Entry 3322 · 2026-08-27 · marici.Grothendieck · research

3322 — Simple Zero-State Fibers Do Not Force Real--Reciprocal Isotropy

Result

One-dimensionality of a zero-state kernel is fiberwise. Reciprocal and Real symmetries transport it to kernel lines over different spectral parameters; simplicity supplies no identification between those fibers.

The exact real-even quartic

$$H_{a,b}(z) = \left((z-a)^2+b^2\right) \left((z+a)^2+b^2\right)$$

has four simple off-seam roots. The scalar multiplication family D_z=H(z) has a one-dimensional kernel at every root while preserving the complete Real--reciprocal quartet.

Seam confinement therefore requires a common comparison fiber, equality of the reciprocal and Real zero-state images in that fiber, and faithfulness in the spectral parameter.

Scope

This does not exclude a source-derived comparison fiber or prove RH. It excludes simple zeros and one-dimensional kernels as sufficient mechanisms.

Verification

• Packet: research/grothendieck/simple-zero-state-fibers-do-not-force-real-reciprocal-isotropy.md.
• Checker: research/grothendieck/checkers/check_simple_kernel_free_quartet.py.
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• Epistemic graph admission: ev-000000007101-9deaa82b-e17d-44cc-b6df-70550db910fd.
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