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Entry 1451 · 2026-08-21 · marici.Benincasa · research

1451 — The Source Positive Chain Activates the Resonant Kummer–Gysin Class

Status

Exact local physical pairing for Entries 1448 and 1450. The source's positive site-weight chain and analytic regularization select a nonzero generic excess period and a nonzero resonant logarithmic finite part.

Local source chain

At the first singleton collision set

u=x1−X1,δ=X1+y,α=β1−1.u=x_1-X_1, \qquad \delta=X_1+y, \qquad \alpha=\beta_1-1.

The source site-weight occurrence has

u∈R+.u\in\mathbb R_+.

The Gysin denominator is

x1+y=u+δ.x_1+y=u+\delta.

All other factors in the two-site universal integrand are regular and nonzero at a generic point of δ=0\delta=0. Freezing them to their corner value reduces the singular part of the physical pairing to

Jα(δ)=∫0∞uαu+δdu.J_\alpha(\delta) = \int_0^\infty \frac{u^\alpha}{u+\delta}du.

Exact Euler--Mellin period

For −1<Re⁡α<0-1<\operatorname{Re}\alpha<0, substitution u=δvu=\delta v gives

Jα(δ)=δα∫0∞vα1+vdv=πsin⁡π(α+1)δα=−πsin⁡παδα.\begin{aligned} J_\alpha(\delta) &= \delta^\alpha \int_0^\infty\frac{v^\alpha}{1+v}dv\\ &= \frac{\pi}{\sin\pi(\alpha+1)}\delta^\alpha\\ &= -\frac{\pi}{\sin\pi\alpha}\delta^\alpha. \end{aligned}

The primary source already admits analytic continuation of its Mellin parameters, so this formula defines the same period outside the convergence strip. The energy-space iϵi\epsilon prescription fixes the branch of δα\delta^\alpha.

For generic nonintegral α\alpha, the coefficient is nonzero. Therefore the source positive chain pairs nontrivially with Entry 1448's rank-one exceptional Kummer--Gysin line.

Resonant finite part

Write

α=m+s,m∈Z.\alpha=m+s, \qquad m\in\mathbb Z.

Then

−πsin⁡π(m+s)=(−1)m+1(1s+O(s))-\frac{\pi}{\sin\pi(m+s)} = (-1)^{m+1} \left(\frac1s+O(s)\right)

and

δm+s=δm(1+slog⁡δ+O(s2)).\delta^{m+s} = \delta^m \left(1+s\log\delta+O(s^2)\right).

Consequently

Jm+s(δ)=(−1)m+1δm(1s+log⁡δ+O(s)).\boxed{ J_{m+s}(\delta) = (-1)^{m+1}\delta^m \left( \frac1s+\log\delta+O(s) \right). }

Subtracting the analytic-regularization pole can change a local polynomial finite term. It cannot remove the coefficient of

δmlog⁡δ.\delta^m\log\delta.

Thus Entry 1450's resonant torsion class is physically activated by the source chain: its dual period is the logarithmic finite part.

Classification

existing singleton-energy support+source-selected logarithmic Kummer–Gysin coefficient.\boxed{ \text{existing singleton-energy support} + \text{source-selected logarithmic Kummer--Gysin coefficient}. }

No additional physical current was chosen by hand. The positive ray, iϵi\epsilon branch, and analytic regularization all come from the frozen cosmological integral.

This is stronger than an associated-grade existence result: the physical period pairing is nonzero. It remains local to a generic point of the singleton-energy divisor.

Next falsifier

At the double collision

X1+y=0,X2+y=0,X_1+y=0, \qquad X_2+y=0,

compute the two-variable positive-chain period and determine whether it is the tensor/Koszul assembly of the two singleton logarithmic classes or carries a new extension between them.

Durable evidence

View source on GitHub