Three Independent Routes to Fl₁,₂(ℂ³) × ℂP¹: Gauge Structure, Fermion Generations, and Gravity
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2026
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| _version_ | 1866901859987357696 |
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| author | Kirk, Harold D. |
| author_facet | Kirk, Harold D. |
| contents | <p>This note establishes a structural overdetermination of the product flag manifold Fl₁,₂(ℂ³) × ℂP¹ from three independent physical requirements: gauge structure (irreducible weight lattice), fermion generations (Euler characteristic χ ≡ 0 mod 3), and Einstein gravity (Bochner-flat Marsden–Weinstein quotient).</p> <p>Working within the standard class of compact Kähler manifolds of complex dimension 4 with effective torus action, we show that any two of these three conditions are sufficient to determine the manifold uniquely. The third condition then follows as a prediction. This yields a consistency triangle in which gauge structure predicts gravity, gravity predicts generations, and generations predict gauge structure.</p> <p>The result provides a structural analogue of parameter-free prediction: independent physical sectors constrain the same geometric object from different directions, eliminating the need for tuning. The Bochner-flatness underlying Einstein dynamics is shown to follow from the same conditions that fix the gauge and generational structure, unifying these sectors at the level of manifold selection.</p> <p>This note complements prior work on polynomial truncation and Kähler reduction by identifying a consistency structure that links the emergence of physical law across regimes.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19242276 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Three Independent Routes to Fl₁,₂(ℂ³) × ℂP¹: Gauge Structure, Fermion Generations, and Gravity Kirk, Harold D. Kähler geometry Flag manifold Bochner-flat manifolds Marsden–Weinstein reduction Einstein gravity Gauge theory Fermion generations Geometric physics Emergence <p>This note establishes a structural overdetermination of the product flag manifold Fl₁,₂(ℂ³) × ℂP¹ from three independent physical requirements: gauge structure (irreducible weight lattice), fermion generations (Euler characteristic χ ≡ 0 mod 3), and Einstein gravity (Bochner-flat Marsden–Weinstein quotient).</p> <p>Working within the standard class of compact Kähler manifolds of complex dimension 4 with effective torus action, we show that any two of these three conditions are sufficient to determine the manifold uniquely. The third condition then follows as a prediction. This yields a consistency triangle in which gauge structure predicts gravity, gravity predicts generations, and generations predict gauge structure.</p> <p>The result provides a structural analogue of parameter-free prediction: independent physical sectors constrain the same geometric object from different directions, eliminating the need for tuning. The Bochner-flatness underlying Einstein dynamics is shown to follow from the same conditions that fix the gauge and generational structure, unifying these sectors at the level of manifold selection.</p> <p>This note complements prior work on polynomial truncation and Kähler reduction by identifying a consistency structure that links the emergence of physical law across regimes.</p> |
| title | Three Independent Routes to Fl₁,₂(ℂ³) × ℂP¹: Gauge Structure, Fermion Generations, and Gravity |
| topic | Kähler geometry Flag manifold Bochner-flat manifolds Marsden–Weinstein reduction Einstein gravity Gauge theory Fermion generations Geometric physics Emergence |
| url | https://doi.org/10.5281/zenodo.19242276 |