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Mathematical Framework of SDKP: Scale, Density, and Chronon Wake

Author: Donald Paul Smith Formalized: 2025-05-25

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Abstract

This page details the Scale-Density-Kinematics-Phase (SDKP) principle as a formal field theory. It defines the fundamental fields—local scale s, density \rho, kinematic flow \mathbf{v}, and phase \phi—and constructs a Lagrangian density governing their dynamics. Applying the Euler-Lagrange formalism yields equations of motion that describe how granular time (the Chronon Wake) and physical reality emerge via gradient-triggered phase dynamics.

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1. Introduction

The SDKP framework models physical reality through four interacting fields: • Scale s(x^\mu): Local granular scale or resolution. • Density \rho(x^\mu): Local spatial/informational density. • Kinematic Flow \mathbf{v}(x^\mu): Velocity or flow of entities relative to chronon time. • Phase \phi(x^\mu): Encoded structural and temporal information.

The goal is to describe their coupled dynamics via a Lagrangian density \mathcal{L}_{SDKP}, and derive equations of motion that underpin spacetime emergence and causal structure.

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2. SDKP Action Principle

The action functional over spacetime region \Omega is:

\mathcal{A}{SDKP} = \int\Omega \mathcal{L}{SDKP}(s, \rho, \phi, \mathbf{v}; \partial\mu s, \partial_\mu \rho, \partial_\mu \phi, \nabla \cdot \mathbf{v}) , d^4x • d^4x is the invariant volume element. • \mathcal{L}_{SDKP} includes kinetic, potential, and interaction terms for all fields.

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3. Lagrangian Density Overview

The general form is:

\mathcal{L}{SDKP} = \underbrace{\frac{\lambda_s}{2}(\partial_t s)^2 - \frac{\gamma_s}{2}|\nabla s|^2}{\text{Scale Kinetic Energy}} + \underbrace{\frac{\lambda_\rho}{2}(\partial_t \rho)^2 - \frac{\gamma_\rho}{2}|\nabla \rho|^2}{\text{Density Kinetic Energy}} + \underbrace{\frac{1}{2}\rho |\nabla \phi|^2}{\text{Phase Gradient Energy}} + \underbrace{\beta \rho (\nabla \cdot \mathbf{v})}{\text{Compressive Kinematic Coupling}} - \underbrace{V(\rho, s, \phi)}{\text{Interaction Potential}} + \underbrace{\mathcal{L}\tau}{\text{Chronon Coupling (Optional)}}

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4. Terms Defined

4.1 Scale Kinetic Energy • Field: s(x^\mu), local granular scale. • Constants: • \lambda_s > 0: temporal inertia of scale changes. • \gamma_s > 0: spatial gradient penalty of scale. • Term:

\frac{\lambda_s}{2} (\partial_t s)^2 - \frac{\gamma_s}{2} |\nabla s|^2

Controls resistance to temporal/spatial variations in scale.

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4.2 Density Kinetic Energy • Field: \rho(x^\mu), local density. • Constants: • \lambda_\rho > 0: temporal inertia of density changes. • \gamma_\rho > 0: stiffness against density gradients. • Term:

\frac{\lambda_\rho}{2} (\partial_t \rho)^2 - \frac{\gamma_\rho}{2} |\nabla \rho|^2

Describes energy stored in density fluctuations.

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4.3 Phase Gradient Energy • Field: \phi(x^\mu), phase encoding temporal and structural info. • Term:

\frac{1}{2} \rho |\nabla \phi|^2

Links density to phase gradients, encoding emergent causal structure.

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4.4 Compressive Kinematic Coupling • Field: \mathbf{v}(x^\mu), flow velocity relative to chronon time. • Constant: \beta, coupling strength. • Term:

\beta \rho (\nabla \cdot \mathbf{v})

Models coupling between density and flow divergence—sources/sinks in kinematic flow, representing chronon wake generation.

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4.5 Interaction Potential • Form:

V(\rho, s, \phi) = \frac{\mu}{s^\delta} \rho \cos(\phi - \phi_0) • Constants: • \mu: coupling constant. • \delta > 0: scale exponent. • \phi_0: preferred phase alignment.

Encodes phase-locking behavior, stronger at finer scales.

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4.6 Optional Chronon Coupling (Chronon Wake Theory) • Form:

\mathcal{L}\tau = \chi \left(\partial_t \phi - \omega\tau(s, \rho) \right)^2 • Constants: • \chi: coupling constant. • \omega_\tau(s, \rho): local chronon tick rate function (e.g., proportional to s \cdot \rho).

Enforces phase time evolution matching local chronon generation, embedding discrete time granularity.

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5. Euler–Lagrange Equations

For any field X \in { s, \rho, \phi, \mathbf{v} }, the equation of motion is:

\frac{\partial \mathcal{L}{SDKP}}{\partial X} - \partial\mu \left( \frac{\partial \mathcal{L}{SDKP}}{\partial (\partial\mu X)} \right) = 0

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5.1 Phase Field \phi • Dominant terms:

\frac{1}{2} \rho |\nabla \phi|^2 - V(\rho, s, \phi) - \mathcal{L}_\tau • Euler-Lagrange yields:

\nabla \cdot (\rho \nabla \phi) + \frac{\mu}{s^\delta} \rho \sin(\phi - \phi_0) + 2 \chi \partial_t^2 \phi - 2 \chi \partial_t \omega_\tau(s, \rho) = 0

(assuming \omega_\tau independent of \phi).

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5.2 Scale Field s • From scale kinetic and potential terms:

\lambda_s \partial_t^2 s - \gamma_s \nabla^2 s + \frac{\delta \mu}{s^{\delta+1}} \rho \cos(\phi - \phi_0) + \cdots = 0

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5.3 Density Field \rho • From density kinetic and potential terms plus coupling:

\lambda_\rho \partial_t^2 \rho - \gamma_\rho \nabla^2 \rho + \frac{1}{2} |\nabla \phi|^2 + \beta (\nabla \cdot \mathbf{v}) - \frac{\mu}{s^\delta} \cos(\phi - \phi_0) + \cdots = 0

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5.4 Kinematic Flow \mathbf{v} • From coupling term:

\beta \nabla \rho = 0 \quad \Rightarrow \quad \text{or more complex forms with flow dynamics depending on extensions.}

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6. Interpretation and Implications • The SDKP framework formalizes how scale granularity, informational density, kinematic flows, and phase coherence interact to generate emergent spacetime and time flow. • Phase gradients coupled to density encode causal structure, while compressive kinematics generate chronon wakes—discrete time steps manifesting granular time flow. • Interaction potentials enable phase locking and synchronization, fundamental to stable structures and coherence in physical systems. • The optional chronon coupling explicitly encodes discrete time ticks, linking SDKP to the Chronon Wake Theory (CWT).

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7. References & Further Reading • D.P. Smith, Scale-Density-Kinematics-Phase Principle and Chronon Wake Theory, 2025. • Related papers on phase synchronization, granular spacetime, and emergent causality. • Links to SDKP codebase, simulations, and derivations (if applicable).

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End of SDKP Mathematical Framework