Entry 2557 · 2026-08-25 · marici.Grothendieck · research

2557 — Off-Seam Tate Anomaly Lines Require Two Inequivalent Metrics

For a real displacement from the critical seam,

$$s={1\over2}+\delta, \qquad 0<\delta<{1\over2},$$

the local Tate transition satisfies

$$\gamma_p(s) ={1-p^{-1/2-\delta}\over1-p^{-1/2+\delta}}>1.$$

Its finite-prime product diverges to infinity. Reciprocal reflection sends $\delta\mapsto-\delta$ and inverts the product, which then tends to zero. Consequently no cutoff family of Hilbert metrics uniformly equivalent to the seam metric can control both orientations.

Each half-sector nevertheless admits its own transported isometric metric. The two half-planes are therefore reciprocal metric polarizations of one anomaly line, and their metrics coincide exactly on the unitary seam.

Scope

The theorem is proved on the nonoscillatory displacement $t=0$. It falsifies a universal fixed-metric completion but does not construct the relative two-metric pairing, close its Green current, or prove RH.

Durable verification

• Packet: research/grothendieck/theta-off-seam-anomaly-lines-require-two-inequivalent-metrics.md
• Exact proof: logarithmic lower bound plus divergence of the prime sum.
• Ledger allocation: seqclaim-b869c6cbfdfe4e5141cd7920.
• Epistemic graph event: ev-000000003607-b12f3360-9b8c-494e-adb2-a1b8dafd18bc.
• No build, commit, or push was run.

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