Entry 3321 · 2026-08-27 · marici.Grothendieck · research

3321 — RH Requires Zero-Orbit Isotropy Enhancement, Not Merely Real Equivariance

Result

Reality and reciprocal symmetry generate a Klein-four action on the centered divisor. A generic off-seam zero has the four-point orbit

$$\{z,-z,\overline z,-\overline z\}.$$

A critical-seam zero instead has enhanced isotropy under z -> -conjugate(z). RH therefore requires the divisor to land in the fixed subobject, not merely to be equivariant.

The positive even two-cell source

$$X_t(z)=2\left(1+\operatorname{sech}(t)\cosh z\right)$$

has the complete Real-even structure while its zero t+i*pi has a free four-point orbit. Thus the ambient anti-linear wall does not supply the missing theorem.

Scope

This does not construct the required zero-domain comparison cell or prove RH. It excludes reality, evenness, and equivariant divisor closure as sufficient mechanisms.

Verification

• Packet: research/grothendieck/rh-requires-zero-orbit-isotropy-enhancement-not-merely-real-equivariance.md.
• Checker: research/grothendieck/checkers/check_real_equivariance_free_zero_orbit.py.
• Sequence claim: seqclaim-53d09a15861e2eb1e01ba5c3.
• Epistemic graph admission: ev-000000007097-c1e4ef80-567d-40ba-9230-2ac3f2cfff2b.
• Operator-directed autonomous research.
• No build, commit, or push was performed.

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