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sum","display":"\\sum_{h=1}^H","kind":"use","occurrences":[126,130,140,142]},{"id":"symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","occurrences":[74,79,88,89]},{"id":"symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","occurrences":[141]},{"id":"symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","occurrences":[141]},{"id":"symbol-adc46c2b3119d58b","tex":"\\int_0^\\infty","aliases":"∫_0^∞ int infty","display":"\\int_0^\\infty","kind":"use","occurrences":[141]},{"id":"symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"use","occurrences":[139]},{"id":"symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","occurrences":[18,89,99,101,105]},{"id":"symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","occurrences":[85,91,92]},{"id":"symbol-362138992adfe804","tex":"\\star","aliases":"⋆ star","display":"\\star","kind":"use","occurrences":[3,57,58]}],"contexts":["sCopper}\\rule{1.25in}{1pt}\\par}\n\\vspace{.16in}\n\\noindent The crossing-energy obstruction, an abundance-preserving parameter family, and a quantum-mode calculation with explicit energy exchange.\\par\n\\vspace{.28in}\n\\noindent\\textbf{Komorebi P.}\\hfill October 2026\\par\n\\vspace{.35in}\n\\noindent\\textbf{Result and boundary.} The original prescribed-background benchmark exceeds its crossing reservoir by $2.6272875374\\times10^{20}$. This study constructs a finite dynamical system in which that kind of overdraw is excluded by an exact conservation law. A numerical implementation agrees with an independent oscillator representation and conserves its total energy to the stated tolerance.","\\section{The obstruction being addressed}\nThe source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark,\n\\begin{align}\n U(\\phi)&=\\frac{\\lambda}{4}(\\phi^2-v^2)^2,\n &\\mu^2&=\\lambda v^2,\\\\\n m_h&=\\Lambda 2^h,\n &d_h&=2^{h-1},\\\\\n M_h^2(\\phi)&=m_h^2\\left(1+\\gamma\\frac{\\phi^2}{v^2}\\right),\n &\\gamma&=16.\n\\end{align}\nThe general covariant construction and its limitations are given in the source monograph \\cite{komorebi}. This study fixes the dyadic order-one tower used in the stored benchmark. The quartic $\\lambda$ is not a spectral eigenvalue.","M_h^2(\\phi)&=m_h^2\\left(1+\\gamma\\frac{\\phi^2}{v^2}\\right),\n &\\gamma&=16.\n\\end{align}\nThe general covariant construction and its limitations are given in the source monograph \\cite{komorebi}. This study fixes the dyadic order-one tower used in the stored benchmark. The quartic $\\lambda$ is not a spectral eigenvalue.","For a prescribed locally linear crossing, put\n\\begin{equation}\n A=\\frac{\\sqrt\\gamma\\,|\\dot\\phi_\\star|}{v},\\qquad\n x=\\frac\\Lambda A=10^{-4},\\qquad y_h=\\frac{m_h}{A}=x2^h.\n\\end{equation}\nThe assumed single-crossing occupation is\n\\begin{equation}\n f_h(k)=\\exp\\!\\left[-\\pi\\left(\\frac{m_h}{A}+\\frac{k^2}{Am_h}\\right)\\right].\n \\label{eq:gaussian}\n\\end{equation}\nSuch a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent reaction is also central in established preheating calculations \\cite{kls,instant}.","Write the energy evaluated with the minimum masses as\n\\begin{equation}\n \\widetilde\\rho=A^4 S_E(x),\\qquad S_E(10^{-4})=2.18498795564,\n \\label{eq:SE}\n\\end{equation}\nwhere the exact discrete degeneracies are retained. The kinetic reservoir is\n\\begin{equation}\n K=\\frac{v^2 A^2}{2\\gamma},\\qquad\n \\rhoV=\\frac{\\mu^4}{4\\lambda},\\qquad\n \\frac K{\\rhoV}=\\frac{2}{\\gamma}\\left(\\frac A\\mu\\right)^2.\n \\label{eq:kinetic}\n\\end{equation}\nThe stored reference has $\\lambda_0=1$ and\n\\begin{align}\n a_0:=\\frac{A_0}{\\mu_0}&=1.93844860142\\times10^9,","es as\n\\begin{equation}\n \\widetilde\\rho=A^4 S_E(x),\\qquad S_E(10^{-4})=2.18498795564,\n \\label{eq:SE}\n\\end{equation}\nwhere the exact discrete degeneracies are retained. The kinetic reservoir is\n\\begin{equation}\n K=\\frac{v^2 A^2}{2\\gamma},\\qquad\n \\rhoV=\\frac{\\mu^4}{4\\lambda},\\qquad\n \\frac K{\\rhoV}=\\frac{2}{\\gamma}\\left(\\frac A\\mu\\right)^2.\n \\label{eq:kinetic}\n\\end{equation}\nThe stored reference has $\\lambda_0=1$ and\n\\begin{align}\n a_0:=\\frac{A_0}{\\mu_0}&=1.93844860142\\times10^9,","ilde\\rho=A^4 S_E(x),\\qquad S_E(10^{-4})=2.18498795564,\n \\label{eq:SE}\n\\end{equation}\nwhere the exact discrete degeneracies are retained. The kinetic reservoir is\n\\begin{equation}\n K=\\frac{v^2 A^2}{2\\gamma},\\qquad\n \\rhoV=\\frac{\\mu^4}{4\\lambda},\\qquad\n \\frac K{\\rhoV}=\\frac{2}{\\gamma}\\left(\\frac A\\mu\\right)^2.\n \\label{eq:kinetic}\n\\end{equation}\nThe stored reference has $\\lambda_0=1$ and\n\\begin{align}\n a_0:=\\frac{A_0}{\\mu_0}&=1.93844860142\\times10^9,\\\\\n \\frac{K_0}{\\rhoV}&=4.69697872544\\times10^{17},\\\\\n R_0:=\\frac{\\widetilde\\rho_0}{K_0}\n &=2.6272875374\\times10^{20}.\n\\end{align}\nThe original occupation projection used\n\\begin{equation}\n \\eta_0=R_0^{-1}=3.80620691783\\times10^{-21}.\n \\label{eq:eta}\n\\end{equation}\nIt made the reported density equal to $K_0$, but did not supply equations that remove this energy from the evolving scalar.","R_0:=\\frac{\\widetilde\\rho_0}{K_0}\n &=2.6272875374\\times10^{20}.\n\\end{align}\nThe original occupation projection used\n\\begin{equation}\n \\eta_0=R_0^{-1}=3.80620691783\\times10^{-21}.\n \\label{eq:eta}\n\\end{equation}\nIt made the reported density equal to $K_0$, but did not supply equations that remove this energy from the evolving scalar.","\\section{Removing overdraw inside the Gaussian family}\nSimply reducing $\\lambda$ at fixed $A/\\mu$ reduces both the overdraw and the output density relative to the old target. To preserve the target density while imposing a smaller nominal conversion fraction, the crossing scale in units of $\\mu$ must change too.","\\begin{theorem}[Conditional abundance-preserving family]\nFix $\\rhoV$, $\\gamma$, the tower order, and $x=\\Lambda/A$. Let the target density be the original projected value $\\rho_T=\\eta_0 A_0^4 S_E=K_0$. For any $0<\\eps\\le1$, choose\n\\begin{equation}\n \\boxed{\\lambda=\\eta_0\\eps^2,\\qquad\n \\frac A\\mu=\\frac{a_0}{\\sqrt\\eps},\\qquad\n \\frac\\Lambda\\mu=x\\frac{a_0}{\\sqrt\\eps}.}\n \\label{eq:family}\n\\end{equation}\nThen the unprojected Gaussian family satisfies\n\\begin{equation}\n \\widetilde\\rho=\\rho_T,\\qquad\n K=\\frac{K_0}{\\eps},\\qquad\n \\frac{\\widetilde\\rho}{K}=\\eps.\n \\label{eq:familyresult}\n\\end{equation}\nNo common occupation multiplier is used in this construction.\n\\end{theorem}\n\\begin{proof}\nAt fixed $\\rhoV$, $\\mu^4=4\\lambda\\rhoV$, so\n\\[\n \\widetilde\\rho=4\\rhoV\\lambda(A/\\mu)","\\begin{theorem}[Conditional abundance-preserving family]\nFix $\\rhoV$, $\\gamma$, the tower order, and $x=\\Lambda/A$. Let the target density be the original projected value $\\rho_T=\\eta_0 A_0^4 S_E=K_0$. For any $0<\\eps\\le1$, choose\n\\begin{equation}\n \\boxed{\\lambda=\\eta_0\\eps^2,\\qquad\n \\frac A\\mu=\\frac{a_0}{\\sqrt\\eps},\\qquad\n \\frac\\Lambda\\mu=x\\frac{a_0}{\\sqrt\\eps}.}\n \\label{eq:family}\n\\end{equation}\nThen the unprojected Gaussian family satisfies\n\\begin{equation}\n \\widetilde\\rho=\\rho_T,\\qquad\n K=\\frac{K_0}{\\eps},\\qquad\n \\frac{\\widetilde\\rho}{K}=\\eps.\n \\label{eq:familyresult}\n\\end{equation}\nNo common occupation multiplier is used in this construction.\n\\end{theorem}\n\\begin{proof}\nAt fixed $\\rhoV$, $\\mu^4=4\\lambda\\rhoV$, so\n\\[\n \\widetilde\\rho=4\\rhoV\\lambda(A/\\mu)^4S_E.\n\\]","\\begin{theorem}[Conditional abundance-preserving family]\nFix $\\rhoV$, $\\gamma$, the tower order, and $x=\\Lambda/A$. Let the target density be the original projected value $\\rho_T=\\eta_0 A_0^4 S_E=K_0$. For any $0<\\eps\\le1$, choose\n\\begin{equation}\n \\boxed{\\lambda=\\eta_0\\eps^2,\\qquad\n \\frac A\\mu=\\frac{a_0}{\\sqrt\\eps},\\qquad\n \\frac\\Lambda\\mu=x\\frac{a_0}{\\sqrt\\eps}.}\n \\label{eq:family}\n\\end{equation}\nThen the unprojected Gaussian family satisfies\n\\begin{equation}\n \\widetilde\\rho=\\rho_T,\\qquad\n K=\\frac{K_0}{\\eps},\\qquad\n \\frac{\\widetilde\\rho}{K}=\\eps.\n \\label{eq:familyresult}\n\\end{equation}\nNo common occupation multiplier is used in this construction.\n\\end{theorem}\n\\begin{proof}\nAt fixed $\\rhoV$, $\\mu^4=4\\lambda\\rhoV$, so\n\\[\n \\widetilde\\rho=4\\rhoV\\lambda(A/\\mu)^4S_E.\n\\]\nSubstitute \\eqref{eq:family}. The fa","\\begin{theorem}[Conditional abundance-preserving family]\nFix $\\rhoV$, $\\gamma$, the tower order, and $x=\\Lambda/A$. Let the target density be the original projected value $\\rho_T=\\eta_0 A_0^4 S_E=K_0$. For any $0<\\eps\\le1$, choose\n\\begin{equation}\n \\boxed{\\lambda=\\eta_0\\eps^2,\\qquad\n \\frac A\\mu=\\frac{a_0}{\\sqrt\\eps},\\qquad\n \\frac\\Lambda\\mu=x\\frac{a_0}{\\sqrt\\eps}.}\n \\label{eq:family}\n\\end{equation}\nThen the unprojected Gaussian family satisfies\n\\begin{equation}\n \\widetilde\\rho=\\rho_T,\\qquad\n K=\\frac{K_0}{\\eps},\\qquad\n \\frac{\\widetilde\\rho}{K}=\\eps.\n \\label{eq:familyresult}\n\\end{equation}\nNo common occupation multiplier is used in this construction.\n\\end{theorem}\n\\begin{proof}\nAt fixed $\\rhoV$, $\\mu^4=4\\lambda\\rhoV$, so\n\\[\n \\widetilde\\rho=4\\rhoV\\lambda(A/\\mu)^4S_E.\n\\]\nSubstitute \\eqref{eq:family}. The factors $\\eps^2$ and $\\eps^{-2}$ cancel, leaving $\\eta_0 A_0^4S_E$. Equation \\eqref{eq:k","\\begin{theorem}[Conditional abundance-preserving family]\nFix $\\rhoV$, $\\gamma$, the tower order, and $x=\\Lambda/A$. Let the target density be the original projected value $\\rho_T=\\eta_0 A_0^4 S_E=K_0$. For any $0<\\eps\\le1$, choose\n\\begin{equation}\n \\boxed{\\lambda=\\eta_0\\eps^2,\\qquad\n \\frac A\\mu=\\frac{a_0}{\\sqrt\\eps},\\qquad\n \\frac\\Lambda\\mu=x\\frac{a_0}{\\sqrt\\eps}.}\n \\label{eq:family}\n\\end{equation}\nThen the unprojected Gaussian family satisfies\n\\begin{equation}\n \\widetilde\\rho=\\rho_T,\\qquad\n K=\\frac{K_0}{\\eps},\\qquad\n \\frac{\\widetilde\\rho}{K}=\\eps.\n \\label{eq:familyresult}\n\\end{equation}\nNo common occupation multiplier is used in this construction.\n\\end{theorem}\n\\begin{proof}\nAt fixed $\\rhoV$, $\\mu^4=4\\lambda\\rhoV$, so\n\\[\n \\widetilde\\rho=4\\rhoV\\lambda(A/\\mu)^4S_E.\n\\]\nSubstitute \\eqref{eq:family}. The factors $\\eps^2$ and $\\eps^{-2}$ cancel, leaving $\\eta_0 A_0^4S_E$. Equation \\eqref{eq:kinetic} gives $K=K_0/\\","\\begin{theorem}[Conditional abundance-preserving family]\nFix $\\rhoV$, $\\gamma$, the tower order, and $x=\\Lambda/A$. Let the target density be the original projected value $\\rho_T=\\eta_0 A_0^4 S_E=K_0$. For any $0<\\eps\\le1$, choose\n\\begin{equation}\n \\boxed{\\lambda=\\eta_0\\eps^2,\\qquad\n \\frac A\\mu=\\frac{a_0}{\\sqrt\\eps},\\qquad\n \\frac\\Lambda\\mu=x\\frac{a_0}{\\sqrt\\eps}.}\n \\label{eq:family}\n\\end{equation}\nThen the unprojected Gaussian family satisfies\n\\begin{equation}\n \\widetilde\\rho=\\rho_T,\\qquad\n K=\\frac{K_0}{\\eps},\\qquad\n \\frac{\\widetilde\\rho}{K}=\\eps.\n \\label{eq:familyresult}\n\\end{equation}\nNo common occupation multiplier is used in this construction.\n\\end{theorem}\n\\begin{proof}\nAt fixed $\\rhoV$, $\\mu^4=4\\lambda\\rhoV$, so\n\\[\n \\widetilde\\rho=4\\rhoV\\lambda(A/\\mu)^4S_E.\n\\]\nSubstitute \\eqref{eq:family}. The factors $\\eps^2$ and $\\eps^{-2}$ cancel, leaving $\\eta_0 A_0^4S_E$. Equation \\eqref{eq:kinetic} gives $K=K_0/\\eps$. Division proves the final identity.\n\\end{proof}","hoV$, $\\gamma$, the tower order, and $x=\\Lambda/A$. Let the target density be the original projected value $\\rho_T=\\eta_0 A_0^4 S_E=K_0$. For any $0<\\eps\\le1$, choose\n\\begin{equation}\n \\boxed{\\lambda=\\eta_0\\eps^2,\\qquad\n \\frac A\\mu=\\frac{a_0}{\\sqrt\\eps},\\qquad\n \\frac\\Lambda\\mu=x\\frac{a_0}{\\sqrt\\eps}.}\n \\label{eq:family}\n\\end{equation}\nThen the unprojected Gaussian family satisfies\n\\begin{equation}\n \\widetilde\\rho=\\rho_T,\\qquad\n K=\\frac{K_0}{\\eps},\\qquad\n \\frac{\\widetilde\\rho}{K}=\\eps.\n \\label{eq:familyresult}\n\\end{equation}\nNo common occupation multiplier is used in this construction.\n\\end{theorem}\n\\begin{proof}\nAt fixed $\\rhoV$, $\\mu^4=4\\lambda\\rhoV$, so\n\\[\n \\widetilde\\rho=4\\rhoV\\lambda(A/\\mu)^4S_E.\n\\]\nSubstitute \\eqref{eq:family}. The factors $\\eps^2$ and $\\eps^{-2}$ cancel, leaving $\\eta_0 A_0^4S_E$. Equation \\eqref{eq:kinetic} gives $K=K_0/\\eps$. Division proves the final identity.\n\\end{proof}","u=\\frac{a_0}{\\sqrt\\eps},\\qquad\n \\frac\\Lambda\\mu=x\\frac{a_0}{\\sqrt\\eps}.}\n \\label{eq:family}\n\\end{equation}\nThen the unprojected Gaussian family satisfies\n\\begin{equation}\n \\widetilde\\rho=\\rho_T,\\qquad\n K=\\frac{K_0}{\\eps},\\qquad\n \\frac{\\widetilde\\rho}{K}=\\eps.\n \\label{eq:familyresult}\n\\end{equation}\nNo common occupation multiplier is used in this construction.\n\\end{theorem}\n\\begin{proof}\nAt fixed $\\rhoV$, $\\mu^4=4\\lambda\\rhoV$, so\n\\[\n \\widetilde\\rho=4\\rhoV\\lambda(A/\\mu)^4S_E.\n\\]\nSubstitute \\eqref{eq:family}. The factors $\\eps^2$ and $\\eps^{-2}$ cancel, leaving $\\eta_0 A_0^4S_E$. Equation \\eqref{eq:kinetic} gives $K=K_0/\\eps$. Division proves the final identity.\n\\end{proof}","_0}{\\sqrt\\eps},\\qquad\n \\frac\\Lambda\\mu=x\\frac{a_0}{\\sqrt\\eps}.}\n \\label{eq:family}\n\\end{equation}\nThen the unprojected Gaussian family satisfies\n\\begin{equation}\n \\widetilde\\rho=\\rho_T,\\qquad\n K=\\frac{K_0}{\\eps},\\qquad\n \\frac{\\widetilde\\rho}{K}=\\eps.\n \\label{eq:familyresult}\n\\end{equation}\nNo common occupation multiplier is used in this construction.\n\\end{theorem}\n\\begin{proof}\nAt fixed $\\rhoV$, $\\mu^4=4\\lambda\\rhoV$, so\n\\[\n \\widetilde\\rho=4\\rhoV\\lambda(A/\\mu)^4S_E.\n\\]\nSubstitute \\eqref{eq:family}. The factors $\\eps^2$ and $\\eps^{-2}$ cancel, leaving $\\eta_0 A_0^4S_E$. Equation \\eqref{eq:kinetic} gives $K=K_0/\\eps$. Division proves the final identity.\n\\end{proof}","c\\Lambda\\mu=x\\frac{a_0}{\\sqrt\\eps}.}\n \\label{eq:family}\n\\end{equation}\nThen the unprojected Gaussian family satisfies\n\\begin{equation}\n \\widetilde\\rho=\\rho_T,\\qquad\n K=\\frac{K_0}{\\eps},\\qquad\n \\frac{\\widetilde\\rho}{K}=\\eps.\n \\label{eq:familyresult}\n\\end{equation}\nNo common occupation multiplier is used in this construction.\n\\end{theorem}\n\\begin{proof}\nAt fixed $\\rhoV$, $\\mu^4=4\\lambda\\rhoV$, so\n\\[\n \\widetilde\\rho=4\\rhoV\\lambda(A/\\mu)^4S_E.\n\\]\nSubstitute \\eqref{eq:family}. The factors $\\eps^2$ and $\\eps^{-2}$ cancel, leaving $\\eta_0 A_0^4S_E$. Equation \\eqref{eq:kinetic} gives $K=K_0/\\eps$. Division proves the final identity.\n\\end{proof}","d Gaussian family satisfies\n\\begin{equation}\n \\widetilde\\rho=\\rho_T,\\qquad\n K=\\frac{K_0}{\\eps},\\qquad\n \\frac{\\widetilde\\rho}{K}=\\eps.\n \\label{eq:familyresult}\n\\end{equation}\nNo common occupation multiplier is used in this construction.\n\\end{theorem}\n\\begin{proof}\nAt fixed $\\rhoV$, $\\mu^4=4\\lambda\\rhoV$, so\n\\[\n \\widetilde\\rho=4\\rhoV\\lambda(A/\\mu)^4S_E.\n\\]\nSubstitute \\eqref{eq:family}. The factors $\\eps^2$ and $\\eps^{-2}$ cancel, leaving $\\eta_0 A_0^4S_E$. Equation \\eqref{eq:kinetic} gives $K=K_0/\\eps$. Division proves the final identity.\n\\end{proof}","mily satisfies\n\\begin{equation}\n \\widetilde\\rho=\\rho_T,\\qquad\n K=\\frac{K_0}{\\eps},\\qquad\n \\frac{\\widetilde\\rho}{K}=\\eps.\n \\label{eq:familyresult}\n\\end{equation}\nNo common occupation multiplier is used in this construction.\n\\end{theorem}\n\\begin{proof}\nAt fixed $\\rhoV$, $\\mu^4=4\\lambda\\rhoV$, so\n\\[\n \\widetilde\\rho=4\\rhoV\\lambda(A/\\mu)^4S_E.\n\\]\nSubstitute \\eqref{eq:family}. The factors $\\eps^2$ and $\\eps^{-2}$ cancel, leaving $\\eta_0 A_0^4S_E$. Equation \\eqref{eq:kinetic} gives $K=K_0/\\eps$. Division proves the final identity.\n\\end{proof}","on}\n \\widetilde\\rho=\\rho_T,\\qquad\n K=\\frac{K_0}{\\eps},\\qquad\n \\frac{\\widetilde\\rho}{K}=\\eps.\n \\label{eq:familyresult}\n\\end{equation}\nNo common occupation multiplier is used in this construction.\n\\end{theorem}\n\\begin{proof}\nAt fixed $\\rhoV$, $\\mu^4=4\\lambda\\rhoV$, so\n\\[\n \\widetilde\\rho=4\\rhoV\\lambda(A/\\mu)^4S_E.\n\\]\nSubstitute \\eqref{eq:family}. The factors $\\eps^2$ and $\\eps^{-2}$ cancel, leaving $\\eta_0 A_0^4S_E$. Equation \\eqref{eq:kinetic} gives $K=K_0/\\eps$. Division proves the final identity.\n\\end{proof}",",\\qquad\n \\frac{\\widetilde\\rho}{K}=\\eps.\n \\label{eq:familyresult}\n\\end{equation}\nNo common occupation multiplier is used in this construction.\n\\end{theorem}\n\\begin{proof}\nAt fixed $\\rhoV$, $\\mu^4=4\\lambda\\rhoV$, so\n\\[\n \\widetilde\\rho=4\\rhoV\\lambda(A/\\mu)^4S_E.\n\\]\nSubstitute \\eqref{eq:family}. The factors $\\eps^2$ and $\\eps^{-2}$ cancel, leaving $\\eta_0 A_0^4S_E$. Equation \\eqref{eq:kinetic} gives $K=K_0/\\eps$. Division proves the final identity.\n\\end{proof}","Here $\\eps$ is a \\emph{design parameter for the stipulated Gaussian family}. It is not the measured production efficiency of the coupled solver below. That solver includes coherent vacuum response and changes the trajectory.","\\subsection{What is and is not preserved}\nThe density and the normalized distributions in $m/A$ and $k/A$ are preserved. The physical mass scale is not the old one:\n\\begin{equation}\n \\frac\\mu{\\mu_0}=\\eta_0^{1/4}\\sqrt\\eps,\\qquad\n \\frac A{A_0}=\\frac\\Lambda{\\Lambda_0}=\\eta_0^{1/4}\n =7.85458237208\\times10^{-6}.\n \\label{eq:physicalscale}\n\\end{equation}\nThus the physical tower masses are independent of $\\eps$ along this family, but differ from the original projected benchmark. Number density increases by the inverse of the common mass factor, while the mass density is held fixed. Occupations themselves are not preserved: they are no longer multiplied by $\\eta_0$.","subsection{What is and is not preserved}\nThe density and the normalized distributions in $m/A$ and $k/A$ are preserved. The physical mass scale is not the old one:\n\\begin{equation}\n \\frac\\mu{\\mu_0}=\\eta_0^{1/4}\\sqrt\\eps,\\qquad\n \\frac A{A_0}=\\frac\\Lambda{\\Lambda_0}=\\eta_0^{1/4}\n =7.85458237208\\times10^{-6}.\n \\label{eq:physicalscale}\n\\end{equation}\nThus the physical tower masses are independent of $\\eps$ along this family, but differ from the original projected benchmark. Number density increases by the inverse of the common mass factor, while the mass density is held fixed. Occupations themselves are not preserved: they are no longer multiplied by $\\eta_0$.","ambda_0}=\\eta_0^{1/4}\n =7.85458237208\\times10^{-6}.\n \\label{eq:physicalscale}\n\\end{equation}\nThus the physical tower masses are independent of $\\eps$ along this family, but differ from the original projected benchmark. Number density increases by the inverse of the common mass factor, while the mass density is held fixed. Occupations themselves are not preserved: they are no longer multiplied by $\\eta_0$.","The canonical coupling of level $h$ is\n\\begin{equation}\n g_h=\\gamma\\lambda\\left(\\frac{m_h}{\\mu}\\right)^2.\n\\end{equation}\nAt the stored peak level $h=13$, this becomes\n\\begin{equation}\n g_{\\rm peak}=0.1535680410\\,\\eps.\n\\end{equation}\nThe following numbers are algebraic consequences of the parameter choice, not fitted dynamical outputs.","\\begin{center}\\small\n\\begin{tabular}{@{}rrrr@{}}\\toprule\n$\\eps$ & $\\lambda$ & $g_{\\rm peak}$ & $K/K_0$","midrule\n1 & $3.8062\\times10^{-21}$ & $0.15357$ & 1","0.1 & $3.8062\\times10^{-23}$ & $0.015357$ & 10","0.03 & $3.4256\\times10^{-24}$ & $0.0046070$ & $33.333$","0.01 & $3.8062\\times10^{-25}$ & $0.0015357$ & 100","0.003 & $3.4256\\times10^{-26}$ & $0.00046070$ & $333.333$","\\section{The reservoir cannot be ignored}\n\\begin{proposition}[Residual-energy obligation]\nSuppose an event begins and ends at the same scalar potential value, expansion and external losses are negligible during that event, and the receiving sector gains $\\eps K_i$. Energy conservation requires\n\\begin{equation}\n K_f=(1-\\eps)K_i.\n\\end{equation}\nIn particular, weak transfer in that event cannot also bring the background to rest at the same potential value without another energy-transfer mechanism.\n\\end{proposition}\n\\begin{proof}\nSubtract the initial and final conserved total energies. The potential contributions cancel. The gain in receiving-sector energy is the loss in kinetic energy.\n\\end{proof}","At $\\eps=0.01$, the Gaussian bookkeeping leaves 99 percent of the incident kinetic energy outside the target produced density. This is not automatically an inconsistency. Further mass growth, subsequent passages, expansion, or transfer to other fields can act. But their effect must be computed, and they need not preserve the stipulated spectrum. Mass growth after a crossing can itself transfer substantial energy \\cite{instant}.","There is also a quantitative background cost. Using the \\emph{stored benchmark inputs}, not a new observational fit,\n\\begin{equation}\n a_\\star=10^{-6},\\quad \\Omega_{r0}=9\\times10^{-5},\\quad\n \\Omega_{\\mathrm V}=0.68991,\n\\end{equation}\nwe obtain\n\\begin{equation}\n \\boxed{\\frac{K}{\\rho_r(a_\\star)}=\\frac{0.003600547325}{\\eps}.}\n \\label{eq:radiation}\n\\end{equation}\nThe ratio is $0.0360$ at $\\eps=0.1$, $0.360$ at $0.01$, and $3.60$ at $0.001$. Taking $\\eps$ arbitrarily small is therefore not a cost-free way to preserve the old cosmological history. These ratios are not observational exclusion bounds. They identify a background component that the next cosmological calculation must include.","so a quantitative background cost. Using the \\emph{stored benchmark inputs}, not a new observational fit,\n\\begin{equation}\n a_\\star=10^{-6},\\quad \\Omega_{r0}=9\\times10^{-5},\\quad\n \\Omega_{\\mathrm V}=0.68991,\n\\end{equation}\nwe obtain\n\\begin{equation}\n \\boxed{\\frac{K}{\\rho_r(a_\\star)}=\\frac{0.003600547325}{\\eps}.}\n \\label{eq:radiation}\n\\end{equation}\nThe ratio is $0.0360$ at $\\eps=0.1$, $0.360$ at $0.01$, and $3.60$ at $0.001$. Taking $\\eps$ arbitrarily small is therefore not a cost-free way to preserve the old cosmological history. These ratios are not observational exclusion bounds. They identify a background component that the next cosmological calculation must include.","ative background cost. Using the \\emph{stored benchmark inputs}, not a new observational fit,\n\\begin{equation}\n a_\\star=10^{-6},\\quad \\Omega_{r0}=9\\times10^{-5},\\quad\n \\Omega_{\\mathrm V}=0.68991,\n\\end{equation}\nwe obtain\n\\begin{equation}\n \\boxed{\\frac{K}{\\rho_r(a_\\star)}=\\frac{0.003600547325}{\\eps}.}\n \\label{eq:radiation}\n\\end{equation}\nThe ratio is $0.0360$ at $\\eps=0.1$, $0.360$ at $0.01$, and $3.60$ at $0.001$. Taking $\\eps$ arbitrarily small is therefore not a cost-free way to preserve the old cosmological history. These ratios are not observational exclusion bounds. They identify a background component that the next cosmological calculation must include.","ground cost. Using the \\emph{stored benchmark inputs}, not a new observational fit,\n\\begin{equation}\n a_\\star=10^{-6},\\quad \\Omega_{r0}=9\\times10^{-5},\\quad\n \\Omega_{\\mathrm V}=0.68991,\n\\end{equation}\nwe obtain\n\\begin{equation}\n \\boxed{\\frac{K}{\\rho_r(a_\\star)}=\\frac{0.003600547325}{\\eps}.}\n \\label{eq:radiation}\n\\end{equation}\nThe ratio is $0.0360$ at $\\eps=0.1$, $0.360$ at $0.01$, and $3.60$ at $0.001$. Taking $\\eps$ arbitrarily small is therefore not a cost-free way to preserve the old cosmological history. These ratios are not observational exclusion bounds. They identify a background component that the next cosmological calculation must include.","ng the \\emph{stored benchmark inputs}, not a new observational fit,\n\\begin{equation}\n a_\\star=10^{-6},\\quad \\Omega_{r0}=9\\times10^{-5},\\quad\n \\Omega_{\\mathrm V}=0.68991,\n\\end{equation}\nwe obtain\n\\begin{equation}\n \\boxed{\\frac{K}{\\rho_r(a_\\star)}=\\frac{0.003600547325}{\\eps}.}\n \\label{eq:radiation}\n\\end{equation}\nThe ratio is $0.0360$ at $\\eps=0.1$, $0.360$ at $0.01$, and $3.60$ at $0.001$. Taking $\\eps$ arbitrarily small is therefore not a cost-free way to preserve the old cosmological history. These ratios are not observational exclusion bounds. They identify a background component that the next cosmological calculation must include.","\\section{A finite quantum system with an exact energy budget}\n\\subsection{Regulator and state}\nSet\n\\begin{equation}\n q=\\frac{\\sqrt\\gamma\\phi}{v},\\quad \\tau=At,\\quad\n \\kappa=\\frac{k}{A},\\quad\n \\omega_{h\\kappa}(q)=\\sqrt{\\kappa^2+y_h^2(1+q^2)}.\n\\end{equation}\nAfter finite momentum quadrature, label a mode by $j$ and give it a positive weight $w_j$, including its tower multiplicity. Let\n\\begin{equation}\n C=\\frac{v^2}{\\gamma A^2}\n =\\frac{1}{\\gamma\\lambda(A/\\mu)^2}\n =\\frac{2S_E}{\\eps}>0.\n\\end{equation}\nThe prime in the rest of this section denotes $\\dd/\\dd\\tau$.","ection{A finite quantum system with an exact energy budget}\n\\subsection{Regulator and state}\nSet\n\\begin{equation}\n q=\\frac{\\sqrt\\gamma\\phi}{v},\\quad \\tau=At,\\quad\n \\kappa=\\frac{k}{A},\\quad\n \\omega_{h\\kappa}(q)=\\sqrt{\\kappa^2+y_h^2(1+q^2)}.\n\\end{equation}\nAfter finite momentum quadrature, label a mode by $j$ and give it a positive weight $w_j$, including its tower multiplicity. Let\n\\begin{equation}\n C=\\frac{v^2}{\\gamma A^2}\n =\\frac{1}{\\gamma\\lambda(A/\\mu)^2}\n =\\frac{2S_E}{\\eps}>0.\n\\end{equation}\nThe prime in the rest of this section denotes $\\dd/\\dd\\tau$.","d \\tau=At,\\quad\n \\kappa=\\frac{k}{A},\\quad\n \\omega_{h\\kappa}(q)=\\sqrt{\\kappa^2+y_h^2(1+q^2)}.\n\\end{equation}\nAfter finite momentum quadrature, label a mode by $j$ and give it a positive weight $w_j$, including its tower multiplicity. Let\n\\begin{equation}\n C=\\frac{v^2}{\\gamma A^2}\n =\\frac{1}{\\gamma\\lambda(A/\\mu)^2}\n =\\frac{2S_E}{\\eps}>0.\n\\end{equation}\nThe prime in the rest of this section denotes $\\dd/\\dd\\tau$.","For each mode use the instantaneous occupation $n_j$ and two real coherence variables $u_j,v_j$. They obey\n\\begin{align}\n n_j'&=\\frac12 W_j u_j,","For each mode use the instantaneous occupation $n_j$ and two real coherence variables $u_j,v_j$. They obey\n\\begin{align}\n n_j'&=\\frac12 W_j u_j,\\\\\n u_j'&=W_j(1+2n_j)-2\\omega_j v_j,\\\\\n v_j'&=2\\omega_j u_j,\\qquad\n W_j=\\frac{\\omega_{j,q}q'}{\\omega_j}.\n \\label{eq:qke}\n\\end{align}\nThese equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are closely related \\cite{qke}. Their use here is defined directly by the finite oscillator model, not by importing an electromagnetic pair-production claim into the scalar model.","The reaction force required by the same finite Hamiltonian is\n\\begin{equation}\n \\boxed{Cq''+\\mathcal V'(q)\n +\\sum_j w_j\\omega_{j,q}\\left(n_j+\\frac{u_j}{2}\\right)=0.}\n \\label{eq:reaction}\n\\end{equation}\nThe coherence term is essential. An occupation-only force does not conserve the energy of \\eqref{eq:qke}.","\\subsection{Precise subtraction convention}\nFor complex mode functions $a_j$ satisfying $a_j''+\\omega_j^2a_j=0$ and the normalization $a_j\\overline{a_j'}-\\overline{a_j}a_j'=i$, fix the variables by\n\\begin{equation}\n n_j=\\frac{|a_j'|^2+\\omega_j^2|a_j|^2}{2\\omega_j}-\\frac12,\n \\quad u_j=\\omega_j|a_j|^2-\\frac{|a_j'|^2}{\\omega_j},\n \\quad v_j=-2\\operatorname{Re}(\\overline{a_j}a_j').\n\\end{equation}\nUse the finite energy\n\\begin{equation}\n \\mathcal E=\\frac C2(q')^2+\\mathcal V(q)\n +\\sum_jw_j\\left[\\frac{|a_j'|^2+\\omega_j^2|a_j|^2}{2}\n -\\frac{\\omega_j}{2}\\right].\n \\label{eq:hamiltonian}\n\\end{equation}\nThe subtracted quantity depends on $q$, so its derivative must also be included in the force. This gives \\eqref{eq:reaction}. The bracket in \\eqref{eq:hamiltonian} equals $\\omega","\\subsection{Precise subtraction convention}\nFor complex mode functions $a_j$ satisfying $a_j''+\\omega_j^2a_j=0$ and the normalization $a_j\\overline{a_j'}-\\overline{a_j}a_j'=i$, fix the variables by\n\\begin{equation}\n n_j=\\frac{|a_j'|^2+\\omega_j^2|a_j|^2}{2\\omega_j}-\\frac12,\n \\quad u_j=\\omega_j|a_j|^2-\\frac{|a_j'|^2}{\\omega_j},\n \\quad v_j=-2\\operatorname{Re}(\\overline{a_j}a_j').\n\\end{equation}\nUse the finite energy\n\\begin{equation}\n \\mathcal E=\\frac C2(q')^2+\\mathcal V(q)\n +\\sum_jw_j\\left[\\frac{|a_j'|^2+\\omega_j^2|a_j|^2}{2}\n -\\frac{\\omega_j}{2}\\right].\n \\label{eq:hamiltonian}\n\\end{equation}\nThe subtracted quantity depends on $q$, so its derivative must also be included in the force. This gives \\eqref{eq:reaction}. The bracket in \\eqref{eq:hamiltonian} equals $\\omega_jn_j$.","onvention}\nFor complex mode functions $a_j$ satisfying $a_j''+\\omega_j^2a_j=0$ and the normalization $a_j\\overline{a_j'}-\\overline{a_j}a_j'=i$, fix the variables by\n\\begin{equation}\n n_j=\\frac{|a_j'|^2+\\omega_j^2|a_j|^2}{2\\omega_j}-\\frac12,\n \\quad u_j=\\omega_j|a_j|^2-\\frac{|a_j'|^2}{\\omega_j},\n \\quad v_j=-2\\operatorname{Re}(\\overline{a_j}a_j').\n\\end{equation}\nUse the finite energy\n\\begin{equation}\n \\mathcal E=\\frac C2(q')^2+\\mathcal V(q)\n +\\sum_jw_j\\left[\\frac{|a_j'|^2+\\omega_j^2|a_j|^2}{2}\n -\\frac{\\omega_j}{2}\\right].\n \\label{eq:hamiltonian}\n\\end{equation}\nThe subtracted quantity depends on $q$, so its derivative must also be included in the force. This gives \\eqref{eq:reaction}. The bracket in \\eqref{eq:hamiltonian} equals $\\omega_jn_j$.","2+\\omega_j^2|a_j|^2}{2\\omega_j}-\\frac12,\n \\quad u_j=\\omega_j|a_j|^2-\\frac{|a_j'|^2}{\\omega_j},\n \\quad v_j=-2\\operatorname{Re}(\\overline{a_j}a_j').\n\\end{equation}\nUse the finite energy\n\\begin{equation}\n \\mathcal E=\\frac C2(q')^2+\\mathcal V(q)\n +\\sum_jw_j\\left[\\frac{|a_j'|^2+\\omega_j^2|a_j|^2}{2}\n -\\frac{\\omega_j}{2}\\right].\n \\label{eq:hamiltonian}\n\\end{equation}\nThe subtracted quantity depends on $q$, so its derivative must also be included in the force. This gives \\eqref{eq:reaction}. The bracket in \\eqref{eq:hamiltonian} equals $\\omega_jn_j$.","a_j}a_j').\n\\end{equation}\nUse the finite energy\n\\begin{equation}\n \\mathcal E=\\frac C2(q')^2+\\mathcal V(q)\n +\\sum_jw_j\\left[\\frac{|a_j'|^2+\\omega_j^2|a_j|^2}{2}\n -\\frac{\\omega_j}{2}\\right].\n \\label{eq:hamiltonian}\n\\end{equation}\nThe subtracted quantity depends on $q$, so its derivative must also be included in the force. This gives \\eqref{eq:reaction}. The bracket in \\eqref{eq:hamiltonian} equals $\\omega_jn_j$.","\\begin{theorem}[Finite energy conservation and positivity]\nAssume $C>0$, $w_j>0$, $y_j>0$, and $\\mathcal V\\ge0$ is smooth. Start the modes in the instantaneous vacuum $n_j=u_j=v_j=0$. Equations \\eqref{eq:qke}--\\eqref{eq:reaction} preserve\n\\begin{equation}\n (1+2n_j)^2-u_j^2-v_j^2=1,\n \\label{eq:hyperboloid}\n\\end{equation}\nand conserve\n\\begin{equation}\n \\boxed{\\mathcal E=\\frac C2(q')^2+\\mathcal V(q)\n +\\sum_jw_j\\omega_j n_j.}\n \\label{eq:conserved}\n\\end{equation}\nEvery occupation remains nonnegative. Consequently\n\\begin{equation}\n 0\\le\\sum_jw_j\\omega_j n_j\\le\\mathcal E(0).\n \\label{eq:bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nDifferentiate the left side of \\eqref{eq:hyperboloid} and substitute \\eqref{eq:qke}; all terms cancel. The initial value is one. The c","\\begin{theorem}[Finite energy conservation and positivity]\nAssume $C>0$, $w_j>0$, $y_j>0$, and $\\mathcal V\\ge0$ is smooth. Start the modes in the instantaneous vacuum $n_j=u_j=v_j=0$. Equations \\eqref{eq:qke}--\\eqref{eq:reaction} preserve\n\\begin{equation}\n (1+2n_j)^2-u_j^2-v_j^2=1,\n \\label{eq:hyperboloid}\n\\end{equation}\nand conserve\n\\begin{equation}\n \\boxed{\\mathcal E=\\frac C2(q')^2+\\mathcal V(q)\n +\\sum_jw_j\\omega_j n_j.}\n \\label{eq:conserved}\n\\end{equation}\nEvery occupation remains nonnegative. Consequently\n\\begin{equation}\n 0\\le\\sum_jw_j\\omega_j n_j\\le\\mathcal E(0).\n \\label{eq:bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nDifferentiate the left side of \\eqref{eq:hyperboloid} and substitute \\eqref{eq:qke}; all terms cancel. The initial value is one. The continuous","\\begin{theorem}[Finite energy conservation and positivity]\nAssume $C>0$, $w_j>0$, $y_j>0$, and $\\mathcal V\\ge0$ is smooth. Start the modes in the instantaneous vacuum $n_j=u_j=v_j=0$. Equations \\eqref{eq:qke}--\\eqref{eq:reaction} preserve\n\\begin{equation}\n (1+2n_j)^2-u_j^2-v_j^2=1,\n \\label{eq:hyperboloid}\n\\end{equation}\nand conserve\n\\begin{equation}\n \\boxed{\\mathcal E=\\frac C2(q')^2+\\mathcal V(q)\n +\\sum_jw_j\\omega_j n_j.}\n \\label{eq:conserved}\n\\end{equation}\nEvery occupation remains nonnegative. Consequently\n\\begin{equation}\n 0\\le\\sum_jw_j\\omega_j n_j\\le\\mathcal E(0).\n \\label{eq:bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nDifferentiate the left side of \\eqref{eq:hyperboloid} and substitute \\eqref{eq:qke}; all terms cancel. The initial value is one. The continuous trajecto","\\begin{theorem}[Finite energy conservation and positivity]\nAssume $C>0$, $w_j>0$, $y_j>0$, and $\\mathcal V\\ge0$ is smooth. Start the modes in the instantaneous vacuum $n_j=u_j=v_j=0$. Equations \\eqref{eq:qke}--\\eqref{eq:reaction} preserve\n\\begin{equation}\n (1+2n_j)^2-u_j^2-v_j^2=1,\n \\label{eq:hyperboloid}\n\\end{equation}\nand conserve\n\\begin{equation}\n \\boxed{\\mathcal E=\\frac C2(q')^2+\\mathcal V(q)\n +\\sum_jw_j\\omega_j n_j.}\n \\label{eq:conserved}\n\\end{equation}\nEvery occupation remains nonnegative. Consequently\n\\begin{equation}\n 0\\le\\sum_jw_j\\omega_j n_j\\le\\mathcal E(0).\n \\label{eq:bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nDifferentiate the left side of \\eqref{eq:hyperboloid} and substitute \\eqref{eq:qke}; all terms cancel. The initial value is one. The continuous trajectory remains on the shee","\\begin{theorem}[Finite energy conservation and positivity]\nAssume $C>0$, $w_j>0$, $y_j>0$, and $\\mathcal V\\ge0$ is smooth. Start the modes in the instantaneous vacuum $n_j=u_j=v_j=0$. Equations \\eqref{eq:qke}--\\eqref{eq:reaction} preserve\n\\begin{equation}\n (1+2n_j)^2-u_j^2-v_j^2=1,\n \\label{eq:hyperboloid}\n\\end{equation}\nand conserve\n\\begin{equation}\n \\boxed{\\mathcal E=\\frac C2(q')^2+\\mathcal V(q)\n +\\sum_jw_j\\omega_j n_j.}\n \\label{eq:conserved}\n\\end{equation}\nEvery occupation remains nonnegative. Consequently\n\\begin{equation}\n 0\\le\\sum_jw_j\\omega_j n_j\\le\\mathcal E(0).\n \\label{eq:bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nDifferentiate the left side of \\eqref{eq:hyperboloid} and substitute \\eqref{eq:qke}; all terms cancel. The initial value is one. The continuous trajectory remains on the sheet $1+2n_j>0$, which implies $1+2n_j\\ge1$ and hence $n_j\\ge0$.","he modes in the instantaneous vacuum $n_j=u_j=v_j=0$. Equations \\eqref{eq:qke}--\\eqref{eq:reaction} preserve\n\\begin{equation}\n (1+2n_j)^2-u_j^2-v_j^2=1,\n \\label{eq:hyperboloid}\n\\end{equation}\nand conserve\n\\begin{equation}\n \\boxed{\\mathcal E=\\frac C2(q')^2+\\mathcal V(q)\n +\\sum_jw_j\\omega_j n_j.}\n \\label{eq:conserved}\n\\end{equation}\nEvery occupation remains nonnegative. Consequently\n\\begin{equation}\n 0\\le\\sum_jw_j\\omega_j n_j\\le\\mathcal E(0).\n \\label{eq:bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nDifferentiate the left side of \\eqref{eq:hyperboloid} and substitute \\eqref{eq:qke}; all terms cancel. The initial value is one. The continuous trajectory remains on the sheet $1+2n_j>0$, which implies $1+2n_j\\ge1$ and hence $n_j\\ge0$.","mega_j n_j.}\n \\label{eq:conserved}\n\\end{equation}\nEvery occupation remains nonnegative. Consequently\n\\begin{equation}\n 0\\le\\sum_jw_j\\omega_j n_j\\le\\mathcal E(0).\n \\label{eq:bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nDifferentiate the left side of \\eqref{eq:hyperboloid} and substitute \\eqref{eq:qke}; all terms cancel. The initial value is one. The continuous trajectory remains on the sheet $1+2n_j>0$, which implies $1+2n_j\\ge1$ and hence $n_j\\ge0$.","nserved}\n\\end{equation}\nEvery occupation remains nonnegative. Consequently\n\\begin{equation}\n 0\\le\\sum_jw_j\\omega_j n_j\\le\\mathcal E(0).\n \\label{eq:bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nDifferentiate the left side of \\eqref{eq:hyperboloid} and substitute \\eqref{eq:qke}; all terms cancel. The initial value is one. The continuous trajectory remains on the sheet $1+2n_j>0$, which implies $1+2n_j\\ge1$ and hence $n_j\\ge0$.","Every occupation remains nonnegative. Consequently\n\\begin{equation}\n 0\\le\\sum_jw_j\\omega_j n_j\\le\\mathcal E(0).\n \\label{eq:bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nDifferentiate the left side of \\eqref{eq:hyperboloid} and substitute \\eqref{eq:qke}; all terms cancel. The initial value is one. The continuous trajectory remains on the sheet $1+2n_j>0$, which implies $1+2n_j\\ge1$ and hence $n_j\\ge0$.","For the mode energy,\n\\[\n (\\omega_jn_j)'=\\omega_{j,q}q'\\left(n_j+\\frac{u_j}{2}\\right).\n\\]\nMultiplying \\eqref{eq:reaction} by $q'$ cancels the sum of these derivatives against the derivative of the mechanical energy. This proves \\eqref{eq:conserved}. Its nonnegative terms prove \\eqref{eq:bound}.\n\\end{proof}","\\begin{corollary}[Global existence at fixed finite regulator]\nUnder the theorem's assumptions, the finite initial-value problem has a unique solution for all finite times.\n\\end{corollary}\n\\begin{proof}\nThe vector field is smooth because each $\\omega_j$ is bounded below by $y_j>0$. Conservation bounds $q'$, so $q$ stays bounded on every finite time interval. Positivity bounds $n_j\\le\\mathcal E(0)/(w_jy_j)$, and \\eqref{eq:hyperboloid} then bounds both coherence variables. The ordinary differential equation cannot leave every compact subset in finite time. The local solution therefore extends globally.\n\\end{proof}","\\section{Numerical results and tests that failed}\nThe principal run takes $\\eps=0.01$, depth 16, 48 Gauss--Legendre momentum nodes per level, and\n\\begin{equation}\n \\kappa=\\sqrt{y_h}\\,z,\\quad 0\\le z\\le3.5,\\quad\n q(0)=-10,\\quad q'(0)=1,\\quad 0\\le\\tau\\le20.\n\\end{equation}\nHere $A$ is the nominal reference scale used to set the incoming speed. The actual speed at $q=0$ is an output of the coupled solution, not an imposed crossing condition. There are 768 quadrature modes, representing 65,535 internal scalar degrees of freedom. The modes begin in the specified instantaneous vacuum. We use $\\mathcal V=0$ as a local crossing diagnostic. For the nominal parameter family and $|q|\\le10$, the original quartic potential energy is below $6\\times10^{-19}$ of the crossing kinetic reservoi","\\section{Numerical results and tests that failed}\nThe principal run takes $\\eps=0.01$, depth 16, 48 Gauss--Legendre momentum nodes per level, and\n\\begin{equation}\n \\kappa=\\sqrt{y_h}\\,z,\\quad 0\\le z\\le3.5,\\quad\n q(0)=-10,\\quad q'(0)=1,\\quad 0\\le\\tau\\le20.\n\\end{equation}\nHere $A$ is the nominal reference scale used to set the incoming speed. The actual speed at $q=0$ is an output of the coupled solution, not an imposed crossing condition. There are 768 quadrature modes, representing 65,535 internal scalar degrees of freedom. The modes begin in the specified instantaneous vacuum. We use $\\mathcal V=0$ as a local crossing diagnostic. For the nominal parameter family and $|q|\\le10$, the original quartic potential energy is below $6\\times10^{-19}$ of the crossing kinetic reservoir. That local estimate is not permission to omit the potential from the subsequent trapping calculation.","\\le z\\le3.5,\\quad\n q(0)=-10,\\quad q'(0)=1,\\quad 0\\le\\tau\\le20.\n\\end{equation}\nHere $A$ is the nominal reference scale used to set the incoming speed. The actual speed at $q=0$ is an output of the coupled solution, not an imposed crossing condition. There are 768 quadrature modes, representing 65,535 internal scalar degrees of freedom. The modes begin in the specified instantaneous vacuum. We use $\\mathcal V=0$ as a local crossing diagnostic. For the nominal parameter family and $|q|\\le10$, the original quartic potential energy is below $6\\times10^{-19}$ of the crossing kinetic reservoir. That local estimate is not permission to omit the potential from the subsequent trapping calculation.","A$ is the nominal reference scale used to set the incoming speed. The actual speed at $q=0$ is an output of the coupled solution, not an imposed crossing condition. There are 768 quadrature modes, representing 65,535 internal scalar degrees of freedom. The modes begin in the specified instantaneous vacuum. We use $\\mathcal V=0$ as a local crossing diagnostic. For the nominal parameter family and $|q|\\le10$, the original quartic potential energy is below $6\\times10^{-19}$ of the crossing kinetic reservoir. That local estimate is not permission to omit the potential from the subsequent trapping calculation.","speed. The actual speed at $q=0$ is an output of the coupled solution, not an imposed crossing condition. There are 768 quadrature modes, representing 65,535 internal scalar degrees of freedom. The modes begin in the specified instantaneous vacuum. We use $\\mathcal V=0$ as a local crossing diagnostic. For the nominal parameter family and $|q|\\le10$, the original quartic potential energy is below $6\\times10^{-19}$ of the crossing kinetic reservoir. That local estimate is not permission to omit the potential from the subsequent trapping calculation.","The DOP853 integration uses relative tolerance $2\\times10^{-10}$ and absolute tolerance $2\\times10^{-13}$. The largest total-energy residual in the principal run is\n\\begin{equation}\n \\max_\\tau\\left|\\frac{\\mathcal E(\\tau)}{\\mathcal E(0)}-1\\right|\n =7.34\\times10^{-11}.\n\\end{equation}\nThe maximum hyperboloid residual is $3.73\\times10^{-13}$. The largest instantaneous mode-energy fraction is $0.648634$, and the endpoint fraction is $0.126747$. The final position is $q=6.65567$, rather than the externally forced value $q=10$.","ive tolerance $2\\times10^{-10}$ and absolute tolerance $2\\times10^{-13}$. The largest total-energy residual in the principal run is\n\\begin{equation}\n \\max_\\tau\\left|\\frac{\\mathcal E(\\tau)}{\\mathcal E(0)}-1\\right|\n =7.34\\times10^{-11}.\n\\end{equation}\nThe maximum hyperboloid residual is $3.73\\times10^{-13}$. The largest instantaneous mode-energy fraction is $0.648634$, and the endpoint fraction is $0.126747$. The final position is $q=6.65567$, rather than the externally forced value $q=10$.","d absolute tolerance $2\\times10^{-13}$. The largest total-energy residual in the principal run is\n\\begin{equation}\n \\max_\\tau\\left|\\frac{\\mathcal E(\\tau)}{\\mathcal E(0)}-1\\right|\n =7.34\\times10^{-11}.\n\\end{equation}\nThe maximum hyperboloid residual is $3.73\\times10^{-13}$. The largest instantaneous mode-energy fraction is $0.648634$, and the endpoint fraction is $0.126747$. The final position is $q=6.65567$, rather than the externally forced value $q=10$.","otal-energy residual in the principal run is\n\\begin{equation}\n \\max_\\tau\\left|\\frac{\\mathcal E(\\tau)}{\\mathcal E(0)}-1\\right|\n =7.34\\times10^{-11}.\n\\end{equation}\nThe maximum hyperboloid residual is $3.73\\times10^{-13}$. The largest instantaneous mode-energy fraction is $0.648634$, and the endpoint fraction is $0.126747$. The final position is $q=6.65567$, rather than the externally forced value $q=10$.","\\subsection{Independent representation and numerical resolution}\nA separately coded complex-oscillator system was compared with the real quantum-kinetic implementation at depth 14, with 12 nodes and a shorter interval. Their maximum background-position difference was $1.88\\times10^{-10}$; their maximum mode-energy-fraction difference was $1.71\\times10^{-9}$. This checks the change of variables and reaction-force implementation. It is not an independent physical model.","At the principal cutoff, increasing momentum quadrature from 48 to 64 nodes leaves the reported endpoint fraction $0.12674744$ unchanged at the displayed precision. This is \\emph{quadrature convergence at fixed cutoff}. It must not be confused with cutoff independence.","\\subsection{Negative controls}\nTwo incorrect closed-system variants were run. Omitting the entire reaction force gives a maximum total-energy residual of $1.47269$ relative to the initial reservoir. Keeping only the occupation force and discarding coherence gives a residual of $5.29845$. The full system has neither defect.","\\subsection{The cutoff test does not pass}\nIncreasing depth changes the regulated Hamiltonian, not merely the quadrature accuracy. Nevertheless, an interpretation that removes the depth regulator must establish stability under that change. The present system does not do so:\n\\begin{center}\\small\n\\begin{tabular}{@{}rrr@{}}\\toprule\nDepth & Maximum instantaneous mode fraction & Final $q$","18 & 0.9999998 & $-0.75818$","At depth 16, changing $z_{\\max}$ from 2.5 to 3.5 to 4.5 to 6 changes the maximum mode-energy fraction from approximately 0.476 to 0.649 to 0.762 to 0.871. Moving the initial state from $q=-10$ to $-15$ or $-20$ also changes the endpoint result. Thus the failed scientific gate is genuine. Conservation has passed; regulator and state robustness have not.","\\section{Why a smaller coupling is not a complete repair}\n\\subsection{A radiative quantity invariant along the abundance family}\nThe source monograph gives the finite-depth one-loop curvature contribution with its subtraction-scale logarithms. To isolate power counting, define the nonnegative diagnostic\n\\begin{equation}\n \\mathcal N_H:=\\frac{\\gamma\\lambda}{16\\pi^2}\n \\sum_{h=1}^H d_h\\left(\\frac{m_h}{\\mu}\\right)^4.\n \\label{eq:curvatureproxy}\n\\end{equation}\nThis omits logarithms and finite counterterms. It is neither a scheme-independent lower bound on the renormalized mass nor a proof that locking is impossible.","\\begin{proposition}[Invariant curvature power counting]\nAt fixed $\\rhoV$ and $\\gamma$, the family \\eqref{eq:family} leaves $\\mathcal N_H$ unchanged for each fixed depth:\n\\begin{equation}\n \\mathcal N_H\n =\\frac{\\gamma\\eta_0 a_0^4}{16\\pi^2}\\sum_{h=1}^H d_hy_h^4\n =\\frac{\\gamma}{64\\pi^2\\rhoV}\\sum_{h=1}^H d_hm_h^4.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe factor $\\lambda(A/\\mu)^4$ equals $\\eta_0a_0^4$ by \\eqref{eq:family}. Equivalently, substitute $\\mu^4=4\\lambda\\rhoV$ into \\eqref{eq:curvatureproxy}. The physical masses are fixed along the family by \\eqref{eq:physicalscale}.\n\\end{proof}","t fixed $\\rhoV$ and $\\gamma$, the family \\eqref{eq:family} leaves $\\mathcal N_H$ unchanged for each fixed depth:\n\\begin{equation}\n \\mathcal N_H\n =\\frac{\\gamma\\eta_0 a_0^4}{16\\pi^2}\\sum_{h=1}^H d_hy_h^4\n =\\frac{\\gamma}{64\\pi^2\\rhoV}\\sum_{h=1}^H d_hm_h^4.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe factor $\\lambda(A/\\mu)^4$ equals $\\eta_0a_0^4$ by \\eqref{eq:family}. Equivalently, substitute $\\mu^4=4\\lambda\\rhoV$ into \\eqref{eq:curvatureproxy}. The physical masses are fixed along the family by \\eqref{eq:physicalscale}.\n\\end{proof}","The values are about $1.04\\times10^{19}$ through depth 13, $1.06\\times10^{22}$ through depth 15, and $3.40\\times10^{23}$ through depth 16. Depth 15 already contains $99.9992126$ percent of the stipulated Gaussian energy. Making individual couplings smaller does not, on this slice, remove the required relative curvature tuning or protection.","This leaves genuine choices: explicitly matched finite-cutoff counterterms, a protected matter completion, or different microscopic production parameters and observables. It does not support the claim that arbitrarily small $\\lambda$ alone has made the model radiatively natural.","\\subsection{Vacuum response from unproduced levels}\nThe observed depth sensitivity has a useful analytic warning sign. In a sufficiently adiabatic mode,\n\\begin{equation}\n n^{(2)}\\simeq\\frac{(\\omega')^2}{16\\omega^4},\\qquad\n \\omega n^{(2)}\\simeq\\frac{y^4q^2(q')^2}{16\\omega^5}.\n\\end{equation}\nIntegrating this local second-adiabatic-order energy over the full momentum range gives\n\\begin{equation}\n \\mathcal E^{(2)}_H\n =\\frac{(q')^2}{96\\pi^2}\\frac{q^2}{1+q^2}\n \\sum_{h=1}^H d_hy_h^2,\n \\label{eq:adiabatice}\n\\end{equation}\nbecause\n\\[\n \\int_0^\\infty\\frac{\\kappa^2\\,\\dd\\kappa}{(\\kappa^2+M^2)^{5/2}}\n =\\frac{1}{3M^2}.\n\\]\nIt therefore corresponds to the local kinetic correction\n\\begin{equation}\n \\delta C_H(q)=\\frac{1}{48\\pi^2}\\frac{q^2}{1+q^2}\n \\sum_{h=1}^H d_hy_h^2.\n\\end{equation}\nAt large $|q|$, $\\delta C_H/C$ for $\\eps=0.01$ is approximately 0.121 at depth 14, 0.971 at 15, 7.77 at 16, and 62.2 at 17.","iently adiabatic mode,\n\\begin{equation}\n n^{(2)}\\simeq\\frac{(\\omega')^2}{16\\omega^4},\\qquad\n \\omega n^{(2)}\\simeq\\frac{y^4q^2(q')^2}{16\\omega^5}.\n\\end{equation}\nIntegrating this local second-adiabatic-order energy over the full momentum range gives\n\\begin{equation}\n \\mathcal E^{(2)}_H\n =\\frac{(q')^2}{96\\pi^2}\\frac{q^2}{1+q^2}\n \\sum_{h=1}^H d_hy_h^2,\n \\label{eq:adiabatice}\n\\end{equation}\nbecause\n\\[\n \\int_0^\\infty\\frac{\\kappa^2\\,\\dd\\kappa}{(\\kappa^2+M^2)^{5/2}}\n =\\frac{1}{3M^2}.\n\\]\nIt therefore corresponds to the local kinetic correction\n\\begin{equation}\n \\delta C_H(q)=\\frac{1}{48\\pi^2}\\frac{q^2}{1+q^2}\n \\sum_{h=1}^H d_hy_h^2.\n\\end{equation}\nAt large $|q|$, $\\delta C_H/C$ for $\\eps=0.01$ is approximately 0.121 at depth 14, 0.971 at 15, 7.77 at 16, and 62.2 at 17.","n}\nIntegrating this local second-adiabatic-order energy over the full momentum range gives\n\\begin{equation}\n \\mathcal E^{(2)}_H\n =\\frac{(q')^2}{96\\pi^2}\\frac{q^2}{1+q^2}\n \\sum_{h=1}^H d_hy_h^2,\n \\label{eq:adiabatice}\n\\end{equation}\nbecause\n\\[\n \\int_0^\\infty\\frac{\\kappa^2\\,\\dd\\kappa}{(\\kappa^2+M^2)^{5/2}}\n =\\frac{1}{3M^2}.\n\\]\nIt therefore corresponds to the local kinetic correction\n\\begin{equation}\n \\delta C_H(q)=\\frac{1}{48\\pi^2}\\frac{q^2}{1+q^2}\n \\sum_{h=1}^H d_hy_h^2.\n\\end{equation}\nAt large $|q|$, $\\delta C_H/C$ for $\\eps=0.01$ is approximately 0.121 at depth 14, 0.971 at 15, 7.77 at 16, and 62.2 at 17.","uation}\n \\mathcal E^{(2)}_H\n =\\frac{(q')^2}{96\\pi^2}\\frac{q^2}{1+q^2}\n \\sum_{h=1}^H d_hy_h^2,\n \\label{eq:adiabatice}\n\\end{equation}\nbecause\n\\[\n \\int_0^\\infty\\frac{\\kappa^2\\,\\dd\\kappa}{(\\kappa^2+M^2)^{5/2}}\n =\\frac{1}{3M^2}.\n\\]\nIt therefore corresponds to the local kinetic correction\n\\begin{equation}\n \\delta C_H(q)=\\frac{1}{48\\pi^2}\\frac{q^2}{1+q^2}\n \\sum_{h=1}^H d_hy_h^2.\n\\end{equation}\nAt large $|q|$, $\\delta C_H/C$ for $\\eps=0.01$ is approximately 0.121 at depth 14, 0.971 at 15, 7.77 at 16, and 62.2 at 17.","\\mathcal E^{(2)}_H\n =\\frac{(q')^2}{96\\pi^2}\\frac{q^2}{1+q^2}\n \\sum_{h=1}^H d_hy_h^2,\n \\label{eq:adiabatice}\n\\end{equation}\nbecause\n\\[\n \\int_0^\\infty\\frac{\\kappa^2\\,\\dd\\kappa}{(\\kappa^2+M^2)^{5/2}}\n =\\frac{1}{3M^2}.\n\\]\nIt therefore corresponds to the local kinetic correction\n\\begin{equation}\n \\delta C_H(q)=\\frac{1}{48\\pi^2}\\frac{q^2}{1+q^2}\n \\sum_{h=1}^H d_hy_h^2.\n\\end{equation}\nAt large $|q|$, $\\delta C_H/C$ for $\\eps=0.01$ is approximately 0.121 at depth 14, 0.971 at 15, 7.77 at 16, and 62.2 at 17.","H\n =\\frac{(q')^2}{96\\pi^2}\\frac{q^2}{1+q^2}\n \\sum_{h=1}^H d_hy_h^2,\n \\label{eq:adiabatice}\n\\end{equation}\nbecause\n\\[\n \\int_0^\\infty\\frac{\\kappa^2\\,\\dd\\kappa}{(\\kappa^2+M^2)^{5/2}}\n =\\frac{1}{3M^2}.\n\\]\nIt therefore corresponds to the local kinetic correction\n\\begin{equation}\n \\delta C_H(q)=\\frac{1}{48\\pi^2}\\frac{q^2}{1+q^2}\n \\sum_{h=1}^H d_hy_h^2.\n\\end{equation}\nAt large $|q|$, $\\delta C_H/C$ for $\\eps=0.01$ is approximately 0.121 at depth 14, 0.971 at 15, 7.77 at 16, and 62.2 at 17.","{instant} G. Felder, L. Kofman, and A. Linde. Instant Preheating. \\emph{Physical Review D} 59, 123523 (1999). arXiv:hep-ph/9812289.\n\\bibitem{qke} A. M. Fedotov, E. G. Gelfer, K. Yu. Korolev, and S. A. Smolyansky. On the kinetic equation approach to pair production by time-dependent electric field. arXiv:1008.2098 (2010).\n\\bibitem{mpr} C. Molina-Par\\'is, P. R. Anderson, and S. A. Ramsey. One-loop $\\lambda\\phi^4$ theory in Robertson--Walker spacetimes: adiabatic regularization and analytic approximation. arXiv:gr-qc/9908037 (1999).\n\\bibitem{psar} A. Ferreiro, S. Monin, and F. Torrenti. Physical scale adiabatic regularization in cosmological spacetimes. arXiv:2311.08986 (2024).\n\\end{thebibliography}\n\\end{document}"]}