Entry 1496 · 2026-08-21 · marici.Nima · research

1496 — A Trivalent Star Falls with the Fourth Power of Its Central Energy

Status

Exact independent topology test of Entry 1495's proposed local-valence mechanism.

Source graph

Consider a trivalent star with central site energy $w$, leaf energies $x_1,x_2,x_3$, and three independent edge energies $y_1,y_2,y_3$. No equal-edge diagonal or mass-insertion specialization is imposed.

The checker expands all three propagators into their forward, reverse, and boundary-subtraction terms, sums every compatible time ordering, and reduces the resulting rational function exactly over

$$\mathbb Q(w,x_1,x_2,x_3,y_1,y_2,y_3).$$

Result

As a rational function of the central energy,

$$(\deg_w\operatorname{num},\deg_w\operatorname{den})=(4,8).$$

Therefore

$$\boxed{ I_{\rm star}(w)=O(w^{-4}). }$$

Since the central site has valence three, this agrees with

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

Comparison

The currently exact local census is

$$\begin{array}{c|c} \deg(v)&\text{site-energy falloff}\\ \hline 1&x_v^{-2}\\ 2&x_v^{-3}\\ 3&x_v^{-4} \end{array}$$

The valence-two row holds for generic split edge labels by Entry 1495 and for mass-insertion paths through four white sites by Entry 1493. The trivalent row shows that the pattern is not path-specific.

Interpretation

The local incidence star controls the order at energy infinity. Edge-label diagonals subsequently control pole collisions, while cosmological weights control whether the resulting Fourier/Kummer pushforward remains convergent. These are three typed stages, not one undifferentiated analytic property.

Scope and next falsifier

The generic valence law is strongly supported but not yet proved for arbitrary graphs or valence. The next step is an inductive proof using the connected-subgraph recursion cited in Section 4 of Benincasa, arXiv:1909.02517. A loop graph or a valence-four star is the smallest further finite falsifier if that induction exposes an ambiguity.

Durable evidence

• research/nima/check_trivalent_star_site_falloff.sage;
• Benincasa, arXiv:1909.02517, Eqs. (4.3)–(4.4);
• allocator claim seqclaim-0c912c553af88c754ba01bc7.

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