The number 168 = |PSL(2,7)| governs the combinatorics of non-associativity in the octonions and the zero-divisor geometry of the sedenions. We establish a mechanism connecting these two manifestations. The collinearity-associativity equivalence in the Fano plane PG(2,2) shows that each of the 168 ordered non-collinear triples corresponds to an anti-associative basis configuration. Under Cayley-Dickson doubling 𝕆 → 𝕊, each such configuration produces a primitive zero-divisor pair via a Sign Cancellation Lemma-an identity on octonion structure constants derived analytically from anti-associativity and the collinear cyclic property, without reference to any specific multiplication table. We give an intrinsic characterisation of primitive zero divisors, prove that each has exactly 4 primitive partners (nullity 4), and establish a natural bijection Φ between ordered non-collinear Fano triples and unordered primitive zero-divisor pairs-both sets having cardinality 168. The bijection is GL(3, 2)-equivariant under the embedding GL(3, 2) ≅ PSL(2, 7) ↪ 𝐺 2 = Aut(𝕆). The ordered pair space 𝒵(𝕊) = {(𝑎, 𝑏) : ‖𝑎‖ = ‖𝑏‖ = 1, 𝑎𝑏 = 0} is isometric to the compact Lie group 𝐺 2 (Reggiani 2024); together with Der(𝕊) = 𝔤 2 (unconditionally, via the Schafer-Wilmot derivation firewall), this yields two independent pathways to 168: a discrete path through Fano combinatorics and a continuous path through PSL(2, 7) ⊂ 𝐺 2 ≅ 𝒵(𝕊). All results are additionally verified by exhaustive computation.

Uncertainty quantification in neural networks typically requires expensive ensemble methods or approximate Bayesian inference. We show that Kolmogorov-Arnold Networks (KANs) with piecewise-linear hat-basis edge activations admit exact first-order uncertainty propagation under the Guide to the Expression of Uncertainty in Measurement (GUM, JCGM 100:2008), the international metrological standard. Our method, E-KAN (Epistemic KAN), propagates coefficient standard uncertainties analytically through each layer using the law of propagation of uncertainty (LPU), producing calibrated confidence intervals in a single forward pass-with no sampling, no ensembles, and no posterior approximation. On three UCI regression benchmarks, E-KAN GUM achieves 90-100% coverage at the 95% confidence level where 5-model deep ensembles achieve 0-76%. Validated against N =2,000 Monte Carlo trials on both a pharmacokinetic ODE system (σ-ratio 0.986, coverage 94.85%) and the E-KAN network itself (coverage 99.8%), GUM propagation is 20× faster than ensembles and provides ISO-traceable uncertainty budgets. We characterize failure modes: GUM breaks on feature interactions (Friedman-1: 10%), out-of-distribution inputs, and heteroscedastic noise.

Structured state space models (SSMs) such as S4 and Mamba rely on associative matrix operations to enable efficient parallel scans over sequences. We propose O-SSM, a state space model whose hidden state evolves via octonion multiplication in O ∼ = R 8 , deliberately exploiting the non-associativity of the octonion algebra. Among the 7 3 = 343 basis triples of the imaginary octonions, exactly 168 = |PSL(2, 7)| produce nonzero associators-creating 168 directions in which sequential state products depend on parenthesization order. Across 15 benchmarks spanning order-dependent, hierarchical, symmetry, and temporal tasks, O-SSM wins 12, including sorting (69.5% vs 35%), LRA-style ListOps (26% vs 15%), and Morse decoding (44.5% vs 14%). Multi-head scaling (4 heads × 8-dim = 32-dim hidden, 640 parameters) further improves sorting accuracy to 72.5% while diagonal SSMs remain at random chance (32.5%). O-SSM also outperforms S4D-Inv initialized diagonal SSMs by 11% on next-token prediction (loss 1.80 vs 2.01), showing that cross-dimensional coupling provides genuine advantage over structured initialization alone. The composition algebra property |xy| = |x| • |y| (Hurwitz's theorem) guarantees norm preservation through time, providing inherent training stability absent in sedenion (dim 16) extensions where zero divisors destroy state information. O-SSM is uniquely positioned at the Cayley-Dickson boundary: the maximal algebra combining non-commutativity, non-associativity, and norm preservation.

We study the sign of mean OllivierRicci curvature (ORC) on random k-regular graphs as a function of the density parameter η = k 2 /N. Exact linear programming computation of the Wasserstein-1 distance across N ∈ {50, 100, 200, 500, 1000} reveals a nite-size crossover at η c (N) = 3.75 − 14.62/ √ N (bootstrap 95% CI: η ∞ c ∈ [3.61, 3.84]; R 2 = 0.995). The transition slope scales as N −0.20 , indicating a smooth crossover rather than a sharp phase transition. Erd®sRényi and BarabásiAlbert graphs exhibit the same phenomenon at shifted η c , with the ordering η c (BA) < η c (ER) < η c (reg) explained by degree heterogeneity. A semi-analytical framework based on the Hehl matching formula reproduces the crossover with MAE = 0.009 and predicts a clustering-dependent phase boundary η c (N, C). Sphere-embedded ORC on the CayleyDickson tower (S 3 through S 127) saturates at a positive asymptote κ∞ > 0 with power-law exponent β = 0.283 ± 0.034, below the Johnson Lindenstrauss bound. A Lean 4 formalization provides machine-checked proofs of regime exclusivity (η < η c ⇒ κ < 0; η > η c ⇒ κ > 0). We validate the crossover on 6 real-world semantic networks spanning 4 languages, achieving 6/6 correct phase classication.

Scientic computing rarely fails because arithmetic is wrong; it fails because epistemic context is lost. Measurements are treated as bare numbers, uncertainty is bolted on post hoc, provenance is externalised into fragile logs, and side eects blur the boundary between inference and action. We present Sounio, a compiled systems language for epistemic scientic computing in which uncertainty, condence, and provenance are rst-class program data.