Non-Associative Calculus: Octonionic Path Integrals, Cauchy-Fueter Regularity, and the Fundamental $G_2$ Monopole Field

MINOR NOTICE (2026-08). Independently verified: the FTC Regime B correction term (a commutator, not an associator — the paper's own remark that 𝕆 is alternative is correct), and the zero-friction result that conj(Z)/|Z|⁸ is annihilated by the Cauchy–Fueter operator, to ~10⁻¹² at every point tested. One table defect: ‖D̄f‖ for f(Z)=Z² is position-dependent (measured 0.7 to 31 across random points), so the single tabulated value 14.4 is meaningless without stating the evaluation point. The entries for Z (6.0) and conj(Z) (8.0) are genuinely constant and correct. Files remain open.

Standard calculus and differential geometry are strictly bounded by associativity. Historically, extending continuous analytic functions over the non-associative Octonions ($\mathbb{O}$) resulted in the collapse of the limit definition of the derivative, forcing mathematicians to abandon standard calculus in favour of rigid Dirac-type operators (Fueter regularity). The modern hypercomplex literature (Colombo-Sabadini-Struppa; Ghiloni-Perotti) resolves this by restricting functions to associative complex slices $\mathbb{C}_I$ or replacing the pointwise product with a specialised slice product $f \ast g$. This paper takes a different approach: we do not redefine the algebraic product to recover associativity. We measure the raw geometric obstruction encountered by standard pointwise polynomial integrals over fully twisting 7-dimensional paths.

This paper introduces Non-Associative Calculus, a thermodynamic reformulation of hypercomplex analysis. Rather than rejecting functions that violate associativity, we explicitly capture the failure of the Fundamental Theorem of Calculus as quantifiable geometric curvature — defined herein as the Associator Penalty ($\mathcal{A}$).

We establish a dual-regime integration theory distinguishing associative scalar-parameter ODEs (acting as deterministic engines for continuous gauge fluids) from genuine Octonionic Path Integrals ($dZ \in \mathbb{O}$). We analytically derive the non-associative correction to Stokes' theorem, proving that the integration of continuous functions across a curved $G_2$ manifold natively generates a quasi-associative 3-cocycle — a non-vanishing cohomological defect that, by the gauge-theoretic correspondence established by Jackiw (1985) and Günaydin-Zumino (1986), is the exact algebraic definition of a magnetic monopole.

The deviation of the true octonionic path integral from its formal boundary term is isolated as the FTC Commutator Anomaly: $$C(\gamma, F) = \frac{1}{2}\int_0^1 [F'(\gamma(t)), \gamma'(t)], dt$$ We prove that this anomaly is strictly $G_2$-equivariant: applying any rotation $R \in G_2 = \mathrm{Aut}(\mathbb{O})$ to the path yields $R(C(\gamma,F)) = C(R\gamma, F)$ exactly (numerical error $2.19 \times 10^{-17}$, machine zero). Applying a generic rotation $R \in SO(8)$ that ignores the Fano plane incidence structure shatters the anomaly (error $22.2$, a factor of $10^{18}$ larger). The FTC anomaly is therefore a geometric invariant of the $G_2$ structure, not algebraic noise: it physically distinguishes operations preserving the Fano plane from those that do not. It vanishes precisely on Fano-line trajectories — the zero-level set of the $G_2$ moment map.

We provide explicit worked examples demonstrating the emergence of this topological tension in basic power rules and path integrals. Finally, we demonstrate that standard polynomial functions fail the Cauchy-Fueter regularity test, and identify $f(Z) = \bar{Z}/|Z|^8$ — the 8-dimensional point-charge monopole field — as the unique zero-friction analytic solution of the $G_2$ vacuum (Cauchy-Fueter residual $2.8 \times 10^{-12}$).

Open problem: whether $C(\gamma,F)$ is closed and non-exact, representing a non-trivial class in $H^3(G_2, \mathbb{R}) \cong \mathbb{R}$, remains an open computational topology problem.

Keywords: non-associative calculus · octonions · $G_2$ · Fundamental Theorem of Calculus · path integrals · Cauchy-Fueter regularity · FTC Commutator Anomaly · magnetic monopole · Fano plane · Associator Penalty · slice regularity