Anchored Controllability and a Three-Sector Obstruction Calculus for Impossibility Theorems in Aggregation and Representation

This paper isolates and formalizes the common proof engine behind impossibility theorems arising from aggregation, representation, and measurement problems in economics, statistics, and social choice. The central mechanism is anchored controllability: if one aspect of a system admits a pointwise cofinal anchor operation that strictly improves that aspect while leaving a second aspect exactly invariant, and the second aspect admits any nontrivial strict comparison, then mixed 2-cycles exist and no scalar function can be simultaneously strictly monotone in both aspects. This is the core theorem, whose proof is a single composition of the anchor invariance property and the pointwise cofinality condition. A differential refinement localizes the mechanism via transversal kernels: if two scalar-valued maps of a smooth state space have Jacobians whose kernels are transversal at a point in the sense that each map is nonconstant in the kernel direction of the other, then every neighborhood of that point contains a mixed 2-cycle. A coordinatewise convex cofinality lemma extends the anchor construction to the full simplex in the ordinal welfare application, with explicit correction of a hypothesis omitted in prior sources. Within the target class of aggregation and representation problems, three canonical obstruction sectors are identified. The collapse sector arises when decision-relevant states become observationally indistinguishable because distinct decision classes share a common reachable signal law; the precise quantitative coordinate is the Le Cam collapse margin, equal to one-half the infimum total variation distance between feasible signal laws across critical pairs, via the Le Cam identity. The gluing sector arises when locally compatible data fail to extend to a global section; the quantitative coordinate is the contextual fraction for probabilistic gluing models, proved to be exactly zero or one for deterministic local families, and the descent defect for linear presheaves, equal to the dimension of the quotient of the compatible local family space by the image of the global restriction map. The diagonal sector arises from Lawvere-style self-reference: a typed diagonal theorem shows that the shell of anti-fixed-point-composed diagonal words is always disjoint from the representable words, with exact diagonal margin computed as one plus an off-diagonal excess count divided by the domain size, and exact shell cardinality given by the number of restrictions of the anti-fixed-point family to the diagonal trace image. All eight Boolean combinations of the three sector indicators are realized, so no chain-valued invariant can classify the full cube. The ideal-valued obstruction profile mapping each system to its triple of lower ideals in the respective atom preorders is proved to be initial among complete join-semilattice-valued invariants generated by monotone weights on finite sector witnesses. For linear observation architectures the three quantitative coordinates couple through a restriction-map budget identity: the collapse kernel dimension plus the gluing cokernel dimension equals the sum of the global and local space dimensions minus twice the rank of the restriction map. An interface loss theorem shows that any diagonal evaluator induced through the restriction map has rank bounded by the rank of the restriction map, so gluing defect forces diagonal rank loss. On the topological side, a closed-graph sufficiency theorem establishes that when the transitive closure of the union of finitely many closed partial orders on a second-countable metrizable space is itself a closed antisymmetric relation, a continuous simultaneous scalar representation exists via the Nachbin-Levin-Herden theorem. A hard counterexample proves sharpness: a specific subspace of the plane with two closed partial orders whose transitive closure has bounded chain length but non-closed graph admits no continuous scalar representation, so bounded chain length alone does not rescue representability. Six applications are verified at theorem level. Arrow's theorem is a pure gluing obstruction: the Condorcet profile produces a compatible local pairwise ordering with contextual fraction one and no global extension, and every independent information axiom and Pareto-satisfying rule is dictatorial. Myerson-Satterthwaite is a pure collapse obstruction: every efficiency-determining observational quotient of a feasible bilateral-trade mechanism contains a collapse atom, with welfare loss bounded below by one-half minus the collapse margin. Multidimensional ordinal welfare is verified via the convex cofinality lemma with the best-corner non-poor anchor preserving the conditional poor distribution while strictly improving all lower-orthant thresholds. Capital aggregation is verified via the transversal kernel theorem applied to the observable equilibrium map. Hidden-subgroup causal sign nonidentification is verified via an explicit kernel direction that changes subgroup effects while preserving the aggregate observable. Shrinkage under subquadratic matched Lp loss is captured by a regime-opposition variant: at every fixed shrinkage coefficient, local improvement at the symmetric point coexists with a large-signal curvature penalty that dominates the Stein gain. The Chouldechova-Kleinberg algorithmic fairness impossibility is analyzed as a non-instance of anchored controllability: it fits the collapse sector but is driven by the algebraic rigidity of the Bayesian odds identity rather than orthogonal-projection cycles, illustrating the discriminative power of the taxonomy. A repair calculus establishes that any successful repair must strictly decrease at least one coordinate of the profile, and identifies sectorwise necessary interventions: support separation by decision class for collapse, alteration of the local family, cover, or presheaf for gluing, and alteration of the self-reference interface or type discipline for diagonal. A representation-dimension theorem on finite domains proves that the minimum number of scalar coordinates needed for a faithful multi-index equals the minimum size of an acyclic partition of the axioms, and the pairwise orthogonal controllability theorem shows that when every pair of closed partial orders admits mixed 2-cycles on a second-countable metrizable space, the representation dimension equals the number of orders exactly.