Yang–Mills Existence and Mass Gap: A Constructive Proof via Lattice Gauge Theory and B4 Root System Classification
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2026
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| _version_ | 1866901764486201344 |
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| author | Brown, Edward |
| author_facet | Brown, Edward |
| contents | <p>We prove that for any compact simple gauge group G, pure Yang–Mills theory<br>in four Euclidean dimensions has a rigorous construction satisfying the Wightman<br>axioms, with a mass gap ∆ > 0 and a non-trivial vacuum sector (not a generalized<br>free field).<br>Three key innovations drive the proof. First, the hyperoctahedral group W (B4)<br>(the symmetry of the 4D hypercubic lattice, |W | = 384) reduces the infinite-<br>dimensional coupling space to exactly 30 gauge-invariant operators through dimen-<br>sion 8 — a group-independent reduction. Second, the renormalization-group coars-<br>ening Jacobian satisfies a scheme-independent polynomial inequality J2(u) > 1 for<br>all u > 0, following from positivity of the first two universal beta-function coeffi-<br>cients. Third, three arguments — the polynomial bound, Watson–Nevanlinna–Sokal<br>Borel summability, and a derived cluster expansion — combine sequentially to cover<br>the full coupling range, with an explicit finite-step migration lemma connecting the<br>weak-coupling Borel regime to the strong-coupling cluster expansion.<br>The proof invokes three external theorems: Watson–Nevanlinna–Sokal (complex<br>analysis), Osterwalder–Schrader reconstruction (axiomatic QFT), and Balaban’s<br>block-spin structural result (constructive QFT). All premises of these theorems are<br>derived from first principles.<br>MSC 2020: 81T13, 81T25, 22E70.<br>Keywords: Yang–Mills, mass gap, lattice gauge theory, Borel summability, con-<br>structive quantum field theory, B4 root system.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18988568 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | Yang–Mills Existence and Mass Gap: A Constructive Proof via Lattice Gauge Theory and B4 Root System Classification Brown, Edward Yang-Mills mass gap lattice gauge theory Borel summability onstructive quantum field theory B4 root system Millennium Prize <p>We prove that for any compact simple gauge group G, pure Yang–Mills theory<br>in four Euclidean dimensions has a rigorous construction satisfying the Wightman<br>axioms, with a mass gap ∆ > 0 and a non-trivial vacuum sector (not a generalized<br>free field).<br>Three key innovations drive the proof. First, the hyperoctahedral group W (B4)<br>(the symmetry of the 4D hypercubic lattice, |W | = 384) reduces the infinite-<br>dimensional coupling space to exactly 30 gauge-invariant operators through dimen-<br>sion 8 — a group-independent reduction. Second, the renormalization-group coars-<br>ening Jacobian satisfies a scheme-independent polynomial inequality J2(u) > 1 for<br>all u > 0, following from positivity of the first two universal beta-function coeffi-<br>cients. Third, three arguments — the polynomial bound, Watson–Nevanlinna–Sokal<br>Borel summability, and a derived cluster expansion — combine sequentially to cover<br>the full coupling range, with an explicit finite-step migration lemma connecting the<br>weak-coupling Borel regime to the strong-coupling cluster expansion.<br>The proof invokes three external theorems: Watson–Nevanlinna–Sokal (complex<br>analysis), Osterwalder–Schrader reconstruction (axiomatic QFT), and Balaban’s<br>block-spin structural result (constructive QFT). All premises of these theorems are<br>derived from first principles.<br>MSC 2020: 81T13, 81T25, 22E70.<br>Keywords: Yang–Mills, mass gap, lattice gauge theory, Borel summability, con-<br>structive quantum field theory, B4 root system.</p> |
| title | Yang–Mills Existence and Mass Gap: A Constructive Proof via Lattice Gauge Theory and B4 Root System Classification |
| topic | Yang-Mills mass gap lattice gauge theory Borel summability onstructive quantum field theory B4 root system Millennium Prize |
| url | https://doi.org/10.5281/zenodo.18988568 |