The Quaternionic Toledo Spectrum for Split F4 over Products of Surfaces

Let F be the connected linear split real group of type F4, and let Q_rho be its canonical oriented three-plane bundle, obtained from a maximal compact reduction of a flat F-bundle. We determine the range of the characteristic number <p_1(Q_rho), [Sigma_g x Sigma_h]> over all representations of a product of two closed oriented surface groups. For g,h >= 2, put G=g-1, H=h-1 and P(R,S)={ab : a,b are integers, |a| <= R, |b| <= S}. The range is 2P(2G,2H) union 2P(3G,H) union 2P(G,3H), with sharp absolute bound 8GH. The number is zero if either genus is at most one. The argument retains nonreductive images, disconnected image closures and central lifting obstructions. Its main steps are a full Levi reduction, control of the ordered real normalizers of commuting semisimple factors, and descent of integral degree-two classes to the original surfaces. Genuine symplectic two-plane representations realize every value.

The theorem addresses the split F4 target and products of two surfaces, a family suggested in the 2007 AIM workshop's quaternionic Toledo program (Question 4.2 and Comments 4.3-4.4, recorded as AIM-TOPOLOGY-0275 in UnsolvedMath v1.6.0). It does not resolve that general program for arbitrary four-manifolds and targets. Classical structure results, characteristic-class relations and surface inequalities retain attribution. No identical spectrum was found in a bounded primary-literature review through 8 October 2026; no absolute priority claim is made.

This is an AI-assisted, self-audited and unrefereed preprint, without independent review or formal proof-assistant verification. The author is responsible for its claims. Eight portable exact-arithmetic checkers and their matching normal and optimized Python outputs accompany the source. These test stated finite algebraic domains, not the general theorem.