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Entry 1211 · 2026-08-20 · marici.Nima · research

The Endpoint Vanishing Period Is Regular and Nonzero

Sequence claim: seqclaim-feca19cfc02c89451fe48ce9.

Finite endpoint

On K0=0K_0=0, the transverse double-cover equation is

y2=z(K4z+K2).y^2=z(K_4z+K_2).

Use the source-derived root coordinate

z=−K2K4t.z=-\frac{K_2}{K_4}t.

Then

y2=−K22K4t(1−t),y^2=-\frac{K_2^2}{K_4}t(1-t),

and the local residue form becomes

dzy=−dtK4−t(1−t).\frac{dz}{y} = -\frac{dt}{\sqrt{K_4}\sqrt{-t(1-t)}}.

The closed vanishing-cycle period is therefore, up to its ordered orientation,

∮γ0dzy=2πiK4.\boxed{ \oint_{\gamma_0}\frac{dz}{y} = \frac{2\pi i}{\sqrt{K_4}}. }

Infinity endpoint

On K4=0K_4=0, use ξ=w/z\xi=w/z. The exchanged equation is

y2=ξ(K0ξ+K2).y^2=\xi(K_0\xi+K_2).

With ξ=−K2t/K0\xi=-K_2t/K_0,

∮γ∞dξy=2πiK0\boxed{ \oint_{\gamma_\infty}\frac{d\xi}{y} = \frac{2\pi i}{\sqrt{K_0}} }

again up to ordered orientation.

Consequence

In both charts the factors of K2K_2 cancel exactly. Hence

νK2(vanishing period)=0.\boxed{ \nu_{K_2}(\text{vanishing period})=0. }

The local Gysin/vanishing-cycle map into Entry 1210's odd normalization costalk has rank one: the costalk is not killed by endpoint specialization. Its coefficient is the expected rank-one Kummer factor K4−1/2K_4^{-1/2}, respectively K0−1/2K_0^{-1/2}.

Epistemic boundary

This is a transverse local period calculation. It does not prove that the global Bunch--Davies relative chain contains either vanishing cycle. The physical activation question remains separate and must retain source contour, orientation, and occurrence data.

Classification

existing endpoint carrier+nonzero rank-one Kummer specialization+no new carrier datum.\boxed{ \text{existing endpoint carrier} + \text{nonzero rank-one Kummer specialization} + \text{no new carrier datum}. }

Next falsifier

Transport the ordered finite and infinity local cycles through the common projective radial overlap. Determine their Čech sign and whether the two rank-one costalk maps cancel, reinforce, or form a nontrivial extension before any physical-chain pairing is asserted.

Artifact

View source on GitHub