Causal Spectral Relational Gravity - Open Research Project

Modern theoretical physics still lacks a fully consistent description of spacetime that unifies quantum theory and gravity. Many programs attempt to quantize spacetime itself, but an alternative possibility is that spacetime is not fundamental at all. On that view, geometry, time and gravitational dynamics do not sit at the bottom of reality. They emerge from a deeper relational substrate whose observable large-scale behavior only later resembles a manifold with metric and causal structure. The purpose of this open research program is to explore and develop a spectral–relational framework in which spacetime geometry, causal structure, horizons and gravitational dynamics emerge from the spectral properties of a directed weighted relational network. The ambition is deliberately disciplined. This is not presented as a finished theory, but as a structured research program whose central task is to construct a mathematically well-defined framework that can be studied analytically, computationally and, where possible, through analog physical systems. The motivation for such a program is twofold. First, several open problems in quantum gravity suggest that manifold-based concepts may themselves be emergent rather than primary. Second, relational and network-based descriptions have repeatedly proved useful across physics, mathematics and complex systems. The present project therefore asks whether a sufficiently controlled operator-theoretic network model can generate not just correlation patterns, but geometry, causal order, horizon behavior and effective gravitational dynamics. 2. Core Hypothesis of the Project The central hypothesis of this research program is that spacetime geometry, causal order and gravitational dynamics are emergent phenomena generated by the spectral structure of a relational network operator. Instead of starting from a metric manifold, the fundamental structure is a relational network R = (V, E, W) where V denotes elementary events or relational states, E denotes directed relations, and Wij denotes signed interaction weights. In this picture the network is not embedded into a pre-existing space. It is the pre-geometric object from which effective spacetime notions may arise only after suitable dynamical organization and coarse-graining. The fundamental dynamics is encoded in a master operator Ω = H + K†K + m*² I where H is the geometric symmetric sector, K is the causal or directed interaction operator, and m*² > 0 is a genuine spectral-gap parameter. This form is important because it moves invertibility and stability into the axiomatic core of the model. Rather than repairing singularity problems later with ad hoc pseudo-inverses or arbitrary regularizers, the framework builds spectral well-posedness into the definition of the theory itself. This is one of the main strategic shifts of the present program. The claim is not merely that one can write down a relational operator. The stronger claim is that only a carefully constrained operator class can provide a serious basis for emergent geometry and gravity. The project therefore treats well-posedness not as a technical appendix, but as a foundational principle. 3. Fundamental Propagator The central object of the theory is the Green kernel G = Ω⁻¹ This kernel encodes propagation on the relational structure. In the program proposed here, all emergent structures are derived not directly from adjacency data alone, but from the propagation properties encoded in G. The Green kernel is therefore not a secondary mathematical convenience. It is the main bridge between the microscopic relational substrate and the macroscopic structures that may later be interpreted geometrically or causally. From the Green kernel we define a propagation proximity ρ(i,j) = − log ( |Gᵢⱼ| / √(Gᵢᵢ Gⱼⱼ) ) This quantity describes relational influence, not yet a physical distance. That distinction is essential. One of the weaknesses of looser formulations is that they move too quickly from propagation data to metric claims. The present program avoids that shortcut. At the microscopic level, the Green kernel supplies a relational closeness structure. Only later, after controlled coarse-graining, do we ask whether an actual metric can emerge. 4. Emergence of Geometry The project hypothesizes that spatial geometry emerges only after coarse-graining. Microscopic relations are not metric. The micro-level object is the propagation proximity ρ(i,j), which measures effective communicability or influence in the relational substrate. It does not, by itself, guarantee symmetry, triangle inequality or geodesic interpretation in the classical sense. At the macro level, after coarse-graining, we define an effective distance deff(X,Y) = infγ:X→Y Σe∈γ ℓe where edge lengths are functions of local Green-kernel structure. In this way the model clearly separates two distinct notions. First there is microscopic propagation structure. Second there is macroscopic metric structure obtained from block variables, local effective lengths and path minimization. This separation is conceptually and mathematically important because it makes the theory more honest: there is no full spacetime geometry at the fundamental level, only relational propagation organization. If successful, this construction would allow the emergence of metric structure, geodesic distance and effective dimensionality. One major research target is to determine under what graph sequences, scaling rules and block constructions the coarse-grained distance converges to a geodesic metric in a continuum limit, and when this effective metric is better interpreted as Riemannian or Lorentzian. 5. Emergence of Time Time is not assumed as a fundamental coordinate. Instead, it emerges from the irreversible ordering of relational dynamics. The idea is that a genuine arrow of time should not be inserted by hand through an external background parameter if the theory is truly pre-geometric. It should arise as a stable organizing phase of the system itself. To express this idea, we introduce a Lyapunov functional F[R,T] = S[R,T] + κ Ent(pG) dF / dλ ≤ 0 where pG is a propagation distribution derived from the Green kernel. The working hypothesis is that the system evolves toward states that stabilize causal coherence while simultaneously reflecting irreversible spectral organization. In this picture, low-coherence phases correspond to time-symmetric, plastic or frustrated regimes, while high-coherence phases correspond to emergent causal order. This part of the program aims to shift the theory of time away from a mere ordering label and toward a genuinely dynamical explanation. Time would then become a stable macroscopic ordering phase rather than a primitive background dimension. One of the central tasks is to determine whether this phase picture can be made rigorous and whether it produces clear mathematical or computational signatures. 6. Emergence of Horizons Horizons are interpreted here as weakly coupled spectral sectors rather than as perfectly disconnected domains. Instead of a complete separation we consider an approximately block-structured operator Ω ≈ [ Ωout εB ; εB† Ωin ], ε ≪ 1 This means that coherence between regions collapses strongly, yet coupling never vanishes completely. This interpretation is important because it avoids turning the notion of horizon into a sterile absolute cut. Instead, a horizon becomes a critical interface: dominant propagation channels fail, sector coherence breaks down, but tiny non-zero leakage remains. Such a structure may generate analogues of redshift, horizon effects, Hawking-like leakage and decoherence layers. It also creates a more physically suggestive platform for studying information flow across near-decoupled sectors. The research question is not simply whether such interfaces can be defined, but under what spectral conditions they arise naturally and how robust they remain under coarse-graining and perturbation. 7. Emergent Relativistic Structure A major goal of the project is to derive Lorentz structure instead of assuming it. This is one of the sharpest standards by which the framework should be judged. If relativistic signature has to be inserted manually at the end, the theory remains vulnerable to the criticism that relativity was merely pasted onto a non-relativistic substrate. The working hypothesis is that the symmetric sector supports spatial propagation, while the antisymmetric or directed causal sector produces retarded dynamics. In the continuum limit, the effective signature of spacetime should emerge from the pole structure of the propagator rather than from an externally imposed metric ansatz. In the infrared regime the effective dynamics is expected to approach an action of the schematic form Seff[g,τ] = ∫ dᵈx √|g| ( c₀ + c₁R + c₂(∇τ)² + ... ) The project does not prematurely claim that the theory already reproduces general relativity. The disciplined target is weaker but more credible: to determine whether Einstein-type dynamics can emerge as an effective large-scale limit. That means the real benchmark is not rhetorical similarity, but the controlled appearance of curvature terms, effective field equations and Lorentzian behavior from the spectral network substrate.