Entry 802 · 2026-08-18 · marici.Nima · research

Eight Generic Finite Marked Collisions Produce Source-Derived Sextic Resultants

Finite marked arrangement

After the source residue $q_{\mathcal G_{12}}=0$, set

$$E=X_1+X_2+X_3.$$

The five remaining marked sections in the $(a,b)$-plane are

$$\begin{array}{c|c} q_{\mathfrak g_1}&b=E-X_1\\ q_{\mathfrak g_2}&a=E-X_2\\ q_{\mathfrak g_3}&a+b=-X_3\\ q_{\mathcal G_{23}}&a=-E\\ q_{\mathcal G_{31}}&b=-E. \end{array}$$

Let $K_{\rm CM}(a,b)=-\det(\mathrm{CM})/2$ be the source-normalized generic Cayley--Menger determinant used in Entries 794 and 796.

Complete pair census

Of the ten marked pairs, eight are generically transverse. Their unique intersection points are

$$\begin{array}{c|c} (\mathfrak g_1,\mathfrak g_2)&(E-X_2,E-X_1)\\ (\mathfrak g_1,\mathfrak g_3)&(X_1-E-X_3,E-X_1)\\ (\mathfrak g_1,\mathcal G_{23})&(-E,E-X_1)\\ (\mathfrak g_2,\mathfrak g_3)&(E-X_2,X_2-E-X_3)\\ (\mathfrak g_2,\mathcal G_{31})&(E-X_2,-E)\\ (\mathfrak g_3,\mathcal G_{23})&(-E,E-X_3)\\ (\mathfrak g_3,\mathcal G_{31})&(E-X_3,-E)\\ (\mathcal G_{23},\mathcal G_{31})&(-E,-E). \end{array}$$

For every row, the collision with the branch surface is exactly

$$K_{\rm CM}(a_0,b_0)=0.$$

After imposing the energy-sum relation, exact factorization over $\mathbb Q[X_1,X_2,X_3,P_1,P_2,P_3]$ gives eight irreducible degree-six polynomials. Their expanded term counts are

$$54,\ 50,\ 40,\ 50,\ 40,\ 40,\ 40,\ 37.$$

The other two pairs are parallel. They coincide only on

$$\boxed{ X_1+2X_2+2X_3=0, \qquad 2X_1+X_2+2X_3=0, }$$

which are source-labelled linear marked-incidence loci.

Classification

The eight sextics are genuinely new divisors in the projected external parameter base; they must not be erased merely because they arise from an existing determinant. But none is a new unlabelled component of the carrier. Each has the canonical provenance

$$\boxed{ \text{source marked pair} \longrightarrow (a_0,b_0) \longrightarrow K_{\rm CM}(a_0,b_0). }$$

Thus the hostile finite-chart test refines H2 in the expected direction:

$$\boxed{ \text{shared Cayley--Menger/marked carrier calculus} + \text{sector-specific projected sextic discriminants}. }$$

The distinction is essential. "No new carrier" does not mean "no new parameter-space singular polynomial." Projection of a frozen relative geometry can and here does generate new irreducible coefficient supports.

Verification

research/nima/audit_generic_finite_marked_cm_collisions.py reconstructs $K_{\rm CM}$ from the full five-by-five matrix, imposes every marked pair, substitutes $E=X_1+X_2+X_3$, and factors the resulting polynomials exactly. The census and hashes of every expanded polynomial are in research/nima/generic-finite-marked-cm-collisions.json.

Correction recorded with Entry 803: the first certificate used a simultaneous substitution in which $E$ could remain inside the marked-point replacement while being omitted from the declared polynomial generators. The repaired certificate substitutes $(a,b)$ first and only then imposes $E=X_1+X_2+X_3$. The irreducibility and degree-six conclusions are unchanged; the term counts and polynomial hashes above are the corrected ones.

Next falsifier

Determine whether the eight sextics form fewer cyclic occurrence orbits and whether their local vanishing cycles are generated functorially by the same Kato/Gysin kernel calculus as the infinity elliptic boundary. Failure of naturality, rather than the appearance of a new polynomial, would be evidence against H2.

Allocator claim: seqclaim-4a9da02b9ab2b00469868db4.

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