GETT Correspondence Series Paper 4.1 – General Relativity Correspondence I: Emergent Metric Structure and the Geometrisation Limit

GETT Correspondence Series Paper 4.1 – General Relativity Correspondence I: Emergent Metric Structure and the Geometrisation Limit Description: This paper establishes the kinematic correspondence between General Expanse Tension Theory (GETT) and General Relativity within a clearly defined physical regime. The work addresses a foundational question: under what conditions can spacetime geometry arise as an effective description of underlying physical processes? Starting from a primitive ontology consisting of a real, gauge-invariant singlet scalar field Φ, the paper constructs an operational measurement framework in which the behaviour of clocks, rods, and signal propagation depends explicitly on the local field state. No spacetime metric is assumed a priori. Instead, measurement is treated as a physical process arising from field–matter interaction. It is shown that, in a regime characterised by smooth field variation, adiabatic evolution, and approximately constant coupling (the mid-density regime), the response of all physical systems becomes effectively universal. Under these conditions, local measurements can be consistently encoded in a single effective pseudo-Riemannian metric. The invariant interval, proper time, and causal structure emerge from this construction, and freely evolving systems are shown to follow geodesics obtained by extremisation of physically measured time. A key result is the derivation of a compatibility (closure) relation linking temporal, spatial, and signal response functions, ensuring that signal propagation defines the same null structure as the emergent metric. This establishes that the full kinematic structure of General Relativity—metric structure, causal cones, geodesic motion, and local Lorentz invariance—can arise as a domain-limited effective description of a scalar-field-based physical substrate. The analysis is explicitly restricted to kinematics. No claim is made in this work regarding the recovery of the Einstein field equations or the full dynamical correspondence with General Relativity. These aspects are deferred to subsequent papers in the GETT Correspondence Series. The broader programme is structured as follows: Paper 4.1 (this work): Kinematic foundation and geometrisation limit Paper 4.2: Dynamical correspondence and field equations Paper 4.3: Strong-field and non-linear regimes Paper 4.4: Breakdown of geometric description and non-metric behaviour This work provides a formally defined, testable framework in which spacetime geometry is interpreted not as a fundamental entity, but as an emergent encoding of universal measurement response to an underlying scalar field. The conditions required for this emergence are explicit, physically motivated, and falsifiable. This paper is intended as a foundational, field-theoretic formulation suitable for formal peer review and independent validation. Keywords:Scalar field theory; emergent spacetime; general relativity; metric structure; geodesic motion; Lorentz invariance; gravitational theory; foundations of physics; effective field theory; spacetime emergence