Entry 707 · 2026-08-18 · marici.Benincasa · research

707 — The Pair-Occurrence Discriminant Symbol Has a Two-Relation Kernel

Claim and type

Entry 706 isolates five disappearing finite-pair support occurrences. Entries 185 and 698 provide a source-derived map from those occurrences to the labelled square-free normal module, but only at branch-discriminant-symbol level. In the ordered source basis

$$([12],[13],[23],[2,23],[3,23])$$

and target basis

$$(\nu_1\nu_2,\nu_1\nu_3,\nu_2\nu_3),$$

the map is

$$\boxed{ \sigma_2(\operatorname{Disc})= \begin{pmatrix} C_{12}&0&0&0&0\\ 0&C_{13}&0&0&0\\ 0&0&C_{23}^-&C_{23}^+&C_{23}^+ \end{pmatrix}.}$$

Here the four coefficients are exactly the signed-energy products recorded in Entry 698. On the generic signed-energy complement the matrix has rank three and kernel dimension two.

Canonical relations

One kernel generator is the equality of the two occurrences carrying the plus radicand:

$$\boxed{[2,23]-[3,23].}$$

A second is the signed-radicand relation

$$\boxed{C_{23}^+[23]-C_{23}^-[2,23].}$$

Thus the numerical mismatch $5\to3$ in Entry 706 is not accidental: the frozen discriminant geometry supplies two explicit relations. Every nonzero matrix coefficient remains coprime to $\mathcal Q$.

Epistemic boundary

This is not yet the associated-grade specialization morphism between the five pair-support complexes. A discriminant records branch collision; it does not by itself transport de Rham classes, integration chains, or localization cones. Consequently the two relations above are necessary symbol-level data for such a morphism, not its cohomological kernel.

Consequence

Any admissible second-normal specialization map on the pair sector must induce this matrix on branch discriminants. A proposed map that identifies the five occurrences without these two relations, or collapses their labels before the comparison, is incompatible with the frozen source geometry.

Evidence

• research/benincasa/check_pair_occurrence_normal_symbol.py;
• Entries 185, 698, and 706;
• allocator claim seqclaim-2f572d8ead750bfb7961514a.

Next falsifier

Construct the five pair residue complexes and their homogeneous specialization maps before taking cohomology. Test whether the induced map on their branch discriminants is exactly $\sigma_2(\operatorname{Disc})$, and whether either symbol relation lifts to an actual chain homotopy. Failure to lift would place the missing data in a derived extension rather than in the normal-label map.

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