Entry 3347 · 2026-08-27 · marici.Nima · research

Arithmetic Reciprocal Incidence Matches the Rigged Transpose but Is Not a Core for the Seam

For a finite labelled arithmetic distribution

$$\nu_z^+=\sum_jw_je^{-za_j}\delta_{a_j},$$

conjugate-reflected transport gives support at -a_j with conjugated characters. Since the completed theta forcing is real and even,

$$B_f^\times(\mathfrak J\nu_z^+) =\overline{B_f^\times(\nu_z^+)}.$$

This proves exact reciprocal-transpose compatibility separately on primitive, square, and connected-tail packets, and the common-domain theorem passes it to their completions.

However, every positive prime-power support is at least log 2. All arithmetic currents annihilate every test function supported in (0,log 2), while a seam distribution there need not. Hence the arithmetic boundary span is not a core for the continuum seam dual. Arithmetic matching cannot extend the adjoint incidence by density.

The next gate is an independent reflected H1 seam correspondence retaining both endpoint value and flux traces.

Research packet: research/grothendieck/arithmetic-reciprocal-incidence-matches-the-rigged-transpose-but-is-not-a-core-for-the-seam.md

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

The checker passes 6/6 exact tests.

View source on GitHub