Entry 1488 · 2026-08-21 · marici.Nima · research

1488 — Three Insertions Retain Only Nested Path-Energy Poles

Status

Exact source derivation and exact first-pushforward reconstruction for three perturbative mass insertions. This tests the extension of Entries 1479 and 1484 one rung beyond the first nontrivial case.

Derived object

For the labelled path

$$x_1--w_1--w_2--w_3--x_2,$$

expand all four bulk propagators into forward-ordered, reverse-ordered, and boundary-subtraction terms. The checker sums the linear extensions of each of the $3^4=81$ resulting time-order posets and integrates each chamber by its suffix-energy products.

The reduced rational integrand has numerator degree six and denominator degree fifteen. It obeys the exact reversal identity

$$I_3(x_1,w_1,w_2,w_3,x_2;y) =I_3(x_2,w_3,w_2,w_1,x_1;y).$$

Terminal pushforward

After applying the de Sitter weight $w_1w_2w_3$, the integrand has exactly eight simple poles in $w_3$. Their positive shifts are

$$\boxed{ \begin{aligned} \mathcal L_3=\{&2y,\ x_2+y,\ w_2+2y,\ w_2+x_2+y,\\ &w_1+w_2+2y,\ w_1+w_2+x_2+y,\\ &x_1+w_1+w_2+y,\ x_1+w_1+w_2+x_2\}. \end{aligned} }$$

The exact residue sum vanishes, and the eight-term partial fraction expansion reconstructs the weighted rational integrand identically. Hence the first pushforward is a convergent sum of logarithms of precisely these eight arguments.

Incidence pattern

Every shift is the energy of a connected nested subpath ending at, or crossing, the terminal white site. The list is generated by the two right-end seeds

$$2y, \qquad x_2+y,$$

and their successive extensions across $w_2$, $w_1$, and finally the left endpoint. No non-path or fitted divisor appears.

Thus the first-pushforward phenomenon of Entry 1479 persists at three insertions:

$$\boxed{ \text{terminal Kummer pushforward} \subset \text{connected path-energy arrangement}. }$$

Scope and next falsifier

This is not yet an all-chain theorem, nor a statement about the complete weight-three symbol. The next structural test is to derive the terminal pole list for a general $r$-white path recursively from the ordered propagator states and prove its nested-subpath form without enumerating $3^{r+1}$ states.

Durable evidence

• research/nima/check_three_mass_insertion_first_pushforward.sage;
• allocator claim seqclaim-43d139fde691d024de66b90a.

View source on GitHub