# Verification of the split F4 spectrum manuscript

The written argument treats every representation of two closed oriented
surface groups into the connected linear split real group F4(4). For
G=g-1 and H=h-1 positive, it gives

    2P(2G,2H) union 2P(3G,H) union 2P(G,3H),

where P(R,S) is the set of products of integers bounded in absolute value
by R and S. The sharp absolute bound is 8GH. If either genus is at most
one, the number vanishes. The broader AIM question is not claimed solved.

## Written proof and objections addressed

The proof uses full real Levi reduction to preserve the original flat
principal bundle and source-labelled commuting reductive factors. It
constructs an ordered complex dual pair, treats common centers by a
canonical real involution, and handles the seven center-free cases by
their full normalizers. Five are contained in canonically recovered
involution centralizers. The G2+A1 and A2+short-A2 cases are calculated
separately with their actual central quotients and real components.

The self-audit checked the following potential failure points:

1. Covering-space division alone does not determine a discrete downstairs
   spectrum. The proof descends an integral degree-two class and its
   separate source coefficients, preserving an integer rank allocation.
2. A component negating a class does not negate its square. Purity and
   the orientation-preserving product deck action together force the
   relevant class itself to vanish rationally.
3. A complex Lie-algebra list does not classify all real components.
   The argument retains full ordered normalizers and actual group forms.
4. Flat real Euler classes need not vanish. Null orthogonal summands
   retain their flat Euler contribution and use the stated Euler theorem.
5. Compact-reduction subbundles are not individually called flat.
   Rational Pontryagin vanishing applies to the whole flat bundle.
6. The symplectic support lemma is used only in rank at most three,
   retaining real, complex and quaternionic tensor types.
7. Upper inclusions are complemented by genuine four-plane
   representations realizing every value on the original source.

These are originating-researcher checks, not independent or formal review.

## Reproducible finite checks

Eight unchanged standalone scripts accompany the paper. Their individual
JSON outputs specify all finite domains and arithmetic conventions.

| Checker | Calculation tested |
| --- | --- |
| check_f4_characteristic_transfer.py | Compact module weights, descent signs and quadratic characteristic coefficients |
| check_f4_split_subgroup.py | Root subsystem, faithful central kernel, exact signatures and four-plane realizations for G,H in 1..3 |
| check_symplectic_product_support.py | Low-rank signed tensor characters, Witt examples, integer rank budgets and cover arithmetic |
| check_f4_peirce_compact.py | Clifford basis relations, even-monomial trace matrix, Jordan derivations and compact characteristic traces |
| check_f4_real_involutions.py | All 15 nonidentity toral involutions, compact Weyl orbits and quaternionic component action |
| check_f4_normalizer_overgroups.py | Spin restrictions, indexed tensor branch and quadratic-cubic fixed-vector normalization |
| check_f4_g2_normalizer.py | Quaternionic Albert witnesses, weight indices and bounded spectrum inclusions |
| check_f4_a2_normalizer.py | Albert basis products, simultaneous duality, character bounds, central descent and bounded spectrum inclusions |

Run reproducibility/run_checks.py normally and with Python's -O option.
Both freshly executed modes passed 177,180 checks, and the saved JSON
outputs agree byte for byte. The checks use exact
integers and rational fractions, not floating-point approximations.
They do not classify all representations, verify imported theorems or
replace the general geometric proof. No formal-verification claim follows
from their PASS results.

## Manuscript and package checks

The standalone source has an inline bibliography with 16 entries.
Compilation, reference consistency, author presentation, disclosure and
PDF rendering are checked separately from the mathematical argument.
The package manifest gives hashes for the PDF, source, reusable bibliography,
metadata, verification documents and all supplementary scripts and outputs.
The ZIP manifest permits each extracted source member to be checked against
the files supplied here. Third-party source PDFs and private research
coordination records are excluded.

This is an English, AI-assisted, self-audited and unrefereed preprint.
Absolute priority, independent review and proof-assistant verification
are not certified by publication or by this package.