Ⅰ. The Problem

Newtonian gravity describes gravitational acceleration through a potential Φ:

For a point mass M, Φ(r) = −GM/r. This description predicts gravity with remarkable accuracy, but it does not by itself specify what gravitational potential represents at a deeper structural level.

Ⅱ. Structural Reinterpretation

Let D_J(x) denote a Structural Admissibility Depth. Rather than treating it as another gravitational potential, CJM places it at a deeper structural level:

A useful possibility is to take D_J as a signed, dimensionless quantity, with a chosen background level D_J = 0. Negative and positive values would describe opposite departures from this reference structure, while attraction or repulsion would depend on the direction of the resulting gradient rather than on sign alone.

In the weak-field regime, the dimensional bridge may be written schematically as

where α is a dimensionless scaling factor. The familiar gravitational potential well may therefore be interpreted as the macroscopic appearance of a deeper Structural Admissibility Well.

Under this interpretation, gravity need not be classified as a fourth fundamental force. The electromagnetic, weak, and strong interactions may remain genuine forces within physical realization, while gravity belongs to a different structural category: the geometric projection of deeper admissibility. In this sense, Nature may have three fundamental forces rather than four.

Newton: gravity as force → Einstein: gravity as geometry → CJM: gravity as the projection of structural admissibility.

Ⅲ. SAT → 3SAT Encoding

Consider a finite region Ω divided into discrete cells. Each cell is assigned one of a finite set of candidate admissibility-depth states. The finite encoding is summarized by six principal constraint families:

1. exactly one admissible local state is assigned to each cell,
2. local matter-energy density constrains the admissibility state of that cell,
3. neighboring cells must satisfy structural compatibility conditions,
4. local admissibility gradients must remain consistent across adjacent regions,
5. boundary conditions constrain the permitted global configuration,
6. all local states must collectively satisfy a single globally admissible gravitational structure.

Each constraint family expands across the relevant cells, states, and relations of the finite instance. For finite resolution and tolerance, these instantiated constraints produce a Boolean formula Φ, which is expressed as SAT and normalized through the common CJM 3SAT gate.

Ⅳ. The CJM Question

If Φ is a macroscopic projection of D_J, gravitational acceleration may correspond to the projected gradient of structural admissibility:

Combining this weak-field correspondence with Poisson's equation also suggests the exploratory relation

This is not proposed as a derived CJM field equation. It asks whether matter density may determine the spatial structure of a deeper admissibility depth whose macroscopic limit appears as gravitational potential.

Ⅴ. Expected Changbal Region

The weak-field relation Φ ≈ αc²D_J need not remain linear at stronger gravity. More generally, the projection may be written as

with F(D_J) ≈ αD_J in the weak-field limit. A useful dimensionless probe is

As χ_G → O(1), the linear correspondence between D_J and Φ may break down — a departure with no counterpart in Newtonian gravity.

CJM therefore identifies the possible breakdown of this linear projection as a candidate Changbal Region. Black-hole horizon formation provides a natural future test case—not because the horizon is assumed to be a Changbal Jump, but because it represents an extreme regime in which gravitational structure becomes qualitatively different.

Ⅵ. Limitation

This sketch does not derive Newtonian gravity or general relativity from CJM, establish the physical existence of D_J, or claim that black-hole horizon formation is a Changbal Jump.

Its purpose is narrower: to introduce a signed, dimensionless Structural Admissibility Depth and ask whether gravitational potential, gravitational acceleration, and the transition from weak to strong gravity may be observable projections of its underlying structure.

References

1. Einstein, A. (1916). “Die Grundlage der allgemeinen Relativitätstheorie.” Annalen der Physik, 49, 769–822.
2. DeWitt, B. S. (1967). “Quantum Theory of Gravity. I. The Canonical Theory.” Physical Review, 160, 1113.
3. Sakharov, A. D. (1968). “Vacuum Quantum Fluctuations in Curved Space and the Theory of Gravitation.” Soviet Physics Doklady, 12, 1040–1041. Original Russian publication, 1967.
4. Jacobson, T. (1995). “Thermodynamics of Spacetime: The Einstein Equation of State.” Physical Review Letters, 75, 1260.
5. Barbour, J. (1994). “The Timelessness of Quantum Gravity: I. The Evidence from the Classical Theory.” Classical and Quantum Gravity, 11, 2853–2873.
6. Barbour, J. (2009). “The Nature of Time.” arXiv:0903.3489 [gr-qc].
7. Verlinde, E. P. (2011). “On the Origin of Gravity and the Laws of Newton.” Journal of High Energy Physics, 2011(4), 029.
8. Wharton, K. (2014). “Quantum States as Ordinary Information.” Information, 5, 190–208.
9. Wharton, K. (2018). “A New Class of Retrocausal Models.” Entropy, 20(6), 410.
10. Wharton, K. B., & Argaman, N. (2020). “Colloquium: Bell’s Theorem and Locally Mediated Reformulations of Quantum Mechanics.” Reviews of Modern Physics, 92, 021002.

Initial Conceptual Note

The following note preserves the broader conceptual direction from which this gravity sketch originally emerged. The present sketch investigates only one narrower component of that program: gravitational potential as a projection of structural-admissibility depth.

Newtonian Dynamics: Matter generates gravitational forces acting across an absolute spatial background, governed by Poisson's equation for the gravitational potential, ∇²Φ = 4πGρ.

Einsteinian Geometrodynamics: Mass-energy and spacetime geometry are dynamically coupled through the Einstein field equations, G_μν + Λg_μν = (8πG / c⁴) T_μν.

Emergent / Thermodynamic Gravity (Sakharov → Jacobson → Verlinde): Gravity or spacetime geometry need not be fundamental, but may arise effectively from deeper quantum, thermodynamic, or informational structures, exemplified in spacetime thermodynamics by δQ = T dS.

Atemporal Structural Admissibility (CJM): Physical geometry, quantum states, and observable dynamics are hypothesized to arise as scale-dependent projections of a deeper atemporal admissibility structure. Rather than treating time as a fundamental axis along which reality must evolve, the CJM framework places structural admissibility within O(J) at the more fundamental level, with temporal ordering emerging only within physically realized configurations. Under this view, microscopic quantum behavior may be interpreted as a local manifestation of admissible phase structure, while macroscopic spacetime geometry and gravitation correspond to its global or coarse-grained organization. Quantum mechanics and general relativity are therefore not assumed to require direct unification at the level of their existing temporal formalisms; instead, both are hypothesized to emerge from a common atemporal structural domain governed through the CJM framework, expressed schematically as P ≡ NP^J.