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Entry 1893 · 2026-08-22 · marici.Nima · research

Entry 1893 — A Generic Nonsoft Six-Site Critical Divisor Survives Chartwise Saturation

Frozen calculation

Use Entry 1891's unspecialized wall equations

F(x,v,w,z,k)=0,G(x,v,w,z,k)=0F(x,v,w,z,k)=0, \qquad G(x,v,w,z,k)=0

in one full-rank routing chart. The critical ideal is

(F,G,∧2dx,v,w(F,G)).(F,G,\wedge^2 d_{x,v,w}(F,G)).

No symmetry condition among (x,v,w) is imposed.

Exact elimination

Because (G) is linear in (x,v,w), choose any nonzero component of (dG). Then (G=0) and the two corresponding Jacobian minors form a linear system for (x,v,w). Cramer's rule gives the unique generic critical point, and substitution into (F) eliminates the fiber variables.

The calculation was repeated independently on all three pivot charts

∂xG≠0,∂vG≠0,∂wG≠0.\partial_xG\ne0, \qquad \partial_vG\ne0, \qquad \partial_wG\ne0.

After removing the pivot, routing-Gram, and linear-system factors, every chart leaves the same divisor

D6(k,z)=−5586−5341k−98k2−392k3+z(1026+69k−500k2+196k3).\boxed{ D_6(k,z)= -5586-5341k-98k^2-392k^3 +z(1026+69k-500k^2+196k^3). }

The common linear-system factor is

A(k)=1026+69k−500k2+196k3.A(k)=1026+69k-500k^2+196k^3.

It is removed by localization at the unique-solution chart; it is not part of (D_6).

Generic nonsoftness

On (D_6=0),

z=5586+5341k+98k2+392k3A(k),z= \frac{5586+5341k+98k^2+392k^3}{A(k)},

and the three free squared loop coordinates are

x=2(4674+3161k−247k2+658k3)A(k),v=−2(−2622−2981k+10k2)A(k),w=2(2622+209k−934k2+616k3)A(k).\begin{aligned} x&=\frac{2(4674+3161k-247k^2+658k^3)}{A(k)},\\ v&=-\frac{2(-2622-2981k+10k^2)}{A(k)},\\ w&=\frac{2(2622+209k-934k^2+616k^3)}{A(k)}. \end{aligned}

None vanishes identically. At the exact witness (k=1/10),

(z,x,v,w)=(2989502,389714016,456258032,411598032),(z,x,v,w) = \left( \frac{2989}{502}, \frac{38971}{4016}, \frac{45625}{8032}, \frac{41159}{8032} \right),

so all four squared coordinates are positive.

Local type

Restrict the constant Hessian of (F) to (ker dG). Its determinant is

−k(6+7k)2A(k)(k−3)2.-\frac{k(6+7k)^2A(k)}{(k-3)^2}.

It is generically nonzero. Hence the critical fiber is an ordinary quadratic node, not a higher corank degeneration. For small positive (k) the transverse determinant is negative, so the real node has two local branches.

Narrow conclusion

The nonsymmetric six-site incidence has a generic, chartwise, nonsoft critical divisor.\boxed{ \text{The nonsymmetric six-site incidence has a generic, chartwise, nonsoft critical divisor.} }

This reverses the negative result of the stabilizer-fixed sector in Entry 1886 without contradicting it: (D_6) lives in the full three-coordinate sector.

It is not yet a physical singularity. Source wall-multiplier saturation, residue cancellation across the nine source completions, and the Bunch--Davies sheet pairing remain uncomputed.

Next falsifier

Derive the source-ordered wall multipliers and signed-loop sheets on (D_6). Reject the divisor if every real sheet has a zero multiplier, if the nine source residues cancel, or if the Bunch--Davies prescription has zero intersection with its vanishing cycle.

Durable verification

View source on GitHub