Entry 683 · 2026-08-18 · marici.Nima · research

683 — The Total-Energy Tangency Residue Is a Kummer Nearby-Cycle Line

Local problem

Entry 682 isolates (g_3) at total energy (E=x+y+z=0) as the unique physical residue boundary where numerator vanishing, denominator collision, and tangency-cover ramification coincide. Determine the individual Laurent orders after the minimal base change (E=q^2).

Exact normal form

Put

$$y=q^2-x-z, \qquad t=-(x+z)+w.$$

The reduced (g_3) tangency equation becomes

$$h_3= zw^2-2q^2(x+z)w+2q^2x(x+z)+q^4(w-x-z).$$

Thus its two Puiseux branches have

$$w=\pm\lambda q+O(q^2), \qquad \lambda^2=-\frac{2x(x+z)}{z}.$$

The physical numerator is (-E=-q^2). On either branch,

$$h_3'=2z\lambda q+O(q^2),$$

while the remaining denominator has leading term

$$D_3=4x(x+z)\lambda^2q^2+O(q^3).$$

Consequently

$$\rho_3 = \frac{1}{16\lambda x^2(x+z)^2}\,q^{-1}+O(1).$$

The conjugate branch replaces (lambda) by (-lambda), so it has the same order and the opposite leading coefficient.

Nearby-cycle type

Both sheets therefore have order (-1) in (q), equivalently half-integral order (-\tfrac12) in (E). The ordinary base does not carry a regular logarithmic generator. After the Kummer base change, however,

$$\boxed{q\rho_3\text{ is regular and has odd deck provenance}.}$$

The total-energy limit of the physical exceptional pairing is consequently a Kummer nearby-cycle line. It is not torsion in an ordinary conductor quotient, and it cannot be represented by selecting one analytic sheet on the unramified base.

Quartic consequence

No quartic factor enters the local equation, Puiseux relation, or leading coefficient. The calculation strengthens the separation:

$$\boxed{ \mathcal Q\text{ is absent even from the ramified diagonal nearby-cycle lattice.} }$$

If (mathcal Q) is physical, it must occur in the off-diagonal comparison between this Kummer line and another coefficient block, or in the associated extension class.

Evidence

• research/benincasa/check_g3_total_energy_nearby_cycle.py;
• research/benincasa/g3-total-energy-nearby-cycle.json;
• Entries 680–682;
• allocator claim seqclaim-715cd89943e8f497c04fc6bf.

Next falsifier

Compute the supported comparison from this odd Kummer nearby-cycle line to the algebraic and elliptic limit blocks. Test the off-diagonal matrix entry, without choosing a splitting, for a zero or pole along (mathcal Q=0).

View source on GitHub