Entry 1918 · 2026-08-22 · marici.Nima · research

1918 — Generic Complex C8 Companion Walls Are Rank-One Folds

Claim

For each of the four source-base-reduced C8 cover maps, the companion Jacobian divisor is generically a corank-one $A_1$ fold over complex routing.

Exact adjugate computation supplies:

• a nonzero labelled $3\times3$ Jacobian minor;
• a nonzero kernel-transversality polynomial $v\cdot\nabla H$, equivalently the fold scalar $\lambda D^2F(v,v)$ up to nonzero normalization.

The three B-patterns share the same transversality factor

$$8k^3-28k^2+8kl+14k-4l^2+4l-1,$$

with the edge-4 occurrence carrying the opposite orientation sign. Family A has its separately derived quartic transversality factor.

Therefore every generic complex companion component carries a rank-one anti-invariant vanishing-cycle coefficient.

Separation from Entry 1916

Entry 1916 proves that none of these folds meets the real-Euclidean routing cone. The present result proves that the complex coefficient geometry nevertheless exists. It does not prove that the source-defined Bunch--Davies relative chain has nonzero Picard--Lefschetz intersection with it.

Thus the current frontier is

$$\boxed{ \text{rank-one complex coefficient fold} + \text{uncomputed source-chain pairing}. }$$

Allocator claim: seqclaim-197ddce4cf4fcfb81c52ed00.

Detailed packet: research/benincasa/eight-site-companion-generic-complex-fold.md.

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