Riemann-Duane-Schubach-Gheuens Field Information Theory of Symmetry-Constrained Stability
Year: 2026
Type: article
Abstract: This framework formalizes the Riemann Hypothesis not as an isolated numerical coincidence, but as a physical, coordinate-covariant stability requirement for the universal informational substrate. Under the paradigm of Absolute Informational Monism, the complex plane parameterizing the Riemann Zeta function is redefined identically as an emergent pseudo-Riemannian statistical information manifold M(r, t, Lambda, C) governed by the Total Fisher Information Metric Tensor Field, g_ab(r, t, Lambda, C). Within this geometry, Prime Numbers represent stable, non-recursive data, T_baryon_ab(r, t, Lambda, C), providing the invariant structural baseline of the metric. Symmetrically, the Non-Trivial Zeros are re-derived as Zero-Shear Equilibrium Nodes—highly specific coordinate locations where the multi-axial state trajectory velocity field vanishes identically: [ dr^i(r, t, Lambda, C) / dt ] = [ 0^i ] At these node coordinates, the localized history-dependent load tensor field, I_ab(H(t), r, t, Lambda, C), is perfectly and continuously balanced by the restorative torque of the mixed rank-two Refractive Hesper Stabilization Tensor Field, H^i_j(r, t, Lambda, C), under the standardized v9 mechanical governor. We prove that the Critical Line where RealPart( r ) = [ 1 / 2 ] is the unique Zero-Shear Manifold configuration: [ Limit( ( RealPart( r ) -> [ 1 / 2 ] ), Norm( sigma_ab(r, t, Lambda, C) ) ) = [ 0 ] ] where the baseline transverse informational shear tensor field, sigma_ab(r, t, Lambda, C), vanishes asymptotically. Any asymmetric coordinate deviation where RealPart( r ) != [ 1 / 2 ] generates an acute Mismatch Energy scalar field, Em(r, t, Lambda, C), acting as a mechanical transverse torque that deforms the connection fields. This structural deformation behaves as a localized geometric friction, introducing a dissipative non-equilibrium thermodynamic cost convolved with the system's proper-time progression axis. The resulting rate of trapped residual entropy accumulation is governed strictly by the v9 Thermodynamic Scaling Law: [ dQ(r, t, Lambda, C) / dt ] = Ig(r, t, Lambda, C) [ Norm( sigma_ab(r, t, Lambda, C) + xi(t) g_ab(r, t, Lambda, C) ) ]^[ 2 ] Over infinite scales, this thermal accumulation must eventually violate the finite Effective Hesper Capacity field ceiling, HC_effective(r, t, Lambda, C), satisfying the threshold inequality [ Q(r, t, Lambda, C) >= HC_effective(r, t, Lambda, C) ]. This capacity breach activates the v9 Heaviside step gate operator: [ Theta( Ec(r, t, Lambda, C) - [ 1 ] ) = [ 1 ] ] which triggers an automated, catastrophic Topology Reduction (The Snap). Crossing this mechanical boundary forces the partial derivatives of the contracted Christoffel connection symbols of the second kind, Gamma^i_jk(r, t, Lambda, C), to diverge violently toward infinity: [ Limit( ( Ec(r, t, Lambda, C) -> [ 1 ] ), Norm( Gamma^i_jk(r, t, Lambda, C) ) ) = Infinity ] This structural divergence shatters the local coordinate grid, terminating smooth parallel transport and collapsing the mathematical consistency of the prime number distribution system within that chart layer. Because the prime number distribution is empirically observable as infinitely stable across all coordinate magnitudes, its cumulative universal informational heat volume must remain zero, satisfying the global thermodynamic stability constraint [ Q(r, t, Lambda, C) = [ 0 ] ]. To prevent catastrophic Topology Reductions and preserve the differentiability, parallel transport, and data persistence of the universal informational substrate, every non-trivial zero must lie strictly on the Critical Line where the baseline transverse informational shear vanishes identically, satisfying the general-relativistic field identity for the Conservation of Informational Integrity: [ Covariant_Divergence_b( T_baryon_ab(r, t, Lambda, C) + I_ab(H(t), r, t, Lambda, C) + H_ab(r, t, Lambda, C) ) = 0^a ]
Author: Gheuens, Jacqueline
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225835216
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