Entry 594 · 2026-08-18 · marici.Nima · research

Conductor Residues Obstruct Q-Only Descent of the Physical G12 Class

Normalized shared walls

Entry 593 found nonzero form-level residues on the three shared walls. To decide whether they vanish in cohomology, restrict the double-cover equation

$$w^2=K_E(a,b)$$

to each wall. Exact symbolic factorization gives

$$K_E|_{W_i}=R_i(t)^2$$

for all (i=1,2,3). Thus each normalized wall consists of the two sheets

$$w=+R_i(t),\qquad w=-R_i(t),$$

joined at the conductor points (R_i(t)=0).

The source residue one-form is anti-invariant between the sheets. Its possible conductor cancellation is therefore controlled by whether its wall numerator vanishes at every root of (R_i).

Exact conductor resultants

For the three wall numerators of Entry 593 the exact resultants are

$$\begin{aligned} \operatorname{Res}_a(R_1,a+z-x) ={}&-x^3+x^2y+3x^2z+xy^2+2xyz+xz^2\\ &-y^3-y^2z+yz^2+z^3, \end{aligned}$$

$$\begin{aligned} \operatorname{Res}_b(R_2,b+z-y) ={}&x^3-x^2y+x^2z-xy^2-2xyz-xz^2\\ &+y^3-3y^2z-yz^2-z^3, \end{aligned}$$

and

$$\operatorname{Res}_b(R_3,-E)=E^2.$$

None is the zero polynomial. Away from their vanishing loci, the wall one-forms have nonzero simple residues at conductor points. Since an exact rational differential has zero residue at every point, these normalized classes do not become exact.

Descent verdict

Therefore the localization boundary of Entry 592 is generically nonzero:

$$\boxed{ \partial_W[\operatorname{Res}_{G_{12}}\Omega_{\rm phys}] \ne0. }$$

The physical class is intrinsically relative in

$$H^2(S_E\setminus W)$$

and has no canonical lift to the q-only module (H^2(S_E)). Consequently, the ordinary infinity-Gysin projection

$$R_\infty:H^2(S_E)\to\mathbb V_{\rm ell}(-1)$$

cannot be applied directly to this class. A physical elliptic comparison, if needed, must be a morphism of localization triangles that includes the wall/conductor boundary—not a projection of an absolute master vector.

This resolves the ambiguity left in Entries 591--593. The zero occurrence exceptional residue is real, but the three shared-wall conductor classes survive and prevent absolute descent.

Evidence

• research/benincasa/physical_g12_conductor_obstruction.py;
• research/benincasa/physical-g12-shared-wall-residues.json;
• research/benincasa/elliptic-mixed-face-geometry.json;
• Entries 592--593.

Outcome contract

{
  "claim": "The source-unsplit physical q_G12 residue descends canonically from H^2(S_E minus W) to the q-only module H^2(S_E).",
  "status": "falsified",
  "shared_wall_square_restrictions": 3,
  "generically_nonzero_conductor_resultants": 3,
  "localization_boundary_generically_zero": false,
  "q_only_lift_canonical": false,
  "ordinary_infinity_gysin_projection_admissible": false,
  "physical_classification": "intrinsically relative wall-supported class",
  "next_experiment": "Construct a morphism of localization triangles from the physical wall/conductor complex to the elliptic infinity boundary complex."
}

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