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Lorentz-Covariant Completion of the Enchan Field and Earth Weak-Field Validation

This work investigates whether the dimensionless Enchan order field S, together with its proposed weak-field effective metric, can be completed as a Lorentz-covariant spacetime description without introducing a fundamental preferred frame.

The order of inference is fixed before comparison with observational data:

field covariance → tensor metric completion → static Earth → rotation → Gravity Probe B.

First, the Enchan scalar field equation is shown to be Lorentz covariant when S and its source terms are defined as scalars. The finite-tension invariant X is a scalar, and the covariant divergence of the nonlinear field current remains scalar under Lorentz transformations. No absolute time derivative or preferred four-velocity is required.

A scalar-only algebraic metric construction is then shown to be insufficient to reproduce both the static isotropic metric structure and the rotational g0i sector. This motivates a symmetric tensor completion. The normalization is not fitted to Earth observations: the pre-existing Enchan value

σvac = c⁴/(4πG)

fixes the corresponding weak-field tensor coefficient and source equation.

In the static high-acceleration projection, the completed system yields

S(r) = GM/(c²r)

and recovers, from the same weak-field geometry:

Newtonian acceleration,

gravitational clock redshift,

light deflection,

and Shapiro-type propagation delay.

The stationary geometry is separated into a scalar lapse governing local clock rate and a synchronization connection governing temporal circulation. This allows static temporal gradients and rotational temporal circulation to be distinguished within the same covariant framework.

For rotating sources, the 0i sector of the tensor equation generates a nonzero rotational metric component for angular momentum J, while recovering zero rotational shift when J = 0. The resulting weak-field rotation scales as

ω ∝ J/r³

without inserting a Kerr metric or fitting an Earth frame-dragging coefficient.

Using independently fixed Earth parameters, the spherical weak-field and slow-rotation calculation predicts for Gravity Probe B:

Geodetic drift: −6620.97 mas/yr

Frame-dragging drift: −40.93 mas/yr

Compared only after the theory-side calculation was frozen, these correspond to deviations of approximately 1.05σ and 0.52σ, respectively, from the reported Gravity Probe B measurements.

At the level tested here, the Earth weak-field and slow-rotation completion passes its stated criteria. The result should not, however, be interpreted as a completed all-regime relativistic theory.

The remaining problem is isolated as an explicit open completion lemma: construct a diffeomorphism-invariant nonlinear action for the metric, Enchan scalar field, and matter such that

its scalar projection reproduces the finite-tension Enchan equation,

its high-acceleration linearization reproduces the tensor equation,

its matter coupling closes consistently with stress-energy conservation,

and its metric variation satisfies the Bianchi identity.

This action-level construction is the remaining step required before extending the same completed tensor structure across screening-transition, galactic, wide-binary, and strong-field regimes.

Open Review: An open review has been requested through PREreview on August 26, 2026.PREreview: https://prereview.org/preprints/doi-10.5281-zenodo.22066883 License: Enchan Research & Verification License v1.0 (custom non-commercial research and verification license). See the deposited LICENSE file for the complete terms. Update — 30 August 2026: The standalone LICENSE file and the LICENSE copy inside the reproducibility ZIP archive were replaced with the complete text of the Enchan Research & Verification License v1.0. No manuscript, data, code, results, or scientific claims were changed.