Matrix Field Theory (MFT) — Conceptual Draft of a Discrete Computational Field Model

Matrix Field Theory (MFT)

Matrix Field Theory (MFT) is a unified discrete ontological framework that explores how spacetime, matter and interactions could emerge from local rules on a three-dimensional binary lattice. The work develops a “field of differences” picture: physical structures arise as patterns of coordination and interfaces in a discrete network, rather than being postulated as independent continuous entities.

Core concepts

The construction starts from a single ontological axiom A0 — to exist is to differ — together with locality and absence of external randomness. Under these premises, MFT uses a binary 3D lattice (6 neighbors) with a local flip rule:

• nodes flip when the local imbalance Eᵢ = sᵢ · (sum of sⱼ over neighbors j) is positive (Eᵢ > 0);
• neutral flips at Eᵢ = 0 are permitted only for a restricted axial pattern of neighbors (type [2,0,−2]).

A global functional H = Σ⟨i,j⟩ (sᵢ · sⱼ) is shown to be a Lyapunov-like quantity: it strictly decreases under E>0 flips and remains constant under neutral ones. Thus, “energy” is not postulated, but emerges as a measure of total local conflict.

Key elements (current status)

• Axioms and locality
  Formal definitions of observable predicates and admissible relations are given under A0 and locality, with direct neighborhood identified as the only admissible relation in a minimal ontology. A 3D binary lattice with local anti-alignment is argued to be the canonical minimal realization of a difference field.
• Dynamics and emergent spacetime
  The model is formulated as an event-driven local dynamics with a limiting front speed c = 1. Local proper time τ is defined as a flip counter, while topological distance Dtop is defined as minimal path length in the coordination graph. After calibration to laboratory units, an effective Minkowski metric of the form ds² = −dτ² + (κ · dDtop)² is obtained.
• Interfaces, particles and invariants
  Long-lived structures at E ≈ 0 (interfaces) are proposed as candidates for particles. Working discrete invariants w (winding), φ (chirality) and π (layering) are introduced to classify interface configurations. A central conjecture is the existence and stability of toroidal defects as electron/positron-like configurations; this remains an open problem.
• Quantum-like behaviour (program)
  MFT emphasizes that multiple causally admissible histories exist even under deterministic local rules, especially around E = 0 interfaces. The work outlines a program to interpret quantum probabilities via a measure or phase functional Φ[ω] on histories (in the spirit of path integrals), with the Born rule and effective unitarity as explicit open targets. These are not yet derived, but formulated as a research agenda.
• Standard Model and cosmology (qualitative mapping)
  Parts VI–VIII and X propose qualitative correspondences between MFT structures (tori, channels, junctions, horizons) and Standard Model particles, interactions, and cosmological phenomena (inflation-like phase, dark matter/energy as coherence regimes). These sections are explicitly marked as heuristic and non-quantitative: they provide an ontological picture and a list of open quantitative problems, but do not currently reproduce SM parameters or ΛCDM fits.

Cosmological scenario (heuristic)

As a simple working scenario for early cosmology, the model considers an almost homogeneous initial field configuration, followed by rapid percolation of E ≈ 0 channels. This can generate structure and an inflation-like phase in terms of coherence growth and interface dynamics. The text stresses that MFT does not postulate a unique initial state and that questions about measures on initial conditions, fine-tuning, and quantitative matching to CMB/BAO data are left as open problems.

Falsifiability and test directions

The framework formulates several families of falsifiable signatures, including:

• possible medium-dependent dispersion of gravitational waves in structured regions (from gradients of n(x) and ρif);
• topological signatures in the CMB (e.g. matching circles, low-ℓ features) tied to global topology and channel identifications;
• threshold nonlinearities in strong fields (interface tunneling and channel rearrangements);
• small deviations of the effective dark-energy equation-of-state parameter w(z) from −1;
• potential signatures of discrete horizon structure (black-hole echoes and related phenomena).

These are partly quantified as working hypotheses (order-of-magnitude targets), not as fully derived predictions, and are intended as concrete directions for future comparison with data.

Scope and status

This version (v0.5) should be read as a conceptual framework with partial formal results and a structured research program, not as a completed fundamental theory:

• rigorous proofs are provided (or sketched) for some key statements (for example, monotonicity of H under E>0 flips);
• other claims (for example, algorithm independence of macroscopic observables, existence and stability of specific defects, derivation of the Born rule, SM parameter values) are clearly labeled as conjectures or open problems;
• numerical simulations and quantitative parameter extraction are planned but not yet performed.

Relation to existing work

MFT connects to several discrete and emergent approaches to fundamental physics, including cellular automata and lattice spin models, causal set and emergent spacetime programs, lattice gauge theories, and gravitational thermodynamics. It aims to provide a unified, ontologically explicit base (local rule plus difference field) on top of which spacetime, quantum behaviour and interactions can be understood in a single framework.

Note: This is theoretical work, not yet peer-reviewed. No code or data are associated with this version. The document is distributed under the CC BY 4.0 license.