V184 — Weil Positivity as a Missing Arithmetic Hodge Index Subject: A Route‑Pruning and Frontier‑Mapping Document for the Riemann Hypothesis

V184 — Weil Positivity as a Missing Arithmetic Hodge Index Subject: A Route‑Pruning and Frontier‑Mapping Document for the Riemann Hypothesis

Overview
This paper marks the V184 stage of the Reburn program. It does not claim a proof of the Riemann Hypothesis (RH). Instead, it identifies the precise obstruction revealed by earlier Toeplitz, prolate, de Branges, and Connes‑Weil analyses. The central thesis is that classical Weil positivity is the analytic shadow of a missing arithmetic Hodge index theorem.

🔹 Function‑Field Case

• In function fields, RH is proved because Frobenius acts on middle cohomology (H^1).
• The Hodge index theorem applies to primitive correspondences on (C \times C), giving the inequality (D^2 \leq 0).
• This geometric positivity forces Frobenius eigenvalues to have absolute value (q^{1/2}).

🔹 Classical Case

• The Weil explicit formula decomposes into archimedean, prime, and pole terms.
• Weil’s positivity criterion states that RH is equivalent to non‑negativity of a quadratic form built from this formula.
• The missing piece: an arithmetic correspondence subject (D_g) on which an arithmetic Hodge index theorem could act.

🔹 Existing Tools

• Arakelov geometry: provides intersection theory and Hodge index theorems, but no RH‑indexed correspondences.
• Connes’ trace formula: realizes zeros spectrally, but lacks positivity.
• Deninger’s program: offers cohomological architecture, but spaces and positivity remain incomplete.
• F1 geometry: proposes a “field with one element” framework, but no Frobenius graph or intersection theory strong enough for RH.
• Reburn project: finite Toeplitz shadow of Weil positivity, useful diagnostically but not a proof.

🔹 Refined Diagnosis

• The Hodge index tool exists (Arakelov, Yuan–Zhang, etc.).
• The RH subject is missing: no defined (D_g), no correspondence category (C_Q), no Frobenius‑like dynamics.
• Without these, positivity cannot be applied non‑circularly.

🔹 Next Frontier

• Construct the correct category of arithmetic correspondences for the Riemann zeta function.
• Define (D_g) functorially from Weil test functions.
• Establish a primitive projection and intersection pairing reproducing the Weil form.
• Prove a Hodge‑index‑type sign theorem acting independently of RH.

Conclusion
V184 sharpens V183’s result. The Reburn model is a finite shadow of Weil positivity. The true obstruction is not the absence of a positivity theorem, but the absence of the arithmetic subject on which such a theorem could act.

👉 Final distilled thesis: The Hodge index tool exists; the RH subject is missing.

📩 Verification Note: This document is a frontier map, not a proof. It isolates the missing object that future research must construct. Independent verification can be requested via 24ping@naver.com.