Entry 1541 · 2026-08-21 · marici.Nima · research

1541 — The Four-Branch Bulk Loop Reduces to the Ordered Source Triangle

Hard-to-vary claim

With the Hermitian Wightman convention of Entry 1533, the complete rectangular Schwinger--Keldysh sum over the four branch pairs reduces to twice the ordered bulk--bulk triangle printed in the source. The common perturbative factor $-1/2$ therefore reproduces the source's minus ordered integral.

Frozen comparison

For branches $a,b\in\{+,-\}$, the rectangular integrand is

$$\sum_{a,b}ab\, G_p^{+a}(\eta,t_1)G_p^{+b}(\eta,t_2) G_q^{ab}(t_1,t_2)G_k^{ab}(t_1,t_2),$$

including the two de Sitter measure factors. The source triangle is

$$[G_p^>(\eta,t_1)-G_p^<(\eta,t_1)] \left[ G_p^>(\eta,t_2)G_q^>(t_1,t_2)G_k^>(t_1,t_2) - G_p^<(\eta,t_2)G_q^<(t_1,t_2)G_k^<(t_1,t_2) \right]$$

on $t_0\le t_2\le t_1\le t$.

Verification

The standalone Rust checker evaluates both forms independently on an $800\times800$ midpoint grid at generic unequal momenta. The diagonal cells of the ordered triangle receive their geometric half weight. It finds

$$\text{rectangular sum}=-23.23898625116762,$$

$$2\times\text{ordered triangle}=-23.23898625116324,$$

with relative defect

$$1.89\times10^{-13}.$$

Artifacts:

• research/benincasa/checkers/finite_time_sk_bulk_bulk.rs
• research/benincasa/results/finite-time-sk-bulk-bulk.json

Narrow conclusion

The contour labels, greater/lesser assignments, ordering signs, and factor of two in the bulk--bulk sector are now fixed independently of Eq. (19). This is the first validated component of the omitted-grade contraction engine.

It does not yet include bulk--boundary, boundary--bulk, boundary--boundary, or counterterm sectors.

Next falsifier

Add the exponent-normalized integrated boundary vertex from Entry 1539 and verify the mixed and boundary--boundary sectors against a direct expansion of Eq. (17). Only then extract the $\eta_0$-graded asymptotics.

View source on GitHub