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% RAJ_v1_05_claims.tex
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\section{CLAIMS}

What is claimed is:

\subsection{Independent Claims}

\vspace{1em}

\noindent\textbf{1.} A computer-implemented memristive virtual machine system comprising:

\begin{enumerate}[label=(\alph*),nosep,leftmargin=2em]
\item a processor;
\item a memory storing a plurality of computational elements, each computational element having an associated execution heat value representing accumulated execution history;
\item execution tracking logic executed by the processor and configured to:
\begin{enumerate}[label=(\roman*),nosep,leftmargin=2em]
\item increment the execution heat value of a computational element upon invocation thereof, and
\item decrement the execution heat value over time according to a decay function;
\end{enumerate}
\item a lookup mechanism wherein lookup latency for a given computational element varies inversely with the execution heat value thereof, thereby creating state-dependent conductance wherein frequently-executed computational elements exhibit reduced latency; and
\item a phase space monitor configured to track a state vector comprising at least execution heat and a performance parameter through configuration space;
\end{enumerate}
wherein the system exhibits memristive dynamics with effective resistance proportional to accumulated execution history, and wherein a phase space trajectory traced by the state vector exhibits hysteresis behavior characterized by non-retracing paths and approximately 180-degree reversals at architectural cache boundaries.

\vspace{1.5em}

\noindent\textbf{2.} A computer-implemented virtual machine system exhibiting universal computational frequency, comprising:

\begin{enumerate}[label=(\alph*),nosep,leftmargin=2em]
\item a processor;
\item a memory storing computational elements organized for execution;
\item a runtime engine configured to execute the computational elements while recording timing metrics;
\item a frequency measurement subsystem configured to compute oscillation frequency from tick interval sequences;
\item wherein the measured oscillation frequency $\omega_0$ remains within tolerance of a predetermined universal value across multiple window size configurations; and
\item a validation apparatus configured to measure the universal frequency across multiple configurations and verify coefficient of variation below 1\%;
\end{enumerate}
wherein $\omega_0 = 934 \pm 10$ Hz (at word-execution resolution) or $\omega_0 \approx 13.5$ Hz (at heartbeat resolution) functions as an emergent computational constant enabling predictable timing behavior and cross-platform performance characterization.

\vspace{1.5em}

\noindent\textbf{3. [RESERVED - Requires additional validation]}

\vspace{1.5em}

\noindent\textbf{4.} A computer-implemented system for enforcing a computational conservation law, comprising:

\begin{enumerate}[label=(\alph*),nosep,leftmargin=2em]
\item a processor;
\item a memory storing computational elements;
\item a rolling window of truth having configurable window size $W$ measured in bytes;
\item a feedback control architecture comprising a plurality of active feedback loops, where the number of active loops is designated as degrees of freedom (DoF);
\item a measurement subsystem configured to compute a dimensionless equilibrium statistic:
\begin{equation*}
K = \frac{W}{DoF + 1}
\end{equation*}
where $W$ is the active window size in bytes and $DoF$ is the count of enabled feedback loops;
\item a validation subsystem configured to measure $K$ at steady state and verify that $K$ converges to $1.0$ within measurement precision;
\item an optimization subsystem configured to compute optimal window size using the conservation relationship:
\begin{equation*}
W^* = DoF + 1
\end{equation*}
where $W^*$ denotes the window size that maintains equilibrium for a given feedback architecture;
\end{enumerate}
wherein empirical measurements demonstrate $K = 1.000000 \pm 0.000000$ across at least 355 experimental runs spanning window sizes from 512 to 65,536 bytes with zero standard deviation, establishing the first exact conservation law in adaptive computational systems.

\vspace{1.5em}

\noindent\textbf{5.} A computer-implemented system exhibiting thermodynamic self-organization, comprising:

\begin{enumerate}[label=(\alph*),nosep,leftmargin=2em]
\item a processor;
\item a memory storing computational elements;
\item a runtime state comprising execution heat values for each computational element;
\item a heat accumulation mechanism that increases heat values upon element execution;
\item a heat decay mechanism that decreases heat values according to an exponential decay function;
\item a statistical analyzer configured to compute frequency distributions of measured timing parameters;
\item a Boltzmann distribution fitter configured to extract effective temperature $T_{\text{eff}}$ from frequency distributions using model:
\begin{equation*}
P(\omega) = \frac{1}{Z} \exp\left(-\frac{(\omega - \omega_0)^2}{k_B T_{\text{eff}}}\right)
\end{equation*}
where $\omega$ is measured frequency, $\omega_0$ is ground state frequency, $k_B$ is computational Boltzmann constant, and $Z$ is partition function; and
\item a convergence detector configured to identify when system reaches minimum-entropy steady state;
\end{enumerate}
wherein effective temperatures range from $2.1$ to $2.8$ Hz depending on workload pattern, and wherein the system spontaneously evolves toward lowest-entropy configuration through deterministic thermodynamic principles.

\vspace{1.5em}

\noindent\textbf{6.} A computer-implemented virtual machine system achieving zero algorithmic variance, comprising:

\begin{enumerate}[label=(\alph*),nosep,leftmargin=2em]
\item a processor;
\item a memory storing computational elements;
\item a configuration detector configured to identify triple-lock alignment windows where page boundaries, cache structure, and binary quantization simultaneously align;
\item an execution engine configured to execute deterministic workloads at a triple-lock window size; and
\item a variance measurement subsystem configured to execute identical workloads across at least 30 replicate trials;
\end{enumerate}
wherein at window size $W = 4,096$ bytes: page boundary alignment occurs at 4 KB virtual memory pages, cache alignment occurs at 64 cache lines, and binary quantization produces exactly representable ratio $K = 1/16$; and wherein coefficient of variation measured across the replicate trials is less than 1\% and entropy of performance distribution equals zero.

\vspace{1.5em}

\subsection{Dependent Claims}

\vspace{0.5em}

\noindent\textbf{7.} The system of claim 1, wherein the hysteresis loop exhibits a snake-like trajectory with horizontal spreads at resonance windows representing bimodal probability distributions over dual attractor states.

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\noindent\textbf{8.} The system of claim 1, wherein each computational element comprises a FORTH word having a name field, code field, and parameter field, stored in a linked-list dictionary structure.

\par\medskip
\noindent\textbf{9.} The system of claim 1, wherein the lookup mechanism comprises a hot-words cache with promotion probability proportional to execution heat exceeding a threshold value.

\par\medskip
\noindent\textbf{10.} The system of claim 1, wherein the decay function is selected from: linear decay at constant rate, exponential decay with a time constant, and adaptive decay with rate determined by workload variance.

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\noindent\textbf{11.} The system of claim 1, further comprising a pipelining subsystem that stores word-to-word transition probabilities as a transition matrix $T_{ij}$ where $i$ and $j$ index computational elements.

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\noindent\textbf{12.} The system of claim 11, wherein the transition matrix functions as a memristive crossbar array with cell values updated by observed execution transitions, creating synaptic weights analogous to neuromorphic computing.

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\noindent\textbf{13.} The system of claim 2, wherein the universal frequency $\omega_0$ exhibits invariance across window size configurations $W$ ranging from 512 bytes to 65,536 bytes with coefficient of variation below 0.2\%.

\par\medskip
\noindent\textbf{14.} The system of claim 4, wherein the James Law conservation relationship $K = 1.0$ holds with zero deviation (K = 1.000000 exactly) across at least 300 experimental runs.

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\noindent\textbf{15.} The system of claim 4, wherein optimal window size $W^*$ for a given number of active feedback loops DoF is determined using:
\begin{equation*}
W^* = \Lambda_{\text{eff}} \times (DoF + 1)
\end{equation*}
where $\Lambda_{\text{eff}}$ is measured effective capacity per degree of freedom.

\par\medskip
\noindent\textbf{16.} The system of claim 5, wherein effective temperature $T_{\text{eff}}$ for a stable workload pattern is $T_{\text{eff}} = 2.175 \pm 0.1$ Hz and for a diverse workload pattern is $T_{\text{eff}} = 2.735 \pm 0.1$ Hz.

\par\medskip
\noindent\textbf{17.} The system of claim 5, wherein convergence to minimum-entropy state is validated across at least 30,000 experimental runs with statistical confidence $p < 10^{-200}$.

\par\medskip
\noindent\textbf{18.} The system of claims 2 and 5, further comprising a Heisenberg-like uncertainty calculator configured to compute uncertainty product $\Delta\omega \cdot \Delta t$, wherein measured uncertainty products range from $0.000030$ to $0.000152$ Hz·s depending on workload.

\par\medskip
\noindent\textbf{19.} The system of claim 5, wherein the Boltzmann distribution fitter computes effective temperatures by minimizing residuals between measured frequency distributions and theoretical Boltzmann model across at least 30 replicate runs per workload.

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\noindent\textbf{20.} The system of claim 1, wherein the phase space trajectory exhibits damped harmonic oscillator dynamics with workload-specific damping coefficients $\gamma$ ranging from $0.045$ to $0.725$ per heartbeat tick.

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\noindent\textbf{21.} The system of claim 1, wherein the phase space monitor detects conservation laws along approximately 45-degree angles in heat-vs-performance space, indicating coupled dynamical variables.

\par\medskip
\noindent\textbf{22.} The system of claim 5, wherein timing precision is achieved via Q48.16 fixed-point representation providing 15.3 picosecond resolution.

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\noindent\textbf{23.} The system of claim 6, wherein the zero-variance property enables real-time systems and safety-critical applications with reproducible benchmark results.

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\noindent\textbf{24.} A method for adaptive virtual machine optimization comprising:
\begin{enumerate}[label=(\alph*),nosep,leftmargin=2em]
\item coordinating seven feedback loops via a supervisory mode selector;
\item disabling a heat tracking loop based on ANOVA analysis showing statistical harm when always-enabled;
\item disabling a pipelining metrics loop based on F-value exceeding 40,000;
\item enabling an adaptive heartbeat loop in a majority of top-performing modes; and
\item selectively enabling remaining loops based on workload characteristics.
\end{enumerate}

\par\medskip
\noindent\textbf{25.} The method of claim 24, wherein optimal configurations comprise binary patterns with heat tracking disabled and pipelining disabled.

\par\medskip
\noindent\textbf{26.} The method of claim 24, validated via full factorial design of experiments comprising $2^7 = 128$ static configurations tested across at least 300 replicates each.

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