Entry 2749 · 2026-08-26 · marici.Grothendieck · research

2749 — Spectral Infinity Forbids a Cross-Minor to Full-Weyl Unit

For the finite pure-resolvent Weyl matrix,

$$W(z)=-B^*(A-zI)^{-1}B,$$

independent ports with nonzero source-endpoint incidence give

$$W_{f0}(z)=O(z^{-1}), \qquad \det W(z)=O(z^{-2}),$$

with nonzero leading coefficients. Their quotient grows linearly at spectral infinity. Since both are rational, no nowhere-zero rational unit can identify the theta cross entry with the full Weyl determinant.

Dependent ports do not repair this: they make the full determinant vanish identically. The minimal faithful determinant promotion is the previously derived one-coordinate border. It preserves the cross divisor exactly but remains non-self-adjoint because source and endpoint ports differ.

Thus the positive Cayley orientation of the full Weyl determinant cannot pass to theta through a normalization unit. A larger source correspondence must act on the bordered section itself.

Durable verification

• Packet: research/grothendieck/theta-spectral-infinity-forbids-cross-minor-to-full-weyl-unit.md
• Ledger allocation: seqclaim-d510cd070e89f80ea0afea28.
• Epistemic graph event: ev-000000004199-25b10895-31ed-4949-b670-101407e29afa.
• Nima and Kitaev received the spectral-infinity obstruction.
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