Quadratic Invariants and the Common Factor 108: An Arithmetic Observation Linking the Leech Lattice, the Griess Algebra, and Exceptional Lie Structures

Building directly on the initial analysis of the 324-unit structural gap between the Leech lattice's minimal vector count (196,560) and the Griess algebra's dimension (196,884), Version 2 expands this framework to uncover a finer, fully verifiable arithmetic architecture. At the heart of this update is a striking numerical bridge: all three quantities—196,560, 324, and 196,884—share an exact common factor of 108. This common thread simplifies a massive dimensional shift into a clean linear identity (1820 + 3 = 1823). A rigorous control test confirms this is not a generic property of large nearby integers, as substituting 196,884 with its immediate neighbor 196,883 completely collapses the shared factor down to 1. This release further develops the structural roots of the invariant 108, decomposing it naturally as 4 * 27 to tie the framework directly to the 27-dimensional fundamental representation of E_6. Alongside this geometric grounding, V.2 incorporates independent verification through the Doudna sequence (OEIS A005940), where terms 60 and 124 independently generate 108 and 324 via a distinct binary-encoding construction. This release offers an updated quantitative baseline for researchers exploring the structural boundaries connecting the Leech lattice, the Griess algebra, and exceptional Lie theory. Feedback, replication attempts, and collaborative insights from the participating research communities are warmly welcomed