120 lines
27 KiB
TeX
120 lines
27 KiB
TeX
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\@writefile{toc}{\contentsline {section}{\numberline {1}Technical Field of the Invention}{5}{section.1}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Overview}{10}{subsection.3.1}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces \textbf {FIG. 1 -- Configuration Space Distribution.} Performance distribution across the $2^7 = 128$ static configuration space, illustrating the wide variance in execution behavior when feedback loops are configured manually. The distribution demonstrates that static configurations produce highly variable performance outcomes, motivating the need for autonomous mode selection.}}{14}{figure.1}\protected@file@percent }
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\newlabel{fig:config_distribution}{{1}{14}{\textbf {FIG. 1 -- Configuration Space Distribution.} Performance distribution across the $2^7 = 128$ static configuration space, illustrating the wide variance in execution behavior when feedback loops are configured manually. The distribution demonstrates that static configurations produce highly variable performance outcomes, motivating the need for autonomous mode selection}{figure.1}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces \textbf {FIG. 2 -- Configuration Ranking.} Ranking of static configurations by mean performance and stability metrics. This analysis identifies candidate high-performance configurations that form the basis for validated execution modes in the Steady State Machine. All top-performing configurations share the pattern L1=0, L4=0.}}{15}{figure.2}\protected@file@percent }
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\newlabel{fig:config_ranking}{{2}{15}{\textbf {FIG. 2 -- Configuration Ranking.} Ranking of static configurations by mean performance and stability metrics. This analysis identifies candidate high-performance configurations that form the basis for validated execution modes in the Steady State Machine. All top-performing configurations share the pattern L1=0, L4=0}{figure.2}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces \textbf {FIG. 3 -- Main Effects Analysis.} Main effects plot showing the influence of individual feedback loops (L1--L7) on overall system performance. ANOVA analysis reveals that L1 (heat tracking) and L4 (pipelining metrics) are statistically harmful when always-enabled ($F > 1000$, $p < 10^{-200}$), while L7 (adaptive heartbeat) appears in 71\% of top-performing modes.}}{16}{figure.3}\protected@file@percent }
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\newlabel{fig:main_effects}{{3}{16}{\textbf {FIG. 3 -- Main Effects Analysis.} Main effects plot showing the influence of individual feedback loops (L1--L7) on overall system performance. ANOVA analysis reveals that L1 (heat tracking) and L4 (pipelining metrics) are statistically harmful when always-enabled ($F > 1000$, $p < 10^{-200}$), while L7 (adaptive heartbeat) appears in 71\% of top-performing modes}{figure.3}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces \textbf {FIG. 4 -- Configuration Runoff Comparison.} Box plot comparison of candidate top-performing configurations under identical workload conditions. This runoff analysis validates that the selected execution modes represent genuine performance optima rather than statistical artifacts. Configuration 100101 emerges as the winner with optimality score 0.018.}}{17}{figure.4}\protected@file@percent }
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\newlabel{fig:runoff}{{4}{17}{\textbf {FIG. 4 -- Configuration Runoff Comparison.} Box plot comparison of candidate top-performing configurations under identical workload conditions. This runoff analysis validates that the selected execution modes represent genuine performance optima rather than statistical artifacts. Configuration 100101 emerges as the winner with optimality score 0.018}{figure.4}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces \textbf {FIG. 5 -- Optimality Analysis.} Multi-objective optimality scores across configuration candidates, balancing throughput, variance reduction, and convergence speed. The Steady State Machine's mode selector uses similar multi-criteria evaluation to select appropriate execution modes dynamically.}}{18}{figure.5}\protected@file@percent }
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\newlabel{fig:optimality}{{5}{18}{\textbf {FIG. 5 -- Optimality Analysis.} Multi-objective optimality scores across configuration candidates, balancing throughput, variance reduction, and convergence speed. The Steady State Machine's mode selector uses similar multi-criteria evaluation to select appropriate execution modes dynamically}{figure.5}{}}
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\newlabel{fig:shape1}{{6}{19}{\textbf {FIG. 6 -- Workload Shape Performance (Set I).} Performance measurements across the first family of workload shapes, including sinusoidal (damped sine), triangular, and square-wave patterns. The Steady State Machine maintains consistent performance regardless of input waveform characteristics, with coefficient of variation below 3.2\% across all shapes}{figure.6}{}}
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\newlabel{fig:shape2}{{7}{20}{\textbf {FIG. 7 -- Workload Shape Performance (Set II).} Performance validation across additional workload waveforms demonstrating shape-invariant behavior. The system achieves stable throughput across all tested patterns with final confirmation at $n=300$ replicates per shape}{figure.7}{}}
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\newlabel{fig:cv_comparison}{{8}{21}{\textbf {FIG. 8 -- Coefficient of Variation Analysis.} Comparison of coefficient of variation (CV) across workload families, demonstrating that the adaptive system maintains low variance regardless of workload shape. Mean CV = 2.09\% across all shapes indicates predictable, stable execution behavior}{figure.8}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces \textbf {FIG. 9 -- Mode Usage Distribution.} Stacked distribution showing how the Jacquard Mode Selector (L8) allocates time across different execution modes for various workload families. The system autonomously selects appropriate modes: Mode 0 (19.3\%), Mode 1 (79.1\%), Mode 2 (1.4\%), Mode 3 (0.2\%), with mean of 1 mode switch per run.}}{22}{figure.9}\protected@file@percent }
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\newlabel{fig:mode_usage}{{9}{22}{\textbf {FIG. 9 -- Mode Usage Distribution.} Stacked distribution showing how the Jacquard Mode Selector (L8) allocates time across different execution modes for various workload families. The system autonomously selects appropriate modes: Mode 0 (19.3\%), Mode 1 (79.1\%), Mode 2 (1.4\%), Mode 3 (0.2\%), with mean of 1 mode switch per run}{figure.9}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces \textbf {FIG. 10 -- Adaptive vs. Static Performance.} Direct comparison between the Steady State Machine's adaptive behavior and equivalent static configurations. The adaptive system matches or exceeds static performance while providing automatic workload adaptation and dramatically lower variance in mixed or unpredictable workloads.}}{23}{figure.10}\protected@file@percent }
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\newlabel{fig:adaptive_vs_static}{{10}{23}{\textbf {FIG. 10 -- Adaptive vs. Static Performance.} Direct comparison between the Steady State Machine's adaptive behavior and equivalent static configurations. The adaptive system matches or exceeds static performance while providing automatic workload adaptation and dramatically lower variance in mixed or unpredictable workloads}{figure.10}{}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {4.5}State Vector Dynamics and K-Signal Physics}{24}{subsection.4.5}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces \textbf {FIG. 11 -- State Vector Trajectory (Snake Path).} Representative trajectory of the runtime state vector through the execution heat--K parameter phase space. The hysteresis-like behavior demonstrates that the system exhibits memory effects: the current state depends not only on instantaneous workload but also on recent execution history. This path-dependent characteristic enables stable convergence to appropriate operating points.}}{24}{figure.11}\protected@file@percent }
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\newlabel{fig:snake_trajectory}{{11}{24}{\textbf {FIG. 11 -- State Vector Trajectory (Snake Path).} Representative trajectory of the runtime state vector through the execution heat--K parameter phase space. The hysteresis-like behavior demonstrates that the system exhibits memory effects: the current state depends not only on instantaneous workload but also on recent execution history. This path-dependent characteristic enables stable convergence to appropriate operating points}{figure.11}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {12}{\ignorespaces \textbf {FIG. 12 -- K-Signal vs. Window Size (James Law).} Relationship between the stability constant $K$ and the observation window size $W$, showing the fundamental scaling law $K = \lambda _0 / W$ where $\lambda _0 = 256$ bytes. The periodic structure reveals fundamental resonances in the adaptive architecture. This relationship---termed James Law---provides predictive capability for system behavior at untested configurations.}}{25}{figure.12}\protected@file@percent }
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\newlabel{fig:k_vs_window}{{12}{25}{\textbf {FIG. 12 -- K-Signal vs. Window Size (James Law).} Relationship between the stability constant $K$ and the observation window size $W$, showing the fundamental scaling law $K = \lambda _0 / W$ where $\lambda _0 = 256$ bytes. The periodic structure reveals fundamental resonances in the adaptive architecture. This relationship---termed James Law---provides predictive capability for system behavior at untested configurations}{figure.12}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {13}{\ignorespaces \textbf {FIG. 13 -- Spectral Analysis.} Fast Fourier Transform spectrum of K-signal residuals (measured $K$ minus baseline $K = 256/W$), revealing the characteristic frequencies and periodic structures inherent in the Steady State Machine's feedback architecture. Dominant spectral peak at natural frequency $f_0 = 0.667 \pm 0.02$ cycles per window doubling, validated with $p < 0.0001$.}}{26}{figure.13}\protected@file@percent }
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\newlabel{fig:fft_spectrum}{{13}{26}{\textbf {FIG. 13 -- Spectral Analysis.} Fast Fourier Transform spectrum of K-signal residuals (measured $K$ minus baseline $K = 256/W$), revealing the characteristic frequencies and periodic structures inherent in the Steady State Machine's feedback architecture. Dominant spectral peak at natural frequency $f_0 = 0.667 \pm 0.02$ cycles per window doubling, validated with $p < 0.0001$}{figure.13}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {14}{\ignorespaces \textbf {FIG. 14 -- Cache Resonance Pattern.} Performance variations showing interference effects at window sizes $W = 3 \times 2^N$. These patterns demonstrate that the system exhibits cache-like resonance behavior. Fibonacci-sequence windows naturally avoid these penalties through harmonic alignment.}}{26}{figure.14}\protected@file@percent }
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\newlabel{fig:golden_ratio}{{14}{26}{\textbf {FIG. 14 -- Cache Resonance Pattern.} Performance variations showing interference effects at window sizes $W = 3 \times 2^N$. These patterns demonstrate that the system exhibits cache-like resonance behavior. Fibonacci-sequence windows naturally avoid these penalties through harmonic alignment}{figure.14}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {15}{\ignorespaces \textbf {FIG. 15 -- Bimodal State Distributions.} Distribution of K-signal measurements at resonance windows $W \in \{6144, 16384\}$ bytes showing bimodal behavior. Under resonance conditions, the system exhibits discrete stable states---a ``locked'' regime ($K \approx 0.04$) and an ``escaped'' regime ($K \rightarrow 1.0$)---with 47--53\% probability splits analogous to bistable physical systems.}}{27}{figure.15}\protected@file@percent }
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\newlabel{fig:bimodal}{{15}{27}{\textbf {FIG. 15 -- Bimodal State Distributions.} Distribution of K-signal measurements at resonance windows $W \in \{6144, 16384\}$ bytes showing bimodal behavior. Under resonance conditions, the system exhibits discrete stable states---a ``locked'' regime ($K \approx 0.04$) and an ``escaped'' regime ($K \rightarrow 1.0$)---with 47--53\% probability splits analogous to bistable physical systems}{figure.15}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {16}{\ignorespaces \textbf {FIG. 16 -- Performance Correlation with K.} Correlation between the stability constant $K$ and measured performance metrics. This relationship enables the mode selector to predict performance outcomes based on state vector measurements and validates that $K$ functions as a meaningful control parameter.}}{28}{figure.16}\protected@file@percent }
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\newlabel{fig:performance_vs_k}{{16}{28}{\textbf {FIG. 16 -- Performance Correlation with K.} Correlation between the stability constant $K$ and measured performance metrics. This relationship enables the mode selector to predict performance outcomes based on state vector measurements and validates that $K$ functions as a meaningful control parameter}{figure.16}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {17}{\ignorespaces \textbf {FIG. 17 -- Performance by Operating Regime.} Performance breakdown across different operating regimes identified by the K-signal: locked regime ($K < 0.1$), transition zone ($0.1 < K < 0.5$), and escaped regime ($K > 0.5$). Each regime corresponds to distinct feedback loop configurations and mode selections optimized for that operating point.}}{29}{figure.17}\protected@file@percent }
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\newlabel{fig:regime_performance}{{17}{29}{\textbf {FIG. 17 -- Performance by Operating Regime.} Performance breakdown across different operating regimes identified by the K-signal: locked regime ($K < 0.1$), transition zone ($0.1 < K < 0.5$), and escaped regime ($K > 0.5$). Each regime corresponds to distinct feedback loop configurations and mode selections optimized for that operating point}{figure.17}{}}
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\newlabel{tab:anova}{{1}{40}{\textbf {TABLE 1 -- ANOVA Results.} Main effects analysis across 38,400 experimental runs. L1 and L4 are statistically harmful when always-enabled, motivating selective activation via the Jacquard Mode Selector}{table.1}{}}
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\newlabel{tab:top_configs}{{2}{41}{\textbf {TABLE 2 -- Top Configurations.} All top-performing configurations share L1=0, L4=0 pattern. Best configuration 0100011 enables L2 (rolling window), L6 (decay inference), and L7 (heartbeat)}{table.2}{}}
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\newlabel{tab:k_signal}{{3}{42}{\textbf {TABLE 3 -- K-Signal Validation.} Measured $K$ values follow James Law ($K = 256/W$) at non-resonance windows. Large positive residuals at $W \in \{6144, 16384\}$ indicate resonance bimodal behavior}{table.3}{}}
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\newlabel{tab:bimodal}{{4}{43}{\textbf {TABLE 4 -- Bimodal Distribution Validation.} Resonance windows exhibit dual-attractor behavior with 47--53\% splits. Quantized state $K = 1.000$ occurs with exactly 3.3\% probability at resonance, zero at anti-resonance}{table.4}{}}
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\newlabel{tab:mode_usage}{{5}{43}{\textbf {TABLE 5 -- Mode Usage by Workload Type.} The Jacquard Mode Selector exhibits deterministic mode selection with mean 1 switch per run and 79.1\% Mode 1 occupancy across all workload families}{table.5}{}}
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\newlabel{tab:shape_invariance}{{6}{44}{\textbf {TABLE 6 -- Shape Invariance Validation.} The Steady State Machine maintains low variance (CV $< 2.5\%$) across all tested workload shapes, demonstrating shape-invariant operation}{table.6}{}}
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