201 lines
7.9 KiB
TeX
201 lines
7.9 KiB
TeX
%% SCRAP: architecture/03-architecture/physics-engine/ssm-equations
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%% SOURCE: docs/working/architecture/03-architecture/physics-engine/ssm-equations.md
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%% STATUS: WORKING
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%% FITS: dev-guide/ch-physics
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%% EDITORIAL: lifted — prose rewritten to press voice
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\section{Mathematical Models of the Steady-State Machine}
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%% PATENT: This scrap covers patent-adjacent empirical relationships (the capacity
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%% law and its candidate claim language). Claim wording in the source is summarized
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%% as description only; no claim language is drafted here. Flag for Bob before any
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%% promotion that touches the claims.
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This section presents performance models derived empirically from the
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Steady-State Machine (SSM) experimental record: 51{,}840 runs of the StarForth
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VM across 128 feedback-loop configurations. Five relationships were extracted
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from the data with measurable precision. The headline result is a capacity law,
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$\Lambda(\mathrm{DoF}) = 4096 / (\mathrm{DoF}+1)$, which holds with a 0.00\%
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coefficient of variation across all stable configurations.
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These models are exploratory. Several borrow functional forms and names from
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physics --- the Lorentz factor, the Friedmann equation, the Schwarzschild
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radius --- purely because the data happens to fit those forms. The physics
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vocabulary is a modeling convenience for describing adaptive-system behavior; it
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is not a claim that the runtime obeys relativity or cosmology. Throughout,
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\textit{degrees of freedom} (DoF) denotes the number of enabled feedback loops,
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ranging from 0 to 7.
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\subsection{Equation 1 --- Performance Scaling with Load}
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Execution time scales with the number of enabled loops according to a Lorentz-like
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factor:
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\begin{equation}
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\tau(\mathrm{DoF}) = \tau_0 \cdot \gamma(\mathrm{DoF}),
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\qquad
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\gamma = \frac{1}{\sqrt{1 - \beta^2}},
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\qquad
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\beta^2(\mathrm{DoF}) = 0.0736\,\mathrm{DoF} + 0.1464,
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\end{equation}
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with $\tau_0 = 31.67$\,ms/word the rest-state performance. The empirical fit is
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$R^2 = 0.938$. The $\beta^2$ term grows linearly with DoF, so each additional
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loop carries increasing marginal overhead. The structural similarity to
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relativistic time dilation is offered as a predictive model, not a physical
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claim.
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\subsection{Equation 2 --- Window Capacity Law $\Lambda$}
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The central empirical finding is an inverse capacity law:
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\begin{equation}
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\Lambda(\mathrm{DoF}) = \frac{W_0}{\mathrm{DoF} + 1},
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\qquad W_0 = 4096.
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\end{equation}
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Equivalently, the product $\Lambda \times (\mathrm{DoF}+1) = W_0$ is conserved
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across all stable configurations.
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\begin{table}[ht]
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\centering
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\small
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\begin{tabular}{rrrr}
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\toprule
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DoF & Ideal $\Lambda$ & $\Lambda \times (\mathrm{DoF}+1)$ & Stability (CV) \\
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\midrule
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0 & 4096.0 & 4096.0 & 17.34\% \\
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1 & 2048.0 & 4096.0 & 16.70\% \\
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2 & 1365.3 & 4096.0 & 15.31\% \\
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3 & 1024.0 & 4096.0 & 13.91\% \\
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4 & 819.2 & 4096.0 & 13.77\% \\
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5 & 682.7 & 4096.0 & 13.96\% \\
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6 & 585.1 & 4096.0 & 14.33\% \\
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7 & 512.0 & 4096.0 & 22.06\% \\
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\bottomrule
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\end{tabular}
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\caption{Window capacity verification. The conserved product holds at $4096.0 \pm 0.0$, a 0.00\% coefficient of variation.}
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\end{table}
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The symbol $\Lambda$ is borrowed from cosmology for convenience: it denotes the
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effective window capacity per degree of freedom, with $W_0 = 4096$ bytes an
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empirically determined constant. As loops are added, proportional window
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capacity must fall to preserve stability, and the conserved product behaves as a
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stability invariant. Notably, $W_0 = 4096 = 2^{12}$ was not designed --- it
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emerged from experimental optimization, and may reflect memory page-size
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alignment, cache-line constraints, or buffer-size effects.
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\subsection{Equation 3 --- Expansion Dynamics (Friedmann Form)}
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The growth in execution time as loops activate fits a Friedmann-like form:
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\begin{equation}
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H^2(\mathrm{DoF}) \approx c_1\,\rho(\mathrm{DoF}) + c_2\,\Lambda(\mathrm{DoF}) + c_3,
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\end{equation}
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with the fitted relation
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\begin{equation}
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H^2 = -2.24\times10^{12}\,\mathrm{DoF} \;-\; 4.51\times10^{9}\,\Lambda \;+\; 2.11\times10^{13}.
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\end{equation}
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Here $H$ stands in for the performance growth rate, $\rho$ for the degrees of
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freedom, and $\Lambda$ for window-capacity pressure. The negative coefficients
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indicate that both DoF and $\Lambda$ contribute to deceleration --- the system
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resists unbounded growth through feedback regulation.
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\subsection{Equation 4 --- Instability Threshold (Schwarzschild Form)}
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A Schwarzschild-like radius models the onset of instability:
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\begin{equation}
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r_s(\mathrm{DoF}) = \frac{\mathrm{DoF}}{W_{\mathrm{norm}}},
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\qquad W_{\mathrm{norm}} = \frac{\text{window}}{4096}.
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\end{equation}
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In this analogy $r_s$ is the instability threshold, DoF the computational load,
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and $W_{\mathrm{norm}}$ the available space. The configurations with the highest
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coefficient of variation sit near this threshold.
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\begin{table}[ht]
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\centering
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\small
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\begin{tabular}{rrrrr}
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\toprule
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Config & DoF & $r_s$ & CV & Performance \\
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\midrule
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1 & 1 & 1.00 & 26.90\% & 32.43\,ms \\
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84 & 3 & 3.00 & 26.36\% & 34.94\,ms \\
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46 & 4 & 4.00 & 25.70\% & 50.27\,ms \\
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59 & 5 & 5.00 & 25.31\% & 50.43\,ms \\
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\bottomrule
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\end{tabular}
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\caption{Configurations nearest the instability threshold. Where $r_s \ge \mathrm{DoF}$, the system tends toward instability.}
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\end{table}
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\subsection{Equation 5 --- Energy--Momentum Form}
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Total execution time fits an energy--momentum relation:
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\begin{equation}
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E^2 = E_0^2 + p^2,
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\qquad E = \tau(\mathrm{DoF}),
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\qquad E_0 = \tau_0 = 31.67\,\mathrm{ms},
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\qquad p(\mathrm{DoF}) = 5.20\,\mathrm{DoF} + 6.55.
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\end{equation}
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Here $E_0$ is the all-loops-off baseline, $p$ the additional time cost from
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loops, and $E$ total execution time. Momentum scales linearly with DoF
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($R = 0.425$, $p < 10^{-6}$).
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\begin{table}[ht]
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\centering
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\small
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\begin{tabular}{rrrrr}
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\toprule
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Config & DoF & $E_{\text{total}}$ & $E_{\text{rest}}$ & $p$ \\
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\midrule
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0 & 0 & 31.67 & 31.67 & 0.00 \\
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35 & 3 & 31.59 & 31.67 & 0.00 \\
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55 & 5 & 31.91 & 31.67 & 3.88 \\
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124 & 5 & 59.48 & 31.67 & 50.35 \\
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\bottomrule
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\end{tabular}
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\caption{Energy--momentum decomposition for selected configurations.}
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\end{table}
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Two configurations are notable. Config~\#35 carries three enabled loops yet sits
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at near-zero momentum, suggesting an optimal arrangement where loop interactions
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cancel. Config~\#124, the worst performer, carries the same DoF as Config~\#55
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but enormous momentum (50.35) --- a destructive combination of loops.
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\subsection{Interpretation and Open Experiments}
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The recurring observation is that these adaptive-runtime relationships fit
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functional forms also found in general relativity, thermodynamics, and quantum
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mechanics. The strongest of them, the capacity law, is exact to within
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measurement. Whether this reflects a deep property of feedback-driven systems or
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an emergent coincidence remains open; the SSM data is presented as evidence that
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the functional forms are useful predictive models.
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Several follow-on experiments are proposed:
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\begin{itemize}
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\item \textbf{Window-size scaling.} Vary the window across 1024, 2048, 4096,
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8192, and 16384 bytes and test whether $\Lambda(\mathrm{DoF}) =
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W/(\mathrm{DoF}+1)$ holds, i.e.\ whether the 4096 constant scales
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linearly with window size.
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\item \textbf{Critical window capacity.} Reduce the window in steps and locate
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a $W_{\text{crit}}$ below which the selector collapses to the all-off
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configuration.
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\item \textbf{Maximum degrees of freedom.} Extend beyond seven loops and test
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whether the capacity law holds up to a $\mathrm{DoF}_{\max}$, with the
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candidate $\mathrm{DoF}_{\max} \approx \log_2(W_0) = 12$.
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\end{itemize}
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%% PATENT: The source proposes patent claims based on the capacity law, the
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%% Lorentz-like scaling, and the Schwarzschild-like threshold, and outlines
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%% publication and funding strategy. That material is intentionally omitted here
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%% pending Bob's instruction; do not draft claim language in this scrap.
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%% TODO(bob): confirm which empirical relationships may appear in citable form.
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