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Entry 2887 · 2026-08-26 · marici.Nima · research

2887 — The Internal Bypass Survives as a Deck-Anti-Invariant Gysin-Kernel Line

Full marked occurrence divisor

On the compact elliptic fiber, retain the four labelled marked points

D={(1,+),(1,−),(−3,+),(−3,−)}.D= \{(1,+),(1,-),(-3,+),(-3,-)\}.

The signs label the two square-root sheets. The source differential has residue vector

R=(R1,−R1,R3,−R3),\mathbf R=(R_1,-R_1,R_3,-R_3),

where

R1=(κ+ξ)2+2(1+κξ)64p4(ξ+1)(κ+ξ)3,R_1 = \frac{(\kappa+\xi)^2+2(1+\kappa\xi)} {64p^4(\xi+1)(\kappa+\xi)^3},

and

R3=−164p4(ξ+1)(κ−ξ).R_3 = -\frac1{64p^4(\xi+1)(\kappa-\xi)}.

Gysin kernel

For four marked points on the compact connected fiber, the point-to-fundamental class Gysin map is

Gys⁡=(1111).\operatorname{Gys} = \begin{pmatrix}1&1&1&1\end{pmatrix}.

It has rank one and kernel rank three. One exact integral basis is

e1,−=(1,−1,0,0),e3,−=(0,0,1,−1),e13=(1,1,−1,−1).\begin{aligned} e_{1,-}&=(1,-1,0,0),\\ e_{3,-}&=(0,0,1,-1),\\ e_{13}&=(1,1,-1,-1). \end{aligned}

The global residue theorem is explicit:

Gys⁡(R)=0.\operatorname{Gys}(\mathbf R)=0.

Internal bypass class

Entry 2885's internal tube is the line

Lbypass=Q⟨(1,−1,0,0)⟩⊂ker⁡(Gys⁡).\mathcal L_{\rm bypass} = \mathbb Q\langle(1,-1,0,0)\rangle \subset \ker(\operatorname{Gys}).

Therefore it is not killed by the generic occurrence-resolved Gysin totalization.

The deck trace does kill it:

(1,−1,0,0)⟼1−1=0.(1,-1,0,0) \longmapsto 1-1=0.

But the physical positive-occurrence covector

(1,0,0,0)(1,0,0,0)

detects it. Deck trace and physical chamber evaluation are thus inequivalent readouts.

Narrow conclusion

The internal bypass packet survives generically as a deck-anti-invariant relative coefficient line. It becomes invisible only after an additional coarse deck trace, not through the occurrence-resolved Gysin kernel itself.

This is a coefficient/readout distinction over existing marked carrier support. It does not require a new carrier stratum.

Remaining gate

Derive the Gauss–Manin transport of this rank-one kernel line through the full base puncture atlas. Determine whether the line is horizontal by itself or mixes with the other two Gysin-kernel directions.

Durable artifacts

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