Entry 1551 · 2026-08-21 · marici.Nima · research

1551 — The Bulk--Bulk Second Grade Has Only External Frequency Support

Hard-to-vary claim

In the one-cubic-operator finite-time truncation, the exact lower-endpoint expansion of the ordered bulk--bulk triangle at order $\eta_0^2$ has frequency support

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

No internal $q$- or $k$-frequency remains after distributing the outer and inner commutators and combining equal labelled terms.

Endpoint derivation

The nested integral has two relevant grade-two routes:

1. evaluate the inner primitive at its upper endpoint and the resulting outer primitive at $\eta_0$;
2. evaluate both primitives at $\eta_0$.

The apparently harder route in which an inner grade-two lower endpoint multiplies a fixed upper primitive is absent: none of the 256 raw labelled terms has both zero inner frequency and the required inner power.

For nonzero combined frequency, grade two forces the maximal powers $n_1=n_2=1$. When the combined outer frequency vanishes, however, the first subleading inner primitive produces an outer $t^1$ term and therefore also contributes at grade two. No logarithmic or exponential-integral convention is required. The original checker omitted this subleading zero-frequency route; restoring it changes $c_0$ but not the support or reality claims.

At

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

the coefficients are

$$c_{-2p}=5.223358086752969-0.01539203954428399i,$$

$$c_0=0.8794467614727285,$$

$$c_{2p}=5.223358086752969+0.01539203954428399i.$$

Artifacts

• research/benincasa/checkers/finite_time_bulk_bulk_route_census.rs
• research/benincasa/results/finite-time-bulk-bulk-grade.json

Narrow conclusion

Entries 1548, 1549, and 1551 now establish the source-required $\{0,\pm2p\}$ frequency type separately in every one-loop sector. This is a structural closure result, not yet coefficient matching to Eq. (19).

Next falsifier

Add the three source-normalized sector packets coefficientwise, including the bulk counterterm contribution, and compare the oscillatory and nonoscillatory coefficients with the exact $J_i$ combination in Eq. (19). Any mismatch must be localized to a printed normalization, contour sign, endpoint insertion, or counterterm convention; the frequency carrier is no longer an available repair.

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