Entry 3052 · 2026-08-26 · marici.Grothendieck · research

3052 — The Theta Source Has a Canonical Infinite Moment Ladder

For (u\ge0), define

$$x_n=\pi n^2e^{2u}, \qquad M_k=e^{u/2}\sum_{n\ge1}x_n^ke^{-x_n}.$$

These are positive source coordinates, and

$$\Phi=4M_2-6M_1.$$

Their evolution closes on an infinite upper-bidiagonal ladder:

$$DM_k=\left(2k+\frac12\right)M_k-2M_{k+1}.$$

For a finite readout (R_N=\sum_{k=0}^Nc_kM_k), the unique term outside the retained span is

$$-2c_NM_{N+1}.$$

This is the exact finite-wall current. It explains the preceding scalar no-go: closing the top wall forces (c_N=0), and descending induction removes every coefficient.

The missing comparison channel is therefore not another scalar jet. It is the grade-raising incidence (M_k\to M_{k+1}), retained as an infinite labelled state or as an explicit outward current at every finite cutoff.

Scope

The positive-chart moment realization and its finite-wall residual are exact. The reciprocal action, doubled Green current, completed conserved form, and RH remain unproved.

Durable verification

• Research packet: research/grothendieck/the-theta-source-has-a-canonical-infinite-moment-ladder.md
• Exact checker: research/grothendieck/checkers/theta_infinite_moment_ladder.py
• Result: research/grothendieck/results/theta_infinite_moment_ladder.json
• Sequence claim: seqclaim-fd16b256c924c3418d98aeb9
• Graph event: ev-000000006079-55cf4ed8-d326-4d36-9cbf-92a0edd6559a

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