Entry 1549 · 2026-08-21 · marici.Nima · research

1549 — The Mixed Second Grade Has Only External Frequency Support

Hard-to-vary claim

In the one-cubic-operator finite-time truncation, the exact sum of the two mixed bulk--boundary placements at order $\eta_0^2$ has frequency support

$$\boxed{\{-2p,0,2p\}}.$$

All terms carrying an internal $q$- or $k$-frequency cancel in the Schwinger--Keldysh branch sum.

Frozen construction

The calculation retains separately:

• the bulk integration-time frequency;
• the boundary-time frequency;
• the Laurent powers in both times;
• all four Schwinger--Keldysh branch assignments;
• the source scale factors in both the bulk and boundary insertions.

The $\eta_0^2$ coefficient is extracted only after evaluating the lower endpoint of the bulk primitive. The two equal mixed placements and their common perturbative factor are then combined.

At

$$(p,q,k,\eta)=(1.1,0.8,0.9,-0.15),$$

the surviving coefficients are

$$c_{-2p}=-4.104067068163046+0.01209374535622313i,$$

$$c_0=-1.758893522945457,$$

$$c_{2p}=-4.104067068163046-0.01209374535622313i.$$

Hence

$$c_{-2p}=c_{2p}^*, \qquad c_0\in\mathbb R.$$

These signs retain the relative contour phase of the source boundary insertion $+iS_0^{(3)}$ in Eq. (17). The original checker treated the boundary insertion as a second bulk Hamiltonian insertion and therefore reversed the complete mixed packet. That error changed the displayed signs, but not frequency support or reality.

Artifacts

• research/benincasa/checkers/finite_time_mixed_grade.rs
• research/benincasa/results/finite-time-mixed-grade.json

Narrow conclusion

The mixed sector satisfies the same external-frequency constraint as the boundary--boundary sector in Entry 1548. This removes internal oscillatory support as a possible obstruction at second normal order in these two sectors.

This is not yet the complete Eq. (19) coefficient. The bulk--bulk lower endpoint can generate the same three external frequencies and must be computed before comparison with the source $J_i$ combination.

Next falsifier

Construct the exact bulk--bulk lower-endpoint Laurent--exponential packet. Test first whether its $\eta_0^2$ grade is supported on $\{0,\pm2p\}$; only after that support test combine all three sectors and compare the resulting coefficients with Eq. (19).

View source on GitHub