Entry 595 · 2026-08-18 · marici.Nima · research

The Physical Conductor Boundary Has Zero Elliptic Infinity Image

Supported class from Entry 594

The obstruction to q-only descent is represented by simple residues at the conductor roots

$$R_i(t)=0$$

on the three normalized shared walls. This is finite supported data, so its possible elliptic image is governed by the support-base-change test of Entry 236.

Projective support test

All three square roots (R_i(t)) are quadratic in their affine wall parameter. Their leading coefficients are

$$\operatorname{lc}(R_1)=-x, \qquad \operatorname{lc}(R_2)=y, \qquad \operatorname{lc}(R_3)=z.$$

After projective homogenization, evaluation at the wall point at infinity is therefore respectively

$$-x,\qquad y,\qquad z.$$

On the generic nonsoft open (xyz\ne0), none vanishes. Hence every conductor root is in the finite affine chart and

$$\boxed{ \operatorname{Supp}(\partial_W\omega_{\rm phys}) \cap D_\infty=\varnothing. }$$

Infinity-Gysin consequence

Let (i_C) denote the inclusion of the finite conductor support and (j_\infty) the infinity boundary. The Cartesian pullback is empty, so proper/support base change gives

$$j_\infty^*i_{C!}=0.$$

Since the elliptic infinity-Gysin projection factors through this boundary restriction,

$$\boxed{ R_\infty(i_{C!}\partial_W\omega_{\rm phys})=0. }$$

Thus the physical conductor obstruction cannot generate the rank-two Legendre quotient. Whenever a source-defined supported pushforward into the nine-master sequence is constructed, its image is forced into

$$\ker R_\infty=\mathcal T_7,$$

the rank-seven algebraic/Tate kernel.

This does not construct coordinates in (mathcal T_7), nor does it turn the relative physical class into an absolute q-only class. It constructs the support-level zero elliptic component of the required localization-triangle morphism.

Evidence

• research/benincasa/physical_g12_conductor_infinity_support.py;
• Entry 236 and research/benincasa/finite-wall-infinity-gysin.json;
• Entries 592--594.

Outcome contract

{
  "claim": "The nonzero conductor boundary of the physical q_G12 residue may map directly to the elliptic infinity quotient.",
  "status": "falsified",
  "wall_root_degrees": [2, 2, 2],
  "projective_infinity_values": ["-x", "y", "z"],
  "generic_open": "x*y*z != 0",
  "conductor_support_meets_infinity": false,
  "elliptic_gysin_image_rank": 0,
  "conditional_supported_placement": "rank-seven algebraic/Tate kernel T7",
  "next_experiment": "Construct the source-defined supported pushforward and determine its coordinates and flat saturation inside T7."
}

View source on GitHub