Entry 1542 · 2026-08-21 · marici.Nima · research

1542 — The Boundary-Containing In-In Sectors Obey Placement and Reality Checks

Hard-to-vary claim

Using the source-derived cubic kernel

$$C_K=\frac1K\left(\frac1{K\eta_0}-i\right)$$

and the exponent-normalized insertion of Entry 1539, the branch-labelled bulk--boundary, boundary--bulk, and boundary--boundary sectors form a consistent real Schwinger--Keldysh contribution.

Frozen branch weights

For a bulk vertex on branch $a\in\{+,-\}$, use

$$w_a^{\rm bulk}(t)=\frac{a}{t^2}.$$

For the integrated boundary vertex, the plus and minus weights are

$$w_+^{\partial}=-a_0^2C_K, \qquad w_-^{\partial}=a_0^2C_K^*, \qquad a_0^2=\frac1{H^2\eta_0^2}.$$

No endpoint delta is numerically integrated; its source-authoritative integrated value is used directly.

Finite tests

At generic unequal momenta, independently compute

$$M_{B\partial} = \sum_{a,b}\int dt\, w_a^{\rm bulk}(t)w_b^\partial G_p^{+a}(\eta,t)G_p^{+b}(\eta,\eta_0) G_q^{ab}(t,\eta_0)G_k^{ab}(t,\eta_0),$$

and the placement-reversed $M_{\partial B}$. The checker finds

The corrected numerical values are recorded in the machine-readable packet.

with relative placement defect

$$\text{below }10^{-14}.$$

Their sum is real. The independently assembled boundary--boundary sector is also real.

Artifacts

• research/benincasa/checkers/finite_time_sk_boundary_sectors.rs
• research/benincasa/results/finite-time-sk-boundary-sectors.json

Narrow conclusion

Together with Entry 1541, every cubic-vertex placement in Eq. (18) now has a branch-labelled computational representation satisfying its first structural checks. This does not yet reproduce Eq. (19), because the momentum integral, bulk counterterm insertion, and $\eta_0$-graded asymptotic reduction remain.

Next falsifier

Assemble the four cubic placement sectors with the corrected bulk counterterms, extract the $\eta_0^2$ asymptotic coefficient, and require exact reproduction of Eq. (19) before calculating the omitted grades.

View source on GitHub