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Entry 1479 · 2026-08-21 · marici.Nima · research

1479 — The Two-Insertion Pushforward Has Only Nested Path-Energy Letters

Status

Exact source-derived integrand and exact first pushforward. The complete twofold integral has not yet been evaluated, so no weight-two closed form is claimed.

Source derivation

Expand every conformal bulk-to-bulk edge propagator into its two ordered bulk terms and its boundary-subtraction term. For each resulting edge-state word, sum all linear extensions of its time-order poset and integrate the ordered times by suffix-energy denominators. This derives the path integrand without fitting a rational ansatz.

As a calibration, the same algorithm applied to one white site, with its de Sitter weight w1w_1, reproduces all four logarithmic coefficients of Eq. (4.7) of Benincasa, arXiv:1909.02517v1 exactly.

Two-white-site integrand

For the path

x1  −−  w1  −−  w2  −−  x2x_1\;--\;w_1\;--\;w_2\;--\;x_2

with common edge energy yy, the derived rational integrand has the ten linear denominator factors

(x2+y)(w2+2y)(w2+x2+y)(w1+2y)×(w1+w2+2y)(w1+w2+x2+y)(x1+y)×(x1+w1+y)(x1+w1+w2+y)(x1+w1+w2+x2),\begin{aligned} &(x_2+y)(w_2+2y)(w_2+x_2+y)(w_1+2y)\\ &\quad\times(w_1+w_2+2y)(w_1+w_2+x_2+y)(x_1+y)\\ &\quad\times(x_1+w_1+y)(x_1+w_1+w_2+y) (x_1+w_1+w_2+x_2), \end{aligned}

and the checker derives and retains its exact cubic numerator. The physical double insertion supplies the measure weight w1w2w_1w_2.

First Kummer pushforward

As a rational function of w2w_2, the weighted integrand has six simple poles. Exact residue reconstruction gives

w1w2I2=∑j=16Aj(w1)w2+Lj(w1),∑jAj=0,w_1w_2 I_2 =\sum_{j=1}^{6}\frac{A_j(w_1)}{w_2+L_j(w_1)}, \qquad \sum_jA_j=0,

with

{Lj}={2y, x2+y, w1+2y, w1+x2+y, x1+w1+y, x1+w1+x2}.\boxed{ \{L_j\} = \{2y,\ x_2+y,\ w_1+2y,\ w_1+x_2+y, \ x_1+w_1+y,\ x_1+w_1+x_2\}. }

Therefore the convergent first pushforward is

−∑jAj(w1)log⁡Lj(w1).-\sum_j A_j(w_1)\log L_j(w_1).

Both the vanishing residue sum and the full rational reconstruction are verified symbolically over Q(x1,w1,w2,x2,y)\mathbb Q(x_1,w_1,w_2,x_2,y).

Interpretation

No new divisor has appeared. Every logarithmic argument is a connected partial-energy sum inherited from the labelled path, and the four w1w_1-dependent letters are precisely its nested subpath sums. Thus the second insertion enlarges the coefficient alphabet by iterated path incidence, not by changing the carrier.

This supports—but does not yet prove—the conjecture that higher mass insertions are iterated Kummer/divided-difference pushforwards on the existing path-energy arrangement.

Next falsifier

Perform the remaining w1w_1 pushforward and compute its symbol. The iterated-incidence hypothesis fails if a symbol letter is not generated by the source partial-energy arrangement, or if a nonremovable mass-difference pole survives.

Durable evidence

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