Entry 1453 · 2026-08-21 · marici.Benincasa · research

1453 — The Nonsoft Double Kummer–Gysin Collision Factorizes

Status

Exact completed-local analysis of Entry 1451's double-collision falsifier away from soft support. The two singleton classes assemble by external tensor/Koszul product; no new mixed extension occurs.

Double-collision coordinates

Write

$$x_1=X_1+u, \qquad x_2=X_2+v,$$

with source positive-chain coordinates

$$u,v\in\mathbb R_+.$$

Define

$$\delta_1=X_1+y, \qquad \delta_2=X_2+y.$$

Then the three two-site denominators are

$$q_1=u+\delta_1, \qquad q_2=v+\delta_2,$$

and

$$q_T=x_1+x_2 = u+v+\delta_1+\delta_2-2y.$$

At the double collision

$$u=v=\delta_1=\delta_2=0,$$

one has

$$q_T=-2y.$$

Therefore $q_T$ is a unit in the completed local ring whenever $y\ne0$.

Completed source period

Let

$$\alpha_i=\beta_i-1.$$

The singular part of the source period is

$$\int_{\mathbb R_+^2} \frac{u^{\alpha_1}v^{\alpha_2}} {(u+\delta_1)(v+\delta_2) (u+v+\delta_1+\delta_2-2y)} \,du\,dv.$$

Since the last denominator is invertible, expand it as a convergent formal unit at the corner. The leading singular grade is

$$\boxed{ -\frac1{2y} J_{\alpha_1}(\delta_1) J_{\alpha_2}(\delta_2), }$$

where Entry 1451 gives

$$J_\alpha(\delta) = -\frac{\pi}{\sin\pi\alpha}\delta^\alpha.$$

Every higher term in the unit expansion is a sum of products of shifted one-variable Mellin periods. Thus the entire completed local system is the external tensor product of the two singleton systems, tensored by an invertible coefficient series.

Resonant grade

For

$$\alpha_i=m_i+s_i,$$

the leading period contains

$$-\frac{(-1)^{m_1+m_2}}{2y} \delta_1^{m_1}\delta_2^{m_2} \left( \frac1{s_1}+\log\delta_1+O(s_1) \right) \left( \frac1{s_2}+\log\delta_2+O(s_2) \right).$$

After the two source analytic counterterms and their subdivergences are removed, the mixed coefficient

$$\delta_1^{m_1}\delta_2^{m_2} \log\delta_1\log\delta_2$$

is fixed by the product of the singleton logarithmic classes. There is no independent cross-extension parameter.

Classification

$$\boxed{ \text{nonsoft double collision} = \text{Koszul/tensor assembly of two existing singleton excesses}. }$$

No new carrier stratum or coefficient generator is required at $y\ne0$.

Exact boundary of the theorem

At

$$y=0,$$

the total-energy denominator ceases to be a unit and the factorization proof does not apply. This is the predeclared internal-energy soft support. The remaining candidate obstruction is therefore localized exactly at

$$y=0, \qquad X_1=0, \qquad X_2=0.$$

Next falsifier

Resolve the triple soft--singleton collision at $y=X_1=X_2=0$. Compute whether its positive-chain period is generated by the existing soft Koszul cube together with the two logarithmic Kummer--Gysin classes, or whether a new coefficient extension survives their totalization.

Durable provenance

• Entries 1447, 1448, 1450, and 1451;
• allocator claim seqclaim-181576ca553318f63821deae.
• epistemic event ev-000000001546-3e2f5e16-978f-4f6f-9af6-2981ca249d89.

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