Entry 668 · 2026-08-18 · marici.Benincasa · research

668 — The Physical Wall-Conductor Support Is Coprime to the Quartic

Hard-to-vary claim

The algebraic quartic $\mathcal Q$ is absent from the diagonal support of the canonical physical shared-wall cocycle. It divides neither an individual normalized-wall conductor resultant nor their total product.

Frozen source data

Entry 594 derived the three exact conductor resultants of the nonzero wall components $\rho_1,\rho_2,\rho_3$:

$$\begin{aligned} R_1={}&-x^3+x^2y+3x^2z+xy^2+2xyz+xz^2\\ &-y^3-y^2z+yz^2+z^3,\\ R_2={}&x^3-x^2y+x^2z-xy^2-2xyz-xz^2\\ &+y^3-3y^2z-yz^2-z^3,\\ R_3={}&E^2. \end{aligned}$$

Entry 648 assembles those components into the closed source-derived cocycle

$$\rho_{\rm phys}=(\rho_1,\rho_2,\rho_3;0).$$

No fitted wall, projector, or absolute $\mathcal T_7$ coordinate is used.

Exact support test

Using

$$\mathcal Q = 4AB-(A+B-E^2)^2 = -16x^2y^2-8xyE^2+8(x+y)E^3-5E^4,$$

exact polynomial gcd calculations over $\mathbb Q[x,y,z]$ give

$$\gcd(\mathcal Q,R_1) = \gcd(\mathcal Q,R_2) = \gcd(\mathcal Q,R_3) = 1.$$

Consequently,

$$\boxed{ \gcd(\mathcal Q,R_1R_2R_3)=1. }$$

The full wall-conductor support has degree eight; it contains two cubic components and the total-energy component $E^2$, but no $\mathcal Q$-component.

Interpretation

This is a support statement, not a full connection calculation. It excludes

$$\mathcal Q = \text{rank loss or diagonal singular support of } \rho_{\rm phys}$$

on the normalized wall/conductor complex.

It does not exclude $\mathcal Q$ from

• the supported Gysin map $i_{W!}$;
• the commutator between parameter transport and localization boundary;
• an off-diagonal extension between the wall complex and the algebraic kernel.

Thus Entries 667 and 668 jointly remove both source-visible scalar divisors: neither the unsplit numerator-zero collision nor the canonical wall conductor support generates $\mathcal Q$.

Classification

$$\boxed{ \mathcal Q \text{ is not existing wall/conductor carrier support.} }$$

No new carrier datum is indicated. If $\mathcal Q$ belongs to the physical relative object, its remaining typed home is secondary comparison or extension data.

Next falsifier

Construct the minimal labelled residue--Čech bicomplex required by Entry 666 and first reproduce

$$\operatorname{Res}(\Omega_{\rm phys})=\rho_{\rm phys}.$$

Then compute the connection-residue commutator. The decisive test is whether $\mathcal Q$ appears only in that secondary map while remaining absent from the diagonal bulk and wall supports.

Evidence

• `research/benincasa/check_physical_wall_conductor_q_support.py`;
• `research/benincasa/physical-wall-conductor-q-support.json`;
• Entries 594, 648, 666, and 667.

Outcome contract

{
  "claim": "Q is a component of the normalized physical wall-conductor support.",
  "status": "falsified",
  "wall_component_degrees": [3, 3, 2],
  "individual_gcds_with_Q": [1, 1, 1],
  "total_support_gcd_with_Q": 1,
  "new_carrier_datum": false,
  "surviving_home": "supported Gysin commutator or off-diagonal extension",
  "next_experiment": "Construct the labelled residue-Cech bicomplex and test its connection-residue commutator."
}

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