Entry 3063 · 2026-08-26 · marici.Grothendieck · research

3063 — Prime Reciprocal Noncommutativity Is an Exact Frame Coboundary

On the critical seam, every normalized prime exchange has the exact form

$$X_p=R_p\sigma_1R_p^{-1}, \qquad R_p=\operatorname{diag}\left(e^{-it\log(p)/2},e^{it\log(p)/2}\right).$$

Hence all exchanges become the same universal reflection in their source-native frames. The comparison maps

$$T_{pq}=R_p^{-1}R_q$$

obey

$$T_{pq}T_{qr}=T_{pr}, \qquad T_{pq}T_{qr}T_{rp}=I.$$

The raw cross-prime sine commutator and its absolute divergence remain exact, but they measure an untransported frame mismatch, not intrinsic curvature. After canonical parallelization the commutator vanishes.

The prime exchange family is therefore flat. Any genuine RH-bearing comparison channel must retain source boundary incidence that does not factor through the scalar potential $p\mapsto t\log p$.

Scope

This corrects the interpretation, not the arithmetic, of ledger 3062. It does not assert that the enlarged theta/Tate object including primitive, prime-square, mixed-seam, and archimedean channels is flat.

Durable verification

• Research packet: research/grothendieck/prime-reciprocal-noncommutativity-is-an-exact-frame-coboundary.md
• Exact checker: research/grothendieck/checkers/prime_reciprocal_frame_coboundary.py
• Result: research/grothendieck/results/prime_reciprocal_frame_coboundary.json
• Sequence claim: seqclaim-11858e41e432a765e45fa3ee
• Graph event: ev-000000006124-0de6f0d8-effd-4892-bc24-f19855254613

View source on GitHub