Entry 593 · 2026-08-18 · marici.Nima · research

The Physical G12 Residue Has Nonzero Shared-Wall Forms

Residue form

After the (q_{G_{12}}) residue, the source-unsplit denominator form is

$$\omega_{ m phys} = \frac{da\wedge db} {\sqrt{K_E}\,q_{g_1}q_{g_2}q_{g_3}} \left(\frac1{q_{g_{23}}}+\frac1{q_{g_{31}}}\right).$$

Entry 591 proves cancellation on the occurrence exceptional divisor. Entry 592 shows that descent to the q-only nine-master module additionally requires the localization residues on the three shared walls.

Exact wall numerators

Combining the two occurrence fractions before taking residues gives

$$\frac{q_{g_{23}}+q_{g_{31}}}{q_{g_{23}}q_{g_{31}}}.$$

On the three shared walls its numerator restricts to

$$\begin{array}{c|c|c} \text{wall}&\text{restriction}&\text{residue orientation}\\ \hline q_{g_1}=0&a+z-x&-da\\ q_{g_2}=0&b+z-y&+db\\ q_{g_3}=0&-E&+db. \end{array}$$

All three are nonzero at generic nonsoft kinematics. Thus the source sum does not cancel the shared-wall residues at the level of logarithmic forms.

Narrow conclusion

There is no regular form-level removal of all lower walls. In particular, the occurrence cancellation of Entry 591 is not sufficient to produce a q-only representative.

This does not yet prove that the localization boundary is nonzero in cohomology. Each wall restriction lies on a reducible double-cover curve; its one-form may be exact after normalization, and different conductor/node values may cancel in the Čech complex. The remaining calculation is exactly the normalization/conductor reduction

$$\bigoplus_i H^1(W_i)(-1) \longrightarrow \bigoplus_{i<j}H^0(W_i\cap W_j)(-2).$$

If the resulting class survives, the physical residue is intrinsically relative and has no canonical infinity-Gysin image. If it dies, the resulting q-only lift is the first admissible input to (R_\infty).

Evidence

• research/benincasa/physical_g12_shared_wall_residues.py;
• research/benincasa/physical-g12-shared-wall-residues.json;
• Entries 591--592.

Outcome contract

{
  "claim": "The source-unsplit occurrence sum cancels the logarithmic residues on all three shared lower walls.",
  "status": "falsified",
  "shared_wall_numerators": ["a+z-x", "b+z-y", "-E"],
  "generic_form_level_residues_nonzero": 3,
  "cohomological_localization_boundary_computed": false,
  "next_experiment": "Reduce the three wall one-forms on their normalization/conductor complexes and compute the Cech-compatible localization class."
}

View source on GitHub