The Communication–Coherence Framework (CCF) treats coherence as a dynamical quantity that can be

transported, regenerated, and dissipated through communication. Coherence C(x,t) is modeled as a density

field describing the degree of ordered structure in a system, while information I(x,t) and entropy S(x,t) capture

informational content and disorder.

1. Central Continuity Relation

At the core of CCF is a continuity-style equation that governs how coherence evolves under the influence of

information, entropy, and energy exchange. Let C, I, S denote coherence, information, and entropy densities,

with associated fluxes JC, JI, JS. The Communication–Coherence continuity relation takes the form:

∂C/∂t + ∇·JC = α ∇·JI − κ ∇·JS + β P

where P represents external power input and α, κ, β are system-specific coupling coefficients. This equation

generalizes non-equilibrium transport laws by treating communication as the flux of coherence in time, with

information flow acting as a source and entropy flow as an effective sink.

Structurally, the CCF continuity law parallels non-equilibrium thermodynamic transport and active-matter

continuity equations, but extends them by treating coherence as the transported quantity and by explicitly

coupling to information and entropy fluxes.

In this manuscript, C, I, and S are treated as densities per unit volume. For concreteness: C is taken as a

dimensionless coherence index per unit volume; I is information density (bits per unit volume); S is entropy

density (bits or joules per kelvin per unit volume). The fluxes JC, JI, JS therefore have units of density per unit

time. The coupling coefficients α, κ, β are chosen so that each term in the continuity equation has dimensions

of C per unit time, with full dimensional analysis provided in Appendix C.2.

The appearance of ∇·JI and ∇·JS as driving terms is a deliberate modeling ansatz: CCF assumes that spatially

structured divergences of information and entropy flux act as effective sources and sinks for coherence, rather

than treating I or S themselves as explicit local production terms. Alternative formulations with production terms

(e.g., +αI − κS) are mathematically possible and may be more appropriate in certain regimes; the present

choice is adopted for consistency with transport-style equations in non-equilibrium statistical mechanics and to

facilitate empirical fitting from measurable flux gradients.

2. Radiative and Scalar Coherence Channels

To distinguish conventional, lossy propagation from hypothetical reduced-loss channels, the coherence and

information fluxes are decomposed into radiative and scalar components:

JC = Jrad

C + Jscalar

C, JI = Jrad

I + Jscalar

I

A scalar-mode participation coefficient χ ∈ [0,1] encodes the effective fraction of coherence exchange occurring

through non-radiative channels. The entropy coupling is promoted to a mode-dependent term κeff(χ), which

decreases as χ → 1, so that in the scalar-dominated limit entropy-driven loss is minimized, whereas χ → 0

recovers the standard radiative, dissipative regime.

Importantly: The scalar channel and χ are introduced as an effective reduced-loss parametrization of

coherence transfer at the coarse-grained level, not as a claim of an additional fundamental physical field

beyond established quantum and classical mechanics. They serve to capture, in phenomenological terms,

regimes where coherence behaves as if transported with suppressed entropy coupling—a useful abstraction for

modeling and prediction, pending empirical validation.

Null hypothesis and reduction: In the limit χ = 0 and with coherence stabilization σC = 0, the framework

reduces to a standard radiative, dissipative transport picture equivalent to conventional decoherence and

open-system treatments, providing a clear baseline against which any putative CCF effects must be tested.

3. Coherence Stabilization and Field Formulation

CCF introduces a coherence stabilization term σC (often written as Γ or an equivalent source term) to represent

active or structural processes that regenerate coherence against decoherence. In field form, a coherence field

Φ is used to represent the unified coherence structure underlying quantum and relativistic descriptions, and an

informational–thermodynamic Lagrangian density is defined so that the Euler–Lagrange equation for Φ yields

local continuity expressions for C and its fluxes. In this formulation, coherence stabilization appears as an

additional source term that can counterbalance entropy-driven degradation in appropriate regimes.

4. Time–Communication Reciprocity

Within CCF, time is treated as an emergent parameter measuring ordered changes in coherence across the

communication manifold. The continuity relation implies that temporal structure arises from coherence flux: in

regions where JC = 0, coherence becomes stationary and time is locally degenerate, whereas increasing

coherence flux differentiates temporal structure. An effective time–communication reciprocity is expressed by

treating temporal continuity and communication as mutually generative operators, such that equilibrium in

coherence flux corresponds to the limit where local time ceases to progress operationally.

5. Minimal Toy Model Implementation

In a minimal toy implementation, consider a 1D lattice or network of N nodes, each with a scalar coherence

index Ci(t) representing local order (e.g., normalized phase coherence of an oscillator cluster). The coherence

flux between neighbouring nodes is modeled as JC,i→i+1

∝ Ci − Ci+1, so that ∇·JC reduces to nearest-neighbour

differences on the graph. Information and entropy fluxes are instantiated as analogous network flows derived

from signal amplitudes or noise levels, and the scalar participation χ enters as a factor that partially redirects

coherence exchange into an effective reduced-loss channel on selected edges. In this discrete setting, the CCF

continuity relation becomes a set of coupled difference equations for Ci(t), allowing numerical exploration of

how varying α, κ, β, χ affects coherence spreading, stabilization, and decay in a controlled, interpretable model.

6. Measurement and Operationalization

To connect CCF to empirical systems, coherence and its flux must be instantiated as measurable quantities in

specific domains:

Engineered Systems (Oscillators, Communication Networks): In oscillator networks or communication

hardware, C can be operationalized as a normalized order parameter (e.g., Kuramoto-type phase coherence or

spectral coherence between channels). The coherence flux JC is estimated from spatial or network gradients in

this order parameter over time, allowing direct comparison of the model to measurable network synchrony

evolution.

Neural Systems: In neural data, C is operationalized via phase-locking values (PLV) or cross-spectral

coherence between brain regions. The coherence flux JC is approximated from changes in coherence along

anatomical or functional connectivity graphs, using time-resolved measures such as sliding-window PLV or

weighted phase-lag index (wPLI). This approach enables testing of CCF predictions against EEG/MEG and

fMRI time series.

7. Scope and Regime of Validity

This core framework is intended as a coarse-grained, phenomenological description. Coherence is not

assumed to be universally conserved; approximate conservation arises only under closed or

symmetry-preserving conditions, and the coupling coefficients α, κ, β and scalar participation χ are understood

as empirically determined parameters, not universal constants. In this way, CCF stays continuous with

established quantum and thermodynamic formalisms while proposing a specific, testable structure for how

communication, information, and entropy jointly govern coherence evolution.

Core Assumptions and Limits

Coarse-grained description: Coherence C, information I, and entropy S are treated as effective densities

and fluxes over coarse-grained volumes, not as microscopic observables for individual particles or degrees

of freedom.

Conditional, not universal, conservation: Coherence is not assumed to be a fundamental conservation

law; the continuity relation allows source and sink terms, and approximate conservation appears only in

effectively closed or symmetry-preserving regimes.

Empirical coupling coefficients: The parameters α, κ, β and the scalar participation coefficient χ are

phenomenological and system-dependent; they must be inferred or fit from experiment rather than

assumed universal constants.

Flux-divergence ansatz: The choice to drive coherence evolution via ∇·JI and ∇·JS is a modeling

assumption; alternative formulations (e.g., with explicit production terms) may be more suitable in some

regimes and should be explored empirically.

Mode decomposition as effective parametrization: The radiative/scalar split and χ-dependent entropy

coupling are effective reduced-loss models, not claims of new fundamental fields; they capture

coarse-grained regimes and are validated by fit to data.

Standard causal structure: The framework assumes a standard relativistic causal structure with no

superluminal signaling; 'retrocausal' behavior arises only as global boundary-condition constraints in the

variational formulation, not as backward-in-time communication.

Time as emergent, operational: Statements about time 'degenerating' when JC = 0 are interpretive and

operational: they concern the absence of measurable ordered change in coherence, not the literal

disappearance of a spacetime dimension.

Domain of application: The continuity equations and stabilization terms are intended as testable models

for quantum, thermodynamic, and neural/engineered systems where coherence measures and fluxes are

experimentally accessible; extensions to cosmology, afterlife, or value are treated as speculative modules

built on top of this core, with clearly marked conceptual gaps.