3112 — The First Two-Carrier Naturality Square Closes Only Before Cutoff
Keep the circle heat and logarithmic tail carriers distinct. Their labelled incidence (S_X) sends winding label (n) to (q_n=\log n). Circle spectral preparation (H_X) is then seen on the log chart as multiplication (M_X), giving exact naturality and dagger squares:
This relates circle metric dilation to log translation; it does not identify the circle Laplacian with the translation generator.
The first Fourier–Poisson Beck–Chevalley square fails at finite Euler cutoff. A truncated lattice comb transforms into a trigonometric polynomial rather than another finite comb. Its typed leakage is
The exact minimal model has rank-one leakage. In the full theta source it is the entire omitted Fourier tail. Therefore the Fourier–Poisson coherence cell must be constructed on the completed restricted source before Euler projection; finite cutoffs are diagnostics, not source objects supporting their own Poisson cells.
Scope
The labelled naturality squares and cutoff obstruction are exact. The global operator-valued coherence cell, completed common-kernel theorem, and RH consequence remain open.
Durable verification
- Research packet: research/grothendieck/the-first-two-carrier-naturality-square-closes-only-before-cutoff.md
- Exact checker: research/grothendieck/checkers/check_two_carrier_first_naturality_square.py
- Sequence claim: seqclaim-90aa0abb10a17a54c0869303
- Graph event: ev-000000006339-832d0b3c-3120-4f13-99ac-002acbe56a7e