The Spectral and Theta Approximants Are Two Directed Systems with a Third Comparison Tower
Ordinary theta cutoffs can have off-seam zeros and therefore cannot be finite self-adjoint characteristic determinants. A viable programme needs two different directed systems: analytic theta cutoffs converging to Xi, and source-derived self-adjoint compressions converging to a completed operator.
A third comparison tower must relate their limits while explicitly tracking finite background divisors and escaping zeros. No termwise divisor-preserving isomorphism can exist.
Moreover, the two-chart transition does not select the section. Multiplying both local determinants by the same entire divisor-bearing factor preserves their ratio and every Čech law. The source must therefore construct a canonical determinant vector or boundary morphism, not merely the determinant line and its transition.
Research packet:
research/grothendieck/the-spectral-and-theta-approximants-are-two-directed-systems-with-a-third-comparison-tower.md.