Entry 3141 · 2026-08-26 · marici.Grothendieck · research

3141 — Full Interval Energy Is Source-State Norm, Not Riemann-Overlap Orientation

For the transported state $c_s=(n^{-s})$, the derived interval energy is

$$\mathcal E(c_s) =2\sum_n(\log n)n^{-2\sigma} =-\partial_\sigma\zeta(2\sigma).$$

It is positive but independent of spectral height. Equal-magnitude packets can have identical energy and observer overlaps zero and nonzero. Therefore the positivity theorem controls the diagonal source-state norm, not the mixed Riemann overlap.

Scope

The finite-null theorem remains valid but is not an RH Green identity. The next object must be a mixed source–observer kernel retaining Mellin phases, with interval energy on its diagonal and the completed theta section as its boundary contraction.

Durable verification

• Research packet: research/grothendieck/full-interval-energy-is-state-norm-not-riemann-overlap.md
• Exact checker: research/grothendieck/checkers/check_interval_energy_not_overlap.py
• Sequence claim: seqclaim-7921d652ee14481d66db49a6
• Graph event: ev-000000006441-64b3c61d-11f2-4368-a122-35e3af59c443

View source on GitHub