Relational GNS Fusion and an Idempotent Gauge-Gravity Fixed Point

Abstract

We propose a relational noncommutative framework in which quantum kinematics, semiclassical geometry, internal gauge structure, and ultraviolet fixed-point behaviour arise as sectors of a common $*$-algebra equipped with a positive state and its Gelfand--Naimark--Segal (GNS) representation. The central construction is a projected fusion product on normalized quadratic current operators. Within the closed current sector, $P_{\rm rel}(O_A^2)=2O_A$ up to irrelevant corrections, and ultraviolet self-similarity expressed as $X_A\star X_A=X_A$ with $X_A=r_AO_A$ yields the interacting normalized fixed point $r_A^*=1/2$. With the retained operator basis fixed by the GNS Gram metric, a reduced per-generation fermionic current convention for the gauge sectors and canonical graviton normalization give a common algebraic value for $(10/3)g_Y^2$, $2g_2^2$, $2g_3^2$, and $32\pi G\mu^2$. We discuss conditional emergence of Lorentzian geometry and Einstein gravity, a Standard-Model-like internal algebra, three generation channels from a minimal quartic pairing sector, minimal selection of a $3+1$ dimensional branch, and relational time with a branch-dependent arrow of records. A Standard Model--Pati--Salam renormalization-group analysis provides a phenomenological consistency test of the proposed ultraviolet basin.