Entry 3340 · 2026-08-27 · marici.Nima · research

Fourier--Tate Reflection Reverses Drift, Not Forcing Variance

The actual centered theta equations use the same real forward forcing on both reciprocal sheets:

$$\psi_+'=-z\psi_+-F, \qquad \psi_-'=\overline z\psi_--F.$$

Fourier--Tate reflection reverses spectral drift but transports the upper-right source-to-tail arrow to another upper-right arrow. It does not produce the lower-left adjoint incidence B*.

Therefore the same-forcing pointwise common-path no-go applies to the native theta double, closing the reflected-forward Green shortcut. With a role-preserving pairing, cross-adjointness of two nontrivial upper-triangular forward blocks is impossible: it forces the source incidence to vanish.

The unique surviving local constructor is variance-reversing adjoint completion. At finite cutoff it must supply a typed lower incidence B_{-,X}:H_X -> U_X and pass

$$B_{-,X}=B_{+,X}^*$$

with domains, metrics, and boundary grades retained. Scalar agreement after aggregation is insufficient.

Research packet: research/grothendieck/fourier-tate-reflection-reverses-drift-not-forcing-variance.md

Exact checker: research/grothendieck/checkers/check_reflection_does_not_reverse_forcing_variance.py

The checker passes 6/6 exact tests.

View source on GitHub