural module is the direct sum of \dim U copies of N, or of a prescribed dual or Frobenius twist of N. The commuting omitted factor changes the multiplicity space, not this formula.

3. (iii)

   Restriction of scalars from \mathbb F_{p^{ab}} to \mathbb F_{p^a}, followed by extension of scalars to k, has the explicit decomposition

   k\otimes_{\mathbb F_{p^a}}\mathbb F_{p^{ab}} \cong\bigoplus_{j=0}^{b-1}k, \qquad x\longmapsto(x,x^{p^a},\ldots,x^{p^{a(b-1)}}).

   Thus its map is the direct sum of the rational representations N^{[aj]}. If a form pairs two summands, the corresponding summand is expressed as the required dual. A subfield inclusion is the same natural matrix map, possibly precomposed with its specified field or diagram automorphism. On natural modules in the ranks in use, a diagram automorphism gives N or N^*; triality has already been discarded with bounded rank.

4. (iv)

   In a twisted tensor orbit the map is

   g\longmapsto \rho_0(F^{a_0}g)\otimes\cdots\otimes\rho_{b-1}(F^{a_{b-1}}g),

   where each \rho_j is natural or dual. This is a rational representation and it is the defining matrix construction of the finite subgroup. Its natural module is \bigotimes_j N_j^{[a_j]}, N_j\in\{N,N^*\}. The orbit lengths, field degrees, and omitted tensor dimensions are bounded by Proposition 4.16.

In characteristic two, two further classical maps must be interpreted literally. The nonsingular-vector stabilizer in a group of type D_{r+1} has connected reduced subgroup of type B_r; its action on the 2r+2-dimensional natural space has, by Lemma 5.7, the invariant filtration

with factors k,N_{B_r},k. Its simply connected cover therefore supplies the required B_r-to-D_{r+1} map. The middle symplectic action is an isogeny from this adjoint B_r stabilizer to \operatorname{Sp}_{2r}. It has no nontrivial kernel on k-points, consistently with Lemma 5.7; it is not a finite unipotent radical. In the following display, instead, \mathbf B_r=\operatorname{Spin}_{2r+1} and \mathbf C_r=\operatorname{Sp}_{2r} denote the simply connected groups. The preceding vector stabilizer is the adjoint quotient of \mathbf B_r; the kernels of its symplectic action and of the simply connected maps are not being identified. For the two special isogenies

one has

by [17, author preprint, §I.3.2]. If the inverse of s on \mathbb F_q-points is needed, it is induced by the rational map tF_p^{f-1}, since q=p^f. No inverse inseparable morphism is asserted. These formulas preserve the claimed description of simple constituents, including the trivial factors of the full odd orthogonal module.

Rational maps for the nonlinear edges. For a defining-characteristic almost-simple edge, inflate its actual finite irreducible representation along \pi_i:U_i\twoheadrightarrow L_i, using the unique linear lift of Lemma 5.3 if the natural action was given projectively. Surjectivity preserves every invariant subspace and every prescribed form. A form similitude multiplier is a character of the perfect group U_i, hence is trivial. Only now apply Steinberg’s restriction Theorem [35, Theorem 43]: the resulting U_i-module is the restriction of its q_i-restricted rational simple module, with the specified field and diagram twist. Here \sigma_i=F_{p^{f_i}}\gamma_i uses f_i ordinary Frobenius digits, including in the unitary case; the unitary natural scalar field \mathbb F_{p^{2f_i}} does not change this restriction parameter. This yields a rational map to \operatorname{GL}(V) whose determinant is trivial: a connected semisimple group has no nontrivial rational character. Thus it maps to a linear target, including the algebraic group underlying a finite unitary target. In a symplectic target or an orthogonal target in odd characteristic, finite self-duality and the injective highest-weight parametrization identify the invariant rational bilinear form with the finite form, by Schur’s Lemma. Its symmetry or alternating property is consequently the prescribed one. For an orthogonal target in characteristic two the remaining nonlinear module satisfies Proposition 5.5, which lifts its actual quadratic form. The natural and special-isogeny cases are the geometric maps already constructed. The image is connected and hence lies in the identity component of the isometry group.

Compatibility through the fixed central covers. These constructions initially give maps into whichever central form of the target is specified by its natural action. A homomorphism from a simply connected semisimple source lifts through a central isogeny of semisimple algebraic groups; see [15, author version, Exercise 6.5.2(iii), p. 236]. Thus they give maps between simply connected covers as claimed. Let \phi_i:\mathbf G_i\to\mathbf G_{i+1} be such a map and let \iota_i:L_i\hookrightarrow L_{i+1} be the actual finite inclusion. By construction their induced maps to the target adjoint group agree on U_i. For u\in U_i, the target adjoint image of \phi_i(u) is fixed by \sigma_{i+1}, so

Centrality gives \delta(uv)=\delta(u)\delta(v). Perfectness of U_i forces \delta=1, so \phi_i(U_i)\leq U_{i+1}, including when the algebraic centre is nonreduced. The two homomorphisms \pi_{i+1}\phi_i|_{U_i} and \iota_i\pi_i to L_{i+1} agree modulo its centre. Their discrepancy is consequently a homomorphism U_i\to Z(L_{i+1}), and again is trivial. Thus we have the exact commutative diagram

This proves compatibility with the actual finite inclusions through the fixed central-cover maps. It gives no uniqueness assertion for rational maps agreeing on finite points.□

Proof. The finite compatibility statements for the edges compose, so the first assertion follows by induction on j-i. Every subspace invariant under the algebraic image is invariant under the contained finite image. Finite irreducibility therefore implies algebraic irreducibility, without a density assertion or a bound on the highest weight of the composite representation. The kernel of a nontrivial map from a connected simple algebraic group is finite as a group scheme: its connected reduced normal subgroup cannot be the whole source. Its image is closed and connected, and is simple up to the allowed central form. Inseparability can interchange the types B and C in characteristic two, but does not change rank. The last assertion follows from the rank inequality.□

Proof. Work over the algebraic closure. Any proper submodule or nontrivial composition filtration for the L-action would be an H-invariant one. Thus the target module is a single nontrivial composition factor for L. Proposition 5.8 expresses it as a tensor product of twists of the source natural module or its dual. If one factor had a nonzero proper H-invariant subspace, tensoring that subspace with all the other factors would give a nonzero proper invariant subspace of the product. Hence each factor is irreducible on H. Taking a dual or applying a field Frobenius to all matrix entries preserves irreducibility, so the source natural module itself is irreducible on H.□

Proof. This is Proposition 4.16(v), with its fixed-spacing thinning already incorporated in the retained chain.□

Proof. Work first over the finite natural scalar field. Put

If V is not semisimple, R_0\ne0, and a simple submodule of R_0 shows that U\ne0. Also W<V: the proper submodule S is contained in a maximal submodule, which necessarily contains R_0. Socles and radicals are intrinsic, so every semilinear normalizer preserves these constructions.

For a linear group without graph operations, U gives an M-normalized parabolic, contrary to Lemma 6.1. Suppose that M also contains duality operations. The side-preserving subgroup has index two. A duality acts on subspaces by a semilinear correlation, and its transport of a module to the dual exchanges the socle with the annihilator of the radical, and the radical with the annihilator of the socle. Consequently it exchanges U and W as members of a flag. Thus the flag U\le W is preserved by all of M; if U=W it is a one-member flag. Both members are nonzero and proper. This again contradicts Lemma 6.1. This argument uses correlations on subspaces and does not treat a graph automorphism as a linear operator on V.

In a bilinear or Hermitian space, identify the module with its appropriate dual using the preserved form. Duality of socles and radicals gives

Hence U is totally isotropic. This supplies a forbidden parabolic in the symplectic, unitary, and odd-characteristic orthogonal cases.

It remains to consider a quadratic form Q_V in characteristic two. On a totally polar-isotropic space U its restriction has the form \ell^2, for a linear functional \ell, since the finite field is perfect. Its zero space is an M-invariant singular subspace. Unless U is a nonsingular line, this zero space is nonzero and contradicts Lemma 6.1. We are therefore left with the nonsingular line R=U.

The space R^\perp/R has a nondegenerate alternating form. If its E-module were not semisimple, the argument with socle and radical would give a nonzero M-invariant totally isotropic subspace A of that quotient. Its inverse image B in R^\perp is totally polar-isotropic and contains R. On B the quadratic form is the square of a linear functional nonzero on R. Its zero hyperplane is therefore a singular complement to R, projecting isomorphically onto A. This complement is M-invariant and again gives a forbidden parabolic in the original quadratic space. The quotient is semisimple.

The flag 0<R<R^\perp<V adds precisely two one-dimensional composition factors to that quotient. Both are trivial for E because E is perfect. By Lemma 5.7, the inner stabilizer has the canonical finite-group isomorphism

Every m\in M induces a semilinear similarity m^\circ on V^\circ. Evaluation on quotient vectors gives

Consequently inverse images under \theta preserve normalization by the actual M, inclusions, and exact commutation of inner subgroups. Since P=M\cap T_j preserves R, it lies in this inner stabilizer. Write P^\circ=\theta(P) and E^\circ=\theta(E); then P^\circ/E^\circ\cong P/E. Every candidate section subgroup is pulled back through \theta before applying a normal test. Every lifted enlargement lies in the proper original line stabilizer, even if its symplectic image is the whole symplectic group. Transport over the original coordinates uses the same normal higher core F_j in (N1) and (N2). There is no additional finite kernel in this inner-group isomorphism.

Finally, extending the finite scalar field to its algebraic closure preserves semisimplicity here. Indeed the semisimple image algebra of the finite group is a product of matrix algebras over finite fields; those fields are separable, and scalar extension preserves semisimplicity of that algebra.□

Proof. Write C=C_{T_j}(E) in the original natural realization. The group MC normalizes E and cannot contain T_j: the proper nontrivial E cannot be normal in the simple ambient group. Hence the full preimage of \overline U=\langle z_i/F_j,\widehat E,\widehat C\rangle is proper in z_j. The two products \widehat E,\widehat C are normal in \overline U and commute exactly. Their full preimages commute modulo F_j\trianglelefteq U, with F_j\le D. The active product \widehat E is not killed, so (N2) kills \widehat C and forces C\le P. Since

C is soluble.

For the symplectic section of Lemma 6.2, put E^\circ=\theta(E) and take the inverse image under \theta of C_{\operatorname{Sp}(V^\circ)}(E^\circ). The M-equivariance and injectivity of \theta show that this inner inverse image and E commute exactly and are normal in their enlargement with M. Their transported products again commute in N_j/F_j, and their original preimages commute modulo the normal subgroup F_j\le D. The enlargement lies in the proper original nonsingular-line stabilizer even if its symplectic image is the full symplectic group. Condition (N2) and the same quotient calculation prove that the symplectic centralizer is soluble.□

Proof. We give the linear-algebra details needed for a bound independent of the fields and constituent dimensions. Let U be a simple module over the finite natural field F. Its endomorphism division ring is a finite field F'. On an isotypic summand U^m the commuting endomorphism algebra is \operatorname{Mat}_m(F'). After scalar extension, U splits into distinct absolute simples, each once; thus m is exactly their common absolute multiplicity.

In the linear case the centralizer contains \operatorname{GL}_m(F'). Its subgroup \operatorname{SL}_m(F') has determinant one on the whole F-space: its determinant there is a power of the norm of its F'-determinant. Thus its derived subgroup survives the ambient special-linear condition.

For a form group, pair the simple isotypic summands by duality. A pair of distinct dual summands again supplies \operatorname{GL}_m(F'), acting on the second summand by the inverse adjoint. On a self-paired summand the adjoint operation on \operatorname{Mat}_m(F') is an involution. If it acts nontrivially on F', the commuting isometries are a unitary group on an m-dimensional multiplicity space. If it fixes F', they preserve a nonsingular symmetric or alternating bilinear form on that multiplicity space. These descriptions follow by writing the invariant pairing as a matrix of pairings between the m copies of U; Schur’s Lemma makes each matrix entry an element of F', and the symmetry of the original form is precisely the adjoint condition on that matrix.

There is one useful even-characteristic qualification. A nonsingular symmetric multiplicity form need not be alternating. Over a perfect finite field its diagonal function is the square of a linear functional. The vector representing that functional allows one to split off a nonsingular subspace of dimension one or two; its perpendicular is alternating and nonsingular. Thus a multiplicity space of dimension m still contains a nonsingular alternating subspace of dimension at least m-2.

If m\ge16, every group just described contains a subgroup isomorphic to \operatorname{SL}_3(F'') for a suitable finite field F''. In a general linear group use a three-dimensional block. In a symplectic, orthogonal, or unitary multiplicity space choose paired totally isotropic three-spaces and let g\in\operatorname{SL}_3(F'') act as g and its inverse adjoint, fixing their perpendicular complement. Finite-field forms of dimension at least 16 have such paired spaces; the anisotropic part has dimension at most two. The preceding paragraph handles the nonalternating symmetric form in characteristic two in the same way.

These embedded groups are perfect and have nonabelian simple projective quotients. Being perfect, their determinants, similitude multipliers, spinor norms where applicable, and Dickson invariants in even-characteristic orthogonal groups are trivial. Therefore they lie in the actual inner classical group, rather than merely its full similitude group. A scalar quotient cannot make their images soluble.

In an orthogonal target of characteristic two, preserving the polar form on a nontrivial isotypic summand also preserves the quadratic form. Indeed two E-invariant quadratic refinements of that polar form differ by \ell^2 for an E-invariant linear functional \ell. On a sum of nontrivial simple E-modules there is no such nonzero functional. A commuting polar isometry consequently preserves the given refinement. On the trivial isotypic space use its actual quadratic isometry group; the same paired-three-space construction applies. The trivial space is nondegenerate, since it is orthogonal to every nontrivial isotypic summand.

We have proved that m\ge16 would put a nonsoluble subgroup in the centralizer. Lemma 6.3 excludes this.□

Proof. Let V_1,V_2 be two such unions, and put all remaining parts in V_3. Let H_1,H_2 be the derived groups of the full linear or isometry groups on V_1,V_2, acting trivially on the other parts. They commute, are normalized by M, and are normal in the enlargement generated by them and M. Its inner part preserves the proper decomposition and cannot contain T_j. Thus \overline U=\langle z_i/F_j,\widehat H_1,\widehat H_2\rangle has proper full preimage in z_j. Both H_1,H_2 are nonsoluble: the explicit \operatorname{SL}_3 construction in the proof of Lemma 6.4 lies in each derived group and in the ambient inner group.

The two transported products are normal in this \overline U and commute exactly; their original preimages commute modulo F_j\trianglelefteq U, with F_j\le D. Condition (N2) kills one product and forces, say, H_1\le P. The kernel of the nontrivial action of E on V_2 is central in E, so H_1\cap E\le Z(E). Hence

would make H_1 soluble, a contradiction. On the symplectic section, take inverse images under the finite M-equivariant isomorphism \theta from Lemma 6.2. They commute exactly and their enlargement lies in the proper original line stabilizer, so the same application uses the original higher core F_j, as in Lemma 6.3.

Suppose that a nontrivial packet has dimension at least 16. If another nontrivial packet occurs, apply the first assertion to the large packet and its entire complement, including the trivial packet. The complement has dimension at most 15. If no other nontrivial packet occurs, the complement is the trivial isotypic space, whose dimension is at most 15 by Lemma 6.4. If no packet has dimension 16, but their total dimension were at least 46, accumulate packets until their dimension first reaches 16. The accumulated dimension is at most 30, leaving at least 16 dimensions in the other union. This contradicts what was just proved. Thus in this case the total is at most 45.□

Proof. We work on simply connected covers and use Frobenius positions modulo the group parameter, rather than the degree of its natural scalar field. Write a classical finite group with parameter p^r as the fixed points of \tau F_p^r, where \tau is a diagram operation. Steinberg’s Restriction and Tensor Product Theorems give the unique parametrization

of its absolute simple modules; see [35, Theorems 41 and 43] and [34]. For a nontrivial simple module define

For an arbitrary module take the union of these supports over its nontrivial composition factors. A field twist by F_p^b translates support by b. Diagram operations and contragredience preserve positions. When a position wraps modulo r, its restricted weight is transformed by \tau^{-1}, which does not change the position.

Consider one edge, with source and target exponents r,t, and let d=\gcd(r,t). Suppose that the nonempty union \mathcal E of source digit supports in the target natural module lies in one residue class modulo d. If a normalizer has target scalar exponent b and source field exponent a_0, its transport of a composition factor has type

where \gamma records the source diagram operation. This formula follows directly from x(mw)=m(\alpha_m^{-1}(x)w) and the p^b-semilinearity of m. It applies to composition factors by transporting a composition series; semisimplicity on an individual edge is not required. An ambient correlation adds contragredience. Inner-diagonal automorphisms do not affect the absolute module type, because they are algebraic inner conjugations on the cover. Normalization therefore implies

Since \mathcal E is nonempty and contained in a single residue class modulo d, we obtain

Assume for the moment that this support condition holds for both edges of the shortcut. Their source and target exponents are f,f' in opposite orders, so their common divisor d is the same integer h. For an element of M, denote its induced intermediate field exponent by a_1. Its existence on the same intermediate component is the normalization conclusion of Proposition 5.8. The natural realization of that automorphism has scalar exponent a_1, with any graph part recorded separately. Applying (6.2) to the two edges gives

Their sum proves (6.1).

To bound a packet, let \Phi denote the operation of field twisting on absolute source types, and let \Gamma be the diagram group. The rank assumption gives |\Gamma|\le2. These operations commute, and \Phi^f=\tau^{-1}. Every stabilizer action lies in \langle\Phi^h,\Gamma\rangle. If \tau=1, that group has order at most 2f/h. If \tau\ne1, the diagram involution already belongs to \langle\Phi^h\rangle, whose order is at most 2f/h. Thus in either case the order is at most 2v. Descent over the upper natural field has exponent f or 2f and belongs to the same group. Duality is the opposition diagram operation, also already included. The packet bound follows.

It remains to verify the support hypothesis for every remaining edge of Proposition 4.16 and Proposition 5.8.

1. (a)

   In a nonlinear edge the natural module is a Frobenius twist of a nonzero restricted module. Its support is a singleton. The starting Frobenius position may be arbitrary; it cancels in (6.2).

2. (b)

   A block, form, or subfield edge has natural or dual-natural nontrivial composition factors in one Frobenius position, allowing repetitions and trivial factors. A tensor edge with one active component has the same property: its other tensor factors are trivial for that perfect component, so they supply repetitions. The characteristic-two nonsingular-line representation has these composition factors as well, with the additional trivial factors already described.

3. (c)

   For restriction of scalars, the source constituents are a direct sum of the Galois conjugates of the natural module. To distinguish group parameters from natural scalar fields, put \epsilon_s,\epsilon_t=2 for a unitary source or target, respectively, and 1 otherwise. A scalar restriction of degree b_{\rm sc} satisfies

   \epsilon_s r=b_{\rm sc}\epsilon_t t.

   Its constituent positions are t_0+j\epsilon_t t, so they all lie in one residue class modulo d=\gcd(r,t). For example, restricting the natural matrix field of a unitary component with parameter p^r to a symplectic target with the same parameter has scalar degree two. Over the algebraic closure its module is

   N\oplus N^{[r]} \cong N\oplus N^*.

   Both positions are zero modulo r; this repetition causes no tensor carry because the operation is a direct sum.

4. (d)

   For a retained natural twisted-tensor edge, let b_{\rm cyc} be the prime tensor-cycle length. Lemma 4.3 gives the exact usual-field relation

   r=b_{\rm cyc}t.

   In the finite realization supplied by Proposition 5.8, the cyclic fixed-point construction gives the following actual finite-source modules, up to a common initial Frobenius twist [25, Proposition 2.8(ii) and Section 4(5), p. 449]. Here W is the source natural module in each case.

   Linear, symplectic, and orthogonal targets. The module is

   \bigotimes_{j=0}^{b_{\rm cyc}-1}W_j^{[jt]},

   where each W_j is W or W^* as prescribed by the diagram operation. Its positions are jt for 0\le j<b_{\rm cyc}; diagram changes affect the weights rather than these positions.

   Unitary target, odd cycle. The source is \operatorname{SU}_m(p^r) and the module is

   \bigotimes_{j=0}^{b_{\rm cyc}-1}W^{[2jt]}.

   Indeed, the cycle initially gives W^{[jt]} for even j and (W^*)^{[jt]} for odd j. On this finite source, W^*\cong W^{[r]}. The integers j for even j and j+b_{\rm cyc} for odd j are precisely 0,2,\ldots,2b_{\rm cyc}-2, giving the displayed module. Its positions 2jt are distinct modulo r because b_{\rm cyc} is odd.

   Unitary target, two-cycle. Since the cycle length is prime, this is the only even case. The source is \operatorname{SL}_m(p^{2t}) and its module is

   W\otimes(W^*)^{[t]},

   with distinct positions 0,t modulo r=2t.

   Thus in every case the positions are distinct modulo r=b_{\rm cyc}t and lie in one residue class modulo d=\gcd(r,t)=t. A common initial twist adds the same t_0 to every position and preserves both assertions. Steinberg’s Tensor Product Theorem therefore gives the required support condition with no coincident-digit carries. This distinctness comes from the exact finite cyclic construction.

Changes between the type B and type C conventions in characteristic two do not alter the numerical field exponent. The special isogenies are defined over \mathbb F_2, commute with Frobenius, and have Frobenius as their composites [17, Section I.3.1]. Thus each edge can use the convention in which its displayed module has the indicated support, while the intermediate field action has the same exponent in either convention.□

Proof. Apply Lemma 6.2. On its semisimple natural section, Lemma 6.4 bounds each absolute multiplicity by 15, and Lemma 6.6 bounds each packet by 2B absolute types.

If there is a large nontrivial packet, it contributes at most 30B factors. All other nontrivial packets have total dimension at most 15 by Lemma 6.5. If no packet is large, their total dimension is at most 45. In either case the number of nontrivial composition factors is at most 30B+45. The trivial isotypic space has dimension at most 15, again by Lemma 6.4. Thus the semisimple section has length at most 30B+60. Adding the possible two factors from the nonsingular-line case gives the displayed bound.

All constants came from the remaining-edge bounds, explicit finite-dimensional linear algebra, and the common cyclic congruence (6.1). They are independent of field exponent, rank, and chain length.□

Proof. Apply the corrected irreducible-triple Theorem to V and use Table 1. Only \mathrm S_6 and \mathrm {MR}_5 require an additional argument. Each of their t blocks contributes a nontrivial constituent of the natural restriction, of dimension 2r. In \mathrm {MR}_5, passing from the odd orthogonal natural module to its simple quotient removes one trivial constituent and leaves all t nontrivial constituents. In \mathrm S_6, there are also two trivial constituents, which do not affect the lower bound t on length. Different Frobenius twists may change the isomorphism types of these constituents, but not their dimensions or their number counted with multiplicity. Thus t\leq c. The image conventions and (7.2) justify the same calculation on the original groups. No complete-reducibility assertion is used.□

Proof. Use the same representation \varphi:Y\longrightarrow\operatorname{SL}(V) throughout the proof. For (Y,H,V), every row of Table 1 outside \mathrm S_6 and \mathrm {MR}_5 gives the conclusion. Suppose one of these two rows occurs. Then p=2, the image of H is B_r, the image of Y is D_m or B_m, and V is a half-spin or spin module, respectively. Let W be the orthogonal natural module of this fixed ambient image, on a cover when necessary. Every nontrivial composition factor of W|_H has dimension 2r.

Since H<X, irreducibility on H implies irreducibility on X. Apply the irreducible-triple Theorem a second time to (Y,X,V), keeping the ambient group and representation unchanged. The theorem requires restrictedness and tensor-indecomposability on Y, so it makes no such assumption about the X-module V|_X. If the ambient image is B_m, the only row with a simple subgroup of rank at least ten is \mathrm {MR}_5. If it is D_m and V is half-spin, the only such rows are \mathrm {IV}_1, \mathrm {IV}'_1, \mathrm {IV}'_2, and \mathrm S_6. In all these cases the image of X is B_s and every nontrivial composition factor of W|_X is a Frobenius twist of N_X, of dimension 2s.

By hypothesis and (7.2), each such constituent remains irreducible on H. Restricting a composition series for W|_X and refining it over H therefore exhibits an H-composition factor of dimension 2s. This contradicts the preceding dimension 2r, since s>r. The argument works for nonsplit natural restrictions as well as for direct sums.

There is no extra case from \mathrm {MR}_4: its ambient image is C_m, whereas the image of the fixed representation here is B_m or D_m. One cannot change that image between the two applications. Equations (7.1)–(7.2) preserve the natural irreducibility hypothesis on passing to these image groups, including an inseparable isogeny. They do not replace the fixed spin representation V by a natural representation.

Finally, if X=Y, the irreducibility of N_Y|_H directly excludes \mathrm S_6 and \mathrm {MR}_5, since their natural restrictions have at least two nontrivial composition factors.□

Proof. Let U_1,\ldots,U_a be the nontrivial L_j-constituents, counted with multiplicity. If U_t|_{L_i} had only trivial composition factors, the image of the perfect group L_i would be conjugate into an upper unitriangular group and hence would be both perfect and solvable. It would be trivial, contrary to the nontriviality hypothesis. Thus, writing a_t,b_t for its nontrivial and trivial composition counts, we have a_t\geq1 and b_t\geq0. Additivity of composition multiplicities along restriction gives

It follows that a_t=1 and b_t=0 for every t, as required.□

Proof. By Theorem 6.7, fix an absolute integer c bounding \ell_{L_i}(N_j) for every retained comparison i<j. The pair of nontrivial and trivial composition counts has at most (c+1)^2 possible values. Finite Ramsey applied to these pair colors gives arbitrarily long subchains with constant counts (a,b). These restrictions are nontrivial, and a\geq1 follows because a perfect group cannot act nontrivially with only trivial composition factors. Moreover, a nontrivial simple representation of a quasisimple group has central kernel. Its restriction along the noncentral earlier component is therefore nontrivial. The hypotheses of Lemma 7.3 hold.

For each j<k, realize every nontrivial constituent U of N_k|_{L_j} as the restriction of the corresponding restricted finite-group highest weight module of \mathbf G_j. Steinberg’s Tensor Product Theorem decomposes that rational simple module into Frobenius twists of p-restricted simple modules; see [35, Theorem 41]. If a restricted factor is tensor decomposable, continue factoring it until its nontrivial factors are tensor-indecomposable. This terminates since every factor has smaller positive dimension. Every resulting factor is p-restricted: its dominant highest weight is a summand of a p-restricted dominant weight.

Color a triple i<j<k red if some such U has a nontrivial terminal factor V, not of natural type, and blue otherwise. In a red triple, Lemma 7.3 says that U|_{L_i} is irreducible. Hence each tensor factor V^{[d]} is irreducible on L_i. Use a path lift from \mathbf G_i to \mathbf G_j and replace its image by its p^d-Frobenius image. The module V is irreducible on this closed connected simple subgroup: any proper rational invariant subspace would also be invariant under the corresponding finite component. The subgroup has rank r_i and is proper because the endpoint ranks increase.

The natural restriction on this subgroup has algebraic composition length at most c. Indeed, restriction from an algebraic group to a finite subgroup is exact and cannot reduce composition length, and Frobenius transport preserves length. The finite restriction here is a twist of N_j|_{L_i}. Lemma 7.1 consequently bounds the number of possible middle ranks r_j by 2c+7, for fixed i.

A red homogeneous chain with more than 2c+9 vertices is impossible: fix its first and last vertices i,k and vary its middle vertex j. The distinct ranks r_j would exceed that bound. A second application of finite Ramsey therefore gives arbitrarily long blue homogeneous subchains.

Fix i<j<k in a blue subchain and choose a nontrivial constituent U of N_k|_{L_j}. There is at least one nontrivial terminal tensor factor, and every such factor is now of natural type. Since U|_{L_i} is irreducible, a natural or dual natural Frobenius factor restricts irreducibly to L_i. Undoing its twist, dual, and the isogenies (7.1) shows that N_j|_{L_i} is irreducible. Delete the last vertex of the blue subchain, so that a suitable k exists for every remaining pair i<j. All modules and restrictions were taken over k, so the resulting finite restrictions are absolutely irreducible.□

Proof. A tensor factor of a simple module is simple, since any proper nonzero submodule of the factor would tensor to one of the whole module. Highest weights add in an irreducible tensor product. The highest weights of the natural-type factors have the form p^a\omega_1, or p^a\omega_r in type A_r for the dual. Equation (7.1) shows explicitly that the special isogenies add no further natural weights. Since their sum is p-restricted and all fundamental-weight coefficients are nonnegative, every exponent a is zero. The terminal factors are therefore N_J or N_J^*.

Any two of these factors have reducible tensor product. For two identical factors, the kernel of the flip minus the identity is a nonzero proper invariant subspace: it contains v\otimes v, while v\otimes w-w\otimes v\ne0 for independent v,w. This also proves properness in characteristic two. For opposite factors, N_J\otimes N_J^*\cong\operatorname{End}(N_J) contains the proper invariant line of scalar endomorphisms. The same arguments apply to dual identical factors. Tensoring such a proper submodule with all remaining factors contradicts simplicity of M. There is thus exactly one terminal factor.□

Proof. Irreducibility on L implies irreducibility on the source component, so the target restriction has a single absolute composition factor. It is nontrivial because the target natural dimension exceeds one. By hypothesis this factor is a tensor product of natural-type modules. Irreducibility of its further restriction forces each factor to restrict irreducibly. Undoing a twist, a dual, and the natural isogenies yields the assertion for N_J. For retained geometric edges this also follows from Corollary 5.11. For the stated one-digit edges it follows from Lemma 7.5, after restoring the common twist.□

Completion of the proof of Theorem 3.2. Suppose tested chains with different endpoint labels have unbounded length. Apply Proposition 3.13 and Proposition 4.16. We obtain arbitrarily long configurations with strictly increasing endpoint ranks, with their minimum rank tending to infinity, and with two-edge shortcuts on all pairs. There are absolute constants c_0>0, C\geq1, and B\geq1 such that the natural endpoint dimensions satisfy

and every retained geometric edge satisfies

Every other edge is a defining-characteristic irreducible representation with one restricted digit after removing a common twist. At least one such nonlinear edge occurs in each shortcut. Proposition 5.8 and Lemma 5.9 supply the actual component homomorphisms and their algebraic path lifts. By Proposition 7.4, pass to arbitrarily long subchains with all natural endpoint restrictions absolutely irreducible.

Color a pair of endpoints red if its chosen shortcut has a one-digit edge with a nontrivial restricted tensor-indecomposable factor not of natural type. Color the pair blue otherwise. Finite Ramsey leaves arbitrarily long homogeneous configurations of one of these two kinds. We exclude both.

First consider a blue configuration. On every one-digit edge, Lemma 7.5 bounds the target natural dimension by the source natural dimension; restoring a Frobenius twist does not change dimension. Thus every edge of every shortcut satisfies (7.5), enlarging C,B if necessary. Composing at most two edges gives the uniform bound

Choose a fixed integer d with 2^d>E. Along d consecutive comparisons of a retained chain, (7.4) gives

whereas the shortcut from its first to its last endpoint gives n_d\leq A n_0^E. These inequalities contradict each other as the minimum rank, and hence n_0, tends to infinity. This proves the bounded-natural-factor case without imposing a bound on the number of Steinberg digits of an arbitrary composite endpoint representation.

Now consider a red homogeneous configuration. Fix an early endpoint H of rank r\geq9. Choose eight later pair comparisons with disjoint ordered index intervals, leaving a gap between successive intervals. For each such comparison, let X be its lower endpoint, and choose the last edge in its shortcut that has a restricted factor not of natural type. Let Y be the algebraic source of that edge and Z its target. Every edge after it has a natural tail in the sense of Lemma 7.6: it is either geometric or a one-digit edge with only natural restricted terminal factors. The upper endpoint natural module is irreducible on the finite component of H. Applying that lemma backwards along the tail shows that N_Z is irreducible on the same component.

Remove the common Frobenius twist of the Y-representation on N_Z, and choose a nontrivial restricted tensor-indecomposable factor V not of natural type. Irreducibility of the whole restriction implies irreducibility of the corresponding twisted factor on the finite H. Compose the actual algebraic maps along the path from H through X to Y, and apply the removed Frobenius morphism to both lower images. Then V is irreducible on the resulting connected simple image of H. Its rank is r, and the resulting image of X has its original rank s>r. Its natural simple module remains irreducible on the image of H, by the endpoint conclusion and (7.2). Finite irreducibility implies each of these rational irreducibility statements, since a rational invariant subspace would also be invariant under the actual finite image.

If the algebraic image of X is proper in Y, apply Lemma 7.2. If that image is all of Y, the map from X to Y is an isogeny. This can occur even when the finite groups are strictly nested, for example on a subfield edge. The pullback of N_Y is a natural simple module or its dual, up to Frobenius twist and the isogenies in (7.2), by the root-datum classification of isogenies [15, Definition 6.1.8, Example 6.1.9 and Theorem 6.1.16(1)]. Indeed, for compatible simple roots an isogeny has a root bijection \delta and powers q_i of the characteristic. Write the Cartan matrices in the row-coroot convention A^X_{ij}=\langle\alpha_j,\alpha_i^\vee\rangle and A^Y_{ab}=\langle\beta_b,\beta_a^\vee\rangle. The compatibility equation is q_j A^X_{ij}=q_i A^Y_{\delta(i)\delta(j)}. Pairing the pullback of the natural fundamental weight with the source simple coroots gives q_i at the node mapped to the natural node and zero elsewhere. In the retained ranks this node is the natural node, or its type-A dual; the only change of root lengths is the characteristic-two B/C case in (7.2). Since N_X is irreducible on the lower H, it follows that N_Y is irreducible on the corresponding image of H. The X=Y case of Lemma 7.2 therefore applies. Thus every one of the eight chosen source ranks belongs to \mathcal R(r). They are nevertheless distinct. Along an algebraic edge the rank cannot decrease, because a maximal torus maps with finite kernel into a torus of the image. Hence each chosen source rank lies between the ranks of the two endpoints of its comparison. The gaps between the ordered comparison intervals, and strict increase of endpoint ranks, make these eight intervals disjoint in rank. This contradicts |\mathcal R(r)|\leq7.

Both homogeneous alternatives are impossible. All Ramsey bounds and all path lengths used here depend only on the absolute constants in the preceding lemmas. In particular only fixed finite paths are needed before taking the large-rank limit. The assumed unbounded family of tested chains cannot exist, proving the asserted absolute chain bound and completing Theorem 3.2.□

Proof. The prescribed order is transitive: every route from an old vertex into the fresh interior passes through a, and every route back passes through b. This proves (8.6) and antisymmetry. For an old vertex v incomparable with a fresh interior vertex w, the cross operations are

where the operations on the right are those of P. In the comparable cases, the smaller and larger vertices give the meet and join. For two fresh vertices, the operations in Q remain their operations: an old common lower bound lies below a, and an old common upper bound lies above b. An analogous argument shows that an old meet or join cannot be changed by a new bound. This proves the lattice assertion.

If a fresh vertex w lies strictly between the old cover c\prec d, then (8.6) gives c\leq a<b\leq d. The cover condition forces a=c and b=d. Conversely, an insertion with those endpoints and a fresh interior vertex destroys the cover.□

Proof. Vertices in different components of the proper comparability graph have meet a and join b. Indeed, a common lower bound strictly above a, or a common upper bound strictly below b, would connect them. For a given interior vertex, choose the two comparable vertices in one of the two specified components different from its own. They are its required complements.

Consider an insertion into [c,d]. A fresh vertex can belong to [a,b] only if a\leq c<d\leq b, by Lemma 8.2. If both c,d are proper vertices of [a,b], they lie in the same comparability component. All comparisons from the new interior to the old proper part then enter that component. The same is true if just one of c,d is proper. If c=a and d=b, the new interior has no comparisons with the old proper part. Thus no insertion merges two old components. The two reserved comparable pairs survive, proving persistence. A fencing insertion supplies two such separate components.□

Proof. Each shortcut edge is a cover immediately after its one-vertex insertion. By the cover criterion in Lemma 8.2, a later shape insertion could destroy it only by using that exact edge as its endpoint pair. The listed pairs never do so. The only apparent possibilities would be [a,v_{0j}], which is excluded from (8.7), and [v_{iN},b], which is excluded from (8.8). This proves (i).

All pair fences are present in the third batch and persist by Lemma 8.3. The chain targets x_j already have the fence on [a,x_j] from that batch; the other upper targets are listed there expressly. Each reserved coatom witness is fresh when inserted and is not an endpoint of a subsequent shape insertion. Its edge to the upper endpoint therefore survives. Its meet with its companion remains the lower endpoint, by Lemma 8.2.

When an M_{16} branch is inserted, its interval [m_u,u] has no other vertices: an old vertex above m_u must be above u. A subsequent shape insertion could add a vertex inside this interval only if both its endpoints lay in [m_u,u]. None of m_u,t_{u,k} is used subsequently as a shape endpoint. The remaining possible endpoint is u, which cannot by itself form a nondegenerate interval inside [m_u,u]. Thus this interval remains exactly M_{16}. These arguments give (ii) and (iii).

For completeness, the numbers of new vertices in the first four batches are respectively

The fifth batch adds at most 2N(N+1) vertices. Their sum is 15N^2+12N-1\leq27N^2.□

Proof. We first record how the two shape constructions are separated. Every fresh vertex of a forward shape construction lies below no proper original vertex. This follows inductively from (8.6): its upper insertion endpoint is either 1 or an earlier fresh forward vertex with the same property. In contrast, every vertex of a reverse shape construction lies in a forward interval [v,u] with 0<v<u<1 original. Thus every vertex of \mathcal R_C lies below a proper original upper bound, and no fresh forward vertex belongs to \mathcal R_C.

These observations also show that the shape constructions do not spoil each other’s designated intervals. Different forward tests have disjoint fresh interiors; their only shared shape endpoints are original vertices and 1. Hence neither uses a fresh endpoint of the other’s designated cover. No reverse shape insertion can use such a forward endpoint as its lower endpoint, since it lies below a proper original upper bound. For reverse tests, two different tests again have disjoint fresh interiors and share only original endpoints. Lemma 8.2 therefore protects their covers. The same argument protects the exact M_{16} intervals: a shape insertion could enter one only if its two endpoints lay inside it, whereas its fresh bottom and atoms are never shape endpoints. Fences persist by Lemma 8.3.

We next check the final batch. A new vertex of [0,t] belongs to no comparison interval [a,b] with a>0, by (8.6). A new vertex of [t,1] can belong to such a comparison only if b=1 and a\leq t. The lower endpoints of the latter insertions are in \mathcal R_C, and consequently lie below proper original upper bounds.

In particular, a reverse fence cannot subdivide a fresh forward coatom edge c\prec1: that would require its lower endpoint to be exactly c, which lies below no proper original vertex. A forward cover with a proper upper endpoint cannot be subdivided by an insertion ending at 1. Thus all forward shortcut edges and all reserved coatom witnesses survive. A forward M_{16} interval with proper upper endpoint cannot be entered by a reverse fence. If its upper endpoint is 1, its fresh bottom lies below no proper original vertex; hence no point of \mathcal R_C lies above that bottom. So this M_{16} interval also remains exact. Forward base insertions cannot enter any of these positive-bottom intervals.

All reverse comparison intervals have two proper original bounds in the forward order. Neither kind of final insertion can add a vertex to such an interval: an insertion beginning at 0 cannot do so, and an insertion ending at 1 cannot do so. Their covers, witnesses and exact M_{16} intervals are therefore unchanged. The final reverse fences label every vertex in each reverse comparison interval by the definition of \mathcal R_C. All required forward principal vertices belong to \mathcal F_C; the sole omitted proper target is the detector X, whose lower interval was fenced at the outset. This proves (i) and (ii).

For (iii), an old vertex in a forward comparison belongs to \mathcal F_C, so its lower interval is fenced, with the stated exception for the global top. The only possible new vertex is an interior vertex w of a final reverse fencing insertion into [s,1]. If x_i\leq w, then x_i\leq s by (8.6). Every vertex below w either lies below s or lies on the same private chain above s. This assertion also includes vertices of other insertions in the final batch: a point from another branch can be below w only by passing through the old endpoint s. Thus s is comparable to every vertex of [0,w]. Moreover s\in\mathcal F_C, since x_i\leq s<1, and its lower interval is fenced. This is exactly alternative (b), including the requirement that the bottleneck lie above the lower chosen chain vertex. It proves closure of the normal-group tests without any new round of insertions.

For (iv), no shape insertion introduces a point strictly below X. Forward fresh points lie below no proper original vertex, and reverse shape insertions in the detector are contained in [X,z]. A final insertion ending strictly below X introduces only points below that endpoint, so it cannot create a new coatom of [0,X]. A final insertion ending above X introduces no new point below X. The only remaining possibility would be an insertion into [0,X], which was omitted. Hence the original covers q_i\prec X are precisely the coatoms of the lower half. Its two separate original branches, and those of the upper half, still give fences by Lemma 8.3.

Finally, Lemma 8.4 bounds the size before the final batch by q=q_0+27hN^2. Each of \mathcal F_C and \mathcal R_C has at most q members, and each fencing insertion adds four vertices. The final size is at most q+4q+4q=9q. This also proves finiteness directly from the displayed batches.□

Proof of Proposition 8.1. Let B be the absolute chain threshold of Theorem 3.2, and choose the single integer N=\max\{2,\lceil B\rceil\}. Construct L=L_N by the finite operations above. This choice is independent of every possible representing group. Finiteness follows from Lemma 8.5 and (8.11); no limiting lattice is used.

All original lattice operations survive by Lemma 8.2. Every proper original Boolean vertex is an endpoint of a forward requested pair and of a reverse requested pair, so its two principal intervals receive the required fences. The detector statements follow from Lemma 8.5. This gives (i) and (ii).

For (iii), take a finite representation [D,G]. By Lemma 2.3, the global fence and the required principal fences provide all forward labels. In every forward requested test, the sequence x_0<\cdots<x_N and its private shortcuts have the shape required by Theorem 3.2, including its nonconsecutive shortcuts. The interval [x_i,x_j] is fenced; every target in (8.7) has its bottom saturation, fence, and exact positive-bottom M_{16} interval; and all the specified principal vertices are meets of coatoms below the largest endpoint, with v_{iN} themselves coatoms. Finally, Lemma 8.5(iii), together with the global fence, verifies condition T5 at every vertex of every comparison: either its lower interval is fenced or it has the specified labelled bottleneck above the lower chain vertex. In the latter case, Corollary 2.4 supplies the required soluble-normal and commuting-normal consequences. All hypotheses of the chain Theorem hold, and N\geq B forces equality of its endpoint types. The forward requested pairs are precisely (8.1) and (8.5), giving the asserted equalities.

For (iv), fix one of the indicated actual upper-filter representations. Its least lattice vertex is the former 1, and its group bottom is A. The entire reverse test for (u,v) lies inside this upper filter. Its principal labels come from intervals [w,1]^{\mathrm{op}}, which are fenced by the final reverse batch. All vertices of its comparison intervals have those fences, by Lemma 8.5(ii). The reverse shape construction supplies all the other hypotheses of the same chain Theorem. Consequently the two endpoint labels, now taken relative to A, are equal. The proof for (z,X) is identical. If the actual intervals are compatible, their common subgroups and their labels are the same on overlaps; the argument makes no additional choice of a representation and thus applies simultaneously.

Finally, the order reversal exchanging the paired components in (8.11) supplies the same Boolean and detector data in the other orientation. This proves the last assertion.□

Proof. An automorphism \sigma centralizing every inner automorphism satisfies c_{\sigma(t)}=c_t for every t\in T. Since Z(T)=1, this implies \sigma(t)=t for every t, so C_{\operatorname{Aut}(T)}(\operatorname{Inn}(T))=1. If K\trianglelefteq H, then

Simplicity makes the intersection either trivial or all of \operatorname{Inn}(T). In the first case the centralizer calculation gives K=1. This proves the assertions.□

Proof. Lemma 2.3 would give a nonabelian simple composition factor of the soluble top group. For the second assertion, quotient by \operatorname{Inn}(T), which is contained in the bottom B. The resulting interval lies in the soluble group \operatorname{Out}(T), by Schreier’s Theorem.□

Proof. Put E=D\cap S. For every Y\in[D,G], the identity G=DS gives

Thus Y\mapsto Y\cap S is an isomorphism from [D,G] to the interval of D-invariant subgroups between E and S; its inverse is K\mapsto DK.

First suppose that E is subdirect in S. By the subdirect-power description in Lemma 2.5, its simple factors are full diagonal strips on a partition \Pi of I. Every subgroup between E and S is subdirect. Its strips split the blocks of \Pi, and their diagonal identifications are forced by their containment of E. Therefore the preceding interval is anti-isomorphic to the lattice of D-invariant equivalence relations refining \Pi.

Fix i_0\in I, let A=D_{i_0}, and let B be the stabilizer in D of the block of \Pi containing i_0. The block correspondence for a transitive action identifies these equivalence relations with [A,B]: the block containing i_0 associated with U\in[A,B] is U i_0, and all its other blocks are its D-translates. Conversely, the setwise stabilizer of the block containing i_0 recovers U. It follows that \mathcal M^{\mathrm{op}}\cong[A,B]. But |B|\le |D|<|G|, contrary to the choice of G over both orientations.

The coordinate projections of E have equal order, since D is transitive on I. If they are proper and nontrivial, Proposition 2.7, applied with the possibly unlabelled lower endpoint D, and the global fence imply that E is the product of those projections. Choose a proper original Boolean vertex Y of the decorated lattice. Its forward label equals the top label, so Y\cap S is subdirect by Proposition 2.6. It is proper, because Y<G.

A product of nontrivial coordinate subgroups cannot lie in a proper subdirect subgroup: the latter has a diagonal link between two coordinates, whereas the former contains an element that is nontrivial in one of those coordinates and trivial in the other. This contradicts E\le Y\cap S. The only remaining possibility is that all coordinate projections of E are trivial, and then E=1.□

Proof. Use left conjugation for the given action and let \pi_0:S\to T be the chosen coordinate projection. The map

identifies S with

Indeed, values on representatives of A\backslash D are independent coordinates of S. Under this identification the action is (g\cdot f)(d)=f(dg).

An extension \beta defines

This is a D-invariant subgroup of \mathcal F_\alpha, with independent values on the cosets in U\backslash D. Consequently \mathcal F_\beta\cong T^{[D:U]}, and it projects onto every coordinate of \mathcal F_\alpha. Set S(U,\beta)=\mathcal F_\beta in these coordinates.

Conversely, let K\le\mathcal F_\alpha be a D-invariant subdirect subgroup. Its diagonal-strip partition is invariant under the transitive action on A\backslash D. The block containing A has stabilizer U for some A\le U\le D. Projection at 1 identifies the diagonal factor on that block with T. The induced action of U on this factor gives a homomorphism \beta:U\to\operatorname{Aut}(T) extending \alpha. For f\in K and u\in U one has f(u)=\beta(u)(f(1)). Applying this identity to d\cdot f gives

Thus K\le\mathcal F_\beta. Both groups have one freely varying diagonal factor for each coset of U, so they are equal.

Containment of two subdirect groups reverses refinement of their strip partitions. Hence S(U,\beta)\le S(V,\gamma) forces V\le U. For v\in V, varying f(1) over T in the two defining equations then gives \beta(v)=\gamma(v). The converse follows immediately from those equations. Finally, every subgroup containing a subdirect subgroup is subdirect, so all members of the stated interval have already been included.□

Proof. Let X be the detector vertex of Proposition 8.1. Its forward type equals T, so E_X=X\cap S is subdirect in S. The action of D on the simple factors of E_X is transitive. Its factor-stabilizer action is \beta_X:U_X\to\operatorname{Aut}(T) by the construction in Proposition 9.4.

Suppose that this action preserved a subgroup 1<P<T. Transporting P among the factors of E_X would give a D-invariant product Q of proper nontrivial coordinate subgroups of E_X. Thus DQ is strictly between D and X. Choose a coatom Y of [D,X] containing DQ. The detector’s coatom property gives a designated vertex d\le Y whose intersection d\cap S is subdirect in the global coordinates. Hence Y\cap S projects onto every simple factor of E_X: projection onto such a factor can be identified with projection onto any one global coordinate in its diagonal support. Thus Y\cap S is a proper subdirect subgroup of E_X containing the product Q, which is impossible by the diagonal-link argument in Proposition 9.3.

The automorphism lemma recalled by Pálfy [28, Lemma 2.11] states that a subgroup of \operatorname{Aut}(T) not containing \operatorname{Inn}(T) preserves some proper nontrivial subgroup of T. It follows that

The reverse detector interval is the actual interval [A,U_X] in (9.3), and it is fenced. Set K=\ker\beta_X. Lemma 2.2 gives either AK=U_X or K\le A. The first possibility gives \beta_X(U_X)=\beta_X(A)=\alpha(A), as required.

In the second possibility K\le C_X=\operatorname{core}_{U_X}(A), and [A,U_X] is represented as

Write H=\beta_X(U_X) and B=\alpha(A). If \operatorname{core}_H(B) is nontrivial, it contains \operatorname{Inn}(T) by Lemma 9.1. Quotienting by that core leaves a soluble top, contradicting Lemma 9.2. Otherwise C_X=K, and U_X/C_X\cong H has socle T with multiplicity one. The reverse detector supplies a strictly smaller labelled domain in [A,U_X] whose simple type is equal to that of U_X/C_X. The equality is supplied by the reverse tests in Proposition 8.1, applied through (9.3). Proposition 2.6 would make its positive socle multiplicity strictly smaller than one. This final contradiction excludes K\le A.□

Proof. The normal-subgroup test gives the asserted supplementation unless \ker\beta\le A. In the latter case [A,U] is isomorphic to [\alpha(A),\beta(U)]. Proposition 9.5 puts the inner group in its bottom, contrary to Lemma 9.2. Supplementation gives the equality of images.□

Proof. Use the minimal representation already chosen. Index the original Boolean component in its reverse order by subsets J of \{1,\ldots,4\}. Let Y_J be the corresponding forward group, so Y_\varnothing=G and Y_{\{1,\ldots,4\}}=D. Every proper nontrivial Y_J has the top forward type and hence a subdirect intersection with S. Proposition 9.4 provides (U_J,\beta_J).

The reverse labels and tests in Proposition 8.1 apply in the actual filters (9.3). They give the fences and equal simple types on comparable proper Boolean nodes. All the nonempty proper nodes therefore have one common simple type R: connect two atoms through their proper two-atom join, and connect any other node to an atom below it. Lemma 2.3 and Proposition 2.6 prove (E1) and the transport assertions in (E2). The order relation and restrictions follow from (9.2).

For the assertion about a proper join, let K be the union of a nonempty family of index sets, and put V=\langle U_J:J\text{ in the family}\rangle. Then A\le V\le U_K, and \beta_K|_V is an extension of all the \beta_J. Its subdirect group is contained in every Y_J\cap S and contains Y_K\cap S. Original Boolean meets are preserved in the decorated lattice, so the intersection of those Y_J is exactly Y_K. The subdirect groups are equal; the dictionary gives V=U_K.

Proposition 9.5 and Lemma 9.6 give (E3). Finally, a common extension for a family with full union would give a nontrivial subdirect subgroup of S contained in all the Y_J\cap S. Their intersection is D\cap S=1, because their reverse Boolean join is the top. This proves (E4).□

Proof. For two normal subgroups,

Induction proves the assertion for the finite family.□

Proof. Write J=\operatorname{Inn}(T) and let \iota:T\to J be the isomorphism t\mapsto c_t. Consider

The family is nonempty. Since J is perfect, Lemma 10.1 shows that its finite intersection still maps over J. The subgroup \alpha^{-1}(J) belongs to the family, so in fact

Thus Q is the least normal subgroup of A with image containing J.

Fix i and put C_i=\operatorname{core}_{U_i}(A). The hypothesis gives Q\le C_i. Make the analogous construction in C_i:

The same intersection argument, using C_i\cap\alpha^{-1}(J), gives \alpha(Q_i)=J. For u\in U_i and c\in C_i,

Since C_i\trianglelefteq U_i and J\trianglelefteq\operatorname{Aut}(T), conjugation by U_i permutes the defining family for Q_i. Thus Q_i\trianglelefteq U_i, and in particular Q_i\trianglelefteq A. Minimality of Q gives Q\le Q_i. Conversely, Q\le C_i, Q\trianglelefteq C_i, and (10.1) give Q_i\le Q. Consequently Q_i=Q for every i.

It follows that Q\trianglelefteq K and Q\le\operatorname{core}_K(A), proving the common-core assertion. Set L=Q\cap\ker\alpha. Since L=Q\cap\ker\beta_i for every i, the subgroup L is normalized by each U_i and therefore by K. The map

is an isomorphism. Transporting the conjugation action of K on Q/L through \vartheta defines a homomorphism \beta:K\to\operatorname{Aut}(T). For u\in U_i and q\in Q,

As \vartheta is onto, this proves \beta|_{U_i}=\beta_i. These restrictions determine \beta uniquely because the U_i generate K.□

Proof. By (E3), U_x=A\ker\beta_x. Since A<U_x and C_x\le A, the kernel is not contained in C_x. The assumption makes \beta_x a homomorphism on U_x/C_x, so its nontrivial kernel there contains the unique minimal normal subgroup N_x/C_x. Consequently \beta_x(N_x)=1.

Let P be the kernel of the action of U_x on I_x. Modulo C_x, the faithful conjugation action on S_x embeds P/N_x into \operatorname{Out}(R)^{I_x}. To justify faithfulness, the centralizer of S_x in U_x/C_x is normal and disjoint from S_x; uniqueness of the nonabelian minimal normal subgroup makes it trivial. Schreier’s Theorem therefore makes P/N_x soluble. Its image under \beta_x is a soluble normal subgroup of H=\alpha(A). Lemma 9.1 makes that image trivial, so P\le\ker\beta_x, as claimed.□

Proof. Fix a proper pair join w=u\cup v. Equal-type transport gives

The first factor on the right acts by inner automorphisms on N_w/C_w, hence preserves every coordinate subproduct there. The second factor satisfies C_u\le C_x\le U_x. Since C_x acts trivially on I_x, equivariance of p_{wx} says that C_u fixes the set F_{w,j}. It too normalizes N_w(F_{w,j}). Thus U_u\cap N_w(F_{w,j}) is normal in N_u and lies in N_u.

In N_w/C_w, the subdirect group (U_u\cap N_w)/C_w has its diagonal strips indexed by I_u. Their supports are the fibres of p_{wu}. Because p_{wx}=p_{ux}p_{wu}, intersecting with the F_{w,j}-support product selects exactly the strips indexed by F_{u,j}. The isomorphism from this subdirect group to N_u/C_u therefore shows that U_u\cap N_w(F_{w,j}) maps onto \prod_{i\in F_{u,j}}S_{u,i} under q_u. The first group in (10.3) is normal in N_u and has the same image. This image is perfect. Lemma 10.1 proves (i).

Every domain occurring in (10.3) contains U_x, and all the factor maps are U_x-equivariant. Its conjugation therefore permutes the whole definition with j, proving (ii). The images in (i) generate N_u/C_u, and C_u\le A; since U_u=AN_u, this proves (iii).

For u\ne v, the two lifts in (iv) lie in disjoint coordinate support products of N_{u\cup v}/C_{u\cup v}. They commute in that quotient. For u=v the identical argument takes place in N_u/C_u. Core inclusion along U_x\le U_u\le U_{u\cup v} gives the asserted containment in C_x.

Finally suppose that \alpha(C_x)=1. Then \beta_u(C_u)=1. The kernel of \beta_u supplements A and is not contained in C_u, so it contains N_u by the unique-minimal-normal-subgroup argument in Lemma 10.3. This proves (v).□

Proof. Let \pi:R_0\to R_0/C be the quotient map. Start with B=R_0. At every stage B will be normal in R_0 and satisfy \pi(B)=R_0/C. For a fixed t, replace B by

The subgroup B\cap D_t still maps onto R_0/C by Lemma 10.1. Put K=B\cap D_t. The subgroup K is normal in D_t, because B\trianglelefteq R_0 and D_t\leq R_0. Since L_t normalizes D_t, every conjugate K^\ell lies in D_t and is normal in D_t. For d\in D_t and \ell\in L_t, the assumption [D_t,L_t]\leq D_t\cap C gives \pi(d^\ell)=\pi(d). Consequently \pi(K^\ell)=\pi(K)=R_0/C for every \ell. This applies the quotient map only to elements of D_t\leq R_0; it does not require C to be normalized by L_t. Applying Lemma 10.1 inside D_t shows that F_t(B) has full image.

The new subgroup is normal in R_0. Indeed, R_0 normalizes B, D_t, and L_t. For r\in R_0 one therefore has (K^\ell)^r=(K^r)^{\ell^r}=K^{\ell^r}, and \ell^r\in L_t. Thus R_0 permutes precisely the conjugates intersected in (10.5). The operation is descending, since the identity conjugate occurs. Apply F_1,\ldots,F_s in repeated rounds. Finiteness forces stabilization, and at a stable round every intermediate inclusion is equality. The resulting subgroup B_* is therefore normalized by each L_t as well as by R_0. Thus B_*\trianglelefteq W and B_*\le R_0, while \pi(B_*)=R_0/C. Since B_*\le\operatorname{core}_W(R_0), the asserted core also has full image.□

Proof. For u\in\mathcal U_x set

The transported subdirect group (U_x\cap N_u)/C_u maps isomorphically onto N_x/C_x. Selecting its strips over k shows that

It is normal in R_k: all elements of R_k\le N_x fix the factor labels I_x, and the support-preimage construction is equivariant.

For j\ne k, both D_{u,k} and L_{u,j} lie in N_u and have disjoint supports modulo C_u. Thus

Since C_u\le D_{u,k}, this also proves that L_{u,j} normalizes D_{u,k}. Moreover R_k normalizes L_{u,j} by Lemma 10.4(ii), since R_k fixes every label of I_x. We can therefore apply Lemma 10.5 to R_k, its normal subgroup C_x, and all the lifts L_{u,j} with j\ne k, using the subgroup D_{u,k} for the lift labelled by (u,j).

Define canonically

That lemma proves (i); the definition proves (ii). Since C_x\le R_k\le W_k, it normalizes B_k. Conjugation by A permutes the pairs (W_k,R_k) and hence their cores, proving (iii). Their images in N_x/C_x are the distinct nontrivial coordinate factors, so the B_k are distinct.□

Proof. Write g as a word whose letters belong to the individual P_\lambda; inverses are allowed because these are subgroups. Interchange adjacent letters with different labels to move the k-labelled letters to the end. Each interchange introduces a commutator in C. Move that error to the right through the remaining word, using

The conjugated letter remains in its original P_\lambda, so its label does not change. Each interchange reduces the number of pairs consisting of a k-letter preceding a non-k-letter. The process terminates with the asserted expression. This is a word calculation; it does not form a quotient by C or require C\trianglelefteq L.□

Proof. The subgroup \alpha(C_x) is normal in \beta_x(U_x)=H. Lemma 9.1 reduces the assertion to excluding \alpha(C_x)=1. Assume that equality and use the lifts and detecting subgroups just constructed. Put

The group A normalizes L.

Let a\in A\cap L. We show that a fixes every member of I_x. For subgroup conjugation use the right-action notation K^g=g^{-1}Kg. If a sent k to h\ne k, equivariance would give B_k^a=B_h. Lemmas 10.4 and 10.7, with C=C_x and the individual L_{u,j} as labelled subgroups, give

The element p normalizes B_k, q normalizes B_h, and c normalizes both. It follows successively that

contradicting Lemma 10.6. Thus a fixes the factor set. Lemma 10.3 now gives \alpha(a)=1.

Consequently A\cap L\le\ker\alpha. Because A normalizes L, the formula

defines a homomorphism. It is well defined by the intersection inclusion; multiplication is respected since L\trianglelefteq AL. For each u\in\mathcal U_x, Lemma 10.4(iii),(v) says that U_u is generated by A and lifts killed by both \theta and \beta_u. Hence \theta|_{U_u}=\beta_u.

The three sets u\in\mathcal U_x have full union \{1,\ldots,4\}. Their generated domain is AL, so this common extension contradicts (E4). The assumption \alpha(C_x)=1 is impossible.□

Proof. The assertion is immediate for an atom, so suppose that |z|\ge2. Every C_x in the intersection contains C_z. Pass to U_z/C_z and use bars for the images of A,U_x,C_x. Put

The transport isomorphism in (E2) gives E_x\cap\overline C_x=1. Both are normal in \overline U_x, so [E_x,\overline C_x]=1.

Since U_z=AN_z and A\le U_x, one has \overline U_x=\overline A E_x. Let M=\langle E_x:x\subseteq z,\ |x|=1\rangle. Each E_x is normalized by \overline A. The compatible-join generation in (E2) therefore gives

so M\trianglelefteq\overline U_z. It is a nontrivial subgroup of the minimal normal subgroup N, and hence M=N. Thus \bigcap_x\overline C_x centralizes N. The centralizer is normal in \overline U_z and intersects N in Z(N)=1. If it were nontrivial it would contain the unique minimal normal subgroup N, a contradiction. The intersection is therefore trivial, which proves the equality of cores.

Each C_x is normal in A and maps over \operatorname{Inn}(T) by Proposition 10.8. Apply Lemma 10.1 with P=\operatorname{Inn}(T) to obtain the image assertion.□

Proof. Take the two proper Boolean nodes

Lemma 10.9 gives \alpha(C_z),\alpha(C_w)\ge\operatorname{Inn}(T). Lemma 10.2 therefore extends \beta_z and \beta_w to \langle U_z,U_w\rangle. Since z\cup w=\{1,\ldots,4\}, this contradicts (E4).□

Proof of Theorem 1.2. Choose the finite decorated lattice of Proposition 8.1. If either it or its dual had a finite-group interval representation, Proposition 9.7 would produce the configuration excluded by Theorem 10.10. Hence this lattice has no such representation, proving Theorem 1.2.□