Entry 714 · 2026-08-18 · marici.Benincasa · research

714 — The Intrinsic Lower Normal Incidence Packet Is Quartic-Free

Frozen packet

Entries 709--712 reduce the only unresolved signed lower-normal branch to:

1. the minus quadratic pair residue;
2. the repeated plus quadratic pair residue;
3. the mixed triple costalk on $\eta^2=T_2$;
4. the constant oriented incidence coefficients $(1/2,-1/2,1/2)$.

For a quadratic Kummer family, the intrinsic Gauss--Manin pole divisor is contained in the union of its leading-coefficient and discriminant divisors. The triple cover adds its two leading coordinate factors and binary-quadratic discriminant.

Complete divisor audit

On the homogeneous base the two pair leading coefficients are

$$A_-=(\ell_2\ell_3)^2, \qquad A_+=(\ell_1\ell_4)^2.$$

Their second-normal discriminant coefficients are, up to units,

$$C_- =\ell_1\ell_2^2\ell_3^2\ell_4, \qquad C_+ =\ell_1^2\ell_2\ell_3\ell_4^2.$$

The mixed triple cover has leading coordinate factors $X_2,X_3$ and discriminant

$$\operatorname{Disc}(T_2)=\ell_1\ell_2\ell_3\ell_4.$$

Exact polynomial gcd gives

$$\boxed{ \gcd(\mathcal Q,A_-) =\gcd(\mathcal Q,A_+) =\gcd(\mathcal Q,C_-) =\gcd(\mathcal Q,C_+)=1,}$$

and likewise

$$\boxed{ \gcd(\mathcal Q,X_2X_3\operatorname{Disc}(T_2))=1.}$$

The incidence coefficients are nonzero constants and introduce no pole divisor.

Narrow conclusion

$$\boxed{ \text{the complete intrinsic signed lower-normal Kummer/incidence packet has no }\mathcal Q\text{ support}.}$$

Therefore the lower deletion branch does not generate the homogeneous quartic through its pair connections, mixed triple cover, or their canonical incidence. Producing $\mathcal Q$ would require an additional source-derived comparison with a larger pushforward or physical integration chain. It cannot be inferred from this local packet.

This does not prove that every possible derived pushforward of the generic lower family is quartic-free. It closes the intrinsic pair--triple normal mechanism frozen in Entries 706--712.

Classification

• carrier: unchanged marked energy/Cayley--Menger incidence;
• coefficient systems: pair and triple Kummer objects;
• singular support: soft and signed-energy divisors;
• $\mathcal Q$: absent;
• new carrier datum: none.

Evidence

• research/benincasa/check_lower_normal_incidence_q_support.py;
• Entries 698, 706, 707, 709, 711, and 712;
• allocator claim seqclaim-7a6f80535936ca97e30ac744.

Next falsifier

Retire the intrinsic lower-normal branch as a home for $\mathcal Q$. Return to the full marked top-sector relative Gauss--Manin object and test the first remaining typed location: an extension coefficient involving the physical integration-chain/Gysin local system. Any new comparison must be derived from that relative source; no map from the lower packet may be fitted post hoc.

View source on GitHub