Relative cohomology, Fargues–Fontaine realizations, and arithmetic positivity: a proof framework for directions C1–C3 in Connes's program

We investigate three proposed routes to Weil positivity in Connes’s program: bivariant arithmetic Riemann–Roch and Hodge index, Fargues–Fontaine realization, and relative cohomology of the arithmetic Picard pair. The paper is an audit rather than a proof of the Riemann Hypothesis. We prove that a positive-semidefinite vector form satisfying scaling covariance and orbit continuity yields an invariant radical, a strongly continuous nor-malized unitary representation on the Hilbert quotient, and the expected integrated adjoint identity. A two-dimensional counterexample shows that these conclusions, even with trace-class descended operators, may erase an off-critical-line summand. The terminal implication therefore requires exact transfer of the unconditional cohomological trace or nonvanishing of every actual zero sector. For the Picard-cohomological route, we prove corrected localization, generalized unit-ball sheaf, divisor-data tensor, completed principal invariance, and scalar section–effectivity results, and exhibit failures of fixed-ball tensoriality, perfectness, and boundary duality. For the Fargues–Fontaine route, we construct the canonical degree-one divisor lift, rank-one modification functor, annular Frobenius eigensheaves, and exact logarithmic degree normalization, while showing that higher rank requires additional lattice data. For the correspondence route, we prove the semilocal graph/measure calculus and a finite equivariant index theorem, but atomic approximation, family effectivity, Hodge index, trace-faithful globalization, and the decisive comparison with the Weil form remain open.