Entry 2892 · 2026-08-26 · marici.Nima · research

2892 — The Opposite Internal Bypass Has an Exact Soft Finite-Part Shift

Rational route coefficient

Let

$$\delta=\xi+\kappa, \qquad a=1-\kappa, \qquad B=\frac{3-\kappa}{(1-\kappa)^2}.$$

The internal marked residue of Entry 2885 has the exact partial fraction decomposition

$$64p^4R = -\frac{B}{\delta+a} +\frac{B}{\delta} -\frac{2}{a\delta^2} +\frac{2(1+\kappa)}{\delta^3}.$$

An exact primitive is

$$F_R(\delta) = \frac1{64p^4} \left[ B\log\!\left(\frac{\delta}{\delta+a}\right) +\frac{2}{a\delta} -\frac{1+\kappa}{\delta^2} \right].$$

Pointed route difference

The basepoint $t=2$, equivalently $\xi=1$, has

$$\delta_0=1+\kappa.$$

Therefore the two globally continued route primitives differ by

$$\Delta P_{\rm route} = 2\pi i \left[ F_R(\delta)-F_R(1+\kappa) \right].$$

Near the soft endpoint $t=\xi+1=0$, the additional logarithmic coefficient, before multiplying by $2\pi i$, is

$$-\frac{B}{64p^4}.$$

After subtracting that logarithm in the same pointed coordinate $t/2$, the finite route shift is

$$\operatorname{FP}_{\rm route} = \frac{2\pi i}{64p^4} \left[ B\log\!\left(-\frac{1-\kappa}{1+\kappa}\right) -\frac{4}{(1-\kappa)^2} \right].$$

The source negative-imaginary boundary value fixes the branch of the displayed logarithm.

Consequence

The opposite bypass is physically distinguishable before any coarse trace. It changes both:

• the soft logarithmic coefficient;
• the pointed finite part.

These changes are predicted by the existing marked Leray packet. They are not free subtraction constants and do not require adding carrier support.

Interpretation

The source $i\epsilon$ prescription now has an explanatory role: it selects one of two globally distinct affine transports whose difference is calculated independently from the internal marked incidence.

Remaining gate

Compose this exact route correction with the complete rank-two elliptic period block and test whether the source-selected branch contributes only to the algebraic/Tate extension or also changes the elliptic quotient.

Durable artifacts

• research/benincasa/check_soft_internal_route_finite_part.py
• research/benincasa/soft-internal-route-finite-part.json

View source on GitHub