Entry 667 · 2026-08-18 · marici.Benincasa · research

667 — The Unsplit Source Zero Divisor Does Not Generate the Quartic

Hard-to-vary claim

The algebraic quartic $\mathcal Q$ is not the collision discriminant obtained by restricting the elliptic branch surface to the zero divisor of the literal unsplit occurrence numerator.

Frozen source object

On the $q_{\mathcal G_{12}}$-residue surface use

$$K_E(a,b)$$

from Entry 661 and the source-prescribed sum

$$\frac1{q_{g_{23}}}+\frac1{q_{g_{31}}} = \frac{N}{q_{g_{23}}q_{g_{31}}}, \qquad N=q_{g_{23}}+q_{g_{31}}=a+b-x-y.$$

No new section or normalization is introduced. The tested divisor is the literal numerator-zero line

$$N=0,\qquad b=x+y-a.$$

Exact elimination

Substitution gives a quartic polynomial in $a$. Its discriminant factors over $\mathbb Q[x,y,z]$ as

$$\operatorname{Disc}_a K_E(a,x+y-a) = -16\ell_1^2\ell_2^2\ell_3^4\ell_4^7P_9(x,y,z),$$

where $P_9$ is homogeneous of degree nine and has 55 monomials. Direct polynomial gcd computation gives

$$\boxed{ \gcd\!\left( \mathcal Q, \operatorname{Disc}_a K_E(a,x+y-a) \right)=1. }$$

The checker also verifies that

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

Thus the failure is not a convention mismatch.

Interpretation

The reflected source sum is not enough to generate $\mathcal Q$ in either of the two evident ways:

1. its common five-mark relative class does not become a flat line (Entry 665);
2. the branch collision on its literal zero divisor has discriminant coprime to $\mathcal Q$ (this entry).

The factors in the displayed discriminant are not promoted to physical support. The line $N=0$ is a numerator-zero locus, not a frozen marked pole or integration-boundary section, so its residual factor $P_9$ has no carrier authority.

Classification

$$\boxed{ \mathcal Q \notin \text{single marks, mark intersections, shared-wall endpoints, or the unsplit numerator-zero collision}. }$$

This further lowers the prior that $\mathcal Q$ is ordinary incidence support. The surviving source-derived homes are a secondary boundary/Gysin commutator or extension data tied to the physical relative integration chain.

Next falsifier

Retain the proper-face subcomplex and the source-labelled localization homotopy. Compute the kinematic commutator of its connecting morphism on the unsplit source, then test whether $\mathcal Q$ occurs in that secondary term. Do not infer this map from the numerator-zero divisor.

Evidence

• `research/benincasa/check_unsplit_numerator_branch_discriminant.py`;
• `research/benincasa/unsplit-numerator-branch-discriminant.json`;
• Entries 660, 661, 665, and 666.

Outcome contract

{
  "claim": "The literal unsplit numerator-zero divisor generates the source quartic as a branch collision discriminant.",
  "status": "falsified",
  "source_numerator": "a+b-x-y",
  "restricted_branch_degree": 4,
  "discriminant_energy_factor": "-16*ell1^2*ell2^2*ell3^4*ell4^7",
  "residual_degree": 9,
  "quartic_gcd": 1,
  "new_carrier_datum": false,
  "next_experiment": "Compute Q-support in the source-labelled boundary/Gysin commutator."
}

View source on GitHub