Resolution, Phase Coupling, and Gravitational Closure

This paper is a major theoretical reconstruction of Resolution, Not Complexity: The Mandelbrot Set as a Minimal Demonstration of Structural Visibility Limits. It relocates the original argument into the Observer Theory branch and develops its conditional connection to phase-based gravity. The Mandelbrot set is used as a controlled mathematical example showing that a finite readout can be non-injective: distinct underlying structures may produce the same finite record, and the boundary of observability need not be the boundary of existence. This mathematical result is not presented as evidence that physical reality is fractal, that the Planck length is a spacetime pixel, or that unresolved physical structure necessarily exists. The paper distinguishes source organisation from accessibility, registration and reconstruction. Established gravity-to-phase results—including neutron and atom interferometry—are kept separate from the proposed reverse direction. Under the explicit hypothesis that gauge-invariant relational phase organisation contributes to gravitational geometry, the coupling required for observation may alter the joint observer–target phase organisation and generate gravitational anchoring. At a derived closure condition, observer and target would cease to remain separately registrable. The Planck regime is therefore proposed conditionally as a possible operational phase-separability boundary, not asserted as an experimentally established minimum length. A five-stage research programme is defined: certified finite-readout collisions in the Mandelbrot system; phase-selective detector accessibility; reproduction of the established gravity-to-phase relation; a matched-source phase-to-gravity discriminator; and derivation of a device-independent closure scale without inserting the Planck length by hand. The phase-to-gravity experiment remains formally NO-GO until the theory supplies a measurable gauge-invariant phase quantity, a covariant and conserved source law, two states with matched complete conventional source predictions, and a fixed non-zero prediction with specified sign, waveform, scaling and minimum effect size. This work presents a bounded, falsifiable theoretical architecture and proposed tests. It reports no new empirical detection of phase-produced gravity or a Planck-scale observability limit.