Qaether Theory(v0.6): A Topologically Protected Information Lattice Framework for Unified Physics

Abstract

Qaether Theory models space–time as a Planck‑scale face‑centred‑cubic (FCC) lattice whose nodes host Qaethers: discrete units that carry an activation bit, an intrinsic orientation, and a continuous U(1) phase. Bonds form only when a three‑fold criterion—activation, lattice directionality, and a \(\mathbb{Z}_6\) phase‑quantisation rule—is satisfied. Missing bonds create voids whose volume defect sources scalar curvature, endowing the lattice with emergent gravitation. Time is defined relationally as the cumulative phase variance of the network. From a microscopic Hamiltonian we derive phase‑oscillator dynamics, normal‑mode spectra, and gauge‑covariant actions for both U(1) and SU(3). Phase locking yields Higgs‑like and gluonic mass matrices, while Jacobson’s thermodynamic argument converts an information stress tensor into Einstein’s equations, completing a gauge–gravity–information duality. A unified conservation law links mass density, curvature, and Shannon entropy, predicting testable correlations in strong‑curvature regimes. We close with a roadmap for resolving outstanding challenges, including chiral fermions, Lorentz restoration, and phenomenology.

 

1 Foundational Postulates

1.1 Discrete Manifold

Spacetime is an FCC lattice

$$\mathcal{L}_{\text{FCC}} = \ell_{p}\,\mathbb{Z}^{3}_{\text{FCC}}$$

characterised by twelve primitive directions

$$D_{\text{FCC}} = \{\vec d_{k}\}_{k=1}^{12}$$

No continuum substrate is assumed.

1.2 Qaether State Vector

At each lattice site i the state

$$\Xi_{i}=(S_{i},\vec Z_{i},\phi_{i})$$

contains an activation bit \(S_{i}\in\{0,1\}\), orientation \(\vec Z_{i}\in\mathbb{S}^{2}\), and phase \(\phi_{i}\in[0,2\pi)\).

1.3 Topological Bond Rule

A bond exists iff

$$A_{ij}=1\;\Longleftrightarrow\; S_{i}=S_{j}=1,\quad \frac{\vec r_{ij}}{\ell_{p}}\in D_{\text{FCC}},\quad \Delta\phi_{ij}\in\mathbb{Z}_{6}\,\frac{\pi}{3}$$

1.4 Relational Time

Define

$$\boxed{\tau=\int\!\Bigl[\tfrac{1}{N}\sum_{(i,j)}\Bigl(\tfrac{\dot\phi_{ij}}{\pi/3}\Bigr)^{2}\Bigr]^{1/2}\,d\tau'}$$

where \(\dot\phi_{ij}\)  denotes the phase‑difference velocity. The definition is scale‑invariant.

1.5 Void–Curvature Correspondence

For a cell with \(m_{c}\) bonds

$$\Delta V(m_{c}) = \alpha\,\ell_{p}^{3}\Bigl(1-\tfrac{m_{c}}{12}\Bigr)^{k},\qquad R(x)=R_{0}+\alpha_{1}\rho_{v}+\alpha_{2}\rho_{v}^{2}$$

1.6 Microscopic Hamiltonian

$$\mathcal{H} = \epsilon_{z}\sum_{\langle i,j\rangle}A_{ij}\bigl(1-f_{ij}\bigr) + \epsilon_{\phi}\sum_{\langle i,j\rangle}A_{ij}\bigl[1-\cos 6\Delta\phi_{ij}\bigr] + \kappa_{v}\sum_{c}\bigl[\Delta V(m_{c})\bigr]^{2}$$

with alignment factor

$$f_{ij}=|\vec Z_{i}\!\cdot\!\vec d_{ij}|\,|\vec Z_{j}\!\cdot\!\vec d_{ji}|$$

 

2 Phase–Oscillator Field Theory

2.1 Non‑linear Dynamics

The Euler–Lagrange equation for the phase field is

$$\frac{d^{2}\phi_{i}}{d\tau^{2}} = 6\epsilon_{\phi}\sum_{j}A_{ij}\sin\bigl[6\,(\phi_{j}-\phi_{i})\bigr]$$

Linearisation about a locked configuration yields a discrete Laplacian spectrum $$\omega_{n}^{2}=36\epsilon_{\phi}\lambda_{n}$$

2.2 Complex Envelope Field

With \(\psi_{i}=A_{i}e^{i\phi_{i}}\) , quadratic fluctuations obey

$$(\partial_{\tau}^{2}+m_{H}^{2})\psi_{i} = -c_{\phi}^{2}\sum_{j}L_{ij}\psi_{j},\qquad m_{H}^{2}=36\epsilon_{\phi}\lambda_{1}$$

2.3 Information Entropy

For probabilities \(p_{\alpha}\,(\alpha=0\dots5)\)

$$S = -\sum_{\alpha}p_{\alpha}\ln p_{\alpha},\qquad \partial_{\tau}S = \alpha\Bigl(1-\frac{S}{\ln 6}\Bigr)$$

The bound S≤ln⁡6S\le\ln 6 establishes a finite‑state arrow of time.

 

3 Gauge‑Covariant Embedding

3.1 U(1)U(1) Electrodynamics

Localising the phase symmetry introduces a gauge field \(A_{\mu}\) and action

$$S_{U(1)}=\int d^{4}x\sqrt{-g}\Bigl[|D_{\mu}\psi|^{2}-\mu^{2}|\psi|^{2}-\tfrac{\lambda}{2}|\psi|^{4}+\tfrac{1}{4}F_{\mu\nu}F^{\mu\nu}\Bigr]$$

A vacuum expectation value \(\langle\psi\rangle=v\) yields \(m_{A}=gv\) and \(m_{H}=\sqrt{2\lambda}v\).

3.2 SU(3) Colour Sector

For \(\psi\in\mathbb{C}^{3}\) and gluons \(A_{\mu}^{a}\)

$$S_{SU(3)}=\int\!\sqrt{-g}\Bigl[(D_{\mu}\psi)^{\dagger}(D^{\mu}\psi)-V(\bar\psi\psi)+\tfrac{1}{4}F_{\mu\nu}^{a}F^{a\mu\nu}\Bigr]$$

Choosing \(\langle\psi\rangle=(v,0,0)^{T}\) breaks \(SU(3)\to SU(2)\), giving six degenerate gluon masses \(g^{2}v^{2}/4\) and leaving two gauge bosons massless.

 

4 Information‑Driven Gravitation

4.1 Information Stress Tensor

$$T_{\mu\nu}^{\text{info}} = \partial_{\mu}\phi\,\partial_{\nu}\phi - \tfrac{1}{2}g_{\mu\nu}(\partial\phi)^{2}$$

Applying Jacobson’s \(\delta Q = T\,\delta S\) to local Rindler horizons reproduces

$$R_{\mu\nu}-\tfrac{1}{2}g_{\mu\nu}R = 8\pi G\,T_{\mu\nu}^{\text{info}}$$

4.2 Mass–Curvature–Entropy Conservation

$$\frac{d}{d\tau}\Bigl[\int \rho_{m}\,d^{3}x + \gamma\int R\,d^{3}x\Bigr] = -T\,\partial_{\tau}S,\qquad \rho_{m}=\beta|\nabla\phi|^{2}$$

An increasing entropy therefore mandates a compensatory decrease in either mass density or curvature.

 

5 Phenomenological Windows

• Early‑Universe Mass Drift High curvature during inflation should transiently suppress Higgs and gauge‑boson masses, leaving signatures in primordial spectra.
• Strong‑Gravity Colliders Near black‑hole horizons the mass–curvature coupling may shift QCD scales, influencing jet‑quenching patterns.
• Quantum‑Information Analogues Entanglement entropy of phase sectors maps onto curvature, enabling superconducting‑qubit emulations on engineered FCC graphs.

 

6 Open Challenges

1. Chiral Fermions Embed staggered or domain‑wall fermions without violating the \(\mathbb{Z}_6\) bond rule.
2. Lorentz Restoration Prove that lattice artefacts decouple below \(\ell_{p}^{-1}\) under renormalisation‑group flow.
3. Gauge‑Invariant Entropy Formulate a local information density invariant under U(1) and SU(3) rotations.
4. Scalable Numerics Compute >\(10^{15}\)‑node configurations while preserving non‑abelian holonomies.
5. Observational Tests Identify astrophysical contexts where the mass–curvature–entropy law yields measurable deviations.

 

7 Conclusion

Qaether Theory re‑imagines geometry, gauge fields, and mass generation as phenomena emergent from a discrete lattice of phase information. Its minimal axioms reproduce quantum kinematics and general‑relativistic dynamics while predicting novel conservation laws intertwining mass, curvature, and entropy. Resolving the highlighted open problems will determine whether Qaether remains a compelling toy model or matures into a viable blueprint for quantum gravity and beyond.