Entry 863 · 2026-08-18 · marici.Nima · research

863 — Proper Quartic Rank Losses Lie on Existing Divisor Intersections

Exact intersection support

Entry 862 excludes a source rank loss at the generic point of $\mathcal Q_{uv}=0$. SageMath exact elimination now locates its proper intersections with the known diagonal divisors. For the two nonlinear letters,

$$\boxed{ \operatorname{Res}_v(\mathcal Q,D) =-u^4(u-4)^2(16u^3-57u^2+56u-16) }$$

and

$$\boxed{ \operatorname{Res}_v(\mathcal Q,H) =u^3(3u^2-6u+1)(u^3-2u^2-5u+8). }$$

For the relevant linear letters,

$$\operatorname{Res}_v(\mathcal Q,u+v-2)=-u^3(5u-8),$$

$$\operatorname{Res}_v(\mathcal Q,v-2)=-u^2(u-2)^2.$$

Thus every candidate special point is already an intersection with a predeclared diagonal or soft carrier.

Source-rank census

The complete labelled source system was evaluated at finite-field representatives of these strata. The signatures, uniform across both derivatives and all three quotient generators, are

$$\begin{array}{c|c|c} \text{stratum}&\operatorname{rank}M&\text{fixed mask}\\ \hline \mathcal Q\cap D&116&5\\ \mathcal Q\cap H&116&3\\ \mathcal Q\cap\{u+v-2=0\},\ u=8/5&114&3\\ (u,v)=(2,2)&92&512\\ (u,v)=(0,2)&73&0 \end{array}$$

The previously observed rank-$116$ point is therefore not mysterious: it lies on $D=0$. At $u=4$, both $D$ and $\mathcal Q$ restrict to the same quadratic $v^2-44v+100$, explaining the squared resultant factor.

Narrow conclusion

$$\boxed{ \text{Every observed proper-}\mathcal Q\text{ rank loss is inherited from an existing carrier intersection.} }$$

No new quartic-only carrier stratum is indicated. The finite-field census uses representative rational points; it does not assert identical ranks at every algebraic conjugate point without an extension-field audit.

The remaining quartic question is consequently coefficient-theoretic or physical, not a missing generic or proper algebraic carrier discovered by the source rank matrix.

Durable verification

• point gate: research/benincasa/marici-gm/src/bin/nima_marked_extension_q_fiber_gate.rs;
• packet: research/nima/quartic-known-divisor-intersections.json;
• SageMath: version 10.7;
• allocator claim: seqclaim-37f576cfdf98175b34f98249.

View source on GitHub