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\newtheorem{definition}{Definition}[section]
\newtheorem{axiom}{Axiom}[section]
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\newtheorem*{scholium}{Scholium}
\title{Spatial Principles of Natural Philosophy}
\author{Mark Rubin}
\date{July 2025}
\begin{document}
\maketitle
%---------------- Preface ----------------
\section*{Preface}
In this treatise, the principles that govern the flow and transformation of energy
are laid out in a geometric framework. As in the manner of Sir Isaac Newton’s
\emph{PhilosophiÆ Naturalis Principia Mathematica}, we begin with definitions,
state the axioms, and proceed by rigorous propositions, corollaries, and scholia.
We also introduce gold as a fourth spatial energy bank, representing the atomic
storage of monetary value in a spinning disk model.
%---------------- Definitions ----------------
\section{Definitions}
\begin{definition}[Spatial Energy Bank]
A \emph{Spatial Energy Bank} is any finite region of space delineated by boundaries through which incoming and outgoing energy fluxes can be precisely measured over time. Such a Bank is characterized by:
\begin{enumerate}
\item A three-dimensional spatial domain whose boundary surfaces intercept energy flow rates (joules per second).
\item Energy inflows and outflows quantified as power (W = J/s), integrating mechanical, chemical, monetary, or atomic streams.
\item Internal stores—mass-value, honey, money, gold—each measured in their joule-equivalent units, whose exchange rates follow the transformation laws.
\item Temporal resolution, ensuring that all flux measurements include the differential element $\mathrm{d}t$ to capture dynamic behavior.
\end{enumerate}
This geometric construct allows rigorous application of the conservation and transformability axioms to any combination of energy forms in motion.
\end{definition}
\begin{definition}[Mass-Value Joule]
The mechanical energy equivalent to one joule of work, stored by mass-value at a specific spatial location. In a Spatial Energy Bank, the Mass-Value Joule store is measured by integrating energy density over the volume. The corresponding mass-value flux (power) across a boundary surface is the rate at which mechanical energy (J/s) enters or leaves via inertial flow, defined by $\dot m\,g\,h + \tfrac12\dot m\,v^{2}$.
\end{definition}
\begin{definition}[Honey-Joule]
The chemical energy equivalent to one joule, stored in honey within a Spatial Energy Bank. Its store is measured in joules of honey energy; the honey-joule flux (power) is the rate (J/s) at which this chemical energy crosses the bank boundary, computed as $\dot H_{\text{kcal}}\times 4184$~J/s when measured in kilocalories per second.
\end{definition}
\begin{definition}[Money-Joule]
The mechanical or service work equivalent of one joule made available by expending one unit of currency within a Spatial Energy Bank. The store of money-joules is measured by converting currency units into joules; the money-joule flux (power) is the rate (J/s) at which economic energy crosses the boundary, given by $\dot M_{\$}\,E_{\$}$ where $E_{\$}$ is joules per dollar.
\end{definition}
\begin{definition}[Gold-Joule]
The energy and monetary value equivalent to one joule stored in a mass of gold in a Spatial Energy Bank. A gold disk stores Gold-Joules in a precise 1:1:1 ratio among its mass, number of atoms, and dollar value. Its flux (power) is the rate (J/s) at which Gold-Joules cross the boundary—e.g., a spinning disk with mass flow, atomic flow, or monetary flow per second.
\end{definition}
\begin{definition}[Dynamic Gold Disk Properties]
A spinning gold disk acts as a dynamic energy bank whose rotating mass, atomic content, angular momentum, and monetary value all flow cohesively:
\begin{enumerate}
\item Mass-Flux: Rotation at angular speed $\omega$ imparts a mass-flow equivalent $\dot m = I\,\frac{d\omega}{dt} / (g\,h_{eff})$, where $I$ is the disk's moment of inertia and $h_{eff}$ an effective lever arm.
\item Atomic-Flux: The number of gold atoms passing a reference radial line per second is $\dot N = N_{tot} \,\omega / (2\pi)$, linking atomic flow directly to spin rate.
\item Momentum-Flux: Angular momentum $L = I\,\omega$ changes at rate $\dot L = I\,\alpha$ (with $\alpha = d\omega/dt$), representing the mechanical work exchange.
\item Value-Flux: Monetary flow is $\dot M_{gold} = \dot N \,E_{atom}$, where $E_{atom}$ is the dollar-joule equivalent per atom, converting atomic motion to economic power.
\end{enumerate}
These relationships make the spinning gold disk a physical scale model: its rotation simultaneously encodes mass-per-second, atoms-per-second, joules-per-second, and dollars-per-second according to unified spatial principles.
\end{definition}
% Additional Newtonian Definitions Adapted
\begin{definition}[Quantity of Mass-Value]
Also known as inertial value, the Quantity of Mass-Value is the measure of stored mechanical energy potential within a Spatial Energy Bank, proportional to mass-value density and volume. It underlies all inertial energy reservoirs in the mass-value joule form.
\end{definition}
\begin{definition}[Quantity of Motion]
Also called momentum, the Quantity of Motion is the product of mass and velocity for moving bodies. In a Spatial Energy Bank, its flux across a boundary is the rate of mechanical energy transfer entering or leaving the region.
\end{definition}
\begin{definition}[Vis Viva (Living Force)]
The sum of the products of the mass of each moving body and the square of its velocity. Vis viva relates to kinetic energy, and in a Spatial Energy Bank equals twice the kinetic portion of mechanical energy, $\sum m v^2$.
\end{definition}
\begin{definition}[Absolute Space]
The infinite, unchanging spatial backdrop within which all bodies and Spatial Energy Banks exist and move. It defines fixed reference coordinates for measuring flux domains.
\end{definition}
\begin{definition}[Absolute Time]
True and mathematical time, which flows uniformly without regard to external conditions. It serves as the universal parameter $t$ for all energy flux measurements (J/s).
\end{definition}
% Additional Newtonian Force Definitions Adapted
\begin{definition}[Impressed Force]
An \emph{Impressed Force} is an action exerted on a body within a Spatial Energy Bank that changes its quantity of motion. It corresponds to an external power flux (J/s) applied across the boundary, altering mass-joule or other energy stores.
\end{definition}
\begin{definition}[Centripetal Force]
A \emph{Centripetal Force} is an impressed force directed toward a center of motion, causing energy flows and bodies to traverse curvilinear paths. In a Spatial Energy Bank, its magnitude equals the radial component of the energy flux required to maintain circular motion.
\end{definition}
\begin{definition}[Gravitational Force]
A \emph{Gravitational Force} is an attractive impressed force between masses in Absolute Space. In a Spatial Energy Bank, it generates potential-joule stores via $mgh$ and follows the inverse-square law of spatial attenuation.
\end{definition}
%---------------- Comparison to Newton's Definitions ----------------
\section{Comparison to Newton's Definitions}
In Newton’s \emph{Principia}, early definitions (e.g., of `quantity of motion`, `vis viva`) are stated in purely geometric and kinematic terms without explicit temporal flux. Our adaptation extends this approach by:
\begin{enumerate}
\item Retaining a clear geometric domain (Spatial Energy Bank) akin to Newton’s `bodies` and `places`.
\item Introducing explicit power units (J/s) to capture \emph{flux rates}, whereas Newton defined only total `quantities`.
\item Defining multiple energy stores (mass, honey, money, gold) in joule-equivalents, analogous to Newton’s single `vis viva`, but generalized across forms.
\item Emphasizing temporal resolution ($\mathrm{d}t$), reflecting modern understanding of dynamic systems absent in the original.
\end{enumerate}
%---------------- Relativistic Considerations ----------------
\section{Relativistic Extension}
\begin{definition}[Relativistic Energy]
In a Spatial Energy Bank, the total energy \((E)\) of a mass includes its rest energy and kinetic energy, given by
\[
E = \gamma m c^2, \quad \gamma = \frac{1}{\sqrt{1 - v^2 / c^2}},
\]
where \(m\) is the invariant mass, \(v\) its velocity relative to Absolute Space, and \(c\) the speed of light. This reduces to Newtonian kinetic energy plus rest energy under low-speed conditions.
\end{definition}
%---------------- Axioms ----------------
\section{Axioms, or Laws of Motion}
\begin{axiom}[Law of Inertia]
A body remains at rest or continues in uniform motion in a straight line unless acted upon by an external impressed force.
\end{axiom}
\begin{axiom}[Law of Acceleration]
The change of motion (momentum) of a body is proportional to the applied impressed force and occurs along the direction of that force. Equivalently,
\[
F = m\,a
\]
where \(F\) is the net impressed force, \(m\) the mass, and \(a\) the acceleration.
\end{axiom}
\begin{axiom}[Law of Action and Reaction]
To every action there is always an equal and opposite reaction; that is, the mutual forces of two bodies on each other are equal in magnitude and opposite in direction.
\end{axiom}
\begin{corollary}[1]
Two bodies of equal mass, when acted upon by equal and opposite forces, will acquire equal and opposite accelerations.
\end{corollary}
\begin{corollary}[2]
The net external force on a closed system of bodies is zero if and only if the total momentum of the system remains constant.
\end{corollary}
\begin{corollary}[3]
The trajectory of a free body in the absence of impressed forces is a straight line in absolute space.
\end{corollary}
\begin{corollary}[4]
For a rotating disk, internal tensions generate centripetal forces that maintain circular motion without net change in momentum direction.
\end{corollary}
\begin{corollary}[5]
Gravitational forces between masses in a spatial energy bank produce potential-joule stores without creating or destroying net energy.
\end{corollary}
\begin{corollary}[6]
The vis viva of a system is conserved in the absence of non-conservative impressed forces.
\end{corollary}
\begin{scholium}
These three laws establish the foundation for all subsequent propositions: energy banks governed by inertia, accelerations driven by impressed forces, and interactions bound by action–reaction symmetry. Each spatial energy form—mass, honey, money, gold—obeys these laws under their equivalent joule representations.
\end{scholium}
%---------------- Book I: Mechanics ----------------
\section*{Book I: Mechanics of Mass-Value}
% Outline of chapters and sections in Book I
\subsection*{Chapter I.1: Kinematics of Mass}
\begin{itemize}
\item Define spatial mass flow and Mass-Joule storage across spatial domains, including honey, money, and gold analogues for moving banks.
\item Describe velocity and position functions for masses, pollen-laden honey flows, currency circulation, and spinning gold disks.
\item Present metric tensors for spatial evaluation of flux rates (J/s) for each energy form.
\end{itemize}
\subsection*{Chapter I.2: Conservation Laws}
\begin{itemize}
\item State conservation of mechanical, chemical, monetary, and atomic energy within Spatial Energy Banks.
\item Prove elastic collision invariants for mass (Newton’s cradle), honey droplet collisions, currency transactions, and gold disk interactions.
\item Derive the global conservation equation incorporating mass, honey, money, and gold terms.
\end{itemize}
\subsection*{Chapter I.3: Forces and Motion}
\begin{itemize}
\item Analyze impressed forces on masses, honey flows (viscosity), market pressures on money flow, and torque on gold disks.
\item Formulate spatially resolved force-induced energy transfers: F = m a, honey viscosity law, economic demand-supply force, and gold disk centripetal requirements.
\item Provide problems on multi-body interactions involving mixed energy banks (e.g., mass-honey coupling).
\end{itemize}
\subsection*{Chapter I.4: Rotational and Frame Effects}
\begin{itemize}
\item Examine angular momentum for masses (rotors), honey swirl in comb cells, currency turnover rates as rotational analogues, and spinning gold disk models.
\item Introduce Coriolis and centrifugal flux components in moving frames for each energy form.
\item Generalize spatial reservoirs under rotation and non-inertial reference frames.
\end{itemize}
\begin{proposition}[I.1]
The mechanical energy of $N$ equal masses in a Newton’s cradle is conserved during each collision.
\end{proposition}
\begin{proof}
Follows from Axiom I applied to mass flow with $H=M=G=0$, giving
\[
\sum_{i=1}^{N} (mgh_i + \tfrac12 m v_i^2) = \text{constant}.
\]
\end{proof}
%---------------- Book II: Honey Engines ----------------
\section*{Book II: Living Engines}
\begin{proposition}[II.1]
Photosynthetic plants convert solar joules into chemical stores at a fixed efficiency, preserving total energy.
\end{proposition}
\begin{proof}
Application of Axiom II with chemical conversion factor $E_{\text{cal}}$.
\end{proof}
\begin{corollary}
Bees may then convert these stores into honey-joules and expend them in hive activities with no net loss.
\end{corollary}
%---------------- Book III: Economic Engines ----------------
\section*{Book III: Economic Engines}
\begin{proposition}[III.1]
A money reservoir, defined in joule-equivalents, abides by Axiom I when incoming and outgoing purchases balance.
\end{proposition}
\begin{proof}
Direct from Axiom I applied to monetary flow $M(t)$ with conversion $E_{\$}$.
\end{proof}
\begin{scholium}
Thus honey-joules, money-joules, and gold-joules serve as interchangeable tokens of the same spatial energy.
\end{scholium}
%---------------- General Scholium ----------------
\section*{General Scholium}
The three books unite under a common geometry: energy moves through space,
through masses, through living stores, through monetary economies,
and through gold reserves without creation or destruction—only transformation and transfer.
%---------------- Detailed Book Outlines ----------------
\section*{Detailed Outline of Books I--III}
\subsection*{Book I: Mechanics of Mass}
\begin{enumerate}[I.1]
\item \textbf{Chapter I.1: Kinematics of Mass}
\begin{enumerate}[a]
\item Definition of mass flow and Mass-Joule storage
\item Spatial Energy Bank volume integrals and surface fluxes
\item Disk and pendulum models: geometric representation
\end{enumerate}
\item \textbf{Chapter I.2: Conservation Laws}
\begin{enumerate}[a]
\item Elastic collision propositions (Newton’s cradle)
\item Impulse and momentum flow across boundaries
\item Relativistic corrections to mechanical energy
\end{enumerate}
\item \textbf{Chapter I.3: Forces and Motion}
\begin{enumerate}[a]
\item Impressed, centripetal, and gravitational forces
\item Spatially resolved force-induced energy transfers
\item Problems on energy redistribution in multi-body systems
\end{enumerate}
\item \textbf{Chapter I.4: Rotational and Frame Effects}
\begin{enumerate}[a]
\item Angular momentum and disk-scale models
\item Coriolis and centrifugal energy flux components
\item Generalized rotational reservoirs in space
\end{enumerate}
\end{enumerate}
\subsection*{Book II: Living Engines}
\begin{enumerate}[II.1]
\item \textbf{Chapter II.1: Photosynthetic Conversion}
\begin{enumerate}[a]
\item Solar flux interception by spatial domains
\item Chemical energy bank in plant tissues
\item Efficiency propositions and proofs
\end{enumerate}
\item \textbf{Chapter II.2: Bee Metabolic Engines}
\begin{enumerate}[a]
\item Honey-Joule production and storage in hives
\item Dynamic honey disk analogues for metabolism
\item Conservation propositions for hive energy use
\end{enumerate}
\item \textbf{Chapter II.3: Ecosystem Energy Transfer}
\begin{enumerate}[a]
\item Predation, decomposition, and energy cascading
\item Spatial banks linking multiple reservoirs
\item Corollaries on sustainability and reserves
\end{enumerate}
\end{enumerate}
\subsection*{Book III: Economic Engines}
\begin{enumerate}[III.1]
\item \textbf{Chapter III.1: Monetary Flow and Markets}
\begin{enumerate}[a]
\item Definition of Money-Joule and market reservoirs
\item Income-velocity propositions for currency flow
\item Balance proofs for open and closed economies
\end{enumerate}
\item \textbf{Chapter III.2: Value Storage and Investment}
\begin{enumerate}[a]
\item Gold-Joule disk as a monetary scale model
\item Atomic count, mass, and dollar value interrelations
\item Propositions on disk stability and storage efficiency
\end{enumerate}
\item \textbf{Chapter III.3: Externalities and Value Extraction}
\begin{enumerate}[a]
\item Exchange of monetary flux for external goods
\item Societal value banks and energy equivalents
\item General scholia on economic sustainability
\end{enumerate}
\end{enumerate}
The three books unite under a common geometry: energy moves through space,
through masses, through living stores, through monetary economies,
and through gold reserves without creation or destruction—only transformation and transfer.
\end{document}