Entry 682 · 2026-08-18 · marici.Benincasa · research

682 — Only Total Energy Ramifies the Physical Tangency Pairing

Hard-to-vary claim

Among the source-derived zero and pole divisors of the six physical reduced-tangency residues, every generic conductor or marked-section collision is étale on its quadratic tangency cover except the $g_3$ total-energy boundary $E=x+y+z=0$. There, physical numerator vanishing and cover ramification coincide.

Sheetwise classification test

For each shared wall, let $h_i(t)=0$ be the reduced quadratic cover. For each irreducible factor $f$ of

$$\operatorname{Res}(h_i,N_i) \quad\text{or}\quad \operatorname{Res}(h_i,D_i),$$

compare $f$ with

$$\Delta_i=\operatorname{Disc}_t(h_i).$$

If $f\nmid\Delta_i$, the cover is étale at the generic point of $f=0$. A resultant zero then belongs to one analytic sheet: one residue has the corresponding zero or pole while the other remains regular. No deck monodromy is generated locally around that component.

Exact result

For $g_1$ and $g_2$, every physical numerator and denominator resultant factor is coprime to the corresponding $\Delta_i$. This includes the physical conductor cubics $R_1,R_2$ and all signed-energy or site-energy denominator collisions.

For $g_3$, all denominator collisions are likewise étale except

$$E=x+y+z=0.$$

The total-energy factor satisfies

$$E\mid\Delta_3, \qquad E^2\mid\operatorname{Res}(h_3,N_3), \qquad E^2\mid\operatorname{Res}(h_3,D_3).$$

Thus

$$\boxed{ E=0 \text{ is the unique tested physical boundary where tangency-sheet ramification and residue collision coincide.} }$$

This is precisely the scattering degeneration already known to require nearby cycles and a choice of physical sheet.

Consequence

Away from $E=0$, extending the physical exceptional pairing across a generic conductor component requires an ordinary sheetwise elementary modification, not a new ramified carrier or nearby-cycle object.

At $E=0$, the two-sheet description itself degenerates. The norm cannot separate the colliding numerator, denominator, and ramification orders. This boundary must be treated by logarithmic nearby cycles before deciding whether the rank-one image remains locally free, becomes torsion, or joins another coefficient block.

The quartic $\mathcal Q$ is absent from every boundary factor in this classification.

Classification

• generic conductor and marked-section boundaries: étale one-sheet coefficient modifications;
• total-energy boundary: ramified nearby-cycle coefficient geometry;
• new carrier datum: none;
• $\mathcal Q$-home: not diagonal sheetwise boundary data.

Next falsifier

Choose a transverse parameter $E$ at generic $X_1X_2\ne0$, normalize the $g_3$ quadratic cover over $\mathbb Q((E^{1/2}))$, and compute the two physical residue Laurent orders and deck action. Compare the resulting nearby-cycle graded line with the known Tate/Kummer scattering limit.

Evidence

• `research/benincasa/classify_physical_tangency_sheet_boundaries.py`;
• `research/benincasa/physical-tangency-sheet-boundaries.json`;
• Entries 673--675, 677, and 680;
• allocator claim `seqclaim-989a28bb34375a3955e84967`.

Outcome contract

{
  "claim": "Generic conductor collisions necessarily ramify the physical tangency double covers.",
  "status": "falsified",
  "generic_conductor_collisions_etale": true,
  "unique_ramified_physical_boundary": "g3 at total energy E=0",
  "total_energy_requires_nearby_cycles": true,
  "Q_is_sheet_boundary_factor": false,
  "new_carrier_datum": false,
  "next_experiment": "Compute g3 residue Laurent orders and deck action over the E^(1/2) nearby-cycle cover."
}

View source on GitHub