The Arithmetic of Order: A Computable Foundation for Physics

ThispaperpresentstheArithmeticofOrder(AoO),aframeworkforfoundationalphysics built onthe singleprinciple that realityemerges fromadiscrete, computable, andnonassociativealgebraicsubstrateovertherationalnumbers.WeformalizeanOrigin–Observation Duality,whereasingletri-octonionstatevectorvgeneratesauniqueHermitianJordanoperatorAthat satisfiesbothanonlinear internal eigen-equation(Av=v[v])andadmitsa classical rational-valuedspectral decomposition(Aw=λw,λ∈Q). Thedual natureof thisspectrumprovidesadeductiveoriginforanemergent3+1Dspacetimewithanintrinsic particle/anti-particlesymmetry.Weapplythis frameworktoreformulateLHCjetproductionas the spectral footprintof aG2 automorphismcascade, anon-perturbativeprocess governedbyoctonionalgebra.Finally,weshowhowthecoreprinciplesofAoOaremirrored inthearchitecturesofmoderngenerativemodels,groundingthetheoryinexperimentaland computationalpractice.