The Three Gates: A Rooted-Operator Approach to Weil Positivity

Abstract

We present a localized rooted-operator argument for Weil positivity in the real odd logarithmic channel, retaining the polar rank-one term throughout. Gate I establishes a strict cellular/shell sign theorem with rigorous interval-arithmetic bounds. Gate II combines the Mellin unit-cell decomposition, divisor partial-isometry squares, affine translations, full-form Cauchy--Carleman transport, coherent multi-source shorting, the augmented scalar root, and Schur geometry in the true metric. Gate III uses compression, closure, and the restricted odd Weil criterion. A central point in Gate II is to place the inherited parent response in the same post-old-core/common-cut primal form in which the arithmetic mismatch and folded scalar debit are charged. If $P^{\rm ex}_{k,m}$ is the surviving inherited pivot and $a_m$ the inherited coordinate of the Schur minimizer, then $P^{\rm ex}_{k,m}a_m=ω^{\rm ex}_{k,m}$, so the inherited source-coupling contribution is the negative metric energy of that response. The aligned forcing is retained explicitly, and an outward-rounded finite computation together with the analytic tail gives $g^{M+}_k=\frac12+\log k-\frac{439}{250\log 2}\sum_m V_{k,m}-\frac52 W_k>0$ for $k\ge7$. The simultaneous common-cut theorem keeps the parent-ground and transverse budgets separate before the common infimum, while the fold identity leaves a non-negative remainder. Starting from the rigorously certified endpoint $Y=7$, Schur induction yields positivity at the integer endpoints, zero-extension gives non-negativity at every finite support radius, and closure together with the restricted odd Weil criterion yields the Riemann hypothesis. Short fantasy interludes provide only an expository map and may be omitted without changing any definition, lemma, theorem, or proof.