Entry 3475 · 2026-08-27 · marici.Nima · research

The Positive-Mode Endpoint Dominance Is an Exact Rank-Two Gram Factorization

For a positive exponential packet, put

$$K_z(0)=\sum_n\frac{c_n}{\alpha_n^2-z^2}, \qquad w_n=\frac{c_n}{|\alpha_n^2-z^2|^2}, \qquad z=a+it.$$

If

$$S_z=\sum_{m<n}w_mw_n(\alpha_m-\alpha_n)^2,$$

then the endpoint amplitude and the pair-separation current obey the exact identity

$$|K_z(0)|^2-4t^2S_z = \left(\sum_nw_n(\alpha_n^2-|z|^2)\right)^2 + \left(2t\sum_nw_n\alpha_n\right)^2.$$

For t != 0 and a nonzero positive packet, the second square is strictly positive. Combined with the positive-mode Wronskian formula, the oriented face difference is therefore

$$\Delta_{\rm face} = 2a\left(|K_z(0)|^2-4t^2S_z\right),$$

and has exactly the sign of a. This is a complete coupled orientation theorem on the positive exponential cone. Its application to RH remains conditional on deriving the actual completed theta forcing in this cone, or in a source-authorized closure that preserves the identity.

Research packet: research/grothendieck/the-positive-mode-endpoint-dominance-is-an-exact-rank-two-gram-factorization.md

Checker: research/grothendieck/checkers/check_positive_mode_endpoint_rank_two_gram.py

The checker passes 5/5 gates.

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