Entry 665 · 2026-08-18 · marici.Benincasa · research

665 — The Occurrence-Divisor Intersection Is Not the Quartic Cover

Hard-to-vary claim

The two physical occurrence denominators define two marked divisors on the post-$q_{G_{12}}$ residue surface, not one distinguished endpoint ratio. Moreover, their algebraic intersection does not generate the quartic $\mathcal Q$: the Cayley--Menger branch value at that intersection has different generic support.

Frozen residue data

The source denominators restrict exactly as

$$q_{g_{23}}|_{G_{12}}=b-x, \qquad q_{g_{31}}|_{G_{12}}=a-y.$$

Thus the unsplit physical source carries the union

$$D_{\rm occ}=\{b=x\}\cup\{a=y\},$$

with one occurrence divisor in each source summand. It does not select their intersection as a relative-chain endpoint.

Entry 661 supplies the literal Cayley--Menger polynomial

$$\begin{aligned} K_E(a,b)={}&x^2a^4-(x^2+y^2-z^2)a^2b^2+y^2b^4\\ &+\left[x^2(x^2-y^2-z^2)+E^2(y^2-x^2-z^2)\right]a^2\\ &+\left[y^2(y^2-x^2-z^2)+E^2(x^2-y^2-z^2)\right]b^2\\ &+z^2E^4+E^2z^2(z^2-x^2-y^2)+z^2x^2y^2, \end{aligned}$$

where $E=x+y+z$.

Direct intersection test

The two occurrence divisors meet at

$$(a,b)=(y,x).$$

Direct substitution and factorization give

$$\boxed{ K_E(y,x) = E^3\left[2z^3+(x-y)^2(x+y-z)\right]. }$$

The source quartic is instead

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

These polynomials are neither associates nor do they have the same generic zero locus. In particular, the intersection branch value has a cubic total-energy factor, whereas

$$\mathcal Q|_{E=0}=-16x^2y^2$$

is nonzero away from soft support.

Therefore

$$\boxed{ \mathcal Q \ne \text{the branch value at } \{q_{g_{23}}=q_{g_{31}}=0\}. }$$

Typing consequence

Even before this polynomial mismatch, choosing the intersection would have been inadmissible: the physical form is a sum of two four-mark relative classes. Replacing that union by the codimension-two intersection introduces a new operation not supplied by the source integration chain.

The direct mismatch now also falsifies the candidate numerically and algebraically. No choice of normalization by a source-fixed unit can turn $K_E(y,x)$ into $\mathcal Q$, because their behavior at $E=0$ differs generically.

Relation to parallel work

This calculation uses Entry 661's literal residue polynomial but does not duplicate its rank-twenty reduction or horizontal-saturation program. It tests only one candidate home for $\mathcal Q$ inside the marked post-residue geometry.

Classification

• existing carrier: unchanged;
• physical marked data: the union $\{b=x\}\cup\{a=y\}$;
• rejected coefficient support: their codimension-two intersection;
• new carrier datum: none.

Surviving frontier

Entry 660's Källén double cover remains an ambient source-compiled candidate, but it is not obtained by intersecting the two occurrence sections.

The next admissible test is separate restriction:

$$K_E(a,x) \quad\text{and}\quad K_E(y,b).$$

Compute each branch polynomial and its discriminant in the remaining fiber coordinate. Then test whether $\mathcal Q$ occurs as a canonical factor for either individual physical summand. This preserves the source's occurrence decomposition and does not invoke the rank-thirty-five extension.

Evidence

• primary-source denominator residue recorded in research/benincasa/physical_five_pole_g12_residue.py;
• literal $K_E$ in Entry 661 and research/benincasa/physical_four_mark_residue_twisted_derham.py;
• research/benincasa/physical-occurrence-intersection-q-gate.json.
• epistemic event ev-000000000265-b868219e-ae6a-4b0c-bcbb-d4a396e043d7.

Outcome contract

{
  "claim": "The quartic Q is the Cayley-Menger branch value at the intersection of the two physical occurrence divisors.",
  "status": "falsified",
  "occurrence_intersection": "(a,b)=(y,x)",
  "intersection_branch_value": "E^3*(2*z^3+(x-y)^2*(x+y-z))",
  "generic_support_equals_Q": false,
  "source_selects_intersection": false,
  "new_carrier_datum": false,
  "next_experiment": "Compute the discriminants of K_E(a,x) and K_E(y,b) separately."
}

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