Reply on RC4

<strong class="journal-contentHeaderColor">Abstract.</strong> <strong>Background:</strong> The true limits to economic growth and how monetary value is determined relative to energy, goods, and services, are primary unresolved questions in economics. <em>Lotka&rsquo;s wheel and the long arm of history</em> (Garrett et al., 2022) presents the conjecture that the past exerts an influence on the present through a ratio of <span style="border-bottom: 1px solid #000; vertical-align: 60%; font-size: .9em; color: #000;">&sum;<em>ⁿ</em>_{<em>i</em> = 0}<em>GW&thinsp;P_i</em></span><span style="margin-left: -3.3em; margin-right: 2.5em; vertical-align: -30%; font-size: .9em; color: #000;"><em>E</em></span> , where <em>E</em> is energy. The numerator of this ratio is a trailing average composed of the sum of gross world product (<em>GW&thinsp;P</em>) over <em>all</em> of human time, given the variable name <em>W</em> . The conjecture presents <span style="border-bottom: 1px solid #000; vertical-align: 50%; font-size: .8em; color: #000;"><em>W</em></span><span style="margin-left: -0.8em; margin-right: .5em; vertical-align: -35%; font-size: .8em; color: #000;"><em>E</em></span> as so close to fixed that a constant <em>w</em> is substituted for it, based on a thermodynamic argument available in past work. The <span style="border-bottom: 1px solid #000; vertical-align: 50%; font-size: .8em; color: #000;"><em>W</em></span><span style="margin-left: -0.8em; margin-right: .5em; vertical-align: -35%; font-size: .8em; color: #000;"><em>E</em></span> conjecture follow-on proposition is that when the first derivative of <em>GW&thinsp;P</em>, which equation includes energy, is in no growth balance, meaning <span style="border-bottom: 1px solid #000; vertical-align: 50%; font-size: .8em; color: #000;"><em>dE</em></span><span style="margin-left: -1.3em; margin-right: .5em; vertical-align: -35%; font-size: .8em; color: #000;"><em>dt</em></span> = 0, then inflation forces real <em>GW&thinsp;P</em> to zero, even when nominal <em>GW&thinsp;P</em> is maintained. <em>Lotka&rsquo;s wheel</em> has a 4 column supplement covering year 1&ndash;2019 CE: A. global <em>GDP</em>; B. <em>W</em> (yearly results of the <em>&sum;ⁿ_i</em>_{= 0}<em> GW&thinsp;P_i</em>); C. population; and D. energy in exajoules <em>E</em>. Replicate datasets were assembled from literature for <em>GW&thinsp;P</em> (<em>GW&thinsp;P_Rep</em>) and <em>E</em> (<em>E_Rep</em>). <strong>Two problems:</strong> First, the equation leading to the central proposition that real gross world product will become zero when <span style="border-bottom: 1px solid #000; vertical-align: 50%; font-size: .8em; color: #000;"><em>dE</em></span><span style="margin-left: -1.3em; margin-right: .5em; vertical-align: -35%; font-size: .8em; color: #000;"><em>dt</em></span> = 0 is based on erroneous use of an approximation constant for a relationship that is not actually a constant. All avenues for support of this proposition fail. Second, prior to 1970 <span style="border-bottom: 1px solid #000; vertical-align: 50%; font-size: .8em; color: #000;"><em>W_Rep</em></span><span style="margin-left: -2.3em; margin-right: .5em; vertical-align: -35%; font-size: .8em; color: #000;"><em>E_Rep</em></span> is radically higher than the supplement's <span style="border-bottom: 1px solid #000; vertical-align: 50%; font-size: .8em; color: #000;"><em>W</em></span><span style="margin-left: -0.8em; margin-right: .5em; vertical-align: -35%; font-size: .8em; color: #000;"><em>E</em></span> because <em>GW&thinsp;P_Rep</em> values are somewhat higher, and <em>E_Rep</em>values are much lower. Key replicate data are high confidence. <strong>Conclusion: </strong>As presented, the long arm of history hypothesis is falsified. However, the thermodynamic argument in prior work is compelling, and in 1970, a radical change in slope of <span style="border-bottom: 1px solid #000; vertical-align: 50%; font-size: .8em; color: #000;"><em>W</em></span><span style="margin-left: -0.8em; margin-right: .5em; vertical-align: -35%; font-size: .8em; color: #000;"><em>E</em></span> occurs. From 1970 forward, the long-arm of history hypothesis as presented appears probable. I believe that prior to 1970, the long-arm of history hypothesis may be true, but there are other factors that are not understood at this time.