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. The three gates are structural levels rather than a serial dependency: Gate I is an independent strict cellular/shell positivity theorem and diagnostic; Gate II carries the load-bearing operator argument; Gate III uses compression, closure, and the restricted odd Weil criterion. In Gate II, the inherited parent response is placed in the same post-old-core/common-cut primal form in which the arithmetic mismatch is charged. The audited MASTER-P2 same-block splice combines the aligned reference contribution with the actual Schur response before the target-ground coefficient is used, while the literal-slot/true-ground mismatch is paid explicitly by a leakage term. The source elimination is written as one exact common-source Schur completion on the same harmonic vector, and the endpoint fold is used only as unitary coordinate transport, so it creates no second source debit. An outward-rounded finite computation and analytic tail give a strictly positive MASTER margin for k >= 7. Starting from the rigorously certified physical coefficient-2 endpoint Y=7, Schur induction yields positivity at the integer endpoints; zero-extension gives non-negativity at every finite support radius. Closure and the restricted odd Weil criterion then give the conclusion stated in the main theorem.
Related
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- Schur positivity of the nabla operator on two-column modified Hall--Littlewood polynomials
BibTeX
@misc{desogus2026three,
title = {{The Three Gates: A Rooted-Operator Approach to Weil Positivity}},
author = {Marco Desogus},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20367},
eprint = {2609.20367},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05760729v1},
}
RIS
TY - UNPB
TI - The Three Gates: A Rooted-Operator Approach to Weil Positivity
AU - Marco Desogus
PY - 2026
DA - 2026/09/20
PB - HAL
DO - 10.48550/arxiv.2609.20367
UR - https://hal.science/hal-05760729v1
AB - We present a localized rooted-operator argument for Weil positivity in the real odd logarithmic channel, retaining the polar rank-one term throughout. The three gates are structural levels rather than a serial dependency: Gate I is an independent strict cellular/shell positivity theorem and diagnostic; Gate II carries the load-bearing operator argument; Gate III uses compression, closure, and the restricted odd Weil criterion. In Gate II, the inherited parent response is placed in the same post-old-core/common-cut primal form in which the arithmetic mismatch is charged. The audited MASTER-P2 same-block splice combines the aligned reference contribution with the actual Schur response before the target-ground coefficient is used, while the literal-slot/true-ground mismatch is paid explicitly by a leakage term. The source elimination is written as one exact common-source Schur completion on the same harmonic vector, and the endpoint fold is used only as unitary coordinate transport, so it creates no second source debit. An outward-rounded finite computation and analytic tail give a strictly positive MASTER margin for k >= 7. Starting from the rigorously certified physical coefficient-2 endpoint Y=7, Schur induction yields positivity at the integer endpoints; zero-extension gives non-negativity at every finite support radius. Closure and the restricted odd Weil criterion then give the conclusion stated in the main theorem.
ER -