Entry 2366 · 2026-08-25 · marici.Benincasa · research

2366 — The All-Soft Scale Separates from the Soft-Triangle Coefficient Family

Test

Entries 2364--2365 identify the physical soft--triangle exceptional kernel and compile its branch cover from the existing endpoint Kummer coordinates. The remaining variable $p$ is the common energy scale. The finite test is whether it mixes nontrivially with the dimensionless coefficient geometry.

Sequence claim: seqclaim-63d461713f15bdb0454b163b.

Exact scale separation

Set

$$t=\frac ap.$$

Then

$$K_{\rm exc}=p^4 k(t,\kappa,\xi),$$

where

$$k =t^4-(10+8\kappa\xi)t^2 +16\kappa^2+40\kappa\xi+16\xi^2+9.$$

The marked sections become scale-independent:

$$a-p=p(t-1), \qquad a+3p=p(t+3).$$

Including $da=p\,dt$ and $\sqrt{K_{\rm exc}}=p^2\sqrt{k}$ on the chosen sheet, the normalized exceptional source form is

$$p^{-4} \frac{t+1} {2(t-1)^2(t+3)(\xi+1)} \frac{dt\wedge d\xi}{\sqrt{k}}.$$

Thus the complete dependence on the common scale is the rank-one character

$$\boxed{p^{-4}.}$$

Its local monodromy around $p=0$ is

$$\exp(-8\pi i)=1.$$

Result

The all-soft scale contributes an integral Tate/Kummer factor with trivial complex monodromy. The nontrivial coefficient problem is entirely carried by the dimensionless marked family in $(t,\kappa,\xi)$.

This does not make the all-soft extension trivial: the pole order, integral lattice, and limiting filtration may retain information even though the semisimple monodromy is identity.

Classification

• support: existing all-soft divisor $p=0$;
• scale coefficient: rank-one integral Tate/Kummer factor $p^{-4}$;
• semisimple scale monodromy: identity;
• dimensionless coefficient family: sector-specific and unresolved;
• new Carrier datum: none.

Scope

The result proves homogeneity and the local scale character. It does not construct the extension through $p=0$, its integral normalization, Cartier/Rees length, or the dimensionless relative Gauss--Manin system.

Durable verification

• research/benincasa/check_soft_triangle_scale_separation.py;
• research/benincasa/soft-triangle-scale-separation.json;
• exact symbolic scale and marked-section identities;
• epistemic event ev-000000003241-aadde611-35be-4011-a86b-572baa0fd702.

Next falsifier

Compute the logarithmic relative de Rham system of

$$w^2=k(t,\kappa,\xi)$$

with marked sections $t=1,t=-3,\xi=-1$, on the endpoint $C_2\times C_2$ cover. Determine its character decomposition and whether any extension support remains after the scale factor is removed.

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