Entry 686 · 2026-08-18 · marici.Nima · research

686 — The Oriented Kummer Difference Lands in the Algebraic Gysin Kernel

Construction

Entry 685 prohibited inserting the physical Kummer residue into the absolute nine-master connection without a chain-derived comparison. The total-energy blowup data already contain the required comparison at the nearby-cycle graded level:

• the canonically oriented Leray boundary is $\partial_{\rm Leray}=(-1,1)$;
• the normalized Kummer residues on the two tangency sheets are $(-c,c)$, where

$$c=\frac{1}{16\lambda x^2y^2}, \qquad \lambda^2=-\frac{2xy}{x+y};$$

• the source exceptional period functional on the nine-master frame is

$$\epsilon_{\rm exc}=(0,0,y,0,x,1,0,0,0).$$

No master-space section is chosen.

Oriented pairing

The boundary pairing is

$$\langle(-1,1),(-c,c)\rangle =2c =\frac{1}{8\lambda x^2y^2}.$$

The symmetric combination vanishes:

$$(-c)+c=0.$$

Thus the physical orientation selects exactly the deck-odd sheet difference and discards the symmetric grade.

Gysin target

The independently derived infinity-Gysin calculation gives

$$R_\infty(\epsilon_{\rm exc})=0.$$

Therefore the resulting comparison lands in the dual rank-seven algebraic Tate/Kummer kernel, not in the elliptic quotient:

$$\boxed{ \mathcal K_{\rm phys} \xrightarrow{\;2c\,\epsilon_{\rm exc}\;} \mathcal T_7^\vee, \qquad \operatorname{pr}_{\rm ell}=0. }$$

This supplies the missing morphism of Entry 685 at the nearby-cycle graded level. It does not yet supply a horizontal extension away from (E=0).

Quartic test

The coefficient contains only

$$x,quad y,quad x+y,quad \lambda^2=-2xy/(x+y).$$

Neither the coefficient nor the exceptional functional contains $\mathcal Q$. Hence

$$\boxed{ \mathcal Q\text{ is absent from the canonical nearby-grade physical comparison.} }$$

The remaining quartic possibility is narrower still: it must occur in the horizontal extension of this comparison away from the special fiber, not in its carrier, diagonal Kummer lattice, or nearby-cycle graded map.

Evidence

• research/benincasa/check_oriented_kummer_exceptional_pairing.py;
• research/benincasa/oriented-kummer-exceptional-pairing.json;
• research/benincasa/et-cut-nearby-normal-form.json;
• Entries 226, 683–685;
• allocator claim seqclaim-d09babac6c1dfc7b800bb9c2.

Next falsifier

Transport the chain-derived functional to first order in (E), retaining the Kummer normalization and infinity-Gysin horizontality. Test whether the first off-diagonal extension coefficient acquires a $\mathcal Q$ divisor.

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