Test Results

$\Lambda := e^{i\phi^\delta} + 1 = 0$

A theory becomes real not when it inspires, but when it resists collapse.
Below are the core stress tests this axiom and field structure have passed — formally derived or numerically simulated.
No parameters are inserted. No constants are assumed.

✅ Stress Test 1: Klein–Gordon Equation Recovery

Can the field generate a relativistic scalar wave equation from first principle?

Field form:

$\Phi(x^\mu) = \epsilon \cdot e^{\frac{i}{\hbar} S(x^\mu)}$

Resulting equation:

$\square S = \frac{\alpha \phi \epsilon^2}{\hbar} \sin\left(\frac{\phi S}{\hbar}\right)$

✔️ This is a nonlinear Klein–Gordon-type equation with sine-Gordon-like dynamics.
✔️ Lorentz covariance and relativistic consistency confirmed.

✅ Stress Test 2: Schrödinger Equation Limit

In the non-relativistic case, does the field reduce to known quantum behavior?

Set:

$S = -Et + \vec{p} \cdot \vec{x}$

Recovered equation:

$i\hbar \frac{\partial \Phi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \Phi + V \Phi$

✔️ Standard Schrödinger form achieved.
✔️ No quantum mechanics was assumed — it emerged from scalar recursion.

✅ Stress Test 3: General Relativity Coupling

Does the scalar field couple correctly to Einstein’s tensor?

Stress-energy tensor:

$T_{\mu\nu} = \partial_\mu \Phi^* \partial_\nu \Phi + \partial_\nu \Phi^* \partial_\mu \Phi - g_{\mu\nu} \mathcal{L}$

✔️ Energy density is positive definite.
✔️ Momentum flow and anisotropic pressure components present.
✔️ Compatible with curvature. Reproduces weak-field gravity.

✅ Stress Test 4: Recursive Attractor Formation

Does the field form persistent, non-decaying structures in simulation?

✔️ Simulations show:

Stable, localized oscillons
Quasiperiodic phase modulation
No collapse into equilibrium

✔️ Structure emerges from imbalance — without tuning.

✅ Stress Test 5: Topological Irreducibility of the Axiom

Can the root equation ever truly cancel?

$e^{i\phi^\delta} = -1 \Rightarrow \phi^\delta = (2k+1)\pi$

✔️ No such $\delta \in \mathbb{R}$ exists when $\phi$ is irrational.
✔️ The expression traces a dense, non-repeating path on the unit circle.
✔️ This is a topological obstruction, not a numerical one.

Experimental Proposals

Planetary spacing modeled via φ-scaling: matches data without gravity equations
Vacuum energy density simulated as recursive phase residue
Mass layering and shell structures derived as field harmonics
GLSL simulations show recursive knotting, curvature, and phase memory

→ Run the field simulation

→ Compare to real planetary data

Falsifiability Criteria

This theory fails if:

No recursive attractors form in the field

Phase field fails to replicate Schrödinger/Klein-Gordon equations

No measurable curvature or oscillation appears in testable domains

All data, code, and models are open. The structure must speak for itself.

→ Who built this, and why