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[3] ai.viXra.org:2608.0003 [pdf] submitted on 2026-08-01 23:42:12
Authors: J. W. McGreevy
Comments: 17 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
We develop a symplectic geometry of atomic shear measures supported on the irregular primes of a modular curve. Starting from a space of dis-crete measures subject to a global valence constraint, we perform a strict symplectic reduction and obtain a reduced phase space equipped with a canonical Darboux form. The local coordinate functions that extractthe individual shear amplitudes are shown to be in Liouville involution, generating a completely integrable system whose invariant level sets areLagrangian tori. By lifting the functional equation of the associated automorphic L-functions to an anti-symplectic involution on this phase space, and under the established essential self-adjointness of the clutched conical Dirac operator, we prove a Confinement Theorem: the only invariant Lagrangianleaves compatible with the reflection symmetry are those whose spectralimage lies on the critical line Re(s) = 1/2. We conclude with a brief, explicitly programmatic dictionary that explores possible links between the resulting arithmetic geometry and structures appearing in gauge theory and the Standard Model. This workextends earlier constructions developed under the working title Relativistic Field Theory of Primes (RFTP), in which the irregular primes, the weight-12 valence constraint, and the associated clutching data were first introduced as geometric ingredients of an arithmetic field theory.
Category: Mathematical Physics
[2] ai.viXra.org:2608.0002 [pdf] submitted on 2026-08-01 23:35:07
Authors: Nigel B. Cook
Comments: 9 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
It is shown that SU(4) and SL(4,R) both offer feasible alternative routes to unification, justifying the twistor unification approach and therefore getting rid of superstring theory. Paper was proof read by Copilot AI to check maths.
Category: Quantum Gravity and String Theory
[1] ai.viXra.org:2608.0001 [pdf] submitted on 2026-08-01 18:50:53
Authors: Lluis Eriksson
Comments: 9 Pages. Lean 4/Mathlib formalization of the finite-SU(2) Pauli/Casimir contraction, trace-skein identity, four oriented local branches with both reconnections, corrected single-crossing closure, and an all-order multitrace-to-single-trace recursion.
Finite-rank Makeenko-Migdal equations generate products of Wilson traces at self-intersections. For SU(2), this apparent multitrace obstruction closes exactly on single traces, but the statement is normalization-sensitive: the traceless Lie algebra contributes a finite-rank correction that disappears for U(2) and must not be dropped. We give a Lean 4/Mathlib formalization of the complete group-algebraic closure mechanism on Mathlib's concrete special unitary matrix group. With normalized trace tau(A)=Tr(A)/2 and normalized anti-Hermitian Pauli directions X_j=i sigma_j/2, the kernel checks the Casimir identity, the rank-two Fierz identity, the induced crossing contraction, and the SU(2) trace-skein identity tau(g)tau(h)=(tau(gh)+tau(gh^{-1}))/2. Consequently, the finite-SU(2) crossing term tau(g)tau(h)-tau(gh)/4 equals tau(gh)/4+tau(gh^{-1})/2. We then formalize a universal local interface with four cyclically ordered branch holonomies, an independent orientation on each branch, the two opposite-strand words, and precisely the two direct/reversed reconnections. Its corrected crossing term closes on those reconnections for every branch assignment and orientation choice. A recursive theorem also extends the reduction to products of arbitrarily many fundamental traces. The identities are classical; the contribution is a concrete, kernel-checked normalization bridge from Pauli contraction to the single-trace closure used in finite-rank loop equations. We do not claim a formal derivation of the Yang-Mills area derivative, planar loop geometry, or the full Makeenko-Migdal equation.
Category: Mathematical Physics