Entry 1512 · 2026-08-21 · marici.Nima · research

1512 — Borderline Sector Weight Exposes a Canonical Infinity Residue

Status

General leading-coefficient consequence of Entry 1509, with exact bivalent and trivalent normalization checks.

Boundary coefficient

For a vertex $v$ of valence $d$, define

$$C_{G,v} =\lim_{x_v\to\infty}x_v^{d+1}I_G.$$

Entry 1509 proves that this limit is finite and generically nonzero. Multiplying the integrand by the borderline sector weight $x_v^d$ gives

$$x_v^dI_G =\frac{C_{G,v}}{x_v}+O(x_v^{-2}).$$

Thus the associated one-form has the canonical infinity residue

$$\boxed{ \operatorname{Res}_{x_v=\infty} \bigl(x_v^dI_G\,dx_v\bigr) =-C_{G,v}. }$$

Incident-edge recursion

Take the leading $x_v$-grade of the source edge-deletion identity. Terms deleting nonincident edges are one order too small. The remaining terms give

$$\boxed{ C_{G,v} =\sum_{e\ni v}C_{G\setminus e,v}^{\rm shifted}. }$$

If deletion disconnects the graph, the coefficient includes the integrands of the components not containing $v$, evaluated with the prescribed endpoint shifts.

All summands are positive in the positive-energy chamber. Hence the borderline logarithmic residue is generically nonzero; it is not merely an allowed boundary that may accidentally vanish.

Exact normalizations

For a generic bivalent star,

$$C_{2} =\frac{2}{(x_1+y_1)(x_2+y_2)}.$$

For a generic trivalent star,

$$C_{3} =\frac{6}{(x_1+y_1)(x_2+y_2)(x_3+y_3)}.$$

More generally the star recursion gives

$$\boxed{ C_{\star_d} =\frac{d!}{\prod_{i=1}^{d}(x_i+y_i)}. }$$

The factorial counts the possible local orders in which incident edges are deleted.

Refined convergence trichotomy

For a sector weight $x_v^m$:

$$\begin{array}{c|c} m<d&\text{no infinity residue}\\ m=d&\text{canonical nonzero logarithmic residue }-C_{G,v}\\ m>d&\text{higher infinity divergence requiring subtraction/support} \end{array}$$

The carrier determines $d$ and $C_{G,v}$; the sector lens chooses $m$. Their comparison determines the required physical readout at infinity.

Durable evidence

• research/nima/check_borderline_weight_infinity_residue.sage;
• Arkani-Hamed, Benincasa, and Postnikov, arXiv:1709.02813, Eq. (2.41);
• allocator claim seqclaim-737c0d6314170d3ca43f9ba1.

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