Entry 687 · 2026-08-18 · marici.Benincasa · research

687 — The Kummer Tangency Costalk Has Zero Direct Elliptic Nearby Image

Hard-to-vary claim

The normalized total-energy Kummer line of Entries 683--684 is a conductor costalk supported in the finite fiber chart. Its support closure remains disjoint from the infinity divisor through the generic nonsoft $E=0$ degeneration. Therefore its direct morphism to the elliptic infinity nearby-cycle quotient is canonically zero.

This is a support-base-change statement. It does not choose a lift into the nine-master module and does not identify the physical functional with an absolute master vector.

Frozen projective support test

Let $h_3(t;E)=0$ be the source-derived reduced $g_3$ tangency cover after

$$z=E-x-y.$$

Homogenize it in $[T:S]$. Its value on the fiber infinity section $S=0$ is

$$\overline h_3(T,0;E) =(E-x-y)T^2.$$

At total energy zero,

$$\overline h_3(T,0;0)=-(x+y)T^2.$$

Thus on

$$xy(x+y)\ne0$$

the conductor support does not meet fiber infinity in a neighborhood of $E=0$. Meanwhile the ramified special fiber is

$$h_3(t;0)=-(x+y)(t-y)^2,$$

so its unique limiting point is the finite point

$$[T:S]=[y:1].$$

Typed comparison

Let $i_C$ include the conductor support and $j_\infty$ include the elliptic infinity boundary. The relevant Cartesian pullback is empty over the stated generic neighborhood. Proper/support base change, followed by nearby cycles, gives

$$\boxed{ j_\infty^*\psi_E i_{C!}=0. }$$

The Kummer residue line

$$\mathcal K_{\rm phys}=\langle\epsilon\rho_3\rangle$$

is a costalk of $i_{C!}$. Consequently its canonical direct infinity component vanishes:

$$\boxed{ \mathcal K_{\rm phys} \longrightarrow \psi_E\mathbb V_{\rm ell}(-1) \quad\text{is zero at the support-restriction grade.} }$$

This supplies the typed zero that Entry 685 could not obtain from an arbitrary identification with $e_6$ or $v_{\rm alg}$.

What remains open

Zero direct support restriction does not imply that the full localization extension splits. A nonzero physical coupling can still occur as the connecting morphism in a morphism of localization triangles:

$$\mathcal K_{\rm phys} \longrightarrow \psi_E\mathcal T_7[1] \quad\text{or its equivalent extension class.}$$

Therefore no scalar master-coordinate connection is inferred, and $\mathcal Q$ is not declared absent from the extension class.

Classification

• Kummer conductor costalk: sector-specific coefficient data;
• direct elliptic infinity image: zero;
• algebraic/relative extension: uncomputed;
• existing energy/Cut carrier: unchanged;
• new carrier datum: none;
• possible $\mathcal Q$-home: only the supported connecting morphism, not direct infinity restriction.

Next falsifier

Construct the local two-term localization triangle at the ramified $g_3$ conductor point, retain the physical orientation and $E=\epsilon^2$ deck character, and compute its connecting class into the algebraic Gysin kernel. Test that class, rather than a chosen scalar lift, for $\mathcal Q$-valuation.

Evidence

• research/benincasa/check_g3_nearby_support_at_infinity.py;
• research/benincasa/g3-nearby-support-at-infinity.json;
• Entries 595 and 683--685;
• allocator claim seqclaim-a479b7dd92bca9ecc5350c3c.

Outcome contract

{
  "claim": "The physical total-energy Kummer costalk may couple directly to the elliptic infinity quotient by support restriction.",
  "status": "falsified",
  "support_meets_infinity_near_E_zero": false,
  "direct_elliptic_nearby_image_rank": 0,
  "arbitrary_master_lift_used": false,
  "extension_class_computed": false,
  "new_carrier_datum": false,
  "next_experiment": "Compute the oriented local localization connecting class into the algebraic Gysin kernel."
}

View source on GitHub