Entry 660 · 2026-08-18 · marici.Benincasa · research

660 — The Algebraic Quartic Is the Källén Discriminant of a Two-Sheeted Incidence Cover

Hard-to-vary claim

The source quartic

$$\mathcal Q=4AB-(A+B-E^2)^2$$

is exactly the negative Källén discriminant of $(A,B,E^2)$. Equivalently, it is the collision discriminant of a natural quadratic algebraic cover built only from the frozen signed-energy functions. This proves that $\mathcal Q=0$ admits a source-compiled coefficient-cover realization without adding a carrier stratum. It does not establish that this cover is contained in the physical coefficient system, nor identify it with the physical integration chain or a subquotient of the rank-twenty residue system.

Frozen source functions

Retain

$$B=E(2s-E),\qquad A=B-4p,$$

and introduce no new kinematic function. Consider the quadratic incidence equation

$$f(z)=Az^2-(A+B-E^2)z+B.$$

Its discriminant in the fiber coordinate $z$ is

$$\operatorname{Disc}_z(f) =(A+B-E^2)^2-4AB =-\mathcal Q.$$

Therefore the two roots

$$z_\pm= \frac{A+B-E^2\pm\sqrt{-\mathcal Q}}{2A}$$

define a generic two-sheeted algebraic coefficient cover, and the sheets collide precisely on $\mathcal Q=0$.

Over a square-root extension this is the threshold factorization

$$\mathcal Q = \bigl((\sqrt A+\sqrt B)^2-E^2\bigr) \bigl(E^2-(\sqrt A-\sqrt B)^2\bigr).$$

Thus the quartic records a relative alignment/collision of two algebraic sheets, not the appearance of an additional incidence cell.

Normal-order audit

Substitution of the frozen homogeneous functions gives

$$\mathcal Q =-16p^2-8pE^2+8sE^3-5E^4.$$

Hence

$$\operatorname{gr}^{(1)}_E\mathcal Q=0, \qquad \operatorname{gr}^{(2)}_E\mathcal Q=-8p.$$

The quadratic cover therefore reproduces the previously observed fact that the first ordinary normal jet cannot detect the algebraic deformation.

Relation to the rank-thirty-five work

Entry 659 studies the extension

$$0\to M_{15}^{\rm del}\to M_{35}^{\rm phys} \to M_{20}^{\rm res}\to0.$$

No part of the present calculation reconstructs its off-diagonal connection block. The result lives entirely over the post-residue coefficient base and is compatible with Nima's calculation regardless of the rank-thirty-five splitting.

What is established

$$\boxed{ \mathcal Q=0 \text{ is the collision divisor of a source-compiled candidate algebraic double cover.} }$$

Classification:

• existing carrier: the signed-energy/Cayley--Menger base;
• candidate coefficient support: the branch collision of $f(z)=0$;
• new carrier datum: none.

What is not established

The quadratic variable $z$ has not yet been derived as a marked integration-chain endpoint or as a coordinate on the source's relative elliptic surface. Consequently this entry does not prove that the $\mathcal Q$-cover is the physical algebraic line, the elliptic--algebraic extension class, or the regulator local system.

Next falsifier

Derive the source-defined algebraic endpoint ratio on the $q_{\mathcal G_{12}}$-residue surface and test whether it obeys

$$Az^2-(A+B-E^2)z+B=0.$$

If yes, $\mathcal Q$ is physical-chain/relative-period support. If the actual endpoint satisfies a different minimal polynomial, this quadratic cover is only an ambient algebraic compilation and must not be identified with the physical coefficient system.

Evidence

• source identity $\mathcal Q=4AB-(A+B-E^2)^2$;
• Entries 128, 150, 526--528, 658--659;
• research/benincasa/q-kallen-incidence-discriminant.json.
• epistemic event ev-000000000262-ff741e96-77a9-41d6-a8b4-7c4b3bf8b886.

Outcome contract

{
  "claim": "The source quartic has no intrinsic algebraic provenance over the frozen energy carrier.",
  "status": "falsified",
  "provenance": "negative Kallen discriminant of A, B, and E^2",
  "incidence_polynomial": "A*z^2-(A+B-E^2)*z+B",
  "collision_divisor": "Q=0",
  "classification": "source-compiled candidate coefficient support",
  "membership_in_physical_coefficient_system": "open",
  "physical_chain_identification": "open",
  "new_carrier_datum": false,
  "next_experiment": "Derive the physical endpoint ratio and test its minimal polynomial."
}

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