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 covariance theorem and its midpoint-shell consequence, with rigorous interval-arithmetic bounds on the finite range used below. 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 inside 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 $\mathfrak g^{\rm M+}k=\frac12+\log k-\frac{439}{250\log 2}\sum_m V{k,m}-\frac52 W_k>0$ 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.