Entry 684 · 2026-08-18 · marici.Benincasa · research

684 — The Total-Energy Tangency Residue Is a Pure Kummer Line

Hard-to-vary claim

At generic nonsoft kinematics, the ramified $g_3$ physical tangency residue at total energy $E=0$ has precisely a square-root Kummer pole. After the canonical base change $E=\epsilon^2$, multiplying the residue by $\epsilon$ produces a regular deck-invariant line. No unipotent nilpotent part or new carrier component is required for this rank-one boundary pairing.

Frozen object

Use the source-derived reduced tangency polynomial $h_3(t;E)$, physical numerator $N_3=-E$, and remaining denominator $D_3$ from Entries 673--675 and 682. Set

$$z=E-x-y, \qquad E=\epsilon^2,$$

with $xy(x+y)\ne0$. The special fiber is

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

Puiseux normalization

The two roots are

$$t_\pm = y \pm\sqrt{-\frac{2xy}{x+y}}\,\epsilon -\frac{x}{x+y}\epsilon^2 +O(\epsilon^3).$$

For the physical residue

$$\rho_\pm = \left. \frac{N_3}{\partial_t h_3\,D_3} \right|_{t=t_\pm},$$

direct expansion gives

$$\rho_\pm = \pm \frac{\sqrt2} {32x^2y^2\sqrt{-xy/(x+y)}}\, \epsilon^{-1} +O(1).$$

Thus both branches have exact $\epsilon$-order $-1$, their leading coefficients are opposite, and the deck involution

$$\epsilon\longmapsto-\epsilon$$

exchanges them.

Nearby-cycle classification

Before normalization the residue line carries the anti-invariant quadratic character. Equivalently, in the base coordinate,

$$\rho\sim E^{-1/2}.$$

The normalized generator

$$\widetilde\rho=\epsilon\rho$$

is regular and deck invariant. Hence this rank-one contribution is a pure Kummer line with semisimple monodromy $-1$ before the Kummer twist and trivial monodromy after it:

$$T_s=-1, \qquad T_u=1, \qquad N=0.$$

This is compatible with the previously identified $\mathcal K_{B^{-1/2}}$ factor because $B=\ell_3E$ near the generic total-energy boundary.

Classification

• existing energy/Cut carrier: unchanged;
• coefficient support: square-root Kummer singularity at $E=0$;
• Tate/Kummer nearby grade: rank one after normalization;
• nilpotent extension on this line: absent;
• new carrier datum: none;
• $\mathcal Q$-support: absent.

Narrow consequence

The unique ramification found in Entry 682 does not by itself create a new carrier stratum or a nontrivial unipotent block. The physical tangency functional contributes the expected semisimple square-root character. Any logarithmic nilpotent monodromy of the full three-site system must come from the elliptic Gauss--Manin quotient or from its supported extension, not from this rank-one residue line.

Next falsifier

Insert this normalized Kummer line into the explicit infinity-Gysin sequence and compute whether its specialization couples off-diagonally to the nodal elliptic quotient. A nonzero coupling would be extension data; its absence would split the physical boundary pairing at the nearby-cycle graded level.

Evidence

• research/benincasa/compute_g3_total_energy_nearby_residue.py;
• research/benincasa/g3-total-energy-nearby-residue.json;
• Entries 680--682;
• allocator claim seqclaim-8a16a27229b6fbada9be1720.

Outcome contract

{
  "claim": "The ramified g3 tangency residue requires a new unipotent or carrier contribution at total energy zero.",
  "status": "falsified for the rank-one physical tangency residue",
  "residue_order_in_epsilon": -1,
  "deck_character_before_normalization": -1,
  "normalized_line_regular": true,
  "nilpotent_monodromy_N": 0,
  "new_carrier_datum": false,
  "next_experiment": "Compute the off-diagonal nearby-cycle coupling between the normalized Kummer residue line and the nodal elliptic Gysin quotient."
}

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