Foundation

Action Primacy: Only Action and Its Flux

Classically, we describe particles moving in spacetime and then define the action \(S\) along their paths. AWS inverts this order. It starts from a single axiom:

\(\text{At the deepest level, there is only Action, and its flux is conserved.}\)

This principle is called Action primacy. From it, AWS builds a precise geometric structure in which everything—spacetime, quantum amplitudes, probabilities—arises from a single real phase field and its conserved flux.

The Fundamental AWS Layer: \((M, g, \Omega, \Phi)\)

Mathematically, AWS lives on a four-dimensional manifold equipped with three linked objects:

\( (M, g, \Omega, \Phi) \)
  • \(\Phi\): a single real scalar phase field (the Action phase);
  • \(\Omega\): a closed two-form encoding conserved Action flux;
  • \(g\): a Lorentzian metric whose light cones are aligned with the phase gradient.

The eikonal one-form \(\kappa = d\Phi\) acts as a generalized momentum–energy covector and is null with respect to the causal class of metrics \([g]\). The flux \(\Omega\) is closed and quantized, so its integrals over closed two-surfaces come in integer multiples of \(2\pi\hbar\).

\( d\Omega = 0, \qquad \frac{1}{2\pi\hbar}[\Omega] \in H^2(M;\mathbb{Z}). \)

AWS is therefore a Lorentzian–symplectic theory: the same phase field \(\Phi\) determines both the causal geometry (via \(g\)) and the symplectic structure (via \(\Omega\)).

Hamilton–Jacobi Compression: 8D Phase Space → 4D Action–Wave Space

Ordinary relativistic mechanics uses the 8D phase space \(T^*M_4\) with coordinates \((x^\mu, p_\mu)\). AWS shows that once Action flux is conserved and quantized, this 8D space collapses to a 4D layer defined entirely by the phase.

The canonical one-form \(\theta = p_\mu\,dx^\mu\) aligns with the phase connection \(\alpha = \hbar^{-1} d\Phi\), which forces an exact Hamilton–Jacobi structure:

\( \theta = dS = \hbar\,d\Phi, \qquad p_\mu = \partial_\mu S. \)

The image \(L = \{(x, \partial S(x))\} \subset T^*M_4\) is a four-dimensional Lagrangian submanifold: the Action–Wave Space layer. Momentum is no longer an independent variable; it is identically the phase gradient. AWS calls this phase–space compression.

Classical Wave Dynamics Instead of Fundamental Operators

On the AWS layer, the phase field \(\Phi\) evolves by a local, diffeomorphism-invariant, shift-symmetric action \(S_{\mathrm{AWS}}[\Phi; g, \Omega]\). Shift symmetry (\(\Phi \mapsto \Phi + \text{const}\)) ensures that only derivatives of \(\Phi\) appear, and the Euler–Lagrange equation is a normally hyperbolic wave equation:

\( E(\Phi; g, \Omega) = 0, \qquad \text{principal part } \propto g^{\mu\nu}\partial_\mu\partial_\nu \Phi. \)

All fundamental dynamics in AWS is therefore classical wave propagation of a single scalar phase. Quantum-looking features—interference, probabilities, “collapse”—do not come from postulated operators. They emerge from the combination of phase–space compression, flux quantization, and a causal projection to spacetime.

Causal Projection: How Spacetime and the Born Rule Emerge

Observable physics takes place on an operational spacetime \(M_4\). AWS relates the deeper Action–Wave layer to spacetime via a smooth projection

\( \pi : M \longrightarrow M_4, \qquad D_x := \pi^{-1}(x), \)

where \(D_x\) is a compact internal fiber attached to each spacetime point \(x\). The spacetime amplitude \(\psi(x)\) is a retarded push-forward of the AWS phase field:

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

Here \(G_R(x; S)\) is the AWS retarded kernel (a causal Green function) and \(A(S)\) is a slowly varying weight. With a canonical, flux-preserving normalization, the associated Gram operator \(K = \Pi^\dagger \Pi\) is an approximate identity, so the conserved phase–flux norm on AWS pushes forward to the usual Born probabilities and POVM measurement operators in spacetime. “Collapse” is just conditioning on an outcome after projection; the underlying phase field evolves smoothly throughout.

Compact Fibers and Topological Sectors

The internal domains \(D_x\) are compact and have fixed topology, selected by a scale-invariant geometric variational principle. The minimal stable sector compatible with finite curvature energy and flux quantization has Betti data

\( (\chi, b_1, b_2) = (2, 1, 3), \quad (b_3 = 1 \text{ in the boundary case}). \)

This means each fiber carries a finite set of harmonic modes of the Action flux. Those discrete, interference-stable modes project to spacetime as distinct particle and field sectors, and they provide a natural geometric ultraviolet scale.

AWS in One Sentence

AWS posits only a scalar phase field \(\Phi\) and its conserved Action flux \(\Omega\) on a 4D Lorentzian–symplectic manifold; everything we see in spacetime—fields, particles, probabilities, and collapse—is the causal projection of this underlying classical phase geometry.

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