A Minimal Compatibility Bridge Between General Relativity and Quantum Phase Dynamics via a Physical Time-Delay Field: Effective Gravity Formulation for SPARC-Style Tests

This manuscript develops a minimal compatibility formulation of the Time Delay Field (TDF) framework. The central purpose is not to replace General Relativity or quantum mechanics, but to define a disciplined bridge between Einstein geometry and quantum phase dynamics through a physical delay-time field (\tau(x,t)).

The paper adopts the conservative TDF positioning established in the canonical scope paper, What the Time Delay Field Framework Is—and Is Not: Scope, Limits, and Scientific Positioning (Zenodo record 20466760). In this formulation, Einstein geometry is preserved as the tested geometric arena, while a (\tau)-sector is introduced as an effective or deeper source contribution to the stress-energy sector.

A key clarification introduced in this manuscript is dimensional consistency. The field (\tau) is treated as a physical delay-time variable with units of seconds. Quantum phase is therefore not identified directly with (\tau), but with the dimensionless projection
[
\theta(x,t)=\Omega_\tau \tau(x,t).
]
This allows delay-time differences (\Delta\tau) to generate quantum phase differences while preserving the standard phase structure of quantum mechanics.

The manuscript also separates two constants that were previously sometimes conflated: (K_g), the gravitational projection coefficient converting (\nabla\tau) into effective acceleration, and (\kappa_\tau), the dynamical stiffness of the (\tau)-field appearing in the mother field equation
[
\kappa_\tau \Box\tau - V'(\tau)=J_\tau.
]

A SPARC-style effective-gravity reconstruction is then formulated through
[
a_{\rm miss}(r)=K_g\frac{d\tau}{dr},
]
providing a clear operational route from rotation-curve residuals to a radial delay-time gradient. The paper emphasizes that rotation-curve reconstruction alone is not sufficient to establish TDF as a physical alternative to dark matter. The stronger falsifiability requirement is to freeze the reconstructed (\tau)-map and use it to predict an independent observable such as lensing or deflection without retuning.

The manuscript also discusses particle-like structures as possible stable phase-locked (\tau)-configurations, but explicitly classifies particle emergence as a Level-C foundational hypothesis rather than an established empirical result. The paper does not claim that dark matter is disproven, that (\Lambda)CDM is replaced, or that quantum gravity has been completed.

Main contributions

• Defines (\tau(x,t)) as a physical delay-time field with units of seconds.

• Introduces the dimensionless quantum phase projection (\theta=\Omega_\tau\tau).

• Separates (K_g) from (\kappa_\tau) to avoid ambiguity between gravitational projection and field stiffness.

• Formulates effective TDF gravity for SPARC-style rotation-curve tests.

• Clarifies how matter may source a surrounding delay-time field that contributes to effective gravity.

• Establishes a claim hierarchy distinguishing testable galaxy-scale reconstruction from speculative particle emergence.

• Uses the TDF scope-and-positioning paper as the canonical reference for claim boundaries and conservative scientific framing.

Keywords
Time Delay Field; TDF; General Relativity; quantum phase; effective gravity; delay-time field; SPARC; missing acceleration; emergent gravity; decoherence; phase-time geometry; dark matter phenomenology; Einstein geometry; tau field.

Version note
This version introduces a revised notation in which (K_g) denotes the gravitational projection coefficient and (\kappa_\tau) denotes the dynamical stiffness of the (\tau)-field. It also corrects the quantum phase representation by using (\theta=\Omega_\tau\tau) rather than treating (\tau) itself as a dimensionless phase.