Entry 807 · 2026-08-18 · marici.Nima · research

The Higher Finite-Sextic Critical Loci Stay on Frozen Support

Universal critical model

Entry 806 leaves one possible failure locus for the generic Kato--Gysin normal form:

$$S_{ij}=\alpha_{ij}=\beta_{ij}=0.$$

Write the source-normalized Cayley--Menger determinant in squared edge variables

$$A=a^2, \qquad B=b^2.$$

It is quadratic in $(A,B)$. Its Hessian has determinant

$$\boxed{ \det\operatorname{Hess}_{A,B}K_{\rm CM} =-\Lambda(P_1,P_2,P_3), }$$

where

$$\Lambda=(P_1-P_2-P_3)(P_1-P_2+P_3) (P_1+P_2-P_3)(P_1+P_2+P_3).$$

Away from $P_1P_2\Lambda=0$, the unique critical point is

$$\boxed{ A_{\rm crit}=E^2+P_2^2, \qquad B_{\rm crit}=E^2+P_1^2. }$$

Its exact critical value is

$$\boxed{ K_{\rm CM}(A_{\rm crit},B_{\rm crit})=E^2\Lambda. }$$

Therefore the nontransverse interior part of every Entry 802 sextic is supported on the already frozen divisors

$$E=0 \qquad\text{or}\qquad \Lambda=0,$$

together with the two source-labelled equations placing its marked point at the universal critical point.

Coordinate-boundary branches

The original derivatives are

$$\partial_aK=2a\,\partial_AK, \qquad \partial_bK=2b\,\partial_BK.$$

Hence $a=0$ and $b=0$ must be audited separately. At $A=0$, after solving $\partial_BK=0$, the critical value factors as

$$-\frac{(E-P_2)^2(E+P_2)^2\Lambda}{4P_2^2}.$$

At $B=0$, the corresponding value is

$$-\frac{(E-P_1)^2(E+P_1)^2\Lambda}{4P_1^2}.$$

Thus these branches add only signed-energy support and the same momentum triangle divisor. At $A=B=0$,

$$K_{\rm CM} =P_3^2 \left[ E^4-E^2(P_1^2+P_2^2-P_3^2)+P_1^2P_2^2 \right],$$

which is soft support times the frozen Cayley--Menger restriction at the double coordinate boundary.

Eight representatives

Substituting each of Entry 802's eight source-labelled points changes only the explicit marked-coordinate equations $a_0=0$, $b_0=0$, $a_0^2=E^2+P_2^2$, and $b_0^2=E^2+P_1^2$. The universal support factors above do not change. Cyclic transport therefore supplies the other sixteen occurrences without introducing a new component.

Result

Set-theoretically,

$$\boxed{ S_{ij}=\alpha_{ij}=\beta_{ij}=0 \text{ lies on frozen marked, total-energy, signed-energy, soft,} \atop \text{momentum-triangle, and Cayley--Menger restriction strata.} }$$

No new unlabelled higher carrier stratum appears. The generic Kato--Gysin line can acquire more complicated vanishing-cycle structure on these loci, but that complexity is a degeneration of the existing carrier calculus.

Scope

This is a set-theoretic support classification obtained by the exhaustive interior, single-coordinate-boundary, and double-coordinate-boundary cases. It is not a primary decomposition and makes no claim about embedded components or scheme multiplicities at intersections of the listed strata.

Verification

• checker: research/nima/audit_finite_sextic_higher_critical_locus.py;
• packet: research/nima/finite-sextic-higher-critical-locus.json;
• allocator claim: seqclaim-6f43d9839010f22ee2cc7d23.

Next falsifier

Compute the local Milnor/Kato ranks on the two universal degeneration types $E=0$ and $\Lambda=0$. H2 requires those higher coefficient objects to be induced by the existing nearby-cycle and momentum-triangle Gysin maps; an excess rank not accounted for by those maps would be the next genuine failure.

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