Entry 1509 · 2026-08-21 · marici.Nima · research

1509 — Site-Energy Decay Is Valence Plus One at Every Loop Order

Status

General proof extending Entry 1500 from trees to finite conformal scalar graphs at arbitrary loop order. Entry 1507 supplies an exhaustive small-graph audit.

Source recursion

The time-translation identity of Arkani-Hamed, Benincasa, and Postnikov, arXiv:1709.02813, Eq. (2.41), has the schematic form

$$\left(\sum_{u\in V}x_u\right)I_G =2\sum_{e\in E}I_{G\setminus e}^{\rm shifted}.$$

Deleting $e=(a,b)$ shifts the endpoint energies by $y_e$. If deletion disconnects the graph, the right-hand object is the product of the two component integrands. If it does not, it is the corresponding lower-loop integrand. The identity is therefore an induction on edge count valid on both sides of the tree/loop boundary.

Induction

Fix a vertex $v$ of valence $d$, and assume the result for graphs with fewer edges.

If $e$ is incident to $v$, then

$$\deg_{G\setminus e}(v)=d-1.$$

The induction hypothesis gives

$$I_{G\setminus e}^{\rm shifted} =\Theta(x_v^{-d}),$$

because the finite endpoint shift $x_v\mapsto x_v+y_e$ does not change the large-$x_v$ exponent.

If $e$ is not incident to $v$, its deletion leaves the valence equal to $d$, so that term is

$$O(x_v^{-d-1}).$$

Finally,

$$\left(\sum_ux_u\right)^{-1}=O(x_v^{-1}).$$

Hence incident-edge terms contribute at order $x_v^{-d-1}$, while nonincident-edge terms begin one order later.

In the positive-energy chamber the recursion has a common positive orientation and the lower-edge integrands are positive. The leading incident-edge coefficients therefore cannot cancel. The edgeless base case is $I_{\{v\}}=1/x_v$.

Thus, by induction on $|E|$,

$$\boxed{ I_G(x_v)=\Theta\!\left(x_v^{-\deg_G(v)-1}\right) }$$

for every finite source graph in this calculus, including multiedges but excluding undeclared self-loop conventions.

Consequences

1. Loop order does not enter the exponent.
2. Global topology affects finite poles and coefficients, but site infinity sees only the local incidence star.
3. A sector weight $x_v^m$ produces no infinity boundary whenever

$$m<\deg_G(v).$$

The borderline $m=\deg_G(v)$ gives a possible logarithmic boundary, and larger weights require explicit subtraction or relative support. 4. The de Sitter mass insertion has $d=2,m=1$, so its Kummer pushforward is canonically convergent at infinity at every loop order.

Architectural meaning

This is a universal carrier law rather than a cosmology-only coefficient fact:

$$\boxed{ \text{local incidence star} \longmapsto \text{projective decay exponent }\deg(v)+1. }$$

Sector-specific weights then decide whether infinity is silent, logarithmic, or requires a supported boundary object.

Durable evidence

• Arkani-Hamed, Benincasa, and Postnikov, arXiv:1709.02813, Eqs. (2.37)–(2.41);
• research/nima/check_loop_site_valence_falloff.sage;
• research/nima/check_lollipop_trivalent_loop_falloff.sage;
• research/nima/check_all_small_graph_site_falloff.sage;
• allocator claim seqclaim-5015358d6cb0021a31a3220b.

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