A Tangent Structure for Injected Data: Feasible Cones, Universality at Corners, and Jets on the Tetrad Scheme

Abstract In the lab for latent actors [1], an adversary injects a share ε ≥0 of records from a law Q into data from a truth P. Every slope, stealth direction and obstruction in that paper is a statement about tangent vectors of the probability simplex. The lab paper only named the tangent-category structure behind them. This companion builds it and checks every axiom on finite examples. The cone structure. Take polyhedra and the lexicographic sets they generate, with affine maps; on simplices the affine maps are exactly the stochastic matrices. On this category the feasible-direction functor T carries a projection, a zero, an addition, a vertical lift and a canonical flip satisfying every equational axiom of Cockett and Cruttwell [2]. Its additive bundles are commutative monoids, which have negatives only at relative-interior points. An injection direction Q−P is exactly a tangent vector. Universality fails exactly at the boundary. The vertical lift is universal at a point if and only if the point lies in the relative interior. At a boundary point the fibre of T(p) over the zero section contains (x,u,0,−u) for every u that enters the interior: a second-order direction may take back mass that the first-order direction has just placed. The fibre equals Cx ×Cx,u, where Cx,u is the cone at “x moved infinitesimally along u”. Second- order functoriality also needs affinity: a polynomial self-map of ∆3 sends a point of T2∆3 outside it. The lab’s maps. On the open simplex the usual tangent category applies. Theorem E’s slopes are T(F)(P,Q−P), and the stealth direction χ spans the vertical bundle of the marginal map. The decision pipeline is a morphism off the coincidence strata, where |Tˆ r| grows like |v2|−1. The value function is a morphism on each chamber; on a wall it fails to preserve addition by exactly the relevance functional |ℓ(ξ)|. Jets on the tetrad scheme. With the Weil-algebra tangent structure the vertical lift is universal everywhere, even at singular points. What fails instead is completing a pair in T2V to a point of T2V. The obstruction is the class [H(u,v)] ∈coker J. It vanishes off the two-coincidence stratum, and on it it has norm |q(u,v)|/√2 for an explicit quadric q. Its diagonal is twice the Andr´e–Quillen obstruction of the lab’s Theorem G(4), so that obstruction is now a statement about T2 . Together these give a dichotomy. Positivity breaks universality of the lift, and the algebra breaks completion to second order. This parallels the lab’s finding that real descent fails only through positivity. Fourteen checks (TG1–TG13 and TG6b) verify everything exactly or to rounding.