Does the index theorem straddle two licence mechanisms, or reduce to one?

Status: open · laid 2026-09-11 · type: derivation (large; laid as its own probe rather than run in a round-2 slot)

Statement. Several results evaluate one quantity by two procedures, and at least three distinct structures license the second evaluation: (i) reordering a double integral — Fubini–Tonelli, precondition σ-finiteness plus absolute convergence, which is free when the integrand is non-negative; (ii) adjointness of d and ∂ under a pairing — Stokes, precondition an orientable manifold and a C¹ form, structural rather than a convergence condition; (iii) coincidence of primal and dual optima — convex duality, precondition convexity plus a constraint qualification. An index theorem equates a spectral quantity (mechanism i, in heat-kernel form) with an integral of a characteristic class (mechanism ii).

Friction. If the three mechanisms are genuinely distinct, a result equating a mechanism-(i) quantity with a mechanism-(ii) quantity is not an instance of either and needs its own account of why the two agree. The alternative is that one of the two mechanisms is not primitive — that the heat-kernel side is secretly a Stokes-type statement, or the characteristic-class side secretly a reordering — in which case the partition into mechanisms is finer than it needs to be. Neither is established: the difficulty of index theorems is consistent with both readings.

Question. Does the index theorem require a fourth licence, or does one of its two sides reduce to the other's mechanism? A satisfactory answer says which, and if a fourth licence, states its precondition in the same form as the three above.