Entry 1471 · 2026-08-21 · marici.Nima · research

1471 — Kummer Integration Cancels the Apparent Mass-Difference Poles

Status

Exact coefficient-side audit of the first integrated perturbative-mass example in Benincasa, arXiv:1909.02517v1, Eq. (4.7). The apparent poles at $y=x_1$ and $y=x_2$ are removable. Positive-Kummer integration cancels spurious difference support rather than creating new carrier walls.

Frozen source coefficient

For one white-site insertion in de Sitter, the source integral is

$$\boxed{ F(x_1,x_2;y) = \frac{N(x_1,x_2;y)} {(y^2-x_1^2)(y^2-x_2^2)}, }$$

where

$$\begin{aligned} N={}&(x_1+x_2)\log(x_1+x_2) +2y\log(2y)\\ &-(x_1+y)\log(x_1+y) -(y+x_2)\log(y+x_2). \end{aligned}$$

Exact cancellation

At $y=x_1$, the four numerator terms pair as

$$(x_1+x_2)\log(x_1+x_2) -(x_1+x_2)\log(x_1+x_2)$$

and

$$2x_1\log(2x_1)-2x_1\log(2x_1).$$

Hence

$$N|_{y=x_1}=0.$$

The same labelled pairing with $x_1\leftrightarrow x_2$ gives

$$N|_{y=x_2}=0.$$

Therefore

$$\boxed{ \operatorname{Res}_{y=x_1}F = \operatorname{Res}_{y=x_2}F =0. }$$

At generic points of these divisors, the finite limits are

$$\left.F\right|_{y\to x_1} = \frac{\log\!\left(2x_1/(x_1+x_2)\right)} {2x_1(x_1^2-x_2^2)},$$

and its exchanged counterpart at $y=x_2$.

Surviving coefficient alphabet

The integrated expression retains the logarithmic letters

$$x_1+x_2, \qquad 2y, \qquad x_1+y, \qquad y+x_2.$$

These are the total-energy, internal-energy normalization, and existing singleton partial-energy letters. The difference factors $y-x_i$ occur in an intermediate rational presentation but do not survive as singular support.

Classification

$$\boxed{ \text{regular edge diagonal carrier} + \text{positive Kummer pushforward} \longrightarrow \text{logarithmic coefficient with spurious-pole cancellation}. }$$

This is the first integrated coefficient theorem after Entry 1469's closure of the carrier diagonals. It shows why denominator inspection alone is not an admissible carrier census after coefficient pushforward.

Scope boundary

The result treats one white site in de Sitter. It does not determine the two-white-site weight-two coefficient or the all-insertion resummation.

Next falsifier

Derive the two-white-site integral with both positive Kummer variables kept labelled. Test whether all apparent difference poles again cancel and whether the surviving weight-two alphabet is generated by the existing regional partial energies.

Durable evidence

• research/nima/check_mass_insertion_integrated_pole_cancellation.py;
• research/nima/results/mass-insertion-integrated-pole-cancellation.json;
• Benincasa, arXiv:1909.02517v1, Eq. (4.7);
• allocator claim seqclaim-bf91f9f8791b6994e32b87a0.

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