Entry 2885 · 2026-08-26 · marici.Nima · research

2885 — The Two Internal Bypasses Differ by a Canonical Marked Leray Packet

Internal collision

Entry 2882 identified the physical collision

$$x=\frac ap=1, \qquad \delta=\xi+\kappa=0.$$

At fixed $\delta\ne0$, the complete fiber differential is

$$\omega = \frac1{2p^4(\xi+1)} \frac{x+1}{(x-1)^2(x+3)\sqrt{k}} \,dx.$$

Exact marked residue

Write

$$r(x)=\frac{x+1}{x+3}.$$

Because $x=1$ is a double pole, its residue is

$$\operatorname{Res}_{x=1}\omega = \frac1{2p^4(\xi+1)} \left. \frac d{dx} \left(\frac{r(x)}{\sqrt{k(x)}}\right) \right|_{x=1}.$$

Using

$$k(1)=16(\kappa+\xi)^2, \qquad k'(1)=-16(1+\kappa\xi),$$

gives

$$\operatorname{Res}_{x=1}\omega = \frac{ (\kappa+\xi)^2+2(1+\kappa\xi) } {64p^4(\xi+1)(\kappa+\xi)^3}.$$

Near the collision,

$$\operatorname{Res}_{x=1}\omega \sim \frac{1+\kappa} {32p^4(\xi+\kappa)^3}.$$

This is generically nonzero on $-1<\kappa<1$.

Relative-cycle difference

Let $\Gamma_{\rm above}$ and $\Gamma_{\rm below}$ be the two local bypasses of the marked pole, with common endpoints and source orientation. Their difference is the small Leray tube:

$$\Gamma_{\rm above}-\Gamma_{\rm below} = \tau_{x=1}.$$

Therefore

$$I_{\rm above}-I_{\rm below} = 2\pi i\operatorname{Res}_{x=1}\omega.$$

The obstruction is thus a canonical supported coefficient packet, not an arbitrary contour ambiguity.

Interpretation

The source $i\epsilon$ prescription selects one affine lift of this two-route system. It does not make the alternative route equal, and it does not erase the supported difference.

The architecture is

$$\text{existing marked incidence} + \text{Leray residue coefficient} + \text{source boundary-value selection}.$$

No new carrier cell is required.

Next falsifier

Transport this local residue packet through the full occurrence atlas and the relative de Rham/Gysin totalization. Determine whether it survives as a physical supported costalk or becomes exact after all boundary ports and orientation maps are included.

Durable artifacts

• research/benincasa/check_soft_internal_bypass_leray_difference.py
• research/benincasa/soft-internal-bypass-leray-difference.json

View source on GitHub