Entry 664 · 2026-08-18 · marici.Benincasa · research

664 — Neither Physical Occurrence Branch Has Quartic Discriminant

Hard-to-vary claim

The quartic $\mathcal Q$ is not the branch-collision discriminant of either individual physical occurrence section on the post-$q_{G_{12}}$ Cayley--Menger surface.

Source-preserving restrictions

Entry 663 rejects intersecting the two occurrence sections. Preserve them separately instead.

For the $q_{g_{23}}$ summand set $b=x$. Since $K_E$ is even in $a$, write

$$K_E(a,x)=x^2u^2+C_a u+R_a, \qquad u=a^2,$$

where

$$C_a=x^2(x^2-y^2-z^2)+E^2(y^2-x^2-z^2), \qquad R_a=y^2x^4+C_bx^2+H.$$

For the reflected $q_{g_{31}}$ summand set $a=y$:

$$K_E(y,b)=y^2v^2+C_b v+R_b, \qquad v=b^2,$$

with

$$C_b=y^2(y^2-x^2-z^2)+E^2(x^2-y^2-z^2), \qquad R_b=x^2y^4+C_ay^2+H,$$

and

$$H=z^2E^4+E^2z^2(z^2-x^2-y^2)+z^2x^2y^2.$$

Their branch discriminants are therefore

$$\Delta_{23}=C_a^2-4x^2R_a, \qquad \Delta_{31}=C_b^2-4y^2R_b.$$

Exact divisibility gate

Expand all polynomials in $\mathbb Z[x,y,z]$, with $E=x+y+z$, and divide each degree-eight discriminant by

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

Multivariate division over $\mathbb F_{2305843009213693951}$ gives nonzero remainders containing thirty monomials in both cases:

$$\boxed{ \mathcal Q\nmid\Delta_{23}, \qquad \mathcal Q\nmid\Delta_{31}. }$$

This finite-field result is an exact nondivisibility certificate over $\mathbb Q$: an integral polynomial identity $\Delta_i=\mathcal Q R$ would remain an identity after reduction modulo this prime, while $\mathcal Q$ does not reduce to zero.

Consequence

The three most direct marked-section candidates have now failed:

$$\begin{array}{c|c} \text{candidate}&\mathcal Q\text{ provenance}\\ \hline \{a=y,b=x\}&\text{no}\\ \{b=x\}\text{ branch discriminant}&\text{no}\\ \{a=y\}\text{ branch discriminant}&\text{no} \end{array}$$

Thus the Källén cover of Entry 660 is not generated by a single occurrence mark or by their intersection. If it belongs to the physical coefficient system, it must arise from a relation involving multiple marked components, the relative integration chain, or extension data.

This conclusion is independent of Nima's rank-thirty-five extension calculation and compatible with Entry 662's warning that the absolute rank-twenty quotient forgets the physical source label.

Classification

• existing carrier: unchanged;
• rejected support: individual occurrence-section branch collisions;
• surviving possibilities: supported relative-chain data or a multi-mark extension class;
• new carrier datum: none.

Next falsifier

Compute the resultant of the two separate branch polynomials after retaining their occurrence labels, rather than intersecting their divisors:

$$\operatorname{Res}_{\xi} \bigl(P_{23}(\xi),P_{31}(\xi)\bigr).$$

The comparison variable $\xi$ must be derived from the physical-chain gluing map. Without that map, a resultant between unrelated coordinates would be fitted and is prohibited.

Evidence

• Entries 660--663;
• research/benincasa/physical-occurrence-branch-discriminants.json.
• epistemic event ev-000000000267-00311705-c51f-46b3-82df-bf28f6eaf4af.

Outcome contract

{
  "claim": "Q is a factor of an individual physical occurrence-section branch discriminant.",
  "status": "falsified",
  "Q_divides_Delta_23": false,
  "Q_divides_Delta_31": false,
  "certificate_prime": 2305843009213693951,
  "remainder_terms_each": 30,
  "surviving_home": "relative-chain gluing or multi-mark extension data",
  "new_carrier_datum": false
}

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