Probability, Context and Differentiation: The Layers Steps 1–4c: the distribution monad, the provenance comonad, the tangent layer, and orders of transitions

Abstract The three companions [1, 2, 3] use three structures side by side: probability laws and kernels, provenance (which source a record came from), and differentiation at the boundary of the simplex. This paper studies how they fit together, one layer at a time. Every ingredient of Step 1 is standard in categorical probability; what this step contributes is a set of exact statements about where the ingredients fail to commute, which the later steps build on. The finite distribution monad Dand the provenance comonad W= L×(−) are related by a distributive law λ: WD⇒DW, (l,p) →δl⊗p. All four axioms are written out with their types and proved. Its two-sided Kleisli category is the category of Markov kernels indexed by a source, with one source held throughout a composite. Marginalisation is total and affine. Event conditioning is partial and nonlinear. It is undefined exactly where the event has probability zero, it cannot be extended continuously to that wall, and its derivative grows like 1/p(A) near it. Conditioning commutes with marginalisation only for events that are cylinders over the kept coordinates. Contextual Bayes decomposition. Conditioning a pooled law equals pooling the conditioned sources with Bayes-updated weights. In the worked example, source weights 1 3 4 , 4 become 8 15 23 , 23 , and keeping the old weights gives the wrong answer ( 63 160 instead of 9 23 ). Forgetting provenance is marginalisation over the sources. It commutes with every window, but it is not injective: two families of sources can share a pooled law and every pooled table while differing in every labelled table. Step 2, the tangent layer. The cone tangent structure of the tangent companion commutes with the provenance comonad. Each source is a face of the labelled simplex, and a feasible direction leaves that face exactly when it injects whole records under another label; after forgetting provenance, an injected source and a drifting source have the same direction. Marginalisation and conditioning map feasible cones onto feasible cones, with explicit lifts; conditioning’s lift shrinks like p(A) near its wall. The cone TkHof the half-line is generated by elementary tensors, so it is closed under products, and pooling is a morphism at every order: the pooling paradox lives in the decision functional, not in pooling. Differentiating the contextual Bayes decomposition splits every sensitivity into a reweighting term and a within-source term; in the worked example the answer 9 16 23 moves at rate− 529 per unit of small-trial share. Strata and jumps. The sources that survive conditioning are exactly those with πlpl(A) > 0. Conditioning maps fibres of forgetting onto fibres: it keeps provenance ambiguity on the event, discards it off the event, and can destroy identifiability but never create it. Inside 1its domain, conditioning moves at rate at most 2/P(A), so nothing jumps and a sudden change needs a rare event. At the wall the limit is the conditional of the path’s leading jet; analytic passages are removable, and jumps need a kink, a non-analytic path, or a jump of the law. Step 3, orders of transitions. On non-negative germs in a Hardy field, order is a semiring homomorphism to (min,+), so every layer operation has an exact order calculus. Each source reaches an event at its own order. The sources of least order survive conditioning, with weights given by leading coefficients; the others vanish at their extinction orders. The two sides of a transition carry independent order data, analytic passages are removable but smooth ones need not be, and flat approaches have limits inside a Hardy field but not outside. A transition of order mis invisible to the tangent tower below level m. At a fixed gap, a higher-order approach switches more gradually. Step 4a–4b, several parameters. When several rates vanish together, orders form a fan: the normal fan of a Newton polyhedron, one cone per bounded face, each with its own surviving sources. The limit set is computed cone by cone. A limit exists when the exponents have a componentwise minimum, and for monomial cells only then. Lines through the limit point are not enough, and degenerate directions hide whole intervals. Iterated limits are the corners of the fan, and they are exactly what the tangent tower computes: the lexicographic lemma reads off one corner, and the canonical flip moves to the other. Five small examples show each case. Step 4c, parameters as an object. Parameters get their own 2-category, the Para con- struction over source-indexed kernels, whose 2-cells are reparametrisations; arcs are 2-cells from the germ line, and the real Puiseux field holds all arcs at once. Every layer operation commutes with reparametrisation. Decisions live on the order fan: along every arc the act is eventually constant, ties at the limit are broken at higher order, and the decision fan transports along monomial reparametrisations. In the worked example, two study designs are two 2-cells, each filling one cone, so each has a limit where the raw race has none. Step 4d, the decorated fan and Theorem J. The order fan carries a decoration, a facewise presheaf of initial forms, and only the pair (a probability-decorated order fan) determines the limits. Pooling’s whole curvature is its exact cross term. With the sample size as a parameter, the lab’s Theorem J becomes an inequality of orders: the band is the leading coefficient of the cross term on the one ray where exposure stays of order one, since exposure sees the share at half weight and impact at full weight. The same fan marks where Gaussian bands are valid, with a Poisson ray and an empirical wall beyond it. On the Poisson ray an exact discrete band replaces J’s, and beyond it a bounded adversary owns the rare event. Step 4e, graded classes. The classes Mk of maps that keep feasibility at order k form a strictly descending chain of subcategories. Every subtraction-free map lies in all of them, so every layer operation does, conditioning included, at every order. Only subtraction (contrasts, as in estimators and decision gaps) breaks feasibility. Step 4f, degenerate directions. A direction needs refining only when the event’s total mass P(A) has a degenerate initial form, because non-negative cells cannot cancel. The resolution is Newton–Puiseux for P(A): finite except along exact common roots, which lie in the wall. Two examples show a limit two refinements deep, and an infinite root cut by a competitor.