Entry 1548 · 2026-08-21 · marici.Nima · research

1548 — The Boundary--Boundary Second Grade Has Only External Frequency Support

Hard-to-vary claim

After restoring the source scale factor $a_0^2$, the exact boundary--boundary sector at order $\eta_0^2$ has frequency support

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

No phase involving the internal momenta $q$ or $k$ survives.

Derivation

Expand each factor as a finite Laurent--exponential sum in $\eta_0$:

• two boundary weights $a_0^2C_K$;
• two external Wightman functions;
• the two equal-time internal Wightman functions.

Multiplication is performed while preserving the integer frequency label. Equal labels are combined only after the complete four-branch sum.

At a generic kinematic sample, the quadratic-grade coefficients are

$$c_{-2p}=-0.439721471588898+0.00129575843102391i,$$

$$c_0=0.879446761472728,$$

$$c_{2p}=-0.439721471588898-0.00129575843102391i.$$

Thus

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

and the sector is real.

These values include the common spatial factor $(p^2+q^2+k^2)^2$ and the boundary--boundary perturbative weight $-\tfrac12$. The original checker version omitted both factors; that omission changed the displayed normalization but not the support or reality claims.

Artifacts

• research/benincasa/checkers/finite_time_boundary_boundary_grade.rs
• research/benincasa/results/finite-time-boundary-boundary-grade.json

Narrow conclusion

The first exact $\eta_0$-graded sector has the frequency type required by Eq. (19). This does not prove the Eq. (19) coefficient: the mixed and bulk--bulk grade packets must still be added, and only their sum can be tested against the $J_i$ combination.

Next falsifier

Compute the two mixed-sector grade packets with the same labelled algebra. Require cancellation of every frequency outside $\{0,\pm2p\}$ before combining with the bulk--bulk endpoint expansion.

View source on GitHub