The Scalar Wall: A Negative Result on Modulo-9 Cellular Automata and the Necessity of Octonionic Operations in Computational Finitism

We document a systematic negative result: scalar Modulo-9 cellular automata, regardless of dimensionality, weighting, or update rule, cannot produce two essential features of the electron: chirality and localized defects. Twenty distinct simulations were run across 1D, 2D, and 4D lattices with Fibonacci-weighted, center-weighted, bounded-saturation, and hierarchical-overflow update rules. Every rule failed in one of two ways. Wall 1: chirality fails because addition is commutative; reordering the neighbors cannot change the sum. Wall 2: localization fails because Modulo-9 arithmetic is not conservative; a bounded field must either explode (wrapping injects energy) or decay (saturating clips the field). We show that these walls are not bugs but mathematical properties of scalar arithmetic. The conclusion is structural: the Finitism substrate is computational, but its primitive operation is algebraic, not scalar. The minimal algebraic structure that is non-commutative, non-associative, multiplicative, and contains the Fano plane is the octonions. This paper documents the failed path so that future work does not repeat it, and clarifies which of the 55+ companion papers are affected and which are not.