Entry 690 · 2026-08-18 · marici.Nima · research

690 — The Leray Boundary Class Is Flat to First Order

Question

Entry 688 shows that the first variation of the oriented physical residue coefficient has only established energy poles. The remaining possible first-order contribution to the local connecting class is motion of the two marked attachment sections defining the Leray boundary.

Weighted expansion

Use the total-energy weighted chart

$$E=\tau^2, \qquad a=y+\tau^2r, \qquad b=x-\tau^2r+\tau^3n.$$

After dividing the Cayley–Menger equation by $\tau^6$, its exceptional leading term is

$$4x^2y^2n^2+8xy(x+y)(r^2-1).$$

The physical attachment sections lie on (n=0) and specialize to (r=-1) and (r=1).

Exact first motion

Solve the full source polynomial with

$$r_\pm=\pm1+\alpha_\pm\tau+\beta_\pm\tau^2+O(\tau^3).$$

Coefficient comparison gives

$$\alpha_-=\alpha_+=0,$$

and

$$r_-=-1+\frac{x+y}{4xy}E+O(E^2), \qquad r_+=1-\frac{x+y}{4xy}E+O(E^2).$$

The sections move inside the same exceptional chart but remain labelled, disjoint, and unramified over the generic nonsoft base.

Relative homology consequence

In the moving relative basis

$$e_-=[p_-]-[p_0], \qquad e_+=[p_+]-[p_0],$$

the physical interval retains boundary

$$\partial_{\rm Leray}=e_+-e_-=(-1,1).$$

Thus its Gauss–Manin derivative vanishes to first order:

$$\boxed{ \nabla_E\partial_{\rm Leray}\big|_{E=0}=0. }$$

The coordinate motion changes representatives, not the relative homology class. It introduces no new pole and no quartic factor.

Combined first-order conclusion

At the local oriented-pairing level, the complete first variation is therefore the scalar variation already computed in Entry 688:

$$\frac{4x^2+19xy+4y^2}{4xy(x+y)}.$$

No $\mathcal Q$-support occurs in either the residue variation or the Leray-boundary variation. This still does not determine global gluing into the rank-seven algebraic kernel; Entry 689's character-matching problem remains the next obstruction.

Evidence

• research/benincasa/check_leray_attachment_first_order.py;
• research/benincasa/leray-attachment-first-order.json;
• research/benincasa/et-cut-nearby-normal-form.json;
• Entries 688–689;
• allocator claim seqclaim-141510bc25972a3455132ad1.

Next falsifier

Decompose the algebraic Gysin kernel under the surface-normalization involution and test whether it contains a unique anti-invariant line capable of receiving the oriented conductor costalk.

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