A Coherence-Based Framework for Emergent Spacetime, Matter, and Cosmological Dynamics

This work presents a speculative but internally consistent conceptual framework in which spacetime, matter, and gravitational dynamics are interpreted as emergent manifestations of a single underlying coherence structure. A scalar mismatch field ΔΦ is introduced, whose nonlinear dynamics admits localized configurations interpreted as particle-like excitations, while its associated energy defines effective mass and sources curvature through an Einstein-type relation. At the cosmological level, the framework extends through a global geometric coupling between complementary descriptions, providing an alternative interpretation of the cosmological constant as an emergent effect rather than a fundamental parameter. The model remains compatible with standard ΛCDM cosmology at the observational level, while suggesting small, potentially testable deviations, including weak environment-dependent variations in inferred expansion rates and subtle correlations with large-scale structure. The work is presented in a progressive “notebook-style” format, documenting its development from initial conceptual hypotheses to semi-formal mathematical structure. This framework is not proposed as a predictive theory, but as a coherent reinterpretation of spacetime, matter, and cosmology based on global consistency conditions. Previous versions of this work were presented under a different title emphasizing the initial mirror-geometry interpretation. (Speculative Model of Coupled Mirror Universes: A Conceptual Framework) Comments and feedback are welcome Contact: tarek. notebook@gmail. com Version 13 — Incremental Structural Consolidation This version represents a significant consolidation of the coherence-based framework, with a focus on dynamical consistency, emergent geometry, and minimal testability. Key Additions Propagation of coherence perturbations (Appendix 39) Formalization of perturbative dynamics leading to wave-like behavior and causal structure. Effective metric and emergent geometry (Appendix 40) Introduction of a geometry derived from propagation properties of the coherence field. Action principle formulation (Appendix 41) Definition of a minimal action and derivation of the associated field equations. Connection to Einstein-like structure (Appendix 41. 9–41. 15) Conservative identification of a structural correspondence with Einstein field equations. Observables and qualitative testability (Appendix 42) Identification of measurable quantities and minimal falsifiability criteria. Constraints and parameter space (Appendix 43) Introduction of consistency conditions, stability requirements, and parameter limitations. Minimal predictions (Appendix 44) Definition of qualitative, structurally testable predictions. Conceptual Refinements Clarification of coherence as a dynamical and propagating quantity Interpretation of geometry as emergent from propagation constraints Introduction of multiscale coherence behavior Identification of extreme regimes as future exploration domains Visual Support Addition of three conceptual infographics: Core framework Coherence propagation Emergent geometry Positioning This version does not claim derivation of established physical theories, but establishes a structurally consistent framework compatible with known dynamical and geometric principles. Status Exploratory but internally coherent. Prepared for future refinement, quantitative development, and structural validation. Version 12 — Discrete Dynamics, Numerical Exploration, and Coherence Visualization Version 12 introduces the first explicit discrete realization of the coherence-based framework, together with numerical exploration and visual representation of its dynamics. This update marks a transition from purely conceptual development toward computationally grounded interpretation. Key developments Discrete coherence dynamics (Appendix 35) A graph-based formulation is introduced, where configurations are represented as pairs of states across underlying structures. A mismatch functional defines coherence, and system evolution is modeled through stochastic transitions favoring reduced mismatch. Numerical exploration and emergent behaviorA minimal computational implementation demonstrates that the system evolves toward coherent configurations while remaining dynamically active. Coherence emerges as an attractor region rather than a fixed state. Visualization of coherence landscapesThe mismatch structure is represented as a landscape, with a clear diagonal coherence region. System trajectories illustrate convergence toward this region and continued exploration within it. Emergent state selection (Appendix 36) The framework distinguishes between possible coherent configurations and dynamically realized states. The realized state depends on accessibility, stability, and dynamical history. Discrete–continuous correspondence (Appendix 37) A conceptual bridge is established between the discrete formulation and the underlying continuous coherence field. The discrete model is interpreted as a coarse-grained realization of continuous dynamics. Conceptual outcome Version 12 supports a unified interpretation in which: coherence defines a space of admissible configurations, dynamics explores and selects within this space, and the realized state reflects accessibility rather than global optimality. Positioning This version represents the first integration of: discrete dynamics, numerical exploration, and visual representation within a coherent and internally consistent framework. Status The framework now includes: geometric construction, field-theoretic interpretation, dynamical formulation, discrete realization, numerical illustration, and visual synthesis. It remains exploratory, but increasingly structured and computationally grounded. Version 11 — Structural Integration and Coherence Field Consolidation Version 11 represents the most structurally integrated stage of the framework developed so far, consolidating the geometric, field-theoretic, and dynamical aspects into a unified formulation. This update introduces a coherent connection between previously developed components, resulting in a self-consistent framework in which spacetime, matter, and dynamics emerge from a single underlying coherence structure. Key developments in Version 11 Appendix 23–25 consolidationThe geometric construction of spacetime as an intersection of higher-dimensional structures is now consistently connected to the definition of the coherence mismatch field ΔΦ ΔΦ and its interpretation as the fundamental degree of freedom. Matter is formalized as localized, finite-energy configurations (solitons) of this field. Emergent causal structure (Appendix 28) The propagation of perturbations of ΔΦ ΔΦ defines an emergent causal structure, including an effective light cone. The framework allows for exact or approximate correspondence with relativistic causality. Particle motion and geodesics (Appendix 29) Localized field configurations are shown to follow effective trajectories that can be interpreted as geodesics of an emergent metric, connecting field dynamics with geometric motion. Energy–momentum and backreaction (Appendix 30) A consistent coupling between the coherence field and spacetime geometry is established. The framework satisfies conservation laws and forms a closed dynamical system, avoiding problematic circularity. Minimal action formulation (Appendix 31) The entire structure is expressed through a unified variational principle, from which field equations, stress–energy tensor, and geometric dynamics are derived. Connection to observables (Appendix 32) A first mapping between the coherence field and effective cosmological observables is introduced, together with order-of-magnitude estimates and constraints. Critical assessment (Appendix 33) The epistemic status, limitations, and scope of the framework are explicitly defined, emphasizing its exploratory and semi-formal nature. Outlook (Appendix 34) Future directions are outlined, including quantitative refinement, observational testing, and deeper theoretical development. Conceptual outcome Version 11 establishes a unified picture in which: spacetime emerges from geometric intersection, matter arises from localized coherence mismatch, gravity corresponds to the geometric response to this mismatch, and causal structure emerges from field propagation. Status of the framework The model remains: exploratory and semi-formal, internally consistent, compatible with established physical theories in known regimes, and open to further refinement and empirical investigation. Positioning This version should be understood as a structured consolidation of the framework rather than a final theory, providing a coherent foundation for future work. Version 10 — Coherence Field Formalization and Emergent Structure Version 10 introduces a consolidated formulation of the framework in which spacetime, matter, and gravitational dynamics are interpreted as emergent manifestations of a single underlying coherence structure. This update formalizes the coherence mismatch field ΔΦ ΔΦ as the central degree of freedom of the model, defined as the difference between complementary underlying descriptions. Its nonlinear dynamics is shown to admit localized, finite-energy configurations, interpreted as particle-like excitations, establishing a direct connection between field structure and effective matter content. The associated energy functional is explicitly developed, allowing mass to be interpreted as the energy of localized coherence structures. A scaling relation between mismatch amplitude and coupling strength is introduced, providing a minimal link between field parameters and effective physical quantities. The framework is further extended