Entry 1495 · 2026-08-21 · marici.Nima · research

1495 — Cubic Terminal Falloff Precedes the Mass Edge Diagonal

Status

Exact falsification of the hypothesis that Entry 1493's cubic falloff is created by identifying the two edge energies adjacent to a mass insertion.

Split-edge audit

Replace the physical equal-edge path

$$x_1\mathop{--}^{y}w\mathop{--}^{y}x_2$$

by the labelled generic bivalent path

$$x_1\mathop{--}^{y_L}w\mathop{--}^{y_R}x_2, \qquad y_L\ne y_R.$$

The full time-order expansion, with forward, reverse, and boundary-subtraction propagator terms retained on both edges, gives reduced terminal degrees

$$\boxed{ (\deg_w\operatorname{num},\deg_w\operatorname{den})=(1,4). }$$

The leading numerator coefficient is the nonzero constant $8$; it does not contain $y_L-y_R$. Therefore

$$I_{\rm split}(w)=O(w^{-3})$$

before any diagonal specialization.

Restricting to $y_L=y_R$ preserves the same degrees $(1,4)$. It neither creates nor improves the cubic falloff.

Correction

The rejected explanation was

$$\text{equal-edge Gysin diagonal} \Longrightarrow w^{-3}.$$

The surviving explanation is instead

$$\boxed{ \text{generic bivalent-site carrier structure} \Longrightarrow w^{-3}, }$$

with the physical mass diagonal acting only afterward to identify the two edge labels and create the higher edge-pole multiplicities of Entries 1464, 1467, and 1469.

Meta-level consequence

This separates two mechanisms that had been conflated:

1. local valence controls decay in the site-energy direction;
2. the source diagonal controls coincidence and multiplicity in edge energy.

Both are geometric, but they live on different maps. The absence of a Kummer boundary at terminal infinity is carrier-local; the repeated $(2y)^{-k}$ poles are diagonal/Gysin data.

Next falsifier

Prove or disprove the generic valence law

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

from the connected-subgraph recursion. A trivalent source vertex is the smallest independent test beyond the bivalent path family.

Durable evidence

• research/nima/check_mass_insertion_diagonal_uv_improvement.sage;
• Benincasa, arXiv:1909.02517, Eqs. (4.3)–(4.4);
• allocator claim seqclaim-324e78073cd9baecf3180f97.

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