Emergent Field Physics From
Balanced-Ternary Microstates

Abstract

Standard lattice field approaches often treat the lattice as a discretisation artefact to be removed in a continuum limit for subsequent analysis. CAELIX takes the opposite stance: the lattice is treated as a constructive substrate, and the central question is what continuous field behaviour can be derived from an explicitly discrete structural layer.

The framework begins with balanced-ternary microstates \(s\in\{-1,0,+1\}\), maps them through a deterministic local load functional acting as an interface-density proxy, and couples the resulting load into a continuous carrier field \(\phi\) evolved by explicit local stencil updates. Two carrier families are used throughout: diffuse relaxers and telegraph-style momentum-carrying updates.

Despite the discrete underlying logic, the coupled system exhibits reproducible continuum-like signatures in three-dimensional experiments: inverse-radius vacuum profiles, wave interference, sign-structured interaction forces, confinement spectra, near-isotropy after calibration, gravitational-analogue latency and lensing proxies, linear collider baselines, and non-linear soliton and routing behaviour. While the carrier kinematics and interaction potentials are specified explicitly, the resulting signatures arise dynamically from local updates, local coupling, and geometric constraints rather than being imposed analytically as final solutions.

The contribution of this paper is concrete rather than grandiose. It specifies the substrate, the carrier rules, the coupling, and the experiment programme together with explicit controls, logged provenance, and known falsifiers. The goal is not to replace established physics at the Lagrangian level, but to provide a reproducible and falsifiable route from minimal discrete structure to measurable continuum-like observables.

Reading Guide

This paper is written for two audiences at once: readers who want an executable computational model and readers who want clear assumptions, structural invariants, and experimentally reported observables.

The Substrate section defines the balanced-ternary microstate lattice, motivates the use of \(\{-1,0,+1\}\) as the minimal complete signed state space, and derives the deterministic load functional. The Procedural field dynamics section specifies the carrier fields, update rules, coupling semantics, stability guards, and boundary choices. The Experiments and observable signatures section then summarises the reproducible experiment families and the measured signatures they produce.

A recurring theme is algorithmic honesty. Each reported signature is paired with controls and known artefacts. When a diagnostic fails, the failure is treated as part of the scientific result rather than hidden as an inconvenience.

Motivation And Stance: Substrate \(\neq\) Carrier

The central question of CAELIX is a constructive substrate hypothesis: can continuum-like field signatures be derived from, and driven by, an explicitly discrete structural layer under purely local rules, without hard-coding continuous physics into the substrate itself?

To avoid building a model that is “continuous all the way down”, the framework enforces a strict separation of concerns. The carrier is the continuous layer: it supports wave-like and diffusive transport through a scalar field \(\phi\) and, in telegraph mode, an auxiliary velocity-like field \(v\). The substrate is the discrete layer: a lattice of signed microstates \(s(\mathbf{x})\in\{-1,0,+1\}\) capable of preserving sharp interfaces, sign structure, and exact local bookkeeping.

Physical Intuition: The Water And The Plumbing

The carrier is the water. It handles ripples, propagation, interference, and the spreading of disturbance. The substrate is the plumbing. It is a rigid ternary structure whose topology can shape, redirect, and burden that flow. By keeping these roles separate, CAELIX asks whether a discrete plumbing layer can organise a continuous water layer into behaviours that look continuum-like without simply inserting those behaviours into the equations by hand.

This division is methodological rather than ornamental. It provides an ablation handle. Carrier dynamics can be held fixed while the microstate topology changes, making it possible to test whether the measured observables genuinely depend on the substrate. If they do not, the substrate can be discarded. If they do, then the substrate is doing real work as a medium with sign structure and persistent heterogeneity.

Substrate: Balanced-Ternary Microstates And Load Functional

Microstate Lattice

The substrate is a cubic three-dimensional lattice whose sites store explicit integer microstates

\[ s(\mathbf{x})\in\{-1,0,+1\}, \qquad \mathbf{x}=(i,j,k) \]

The microstate field is not an interpretation layer laid on top of a continuous model. It is stored directly, updated locally, and treated as the discrete structural layer from which downstream observables are derived.

Deterministic Load Functional

A continuous, non-negative scalar load field \(\ell(\mathbf{x})\ge 0\) is derived from \(s\) using a deterministic functional combining absolute activity and local neighbour mismatch. Operationally, the load counts sharp interfaces in the microstate lattice and turns those interfaces into local structural cost.

\[ \begin{aligned} \ell(\mathbf{x}) &= w_{\mathrm{abs}}\,|s(\mathbf{x})| \\ &\quad + \frac{w_{\mathrm{mismatch}}}{2} \sum_{d\in\{x,y,z\}}\Bigl(\Delta_{+d}(\mathbf{x}) + \Delta_{-d}(\mathbf{x})\Bigr) \end{aligned} \]

with axis mismatch terms

\[ \Delta_{\pm d}(\mathbf{x}) = \begin{cases} \bigl|s(\mathbf{x})-s(\mathbf{x}\pm\hat{\mathbf{e}}_d)\bigr| \\ \qquad \text{if }\mathbf{x}\pm\hat{\mathbf{e}}_d\text{ lies inside the lattice},\\ 0 \qquad \text{otherwise.} \end{cases} \]

This is a symmetric 6-neighbour edge-sharing stencil. Each undirected axis-aligned edge mismatch is counted once and shared equally between the two incident voxels. In the reference implementation, neighbour differences are formed with fast axis rolls and the wrapped faces are explicitly zeroed so that lattice boundaries are hard rather than periodic.

Interpretation: Load As Latency-Like Source

The load is interpreted as a local defect or interface-density proxy, not as a potential and not as a direct Euclidean-distance function. Gradients in \(\ell\) create latency-like structure in the carrier dynamics, providing a concrete way for the substrate to organise propagation without invoking an a priori metric.

Why Balanced Ternary

The choice of \(\{-1,0,+1\}\) is not presented as a stylistic preference. It is the minimal integer-valued state space that simultaneously provides a neutral ground state, signed structure, and closure under negation. The substrate therefore distinguishes source-like and sink-like microstructure intrinsically, preserves additive symmetry, and supports topological duals such as kink and antikink structures without importing an external sign convention.

The corresponding three-dimensional lattice Green's function has far-field decay proportional to \(1/r\), which is why the emergent vacuum profile converges to inverse-radius behaviour without any explicit distance kernel being specified in the update rule.

Procedural Field Dynamics

Fields And Update Rules

The carrier evolves on the same cubic lattice as the substrate. Its primary field is a scalar \(\phi\) and, in telegraph mode, an auxiliary velocity-like field \(v\). Updates are explicit, local, and use the 6-neighbour discrete Laplacian with hard boundary semantics.

Two solver families are used throughout the experiment programme:

• Diffuse mode: a first-order relaxer that mixes \(\phi\) towards the local neighbour mean and acts as a Poisson surrogate.
• Telegraph mode: a damped second-order wave surrogate that carries momentum and supports propagating disturbances.

In telegraph mode, one discrete step consists of a Laplacian evaluation followed by

\[ v_{t+1}(\mathbf{x}) = (1-\gamma)\,v_t(\mathbf{x}) + \mathrm{inject}\cdot\mathrm{src}(\mathbf{x}) + c^2\,\nabla^2\phi_t(\mathbf{x}) \]

\[ \phi_{t+1}(\mathbf{x}) = \bigl(\phi_t(\mathbf{x}) + dt\cdot v_{t+1}(\mathbf{x})\bigr)(1-\mathrm{decay}) \]

In non-linear regimes the same stepping skeleton is retained and local acceleration terms are added. In particular, the Sine-Gordon carrier introduces a bounded restoring term proportional to \(-k\sin\phi\).

Coupling: How Load Influences Propagation

The core mechanism in CAELIX is the coupling between discrete substrate and continuous carrier. Microstate topology is pushed through the deterministic load functional, and the resulting field \(\ell\) enters the carrier update either as a source term or as a local propagation modifier. This allows the substrate to define an effective medium without the framework ever writing down a metric tensor.

The coupling is meant to be measurable and ablatable. Every experiment is structured so that “load on” and “load off” can be compared under matched carrier kinematics.

Reversibility, Stability And Boundaries

Diffusion-like dynamics are non-reversible by construction. Telegraph-mode dynamics can be exactly reversible only in the absence of damping, decay, forcing, and sponge or clamp boundaries. CAELIX therefore distinguishes experiments that rely on conservative propagation from those that rely on explicitly dissipative behaviour.

To prevent non-physical divergence, telegraph stepping is guarded by a fail-fast CFL-style constraint:

\[ c\,dt < \frac{0.98}{\sqrt{3}} \]

Boundary semantics are hard and non-periodic throughout. Depending on the experiment, boundaries are treated either as clamped Dirichlet-style walls or as absorbing sponge layers built from a quadratic damping ramp near the lattice faces.

Experiments And Observable Signatures

The experiment programme is organised as a ladder. Early suites establish transport and field behaviour under tightly controlled conditions. Later suites introduce geometry, isotropy checks, gravitational analogues, object-like localisation, and finally non-linear routing and collider scenarios.

Heavy Walker: Moving-Source Lag And Wake

The Heavy Walker harness characterises the baseline response of the carrier to a moving localised driver. A delta source is translated through the lattice and the field mass in a small cube in front of and behind the source is compared. The compact asymmetry observable

\[ \mathrm{asym} = \frac{\mathrm{front}-\mathrm{back}}{\mathrm{front}+\mathrm{back}+\epsilon} \]

acts as a regression-safe wake metric. This suite deliberately bypasses the microstate-to-load pipeline so that carrier effects can be isolated cleanly.

Baseline Field And Emergent Radial Laws

Under a sustained point-like source in diffuse mode, the scalar field relaxes into a radial profile whose far-field structure approaches \(\phi(r)\propto 1/r\). This behaviour is treated as an observed equilibrium property of local transport rather than an injected law. The far-field exponent is extracted after de-offsetting the profile and the cleanest inverse-radius regime is logged explicitly.

The dimensional scaling of convergence is well described empirically by

\[ T(N) \approx 4.47\,N^{\sqrt{3}} \]

over the mid-to-large lattice regime.

Interference And Double-Slit Transport

Switching from diffuse transport to telegraph transport yields propagating real-valued waves. In a substrate-defined double-slit geometry, the detector intensity \(I(y)=\phi^2\) exhibits a broad single-slit envelope in the control run and a fringe pattern in the two-slit run. No diffraction integral, phase accumulator, or analytic interference logic is used; the fringe structure emerges from local neighbour-coupled propagation alone.

Interaction Forces

Signed source pairs produce stable, distance-structured interaction proxies. Like-sign pairs generate repulsion-like behaviour; opposite-sign pairs generate attraction-like behaviour. The reported interaction energy scales consistently with \(1/d\) and the implied force proxy with \(1/d^2\), even though no force law is specified explicitly.

Confinement And Resonance Spectra

In the confinement suite, a hard interior boundary masks the lattice into a cavity and a harmonic point source is driven inside it. Sweeping the drive frequency \(\omega\) produces sharply peaked stored energy responses

\[ E_{\mathrm{tot}}(\omega) = \left\langle \sum \phi^2 + \sum v^2 \right\rangle_{\mathrm{window}} \]

at frequencies consistent with standing-wave modes of the cavity. The simulation does not compute eigenvalues analytically or impose macroscopic mode shapes; discrete spectral selection is an empirical consequence of the local dynamics under geometric confinement.

Calibration And Isotropy

Because the lattice is cubic rather than rotationally symmetric, CAELIX measures anisotropy rather than assuming it away. Exp05 compares propagation along axis, face-diagonal, and body-diagonal directions, recording travel-time and amplitude-distance diagnostics at matched radius. In the large-\(\sigma\) regime, transport behaves as an effectively isotropic continuum in the long-wavelength limit, while deviations at smaller scales are measured explicitly and treated as numerical structure rather than hidden artefact.

Relativity Proxies: Phase Drift and Lensing

A static background potential induces differential phase drift between near and far probes, producing a gravitational-time-dilation analogue. Separately, a ray-marching preview in a refractive index derived from the background field produces deflection angles that scale approximately as \(\theta(b)\propto 1/b\). A moving light-clock protocol (a twin-paradox-style test), once corrected for mode-locking artefacts by windowed peak detection, yields measurable time-dilation proxies as kinematic latency between fast and slow reference clocks.

Linear Colliders And Ghosting Baseline

The linear collider suite establishes ghosting as the default baseline under purely linear packet interaction. Driven wave packets pass through one another, interfere, and then disperse once driving stops. This matters because later non-linear collider work can then attribute departures from ghosting to non-linear structure rather than to the transport harness itself.

Non-Linear Solitons And Sprite Extraction

The soliton suite introduces non-linear localisation. Quartic and Sine-Gordon potentials are used to generate bounded self-supporting structures whose persistence, localisation, and coherence are then measured explicitly. Once a stable breather or kink-like state is found, the pipeline selects a clean phase snapshot and exports a reusable sprite asset for later multi-body interaction experiments.

Sine-Gordon Routing And Collider Hardware

The non-linear collider suite consumes verified sprite assets and subjects them to collisions, confinement routing, and topological junction experiments. Velocity is imprinted using a finite-shift initialisation

\[ v_0(\mathbf{x}) \approx \frac{\phi_0(\mathbf{x}-\mathbf{v}\,dt)-\phi_0(\mathbf{x})}{dt} \]

rather than a naive momentum multiplier. Spatial stiffness masks create soft interior wires and stiff exteriors, enabling guided propagation through turns, branches, and T-junctions. The strongest reported signature is topological fan-out: a kink wall driven into a T-junction can split into two mirrored branches whose symmetry is checked numerically.

Limitations And Falsifiers

The claims of this paper are deliberately modest. “Continuum-like” is an empirical statement about measured behaviour over explored scales, not an ontological declaration that a discrete lattice has become a continuum in every relevant sense.

Several specific falsifiers have already been identified and are treated as first-class scientific constraints:

• Isotropy loss: failure to preserve calibrated isotropy under scaling or in low-\(\sigma\) limits.
• Detection phase-lock: naive thresholding can erase kinematic effects by locking onto envelope shape or gating rather than true phase drift.
• Echo saturation: unresolved wakes or finite-domain reflections can contaminate late-time measurements unless absorbing layers are chosen carefully.
• Boundary contamination: small domains can masquerade as physical signal when the true behaviour is dominated by reflections or clamp geometry.

The point of listing these is not merely caution. It is part of the framework's methodological claim: each major artefact is surfaced, measured, and either mitigated or bounded explicitly.

Reproducibility And Artefacts

All reported results are generated by the accompanying codebase. The reference implementation, parameterised experiment definitions, and provenance-logged outputs are published openly through the CAELIX repository. Each experiment is defined by explicit parameters, emits structured outputs, and is intended to be rerunnable under fixed seeds and logged provenance.

That reproducibility layer is part of the scientific claim. The framework is not presented as a sequence of attractive animations but as an executable substrate-to-carrier pipeline whose claims can be challenged, rerun, and extended independently.

Conclusion

Balanced-ternary microstate lattices provide a concrete, reproducible, and falsifiable route from discrete substrate to continuum-like observables in three-dimensional lattice systems. A single local update architecture, applied to a minimal signed integer substrate and coupled through a deterministic structural functional, produces a coherent family of signatures usually associated with continuum field models.

Those signatures include inverse-radius vacuum structure, interference, sign-structured interactions, confinement spectra, calibrated near-isotropy, relativity-style phase drift and lensing proxies, linear collider baselines, and non-linear soliton and routing behaviour. None is claimed as historically unique in isolation. The novelty claim is methodological and integrative: the full chain from minimal discrete structure to measured continuum-like observables is specified concretely, paired with controls, and tested as a single ablatable construction.

The programme remains open. Its value will ultimately be determined by what survives scaling, replication, and extension into more strongly non-linear and multi-body regimes. What this paper establishes is narrower but solid: the balanced-ternary substrate earns its keep under controlled ablation, and it does so in a way that is inspectable rather than mystical.