BFMS Lemma 3.5: proposed completion of the compatibility proof

21 October 2026. Coordinator mathematical reconstruction, with exact local matrix checks. This is not an accepted independent arbitration, an official corrigendum, a formal certificate, or a verification of the main arithmeticity theorem. The argument below supplies the omitted parabolic analysis and makes the normality and torus steps explicit. Independent checking of the assembled replacement remains desirable; the decision graph conservatively retains the original-proof defect.

Source: the frozen arXiv 1903.08467v5 native source, the native manuscript in the reading packet, Lemma lemma:compatible; compare Annals 193 (2021), pp. 851–852. The original lemma's hypotheses are retained. This reconstruction uses characteristic-zero algebraic-group structure, not the later complex-hyperbolic sequel as a proof input. The supplied Astra and Fable reports motivated the exceptional-case check; the dimension argument below replaces the disputed radical/normality steps.

Setup and standard prerequisites

Write d=n−1 and P=MAU, with U=R^d, M=SO(d,R), and A the positive scalar homotheties. Work first over C, in the connected adjoint group H=PSO(d+2,C); for d=2 this is PGL2(C)×PGL2(C). This specifies the identity-component convention obscured by the paper's notation PO. Let J be a nontrivial algebraic subgroup and τ:P→N_H(J)/J continuous. We must show that every nontrivial subgroup U′<U has the same image Zariski closure as U.

We use the following standard characteristic-zero structure facts: connected solvable algebraic groups have unipotent derived subgroup; connected rank-zero linear algebraic groups are unipotent; rank is additive in exact sequences of linear algebraic groups; Levi decomposition and conjugacy of reductive subgroups into a parabolic Levi; and the classical orthogonal parabolic/Levi description. These are algebraic-group inputs, not consequences of the finite matrix computations. Borel, Linear Algebraic Groups, second edition (1991), is a standard source for this structure theory. The paper itself cites Borel–Tits, Éléments unipotents et sous-groupes paraboliques de groupes réductifs I (1971), §3, for containment in a parabolic. An elementary orthogonal fixed-space argument for that particular containment is given below. We have not independently reproduced the general structure theorems.

The image dimension and full rank

If τ(U)=1 there is nothing to show. Otherwise let Ubar be its Zariski closure and Pbar the Zariski closure of τ(P). These groups are connected. Since U=[AU,AU] and AU is solvable, Ubar is unipotent; it is abelian as well. Taking logarithms makes log∘τ|U a real-linear map. Its complexification is an M-equivariant surjection

For d≥3 the complex standard SO(d)-module is irreducible, including d=4. Thus this map is an isomorphism and dim(Ubar)=d. For d=2, dim(Ubar) is either one or two. In either case τ|U is injective: its logarithmic kernel is a real subspace invariant under the rotations M and scalars A, and a nonzero such map cannot have a nonzero kernel.

Whenever dim(Ubar)=d, the conjugation representation of Pbar on Lie(Ubar) contains the standard complex orthogonal group and scalar torus. Its image therefore has rank floor(d/2)+1=rank(H). Let Ptilde and V be the preimages of Pbar and Ubar in N_H(J). Rank additivity and Ptilde⊂H force

rank(Ptilde)=rank(Pbar)=rank(H), rank(J)=rank(V)=0.

Consequently J⁰ and V⁰ are connected normal unipotent subgroups of Ptilde, and

J⁰ ⊊ V⁰, dim(V⁰)=dim(J⁰)+d.

This argument avoids treating SO(4,C) as almost simple.

The parabolic list for d≥3

Ptilde lies in a proper parabolic. Here is the fixed-space justification in this orthogonal setting. Lift the connected unipotent group V⁰ through SO(d+2,C)→H; a central isogeny identifies the unique connected unipotent lift with V⁰. Let W be its full fixed space in the standard orthogonal module. A unipotent group has a nonzero fixed vector, so W≠0; since V⁰≠1, W is proper. If W were nondegenerate, its invariant orthogonal complement would have a nonzero fixed vector too, contradicting the definition of W. Hence I=W∩W⊥ is a nonzero isotropic subspace. The preimage of Ptilde normalizes V⁰, so preserves I. Its stabilizer is proper parabolic; enlarge to a maximal parabolic Q.

A Levi decomposition lifts the semisimple image of M to a semisimple subgroup M′⊂Ptilde, locally SO(d,C). Conjugate it into a Levi L of Q. If Q stabilizes an isotropic a-plane, the derived Levi is locally

SL_a(C) × SO(d+2−2a,C).

For a≥2 and d≠4, so_d is simple. Its projection into the smaller orthogonal factor is zero by dimension, so it embeds into sl_a. Rank gives floor(d/2)≤a−1, while a≤floor((d+2)/2); hence a=floor(d/2)+1. If d=2r+1, the dimension inequality r(2r+1)≤r²+2r gives r≤1: only d=3,a=2. If d=2r and so_d is simple, r(2r−1)≤r²+2r gives r≤3: only d=6,a=4 (the smaller even simple cases are excluded by their dimensions). For d=4, so_4=sl_2⊕sl_2: the a=2 derived Levi has rank one, and a=3 would require a faithful three-dimensional representation of both sl_2 factors. Such a representation has dimension at least four, so neither occurs.

Thus the relevant list is:

For each case the connected derived Levi S is locally SO(d,C), so M′=S after the chosen conjugacy. This retains the triality parabolic rather than incorrectly excluding it.

Why both normal unipotent groups lie in the radical

Let π:Q→L be the Levi quotient. The subgroup π(Ptilde) contains S. Its rank is full because the kernel lies in the unipotent radical and Ptilde has rank(H)=rank(L). Since Q is maximal, L/S is a one-dimensional torus. These two facts force π(Ptilde)=L. This is the missing central-torus argument; containing S and R alone would not suffice.

The images π(J⁰) and π(V⁰) are connected normal unipotent subgroups of the reductive group L, hence trivial. Thus

J⁰ ⊊ V⁰ ⊂ R.

This uses normality and full Levi projection, not the false assertion that all unipotent elements of a parabolic belong to its radical. Since dim(R)=d in all three cases,

It follows that J⁰=1 and V⁰=R. In the Heisenberg case this is impossible: V⁰ maps onto the abelian Ubar with finite kernel, which would identify their Lie algebras, whereas Lie(R) is nonabelian. Thus the n=4 exception is excluded.

In the line and triality cases, Ptilde contains R and surjects onto L, so Ptilde=Q. Now J is finite and normal in the connected Q, hence central. A parabolic of a connected adjoint semisimple group has trivial center, so J=1, a contradiction. This completes the d≥3 case without relying on transitivity on complex projective space or on an unproved radical-containment assertion.

The d=2 case

If dim(Ubar)=2, the same full-rank argument gives V⁰ unipotent and dim(V⁰)=dim(J⁰)+2. A maximal unipotent subgroup of PGL2×PGL2 has dimension two, so J⁰=1 and V⁰ is maximal unipotent R. Ptilde normalizes R, hence lies in its Borel normalizer B. It contains R and has rank two, forcing surjectivity onto B/R and therefore Ptilde=B. The finite normal subgroup J must be central in B, whose center is trivial. Again J=1, a contradiction.

Therefore dim(Ubar)=1. The injectivity of τ|U proved above makes τ(u) nontrivial for every u≠0. Every nontrivial subgroup of the characteristic-zero additive algebraic group G_a is Zariski dense. In particular the cyclic subgroup generated by τ(u), for any nonzero u∈U′, is dense in Ubar. The required image closures coincide.

For a nonarchimedean local field, connected U maps trivially to the totally disconnected target. For a real local field, base change to C as in the paper preserves subgroup nontriviality and equality of the image closures descends. This finishes the proposed local completion.

Executable checks and remaining scope

exact-check.py uses exact rational matrices. In the n=4 case it verifies orthogonality of all three radical generators and the bracket [u₁,u₂]=−z≠0, with z central. In the n=7 case it verifies all six skew-matrix radical generators, their zero brackets, and the Levi action XZ+ZXᵗ, the exterior-square action. Distinct torus weights and connected root action also check irreducibility, although the dimension argument above does not need it. Representative line radicals are checked too; the displayed general block formula explains their commutativity in all dimensions.

The matrix results are saved separately with the source and checker hashes. They establish those local structures, not the structure theorems or the whole manuscript. Still outstanding at campaign level are the n=3 lifting/descent/subdirectness justification, the distinct-place deduction, the normal-subgroup-chain initiality argument, and further essential dynamical reference checks. This note does not turn any of those into verified inputs.