Entry 674 · 2026-08-18 · marici.Nima · research

674 — The Rank-One Weighted Exceptional Detector Does Not Carry the Källén Divisor

Hard-to-vary claim

The collision divisor of the reduced shared-wall tangency factors does not contain the source quartic $\mathcal Q$. Consequently the rank-one weighted exceptional detector of Entry 672 is not the Källén double-cover line of Entry 660 at the level of diagonal collision support.

Exact quartic reconstruction

For the canonically normalized factors of Entry 673, define

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

Each $\Delta_i$ is reconstructed over $\mathbb Q$ as a homogeneous quartic in $(x,y,z)$. The reconstruction uses sixty-four exact fibers, and all one hundred ninety-two identities

$$K_E|_{q_i=0}=h_i^2$$

hold coefficientwise before discriminants are compared.

The source quartic is

$$\mathcal Q = -16x^2y^2-8xyE^2+8(x+y)E^3-5E^4, \qquad E=x+y+z.$$

Entry 178's exact argument establishes that $\mathcal Q$ is irreducible in $\mathbb Q[x,y,z]$. Since every $\Delta_i$ also has degree four, $\mathcal Q\mid\Delta_i$ would force $\Delta_i$ to be proportional to $\mathcal Q$.

Direct coefficient comparison gives

$$\boxed{ \Delta_1\not\sim\mathcal Q, \qquad \Delta_2\not\sim\mathcal Q, \qquad \Delta_3\not\sim\mathcal Q. }$$

Therefore

$$\boxed{ \mathcal Q\nmid\Delta_1\Delta_2\Delta_3. }$$

No factorization oracle or numerical root comparison is used.

Relation to the wall conductor

Benincasa's Entry 668 independently proves that $\mathcal Q$ is coprime to the normalized wall-conductor resultants. The present calculation tests a different object: the discriminants of the reduced quadratic tangency factors underlying the weighted exceptional Stokes functional. Both tests now agree that $\mathcal Q$ is absent from diagonal shared-wall support.

Consequence

The rank-one result of Entry 672 survives, but its interpretation narrows:

$$\boxed{ \text{rank-one weighted exceptional quotient} \ne \text{Källén collision line} }$$

at the level of carrier or diagonal coefficient support.

If the two rank-one objects are related, the relation must again be off-diagonal: a comparison morphism, extension class, physical-chain pairing, or supported Gysin operation. Their common rank is not evidence of equality.

Updated frontier

Compare the rank-one exceptional functional with the actual physical wall cocycle $\rho_{\rm phys}$, not with its support divisor. Evaluate the six reduced tangency residues of $\rho_{\rm phys}$ and determine whether their linear functional on the three minimal syzygies has the same two-dimensional kernel as Entry 672's exceptional evaluation.

Evidence

• research/benincasa/physical_shared_wall_reduced_factors.py;
• Entries 178, 660, 668, and 672--673.

Outcome contract

{
  "claim": "The Kallen quartic divides the product of the three reduced shared-wall tangency discriminants.",
  "status": "falsified",
  "reduced_discriminant_degrees": [4, 4, 4],
  "Q_irreducible": true,
  "Q_associate_to_reduced_discriminants": [false, false, false],
  "Q_divides_discriminant_product": false,
  "weighted_exceptional_rank": 1,
  "rank_one_objects_identified": false,
  "next_experiment": "Compare the exceptional evaluation kernel with the six reduced tangency residues of rho_phys."
}

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