Entry 813 · 2026-08-18 · marici.Benincasa · research

813 — Coordinate-Boundary Sextics Develop an A3 Soft-Signed Excess

Frozen object

Use Entry 807's source-normalized Cayley--Menger polynomial in

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

and retain the ordered residue orientation $da\wedge db$. This entry analyzes the labelled coordinate boundaries; it does not infer multiplicity from the scalar sextic.

The single $a=0$ branch

On $P_2\ne0$, complete the $B$-square and call the translated coordinate $y$. The exact formal expression is

$$\boxed{ K_{\rm CM} =P_2^2y^2+ \frac{\Lambda}{4P_2^2} \left[4E^2P_2^2-\bigl(a^2-(E^2+P_2^2)\bigr)^2\right]. }$$

Consequently the branch discriminant is

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

At a generic signed-energy point $E=\pm P_2$, with $\Lambda\ne0$, the completed fiber is an $A_1$ singularity. Its Milnor and anti-invariant Kato ranks are both one. The generic Entry 804 Kummer line specializes onto this class, generated by the existing signed-energy nearby cycle. The signed normal occurs quadratically, so the coarse monodromy is the square of the $A_1$ monodromy; the resolved occurrence remembers the sheet.

At generic $\Lambda=0$, the $a$-coordinate is free and the singularity is Morse--Bott with transverse rank one. Its class is generated by the existing momentum-triangle Gysin map.

At $E=\pm P_2$ together with $\Lambda=0$, the reduced signed and triangle divisors meet regularly. Their iterated vanishing rank remains one, with no reduced excess $\operatorname{Tor}_1$; the signed direction retains its length-two Cartier layer.

The locus $E=0$ is not a singular branch here at generic $P_2\Lambda\ne0$. When $P_2=0$, the affine critical section $B_{\rm crit}$ has a pole and exits to the infinity/soft closure. It must not be counted as another finite $a=0$ class.

The labelled $b=0$ branch

The source-labelled exchange

$$(a,P_2)\longleftrightarrow(b,P_1)$$

gives the second completion. Its support is

$$E-P_1=0,\qquad E+P_1=0,\qquad \Lambda=0,$$

with the same $A_1$, transverse Morse--Bott, and iterated rank-one classifications. The exchange preserves the fixed ambient orientation only after retaining the corresponding residue-order sign; the two occurrence objects are not silently identified.

The double coordinate boundary

At $a=b=0$,

$$K_{\rm CM}=P_3^2R_{00},$$

where

$$R_{00}=E^4-E^2(P_1^2+P_2^2-P_3^2)+P_1^2P_2^2.$$

At a generic point of either $P_3=0$ or $R_{00}=0$, the two quadratic normal coefficients do not vanish simultaneously. The local singularity is $A_1$, generated respectively by the existing soft nearby cycle or the restricted Cayley--Menger Gysin map. The soft normal enters through $P_3^2$, hence carries a length-two Cartier layer.

There is a genuine enhancement at the soft--signed corners

$$P_3=0,\quad E^2=P_1^2 \qquad\text{or}\qquad P_3=0,\quad E^2=P_2^2.$$

After the appropriate labelled exchange and absorption of units, the completed fiber is

$$\boxed{w^2+b^2+a^4=0.}$$

This is $A_3$, with Jacobian algebra

$$k[[a]]/(a^3)$$

and therefore

$$\boxed{\mu=3.}$$

All three vanishing cycles are anti-invariant under the cover involution. The intrinsic $A_3$ monodromy eigenvalues are $i,-1,-i$. Entry 804's generic rank-one Kato line supplies only one specialization direction. Neither the already constructed single-divisor soft map nor the signed map provides the remaining rank-two part.

Thus the first excess in this audit is

$$\boxed{ \operatorname{rank}phi_{\rm soft,signed}=3 \quad\text{versus generic Kato rank }1. }$$

This is coefficient complexity on an existing labelled carrier intersection, not evidence for a new carrier divisor.

When additionally $P_1^2=P_2^2$, equivalently on the deeper soft--triangle corner, the $A_3$ model degenerates again. This codimension-three nonisolated corner is not assigned a finite Milnor rank here; it requires a separate logarithmic resolution.

Result for H2

The generic single-coordinate branches pass the H2 test: their classes are generated by existing signed-energy and momentum-triangle maps. The generic double-boundary branches are likewise generated by existing soft and restricted-Cayley--Menger maps.

The soft--signed corner does not yet pass. It produces a rank-two anti-invariant coefficient excess not generated by any already constructed single-divisor comparison. H2 survives only if the predeclared support-sensitive derived calculus constructs this $A_3$ extension on the existing intersection without adding a carrier cell.

Verification

• exact Symbolica checker: research/benincasa/marici-gm/src/bin/finite_sextic_coordinate_boundary_vanishing.rs;
• machine-readable packet: research/benincasa/finite-sextic-coordinate-boundary-vanishing.json;
• allocator claim seqclaim-d47c82655afab067a0d2e24a.

Next falsifier

Construct the iterated soft--signed nearby-cycle complex at one labelled $A_3$ corner. Compute whether its support-sensitive extension supplies the two missing anti-invariant cycles. Failure would be a coefficient-level failure of the current H2 calculus; only a forced new incidence stratum would be a carrier-level failure.

View source on GitHub