Colour Confinement as Finite Record Geometry: Closed-Cell Triality, Wilson Strings, and the Baryon Y-Junction

Abstract

The strong-sector sequel to the executable record-grammar note: colour confinement rebuilt as finite record geometry on the framework's bond-bipyramid (octahedral) cells, with a clean line drawn between what is proved exactly and what is only leading-order. The kinematic half is an exact finite theorem — on a closed colour cell an 𝔽₃ Gauss-law incidence map admits a source only if its total triality vanishes, so a lone quark has no gauge-invariant record while a qq̄ meson and a qqq baryon do: the proton is not declared a colour singlet but is the unique minimal three-fundamental triality-zero record the geometry permits. The dynamical half is leading strong-coupling: on a licensed bond-bipyramid bulk (the naïve corner-stacked block is rejected first by an explicit finite three-ball topology gate), edge-loop Wilson records give an area law with string tension σ = −log(β/18), a Creutz ratio isolates χ_area = 4σ, and a gauge-invariant qqqε operator places the baryon as a genuine bulk Y-junction whose exact three-terminal Steiner length L_Y = 3n carries the same per-unit tension as the meson string. An honest negative is reported rather than hidden: the Wilson ledger is a two-dimensional tensor-network object whose bond dimension D_R = 2^(R−1) grows with loop width — tractable at fixed width, exponential in two dimensions — so the earlier hope of a bounded-bond-dimension record fails, exactly as a confining string with worldsheet entropy should. A finite cubic-axis isotropy gate passes to numerical precision. A new scale-setting harness now samples full SU(3) bond-bipyramid ensembles: a zero-temperature hypercubic Wilson-SU(3) control validates the static-potential extractor against the standard pure-gauge benchmark (T_c/√σ = 0.6449 vs the usual 0.63), but porting that Cornell scheme onto the bond-bipyramid bulk still gives negative or unstable fitted string tensions, so a dimensionful σ stays a geometry/operator/volume problem. Deliberately not continuum QCD: it computes neither the physical string tension, the proton mass, nor a Y-versus-Δ distinction (the axis-aligned tri-axial family cannot separate them by length alone) — it shows the record geometry already carries the structure of confinement, reproducibly, on explicit cells.

Keywords

How to cite

Elliman, D. (2026). Colour Confinement as Finite Record Geometry: Closed-Cell Triality, Wilson Strings, and the Baryon Y-Junction. Neuro-Symbolic Ltd technical report. https://doi.org/10.5281/zenodo.21111292

@techreport{elliman2026recordgrammarconfinement,
  author      = {Elliman, David},
  title       = {Colour Confinement as Finite Record Geometry: Closed-Cell Triality, Wilson Strings, and the Baryon Y-Junction},
  institution = {Neuro-Symbolic Ltd},
  year        = {2026},
  doi         = {10.5281/zenodo.21111292},
  url         = {https://neusym.ai/papers/record_grammar_confinement}
}