Postulates – AWS Theory

The Foundational Postulates of AWS

Action-Wave Space (AWS) is a geometric framework in which all quantum, gravitational, and field-theoretic structures emerge from a single real scalar phase field defined on a Lorentzian symplectic manifold. Spacetime, particles, and forces are not fundamental—they are projections of coherent phase structures in action-space. AWS replaces operator-based quantum theory with variational scalar dynamics, topological quantization, and causal projection.

Axiom 1: Physics Emerges from a Lorentzian Symplectic Manifold of Actions

All physical dynamics are encoded in a four-dimensional manifold 𝕍_AWS with local coordinates S^μ representing generalized action (not position). The manifold is equipped with a Lorentzian metric and a symplectic 2-form derived from a real scalar phase field Φ(S):

\(g_{\mu\nu}(S), \quad \Omega_{\mu\nu}(S) = 2 \nabla_{[\mu} \partial_{\nu]} \Phi = \partial_\mu \partial_\nu \Phi – \partial_\nu \partial_\mu \Phi\)

This scalar field defines both geometry and dynamics. There is no background spacetime: physical evolution occurs within 𝕍_AWS.

Axiom 2: Scalar Dynamics Follow from a Variational Principle

The scalar field obeys a covariant wave equation derived from an action principle. The fundamental action is:

\(\mathcal{S}[\Phi, g] = \int_{\mathcal{M}_{\text{AWS}}} \frac{1}{2} g^{\mu\nu} \partial_\mu \Phi \, \partial_\nu \Phi \, \sqrt{-g} \, d^4S\)

Variation yields the scalar wave equation:

\(\nabla^\mu \nabla_\mu \Phi = 0\)

This ensures local energy conservation and defines the stress-energy tensor:

\(T_{\mu\nu}^{(\Phi)} = \partial_\mu \Phi \, \partial_\nu \Phi – \frac{1}{2} g_{\mu\nu} \, \partial^\alpha \Phi \, \partial_\alpha \Phi\)

Axiom 3: Quantization from Topological Phase Closure

Quantization is not imposed via operator algebra. It arises from global phase coherence along closed geodesics in action-space. Physically admissible scalar configurations must obey:

\(\oint_{\gamma} \partial_\mu \Phi \, dS^\mu = 2\pi n\hbar, \quad n \in \mathbb{Z}\)

This topological phase quantization replaces canonical commutators. Only globally coherent wave cycles survive projection.

Axiom 4: Observable Physics Emerges via Causal Scalar Projection

Spacetime and all observables are projections from compact domains D_x = π^-1(x) ⊂ 𝕍_AWS. The scalar field projects onto a local spacetime amplitude:

\(\psi(x) = \int_{D_x} G_R(x, S) \, A(S) e^{i \Phi(S)/\hbar} \, \sqrt{-g} \, d^4S\)

The kernel G_R(x, S) enforces causality, curvature regularity, and phase alignment. Only domains with topology (χ, b₁, b₂) = (2,1,3) yield UV-finite, interference-stable projections. These scalartons define the minimal units of coherent physical reality.

Axiom 5: Spacetime Coordinates Emerge from Phase Centers

Spacetime labels x^μ are not fundamental. They emerge as phase-weighted centers of projection domains:

\(x^\mu = \frac{\int_{D_x} S^\mu \, |G_R \Psi|^2 \sqrt{-g} \, d^4S}{\int_{D_x} |G_R \Psi|^2 \sqrt{-g} \, d^4S}\)

This recovers classical spacetime as a mean-field label over underlying scalar phase interference.