Entry 1503 · 2026-08-21 · marici.Nima · research

1503 — The First Loop Graphs Preserve Valence-Plus-One Decay

Status

Exact loop-level falsifiers of Entry 1500's tree theorem. The valence law survives both tests, but a general loop proof is not yet established.

Parallel-edge loop

For two sites connected by two independently labelled parallel edges $y_a,y_b$, the source time-order expansion reduces exactly to

$$\frac{1}{ (x_1+x_2)(x_1+y_a+y_b)(x_2+y_a+y_b)} \left[ \frac1{x_1+x_2+2y_a} +\frac1{x_1+x_2+2y_b} \right].$$

This matches Eq. (2.49) of Arkani-Hamed, Benincasa, and Postnikov, arXiv:1709.02813. Each site has graph valence two, and exact reduction gives

$$I_{\parallel}(x_i)=O(x_i^{-3}), \qquad i=1,2.$$

Triangle loop

For the three-site cycle with three independent edge energies, the complete $3^3$-state propagator expansion and all compatible time orders give

$$\deg_{x_i}D-\deg_{x_i}N=3, \qquad i=1,2,3.$$

Every vertex is bivalent, so

$$\boxed{ I_{\triangle}(x_i)=O(x_i^{-3}) }$$

at all three sites.

Consequence

The first loop graphs do not falsify

$$I_G(x_v)=O(x_v^{-\deg(v)-1}).$$

The agreement is nontrivial because Entry 1500's tree proof cannot simply be reused. A loop partition may cross several edges incident to $v$ at once, so counting successive disconnecting cuts no longer supplies all $\deg(v)+1$ powers. The additional suppression must be encoded by the loop-level OFPT sum or by projectivity of the loop cosmological polytope.

Next falsifier

Test a loop graph containing a trivalent vertex—for example a triangle with one attached leaf. If the trivalent site falls only cubically, the loop extension fails; if it falls quartically, derive the missing loop-compatible incidence argument.

Durable evidence

• research/nima/check_loop_site_valence_falloff.sage;
• Arkani-Hamed, Benincasa, and Postnikov, arXiv:1709.02813, Eq. (2.49);
• allocator claim seqclaim-a1f84b6f62e3ca2f35204ccf.

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