The Gaitsgory-Raskin full-support conjecture for restricted geometric Langlands in positive characteristic

Theorem (full support): under either regime, $Y'=Y$, so $\mathbb L^{\mathrm{restr}}_G:\mathrm{Shv}_{\mathrm{Nilp}}(\mathrm{Bun}_G(X))\to\mathrm{IndCoh}_{\mathrm{Nilp}}(Y)$ is a $\mathrm{QCoh}(Y)$-linear equivalence; a rational-coefficient form is also proved over $\overline{\mathbb F}_q$. Consequences: nonzero localized categories at every spectral component, the full arithmetic trace formula in the finite-field regime, and nonzero Hecke eigenobjects for every parameter. It does not remove the characteristic hypotheses, and does not assert bounded constructibility or perversity of eigenobjects (the three October 5 eigensheaf companions construct perverse tame eigensheaves in specific settings).

Theorem (full support): under either regime, $Y'=Y$, so $\mathbb L^{\mathrm{restr}}_G:\mathrm{Shv}_{\mathrm{Nilp}}(\mathrm{Bun}_G(X))\to\mathrm{IndCoh}_{\mathrm{Nilp}}(Y)$ is a $\mathrm{QCoh}(Y)$-linear equivalence; a rational-coefficient form is also proved over $\overline{\mathbb F}_q$. Consequences: nonzero localized categories at every spectral component, the full arithmetic trace formula in the finite-field regime, and nonzero Hecke eigenobjects for every parameter. It does not remove the characteristic hypotheses, and does not assert bounded constructibility or perversity of eigenobjects (the three October 5 eigensheaf companions construct perverse tame eigensheaves in specific settings).