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longrightarrow","display":"\\longrightarrow","kind":"use","occurrences":[218,416,475,594,1022,1161,1162,1277]},{"id":"symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","occurrences":[814,1246]}],"contexts":["Let\n\\[\nX\\sim N(0,\\Sigma),\n\\qquad\nM\\succeq0,\n\\]\nand write \\(S=\\Sigma^{1/2}\\).  With \\(X=SZ\\), \\(Z\\sim N(0,I)\\),\n\\[\nQ=X^\\top MX\n=\nZ^\\top(SMS)Z.\n\\]\nIf the nonzero eigenvalues of the positive semidefinite operator\n\\[\nH=SMS\n\\]\nare \\(\\lambda_1,\\ldots,\\lambda_r\\), then\n\\[\n\\boxed{\nQ\\overset d=\n\\sum_{j=1}^r\\lambda_jZ_j^2.\n}\n\\]","Here and throughout the central reductions, the variables $Z_j$ are independent\nstandard normal variables. The numerical tail problem therefore depends only\non a positive spectrum, regardless of how that spectrum was generated.\nIf there are no nonzero coefficients, then $Q=0$ almost surely and\n$P(Q\\ge q)=1$ for $q\\le0$, while $P(Q\\ge q)=0$ for $q>0$.\nThis deterministic case is handled directly, without a saddlepoint or an\ninversion calculation.","\\begin{proposition}[Representation equivalence]\\label{claim:final-2-1}\nThe same central quadratic-form law may be supplied by any of the following:\n\\begin{enumerate}\n\\item a nonnegative spectrum \\(\\{\\lambda_j\\}\\).\n\\item a positive semidefinite matrix \\(H\\).\n\\item a factor \\(A\\) with \\(H=AA^\\top\\) or \\(H=A^\\top A\\).\n\\item a replayable streamed factor\n\\[\nA=[A_1\\ \\cdots\\ A_B].\n\\]\n\\end{enumerate}\nThe nonzero spectra of \\(AA^\\top\\) and \\(A^\\top A\\) are identical, and\n\\[\nAA^\\top=\\sum_bA_bA_b^\\top.\n\\]\n\\end{proposition}","epresentation equivalence]\\label{claim:final-2-1}\nThe same central quadratic-form law may be supplied by any of the following:\n\\begin{enumerate}\n\\item a nonnegative spectrum \\(\\{\\lambda_j\\}\\).\n\\item a positive semidefinite matrix \\(H\\).\n\\item a factor \\(A\\) with \\(H=AA^\\top\\) or \\(H=A^\\top A\\).\n\\item a replayable streamed factor\n\\[\nA=[A_1\\ \\cdots\\ A_B].\n\\]\n\\end{enumerate}\nThe nonzero spectra of \\(AA^\\top\\) and \\(A^\\top A\\) are identical, and\n\\[\nAA^\\top=\\sum_bA_bA_b^\\top.\n\\]\n\\end{proposition}","uivalence]\\label{claim:final-2-1}\nThe same central quadratic-form law may be supplied by any of the following:\n\\begin{enumerate}\n\\item a nonnegative spectrum \\(\\{\\lambda_j\\}\\).\n\\item a positive semidefinite matrix \\(H\\).\n\\item a factor \\(A\\) with \\(H=AA^\\top\\) or \\(H=A^\\top A\\).\n\\item a replayable streamed factor\n\\[\nA=[A_1\\ \\cdots\\ A_B].\n\\]\n\\end{enumerate}\nThe nonzero spectra of \\(AA^\\top\\) and \\(A^\\top A\\) are identical, and\n\\[\nAA^\\top=\\sum_bA_bA_b^\\top.\n\\]\n\\end{proposition}","}\nThe same central quadratic-form law may be supplied by any of the following:\n\\begin{enumerate}\n\\item a nonnegative spectrum \\(\\{\\lambda_j\\}\\).\n\\item a positive semidefinite matrix \\(H\\).\n\\item a factor \\(A\\) with \\(H=AA^\\top\\) or \\(H=A^\\top A\\).\n\\item a replayable streamed factor\n\\[\nA=[A_1\\ \\cdots\\ A_B].\n\\]\n\\end{enumerate}\nThe nonzero spectra of \\(AA^\\top\\) and \\(A^\\top A\\) are identical, and\n\\[\nAA^\\top=\\sum_bA_bA_b^\\top.\n\\]\n\\end{proposition}","\\begin{proof}\nIf $AA^\\top u=\\lambda u$ with $\\lambda>0$, then $A^\\top u\\ne0$ and\n$A^\\top A(A^\\top u)=\\lambda A^\\top u$. Conversely, an eigenvector\n$v$ of $A^\\top A$ with positive eigenvalue maps to the eigenvector $Av$\nof $AA^\\top$. These maps restrict to inverse isomorphisms of the\ncorresponding eigenspaces after scaling, so multiplicities agree.\nHorizontal concatenation gives $AA^\\top=\\sum_b A_bA_b^\\top$ by direct\nmatrix multiplication. Orthogonal diagonalization then identifies the\nassociated Gaussian quadratic-form laws.\n\\end{proof}","The positive case is not the natural boundary of the central theory.  Let\n\\[\nX\\sim N(0,\\Sigma),\n\\qquad\nM=M^\\top,\n\\]\nwhere $M$ may be indefinite, and again write $S=\\Sigma^{1/2}$.  The symmetric\noperator\n\\[\nH=SMS\n\\]\nhas a real signed spectrum.","\\begin{theorem}[Exact central signed reduction]\\label{claim:final-2-2}\nIf $\\lambda_1,\\ldots,\\lambda_r$ are the nonzero eigenvalues of $H$, then\n\\[\n\\boxed{\nX^\\top M X\n\\overset d=\n\\sum_{j=1}^r\\lambda_jZ_j^2.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nWrite $X=SZ$ with $Z\\sim N(0,I)$ and diagonalize\n\\[\nSMS=V\\Lambda V^\\top.\n\\]\nRotational invariance of the standard normal gives\n\\[\nZ^\\top V\\Lambda V^\\top Z\n\\overset d=\n\\sum_j\\lambda_jZ_j^2.\n\\]\nNo positivity assumption on $M$ is needed for this central reduction.\n\\end{proof}","The sign pattern determines the support.  If every $\\lambda_j\\ge0$, the lower\nendpoint is zero.  If every $\\lambda_j\\le0$, the upper endpoint is zero.  If\nboth signs occur, the distribution is unbounded in both directions.","For the right-tail path, the cumulant generating function is\n\\[\n\\kappa(s)\n=\n-\\frac12\\sum_j\\log(1-2s\\lambda_j),\n\\]\nwith real domain\n\\[\n\\boxed{\n\\max_{\\lambda_j<0}\\frac{1}{2\\lambda_j}\n< s <\n\\min_{\\lambda_j>0}\\frac{1}{2\\lambda_j}.\n}\n\\]\nHere a missing negative or positive spectral set gives a domain endpoint\n\\(-\\infty\\) or \\(+\\infty\\), respectively.\nWhen $q>E[Q]=\\sum_j\\lambda_j$, the saddlepoint is the unique positive solution\nof\n\\[\n\\kappa'(\\theta)=q\n\\]\ninside this domain whenever a positive spectral component is present.  The\nnegative-only case with at least one nonzero coefficient has a positive\nsaddlepoint only for $E[Q]<q<0$. Indeed, $\\kappa'(\\theta)$ increases from\n$E[Q]$ to zero as $\\theta\\to\\infty$. For $q\\ge0$ its survival\nprobability is zero, and no finite positive saddlepoint solves the query.","mulant generating function is\n\\[\n\\kappa(s)\n=\n-\\frac12\\sum_j\\log(1-2s\\lambda_j),\n\\]\nwith real domain\n\\[\n\\boxed{\n\\max_{\\lambda_j<0}\\frac{1}{2\\lambda_j}\n< s <\n\\min_{\\lambda_j>0}\\frac{1}{2\\lambda_j}.\n}\n\\]\nHere a missing negative or positive spectral set gives a domain endpoint\n\\(-\\infty\\) or \\(+\\infty\\), respectively.\nWhen $q>E[Q]=\\sum_j\\lambda_j$, the saddlepoint is the unique positive solution\nof\n\\[\n\\kappa'(\\theta)=q\n\\]\ninside this domain whenever a positive spectral component is present.  The\nnegative-only case with at least one nonzero coefficient has a positive\nsaddlepoint only for $E[Q]<q<0$. Indeed, $\\kappa'(\\theta)$ increases from\n$E[Q]$ to zero as $\\theta\\to\\infty$. For $q\\ge0$ its survival\nprobability is zero, and no finite positive saddlepoint solves the query.","_j}.\n}\n\\]\nHere a missing negative or positive spectral set gives a domain endpoint\n\\(-\\infty\\) or \\(+\\infty\\), respectively.\nWhen $q>E[Q]=\\sum_j\\lambda_j$, the saddlepoint is the unique positive solution\nof\n\\[\n\\kappa'(\\theta)=q\n\\]\ninside this domain whenever a positive spectral component is present.  The\nnegative-only case with at least one nonzero coefficient has a positive\nsaddlepoint only for $E[Q]<q<0$. Indeed, $\\kappa'(\\theta)$ increases from\n$E[Q]$ to zero as $\\theta\\to\\infty$. For $q\\ge0$ its survival\nprobability is zero, and no finite positive saddlepoint solves the query.","sing negative or positive spectral set gives a domain endpoint\n\\(-\\infty\\) or \\(+\\infty\\), respectively.\nWhen $q>E[Q]=\\sum_j\\lambda_j$, the saddlepoint is the unique positive solution\nof\n\\[\n\\kappa'(\\theta)=q\n\\]\ninside this domain whenever a positive spectral component is present.  The\nnegative-only case with at least one nonzero coefficient has a positive\nsaddlepoint only for $E[Q]<q<0$. Indeed, $\\kappa'(\\theta)$ increases from\n$E[Q]$ to zero as $\\theta\\to\\infty$. For $q\\ge0$ its survival\nprobability is zero, and no finite positive saddlepoint solves the query.","l set gives a domain endpoint\n\\(-\\infty\\) or \\(+\\infty\\), respectively.\nWhen $q>E[Q]=\\sum_j\\lambda_j$, the saddlepoint is the unique positive solution\nof\n\\[\n\\kappa'(\\theta)=q\n\\]\ninside this domain whenever a positive spectral component is present.  The\nnegative-only case with at least one nonzero coefficient has a positive\nsaddlepoint only for $E[Q]<q<0$. Indeed, $\\kappa'(\\theta)$ increases from\n$E[Q]$ to zero as $\\theta\\to\\infty$. For $q\\ge0$ its survival\nprobability is zero, and no finite positive saddlepoint solves the query.","in endpoint\n\\(-\\infty\\) or \\(+\\infty\\), respectively.\nWhen $q>E[Q]=\\sum_j\\lambda_j$, the saddlepoint is the unique positive solution\nof\n\\[\n\\kappa'(\\theta)=q\n\\]\ninside this domain whenever a positive spectral component is present.  The\nnegative-only case with at least one nonzero coefficient has a positive\nsaddlepoint only for $E[Q]<q<0$. Indeed, $\\kappa'(\\theta)$ increases from\n$E[Q]$ to zero as $\\theta\\to\\infty$. For $q\\ge0$ its survival\nprobability is zero, and no finite positive saddlepoint solves the query.","or \\(+\\infty\\), respectively.\nWhen $q>E[Q]=\\sum_j\\lambda_j$, the saddlepoint is the unique positive solution\nof\n\\[\n\\kappa'(\\theta)=q\n\\]\ninside this domain whenever a positive spectral component is present.  The\nnegative-only case with at least one nonzero coefficient has a positive\nsaddlepoint only for $E[Q]<q<0$. Indeed, $\\kappa'(\\theta)$ increases from\n$E[Q]$ to zero as $\\theta\\to\\infty$. For $q\\ge0$ its survival\nprobability is zero, and no finite positive saddlepoint solves the query.","The packet representation is sign aware: positive and negative eigenvalues are\nclustered separately, and each same-sign packet stores a center $\\mu$ and\ncentered moments.  The local expansion remains\n\\[\n-\\frac12\\log(1-2s(\\mu+\\delta))\n=\n-\\frac12\\log(1-2s\\mu)\n+\n\\frac12\\sum_{n\\ge1}\n\\frac{\\alpha(s)^n\\delta^n}{n},\n\\qquad\n\\alpha(s)=\\frac{2s}{1-2s\\mu},\n\\]\nwith the same geometric remainder condition\n\\[\n|\\alpha(s)|\\max|\\delta|<1.\n\\]","A useful distinction appears in the Fourier tail.  Under the saddlepoint tilt,\n\\[\n\\left|\\phi_\\theta(t)\\right|\n=\n\\prod_j\n\\left[\n1+\n\\left(\n\\frac{2\\lambda_j t}{1-2\\theta\\lambda_j}\n\\right)^2\n\\right]^{-1/4}.\n\\]\nIf $r$ coefficients are nonzero, then asymptotically\n\\[\n|\\phi_\\theta(t)|=O(|t|^{-r/2}).\n\\]\nThus certification difficulty depends strongly on spectral rank, not only on\nhow small the requested probability is.","Let\n\\[\nX\\sim N(\\mu,\\Sigma),\n\\qquad\nM\\succeq0.\n\\]\nAgain set \\(S=\\Sigma^{1/2}\\) and diagonalize\n\\[\nSMS=V\\Lambda V^\\top.\n\\]\nWriting\n\\[\nc=V^\\top SM\\mu,\n\\]\neach positive-eigenvalue direction satisfies\n\\[\n\\lambda_iY_i^2+2c_iY_i\n=\n\\lambda_i\n\\left(\nY_i+\\frac{c_i}{\\lambda_i}\n\\right)^2\n-\n\\frac{c_i^2}{\\lambda_i}.\n\\]","\\begin{theorem}[Exact noncentral reduction]\\label{claim:final-2-3}\nFor the positive eigenvalues \\(\\lambda_i>0\\),\n\\[\n\\boxed{\nQ\n\\overset d=\nq_0\n+\n\\sum_i\n\\lambda_i\n\\chi^2_1(\\delta_i),\n}\n\\]\nwhere\n\\[\n\\delta_i=\n\\left(\\frac{c_i}{\\lambda_i}\\right)^2\n\\]\nand\n\\[\nq_0\n=\n\\mu^\\top M\\mu\n-\n\\sum_i\\frac{c_i^2}{\\lambda_i}\n\\ge0.\n\\]\nThe noncentral chi-square components are independent. To state the\nzero-shift condition precisely, put $T=M^{1/2}S$, $b=M^{1/2}\\mu$,\nand let $P_T$ be the orthogonal projector onto $\\operatorname{range}(T)$.\nThen\n\\[\nq_0=\\|(I-P_T)b\\|^2.\n\\]\nThus $q_0=0$ exactly when $M^{1/2}\\mu\\in\\operatorname{range}(M^{1/2}S)$.\nIn particular, $\\Sigma\\succ0$ is sufficient. For singular covariance the\nshift is the squared deterministic component perpendicular to the random\nrange in these weighted coordinates.\n\\e","\\begin{theorem}[Exact noncentral reduction]\\label{claim:final-2-3}\nFor the positive eigenvalues \\(\\lambda_i>0\\),\n\\[\n\\boxed{\nQ\n\\overset d=\nq_0\n+\n\\sum_i\n\\lambda_i\n\\chi^2_1(\\delta_i),\n}\n\\]\nwhere\n\\[\n\\delta_i=\n\\left(\\frac{c_i}{\\lambda_i}\\right)^2\n\\]\nand\n\\[\nq_0\n=\n\\mu^\\top M\\mu\n-\n\\sum_i\\frac{c_i^2}{\\lambda_i}\n\\ge0.\n\\]\nThe noncentral chi-square components are independent. To state the\nzero-shift condition precisely, put $T=M^{1/2}S$, $b=M^{1/2}\\mu$,\nand let $P_T$ be the orthogonal projector onto $\\operatorname{range}(T)$.\nThen\n\\[\nq_0=\\|(I-P_T)b\\|^2.\n\\]\nThus $q_0=0$ exactly when $M^{1/2}\\mu\\in\\operatorname{range}(M^{1/2}S)$.\nIn particular, $\\Sigma\\succ0$ is sufficient. For singular covariance the\nshift is the squared deterministic component perpendicular to the random\nrange in these weighted coordinates.\n\\end{theorem}","xact noncentral reduction]\\label{claim:final-2-3}\nFor the positive eigenvalues \\(\\lambda_i>0\\),\n\\[\n\\boxed{\nQ\n\\overset d=\nq_0\n+\n\\sum_i\n\\lambda_i\n\\chi^2_1(\\delta_i),\n}\n\\]\nwhere\n\\[\n\\delta_i=\n\\left(\\frac{c_i}{\\lambda_i}\\right)^2\n\\]\nand\n\\[\nq_0\n=\n\\mu^\\top M\\mu\n-\n\\sum_i\\frac{c_i^2}{\\lambda_i}\n\\ge0.\n\\]\nThe noncentral chi-square components are independent. To state the\nzero-shift condition precisely, put $T=M^{1/2}S$, $b=M^{1/2}\\mu$,\nand let $P_T$ be the orthogonal projector onto $\\operatorname{range}(T)$.\nThen\n\\[\nq_0=\\|(I-P_T)b\\|^2.\n\\]\nThus $q_0=0$ exactly when $M^{1/2}\\mu\\in\\operatorname{range}(M^{1/2}S)$.\nIn particular, $\\Sigma\\succ0$ is sufficient. For singular covariance the\nshift is the squared deterministic component perpendicular to the random\nrange in these weighted coordinates.\n\\end{theorem}","l reduction]\\label{claim:final-2-3}\nFor the positive eigenvalues \\(\\lambda_i>0\\),\n\\[\n\\boxed{\nQ\n\\overset d=\nq_0\n+\n\\sum_i\n\\lambda_i\n\\chi^2_1(\\delta_i),\n}\n\\]\nwhere\n\\[\n\\delta_i=\n\\left(\\frac{c_i}{\\lambda_i}\\right)^2\n\\]\nand\n\\[\nq_0\n=\n\\mu^\\top M\\mu\n-\n\\sum_i\\frac{c_i^2}{\\lambda_i}\n\\ge0.\n\\]\nThe noncentral chi-square components are independent. To state the\nzero-shift condition precisely, put $T=M^{1/2}S$, $b=M^{1/2}\\mu$,\nand let $P_T$ be the orthogonal projector onto $\\operatorname{range}(T)$.\nThen\n\\[\nq_0=\\|(I-P_T)b\\|^2.\n\\]\nThus $q_0=0$ exactly when $M^{1/2}\\mu\\in\\operatorname{range}(M^{1/2}S)$.\nIn particular, $\\Sigma\\succ0$ is sufficient. For singular covariance the\nshift is the squared deterministic component perpendicular to the random\nrange in these weighted coordinates.\n\\end{theorem}",":final-2-3}\nFor the positive eigenvalues \\(\\lambda_i>0\\),\n\\[\n\\boxed{\nQ\n\\overset d=\nq_0\n+\n\\sum_i\n\\lambda_i\n\\chi^2_1(\\delta_i),\n}\n\\]\nwhere\n\\[\n\\delta_i=\n\\left(\\frac{c_i}{\\lambda_i}\\right)^2\n\\]\nand\n\\[\nq_0\n=\n\\mu^\\top M\\mu\n-\n\\sum_i\\frac{c_i^2}{\\lambda_i}\n\\ge0.\n\\]\nThe noncentral chi-square components are independent. To state the\nzero-shift condition precisely, put $T=M^{1/2}S$, $b=M^{1/2}\\mu$,\nand let $P_T$ be the orthogonal projector onto $\\operatorname{range}(T)$.\nThen\n\\[\nq_0=\\|(I-P_T)b\\|^2.\n\\]\nThus $q_0=0$ exactly when $M^{1/2}\\mu\\in\\operatorname{range}(M^{1/2}S)$.\nIn particular, $\\Sigma\\succ0$ is sufficient. For singular covariance the\nshift is the squared deterministic component perpendicular to the random\nrange in these weighted coordinates.\n\\end{theorem}","s \\(\\lambda_i>0\\),\n\\[\n\\boxed{\nQ\n\\overset d=\nq_0\n+\n\\sum_i\n\\lambda_i\n\\chi^2_1(\\delta_i),\n}\n\\]\nwhere\n\\[\n\\delta_i=\n\\left(\\frac{c_i}{\\lambda_i}\\right)^2\n\\]\nand\n\\[\nq_0\n=\n\\mu^\\top M\\mu\n-\n\\sum_i\\frac{c_i^2}{\\lambda_i}\n\\ge0.\n\\]\nThe noncentral chi-square components are independent. To state the\nzero-shift condition precisely, put $T=M^{1/2}S$, $b=M^{1/2}\\mu$,\nand let $P_T$ be the orthogonal projector onto $\\operatorname{range}(T)$.\nThen\n\\[\nq_0=\\|(I-P_T)b\\|^2.\n\\]\nThus $q_0=0$ exactly when $M^{1/2}\\mu\\in\\operatorname{range}(M^{1/2}S)$.\nIn particular, $\\Sigma\\succ0$ is sufficient. For singular covariance the\nshift is the squared deterministic component perpendicular to the random\nrange in these weighted coordinates.\n\\end{theorem}","overset d=\nq_0\n+\n\\sum_i\n\\lambda_i\n\\chi^2_1(\\delta_i),\n}\n\\]\nwhere\n\\[\n\\delta_i=\n\\left(\\frac{c_i}{\\lambda_i}\\right)^2\n\\]\nand\n\\[\nq_0\n=\n\\mu^\\top M\\mu\n-\n\\sum_i\\frac{c_i^2}{\\lambda_i}\n\\ge0.\n\\]\nThe noncentral chi-square components are independent. To state the\nzero-shift condition precisely, put $T=M^{1/2}S$, $b=M^{1/2}\\mu$,\nand let $P_T$ be the orthogonal projector onto $\\operatorname{range}(T)$.\nThen\n\\[\nq_0=\\|(I-P_T)b\\|^2.\n\\]\nThus $q_0=0$ exactly when $M^{1/2}\\mu\\in\\operatorname{range}(M^{1/2}S)$.\nIn particular, $\\Sigma\\succ0$ is sufficient. For singular covariance the\nshift is the squared deterministic component perpendicular to the random\nrange in these weighted coordinates.\n\\end{theorem}","a_i\n\\chi^2_1(\\delta_i),\n}\n\\]\nwhere\n\\[\n\\delta_i=\n\\left(\\frac{c_i}{\\lambda_i}\\right)^2\n\\]\nand\n\\[\nq_0\n=\n\\mu^\\top M\\mu\n-\n\\sum_i\\frac{c_i^2}{\\lambda_i}\n\\ge0.\n\\]\nThe noncentral chi-square components are independent. To state the\nzero-shift condition precisely, put $T=M^{1/2}S$, $b=M^{1/2}\\mu$,\nand let $P_T$ be the orthogonal projector onto $\\operatorname{range}(T)$.\nThen\n\\[\nq_0=\\|(I-P_T)b\\|^2.\n\\]\nThus $q_0=0$ exactly when $M^{1/2}\\mu\\in\\operatorname{range}(M^{1/2}S)$.\nIn particular, $\\Sigma\\succ0$ is sufficient. For singular covariance the\nshift is the squared deterministic component perpendicular to the random\nrange in these weighted coordinates.\n\\end{theorem}","),\n}\n\\]\nwhere\n\\[\n\\delta_i=\n\\left(\\frac{c_i}{\\lambda_i}\\right)^2\n\\]\nand\n\\[\nq_0\n=\n\\mu^\\top M\\mu\n-\n\\sum_i\\frac{c_i^2}{\\lambda_i}\n\\ge0.\n\\]\nThe noncentral chi-square components are independent. To state the\nzero-shift condition precisely, put $T=M^{1/2}S$, $b=M^{1/2}\\mu$,\nand let $P_T$ be the orthogonal projector onto $\\operatorname{range}(T)$.\nThen\n\\[\nq_0=\\|(I-P_T)b\\|^2.\n\\]\nThus $q_0=0$ exactly when $M^{1/2}\\mu\\in\\operatorname{range}(M^{1/2}S)$.\nIn particular, $\\Sigma\\succ0$ is sufficient. For singular covariance the\nshift is the squared deterministic component perpendicular to the random\nrange in these weighted coordinates.\n\\end{theorem}","2\n\\]\nand\n\\[\nq_0\n=\n\\mu^\\top M\\mu\n-\n\\sum_i\\frac{c_i^2}{\\lambda_i}\n\\ge0.\n\\]\nThe noncentral chi-square components are independent. To state the\nzero-shift condition precisely, put $T=M^{1/2}S$, $b=M^{1/2}\\mu$,\nand let $P_T$ be the orthogonal projector onto $\\operatorname{range}(T)$.\nThen\n\\[\nq_0=\\|(I-P_T)b\\|^2.\n\\]\nThus $q_0=0$ exactly when $M^{1/2}\\mu\\in\\operatorname{range}(M^{1/2}S)$.\nIn particular, $\\Sigma\\succ0$ is sufficient. For singular covariance the\nshift is the squared deterministic component perpendicular to the random\nrange in these weighted coordinates.\n\\end{theorem}","\\begin{proof}\nExpand\n\\[\n(\\mu+SZ)^\\top M(\\mu+SZ)\n=\nZ^\\top SMSZ\n+\n2\\mu^\\top MSZ\n+\n\\mu^\\top M\\mu\n\\]\nand rotate by \\(V\\).  Completing the square gives the displayed\nnoncentral components.  For a null vector of \\(SMS\\),\n\\[\nM^{1/2}Sv=0,\n\\]\nhence\n\\[\nv^\\top SM\\mu\n=\n(M^{1/2}Sv)^\\top M^{1/2}\\mu\n=\n0,\n\\]\nso there is no unresolved stochastic linear term in a zero-eigenvalue\ndirection. For the positive eigenvectors $v_i$ of $T^\\top T=SMS$,\nthe vectors $Tv_i/\\sqrt{\\lambda_i}$ form an orthonormal basis of\n$\\operatorname{range}(T)$. Their inner products with $b$ are\n$c_i/\\sqrt{\\lambda_i}$. Hence the subtracted sum equals $\\|P_Tb\\|^2$,\nleaving $q_0=\\|(I-P_T)b\\|^2\\ge0$. Orthogonal rotation preserves\nindependence of the normal coordinates, so the squared components are\nindependent as stated.\n\\end{proof}","{proof}\nExpand\n\\[\n(\\mu+SZ)^\\top M(\\mu+SZ)\n=\nZ^\\top SMSZ\n+\n2\\mu^\\top MSZ\n+\n\\mu^\\top M\\mu\n\\]\nand rotate by \\(V\\).  Completing the square gives the displayed\nnoncentral components.  For a null vector of \\(SMS\\),\n\\[\nM^{1/2}Sv=0,\n\\]\nhence\n\\[\nv^\\top SM\\mu\n=\n(M^{1/2}Sv)^\\top M^{1/2}\\mu\n=\n0,\n\\]\nso there is no unresolved stochastic linear term in a zero-eigenvalue\ndirection. For the positive eigenvectors $v_i$ of $T^\\top T=SMS$,\nthe vectors $Tv_i/\\sqrt{\\lambda_i}$ form an orthonormal basis of\n$\\operatorname{range}(T)$. Their inner products with $b$ are\n$c_i/\\sqrt{\\lambda_i}$. Hence the subtracted sum equals $\\|P_Tb\\|^2$,\nleaving $q_0=\\|(I-P_T)b\\|^2\\ge0$. Orthogonal rotation preserves\nindependence of the normal coordinates, so the squared components are\nindependent as stated.\n\\end{proof}","xpand\n\\[\n(\\mu+SZ)^\\top M(\\mu+SZ)\n=\nZ^\\top SMSZ\n+\n2\\mu^\\top MSZ\n+\n\\mu^\\top M\\mu\n\\]\nand rotate by \\(V\\).  Completing the square gives the displayed\nnoncentral components.  For a null vector of \\(SMS\\),\n\\[\nM^{1/2}Sv=0,\n\\]\nhence\n\\[\nv^\\top SM\\mu\n=\n(M^{1/2}Sv)^\\top M^{1/2}\\mu\n=\n0,\n\\]\nso there is no unresolved stochastic linear term in a zero-eigenvalue\ndirection. For the positive eigenvectors $v_i$ of $T^\\top T=SMS$,\nthe vectors $Tv_i/\\sqrt{\\lambda_i}$ form an orthonormal basis of\n$\\operatorname{range}(T)$. Their inner products with $b$ are\n$c_i/\\sqrt{\\lambda_i}$. Hence the subtracted sum equals $\\|P_Tb\\|^2$,\nleaving $q_0=\\|(I-P_T)b\\|^2\\ge0$. Orthogonal rotation preserves\nindependence of the normal coordinates, so the squared components are\nindependent as stated.\n\\end{proof}","+SZ)\n=\nZ^\\top SMSZ\n+\n2\\mu^\\top MSZ\n+\n\\mu^\\top M\\mu\n\\]\nand rotate by \\(V\\).  Completing the square gives the displayed\nnoncentral components.  For a null vector of \\(SMS\\),\n\\[\nM^{1/2}Sv=0,\n\\]\nhence\n\\[\nv^\\top SM\\mu\n=\n(M^{1/2}Sv)^\\top M^{1/2}\\mu\n=\n0,\n\\]\nso there is no unresolved stochastic linear term in a zero-eigenvalue\ndirection. For the positive eigenvectors $v_i$ of $T^\\top T=SMS$,\nthe vectors $Tv_i/\\sqrt{\\lambda_i}$ form an orthonormal basis of\n$\\operatorname{range}(T)$. Their inner products with $b$ are\n$c_i/\\sqrt{\\lambda_i}$. Hence the subtracted sum equals $\\|P_Tb\\|^2$,\nleaving $q_0=\\|(I-P_T)b\\|^2\\ge0$. Orthogonal rotation preserves\nindependence of the normal coordinates, so the squared components are\nindependent as stated.\n\\end{proof}","and rotate by \\(V\\).  Completing the square gives the displayed\nnoncentral components.  For a null vector of \\(SMS\\),\n\\[\nM^{1/2}Sv=0,\n\\]\nhence\n\\[\nv^\\top SM\\mu\n=\n(M^{1/2}Sv)^\\top M^{1/2}\\mu\n=\n0,\n\\]\nso there is no unresolved stochastic linear term in a zero-eigenvalue\ndirection. For the positive eigenvectors $v_i$ of $T^\\top T=SMS$,\nthe vectors $Tv_i/\\sqrt{\\lambda_i}$ form an orthonormal basis of\n$\\operatorname{range}(T)$. Their inner products with $b$ are\n$c_i/\\sqrt{\\lambda_i}$. Hence the subtracted sum equals $\\|P_Tb\\|^2$,\nleaving $q_0=\\|(I-P_T)b\\|^2\\ge0$. Orthogonal rotation preserves\nindependence of the normal coordinates, so the squared components are\nindependent as stated.\n\\end{proof}","e displayed\nnoncentral components.  For a null vector of \\(SMS\\),\n\\[\nM^{1/2}Sv=0,\n\\]\nhence\n\\[\nv^\\top SM\\mu\n=\n(M^{1/2}Sv)^\\top M^{1/2}\\mu\n=\n0,\n\\]\nso there is no unresolved stochastic linear term in a zero-eigenvalue\ndirection. For the positive eigenvectors $v_i$ of $T^\\top T=SMS$,\nthe vectors $Tv_i/\\sqrt{\\lambda_i}$ form an orthonormal basis of\n$\\operatorname{range}(T)$. Their inner products with $b$ are\n$c_i/\\sqrt{\\lambda_i}$. Hence the subtracted sum equals $\\|P_Tb\\|^2$,\nleaving $q_0=\\|(I-P_T)b\\|^2\\ge0$. Orthogonal rotation preserves\nindependence of the normal coordinates, so the squared components are\nindependent as stated.\n\\end{proof}","yed\nnoncentral components.  For a null vector of \\(SMS\\),\n\\[\nM^{1/2}Sv=0,\n\\]\nhence\n\\[\nv^\\top SM\\mu\n=\n(M^{1/2}Sv)^\\top M^{1/2}\\mu\n=\n0,\n\\]\nso there is no unresolved stochastic linear term in a zero-eigenvalue\ndirection. For the positive eigenvectors $v_i$ of $T^\\top T=SMS$,\nthe vectors $Tv_i/\\sqrt{\\lambda_i}$ form an orthonormal basis of\n$\\operatorname{range}(T)$. Their inner products with $b$ are\n$c_i/\\sqrt{\\lambda_i}$. Hence the subtracted sum equals $\\|P_Tb\\|^2$,\nleaving $q_0=\\|(I-P_T)b\\|^2\\ge0$. Orthogonal rotation preserves\nindependence of the normal coordinates, so the squared components are\nindependent as stated.\n\\end{proof}",",\n\\[\nM^{1/2}Sv=0,\n\\]\nhence\n\\[\nv^\\top SM\\mu\n=\n(M^{1/2}Sv)^\\top M^{1/2}\\mu\n=\n0,\n\\]\nso there is no unresolved stochastic linear term in a zero-eigenvalue\ndirection. For the positive eigenvectors $v_i$ of $T^\\top T=SMS$,\nthe vectors $Tv_i/\\sqrt{\\lambda_i}$ form an orthonormal basis of\n$\\operatorname{range}(T)$. Their inner products with $b$ are\n$c_i/\\sqrt{\\lambda_i}$. Hence the subtracted sum equals $\\|P_Tb\\|^2$,\nleaving $q_0=\\|(I-P_T)b\\|^2\\ge0$. Orthogonal rotation preserves\nindependence of the normal coordinates, so the squared components are\nindependent as stated.\n\\end{proof}","ence\n\\[\nv^\\top SM\\mu\n=\n(M^{1/2}Sv)^\\top M^{1/2}\\mu\n=\n0,\n\\]\nso there is no unresolved stochastic linear term in a zero-eigenvalue\ndirection. For the positive eigenvectors $v_i$ of $T^\\top T=SMS$,\nthe vectors $Tv_i/\\sqrt{\\lambda_i}$ form an orthonormal basis of\n$\\operatorname{range}(T)$. Their inner products with $b$ are\n$c_i/\\sqrt{\\lambda_i}$. Hence the subtracted sum equals $\\|P_Tb\\|^2$,\nleaving $q_0=\\|(I-P_T)b\\|^2\\ge0$. Orthogonal rotation preserves\nindependence of the normal coordinates, so the squared components are\nindependent as stated.\n\\end{proof}","The numerical interface implements this reduction directly.  In an isotropic\ncontrol,\n\\[\n\\Sigma=I,\\qquad M=aI,\\qquad a>0,\n\\]\nthe law collapses to the exact scaled noncentral chi-square distribution\n\\[\nQ/a\\sim\\chi^2_d(\\|\\mu\\|^2).\n\\]","The solver separates four logically different representations:\n\\[\n\\boxed{\n\\text{storage blocks}\n\\circ\n\\text{structural phases}\n\\circ\n\\text{Krylov/range compression}\n\\circ\n\\text{spectral tail packets}.\n}\n\\]\nStorage blocks answer where data reside.  Structural phases answer how the\noperator propagates.  Krylov/range compression answers which subspace has\nactually been resolved.  Spectral packets answer how the final transform is\nevaluated.  These decompositions should not be conflated.","\\section{A factor, a spectrum, and an exact law}\\label{sec:worked-law}\nWe will follow one small example through the remaining parts of the book. Its entries are exact binary fractions, so the same matrix can be used for symbolic algebra and for a binary64 input without introducing an input-rounding ambiguity. Set\n\\[\nA=\\frac12\n\\begin{pmatrix}\n 1& 1& 1& 1& 1& 1\\\\\n 1&-1& 1&-1& 1&-1\\\\\n 1& 1&-1&-1&-1&-1\\\\\n 1&-1&-1& 1&-1& 1\n\\end{pmatrix}.\n\\]\nMultiplication gives\n\\[\nK=AA^\\top=\\frac12\n\\begin{pmatrix}\n3&0&-1&0","1&-1&-1& 1&-1& 1\n\\end{pmatrix}.\n\\]\nMultiplication gives\n\\[\nK=AA^\\top=\\frac12\n\\begin{pmatrix}\n3&0&-1&0\\\\0&3&0&-1\\\\-1&0&3&0\\\\0&-1&0&3\n\\end{pmatrix}.\n\\]\nThe orthogonal matrix\n\\[\nV=\\frac12\n\\begin{pmatrix}\n1&1&1&1","0&-1&0&3\n\\end{pmatrix}.\n\\]\nThe orthogonal matrix\n\\[\nV=\\frac12\n\\begin{pmatrix}\n1&1&1&1\\\\1&-1&1&-1\\\\1&1&-1&-1\\\\1&-1&-1&1\n\\end{pmatrix}\n\\]\nsatisfies $V^\\top K V=\\operatorname{diag}(1,1,2,2)$. For $Z\\sim N(0,I_4)$,\n\\[\nQ=Z^\\top KZ\\ \\overset d=\\ Y+2W,\n\\qquad Y,W\\text{ independent with law }\\chi^2_2.\n\\]\nThe spectrum is therefore $(1,1,2,2)$ and $\\mathbb E Q=6$. The six-dimensional feature Gram $A^\\top A$ has these same four nonzero eigenvalues and two additional zeros.","1&-1&-1&1\n\\end{pmatrix}\n\\]\nsatisfies $V^\\top K V=\\operatorname{diag}(1,1,2,2)$. For $Z\\sim N(0,I_4)$,\n\\[\nQ=Z^\\top KZ\\ \\overset d=\\ Y+2W,\n\\qquad Y,W\\text{ independent with law }\\chi^2_2.\n\\]\nThe spectrum is therefore $(1,1,2,2)$ and $\\mathbb E Q=6$. The six-dimensional feature Gram $A^\\top A$ has these same four nonzero eigenvalues and two additional zeros.","Here the probability law also has a closed form. The densities of $Y$ and $2W$ are $\\tfrac12e^{-x/2}$ and $\\tfrac14e^{-x/4}$ on the nonnegative half-line. Their convolution is\n\\[\nf_Q(x)=\\int_0^x\\frac12e^{-y/2}\\frac14e^{-(x-y)/4}\\,dy\n       =\\frac12\\bigl(e^{-x/4}-e^{-x/2}\\bigr).\n\\]\nIntegrating from $q$ to infinity gives\n\\begin{equation}\nP(Q\\ge q)=2e^{-q/4}-e^{-q/2},\\qquad q\\ge0.\n\\label{eq:worked-exact-tail}\n\\end{equation}\nAt $q=40$ the probability is approximately $9.079779837\\times10^{-5}$. This formula will be used as an independent control, not as a shortcut i","Here the probability law also has a closed form. The densities of $Y$ and $2W$ are $\\tfrac12e^{-x/2}$ and $\\tfrac14e^{-x/4}$ on the nonnegative half-line. Their convolution is\n\\[\nf_Q(x)=\\int_0^x\\frac12e^{-y/2}\\frac14e^{-(x-y)/4}\\,dy\n       =\\frac12\\bigl(e^{-x/4}-e^{-x/2}\\bigr).\n\\]\nIntegrating from $q$ to infinity gives\n\\begin{equation}\nP(Q\\ge q)=2e^{-q/4}-e^{-q/2},\\qquad q\\ge0.\n\\label{eq:worked-exact-tail}\n\\end{equation}\nAt $q=40$ the probability is approximately $9.079779837\\times10^{-5}$. This formula will be used as an independent control, not as a shortcut inside the","Here the probability law also has a closed form. The densities of $Y$ and $2W$ are $\\tfrac12e^{-x/2}$ and $\\tfrac14e^{-x/4}$ on the nonnegative half-line. Their convolution is\n\\[\nf_Q(x)=\\int_0^x\\frac12e^{-y/2}\\frac14e^{-(x-y)/4}\\,dy\n       =\\frac12\\bigl(e^{-x/4}-e^{-x/2}\\bigr).\n\\]\nIntegrating from $q$ to infinity gives\n\\begin{equation}\nP(Q\\ge q)=2e^{-q/4}-e^{-q/2},\\qquad q\\ge0.\n\\label{eq:worked-exact-tail}\n\\end{equation}\nAt $q=40$ the probability is approximately $9.079779837\\times10^{-5}$. This formula will be used as an independent control, not as a shortcut inside the COS calculation. The n","Here the probability law also has a closed form. The densities of $Y$ and $2W$ are $\\tfrac12e^{-x/2}$ and $\\tfrac14e^{-x/4}$ on the nonnegative half-line. Their convolution is\n\\[\nf_Q(x)=\\int_0^x\\frac12e^{-y/2}\\frac14e^{-(x-y)/4}\\,dy\n       =\\frac12\\bigl(e^{-x/4}-e^{-x/2}\\bigr).\n\\]\nIntegrating from $q$ to infinity gives\n\\begin{equation}\nP(Q\\ge q)=2e^{-q/4}-e^{-q/2},\\qquad q\\ge0.\n\\label{eq:worked-exact-tail}\n\\end{equation}\nAt $q=40$ the probability is approximately $9.079779837\\times10^{-5}$. This formula will be used as an independent control, not as a shortcut inside the COS calculation. The next part begins with th","Here the probability law also has a closed form. The densities of $Y$ and $2W$ are $\\tfrac12e^{-x/2}$ and $\\tfrac14e^{-x/4}$ on the nonnegative half-line. Their convolution is\n\\[\nf_Q(x)=\\int_0^x\\frac12e^{-y/2}\\frac14e^{-(x-y)/4}\\,dy\n       =\\frac12\\bigl(e^{-x/4}-e^{-x/2}\\bigr).\n\\]\nIntegrating from $q$ to infinity gives\n\\begin{equation}\nP(Q\\ge q)=2e^{-q/4}-e^{-q/2},\\qquad q\\ge0.\n\\label{eq:worked-exact-tail}\n\\end{equation}\nAt $q=40$ the probability is approximately $9.079779837\\times10^{-5}$. This formula will be used as an independent control, not as a shortcut inside the COS calculation. The next part begins with the factor $A$ and asks which operations may change its storage or basis without changing this law.","has a closed form. The densities of $Y$ and $2W$ are $\\tfrac12e^{-x/2}$ and $\\tfrac14e^{-x/4}$ on the nonnegative half-line. Their convolution is\n\\[\nf_Q(x)=\\int_0^x\\frac12e^{-y/2}\\frac14e^{-(x-y)/4}\\,dy\n       =\\frac12\\bigl(e^{-x/4}-e^{-x/2}\\bigr).\n\\]\nIntegrating from $q$ to infinity gives\n\\begin{equation}\nP(Q\\ge q)=2e^{-q/4}-e^{-q/2},\\qquad q\\ge0.\n\\label{eq:worked-exact-tail}\n\\end{equation}\nAt $q=40$ the probability is approximately $9.079779837\\times10^{-5}$. This formula will be used as an independent control, not as a shortcut inside the COS calculation. The next part begins with the factor $A$ and asks which operations may change its storage or basis without changing this law.","and $2W$ are $\\tfrac12e^{-x/2}$ and $\\tfrac14e^{-x/4}$ on the nonnegative half-line. Their convolution is\n\\[\nf_Q(x)=\\int_0^x\\frac12e^{-y/2}\\frac14e^{-(x-y)/4}\\,dy\n       =\\frac12\\bigl(e^{-x/4}-e^{-x/2}\\bigr).\n\\]\nIntegrating from $q$ to infinity gives\n\\begin{equation}\nP(Q\\ge q)=2e^{-q/4}-e^{-q/2},\\qquad q\\ge0.\n\\label{eq:worked-exact-tail}\n\\end{equation}\nAt $q=40$ the probability is approximately $9.079779837\\times10^{-5}$. This formula will be used as an independent control, not as a shortcut inside the COS calculation. The next part begins with the factor $A$ and asks which operations may change its storage or basis without changing this law.","=\\frac12\\bigl(e^{-x/4}-e^{-x/2}\\bigr).\n\\]\nIntegrating from $q$ to infinity gives\n\\begin{equation}\nP(Q\\ge q)=2e^{-q/4}-e^{-q/2},\\qquad q\\ge0.\n\\label{eq:worked-exact-tail}\n\\end{equation}\nAt $q=40$ the probability is approximately $9.079779837\\times10^{-5}$. This formula will be used as an independent control, not as a shortcut inside the COS calculation. The next part begins with the factor $A$ and asks which operations may change its storage or basis without changing this law.","Let\n\\[\nG\\in\\mathbb R^{n\\times m},\n\\qquad\nW=\\operatorname{diag}(w_1,\\ldots,w_m),\n\\quad w_j\\ge 0,\n\\]\nand let \\(Q_X\\in\\mathbb R^{n\\times p}\\) have orthonormal columns spanning\nthe null-model covariates.  Define\n\\[\nM_X=I-Q_XQ_X^\\top,\n\\qquad\nA=M_XGW^{1/2},\n\\qquad\nK=AA^\\top.\n\\]","The central implementation difficulty is not the algebraic size of the\nmodel but the mismatch between the total factor size \\(n\\times m\\) and the\nworking memory of an accelerator.  The key point of this book is that\nseveral decompositions act on different mathematical coordinates and therefore\ncompose:\n\\[\n\\boxed{\n\\text{storage blocks}\n\\circ\n\\text{structural phases}\n\\circ\n\\text{Krylov/range compression}\n\\circ\n\\text{spectral tail packets}.\n}\n\\]","Storage blocking controls residence.  Structural phases encode linkage-disequilibrium (LD) windows,\nannotations, frequency bands, or other coordinate groups.  Krylov methods\ncompress the operator to the rank actually requested.  Spectral packets\ncompress the final generalized-\\(\\chi^2\\) transform.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\linewidth]{figures_production/figure_01_residualization_geometry.pdf}\n\\caption{Orthogonal residualization separates a feature into a component in the covariate span and a component retained by $M_X$. The diagram is an illustrative coordinate model. The identity behind it is the exact projector decomposition.}\n\\label{fig:residualization-geometry}\n\\end{figure}","Partition the variant indices into disjoint blocks\n\\[\n\\{1,\\ldots,m\\}\n=\nJ_1\\sqcup\\cdots\\sqcup J_B.\n\\]\nLet\n\\[\nG_b=G_{[:,J_b]},\n\\qquad\nW_b=W_{J_b,J_b},\n\\qquad\nA_b=M_XG_bW_b^{1/2}.\n\\]\nThen\n\\[\nA=[A_1\\ \\cdots\\ A_B].\n\\]","\\begin{theorem}[Additive block Gram identity]\\label{claim:final-8-1}\n\\[\n\\boxed{\nK\n=\nAA^\\top\n=\n\\sum_{b=1}^B A_bA_b^\\top\n=\nM_X\n\\left(\n\\sum_{b=1}^B G_bW_bG_b^\\top\n\\right)\nM_X.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nHorizontal concatenation gives\n\\[\n[A_1\\ \\cdots\\ A_B]\n\\begin{bmatrix}\nA_1^\\top\\\\\n\\vdots\\\\\nA_B^\\top\n\\end{bmatrix}\n=\n\\sum_bA_bA_b^\\top.\n\\]\nThe final form follows from\n\\(A_b=M_XG_bW_b^{1/2}\\) and\n\\(M_X^\\top=M_X=M_X^2\\).\n\\end{proof}","A_B^\\top\n\\end{bmatrix}\n=\n\\sum_bA_bA_b^\\top.\n\\]\nThe final form follows from\n\\(A_b=M_XG_bW_b^{1/2}\\) and\n\\(M_X^\\top=M_X=M_X^2\\).\n\\end{proof}","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\linewidth]{figures_production/figure_02_exact_block_kernel_assembly.pdf}\n\\caption{Each resident column block contributes a complete sample-space Gram summand $A_bA_b^{\\mathsf T}$. Adding the summands recovers $AA^{\\mathsf T}$ exactly. Storage partitioning changes residence and evaluation order, not the represented operator.}\n\\label{fig:block-kernel-assembly}\n\\end{figure}","\\begin{theorem}[One-pass sample-space streaming action]\\label{claim:final-9-1}\nFor every block vector \\(V\\in\\mathbb R^{n\\times s}\\),\n\\[\n\\boxed{\nKV\n=\nM_X\n\\sum_{b=1}^B\nG_bW_bG_b^\\top\nM_XV.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nApply the additive block Gram identity to \\(KV\\), then associate each block\nproduct as\n\\[\nG_bW_b(G_b^\\top M_XV).\n\\]\n\\end{proof}","\\begin{algorithm}[Streamed \\(KV\\)]\\label{claim:final-9-2}\n\\begin{enumerate}[label=\\arabic*.]\n\\item Compute \\(U=M_XV\\).\n\\item Set \\(Y=0\\).\n\\item For each resident block \\(b\\), compute\n\\[\nT_b=G_b^\\top U,\n\\qquad\nY\\leftarrow Y+G_b(W_bT_b).\n\\]\n\\item Return \\(M_XY\\).\n\\end{enumerate}\n\\end{algorithm}","\\begin{corollary}[Working memory independent of total variant count]\\label{claim:final-9-3}\nIf at most \\(c\\) variants are resident and the block-vector width is \\(s\\),\nthen the algebraic working set is\n\\[\n\\boxed{\nO(nc+ns+cs+np),\n}\n\\]\nindependent of total variant count \\(m\\).\n\\end{corollary}","Let \\(V\\in\\mathbb R^{m\\times s}\\) be partitioned conformally:\n\\[\nV=\n\\begin{bmatrix}\nV_1","Let \\(V\\in\\mathbb R^{m\\times s}\\) be partitioned conformally:\n\\[\nV=\n\\begin{bmatrix}\nV_1\\\\\n\\vdots\\\\\nV_B\n\\end{bmatrix}.\n\\]","\\begin{theorem}[Streamed \\(AV\\)]\\label{claim:final-10-1}\n\\[\n\\boxed{\nAV\n=\nM_X\n\\sum_{b=1}^B\nG_bW_b^{1/2}V_b.\n}\n\\]\n\\end{theorem}","\\begin{theorem}[Streamed \\(A^\\top U\\)]\\label{claim:final-10-2}\nFor \\(U\\in\\mathbb R^{n\\times s}\\),\n\\[\n\\boxed{\n(A^\\top U)_b\n=\nW_b^{1/2}G_b^\\top M_XU.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nBoth identities follow by substituting\n\\(A=M_XGW^{1/2}\\) and partitioning the variant coordinate.\n\\end{proof}","\\begin{corollary}[Streaming Golub--Kahan equivalence]\\label{claim:final-10-3}\nWith the same initial state, algorithmic choices, and random draws, any\nexact-arithmetic algorithm whose access to \\(A\\) occurs only through\n\\(AV\\) and \\(A^\\top U\\) produces the same iterates when those products are\nimplemented by the preceding streamed formulas.\n\\end{corollary}","The variant-space normal operator is\n\\[\nH=A^\\top A.\n\\]","\\begin{theorem}[Two-pass streamed dual action]\\label{claim:final-11-1}\nFor every \\(V\\in\\mathbb R^{m\\times s}\\),\n\\[\n\\boxed{\nHV=A^\\top(AV).\n}\n\\]\nThe product is evaluated exactly by a streamed pass forming\n\\(Y=AV\\in\\mathbb R^{n\\times s}\\), followed by a second streamed pass\nemitting the blocks\n\\[\n(HV)_b=W_b^{1/2}G_b^\\top M_XY.\n\\]\n\\end{theorem}","A variant block may itself exceed device memory.  Let\n\\[\nH_b=G_bW_b^{1/2}\n\\]\nand partition samples into row tiles \\(I_1,\\ldots,I_R\\).","\\begin{theorem}[Two-pass residualized tile construction]\\label{claim:final-12-1}\nDefine\n\\[\nR_b=Q_X^\\top H_b.\n\\]\nThen\n\\[\n\\boxed{\nR_b\n=\n\\sum_r\nQ_{X,I_r}^\\top H_{b,I_r},\n}\n\\]\nand after \\(R_b\\) is known,\n\\[\n\\boxed{\nA_{b,I_r}\n=\nH_{b,I_r}-Q_{X,I_r}R_b.\n}\n\\]\nConsequently,\n\\[\n\\boxed{\nK_{I_r,I_s}\n=\n\\sum_b\nA_{b,I_r}A_{b,I_s}^\\top.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nThe first identity is the row-partitioned matrix product\n\\(Q_X^\\top H_b\\).  The second is the definition\n\\(M_XH_b=H_b-Q_XQ_X^\\top H_b\\).  The last identity is the corresponding\nsample-tile expansion of \\(AA^\\top\\).\n\\end{proof}","With sample tile height \\(t\\) and resident variant width \\(c\\), the dominant\nworking storage, including a resident covariate tile, is\n\\[\nO(tc+pc+tp+t^2).\n\\]\nThis count excludes the backing input store, completed output blocks,\nallocator overhead, and library workspaces.","\\begin{theorem}[Partition-refinement invariance]\\label{claim:final-13-1}\nLet \\(\\mathcal J\\) be any partition of the variants and let\n\\(\\mathcal J'\\) refine \\(\\mathcal J\\).  Then\n\\[\n\\boxed{\n\\sum_{J\\in\\mathcal J}A_JA_J^\\top\n=\n\\sum_{J'\\in\\mathcal J'}A_{J'}A_{J'}^\\top\n=\nAA^\\top.\n}\n\\]\nThe analogous streamed operator action is also invariant.\n\\end{theorem}","\\begin{proposition}[Variant permutation invariance]\\label{claim:final-13-3}\nIf the columns of \\(G\\) and the diagonal entries of \\(W\\) are permuted by the\nsame permutation, \\(K\\) and every streamed sample-space action are unchanged.\n\\end{proposition}","Let \\(d\\) be the number of bytes per floating scalar, \\(c\\) the resident\nvariant count, \\(s\\) the Krylov width, and \\(p\\) the covariate rank.\nIgnoring library workspace and allocator overhead, a streamed sample-space\naction has working memory on the order of\n\\[\n\\boxed{\nd\\,[nc+2ns+cs+np+O(ps+c)].\n}\n\\]","The full factor costs\n\\[\ndnm.\n\\]","For fixed \\(n,p,s,c\\),\n\\[\n\\boxed{\n\\frac{M_{\\rm stream}}{M_{\\rm full}}\n=\nO\\!\\left(\n\\frac{c}{m}\n+\n\\frac{s}{m}\n+\n\\frac{p}{m}\n\\right),\n}\n\\]\nso device memory remains bounded as total variant count grows.","\\begin{table}[htbp]\n\\centering\n\\caption{Algebraic float64 memory model for \\(n=2000\\), \\(p=4\\),\nKrylov width \\(s=16\\), resident variant block \\(c=1024\\), and\nsample tile height \\(t=256\\).}\n\\begin{tabular}{rrrr}\n\\toprule\n\\(m\\) & full factor (MiB) & streamed \\(KV\\) (MiB) & tiled Gram (MiB)","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\linewidth]{figures_production/figure_03_memory_law.pdf}\n\\caption{Binary64 payload as total factor width increases. The complete factor grows with $m$, while the streamed and tiled payloads remain constant at fixed resident dimensions. These are algebraic sizing data, not measurements of total device allocation.}\n\\label{fig:memory-law}\n\\end{figure}","\\section{The running example in three storage blocks}\\label{sec:worked-stream}\nPartition the factor of Section~\\ref{sec:worked-law} into consecutive pairs of columns,\n\\[\nA=[A_1\\ A_2\\ A_3],\\qquad A_i\\in\\mathbb R^{4\\times2}.\n\\]\nAlthough $A_2=A_3$, both column groups must be counted. Their Gram contributions are\n\\[\nA_1A_1^\\top=\\frac12\n\\begin{pmatrix}1&0&1&0","\\section{The running example in three storage blocks}\\label{sec:worked-stream}\nPartition the factor of Section~\\ref{sec:worked-law} into consecutive pairs of columns,\n\\[\nA=[A_1\\ A_2\\ A_3],\\qquad A_i\\in\\mathbb R^{4\\times2}.\n\\]\nAlthough $A_2=A_3$, both column groups must be counted. Their Gram contributions are\n\\[\nA_1A_1^\\top=\\frac12\n\\begin{pmatrix}1&0&1&0\\\\0&1&0&1\\\\1&0&1&0\\\\0&1&0&1\\end{pmatrix},\\qquad\nA_2A_2^\\top=A_3A_3^\\top=\\frac12\n\\begin{pmatrix}1&0&-1&0\\\\0&1&0&-1\\\\-1&0&1&0\\\\0&-1&0&1\\end{pmatrix}.\n\\]\nThe sum is $K$. Merging the last two blocks preserves it. Discarding $A_3$ instead changes the operator and its probability law.","0&-1&0&1\\end{pmatrix}.\n\\]\nThe sum is $K$. Merging the last two blocks preserves it. Discarding $A_3$ instead changes the operator and its probability law.","For the declared vector $v=(1,2,-1,0)^\\top$, the action can be evaluated without forming any block Gram matrix:\n\\[\n\\sum_{j=1}^3 A_j(A_j^\\top v)=Kv=(2,3,-2,-1)^\\top.\n\\]\nThe symbolic check confirms both storage partitions. Every column remains represented. Dropping an operator coupling instead changes the operator and requires the error analysis developed next.","Let the phase projectors form an orthogonal decomposition of variant space:\n\\[\nP_h^\\top=P_h=P_h^2,\\qquad P_hP_g=0\\ (h\\ne g),\\qquad \\sum_hP_h=I.\n\\]\nThus $P_h$ projects onto variant phase $h$. Again set\n\\[\nH=A^\\top A.\n\\]\nDefine the exact phase graph by\n\\[\ng\\to h\n\\quad\\Longleftrightarrow\\quad\nP_hHP_g\\ne0.\n\\]","\\begin{theorem}[Polyphase light cone]\\label{claim:final-14-1}\nIf the graph distance from phase \\(g\\) to phase \\(h\\) exceeds \\(k\\), then\n\\[\n\\boxed{\nP_hH^kP_g=0.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nInsert the projector resolution\n\\(\\sum_jP_j=I\\) between successive factors of \\(H\\).  Every nonzero term in\n\\(P_hH^kP_g\\) corresponds to a phase path of length \\(k\\).  If no such path\nexists, every term vanishes.\n\\end{proof}","\\begin{theorem}[Streaming-polyphase composition]\\label{claim:final-14-2}\nAt Krylov depth \\(k\\), an exact matrix-free iteration may omit every phase\noutside the exact light cone.  Within each active phase, arbitrary storage\nrefinement may be used.\n\\end{theorem}","This is the key composition law:\n\\[\n\\boxed{\n\\text{mathematical locality}\n\\longrightarrow\n\\text{active phases}\n\\longrightarrow\n\\text{active storage blocks}.\n}\n\\]","Let the phase projectors be $P_1,\\ldots,P_q$, and let $X$ be a vector or block\nvector.  Define its phase-norm profile\n\\[\n\\rho(X)_g=\\lVert P_gX\\rVert_F,\n\\]\nand define the nonnegative phase-transfer matrix\n\\[\n\\mathcal B_{hg}=\\lVert P_hHP_g\\rVert_2.\n\\]\nThe matrix $\\mathcal B$ records the strength of every one-step phase transfer\nwithout assuming that weak couplings vanish.","\\begin{theorem}[Weighted phase-transfer envelope]\n\\label{thm:weighted-transfer}\nFor every block vector $X$,\n\\[\n\\boxed{\n\\rho(HX)\\leq \\mathcal B\\rho(X)\n}\n\\]\ncomponentwise.  Consequently, for every integer $k\\geq0$,\n\\[\n\\boxed{\n\\rho(H^kX)\\leq \\mathcal B^k\\rho(X).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nUsing $\\sum_gP_g=I$ and submultiplicativity,\n\\[\n\\begin{aligned}\n\\lVert P_hHX\\rVert_F\n&=\\left\\lVert\\sum_gP_hHP_gX\\right\\rVert_F","\\begin{proof}\nUsing $\\sum_gP_g=I$ and submultiplicativity,\n\\[\n\\begin{aligned}\n\\lVert P_hHX\\rVert_F\n&=\\left\\lVert\\sum_gP_hHP_gX\\right\\rVert_F\\\\\n&\\leq\\sum_g\\lVert P_hHP_g\\rVert_2\\lVert P_gX\\rVert_F\\\\\n&=(\\mathcal B\\rho(X))_h.\n\\end{aligned}\n\\]\nIteration gives the power bound.\n\\end{proof}","Now choose a symmetric set of retained phase blocks and define\n\\[\nH_0=\\sum_{(h,g)\\in\\mathcal E_0}P_hHP_g.\n\\]\nLet\n\\[\n(\\mathcal B_0)_{hg}=\n\\begin{cases}\n\\mathcal B_{hg},&(h,g)\\in\\mathcal E_0,","Now choose a symmetric set of retained phase blocks and define\n\\[\nH_0=\\sum_{(h,g)\\in\\mathcal E_0}P_hHP_g.\n\\]\nLet\n\\[\n(\\mathcal B_0)_{hg}=\n\\begin{cases}\n\\mathcal B_{hg},&(h,g)\\in\\mathcal E_0,\\\\\n0,&\\text{otherwise},\n\\end{cases}\n\\qquad\n\\mathcal R=\\mathcal B-\\mathcal B_0.\n\\]\nSymmetric pair retention preserves self-adjointness of $H_0$, although the\nfollowing error theorem does not require positive semidefiniteness of $H_0$.","0,&\\text{otherwise},\n\\end{cases}\n\\qquad\n\\mathcal R=\\mathcal B-\\mathcal B_0.\n\\]\nSymmetric pair retention preserves self-adjointness of $H_0$, although the\nfollowing error theorem does not require positive semidefiniteness of $H_0$.","\\begin{theorem}[Thresholded-power certificate]\n\\label{thm:thresholded-power}\nFor every $k\\geq1$,\n\\[\n\\boxed{\n\\rho\\bigl((H^k-H_0^k)X\\bigr)\n\\leq\n(\\mathcal B^k-\\mathcal B_0^k)\\rho(X).\n}\n\\]\nIn particular,\n\\[\n\\boxed{\n\\lVert(H^k-H_0^k)X\\rVert_F\n\\leq\n\\left\\lVert(\\mathcal B^k-\\mathcal B_0^k)\\rho(X)\\right\\rVert_2.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nWrite $E=H-H_0$.  The telescoping identity gives\n\\[\nH^k-H_0^k\n=\n\\sum_{j=0}^{k-1}H^{k-1-j}EH_0^j.\n\\]\nThe block-norm matrix of $E$ is $\\mathcal R$.  Applying\nTheorem~\\ref{thm:weighted-transfer} successively to each term gives\n\\[\n\\rho\\bigl((H^k-H_0^k)X\\bigr)\n\\leq\n\\sum_{j=0}^{k-1}\n\\mathcal B^{k-1-j}\\mathcal R\\mathcal B_0^j\\rho(X).\n\\]\nThe matrix telescoping identity\n\\[\n\\mathcal B^k-\\mathcal B_0^k\n=\n\\sum_{j=0}^{k-1}\n\\mathcal B^{k-1-j}(\\mathcal B-\\mathcal B_0)\\mathcal B_0^j\n\\]\nyields the phasewise formula.  Orthogonality of the phase projectors gives\n\\[\n\\lVert Y\\rVert_F^2=\\sum_h\\lVert P_hY\\rVert_F^2,\n\\]\nwhich proves the global bound.\n\\end{proof}","ing\nTheorem~\\ref{thm:weighted-transfer} successively to each term gives\n\\[\n\\rho\\bigl((H^k-H_0^k)X\\bigr)\n\\leq\n\\sum_{j=0}^{k-1}\n\\mathcal B^{k-1-j}\\mathcal R\\mathcal B_0^j\\rho(X).\n\\]\nThe matrix telescoping identity\n\\[\n\\mathcal B^k-\\mathcal B_0^k\n=\n\\sum_{j=0}^{k-1}\n\\mathcal B^{k-1-j}(\\mathcal B-\\mathcal B_0)\\mathcal B_0^j\n\\]\nyields the phasewise formula.  Orthogonality of the phase projectors gives\n\\[\n\\lVert Y\\rVert_F^2=\\sum_h\\lVert P_hY\\rVert_F^2,\n\\]\nwhich proves the global bound.\n\\end{proof}","\\begin{corollary}[Certified approximate light cone]\n\\label{cor:approx-lightcone}\nAssume $X$ is supported on a set of start phases $S$.  If phase $h$ is not\nreachable from $S$ by a retained path of length at most $k$, then\n\\[\nP_hH_0^kX=0\n\\]\nand\n\\[\n\\boxed{\n\\lVert P_hH^kX\\rVert_F\n\\leq\n\\bigl[(\\mathcal B^k-\\mathcal B_0^k)\\rho(X)\\bigr]_h.\n}\n\\]\nThus an exact zero outside the retained light cone is replaced by a certified\nnonzero leakage budget.\n\\end{corollary}","\\begin{corollary}[Scalar threshold law]\\label{claim:final-15-4}\nFor any induced matrix norm, let\n\\[\nM=\\max\\{\\lVert\\mathcal B\\rVert,\\lVert\\mathcal B_0\\rVert\\}.\n\\]\nThen\n\\[\n\\boxed{\n\\lVert\\mathcal B^k-\\mathcal B_0^k\\rVert\n\\leq\nkM^{k-1}\\lVert\\mathcal R\\rVert.\n}\n\\]\nHence weak discarded coupling produces a first-order error bound proportional\nto the discarded transfer norm and to the Krylov depth.\n\\end{corollary}","\\begin{theorem}[Krylov-polynomial certificate]\n\\label{thm:polynomial-certificate}\nLet\n\\[\np(t)=\\sum_{j=0}^d a_jt^j.\n\\]\nThen\n\\[\n\\boxed{\n\\rho\\bigl((p(H)-p(H_0))X\\bigr)\n\\leq\n\\sum_{j=1}^d\n|a_j|(\\mathcal B^j-\\mathcal B_0^j)\\rho(X).\n}\n\\]\n\\end{theorem}","The theorem compares the same polynomial $p$ and the same starting block\n$X$ under the two operators. It therefore covers fixed polynomial filters\nand fixed linear combinations of Krylov powers, not only $H^kX$.\nIf an adaptive iteration selects different coefficients or restart vectors\nunder $H$ and $H_0$, their discrepancy requires a separate bound. The\nfixed-polynomial certificate alone does not control that additional change.","\\begin{corollary}[Certified phase pruning]\\label{claim:final-15-6}\nIf a thresholded phase graph is chosen so that\n\\[\n\\left\\lVert\n(\\mathcal B^k-\\mathcal B_0^k)\\rho(X)\n\\right\\rVert_2\n\\leq\\tau,\n\\]\nthen replacing $H^kX$ by $H_0^kX$ incurs Frobenius error at most $\\tau$.\nThe same statement holds for a Krylov polynomial with the certificate in\nTheorem~\\ref{thm:polynomial-certificate}.\n\\end{corollary}","The validation operator has twelve phases of width four.  A block-bidiagonal\nfactor generates a positive-definite local operator $H_0$, and weak symmetric\nlong-range blocks are then added to obtain $H$.  Both normalized operators\nremain positive definite.  A three-column start block is supported in phase\nzero.","For depths one through eight:\n\\begin{itemize}\n\\item every observed phase norm lay below $\\mathcal B^k\\rho(X)$.\n\\item every observed thresholding error lay below\n$(\\mathcal B^k-\\mathcal B_0^k)\\rho(X)$.\n\\item the largest total observed-error-to-certificate ratio was approximately\n$0.669$.\n\\end{itemize}","A separate hard-threshold sweep retained between $22$ and $132$ directed\noff-diagonal phase blocks.  Every measured depth-five error remained below its\ncertificate. The largest observed-error-to-certificate ratio was approximately\n$0.376$.  Finally, a degree-four signed polynomial in $H$ satisfied the\nphasewise polynomial certificate in all twelve phases, with maximum ratio\napproximately $0.068$.","The preceding section answers a validation question: given a retained phase\noperator $H_0$, how large can the Krylov error be?  This section turns that\ncertificate into a selection rule.  The goal is to remove as much phase\ncommunication as possible while remaining inside a prescribed error budget.","Let $\\mathcal E$ denote the set of undirected off-diagonal phase pairs.  Because\n$H$ is self-adjoint, $\\mathcal B$ is symmetric.  For a phase pair\n$e=\\{h,g\\}$ define the nonnegative matrix $\\Delta_e$ by\n\\[\n(\\Delta_e)_{hg}=\\mathcal B_{hg},\\qquad\n(\\Delta_e)_{gh}=\\mathcal B_{gh},\n\\]\nwith every other entry zero.  If $\\mathcal O\\subseteq\\mathcal E$ is the set\nof omitted pairs, define\n\\[\n\\mathcal B_{\\mathcal O}\n=\\mathcal B-\\sum_{e\\in\\mathcal O}\\Delta_e.\n\\]\nThe diagonal is always retained. The corresponding pruned operator is\n\\[\nH_{\\mathcal O}\n=H-\\sum_{\\{h,g\\}\\in\\mathcal O}\n\\bigl(P_hHP_g+P_gHP_h\\bigr).\n\\]\nThus $\\mathcal B_{\\mathcal O}$ is its block-norm transfer matrix.",".  For a phase pair\n$e=\\{h,g\\}$ define the nonnegative matrix $\\Delta_e$ by\n\\[\n(\\Delta_e)_{hg}=\\mathcal B_{hg},\\qquad\n(\\Delta_e)_{gh}=\\mathcal B_{gh},\n\\]\nwith every other entry zero.  If $\\mathcal O\\subseteq\\mathcal E$ is the set\nof omitted pairs, define\n\\[\n\\mathcal B_{\\mathcal O}\n=\\mathcal B-\\sum_{e\\in\\mathcal O}\\Delta_e.\n\\]\nThe diagonal is always retained. The corresponding pruned operator is\n\\[\nH_{\\mathcal O}\n=H-\\sum_{\\{h,g\\}\\in\\mathcal O}\n\\bigl(P_hHP_g+P_gHP_h\\bigr).\n\\]\nThus $\\mathcal B_{\\mathcal O}$ is its block-norm transfer matrix.","a_e)_{hg}=\\mathcal B_{hg},\\qquad\n(\\Delta_e)_{gh}=\\mathcal B_{gh},\n\\]\nwith every other entry zero.  If $\\mathcal O\\subseteq\\mathcal E$ is the set\nof omitted pairs, define\n\\[\n\\mathcal B_{\\mathcal O}\n=\\mathcal B-\\sum_{e\\in\\mathcal O}\\Delta_e.\n\\]\nThe diagonal is always retained. The corresponding pruned operator is\n\\[\nH_{\\mathcal O}\n=H-\\sum_{\\{h,g\\}\\in\\mathcal O}\n\\bigl(P_hHP_g+P_gHP_h\\bigr).\n\\]\nThus $\\mathcal B_{\\mathcal O}$ is its block-norm transfer matrix.","For depth $k\\ge1$ and starting phase profile $\\rho_0=\\rho(X)$, define the\n\\emph{edge risk vector}\n\\[\n\\boxed{\nq_e^{(k)}\n=\n\\sum_{j=0}^{k-1}\n\\mathcal B^{k-1-j}\\Delta_e\\mathcal B^j\\rho_0\n}\n\\]\nand its scalar risk\n\\[\n\\boxed{\n\\beta_e^{(k)}=\\lVert q_e^{(k)}\\rVert_2.\n}\n\\]\nUnlike the raw block norm $\\mathcal B_{hg}$, this quantity depends on the\nstart profile and on Krylov depth.","\\begin{theorem}[Edge-additive leakage majorant]\n\\label{thm:edge-additive-majorant}\nFor every omitted set $\\mathcal O\\subseteq\\mathcal E$,\n\\[\n\\boxed{\n0\\le\n(\\mathcal B^k-\\mathcal B_{\\mathcal O}^k)\\rho_0\n\\le\n\\sum_{e\\in\\mathcal O}q_e^{(k)}\n}\n\\]\ncomponentwise.  Consequently,\n\\[\n\\boxed{\n\\lVert(H^k-H_{\\mathcal O}^k)X\\rVert_F\n\\le\n\\left\\lVert\n(\\mathcal B^k-\\mathcal B_{\\mathcal O}^k)\\rho_0\n\\right\\rVert_2\n\\le\n\\sum_{e\\in\\mathcal O}\\beta_e^{(k)}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nThe first inequality in the second display is\nTheorem~\\ref{thm:thresholded-power}.  For the transfer matrices, the exact\nnoncommutative telescoping identity is\n\\[\n\\mathcal B^k-\\mathcal B_{\\mathcal O}^k\n=\n\\sum_{j=0}^{k-1}\n\\mathcal B^{k-1-j}\n(\\mathcal B-\\mathcal B_{\\mathcal O})\n\\mathcal B_{\\mathcal O}^j.\n\\]\nAll matrices and $\\rho_0$ are nonnegative, and\n$0\\le\\mathcal B_{\\mathcal O}\\le\\mathcal B$ entrywise.  Hence\n\\[\n\\mathcal B_{\\mathcal O}^j\\rho_0\n\\le\n\\mathcal B^j\\rho_0.\n\\]\nSubstituting\n\\[\n\\mathcal B-\\mathcal B_{\\mathcal O}\n=\n\\sum_{e\\in\\mathcal O}\\Delta_e\n\\]\nand distributing the sum gives the componentwise majorant.  The final scalar\nbound follows from the Euclidean triangle inequality.\n\\end{proof}","ity in the second display is\nTheorem~\\ref{thm:thresholded-power}.  For the transfer matrices, the exact\nnoncommutative telescoping identity is\n\\[\n\\mathcal B^k-\\mathcal B_{\\mathcal O}^k\n=\n\\sum_{j=0}^{k-1}\n\\mathcal B^{k-1-j}\n(\\mathcal B-\\mathcal B_{\\mathcal O})\n\\mathcal B_{\\mathcal O}^j.\n\\]\nAll matrices and $\\rho_0$ are nonnegative, and\n$0\\le\\mathcal B_{\\mathcal O}\\le\\mathcal B$ entrywise.  Hence\n\\[\n\\mathcal B_{\\mathcal O}^j\\rho_0\n\\le\n\\mathcal B^j\\rho_0.\n\\]\nSubstituting\n\\[\n\\mathcal B-\\mathcal B_{\\mathcal O}\n=\n\\sum_{e\\in\\mathcal O}\\Delta_e\n\\]\nand distributing the sum gives the componentwise majorant.  The final scalar\nbound follows from the Euclidean triangle inequality.\n\\end{proof}","transfer matrices, the exact\nnoncommutative telescoping identity is\n\\[\n\\mathcal B^k-\\mathcal B_{\\mathcal O}^k\n=\n\\sum_{j=0}^{k-1}\n\\mathcal B^{k-1-j}\n(\\mathcal B-\\mathcal B_{\\mathcal O})\n\\mathcal B_{\\mathcal O}^j.\n\\]\nAll matrices and $\\rho_0$ are nonnegative, and\n$0\\le\\mathcal B_{\\mathcal O}\\le\\mathcal B$ entrywise.  Hence\n\\[\n\\mathcal B_{\\mathcal O}^j\\rho_0\n\\le\n\\mathcal B^j\\rho_0.\n\\]\nSubstituting\n\\[\n\\mathcal B-\\mathcal B_{\\mathcal O}\n=\n\\sum_{e\\in\\mathcal O}\\Delta_e\n\\]\nand distributing the sum gives the componentwise majorant.  The final scalar\nbound follows from the Euclidean triangle inequality.\n\\end{proof}","\\begin{corollary}[Polynomial edge-risk certificate]\\label{claim:final-16-2}\nLet\n\\[\np(t)=\\sum_{d=0}^{D}a_dt^d.\n\\]\nDefine\n\\[\nq_e^{(p)}\n=\n\\sum_{d=1}^{D}|a_d|q_e^{(d)},\n\\qquad\n\\beta_e^{(p)}=\\lVert q_e^{(p)}\\rVert_2.\n\\]\nThen\n\\[\n\\boxed{\n\\lVert(p(H)-p(H_{\\mathcal O}))X\\rVert_F\n\\le\n\\sum_{e\\in\\mathcal O}\\beta_e^{(p)}.\n}\n\\]\nThus the same scheduling machinery applies to finite Krylov combinations and\npolynomial filters.\n\\end{corollary}","\\begin{corollary}[Query-invisible phase pair]\\label{claim:final-16-3}\nIf\n\\[\n\\Delta_e\\mathcal B^j\\rho_0=0\n\\qquad\n\\text{for every }0\\le j<k,\n\\]\nthen\n\\[\n\\beta_e^{(k)}=0.\n\\]\nTherefore a phase pair can be large in operator norm and still have zero\nrisk for a particular finite-depth query.  A magnitude-only threshold cannot\nrepresent this distinction.\n\\end{corollary}","Suppose each retained phase pair has the same communication or scheduling\ncost, and fix a budget $\\tau\\ge0$. Under the additive certificate, a set of\nomitted pairs is feasible if\n\\[\n\\sum_{e\\in\\mathcal O}\\beta_e^{(k)}\\le\\tau.\n\\]\nSort the risks as\n\\[\n\\beta_{(1)}\\le\\beta_{(2)}\\le\\cdots\\le\\beta_{(N)}.\n\\]","\\begin{theorem}[Cardinality-optimal certified pruning]\n\\label{thm:unit-cost-optimal}\nLet $r\\in\\{0,\\ldots,N\\}$ be the largest integer satisfying\n\\[\n\\sum_{i=1}^{r}\\beta_{(i)}\\le\\tau.\n\\]\nThen omitting the $r$ least-risk phase pairs is feasible, and no feasible\nomitted set can contain more than $r$ pairs under the scalar additive\ncertificate.\n\\end{theorem}","\\begin{proof}\nIf $r=N$, every candidate pair can be omitted and optimality is immediate.\nOtherwise, among all subsets of cardinality $r+1$, the smallest possible sum\nof risks is\n\\[\n\\sum_{i=1}^{r+1}\\beta_{(i)}.\n\\]\nBy maximality of $r$, that sum exceeds $\\tau$.  Hence no set of $r+1$ omitted\npairs is feasible.  The prefix of length $r$ is feasible by construction.\n\\end{proof}","After the cardinality-optimal additive schedule is found, recomputing\n\\[\n\\left\\lVert\n(\\mathcal B^k-\\mathcal B_{\\mathcal O}^k)\\rho_0\n\\right\\rVert_2\n\\]\nmay reveal unused budget.  Additional pairs can then be removed one at a time,\nprovided the exact transfer envelope remains below $\\tau$.","\\begin{proposition}[Certified refinement]\\label{claim:final-16-5}\nAny sequence of additional omissions accepted only when\n\\[\n\\left\\lVert\n(\\mathcal B^k-\\mathcal B_{\\mathcal O}^k)\\rho_0\n\\right\\rVert_2\\le\\tau\n\\]\nremains a valid $\\tau$-certificate for the original operator error.\n\\end{proposition}","Let $c_e\\ge0$ denote the communication or storage cost saved by omitting pair\n$e$.  Under the scalar additive certificate, the exact discrete selection\nproblem is\n\\[\n\\boxed{\n\\max_{x_e\\in\\{0,1\\}}\n\\sum_e c_ex_e\n\\quad\\text{subject to}\\quad\n\\sum_e\\beta_e^{(k)}x_e\\le\\tau.\n}\n\\]\nThis is the $0$--$1$ knapsack problem.  Hence unequal-cost scheduling is\ncombinatorial in general.  The implementation supplies a safe\ncost-to-risk greedy heuristic. Its output remains certified because it never\nexceeds the additive risk budget.  A fractional-knapsack relaxation supplies\nan upper bound on the maximum possible saved cost.","The twelve-phase operator from Section~\\ref{sec:approx-lightcone} was reused at\nKrylov depth five.  There are $66$ undirected off-diagonal phase pairs.\nFor error budgets $10^{-3}$, $2\\times10^{-3}$, $5\\times10^{-3}$, and\n$10^{-2}$, the additive-optimal scheduler retained respectively\n$61$, $56$, $45$, and $36$ pairs.  Exact-envelope refinement reduced those\ncounts further to\n$58$, $50$, $36$, and $30$ pairs while remaining inside budget.","he twelve-phase operator from Section~\\ref{sec:approx-lightcone} was reused at\nKrylov depth five.  There are $66$ undirected off-diagonal phase pairs.\nFor error budgets $10^{-3}$, $2\\times10^{-3}$, $5\\times10^{-3}$, and\n$10^{-2}$, the additive-optimal scheduler retained respectively\n$61$, $56$, $45$, and $36$ pairs.  Exact-envelope refinement reduced those\ncounts further to\n$58$, $50$, $36$, and $30$ pairs while remaining inside budget.","In the same test, the best single global threshold on the raw block norm\nretained $65$ pairs at every one of those four budgets.  The difference is\nstructural: the global threshold is blind to start phase and Krylov depth.","A separate six-phase problem contains $15$ candidate phase pairs, so all\n$2^{15}=32768$ subsets can be enumerated.  At four distinct budgets, the\nsorted-risk scheduler matched the brute-force cardinality optimum under the\nadditive certificate in every case.  The exact-envelope greedy refinement also\nmatched the brute-force exact-envelope optimum in cardinality for all four\nbudgets in this finite audit.  The latter agreement is empirical and is not\npromoted to a theorem.","For an unequal-cost eight-phase test, costs were proportional to the bytes in\nsymmetric dense phase blocks.  The cost-to-risk heuristic saved $5408$ bytes\nout of a $9472$-byte communication proxy while using additive risk $0.011632$\nunder budget $0.012$.  The fractional relaxation upper bound was $5647.8$\nbytes, placing the greedy schedule within about $4.25\\%$ of that relaxation\nbound on this instance.","The mathematical streaming identities do not require transfers and GEMMs to be\nserial.  Consider $B$ storage blocks.  Let $t_b$ be the transfer time for block\n$b$ and $c_b$ its device compute time.","\\begin{proposition}[Ideal two-buffer pipeline]\\label{claim:final-17-1}\nIn a model with perfect asynchronous overlap, transfer of block $b+1$ may run\nwhile block $b$ is being processed.  The serial one-buffer time is\n\\[\nT_{\\rm serial}=\\sum_{b=1}^{B}(t_b+c_b),\n\\]\nwhereas the two-buffer pipeline time is\n\\[\n\\boxed{\nT_{\\rm pipe}\n=\nt_1\n+\n\\sum_{b=1}^{B-1}\\max\\{c_b,t_{b+1}\\}\n+\nc_B.\n}\n\\]\nFor uniform times $t_b=t$ and $c_b=c$,\n\\[\nT_{\\rm pipe}=t+c+(B-1)\\max\\{t,c\\}.\n\\]\n\\end{proposition}","begin{proposition}[Ideal two-buffer pipeline]\\label{claim:final-17-1}\nIn a model with perfect asynchronous overlap, transfer of block $b+1$ may run\nwhile block $b$ is being processed.  The serial one-buffer time is\n\\[\nT_{\\rm serial}=\\sum_{b=1}^{B}(t_b+c_b),\n\\]\nwhereas the two-buffer pipeline time is\n\\[\n\\boxed{\nT_{\\rm pipe}\n=\nt_1\n+\n\\sum_{b=1}^{B-1}\\max\\{c_b,t_{b+1}\\}\n+\nc_B.\n}\n\\]\nFor uniform times $t_b=t$ and $c_b=c$,\n\\[\nT_{\\rm pipe}=t+c+(B-1)\\max\\{t,c\\}.\n\\]\n\\end{proposition}","ition}[Ideal two-buffer pipeline]\\label{claim:final-17-1}\nIn a model with perfect asynchronous overlap, transfer of block $b+1$ may run\nwhile block $b$ is being processed.  The serial one-buffer time is\n\\[\nT_{\\rm serial}=\\sum_{b=1}^{B}(t_b+c_b),\n\\]\nwhereas the two-buffer pipeline time is\n\\[\n\\boxed{\nT_{\\rm pipe}\n=\nt_1\n+\n\\sum_{b=1}^{B-1}\\max\\{c_b,t_{b+1}\\}\n+\nc_B.\n}\n\\]\nFor uniform times $t_b=t$ and $c_b=c$,\n\\[\nT_{\\rm pipe}=t+c+(B-1)\\max\\{t,c\\}.\n\\]\n\\end{proposition}","al two-buffer pipeline]\\label{claim:final-17-1}\nIn a model with perfect asynchronous overlap, transfer of block $b+1$ may run\nwhile block $b$ is being processed.  The serial one-buffer time is\n\\[\nT_{\\rm serial}=\\sum_{b=1}^{B}(t_b+c_b),\n\\]\nwhereas the two-buffer pipeline time is\n\\[\n\\boxed{\nT_{\\rm pipe}\n=\nt_1\n+\n\\sum_{b=1}^{B-1}\\max\\{c_b,t_{b+1}\\}\n+\nc_B.\n}\n\\]\nFor uniform times $t_b=t$ and $c_b=c$,\n\\[\nT_{\\rm pipe}=t+c+(B-1)\\max\\{t,c\\}.\n\\]\n\\end{proposition}","If one resident block uses $M_{\\rm block}$ bytes and all other fixed working\nstate uses $M_{\\rm fixed}$ bytes, the corresponding memory models are\n\\[\nM_{\\rm single}=M_{\\rm fixed}+M_{\\rm block},\n\\qquad\n\\boxed{\nM_{\\rm double}=M_{\\rm fixed}+2M_{\\rm block}.\n}\n\\]\nThe proposition is a scheduling model, not a hardware timing claim.  PCIe or\nNVLink contention, kernel launch overhead, synchronization, page pinning, and\nlibrary workspaces can reduce realized overlap.","Let \\(R\\) be a newly generated Krylov residual and\n\\[\nR_h=P_hR.\n\\]","\\begin{theorem}[Minimal exact phase width]\\label{claim:final-18-1}\nLet\n\\[\nr_h=\\operatorname{rank}(R_h).\n\\]\nEvery phase-supported basis spanning \\(\\operatorname{range}(R_h)\\) has at\nleast \\(r_h\\) vectors, and a rank-revealing QR or compact SVD attains exactly\n\\(r_h\\).\n\\end{theorem}","If an inter-phase coupling has exact factorization\n\\[\nP_hHP_g\n=\nU_{hg}V_{hg}^\\top\n\\]\nof rank \\(r_{hg}\\), then\n\\[\n\\boxed{\nP_hHP_gx\n=\nU_{hg}(V_{hg}^\\top x).\n}\n\\]","Suppose\n\\[\nA=QR,\n\\qquad\nQ^\\top Q=I,\n\\]\nand append a new block \\(U\\).  Set\n\\[\nC=Q^\\top U,\n\\qquad\nF=U-QC,\n\\qquad\nF=Q_\\perp R_\\perp,\n\\qquad Q_\\perp^\\top Q_\\perp=I,\n\\qquad Q^\\top Q_\\perp=0.\n\\]\nHere the columns of \\(Q_\\perp\\) form an orthonormal basis of the nonzero\nresidual range. Zero residual directions are omitted.","\\begin{theorem}[Exact QR-core update]\\label{claim:final-20-1}\n\\[\n[A,U]\n=\n[Q,Q_\\perp]\n\\begin{pmatrix}\nR&C\\\\\n0&R_\\perp\n\\end{pmatrix}.\n\\]\nTherefore the nonzero eigenvalues of\n\\([A,U][A,U]^\\top\\) are exactly the nonzero eigenvalues of the small\ncore Gram, with multiplicity. If $\\operatorname{range}(Q)=\\operatorname{range}(A)$,\nthe compact residual update also spans exactly $\\operatorname{range}([A,U])$.\nIn that case the core has full row rank and its Gram matrix has no additional\nzero eigenvalues.\n\\end{theorem}","0&R_\\perp\n\\end{pmatrix}.\n\\]\nTherefore the nonzero eigenvalues of\n\\([A,U][A,U]^\\top\\) are exactly the nonzero eigenvalues of the small\ncore Gram, with multiplicity. If $\\operatorname{range}(Q)=\\operatorname{range}(A)$,\nthe compact residual update also spans exactly $\\operatorname{range}([A,U])$.\nIn that case the core has full row rank and its Gram matrix has no additional\nzero eigenvalues.\n\\end{theorem}","\\begin{proof}\nThe first block product is \\(QR=A\\). The second is\n\\(QC+Q_\\perp R_\\perp=QC+F=U\\).  The stated orthogonality\nconditions make \\([Q,Q_\\perp]\\) an isometry on the compact core\nspace.  Left multiplication by this isometry preserves all nonzero singular\nvalues of the core, and squaring them yields the nonzero Gram eigenvalues.\nIf $Q$ spans exactly the old range and $Q_\\perp$ spans exactly the residual\nrange, their combined range is precisely that of the appended factor.\nThe core then has full row rank. Without this exact-range hypothesis, unused\nbasis directions can introduce zero eigenvalues into the core Gram.\n\\end{proof}","Suppose an orthonormal basis \\(Q_r\\) captures the factor range:\n\\[\n\\operatorname{range}(A)\\subseteq\\operatorname{range}(Q_r).\n\\]\nThen\n\\[\nA=Q_rB,\n\\qquad\nB=Q_r^\\top A.\n\\]","\\begin{theorem}[Exact range core]\\label{claim:final-21-1}\nThe nonzero eigenvalues of\n\\[\nK=AA^\\top\n\\]\nare exactly the nonzero eigenvalues of\n\\[\nBB^\\top,\n\\]\nwith multiplicity. If $\\operatorname{range}(Q_r)=\\operatorname{range}(A)$,\nthen $BB^\\top$ is positive definite on the core space and its entire spectrum\nis the nonzero spectrum of $K$.\n\\end{theorem}","\\begin{proof}\nSince $A=Q_rB$ and $Q_r^\\top Q_r=I$, one has\n$AA^\\top=Q_r(BB^\\top)Q_r^\\top$. The map $x\\mapsto Q_rx$ is an isometry\nfrom the core space onto $\\operatorname{range}(Q_r)$. On that range the two\noperators are conjugate, and $AA^\\top$ vanishes on its orthogonal complement.\nHence their nonzero eigenvalues agree with multiplicity. If the captured\nrange equals $\\operatorname{range}(A)$, then $B$ has full row rank, so\n$BB^\\top$ is positive definite. Without that equality, the core may retain\nadditional zero eigenvalues.\n\\end{proof}","gin{proof}\nSince $A=Q_rB$ and $Q_r^\\top Q_r=I$, one has\n$AA^\\top=Q_r(BB^\\top)Q_r^\\top$. The map $x\\mapsto Q_rx$ is an isometry\nfrom the core space onto $\\operatorname{range}(Q_r)$. On that range the two\noperators are conjugate, and $AA^\\top$ vanishes on its orthogonal complement.\nHence their nonzero eigenvalues agree with multiplicity. If the captured\nrange equals $\\operatorname{range}(A)$, then $B$ has full row rank, so\n$BB^\\top$ is positive definite. Without that equality, the core may retain\nadditional zero eigenvalues.\n\\end{proof}","and $Q_r^\\top Q_r=I$, one has\n$AA^\\top=Q_r(BB^\\top)Q_r^\\top$. The map $x\\mapsto Q_rx$ is an isometry\nfrom the core space onto $\\operatorname{range}(Q_r)$. On that range the two\noperators are conjugate, and $AA^\\top$ vanishes on its orthogonal complement.\nHence their nonzero eigenvalues agree with multiplicity. If the captured\nrange equals $\\operatorname{range}(A)$, then $B$ has full row rank, so\n$BB^\\top$ is positive definite. Without that equality, the core may retain\nadditional zero eigenvalues.\n\\end{proof}","Thus the computational pipeline can terminate its data-streaming stage after\nthe range is captured:\n\\[\n\\boxed{\n\\text{streamed data}\n\\to\n\\text{small spectral core}\n\\to\n\\text{tail evaluation}.\n}\n\\]","\\clearpage\n\\section{An exact core calculation}\nConsider the orthonormal range basis and core factor\n\\[\nU=\\begin{pmatrix}1&0\\\\0&1/\\sqrt2\\\\0&1/\\sqrt2\\end{pmatrix},\n\\qquad\nB=\\begin{pmatrix}1&0&1\\\\0&2&0\\end{pmatrix},\n\\qquad A=UB.\n\\]\nDirect multiplication gives\n\\[\nBB^\\top=\\begin{pmatrix}2&0","0&2&0\\end{pmatrix},\n\\qquad A=UB.\n\\]\nDirect multiplication gives\n\\[\nBB^\\top=\\begin{pmatrix}2&0\\\\0&4\\end{pmatrix},\n\\qquad\nAA^\\top=\\begin{pmatrix}2&0&0\\\\0&2&2\\\\0&2&2\\end{pmatrix}.\n\\]\nThe large operator has eigenvalues $4,2,0$. Its two-dimensional core has\nexactly the nonzero eigenvalues $4,2$. The Gaussian quadratic form therefore\nhas the law $4Z_1^2+2Z_2^2$, whether it is computed from the full factor, the\nfull Gram matrix, or the core.","0&2&2\\end{pmatrix}.\n\\]\nThe large operator has eigenvalues $4,2,0$. Its two-dimensional core has\nexactly the nonzero eigenvalues $4,2$. The Gaussian quadratic form therefore\nhas the law $4Z_1^2+2Z_2^2$, whether it is computed from the full factor, the\nfull Gram matrix, or the core.","\\subsection*{Why the nonzero-spectrum qualification matters}\nTake instead\n\\[\nA=\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix},\\qquad Q_r=I_2,\\qquad B=Q_r^\\top A=A.\n\\]\nThe captured subspace contains the range of $A$, but is larger than that range.\nBoth Gram matrices have spectrum $\\{1,0\\}$. Their nonzero spectrum is\n$\\{1\\}$. Thus the nonzero spectrum of $AA^\\top$ cannot be identified with the\n\\emph{entire} spectrum of $BB^\\top$. Choosing the compact basis\n$Q_r=(1,0)^\\top$ instead gives $B=(1,0)$ and the one-dimensional core Gram\n$(1)$. The extra zero records an unused basis direction, not an additional\nrandom contribution. This is why range containment and exact-range equality\nhave different conclusions in Theorem~\\ref{claim:final-21-1}.","0&0\\end{pmatrix},\\qquad Q_r=I_2,\\qquad B=Q_r^\\top A=A.\n\\]\nThe captured subspace contains the range of $A$, but is larger than that range.\nBoth Gram matrices have spectrum $\\{1,0\\}$. Their nonzero spectrum is\n$\\{1\\}$. Thus the nonzero spectrum of $AA^\\top$ cannot be identified with the\n\\emph{entire} spectrum of $BB^\\top$. Choosing the compact basis\n$Q_r=(1,0)^\\top$ instead gives $B=(1,0)$ and the one-dimensional core Gram\n$(1)$. The extra zero records an unused basis direction, not an additional\nrandom contribution. This is why range containment and exact-range equality\nhave different conclusions in Theorem~\\ref{claim:final-21-1}.","r=I_2,\\qquad B=Q_r^\\top A=A.\n\\]\nThe captured subspace contains the range of $A$, but is larger than that range.\nBoth Gram matrices have spectrum $\\{1,0\\}$. Their nonzero spectrum is\n$\\{1\\}$. Thus the nonzero spectrum of $AA^\\top$ cannot be identified with the\n\\emph{entire} spectrum of $BB^\\top$. Choosing the compact basis\n$Q_r=(1,0)^\\top$ instead gives $B=(1,0)$ and the one-dimensional core Gram\n$(1)$. The extra zero records an unused basis direction, not an additional\nrandom contribution. This is why range containment and exact-range equality\nhave different conclusions in Theorem~\\ref{claim:final-21-1}.","The previous sections certify the effect of phase pruning on Krylov\npropagation.  A SKAT-like test ultimately depends on the spectrum itself, so\none more bridge is needed:\n\\[\n\\text{phase pruning}\n\\longrightarrow\n\\text{operator error}\n\\longrightarrow\n\\text{spectral error}\n\\longrightarrow\n\\text{tail-probability error}.\n\\]","Let\n\\[\nE=H-H_0\n\\]\nbe the self-adjoint error introduced by phase pruning.  Using the same\northogonal phase projectors \\(P_h\\), define\n\\[\n\\mathcal C_{hg}\n=\n\\|P_hEP_g\\|_2.\n\\]","\\begin{theorem}[Block-norm operator certificate]\\label{claim:final-22-1}\n\\[\n\\boxed{\n\\|E\\|_2\n\\le\n\\|\\mathcal C\\|_2.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nFor any vector \\(x\\), let\n\\[\nr_g=\\|P_gx\\|_2.\n\\]\nThen\n\\[\n\\|P_hEx\\|_2\n\\le\n\\sum_g\n\\|P_hEP_g\\|_2\\,\\|P_gx\\|_2\n=\n(\\mathcal C r)_h.\n\\]\nOrthogonality of the phase decomposition gives\n\\[\n\\|Ex\\|_2^2\n=\n\\sum_h\\|P_hEx\\|_2^2\n\\le\n\\|\\mathcal C r\\|_2^2\n\\le\n\\|\\mathcal C\\|_2^2\\|r\\|_2^2\n=\n\\|\\mathcal C\\|_2^2\\|x\\|_2^2.\n\\]\nTaking the supremum over unit vectors proves the claim.\n\\end{proof}","The important computational point is that \\(\\mathcal C\\) is only a\nphase-by-phase matrix.  The same block norms already used to construct the\npolyphase scheduler therefore provide a global operator certificate.","Assume\n\\[\nH\\succeq0\n\\]\nand let the eigenvalues of \\(H\\) and \\(H_0\\), in decreasing order, be\n\\[\n\\lambda_i\n\\qquad\\text{and}\\qquad\n\\mu_i.\n\\]\nIf\n\\[\n\\|H-H_0\\|_2\\le\\varepsilon,\n\\]\nWeyl's inequality gives\n\\[\n|\\lambda_i-\\mu_i|\\le\\varepsilon.\n\\]","\\begin{corollary}[Positive spectral interval]\\label{claim:final-22-2}\nFor every \\(i\\),\n\\[\n\\boxed{\n\\ell_i\n=\n\\max\\{0,\\mu_i-\\varepsilon\\}\n\\le\n\\lambda_i\n\\le\nu_i\n=\n\\max\\{0,\\mu_i+\\varepsilon\\}.\n}\n\\]\n\\end{corollary}","This remains valid even if symmetric phase pruning makes \\(H_0\\) slightly\nindefinite.","\\subsection{Certified generalized-\\texorpdfstring{\\(\\chi^2\\)}{chi-square} tail bracket}","For independent standard normal variables \\(Z_i\\), define\n\\[\nQ_-=\\sum_i\\ell_iZ_i^2,\n\\qquad\nQ=\\sum_i\\lambda_iZ_i^2,\n\\qquad\nQ_+=\\sum_i u_iZ_i^2.\n\\]","\\begin{theorem}[Stochastic tail enclosure]\\label{claim:final-22-3}\nFor every threshold \\(q\\),\n\\[\n\\boxed{\n\\Pr(Q_-\\ge q)\n\\le\n\\Pr(Q\\ge q)\n\\le\n\\Pr(Q_+\\ge q).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nCouple all three quadratic forms using the same normal variables \\(Z_i\\).\nSince\n\\[\n0\\le\\ell_i\\le\\lambda_i\\le u_i\n\\]\nfor every \\(i\\),\n\\[\nQ_-\\le Q\\le Q_+\n\\]\nalmost surely.  Taking upper-tail events yields the result.\n\\end{proof}","An execution schedule should therefore enforce\n\\[\n\\boxed{\n\\sum_{e\\in\\mathcal O}\\beta_e^{(k)}\n\\le\n\\tau_K,\n\\qquad\n\\|\\mathcal C_{\\mathcal O}\\|_2\n\\le\n\\tau_\\lambda.\n}\n\\]","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\linewidth]{figures_production/figure_14_dual_budget_tail_bridge.pdf}\n\\caption{Tail brackets under a fixed query budget and varying operator budgets. The dotted reference is at $\\log_{10}p=-8$. A sufficiently loose operator budget admits a strong coupling and widens the bracket even though the query-local constraint still passes.}\n\\label{fig:dual-budget-tail}\n\\end{figure}","In the deterministic ten-phase experiment, fixing the query budget at\n\\(0.05\\) and the operator budget at \\(0.005\\) removed \\(23\\) of \\(45\\)\nundirected phase pairs.  At a reference probability near \\(10^{-8}\\), the analytic\ntail bracket was\n\\[\n10^{-8.11143}\n\\le\np\n\\le\n10^{-7.89036},\n\\]\na width of about \\(0.221\\) decades. These endpoints summarize the\nreal-arithmetic error analysis evaluated in floating point, not a directed\nrounding enclosure. With an operator budget of \\(0.0075\\),\n\\(34\\) pairs were removed and the interval widened to about \\(0.320\\)\ndecades.  By contrast, allowing a strong coupling to be removed under a very\nloose operator budget produced a multi-decade interval.","\\) and the operator budget at \\(0.005\\) removed \\(23\\) of \\(45\\)\nundirected phase pairs.  At a reference probability near \\(10^{-8}\\), the analytic\ntail bracket was\n\\[\n10^{-8.11143}\n\\le\np\n\\le\n10^{-7.89036},\n\\]\na width of about \\(0.221\\) decades. These endpoints summarize the\nreal-arithmetic error analysis evaluated in floating point, not a directed\nrounding enclosure. With an operator budget of \\(0.0075\\),\n\\(34\\) pairs were removed and the interval widened to about \\(0.320\\)\ndecades.  By contrast, allowing a strong coupling to be removed under a very\nloose operator budget produced a multi-decade interval.","perator budget at \\(0.005\\) removed \\(23\\) of \\(45\\)\nundirected phase pairs.  At a reference probability near \\(10^{-8}\\), the analytic\ntail bracket was\n\\[\n10^{-8.11143}\n\\le\np\n\\le\n10^{-7.89036},\n\\]\na width of about \\(0.221\\) decades. These endpoints summarize the\nreal-arithmetic error analysis evaluated in floating point, not a directed\nrounding enclosure. With an operator budget of \\(0.0075\\),\n\\(34\\) pairs were removed and the interval widened to about \\(0.320\\)\ndecades.  By contrast, allowing a strong coupling to be removed under a very\nloose operator budget produced a multi-decade interval.","ted phase pairs.  At a reference probability near \\(10^{-8}\\), the analytic\ntail bracket was\n\\[\n10^{-8.11143}\n\\le\np\n\\le\n10^{-7.89036},\n\\]\na width of about \\(0.221\\) decades. These endpoints summarize the\nreal-arithmetic error analysis evaluated in floating point, not a directed\nrounding enclosure. With an operator budget of \\(0.0075\\),\n\\(34\\) pairs were removed and the interval widened to about \\(0.320\\)\ndecades.  By contrast, allowing a strong coupling to be removed under a very\nloose operator budget produced a multi-decade interval.","There is a second route that does not use the pruned operator as the final\nspectral approximation.  Let \\(U\\) be any orthonormal sample-space basis,\npossibly generated cheaply from a pruned or approximate Krylov process, and\ndefine\n\\[\nB=U^\\top A,\n\\qquad\nF=(I-UU^\\top)A,\n\\qquad\n\\delta=\\|F\\|_2.\n\\]","\\begin{theorem}[Range-core singular-value certificate]\\label{claim:final-22-4}\nFor $A\\in\\mathbb R^{n\\times m}$, list the singular values of $A$ and\n$B$ in decreasing order and pad both lists with zeros to length $n$.\nWrite these lists as $\\sigma_i(A)$ and $\\widetilde\\sigma_i(B)$,\nrespectively. They are the square roots of the $n$ eigenvalues of\n$AA^\\top$ and $(UB)(UB)^\\top$. Then, for $1\\le i\\le n$,\n\\[\n\\boxed{\n\\left|\n\\sigma_i(A)\n-\n\\widetilde\\sigma_i(B)\n\\right|\n\\le\n\\delta.\n}\n\\]\nConsequently,\n\\[\n\\max\\{0,\\widetilde\\sigma_i(B)-\\delta\\}^2\n\\le\n\\lambda_i(AA^\\top)\n\\le\n\\left(\n\\widetilde\\sigma_i(B)+\\delta\n\\right)^2.\n\\]\n\\end{theorem}","n{theorem}[Range-core singular-value certificate]\\label{claim:final-22-4}\nFor $A\\in\\mathbb R^{n\\times m}$, list the singular values of $A$ and\n$B$ in decreasing order and pad both lists with zeros to length $n$.\nWrite these lists as $\\sigma_i(A)$ and $\\widetilde\\sigma_i(B)$,\nrespectively. They are the square roots of the $n$ eigenvalues of\n$AA^\\top$ and $(UB)(UB)^\\top$. Then, for $1\\le i\\le n$,\n\\[\n\\boxed{\n\\left|\n\\sigma_i(A)\n-\n\\widetilde\\sigma_i(B)\n\\right|\n\\le\n\\delta.\n}\n\\]\nConsequently,\n\\[\n\\max\\{0,\\widetilde\\sigma_i(B)-\\delta\\}^2\n\\le\n\\lambda_i(AA^\\top)\n\\le\n\\left(\n\\widetilde\\sigma_i(B)+\\delta\n\\right)^2.\n\\]\n\\end{theorem}","{n\\times m}$, list the singular values of $A$ and\n$B$ in decreasing order and pad both lists with zeros to length $n$.\nWrite these lists as $\\sigma_i(A)$ and $\\widetilde\\sigma_i(B)$,\nrespectively. They are the square roots of the $n$ eigenvalues of\n$AA^\\top$ and $(UB)(UB)^\\top$. Then, for $1\\le i\\le n$,\n\\[\n\\boxed{\n\\left|\n\\sigma_i(A)\n-\n\\widetilde\\sigma_i(B)\n\\right|\n\\le\n\\delta.\n}\n\\]\nConsequently,\n\\[\n\\max\\{0,\\widetilde\\sigma_i(B)-\\delta\\}^2\n\\le\n\\lambda_i(AA^\\top)\n\\le\n\\left(\n\\widetilde\\sigma_i(B)+\\delta\n\\right)^2.\n\\]\n\\end{theorem}","\\begin{proof}\nBecause \\(U\\) has orthonormal columns,\n\\[\nA=UB+F\n\\]\nand\n\\[\n\\|A-UB\\|_2=\\|F\\|_2=\\delta.\n\\]\nThe nonzero singular values of $UB$ and $B$ agree, with multiplicity,\nand their zero-padded lists agree in the stated ambient convention.\nThe standard singular-value perturbation inequality gives the stated bound.\nSquaring the nonnegative endpoints gives the eigenvalue interval.\n\\end{proof}","This provides a useful architecture:\n\\[\n\\boxed{\n\\begin{gathered}\n\\text{pruned operator for basis discovery}\\\\\n\\longrightarrow\\ \\text{exact streamed factor check}\\\\\n\\longrightarrow\\ \\text{certified spectral core}\n\\end{gathered}\n}\n\\]","In a full-rank \\(50\\times50\\) stress test, a pruned Krylov basis of dimensions\n\\(12\\) through \\(48\\) left range residuals between roughly \\(0.87\\) and\n\\(0.99\\) times the leading singular value.  The corresponding \\(10^{-8}\\)\ntail intervals were therefore vacuous or thousands of decades wide.  At\ndimension \\(50\\), the range residual fell to approximately\n\\(1.2\\times10^{-15}\\), and the tail interval collapsed to the exact value.","Let \\(U\\in\\mathbb R^{n\\times r}\\) have orthonormal columns and define\n\\[\nB=U^\\top A,\n\\qquad\nF=(I-UU^\\top)A.\n\\]\nThen\n\\[\nA=UB+F,\n\\qquad\nU^\\top F=0.\n\\]","\\begin{theorem}[Exact core--residual decomposition]\\label{claim:final-23-1}\n\\[\n\\boxed{\nA^\\top A\n=\nB^\\top B+F^\\top F.\n}\n\\]\nIn particular the unresolved spectral term\n\\[\nR=F^\\top F\n\\]\nis positive semidefinite.\n\\end{theorem}","\\begin{proof}\nExpand \\((UB+F)^\\top(UB+F)\\).  The two cross terms vanish because\n\\(U^\\top F=0\\), while \\(U^\\top U=I\\).\n\\end{proof}","This changes the certification problem.  Instead of an arbitrary Hermitian\nerror, the missing term is a positive residual with measurable spectral\nmoments.  Define\n\\[\nt=\\operatorname{tr}R,\n\\qquad\nv=\\operatorname{tr}(R^2),\n\\qquad\n\\rho=\\|R\\|_2.\n\\]\nThen\n\\[\n\\boxed{\n t=\\|F\\|_F^2,\n \\qquad\n v=\\|FF^\\top\\|_F^2,\n \\qquad\n \\rho=\\|F\\|_2^2.\n}\n\\]\nThe first quantity is the exact missing mean of the Gaussian quadratic form.\nThe second records how concentrated the residual spectrum is. The third is\nits largest coefficient.","Set\n\\[\nC=B^\\top B,\n\\qquad\nH=C+R,\n\\qquad\nc=\\lambda_{\\max}(C).\n\\]\nFor \\(0<2\\theta(c+\\rho)<1\\), put\n\\[\n\\alpha=(1-2\\theta c)^{-1}.\n\\]","\\begin{theorem}[Trace--Frobenius--cap log-MGF certificate]\\label{claim:final-23-2}\nFor the Gaussian quadratic form with coefficient matrix \\(H\\),\n\\[\n\\kappa_H(\\theta)\n=-\\frac12\\log\\det(I-2\\theta H)\n\\]\nsatisfies\n\\[\n\\boxed{\n\\kappa_H(\\theta)\n\\le\n\\kappa_C(\\theta)\n+\n\\theta\\alpha t\n+\n\\frac{\\theta^2\\alpha^2v}\n     {1-2\\theta\\alpha\\rho}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nFactor the determinant through the known core:\n\\[\nI-2\\theta(C+R)\n=\n(I-2\\theta C)^{1/2}\n(I-2\\theta T)\n(I-2\\theta C)^{1/2},\n\\]\nwhere\n\\[\nT=(I-2\\theta C)^{-1/2}\nR\n(I-2\\theta C)^{-1/2}.\n\\]\nSince\n\\[\n\\|(I-2\\theta C)^{-1/2}\\|_2^2=\\alpha,\n\\]\nwe have\n\\[\n\\operatorname{tr}T\\le\\alpha t,\n\\qquad\n\\operatorname{tr}(T^2)\\le\\alpha^2v,\n\\qquad\n\\|T\\|_2\\le\\alpha\\rho.\n\\]\nFor every eigenvalue \\(x\\) of \\(T\\),\n\\[\n-\\frac12\\log(1-2\\theta x)\n\\le\n\\theta x\n+\n\\frac{\\theta^2x^2}\n     {1-2\\theta\\alpha\\rho}.\n\\]\nThe scalar inequality follows from the power series of \\(-\\log(1-u)\\) and a\ngeometric bound on all terms of degree at least two.  Summing over the\neigenvalues of \\(T\\) proves the result.\n\\end{proof}","gin{proof}\nFactor the determinant through the known core:\n\\[\nI-2\\theta(C+R)\n=\n(I-2\\theta C)^{1/2}\n(I-2\\theta T)\n(I-2\\theta C)^{1/2},\n\\]\nwhere\n\\[\nT=(I-2\\theta C)^{-1/2}\nR\n(I-2\\theta C)^{-1/2}.\n\\]\nSince\n\\[\n\\|(I-2\\theta C)^{-1/2}\\|_2^2=\\alpha,\n\\]\nwe have\n\\[\n\\operatorname{tr}T\\le\\alpha t,\n\\qquad\n\\operatorname{tr}(T^2)\\le\\alpha^2v,\n\\qquad\n\\|T\\|_2\\le\\alpha\\rho.\n\\]\nFor every eigenvalue \\(x\\) of \\(T\\),\n\\[\n-\\frac12\\log(1-2\\theta x)\n\\le\n\\theta x\n+\n\\frac{\\theta^2x^2}\n     {1-2\\theta\\alpha\\rho}.\n\\]\nThe scalar inequality follows from the power series of \\(-\\log(1-u)\\) and a\ngeometric bound on all terms of degree at least two.  Summing over the\neigenvalues of \\(T\\) proves the result.\n\\end{proof}","-2\\theta C)^{-1/2}\nR\n(I-2\\theta C)^{-1/2}.\n\\]\nSince\n\\[\n\\|(I-2\\theta C)^{-1/2}\\|_2^2=\\alpha,\n\\]\nwe have\n\\[\n\\operatorname{tr}T\\le\\alpha t,\n\\qquad\n\\operatorname{tr}(T^2)\\le\\alpha^2v,\n\\qquad\n\\|T\\|_2\\le\\alpha\\rho.\n\\]\nFor every eigenvalue \\(x\\) of \\(T\\),\n\\[\n-\\frac12\\log(1-2\\theta x)\n\\le\n\\theta x\n+\n\\frac{\\theta^2x^2}\n     {1-2\\theta\\alpha\\rho}.\n\\]\nThe scalar inequality follows from the power series of \\(-\\log(1-u)\\) and a\ngeometric bound on all terms of degree at least two.  Summing over the\neigenvalues of \\(T\\) proves the result.\n\\end{proof}","operatorname{tr}T\\le\\alpha t,\n\\qquad\n\\operatorname{tr}(T^2)\\le\\alpha^2v,\n\\qquad\n\\|T\\|_2\\le\\alpha\\rho.\n\\]\nFor every eigenvalue \\(x\\) of \\(T\\),\n\\[\n-\\frac12\\log(1-2\\theta x)\n\\le\n\\theta x\n+\n\\frac{\\theta^2x^2}\n     {1-2\\theta\\alpha\\rho}.\n\\]\nThe scalar inequality follows from the power series of \\(-\\log(1-u)\\) and a\ngeometric bound on all terms of degree at least two.  Summing over the\neigenvalues of \\(T\\) proves the result.\n\\end{proof}","\\begin{corollary}[Trace-aware Chernoff upper tail]\\label{claim:final-23-3}\nFor \\(Q=Z^\\top HZ\\) with \\(Z\\sim N(0,I)\\),\n\\[\n\\boxed{\n\\Pr(Q\\ge q)\n\\le\n\\inf_{0<2\\theta(c+\\rho)<1}\n\\exp\\{\\widehat\\kappa_H(\\theta)-\\theta q\\},\n}\n\\]\nwhere \\(\\widehat\\kappa_H\\) is the preceding MGF upper bound.\n\\end{corollary}","A complementary lower endpoint is immediate from\n\\[\nH=C+R\\succeq C.\n\\]\nUsing the same normal vector,\n\\[\nZ^\\top HZ\\ge Z^\\top CZ\n\\]\nalmost surely, and therefore\n\\[\n\\boxed{\n\\Pr(Z^\\top CZ\\ge q)\n\\le\n\\Pr(Z^\\top HZ\\ge q).\n}\n\\]\nThe lower endpoint can be evaluated by K-COS on the captured core spectrum,\nwhile the upper endpoint is obtained from the optimized trace-aware Chernoff\nbound.","The previous coefficient-wise Weyl interval was intentionally conservative.\nOn the same full-rank, 50-dimensional stress test used in the preceding\nsection, its \\(10^{-8}\\) tail interval remained thousands of decades wide\nuntil essentially the complete range had been captured.  Preserving the exact\nresidual trace and higher residual moments changes that behavior substantially.","\\begin{table}[htbp]\n\\centering\n\\caption{Tail-bracket width in decades at a reference probability near \\(10^{-8}\\).\nThe trace/moment bound is conservative but substantially narrower than\ncoefficient-wise Weyl inflation before full range capture.}\n\\begin{tabular}{rrrr}\n\\toprule\nKrylov depth & basis dimension & Weyl width & moment width","The same invariants determine when it is better to stop approximating the\nmissing spectrum and complete it exactly.  Let\n\\[\ns=\\operatorname{rank}F=\\operatorname{rank}R.\n\\]\nIf the nonzero eigenvalues of \\(R\\) are \\(r_1,\\ldots,r_s\\), then\n\\[\nt=\\sum_i r_i,\n\\qquad\nv=\\sum_i r_i^2,\n\\qquad\nr_i\\le\\rho.\n\\]","\\begin{theorem}[Residual-rank bracket]\\label{claim:final-23-4}\n\\[\n\\boxed{\n\\left\\lceil\n\\max\\left\\{\n\\frac{t}{\\rho},\n\\frac{t^2}{v}\n\\right\\}\n\\right\\rceil\n\\le\n\\operatorname{rank}F\n\\le\nn-r.\n}\n\\]\nThe zero-residual case is interpreted separately as rank zero.\n\\end{theorem}","\\begin{proof}\nThe inequality \\(t\\le s\\rho\\) gives \\(s\\ge t/\\rho\\).  Cauchy--Schwarz gives\n\\[\nt^2\n=\n\\left(\\sum_{i=1}^s r_i\\right)^2\n\\le\ns\\sum_{i=1}^s r_i^2\n=sv,\n\\]\nso \\(s\\ge t^2/v\\).  Finally, every column of \\(F\\) lies in\n\\(\\operatorname{range}(U)^\\perp\\), whose dimension is \\(n-r\\).\n\\end{proof}","Whenever the integer lower bound equals \\(n-r\\), the missing rank is certified\nexactly without computing a residual SVD.","Suppose a compact factorization of the residual is\n\\[\nF=WD,\n\\]\nwhere the columns of $W$ form an orthonormal basis of\n$\\operatorname{range}(F)$. Zero residual directions are omitted.\nSince this range lies in the orthogonal complement of $U$,\n\\[\nU^\\top W=0.\n\\]","\\begin{theorem}[Exact residual completion]\\label{claim:final-23-5}\n\\[\n\\boxed{\nA\n=\n[U\\;W]\n\\begin{bmatrix}\nB\\\\D\n\\end{bmatrix}.\n}\n\\]\nConsequently, the nonzero singular values of $A$ are exactly the nonzero\nsingular values, with multiplicity, of the stacked completion core\n\\[\n\\mathcal C_{\\rm complete}\n=\n\\begin{bmatrix}\nB","D\n\\end{bmatrix}.\n}\n\\]\nConsequently, the nonzero singular values of $A$ are exactly the nonzero\nsingular values, with multiplicity, of the stacked completion core\n\\[\n\\mathcal C_{\\rm complete}\n=\n\\begin{bmatrix}\nB","D\n\\end{bmatrix}.\n}\n\\]\nConsequently, the nonzero singular values of $A$ are exactly the nonzero\nsingular values, with multiplicity, of the stacked completion core\n\\[\n\\mathcal C_{\\rm complete}\n=\n\\begin{bmatrix}\nB\\\\D\n\\end{bmatrix}.\n\\]\n\\end{theorem}","\\begin{proof}\nThe decomposition $A=UB+F$ and the identity $F=WD$ give the factorization.\nThe matrix $[U\\;W]$ has orthonormal columns, so\n$A^\\top A=\\mathcal C_{\\rm complete}^\\top\\mathcal C_{\\rm complete}$.\nThe nonzero singular values therefore agree with multiplicity. If the\ncombined basis contains directions outside $\\operatorname{range}(A)$,\nits row Gram may include additional zeros. Its entire spectrum is the\nnonzero spectrum of $AA^\\top$ only when the combined basis spans exactly\n$\\operatorname{range}(A)$.\n\\end{proof}","B+F$ and the identity $F=WD$ give the factorization.\nThe matrix $[U\\;W]$ has orthonormal columns, so\n$A^\\top A=\\mathcal C_{\\rm complete}^\\top\\mathcal C_{\\rm complete}$.\nThe nonzero singular values therefore agree with multiplicity. If the\ncombined basis contains directions outside $\\operatorname{range}(A)$,\nits row Gram may include additional zeros. Its entire spectrum is the\nnonzero spectrum of $AA^\\top$ only when the combined basis spans exactly\n$\\operatorname{range}(A)$.\n\\end{proof}","The matrix $[U\\;W]$ has orthonormal columns, so\n$A^\\top A=\\mathcal C_{\\rm complete}^\\top\\mathcal C_{\\rm complete}$.\nThe nonzero singular values therefore agree with multiplicity. If the\ncombined basis contains directions outside $\\operatorname{range}(A)$,\nits row Gram may include additional zeros. Its entire spectrum is the\nnonzero spectrum of $AA^\\top$ only when the combined basis spans exactly\n$\\operatorname{range}(A)$.\n\\end{proof}","In the deterministic stress test, exact residual completion reproduced the\nfull singular spectrum at relative errors between approximately\n\\(2\\times10^{-16}\\) and \\(6\\times10^{-16}\\) for all tested incomplete bases.\nThe resulting \\(10^{-8}\\) tail differed from the full-spectrum reference by\nless than \\(10^{-14}\\) decades.  At depth 8 the candidate range had dimension\n48, the missing rank was certified as exactly two, and appending those two\nresidual directions closed the spectrum without another approximate Krylov\nstage.","Residual information supplies a stopping rule for approximate iteration and\na handoff to exact completion:\n\\[\n\\boxed{\n\\begin{array}{c}\n\\text{streamed / pruned Krylov}\\\\[2mm]\n\\downarrow\\\\[-1mm]\n\\text{residual moments }(t,v,\\rho)\\\\[2mm]\n\\downarrow\\\\[-1mm]\n\\begin{cases}\n\\text{continue Krylov}, & \\text{residual range still large},\\\\\n\\text{exact streamed residual completion}, & \\text{completion rank affordable}.\n\\end{cases}\n\\end{array}\n}\n\\]","Let\n\\[\nA\\in\\mathbb R^{n\\times m}\n\\]\nbe supplied as an IEEE binary64 array.  Each stored entry is treated as its\nexact dyadic rational value.  The target operator is therefore the exact Gram\nmatrix\n\\[\nH=AA^T\n\\]\ndefined by those binary64 values, not an unspecified nearby real matrix.","A high-precision symmetric eigensolver is used only to generate candidate\nquantities\n\\[\nQ\\in\\mathbb R^{n\\times n},\n\\qquad\nD=\\operatorname{diag}(d_1,\\ldots,d_n),\n\\qquad\nd_1\\ge\\cdots\\ge d_n\\ge0.\n\\]\nNo correctness claim depends on trusting this eigensolver.","\\begin{theorem}[Candidate eigensystem post-certification]\\label{claim:final-6-1}\nLet\n\\[\nB=QDQ^T.\n\\]\nSuppose directed-rounding computation proves\n\\[\n\\|Q^TQ-I\\|_2\\le\\eta<1\n\\]\nand\n\\[\n\\|H-B\\|_2\\le\\rho.\n\\]\nThen the ordered eigenvalues of the positive semidefinite matrix \\(H\\) satisfy\n\\[\n\\boxed{\n\\max\\{0,(1-\\eta)d_i-\\rho\\}\n\\le\n\\lambda_i(H)\n\\le\n(1+\\eta)d_i+\\rho\n}\n\\]\nfor every \\(i\\).\n\\end{theorem}","\\begin{proof}\nSet\n\\[\nS=Q^TQ.\n\\]\nThe first hypothesis gives the Loewner inequalities\n\\[\n(1-\\eta)I\\preceq S\\preceq(1+\\eta)I.\n\\]\nThe eigenvalues of\n\\[\nB=(QD^{1/2})(QD^{1/2})^T\n\\]\nare the same as those of\n\\[\nD^{1/2}SD^{1/2}.\n\\]\nHence\n\\[\n(1-\\eta)D\n\\preceq\nD^{1/2}SD^{1/2}\n\\preceq\n(1+\\eta)D,\n\\]\nand eigenvalue monotonicity yields\n\\[\n(1-\\eta)d_i\n\\le\n\\lambda_i(B)\n\\le\n(1+\\eta)d_i.\n\\]\nWeyl's perturbation inequality and\n\\(\\|H-B\\|_2\\le\\rho\\) then give\n\\[\n|\\lambda_i(H)-\\lambda_i(B)|\\le\\rho.\n\\]\nCombining the two bounds and using \\(H\\succeq0\\) proves the claim.\n\\end{proof}","Set\n\\[\nS=Q^TQ.\n\\]\nThe first hypothesis gives the Loewner inequalities\n\\[\n(1-\\eta)I\\preceq S\\preceq(1+\\eta)I.\n\\]\nThe eigenvalues of\n\\[\nB=(QD^{1/2})(QD^{1/2})^T\n\\]\nare the same as those of\n\\[\nD^{1/2}SD^{1/2}.\n\\]\nHence\n\\[\n(1-\\eta)D\n\\preceq\nD^{1/2}SD^{1/2}\n\\preceq\n(1+\\eta)D,\n\\]\nand eigenvalue monotonicity yields\n\\[\n(1-\\eta)d_i\n\\le\n\\lambda_i(B)\n\\le\n(1+\\eta)d_i.\n\\]\nWeyl's perturbation inequality and\n\\(\\|H-B\\|_2\\le\\rho\\) then give\n\\[\n|\\lambda_i(H)-\\lambda_i(B)|\\le\\rho.\n\\]\nCombining the two bounds and using \\(H\\succeq0\\) proves the claim.\n\\end{proof}","pothesis gives the Loewner inequalities\n\\[\n(1-\\eta)I\\preceq S\\preceq(1+\\eta)I.\n\\]\nThe eigenvalues of\n\\[\nB=(QD^{1/2})(QD^{1/2})^T\n\\]\nare the same as those of\n\\[\nD^{1/2}SD^{1/2}.\n\\]\nHence\n\\[\n(1-\\eta)D\n\\preceq\nD^{1/2}SD^{1/2}\n\\preceq\n(1+\\eta)D,\n\\]\nand eigenvalue monotonicity yields\n\\[\n(1-\\eta)d_i\n\\le\n\\lambda_i(B)\n\\le\n(1+\\eta)d_i.\n\\]\nWeyl's perturbation inequality and\n\\(\\|H-B\\|_2\\le\\rho\\) then give\n\\[\n|\\lambda_i(H)-\\lambda_i(B)|\\le\\rho.\n\\]\nCombining the two bounds and using \\(H\\succeq0\\) proves the claim.\n\\end{proof}",".\n\\]\nThe eigenvalues of\n\\[\nB=(QD^{1/2})(QD^{1/2})^T\n\\]\nare the same as those of\n\\[\nD^{1/2}SD^{1/2}.\n\\]\nHence\n\\[\n(1-\\eta)D\n\\preceq\nD^{1/2}SD^{1/2}\n\\preceq\n(1+\\eta)D,\n\\]\nand eigenvalue monotonicity yields\n\\[\n(1-\\eta)d_i\n\\le\n\\lambda_i(B)\n\\le\n(1+\\eta)d_i.\n\\]\nWeyl's perturbation inequality and\n\\(\\|H-B\\|_2\\le\\rho\\) then give\n\\[\n|\\lambda_i(H)-\\lambda_i(B)|\\le\\rho.\n\\]\nCombining the two bounds and using \\(H\\succeq0\\) proves the claim.\n\\end{proof}","The implementation does not attempt to evaluate the spectral norms above\nexactly.  Instead it encloses every entry of\n\\[\nQ^TQ-I\n\\quad\\text{and}\\quad\nH-QDQ^T\n\\]\nwith directed-rounding interval arithmetic and uses\n\\[\n\\|M\\|_2\\le\\|M\\|_F.\n\\]\nThus the reported \\(\\eta\\) and \\(\\rho\\) are conservative certified upper\nbounds.","Let the spectral theorem produce\n\\[\n\\ell_i\\le\\lambda_i(H)\\le u_i.\n\\]\nUsing the same independent standard normals in all three sums gives\n\\[\n\\sum_i\\ell_iZ_i^2\n\\le\n\\sum_i\\lambda_iZ_i^2\n\\le\n\\sum_i u_iZ_i^2\n\\qquad\\text{almost surely}.\n\\]\nTherefore\n\\[\n\\boxed{\nP\\!\\left(\\sum_i\\ell_iZ_i^2\\ge q\\right)\n\\le\nP(Q\\ge q)\n\\le\nP\\!\\left(\\sum_i u_iZ_i^2\\ge q\\right).\n}\n\\]\nThe lower endpoint is evaluated by the rigorous positive-tail backend on the\nlower spectrum, and the upper endpoint by the same backend on the upper\nspectrum.  The controller requires more than two strictly positive\ncoefficients in each endpoint spectrum and a threshold above both endpoint\nmeans.  An endpoint spectrum outside these implemented conditions is rejected\nor reported as unsupported. Monotone coupl","Let the spectral theorem produce\n\\[\n\\ell_i\\le\\lambda_i(H)\\le u_i.\n\\]\nUsing the same independent standard normals in all three sums gives\n\\[\n\\sum_i\\ell_iZ_i^2\n\\le\n\\sum_i\\lambda_iZ_i^2\n\\le\n\\sum_i u_iZ_i^2\n\\qquad\\text{almost surely}.\n\\]\nTherefore\n\\[\n\\boxed{\nP\\!\\left(\\sum_i\\ell_iZ_i^2\\ge q\\right)\n\\le\nP(Q\\ge q)\n\\le\nP\\!\\left(\\sum_i u_iZ_i^2\\ge q\\right).\n}\n\\]\nThe lower endpoint is evaluated by the rigorous positive-tail backend on the\nlower spectrum, and the upper endpoint by the same backend on the upper\nspectrum.  The controller requires more than two strictly positive\ncoefficients in each endpoint spectrum and a threshold above both endpoint\nmeans.  An endpoint spectrum outside these implemented conditions is rejected\nor reported as unsupported. Monotone coupling itself remains valid.  The final matrix-defined probability interval is accepted only when\nits multiplicative ratio satisfies the requested accuracy.","\\midrule\n$12\\times18$ & yes & $3.36\\times10^{-58}$ & 0.09","$20\\times30$ & yes & $1.44\\times10^{-57}$ & 0.35","$28\\times36$ & yes & $1.20\\times10^{-56}$ & 0.80","\\subsection{End-to-end matrix-defined tail near \\texorpdfstring{\\(10^{-16}\\)}{1e-16}}","A dense \\(64\\times64\\) exact-binary factor was constructed from a normalized\nHadamard basis with four singular-value groups.  The stored binary64 entries\nare exact dyadic rationals.  The spectral postcheck returned\n\\[\n\\eta\\le1.65\\times10^{-49},\n\\qquad\n\\rho\\le9.18\\times10^{-50}.\n\\]\nThe leading coefficient, whose exact value is one in this control,\nwas enclosed with a width of order $10^{-49}$. The full outward-rounded\nendpoints are preserved in the spectral record. Ellipsized decimal strings\nare not used as interval endpoints.","At\n\\[\nq=131.33201485100517,\n\\]\nthe tail lies near \\(10^{-16}\\).  Starting from the factor matrix itself:","\\midrule\n$10^{-10}$ & $2.60\\times10^{-12}$ & yes","$10^{-12}$ & $4.09\\times10^{-15}$ & yes","For the \\(10^{-12}\\) request, the final enclosure was approximately\n\\[\n\\log_{10}p\n\\in\n[-15.99999999999996625,\n -15.99999999999996447].\n\\]","This closes the basic numerical chain\n\\[\n\\boxed{\n\\text{binary64 factor}\n\\longrightarrow\n\\text{certified spectrum}\n\\longrightarrow\n\\text{certified deep-tail interval}\n}\n\\]\nat moderate matrix dimension.","The theorem applies in every finite dimension, but the present verifier forms a full candidate\nbasis \\(Q\\) and postchecks dense matrices.  It therefore closes the correctness\nboundary before it closes the large-scale memory boundary.  For very large streamed factors, a memory-bounded guarantee would require a\nverified range-core/residual certificate and an interval completion step in\nplace of the full-basis postcheck. Those extensions are not part of the\nimplemented dense verifier.","Let\n\\begin{equation}\nQ=\\sum_{j=1}^{r}\\lambda_j Z_j^2,\n\\qquad \\lambda_j>0,\\quad r\\ge1.\n\\label{eq:Q}\n\\end{equation}\nThe variables $Z_j$ are independent standard normal variables.\nThe moment generating function exists for\n\\[\n\\Re s < \\frac{1}{2\\lambda_{\\max}},\n\\qquad\n\\lambda_{\\max}=\\max_j\\lambda_j,\n\\]\nand its cumulant generating function is\n\\begin{equation}\n\\kappa(s)=\\log \\E e^{sQ}\n=-\\frac12\\sum_{j=1}^{r}\\log(1-2s\\lambda_j).\n\\label{eq:cgf}\n\\end{equation}\nFor complex $s$, $\\kappa$ denotes the analytic logarithm normalized by\n$\\kappa(0)=0$. Each factor $1-2s\\lambda_j$ has positive real part on this\ndomain, so its logarithm in the sum is principal. The sum need not equal the\nprincipal logarithm of the product, whose argument can wrap by $2\\pi$.\nFor real $\\theta<1/(2\\lambda_{\\max})$,\n\\begin{equat","Let\n\\begin{equation}\nQ=\\sum_{j=1}^{r}\\lambda_j Z_j^2,\n\\qquad \\lambda_j>0,\\quad r\\ge1.\n\\label{eq:Q}\n\\end{equation}\nThe variables $Z_j$ are independent standard normal variables.\nThe moment generating function exists for\n\\[\n\\Re s < \\frac{1}{2\\lambda_{\\max}},\n\\qquad\n\\lambda_{\\max}=\\max_j\\lambda_j,\n\\]\nand its cumulant generating function is\n\\begin{equation}\n\\kappa(s)=\\log \\E e^{sQ}\n=-\\frac12\\sum_{j=1}^{r}\\log(1-2s\\lambda_j).\n\\label{eq:cgf}\n\\end{equation}\nFor complex $s$, $\\kappa$ denotes the analytic logarithm normalized by\n$\\kappa(0)=0$. Each factor $1-2s\\lambda_j$ has positive real part on this\ndomain, so its logarithm in the sum is principal. The sum need not equal the\nprincipal logarithm of the product, whose argument can wrap by $2\\pi$.\nFor real $\\theta<1/(2\\lambda_{\\max})$,\n\\begin{equation}\n\\kappa^{(m)}(\\theta)\n=2^{m-1","Let\n\\begin{equation}\nQ=\\sum_{j=1}^{r}\\lambda_j Z_j^2,\n\\qquad \\lambda_j>0,\\quad r\\ge1.\n\\label{eq:Q}\n\\end{equation}\nThe variables $Z_j$ are independent standard normal variables.\nThe moment generating function exists for\n\\[\n\\Re s < \\frac{1}{2\\lambda_{\\max}},\n\\qquad\n\\lambda_{\\max}=\\max_j\\lambda_j,\n\\]\nand its cumulant generating function is\n\\begin{equation}\n\\kappa(s)=\\log \\E e^{sQ}\n=-\\frac12\\sum_{j=1}^{r}\\log(1-2s\\lambda_j).\n\\label{eq:cgf}\n\\end{equation}\nFor complex $s$, $\\kappa$ denotes the analytic logarithm normalized by\n$\\kappa(0)=0$. Each factor $1-2s\\lambda_j$ has positive real part on this\ndomain, so its logarithm in the sum is principal. The sum need not equal the\nprincipal logarithm of the product, whose argument can wrap by $2\\pi$.\nFor real $\\theta<1/(2\\lambda_{\\max})$,\n\\begin{equation}\n\\kappa^{(m)}(\\theta)\n=2^{m-1}(m-1)!\\sum_{j=1}^{r}\n\\frac{\\lambda_j^m}{(1-2\\theta\\lambda_j)^m},\n\\qquad m\\ge1.\n\\label{eq:derivatives}\n\\end{equation}\nAt $\\theta=0$, this recovers the standard cumu","Let\n\\begin{equation}\nQ=\\sum_{j=1}^{r}\\lambda_j Z_j^2,\n\\qquad \\lambda_j>0,\\quad r\\ge1.\n\\label{eq:Q}\n\\end{equation}\nThe variables $Z_j$ are independent standard normal variables.\nThe moment generating function exists for\n\\[\n\\Re s < \\frac{1}{2\\lambda_{\\max}},\n\\qquad\n\\lambda_{\\max}=\\max_j\\lambda_j,\n\\]\nand its cumulant generating function is\n\\begin{equation}\n\\kappa(s)=\\log \\E e^{sQ}\n=-\\frac12\\sum_{j=1}^{r}\\log(1-2s\\lambda_j).\n\\label{eq:cgf}\n\\end{equation}\nFor complex $s$, $\\kappa$ denotes the analytic logarithm normalized by\n$\\kappa(0)=0$. Each factor $1-2s\\lambda_j$ has positive real part on this\ndomain, so its logarithm in the sum is principal. The sum need not equal the\nprincipal logarithm of the product, whose argument can wrap by $2\\pi$.\nFor real $\\theta<1/(2\\lambda_{\\max})$,\n\\begin{equation}\n\\kappa^{(m)}(\\theta)\n=2^{m-1}(m-1)!\\sum_{j=1}^{r}\n\\frac{\\lambda_j^m}{(1-2\\theta\\lambda_j)^m},\n\\qquad m\\ge1.\n\\label{eq:derivatives}\n\\end{equation}\nAt $\\theta=0$, this recovers the standard cumulant identity\n\\[\n\\operatorname{cum}_m(Q)=2^{m-1}(m-1)!\\sum_j\\lambda_j^m.\n\\]","_j>0,\\quad r\\ge1.\n\\label{eq:Q}\n\\end{equation}\nThe variables $Z_j$ are independent standard normal variables.\nThe moment generating function exists for\n\\[\n\\Re s < \\frac{1}{2\\lambda_{\\max}},\n\\qquad\n\\lambda_{\\max}=\\max_j\\lambda_j,\n\\]\nand its cumulant generating function is\n\\begin{equation}\n\\kappa(s)=\\log \\E e^{sQ}\n=-\\frac12\\sum_{j=1}^{r}\\log(1-2s\\lambda_j).\n\\label{eq:cgf}\n\\end{equation}\nFor complex $s$, $\\kappa$ denotes the analytic logarithm normalized by\n$\\kappa(0)=0$. Each factor $1-2s\\lambda_j$ has positive real part on this\ndomain, so its logarithm in the sum is principal. The sum need not equal the\nprincipal logarithm of the product, whose argument can wrap by $2\\pi$.\nFor real $\\theta<1/(2\\lambda_{\\max})$,\n\\begin{equation}\n\\kappa^{(m)}(\\theta)\n=2^{m-1}(m-1)!\\sum_{j=1}^{r}\n\\frac{\\lambda_j^m}{(1-2\\theta\\lambda_j)^m},\n\\qquad m\\ge1.\n\\label{eq:derivatives}\n\\end{equation}\nAt $\\theta=0$, this recovers the standard cumulant identity\n\\[\n\\operatorname{cum}_m(Q)=2^{m-1}(m-1)!\\sum_j\\lambda_j^m.\n\\]","\\quad r\\ge1.\n\\label{eq:Q}\n\\end{equation}\nThe variables $Z_j$ are independent standard normal variables.\nThe moment generating function exists for\n\\[\n\\Re s < \\frac{1}{2\\lambda_{\\max}},\n\\qquad\n\\lambda_{\\max}=\\max_j\\lambda_j,\n\\]\nand its cumulant generating function is\n\\begin{equation}\n\\kappa(s)=\\log \\E e^{sQ}\n=-\\frac12\\sum_{j=1}^{r}\\log(1-2s\\lambda_j).\n\\label{eq:cgf}\n\\end{equation}\nFor complex $s$, $\\kappa$ denotes the analytic logarithm normalized by\n$\\kappa(0)=0$. Each factor $1-2s\\lambda_j$ has positive real part on this\ndomain, so its logarithm in the sum is principal. The sum need not equal the\nprincipal logarithm of the product, whose argument can wrap by $2\\pi$.\nFor real $\\theta<1/(2\\lambda_{\\max})$,\n\\begin{equation}\n\\kappa^{(m)}(\\theta)\n=2^{m-1}(m-1)!\\sum_{j=1}^{r}\n\\frac{\\lambda_j^m}{(1-2\\theta\\lambda_j)^m},\n\\qquad m\\ge1.\n\\label{eq:derivatives}\n\\end{equation}\nAt $\\theta=0$, this recovers the standard cumulant identity\n\\[\n\\operatorname{cum}_m(Q)=2^{m-1}(m-1)!\\sum_j\\lambda_j^m.\n\\]","$Z_j$ are independent standard normal variables.\nThe moment generating function exists for\n\\[\n\\Re s < \\frac{1}{2\\lambda_{\\max}},\n\\qquad\n\\lambda_{\\max}=\\max_j\\lambda_j,\n\\]\nand its cumulant generating function is\n\\begin{equation}\n\\kappa(s)=\\log \\E e^{sQ}\n=-\\frac12\\sum_{j=1}^{r}\\log(1-2s\\lambda_j).\n\\label{eq:cgf}\n\\end{equation}\nFor complex $s$, $\\kappa$ denotes the analytic logarithm normalized by\n$\\kappa(0)=0$. Each factor $1-2s\\lambda_j$ has positive real part on this\ndomain, so its logarithm in the sum is principal. The sum need not equal the\nprincipal logarithm of the product, whose argument can wrap by $2\\pi$.\nFor real $\\theta<1/(2\\lambda_{\\max})$,\n\\begin{equation}\n\\kappa^{(m)}(\\theta)\n=2^{m-1}(m-1)!\\sum_{j=1}^{r}\n\\frac{\\lambda_j^m}{(1-2\\theta\\lambda_j)^m},\n\\qquad m\\ge1.\n\\label{eq:derivatives}\n\\end{equation}\nAt $\\theta=0$, this recovers the standard cumulant identity\n\\[\n\\operatorname{cum}_m(Q)=2^{m-1}(m-1)!\\sum_j\\lambda_j^m.\n\\]","dard normal variables.\nThe moment generating function exists for\n\\[\n\\Re s < \\frac{1}{2\\lambda_{\\max}},\n\\qquad\n\\lambda_{\\max}=\\max_j\\lambda_j,\n\\]\nand its cumulant generating function is\n\\begin{equation}\n\\kappa(s)=\\log \\E e^{sQ}\n=-\\frac12\\sum_{j=1}^{r}\\log(1-2s\\lambda_j).\n\\label{eq:cgf}\n\\end{equation}\nFor complex $s$, $\\kappa$ denotes the analytic logarithm normalized by\n$\\kappa(0)=0$. Each factor $1-2s\\lambda_j$ has positive real part on this\ndomain, so its logarithm in the sum is principal. The sum need not equal the\nprincipal logarithm of the product, whose argument can wrap by $2\\pi$.\nFor real $\\theta<1/(2\\lambda_{\\max})$,\n\\begin{equation}\n\\kappa^{(m)}(\\theta)\n=2^{m-1}(m-1)!\\sum_{j=1}^{r}\n\\frac{\\lambda_j^m}{(1-2\\theta\\lambda_j)^m},\n\\qquad m\\ge1.\n\\label{eq:derivatives}\n\\end{equation}\nAt $\\theta=0$, this recovers the standard cumulant identity\n\\[\n\\operatorname{cum}_m(Q)=2^{m-1}(m-1)!\\sum_j\\lambda_j^m.\n\\]","egin{equation}\n\\kappa(s)=\\log \\E e^{sQ}\n=-\\frac12\\sum_{j=1}^{r}\\log(1-2s\\lambda_j).\n\\label{eq:cgf}\n\\end{equation}\nFor complex $s$, $\\kappa$ denotes the analytic logarithm normalized by\n$\\kappa(0)=0$. Each factor $1-2s\\lambda_j$ has positive real part on this\ndomain, so its logarithm in the sum is principal. The sum need not equal the\nprincipal logarithm of the product, whose argument can wrap by $2\\pi$.\nFor real $\\theta<1/(2\\lambda_{\\max})$,\n\\begin{equation}\n\\kappa^{(m)}(\\theta)\n=2^{m-1}(m-1)!\\sum_{j=1}^{r}\n\\frac{\\lambda_j^m}{(1-2\\theta\\lambda_j)^m},\n\\qquad m\\ge1.\n\\label{eq:derivatives}\n\\end{equation}\nAt $\\theta=0$, this recovers the standard cumulant identity\n\\[\n\\operatorname{cum}_m(Q)=2^{m-1}(m-1)!\\sum_j\\lambda_j^m.\n\\]","appa(s)=\\log \\E e^{sQ}\n=-\\frac12\\sum_{j=1}^{r}\\log(1-2s\\lambda_j).\n\\label{eq:cgf}\n\\end{equation}\nFor complex $s$, $\\kappa$ denotes the analytic logarithm normalized by\n$\\kappa(0)=0$. Each factor $1-2s\\lambda_j$ has positive real part on this\ndomain, so its logarithm in the sum is principal. The sum need not equal the\nprincipal logarithm of the product, whose argument can wrap by $2\\pi$.\nFor real $\\theta<1/(2\\lambda_{\\max})$,\n\\begin{equation}\n\\kappa^{(m)}(\\theta)\n=2^{m-1}(m-1)!\\sum_{j=1}^{r}\n\\frac{\\lambda_j^m}{(1-2\\theta\\lambda_j)^m},\n\\qquad m\\ge1.\n\\label{eq:derivatives}\n\\end{equation}\nAt $\\theta=0$, this recovers the standard cumulant identity\n\\[\n\\operatorname{cum}_m(Q)=2^{m-1}(m-1)!\\sum_j\\lambda_j^m.\n\\]","}\\log(1-2s\\lambda_j).\n\\label{eq:cgf}\n\\end{equation}\nFor complex $s$, $\\kappa$ denotes the analytic logarithm normalized by\n$\\kappa(0)=0$. Each factor $1-2s\\lambda_j$ has positive real part on this\ndomain, so its logarithm in the sum is principal. The sum need not equal the\nprincipal logarithm of the product, whose argument can wrap by $2\\pi$.\nFor real $\\theta<1/(2\\lambda_{\\max})$,\n\\begin{equation}\n\\kappa^{(m)}(\\theta)\n=2^{m-1}(m-1)!\\sum_{j=1}^{r}\n\\frac{\\lambda_j^m}{(1-2\\theta\\lambda_j)^m},\n\\qquad m\\ge1.\n\\label{eq:derivatives}\n\\end{equation}\nAt $\\theta=0$, this recovers the standard cumulant identity\n\\[\n\\operatorname{cum}_m(Q)=2^{m-1}(m-1)!\\sum_j\\lambda_j^m.\n\\]","1-2s\\lambda_j$ has positive real part on this\ndomain, so its logarithm in the sum is principal. The sum need not equal the\nprincipal logarithm of the product, whose argument can wrap by $2\\pi$.\nFor real $\\theta<1/(2\\lambda_{\\max})$,\n\\begin{equation}\n\\kappa^{(m)}(\\theta)\n=2^{m-1}(m-1)!\\sum_{j=1}^{r}\n\\frac{\\lambda_j^m}{(1-2\\theta\\lambda_j)^m},\n\\qquad m\\ge1.\n\\label{eq:derivatives}\n\\end{equation}\nAt $\\theta=0$, this recovers the standard cumulant identity\n\\[\n\\operatorname{cum}_m(Q)=2^{m-1}(m-1)!\\sum_j\\lambda_j^m.\n\\]","ts logarithm in the sum is principal. The sum need not equal the\nprincipal logarithm of the product, whose argument can wrap by $2\\pi$.\nFor real $\\theta<1/(2\\lambda_{\\max})$,\n\\begin{equation}\n\\kappa^{(m)}(\\theta)\n=2^{m-1}(m-1)!\\sum_{j=1}^{r}\n\\frac{\\lambda_j^m}{(1-2\\theta\\lambda_j)^m},\n\\qquad m\\ge1.\n\\label{eq:derivatives}\n\\end{equation}\nAt $\\theta=0$, this recovers the standard cumulant identity\n\\[\n\\operatorname{cum}_m(Q)=2^{m-1}(m-1)!\\sum_j\\lambda_j^m.\n\\]","\\begin{theorem}[Unique upper-tail saddlepoint]\n\\label{thm:saddle}\nIf $q>\\E Q=\\sum_j\\lambda_j$, there exists a unique\n\\[\n\\theta_*=\\theta(q)\\in\\left(0,\\frac{1}{2\\lambda_{\\max}}\\right)\n\\]\nsatisfying\n\\begin{equation}\n\\kappa'(\\theta_*)=q.\n\\label{eq:saddle}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nBy \\eqref{eq:derivatives},\n\\[\n\\kappa'(\\theta)=\\sum_j\\frac{\\lambda_j}{1-2\\theta\\lambda_j}\n\\]\nis continuous and strictly increasing on the stated interval. It equals $\\E Q$ at $\\theta=0$ and diverges as $\\theta\\uparrow1/(2\\lambda_{\\max})$. The intermediate value theorem and strict monotonicity give existence and uniqueness.\n\\end{proof}","Define the exponentially tilted law by\n\\begin{equation}\nf_{\\theta}(x)=e^{\\theta x-\\kappa(\\theta)}f(x).\n\\label{eq:tilt-density}\n\\end{equation}\nIts characteristic function is\n\\begin{equation}\n\\phi_{\\theta}(t)\n=\\exp\\{\\kappa(\\theta+\\ii t)-\\kappa(\\theta)\\},\n\\label{eq:tilt-cf}\n\\end{equation}\nand $\\E_{\\theta}Q=\\kappa'(\\theta)$. Therefore the choice \\eqref{eq:saddle} moves the query $q$ to the mean of the tilted law.","For any $\\theta>0$ in the transform domain,\n\\begin{equation}\n\\Pp(Q\\ge q)\n=e^{\\kappa(\\theta)-\\theta q}\n\\E_{\\theta}\\!\\left[e^{-\\theta(Q-q)}\\mathbf 1_{\\{Q\\ge q\\}}\\right].\n\\label{eq:tilt-tail}\n\\end{equation}\nThis classical exponential-tilting identity lets K-COS evaluate the upper tail\nwithout subtracting a cumulative distribution function (CDF) value close to one.","Let a nonempty packet $P\\subset\\{1,\\ldots,r\\}$ contain $n_P\\ge1$ eigenvalues. Define its center\n\\[\n\\mu_P=\\frac{1}{n_P}\\sum_{j\\in P}\\lambda_j,\n\\qquad\n\\delta_j=\\lambda_j-\\mu_P,\n\\]\nand centered moments\n\\begin{equation}\nM_{P,m}=\\sum_{j\\in P}\\delta_j^m,\n\\qquad m\\ge0.\n\\label{eq:centeredmoments}\n\\end{equation}\nThus $M_{P,0}=n_P$ and $M_{P,1}=0$.","\\begin{theorem}[Centered packet expansion]\n\\label{thm:packet}\nFix $s\\in\\C$ with $\\Re s<1/(2\\lambda_{\\max})$, use the principal logarithm, and define\n\\begin{equation}\n\\alpha_P(s)=\\frac{2s}{1-2s\\mu_P},\n\\qquad\n\\rho_P(s)=|\\alpha_P(s)|\\max_{j\\in P}|\\delta_j|.\n\\label{eq:alpha-rho}\n\\end{equation}\nFor a nonnegative integer expansion order $p$, if $\\rho_P(s)<1$, then the packet contribution to \\eqref{eq:cgf} is\n\\begin{align}\n\\kappa_P(s)\n&=-\\frac{n_P}{2}\\log(1-2s\\mu_P)\n+\\frac12\\sum_{m=1}^{p}\n\\frac{\\alpha_P(s)^m}{m}M_{P,m}\n+R_{P,p}(s),\n\\label{eq:packet-expansion}\n\\end{align}\nwhere\n\\begin{equation}\n|R_{P,p}(s)|\n\\le\n\\frac{n_P}{2}\n\\frac{\\rho_P(s)^{p+1}}{(p+1)(1-\\rho_P(s))}.\n\\label{eq:packet-remainder}\n\\end{equation}\n\\end{theorem}","tered packet expansion]\n\\label{thm:packet}\nFix $s\\in\\C$ with $\\Re s<1/(2\\lambda_{\\max})$, use the principal logarithm, and define\n\\begin{equation}\n\\alpha_P(s)=\\frac{2s}{1-2s\\mu_P},\n\\qquad\n\\rho_P(s)=|\\alpha_P(s)|\\max_{j\\in P}|\\delta_j|.\n\\label{eq:alpha-rho}\n\\end{equation}\nFor a nonnegative integer expansion order $p$, if $\\rho_P(s)<1$, then the packet contribution to \\eqref{eq:cgf} is\n\\begin{align}\n\\kappa_P(s)\n&=-\\frac{n_P}{2}\\log(1-2s\\mu_P)\n+\\frac12\\sum_{m=1}^{p}\n\\frac{\\alpha_P(s)^m}{m}M_{P,m}\n+R_{P,p}(s),\n\\label{eq:packet-expansion}\n\\end{align}\nwhere\n\\begin{equation}\n|R_{P,p}(s)|\n\\le\n\\frac{n_P}{2}\n\\frac{\\rho_P(s)^{p+1}}{(p+1)(1-\\rho_P(s))}.\n\\label{eq:packet-remainder}\n\\end{equation}\n\\end{theorem}","mu_P},\n\\qquad\n\\rho_P(s)=|\\alpha_P(s)|\\max_{j\\in P}|\\delta_j|.\n\\label{eq:alpha-rho}\n\\end{equation}\nFor a nonnegative integer expansion order $p$, if $\\rho_P(s)<1$, then the packet contribution to \\eqref{eq:cgf} is\n\\begin{align}\n\\kappa_P(s)\n&=-\\frac{n_P}{2}\\log(1-2s\\mu_P)\n+\\frac12\\sum_{m=1}^{p}\n\\frac{\\alpha_P(s)^m}{m}M_{P,m}\n+R_{P,p}(s),\n\\label{eq:packet-expansion}\n\\end{align}\nwhere\n\\begin{equation}\n|R_{P,p}(s)|\n\\le\n\\frac{n_P}{2}\n\\frac{\\rho_P(s)^{p+1}}{(p+1)(1-\\rho_P(s))}.\n\\label{eq:packet-remainder}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nFor $j\\in P$,\n\\[\n1-2s\\lambda_j\n=(1-2s\\mu_P)(1-\\alpha_P(s)\\delta_j).\n\\]\nThe transform-domain condition gives positive real part for each\n\\(1-2s\\lambda_j\\) and for \\(1-2s\\mu_P\\).  The disk condition gives\npositive real part for \\(1-\\alpha_P\\delta_j\\).  The two factor arguments\ntherefore sum inside \\((-\\pi,\\pi)\\), so their principal logarithms add\nwithout a branch ambiguity.  Hence\n\\[\n-\\frac12\\log(1-2s\\lambda_j)\n=-\\frac12\\log(1-2s\\mu_P)\n-\\frac12\\log(1-\\alpha_P\\delta_j).\n\\]\nWhen $|\\alpha_P\\delta_j|<1$,\n\\[\n-\\log(1-z)=\\sum_{m\\ge1}\\frac{z^m}{m}.\n\\]\nSumming over $j\\in P$ gives \\eqref{eq:packet-expansion}. The remainder satisfies\n\\[\n\\left|\\sum_{m\\ge p+1}\\frac{z^m}{m}\\right|\n\\le\n\\sum_{m\\ge p+1}\\frac{|z|^m}{p+1}","\\begin{proof}\nFor $j\\in P$,\n\\[\n1-2s\\lambda_j\n=(1-2s\\mu_P)(1-\\alpha_P(s)\\delta_j).\n\\]\nThe transform-domain condition gives positive real part for each\n\\(1-2s\\lambda_j\\) and for \\(1-2s\\mu_P\\).  The disk condition gives\npositive real part for \\(1-\\alpha_P\\delta_j\\).  The two factor arguments\ntherefore sum inside \\((-\\pi,\\pi)\\), so their principal logarithms add\nwithout a branch ambiguity.  Hence\n\\[\n-\\frac12\\log(1-2s\\lambda_j)\n=-\\frac12\\log(1-2s\\mu_P)\n-\\frac12\\log(1-\\alpha_P\\delta_j).\n\\]\nWhen $|\\alpha_P\\delta_j|<1$,\n\\[\n-\\log(1-z)=\\sum_{m\\ge1}\\frac{z^m}{m}.\n\\]\nSumming over $j\\in P$ gives \\eqref{eq:packet-expansion}. The remainder satisfies\n\\[\n\\left|\\sum_{m\\ge p+1}\\frac{z^m}{m}\\right|\n\\le\n\\sum_{m\\ge p+1}\\frac{|z|^m}{p+1}\n=\\frac{|z|^{p+1}}{(p+1)(1-|z|)}.\n\\]\nUsing $|z|\\le\\rho_P(s","\\begin{proof}\nFor $j\\in P$,\n\\[\n1-2s\\lambda_j\n=(1-2s\\mu_P)(1-\\alpha_P(s)\\delta_j).\n\\]\nThe transform-domain condition gives positive real part for each\n\\(1-2s\\lambda_j\\) and for \\(1-2s\\mu_P\\).  The disk condition gives\npositive real part for \\(1-\\alpha_P\\delta_j\\).  The two factor arguments\ntherefore sum inside \\((-\\pi,\\pi)\\), so their principal logarithms add\nwithout a branch ambiguity.  Hence\n\\[\n-\\frac12\\log(1-2s\\lambda_j)\n=-\\frac12\\log(1-2s\\mu_P)\n-\\frac12\\log(1-\\alpha_P\\delta_j).\n\\]\nWhen $|\\alpha_P\\delta_j|<1$,\n\\[\n-\\log(1-z)=\\sum_{m\\ge1}\\frac{z^m}{m}.\n\\]\nSumming over $j\\in P$ gives \\eqref{eq:packet-expansion}. The remainder satisfies\n\\[\n\\left|\\sum_{m\\ge p+1}\\frac{z^m}{m}\\right|\n\\le\n\\sum_{m\\ge p+1}\\frac{|z|^m}{p+1}\n=\\frac{|z|^{p+1}}{(p+1)(1-|z|)}.\n\\]\nUsing $|z|\\le\\rho_P(s)$ and summing $n_P$ terms yields \\eqref{eq:packet-remainder}.\n\\end{proof}","orm-domain condition gives positive real part for each\n\\(1-2s\\lambda_j\\) and for \\(1-2s\\mu_P\\).  The disk condition gives\npositive real part for \\(1-\\alpha_P\\delta_j\\).  The two factor arguments\ntherefore sum inside \\((-\\pi,\\pi)\\), so their principal logarithms add\nwithout a branch ambiguity.  Hence\n\\[\n-\\frac12\\log(1-2s\\lambda_j)\n=-\\frac12\\log(1-2s\\mu_P)\n-\\frac12\\log(1-\\alpha_P\\delta_j).\n\\]\nWhen $|\\alpha_P\\delta_j|<1$,\n\\[\n-\\log(1-z)=\\sum_{m\\ge1}\\frac{z^m}{m}.\n\\]\nSumming over $j\\in P$ gives \\eqref{eq:packet-expansion}. The remainder satisfies\n\\[\n\\left|\\sum_{m\\ge p+1}\\frac{z^m}{m}\\right|\n\\le\n\\sum_{m\\ge p+1}\\frac{|z|^m}{p+1}\n=\\frac{|z|^{p+1}}{(p+1)(1-|z|)}.\n\\]\nUsing $|z|\\le\\rho_P(s)$ and summing $n_P$ terms yields \\eqref{eq:packet-remainder}.\n\\end{proof}","s positive real part for each\n\\(1-2s\\lambda_j\\) and for \\(1-2s\\mu_P\\).  The disk condition gives\npositive real part for \\(1-\\alpha_P\\delta_j\\).  The two factor arguments\ntherefore sum inside \\((-\\pi,\\pi)\\), so their principal logarithms add\nwithout a branch ambiguity.  Hence\n\\[\n-\\frac12\\log(1-2s\\lambda_j)\n=-\\frac12\\log(1-2s\\mu_P)\n-\\frac12\\log(1-\\alpha_P\\delta_j).\n\\]\nWhen $|\\alpha_P\\delta_j|<1$,\n\\[\n-\\log(1-z)=\\sum_{m\\ge1}\\frac{z^m}{m}.\n\\]\nSumming over $j\\in P$ gives \\eqref{eq:packet-expansion}. The remainder satisfies\n\\[\n\\left|\\sum_{m\\ge p+1}\\frac{z^m}{m}\\right|\n\\le\n\\sum_{m\\ge p+1}\\frac{|z|^m}{p+1}\n=\\frac{|z|^{p+1}}{(p+1)(1-|z|)}.\n\\]\nUsing $|z|\\le\\rho_P(s)$ and summing $n_P$ terms yields \\eqref{eq:packet-remainder}.\n\\end{proof}","\\(1-2s\\mu_P\\).  The disk condition gives\npositive real part for \\(1-\\alpha_P\\delta_j\\).  The two factor arguments\ntherefore sum inside \\((-\\pi,\\pi)\\), so their principal logarithms add\nwithout a branch ambiguity.  Hence\n\\[\n-\\frac12\\log(1-2s\\lambda_j)\n=-\\frac12\\log(1-2s\\mu_P)\n-\\frac12\\log(1-\\alpha_P\\delta_j).\n\\]\nWhen $|\\alpha_P\\delta_j|<1$,\n\\[\n-\\log(1-z)=\\sum_{m\\ge1}\\frac{z^m}{m}.\n\\]\nSumming over $j\\in P$ gives \\eqref{eq:packet-expansion}. The remainder satisfies\n\\[\n\\left|\\sum_{m\\ge p+1}\\frac{z^m}{m}\\right|\n\\le\n\\sum_{m\\ge p+1}\\frac{|z|^m}{p+1}\n=\\frac{|z|^{p+1}}{(p+1)(1-|z|)}.\n\\]\nUsing $|z|\\le\\rho_P(s)$ and summing $n_P$ terms yields \\eqref{eq:packet-remainder}.\n\\end{proof}","alpha_P\\delta_j\\).  The two factor arguments\ntherefore sum inside \\((-\\pi,\\pi)\\), so their principal logarithms add\nwithout a branch ambiguity.  Hence\n\\[\n-\\frac12\\log(1-2s\\lambda_j)\n=-\\frac12\\log(1-2s\\mu_P)\n-\\frac12\\log(1-\\alpha_P\\delta_j).\n\\]\nWhen $|\\alpha_P\\delta_j|<1$,\n\\[\n-\\log(1-z)=\\sum_{m\\ge1}\\frac{z^m}{m}.\n\\]\nSumming over $j\\in P$ gives \\eqref{eq:packet-expansion}. The remainder satisfies\n\\[\n\\left|\\sum_{m\\ge p+1}\\frac{z^m}{m}\\right|\n\\le\n\\sum_{m\\ge p+1}\\frac{|z|^m}{p+1}\n=\\frac{|z|^{p+1}}{(p+1)(1-|z|)}.\n\\]\nUsing $|z|\\le\\rho_P(s)$ and summing $n_P$ terms yields \\eqref{eq:packet-remainder}.\n\\end{proof}","a branch ambiguity.  Hence\n\\[\n-\\frac12\\log(1-2s\\lambda_j)\n=-\\frac12\\log(1-2s\\mu_P)\n-\\frac12\\log(1-\\alpha_P\\delta_j).\n\\]\nWhen $|\\alpha_P\\delta_j|<1$,\n\\[\n-\\log(1-z)=\\sum_{m\\ge1}\\frac{z^m}{m}.\n\\]\nSumming over $j\\in P$ gives \\eqref{eq:packet-expansion}. The remainder satisfies\n\\[\n\\left|\\sum_{m\\ge p+1}\\frac{z^m}{m}\\right|\n\\le\n\\sum_{m\\ge p+1}\\frac{|z|^m}{p+1}\n=\\frac{|z|^{p+1}}{(p+1)(1-|z|)}.\n\\]\nUsing $|z|\\le\\rho_P(s)$ and summing $n_P$ terms yields \\eqref{eq:packet-remainder}.\n\\end{proof}","-\\frac12\\log(1-2s\\lambda_j)\n=-\\frac12\\log(1-2s\\mu_P)\n-\\frac12\\log(1-\\alpha_P\\delta_j).\n\\]\nWhen $|\\alpha_P\\delta_j|<1$,\n\\[\n-\\log(1-z)=\\sum_{m\\ge1}\\frac{z^m}{m}.\n\\]\nSumming over $j\\in P$ gives \\eqref{eq:packet-expansion}. The remainder satisfies\n\\[\n\\left|\\sum_{m\\ge p+1}\\frac{z^m}{m}\\right|\n\\le\n\\sum_{m\\ge p+1}\\frac{|z|^m}{p+1}\n=\\frac{|z|^{p+1}}{(p+1)(1-|z|)}.\n\\]\nUsing $|z|\\le\\rho_P(s)$ and summing $n_P$ terms yields \\eqref{eq:packet-remainder}.\n\\end{proof}","\\begin{remark}[What is compressed]\\label{claim:final-26-3}\nA packet of $n_P$ eigenvalues is replaced by its center and $p$ centered moments. Large spectral spikes may be retained exactly. If the representation contains $s$ exact eigenvalues and $R$ packets of order $p$, evaluation of the CGF on $N$ frequencies costs $O(N(s+R(p+1)))$ arithmetic operations after compilation, instead of $O(Nr)$ direct eigenvalue-transform evaluations.\n\\end{remark}","\\begin{remark}[Query dependence]\\label{claim:final-26-4}\nThe relevant packet condition is not merely geometric closeness of eigenvalues. It is the complex-grid quantity $\\rho_P(\\theta_*+\\ii u_k)$. Thus a partition suitable for one tail query or one COS grid need not be suitable for another.\n\\end{remark}","\\section{The running example under an exponential tilt}\\label{sec:worked-tilt}\nFor spectrum $(1,1,2,2)$, the cumulant generator is\n$\\kappa(s)=-\\log(1-2s)-\\log(1-4s)$ on $\\Re s<1/4$.\nAt $q=40$, the saddlepoint equation reduces to\n$160\\theta^2-112\\theta+17=0$, with\n\\[\n\\frac{2}{1-2\\theta}+\\frac{4}{1-4\\theta}=40,\n\\qquad\n\\theta_* = \\frac{14-\\sqrt{26}}{40}\\approx0.2225245122.\n\\]\nThe other root lies outside the transform domain. The tilted mean is $40$,\nwhile the likelihood-ratio identity restores the original tail probability.","ial tilt}\\label{sec:worked-tilt}\nFor spectrum $(1,1,2,2)$, the cumulant generator is\n$\\kappa(s)=-\\log(1-2s)-\\log(1-4s)$ on $\\Re s<1/4$.\nAt $q=40$, the saddlepoint equation reduces to\n$160\\theta^2-112\\theta+17=0$, with\n\\[\n\\frac{2}{1-2\\theta}+\\frac{4}{1-4\\theta}=40,\n\\qquad\n\\theta_* = \\frac{14-\\sqrt{26}}{40}\\approx0.2225245122.\n\\]\nThe other root lies outside the transform domain. The tilted mean is $40$,\nwhile the likelihood-ratio identity restores the original tail probability.","\\begin{figure}[!htbp]\n\\centering\\includegraphics[width=\\linewidth]{figures_production/worked_tilt.pdf}\n\\caption{The running example before and after exponential tilting. The original mean is $6$, while the reweighted mean is the query $q=40$. Shading identifies the tail under the tilted law. Both curves are point evaluations of explicit densities, not numerical enclosures.}\n\\label{fig:worked-tilt}\n\\end{figure}","The two copies of $1$ form one exact packet, and the two copies of $2$ form another. All centered deviations vanish, so the packet remainder is zero on every admissible grid. The numerical enclosure still has support, Fourier-mode, and arithmetic errors.","\\clearpage\n\\section{A packet with a nonzero remainder}\\label{sec:worked-packet}\nTo see the approximation itself, take a separate declared packet\n\\[\n\\{1.96,1.98,2.02,2.04\\},\\qquad \\mu=2,\n\\]\nand evaluate its contribution at $s=0.1+0.4\\,\\mathrm i$. The deviations are $-0.04,-0.02,0.02,0.04$. In particular,\n\\[\nM_1=M_3=0,\\quad M_2=0.004,\\quad M_4=0.00000544.\n\\]\nThe factor $\\alpha=2s/(1-4s)$ gives $\\rho=0.04|\\alpha|<1$, so Theorem~\\ref{thm:packet} applies. Retaining successive even moment orders gives the values below. Both columns are rounded point evaluations. The theorem, not these rounded comparisons, is the reason the remainder is bounded.","\\begin{figure}[!htbp]\n\\centering\\includegraphics[width=\\linewidth]{figures_production/worked_packet_error.pdf}\n\\caption{Pointwise transform error and its geometric remainder for the declared packet $(1.96,1.98,2.02,2.04)$ at $s=0.1+0.4i$. Increasing the retained moment order contracts both quantities. The shaded gap separates the observed discrepancy from the analytic bound.}\n\\label{fig:spectral-packets}\n\\end{figure}","Because $Q\\ge0$, K-COS uses the interval $[0,b]$.  Throughout the\nfollowing functional and packet-error results, assume $b>q\\ge0$, $b>0$,\n$0<\\theta<1/(2\\lambda_{\\max})$, and an integer mode count $N\\ge1$.  Let\n\\[\nu_k=\\frac{k\\pi}{b},\n\\qquad k=0,1,\\ldots,\n\\]\nand define the exponentially weighted cosine integrals\n\\begin{equation}\nJ_k(q,b,\\theta)\n=\\int_q^b e^{-\\theta(x-q)}\\cos(u_kx)\\,dx.\n\\label{eq:J-def}\n\\end{equation}\nThey admit the closed form\n\\begin{equation}\nJ_k\n=\\Re\\left[\n e^{\\ii u_k q}\n \\frac{1-e^{(-\\theta+\\ii u_k)(b-q)}}{\\theta-\\ii u_k}\n\\right].\n\\label{eq:J-formula}\n\\end{equation}\nThe prime on a cosine sum denotes half weight for $k=0$.","the\nfollowing functional and packet-error results, assume $b>q\\ge0$, $b>0$,\n$0<\\theta<1/(2\\lambda_{\\max})$, and an integer mode count $N\\ge1$.  Let\n\\[\nu_k=\\frac{k\\pi}{b},\n\\qquad k=0,1,\\ldots,\n\\]\nand define the exponentially weighted cosine integrals\n\\begin{equation}\nJ_k(q,b,\\theta)\n=\\int_q^b e^{-\\theta(x-q)}\\cos(u_kx)\\,dx.\n\\label{eq:J-def}\n\\end{equation}\nThey admit the closed form\n\\begin{equation}\nJ_k\n=\\Re\\left[\n e^{\\ii u_k q}\n \\frac{1-e^{(-\\theta+\\ii u_k)(b-q)}}{\\theta-\\ii u_k}\n\\right].\n\\label{eq:J-formula}\n\\end{equation}\nThe prime on a cosine sum denotes half weight for $k=0$.","0,1,\\ldots,\n\\]\nand define the exponentially weighted cosine integrals\n\\begin{equation}\nJ_k(q,b,\\theta)\n=\\int_q^b e^{-\\theta(x-q)}\\cos(u_kx)\\,dx.\n\\label{eq:J-def}\n\\end{equation}\nThey admit the closed form\n\\begin{equation}\nJ_k\n=\\Re\\left[\n e^{\\ii u_k q}\n \\frac{1-e^{(-\\theta+\\ii u_k)(b-q)}}{\\theta-\\ii u_k}\n\\right].\n\\label{eq:J-formula}\n\\end{equation}\nThe prime on a cosine sum denotes half weight for $k=0$.","If the exact tilted characteristic function is used, the $N$-mode finite-window expression is\n\\begin{equation}\nT_{N,b}\n=\\frac{2}{b}\\sum_{k=0}^{N-1}{}'\nJ_k\\,\n\\Re\\left[e^{\\kappa(\\theta+\\ii u_k)-\\theta q}\\right].\n\\label{eq:tail-cos}\n\\end{equation}\nThe cancellation of $\\kappa(\\theta)$ in \\eqref{eq:tail-cos} is useful numerically: the likelihood-ratio prefactor from \\eqref{eq:tilt-tail} and the normalization in \\eqref{eq:tilt-cf} do not need to be formed separately inside each mode.","Let $\\widehat\\kappa(s_k)$ be the packet approximation at\n\\[\ns_k=\\theta+\\ii u_k,\n\\]\nand suppose\n\\begin{equation}\n|\\widehat\\kappa(s_k)-\\kappa(s_k)|\\le B_k.\n\\label{eq:Bk}\n\\end{equation}\nFor disjoint packets, $B_k$ can be chosen as the sum of the packet bounds \\eqref{eq:packet-remainder}.","\\begin{theorem}[Finite-grid packet-to-tail bound]\n\\label{thm:packet-tail}\nLet $\\widehat T_{N,b}$ be obtained from \\eqref{eq:tail-cos} by replacing $\\kappa(s_k)$ by $\\widehat\\kappa(s_k)$. Then\n\\begin{equation}\n|\\widehat T_{N,b}-T_{N,b}|\n\\le \\eps_{\\mathrm{pkt}},\n\\label{eq:packet-tail-bound}\n\\end{equation}\nwhere\n\\begin{equation}\n\\eps_{\\mathrm{pkt}}\n=\n\\frac{2}{b}\n e^{\\kappa(\\theta)-\\theta q}\n\\sum_{k=0}^{N-1}{}'\n|J_k|\\bigl(e^{B_k}-1\\bigr).\n\\label{eq:eps-pkt}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nWrite $\\Delta_k=\\widehat\\kappa(s_k)-\\kappa(s_k)$. Then\n\\[\n|e^{\\widehat\\kappa(s_k)-\\theta q}-e^{\\kappa(s_k)-\\theta q}|\n\\le\n|e^{\\kappa(s_k)-\\theta q}|\\,(e^{|\\Delta_k|}-1).\n\\]\nMoreover\n\\[\n\\Re\\kappa(\\theta+\\ii u_k)-\\kappa(\\theta)\n=\\log|\\phi_{\\theta}(u_k)|\\le0,\n\\]\nso\n\\[\n|e^{\\kappa(s_k)-\\theta q}|\n\\le e^{\\kappa(\\theta)-\\theta q}.\n\\]\nSubstitute this into the finite cosine sum and apply the triangle inequality.\n\\end{proof}","Using full characteristic-function coefficients on $[0,b]$ reconstructs the even $2b$-periodization of the tilted density rather than the strictly truncated density. For a nonnegative density supported on $[0,\\infty)$, the total mass folded back into $[0,b]$ equals the tilted mass above $b$. This gives a clean error control for the positive quadratic-form setting.","\\begin{lemma}[Integrated cosine identity by folding]\n\\label{lem:folding}\nLet \\(\\nu_\\theta\\) be the tilted probability measure and let\n\\(F_b:[0,\\infty)\\to[0,b]\\) fold each interval of length \\(2b\\) by\nreflection.  Then\n\\(\\cos(k\\pi F_b(x)/b)=\\cos(k\\pi x/b)\\).\nFor \\(h(x)=e^{-\\theta(x-q)}\\mathbf1_{[q,b]}(x)\\),\n\\[\n\\int h\\,d(F_{b\\#}\\nu_\\theta)\n=\\frac2b\\sum_{k=0}^{\\infty}{}'\nJ_k\\,\\Re\\phi_\\theta(k\\pi/b).\n\\]\nThe series is absolutely convergent for every positive rank \\(r\\ge1\\).\n\\end{lemma}\n\\begin{proof}\nA positive weighted chi-square sum has an absolutely continuous law. Its\nfolded measure therefore has no atom at the jump of \\(h\\).  The even periodic\nextension of \\(h\\) has bounded Fej\\'er means, which converge at every continuity\npoint.  Dominated convergence identifies the integral of","\\begin{lemma}[Integrated cosine identity by folding]\n\\label{lem:folding}\nLet \\(\\nu_\\theta\\) be the tilted probability measure and let\n\\(F_b:[0,\\infty)\\to[0,b]\\) fold each interval of length \\(2b\\) by\nreflection.  Then\n\\(\\cos(k\\pi F_b(x)/b)=\\cos(k\\pi x/b)\\).\nFor \\(h(x)=e^{-\\theta(x-q)}\\mathbf1_{[q,b]}(x)\\),\n\\[\n\\int h\\,d(F_{b\\#}\\nu_\\theta)\n=\\frac2b\\sum_{k=0}^{\\infty}{}'\nJ_k\\,\\Re\\phi_\\theta(k\\pi/b).\n\\]\nThe series is absolutely convergent for every positive rank \\(r\\ge1\\).\n\\end{lemma}\n\\begin{proof}\nA positive weighted chi-square sum has an absolutely continuous law. Its\nfolded measure therefore has no atom at the jump of \\(h\\).  The even periodic\nextension of \\(h\\) has bounded Fej\\'er means, which converge at every continuity\npoint.  Dominated convergence identifies the integral of those means with\n\\(\\int h\\,d(F_{b\\#}\\nu_\\theta)\\).  Their cosine coef","\\begin{lemma}[Integrated cosine identity by folding]\n\\label{lem:folding}\nLet \\(\\nu_\\theta\\) be the tilted probability measure and let\n\\(F_b:[0,\\infty)\\to[0,b]\\) fold each interval of length \\(2b\\) by\nreflection.  Then\n\\(\\cos(k\\pi F_b(x)/b)=\\cos(k\\pi x/b)\\).\nFor \\(h(x)=e^{-\\theta(x-q)}\\mathbf1_{[q,b]}(x)\\),\n\\[\n\\int h\\,d(F_{b\\#}\\nu_\\theta)\n=\\frac2b\\sum_{k=0}^{\\infty}{}'\nJ_k\\,\\Re\\phi_\\theta(k\\pi/b).\n\\]\nThe series is absolutely convergent for every positive rank \\(r\\ge1\\).\n\\end{lemma}\n\\begin{proof}\nA positive weighted chi-square sum has an absolutely continuous law. Its\nfolded measure therefore has no atom at the jump of \\(h\\).  The even periodic\nextension of \\(h\\) has bounded Fej\\'er means, which converge at every continuity\npoint.  Dominated convergence identifies the integral of those means with\n\\(\\int h\\,d(F_{b\\#}\\nu_\\theta)\\).  Their cosine coefficients are the Ces\\`aro\nweighted c","\\begin{lemma}[Integrated cosine identity by folding]\n\\label{lem:folding}\nLet \\(\\nu_\\theta\\) be the tilted probability measure and let\n\\(F_b:[0,\\infty)\\to[0,b]\\) fold each interval of length \\(2b\\) by\nreflection.  Then\n\\(\\cos(k\\pi F_b(x)/b)=\\cos(k\\pi x/b)\\).\nFor \\(h(x)=e^{-\\theta(x-q)}\\mathbf1_{[q,b]}(x)\\),\n\\[\n\\int h\\,d(F_{b\\#}\\nu_\\theta)\n=\\frac2b\\sum_{k=0}^{\\infty}{}'\nJ_k\\,\\Re\\phi_\\theta(k\\pi/b).\n\\]\nThe series is absolutely convergent for every positive rank \\(r\\ge1\\).\n\\end{lemma}\n\\begin{proof}\nA positive weighted chi-square sum has an absolutely continuous law. Its\nfolded measure therefore has no atom at the jump of \\(h\\).  The even periodic\nextension of \\(h\\) has bounded Fej\\'er means, which converge at every continuity\npoint.  Dominated convergence identifies the integral of those means with\n\\(\\int h\\,d(F_{b\\#}\\nu_\\theta)\\).  Their cosine coefficients are the Ces\\`aro\nweighted coefficients \\(2J_k/b\\), and folding preserves the cosine mom","\\begin{lemma}[Integrated cosine identity by folding]\n\\label{lem:folding}\nLet \\(\\nu_\\theta\\) be the tilted probability measure and let\n\\(F_b:[0,\\infty)\\to[0,b]\\) fold each interval of length \\(2b\\) by\nreflection.  Then\n\\(\\cos(k\\pi F_b(x)/b)=\\cos(k\\pi x/b)\\).\nFor \\(h(x)=e^{-\\theta(x-q)}\\mathbf1_{[q,b]}(x)\\),\n\\[\n\\int h\\,d(F_{b\\#}\\nu_\\theta)\n=\\frac2b\\sum_{k=0}^{\\infty}{}'\nJ_k\\,\\Re\\phi_\\theta(k\\pi/b).\n\\]\nThe series is absolutely convergent for every positive rank \\(r\\ge1\\).\n\\end{lemma}\n\\begin{proof}\nA positive weighted chi-square sum has an absolutely continuous law. Its\nfolded measure therefore has no atom at the jump of \\(h\\).  The even periodic\nextension of \\(h\\) has bounded Fej\\'er means, which converge at every continuity\npoint.  Dominated convergence identifies the integral of those means with\n\\(\\int h\\,d(F_{b\\#}\\nu_\\theta)\\).  Their cosine coefficients are the Ces\\`aro\nweighted coefficients \\(2J_k/b\\), and folding preserves the cosine moments\n\\(\\Re\\phi_\\theta(k\\pi/b)\\).  Finally\n\\(J_k=O(","\\begin{lemma}[Integrated cosine identity by folding]\n\\label{lem:folding}\nLet \\(\\nu_\\theta\\) be the tilted probability measure and let\n\\(F_b:[0,\\infty)\\to[0,b]\\) fold each interval of length \\(2b\\) by\nreflection.  Then\n\\(\\cos(k\\pi F_b(x)/b)=\\cos(k\\pi x/b)\\).\nFor \\(h(x)=e^{-\\theta(x-q)}\\mathbf1_{[q,b]}(x)\\),\n\\[\n\\int h\\,d(F_{b\\#}\\nu_\\theta)\n=\\frac2b\\sum_{k=0}^{\\infty}{}'\nJ_k\\,\\Re\\phi_\\theta(k\\pi/b).\n\\]\nThe series is absolutely convergent for every positive rank \\(r\\ge1\\).\n\\end{lemma}\n\\begin{proof}\nA positive weighted chi-square sum has an absolutely continuous law. Its\nfolded measure therefore has no atom at the jump of \\(h\\).  The even periodic\nextension of \\(h\\) has bounded Fej\\'er means, which converge at every continuity\npoint.  Dominated convergence identifies the integral of those means with\n\\(\\int h\\,d(F_{b\\#}\\nu_\\theta)\\).  Their cosine coefficients are the Ces\\`aro\nweighted coefficients \\(2J_k/b\\), and folding preserves the cosine moments\n\\(\\Re\\phi_\\theta(k\\pi/b)\\).  Finally\n\\(J_k=O(k^{-1})\\) and\n\\(|\\phi_\\theta(k\\pi/b)|=O(k^{-r/2})\\), as follows directly from the product\nformul","olding}\nLet \\(\\nu_\\theta\\) be the tilted probability measure and let\n\\(F_b:[0,\\infty)\\to[0,b]\\) fold each interval of length \\(2b\\) by\nreflection.  Then\n\\(\\cos(k\\pi F_b(x)/b)=\\cos(k\\pi x/b)\\).\nFor \\(h(x)=e^{-\\theta(x-q)}\\mathbf1_{[q,b]}(x)\\),\n\\[\n\\int h\\,d(F_{b\\#}\\nu_\\theta)\n=\\frac2b\\sum_{k=0}^{\\infty}{}'\nJ_k\\,\\Re\\phi_\\theta(k\\pi/b).\n\\]\nThe series is absolutely convergent for every positive rank \\(r\\ge1\\).\n\\end{lemma}\n\\begin{proof}\nA positive weighted chi-square sum has an absolutely continuous law. Its\nfolded measure therefore has no atom at the jump of \\(h\\).  The even periodic\nextension of \\(h\\) has bounded Fej\\'er means, which converge at every continuity\npoint.  Dominated convergence identifies the integral of those means with\n\\(\\int h\\,d(F_{b\\#}\\nu_\\theta)\\).  Their cosine coefficients are the Ces\\`aro\nweighted coefficients \\(2J_k/b\\), and folding preserves the cosine moments\n\\(\\Re\\phi_\\theta(k\\pi/b)\\).  Finally\n\\(J_k=O(k^{-1})\\) and\n\\(|\\phi_\\theta(k\\pi/b)|=O(k^{-r/2})\\), as follows directly from the product\nformula below.  Thus the coefficient products are\n\\(O(k^{-1-r/2})\\). Th…","pi F_b(x)/b)=\\cos(k\\pi x/b)\\).\nFor \\(h(x)=e^{-\\theta(x-q)}\\mathbf1_{[q,b]}(x)\\),\n\\[\n\\int h\\,d(F_{b\\#}\\nu_\\theta)\n=\\frac2b\\sum_{k=0}^{\\infty}{}'\nJ_k\\,\\Re\\phi_\\theta(k\\pi/b).\n\\]\nThe series is absolutely convergent for every positive rank \\(r\\ge1\\).\n\\end{lemma}\n\\begin{proof}\nA positive weighted chi-square sum has an absolutely continuous law. Its\nfolded measure therefore has no atom at the jump of \\(h\\).  The even periodic\nextension of \\(h\\) has bounded Fej\\'er means, which converge at every continuity\npoint.  Dominated convergence identifies the integral of those means with\n\\(\\int h\\,d(F_{b\\#}\\nu_\\theta)\\).  Their cosine coefficients are the Ces\\`aro\nweighted coefficients \\(2J_k/b\\), and folding preserves the cosine moments\n\\(\\Re\\phi_\\theta(k\\pi/b)\\).  Finally\n\\(J_k=O(k^{-1})\\) and\n\\(|\\phi_\\theta(k\\pi/b)|=O(k^{-r/2})\\), as follows directly from the product\nformula below.  Thus the coefficient products are\n\\(O(k^{-1-r/2})\\). Their absolute sum converges.  The Ces\\`aro sums and ordinary\nsum have the same limit, proving the displayed identity.  This argument\njustifies the weighted functio…","x)=e^{-\\theta(x-q)}\\mathbf1_{[q,b]}(x)\\),\n\\[\n\\int h\\,d(F_{b\\#}\\nu_\\theta)\n=\\frac2b\\sum_{k=0}^{\\infty}{}'\nJ_k\\,\\Re\\phi_\\theta(k\\pi/b).\n\\]\nThe series is absolutely convergent for every positive rank \\(r\\ge1\\).\n\\end{lemma}\n\\begin{proof}\nA positive weighted chi-square sum has an absolutely continuous law. Its\nfolded measure therefore has no atom at the jump of \\(h\\).  The even periodic\nextension of \\(h\\) has bounded Fej\\'er means, which converge at every continuity\npoint.  Dominated convergence identifies the integral of those means with\n\\(\\int h\\,d(F_{b\\#}\\nu_\\theta)\\).  Their cosine coefficients are the Ces\\`aro\nweighted coefficients \\(2J_k/b\\), and folding preserves the cosine moments\n\\(\\Re\\phi_\\theta(k\\pi/b)\\).  Finally\n\\(J_k=O(k^{-1})\\) and\n\\(|\\phi_\\theta(k\\pi/b)|=O(k^{-r/2})\\), as follows directly from the product\nformula below.  Thus the coefficient products are\n\\(O(k^{-1-r/2})\\). Their absolute sum converges.  The Ces\\`aro sums and ordinary\nsum have the same limit, proving the displayed identity.  This argument\njustifies the weighted functional without requiring pointwise converg…","eries is absolutely convergent for every positive rank \\(r\\ge1\\).\n\\end{lemma}\n\\begin{proof}\nA positive weighted chi-square sum has an absolutely continuous law. Its\nfolded measure therefore has no atom at the jump of \\(h\\).  The even periodic\nextension of \\(h\\) has bounded Fej\\'er means, which converge at every continuity\npoint.  Dominated convergence identifies the integral of those means with\n\\(\\int h\\,d(F_{b\\#}\\nu_\\theta)\\).  Their cosine coefficients are the Ces\\`aro\nweighted coefficients \\(2J_k/b\\), and folding preserves the cosine moments\n\\(\\Re\\phi_\\theta(k\\pi/b)\\).  Finally\n\\(J_k=O(k^{-1})\\) and\n\\(|\\phi_\\theta(k\\pi/b)|=O(k^{-r/2})\\), as follows directly from the product\nformula below.  Thus the coefficient products are\n\\(O(k^{-1-r/2})\\). Their absolute sum converges.  The Ces\\`aro sums and ordinary\nsum have the same limit, proving the displayed identity.  This argument\njustifies the weighted functional without requiring pointwise convergence of\nthe reconstructed density at every boundary.\n\\end{proof}","ve weighted chi-square sum has an absolutely continuous law. Its\nfolded measure therefore has no atom at the jump of \\(h\\).  The even periodic\nextension of \\(h\\) has bounded Fej\\'er means, which converge at every continuity\npoint.  Dominated convergence identifies the integral of those means with\n\\(\\int h\\,d(F_{b\\#}\\nu_\\theta)\\).  Their cosine coefficients are the Ces\\`aro\nweighted coefficients \\(2J_k/b\\), and folding preserves the cosine moments\n\\(\\Re\\phi_\\theta(k\\pi/b)\\).  Finally\n\\(J_k=O(k^{-1})\\) and\n\\(|\\phi_\\theta(k\\pi/b)|=O(k^{-r/2})\\), as follows directly from the product\nformula below.  Thus the coefficient products are\n\\(O(k^{-1-r/2})\\). Their absolute sum converges.  The Ces\\`aro sums and ordinary\nsum have the same limit, proving the displayed identity.  This argument\njustifies the weighted functional without requiring pointwise convergence of\nthe reconstructed density at every boundary.\n\\end{proof}","us law. Its\nfolded measure therefore has no atom at the jump of \\(h\\).  The even periodic\nextension of \\(h\\) has bounded Fej\\'er means, which converge at every continuity\npoint.  Dominated convergence identifies the integral of those means with\n\\(\\int h\\,d(F_{b\\#}\\nu_\\theta)\\).  Their cosine coefficients are the Ces\\`aro\nweighted coefficients \\(2J_k/b\\), and folding preserves the cosine moments\n\\(\\Re\\phi_\\theta(k\\pi/b)\\).  Finally\n\\(J_k=O(k^{-1})\\) and\n\\(|\\phi_\\theta(k\\pi/b)|=O(k^{-r/2})\\), as follows directly from the product\nformula below.  Thus the coefficient products are\n\\(O(k^{-1-r/2})\\). Their absolute sum converges.  The Ces\\`aro sums and ordinary\nsum have the same limit, proving the displayed identity.  This argument\njustifies the weighted functional without requiring pointwise convergence of\nthe reconstructed density at every boundary.\n\\end{proof}","has no atom at the jump of \\(h\\).  The even periodic\nextension of \\(h\\) has bounded Fej\\'er means, which converge at every continuity\npoint.  Dominated convergence identifies the integral of those means with\n\\(\\int h\\,d(F_{b\\#}\\nu_\\theta)\\).  Their cosine coefficients are the Ces\\`aro\nweighted coefficients \\(2J_k/b\\), and folding preserves the cosine moments\n\\(\\Re\\phi_\\theta(k\\pi/b)\\).  Finally\n\\(J_k=O(k^{-1})\\) and\n\\(|\\phi_\\theta(k\\pi/b)|=O(k^{-r/2})\\), as follows directly from the product\nformula below.  Thus the coefficient products are\n\\(O(k^{-1-r/2})\\). Their absolute sum converges.  The Ces\\`aro sums and ordinary\nsum have the same limit, proving the displayed identity.  This argument\njustifies the weighted functional without requiring pointwise convergence of\nthe reconstructed density at every boundary.\n\\end{proof}","p of \\(h\\).  The even periodic\nextension of \\(h\\) has bounded Fej\\'er means, which converge at every continuity\npoint.  Dominated convergence identifies the integral of those means with\n\\(\\int h\\,d(F_{b\\#}\\nu_\\theta)\\).  Their cosine coefficients are the Ces\\`aro\nweighted coefficients \\(2J_k/b\\), and folding preserves the cosine moments\n\\(\\Re\\phi_\\theta(k\\pi/b)\\).  Finally\n\\(J_k=O(k^{-1})\\) and\n\\(|\\phi_\\theta(k\\pi/b)|=O(k^{-r/2})\\), as follows directly from the product\nformula below.  Thus the coefficient products are\n\\(O(k^{-1-r/2})\\). Their absolute sum converges.  The Ces\\`aro sums and ordinary\nsum have the same limit, proving the displayed identity.  This argument\njustifies the weighted functional without requiring pointwise convergence of\nthe reconstructed density at every boundary.\n\\end{proof}","ted convergence identifies the integral of those means with\n\\(\\int h\\,d(F_{b\\#}\\nu_\\theta)\\).  Their cosine coefficients are the Ces\\`aro\nweighted coefficients \\(2J_k/b\\), and folding preserves the cosine moments\n\\(\\Re\\phi_\\theta(k\\pi/b)\\).  Finally\n\\(J_k=O(k^{-1})\\) and\n\\(|\\phi_\\theta(k\\pi/b)|=O(k^{-r/2})\\), as follows directly from the product\nformula below.  Thus the coefficient products are\n\\(O(k^{-1-r/2})\\). Their absolute sum converges.  The Ces\\`aro sums and ordinary\nsum have the same limit, proving the displayed identity.  This argument\njustifies the weighted functional without requiring pointwise convergence of\nthe reconstructed density at every boundary.\n\\end{proof}","\\begin{proposition}[Support and periodization bound]\n\\label{prop:support}\nLet the finite positive spectrum have rank \\(r\\ge1\\), and let\n\\(b>q\\ge0\\), \\(b>0\\), and \\(0<\\theta<1/(2\\lambda_{\\max})\\).  Let\n\\[\n\\tau_{\\theta}(b)=\\Pp_{\\theta}(Q>b).\n\\]\nThen the combined error from (i) replacing the tilted density on $[0,b]$ by its full-coefficient cosine periodization and (ii) omitting the original integral over $(b,\\infty)$ is bounded by\n\\begin{equation}\n\\eps_{\\mathrm{sup}}\n\\le\ne^{\\kappa(\\theta)-\\theta q}\\,\n\\tau_{\\theta}(b)\n\\left(1+e^{-\\theta(b-q)}\\right).\n\\label{eq:eps-support}\n\\end{equation}\n\\end{proposition}","proposition}[Support and periodization bound]\n\\label{prop:support}\nLet the finite positive spectrum have rank \\(r\\ge1\\), and let\n\\(b>q\\ge0\\), \\(b>0\\), and \\(0<\\theta<1/(2\\lambda_{\\max})\\).  Let\n\\[\n\\tau_{\\theta}(b)=\\Pp_{\\theta}(Q>b).\n\\]\nThen the combined error from (i) replacing the tilted density on $[0,b]$ by its full-coefficient cosine periodization and (ii) omitting the original integral over $(b,\\infty)$ is bounded by\n\\begin{equation}\n\\eps_{\\mathrm{sup}}\n\\le\ne^{\\kappa(\\theta)-\\theta q}\\,\n\\tau_{\\theta}(b)\n\\left(1+e^{-\\theta(b-q)}\\right).\n\\label{eq:eps-support}\n\\end{equation}\n\\end{proposition}","nd]\n\\label{prop:support}\nLet the finite positive spectrum have rank \\(r\\ge1\\), and let\n\\(b>q\\ge0\\), \\(b>0\\), and \\(0<\\theta<1/(2\\lambda_{\\max})\\).  Let\n\\[\n\\tau_{\\theta}(b)=\\Pp_{\\theta}(Q>b).\n\\]\nThen the combined error from (i) replacing the tilted density on $[0,b]$ by its full-coefficient cosine periodization and (ii) omitting the original integral over $(b,\\infty)$ is bounded by\n\\begin{equation}\n\\eps_{\\mathrm{sup}}\n\\le\ne^{\\kappa(\\theta)-\\theta q}\\,\n\\tau_{\\theta}(b)\n\\left(1+e^{-\\theta(b-q)}\\right).\n\\label{eq:eps-support}\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nBy Lemma~\\ref{lem:folding}, the infinite cosine functional integrates\nthe folded measure.  The part of that measure originating from \\((b,\\infty)\\)\nis nonnegative and has total mass $\\tau_{\\theta}(b)$. On $[q,b]$, the weight $e^{-\\theta(x-q)}$ is at most one, so its contribution to the weighted functional is at most $\\tau_{\\theta}(b)$. For the omitted original tail, $x\\ge b$ implies\n\\[\ne^{-\\theta(x-q)}\\le e^{-\\theta(b-q)},\n\\]\nso that term is at most $e^{-\\theta(b-q)}\\tau_{\\theta}(b)$. Multiplication by the likelihood-ratio prefactor in \\eqref{eq:tilt-tail} gives \\eqref{eq:eps-support}.\n\\end{proof}","egin{proof}\nBy Lemma~\\ref{lem:folding}, the infinite cosine functional integrates\nthe folded measure.  The part of that measure originating from \\((b,\\infty)\\)\nis nonnegative and has total mass $\\tau_{\\theta}(b)$. On $[q,b]$, the weight $e^{-\\theta(x-q)}$ is at most one, so its contribution to the weighted functional is at most $\\tau_{\\theta}(b)$. For the omitted original tail, $x\\ge b$ implies\n\\[\ne^{-\\theta(x-q)}\\le e^{-\\theta(b-q)},\n\\]\nso that term is at most $e^{-\\theta(b-q)}\\tau_{\\theta}(b)$. Multiplication by the likelihood-ratio prefactor in \\eqref{eq:tilt-tail} gives \\eqref{eq:eps-support}.\n\\end{proof}","nal integrates\nthe folded measure.  The part of that measure originating from \\((b,\\infty)\\)\nis nonnegative and has total mass $\\tau_{\\theta}(b)$. On $[q,b]$, the weight $e^{-\\theta(x-q)}$ is at most one, so its contribution to the weighted functional is at most $\\tau_{\\theta}(b)$. For the omitted original tail, $x\\ge b$ implies\n\\[\ne^{-\\theta(x-q)}\\le e^{-\\theta(b-q)},\n\\]\nso that term is at most $e^{-\\theta(b-q)}\\tau_{\\theta}(b)$. Multiplication by the likelihood-ratio prefactor in \\eqref{eq:tilt-tail} gives \\eqref{eq:eps-support}.\n\\end{proof}","The tilted tail probability itself can be bounded by Chernoff's inequality. For $\\eta>\\theta$ inside the transform domain,\n\\begin{equation}\n\\tau_{\\theta}(b)\n\\le\n\\exp\\left\\{\n\\kappa(\\eta)-\\kappa(\\theta)-(\\eta-\\theta)b\n\\right\\}.\n\\label{eq:tilted-chernoff}\n\\end{equation}\nWhen $b>\\kappa'(\\theta)$, the minimizing $\\eta$ is the unique solution of\n\\begin{equation}\n\\kappa'(\\eta)=b.\n\\label{eq:eta}\n\\end{equation}\nFor the saddlepoint tilt, $\\kappa'(\\theta_*)=q<b$, so such an $\\eta$ exists.","f can be bounded by Chernoff's inequality. For $\\eta>\\theta$ inside the transform domain,\n\\begin{equation}\n\\tau_{\\theta}(b)\n\\le\n\\exp\\left\\{\n\\kappa(\\eta)-\\kappa(\\theta)-(\\eta-\\theta)b\n\\right\\}.\n\\label{eq:tilted-chernoff}\n\\end{equation}\nWhen $b>\\kappa'(\\theta)$, the minimizing $\\eta$ is the unique solution of\n\\begin{equation}\n\\kappa'(\\eta)=b.\n\\label{eq:eta}\n\\end{equation}\nFor the saddlepoint tilt, $\\kappa'(\\theta_*)=q<b$, so such an $\\eta$ exists.","ality. For $\\eta>\\theta$ inside the transform domain,\n\\begin{equation}\n\\tau_{\\theta}(b)\n\\le\n\\exp\\left\\{\n\\kappa(\\eta)-\\kappa(\\theta)-(\\eta-\\theta)b\n\\right\\}.\n\\label{eq:tilted-chernoff}\n\\end{equation}\nWhen $b>\\kappa'(\\theta)$, the minimizing $\\eta$ is the unique solution of\n\\begin{equation}\n\\kappa'(\\eta)=b.\n\\label{eq:eta}\n\\end{equation}\nFor the saddlepoint tilt, $\\kappa'(\\theta_*)=q<b$, so such an $\\eta$ exists.","\\begin{lemma}[Tilted characteristic-function modulus]\n\\label{lem:modulus}\nDefine\n\\begin{equation}\na_j=\\frac{2\\lambda_j}{1-2\\theta\\lambda_j}>0.\n\\label{eq:aj}\n\\end{equation}\nThen\n\\begin{equation}\n|\\phi_{\\theta}(t)|\n=\\prod_{j=1}^{r}\\left(1+(a_jt)^2\\right)^{-1/4}.\n\\label{eq:phi-modulus}\n\\end{equation}\nFor $t>0$, define the logarithmic decay rate\n\\begin{equation}\n\\beta(t)\n=-\\frac{d\\log|\\phi_{\\theta}(t)|}{d\\log t}\n=\\frac12\\sum_{j=1}^{r}\n\\frac{(a_jt)^2}{1+(a_jt)^2}.\n\\label{eq:beta}\n\\end{equation}\nSet $\\beta(0)=0$ by continuous extension. Then $\\beta(t)$ is nondecreasing for $t\\ge0$.\n\\end{lemma}","j=\\frac{2\\lambda_j}{1-2\\theta\\lambda_j}>0.\n\\label{eq:aj}\n\\end{equation}\nThen\n\\begin{equation}\n|\\phi_{\\theta}(t)|\n=\\prod_{j=1}^{r}\\left(1+(a_jt)^2\\right)^{-1/4}.\n\\label{eq:phi-modulus}\n\\end{equation}\nFor $t>0$, define the logarithmic decay rate\n\\begin{equation}\n\\beta(t)\n=-\\frac{d\\log|\\phi_{\\theta}(t)|}{d\\log t}\n=\\frac12\\sum_{j=1}^{r}\n\\frac{(a_jt)^2}{1+(a_jt)^2}.\n\\label{eq:beta}\n\\end{equation}\nSet $\\beta(0)=0$ by continuous extension. Then $\\beta(t)$ is nondecreasing for $t\\ge0$.\n\\end{lemma}","\\label{eq:aj}\n\\end{equation}\nThen\n\\begin{equation}\n|\\phi_{\\theta}(t)|\n=\\prod_{j=1}^{r}\\left(1+(a_jt)^2\\right)^{-1/4}.\n\\label{eq:phi-modulus}\n\\end{equation}\nFor $t>0$, define the logarithmic decay rate\n\\begin{equation}\n\\beta(t)\n=-\\frac{d\\log|\\phi_{\\theta}(t)|}{d\\log t}\n=\\frac12\\sum_{j=1}^{r}\n\\frac{(a_jt)^2}{1+(a_jt)^2}.\n\\label{eq:beta}\n\\end{equation}\nSet $\\beta(0)=0$ by continuous extension. Then $\\beta(t)$ is nondecreasing for $t\\ge0$.\n\\end{lemma}","n\n\\begin{equation}\n|\\phi_{\\theta}(t)|\n=\\prod_{j=1}^{r}\\left(1+(a_jt)^2\\right)^{-1/4}.\n\\label{eq:phi-modulus}\n\\end{equation}\nFor $t>0$, define the logarithmic decay rate\n\\begin{equation}\n\\beta(t)\n=-\\frac{d\\log|\\phi_{\\theta}(t)|}{d\\log t}\n=\\frac12\\sum_{j=1}^{r}\n\\frac{(a_jt)^2}{1+(a_jt)^2}.\n\\label{eq:beta}\n\\end{equation}\nSet $\\beta(0)=0$ by continuous extension. Then $\\beta(t)$ is nondecreasing for $t\\ge0$.\n\\end{lemma}","\\begin{proof}\nTaking the modulus of each factor in \\eqref{eq:tilt-cf} gives \\eqref{eq:phi-modulus}. Differentiating its logarithm gives \\eqref{eq:beta}. Each summand $x/(1+x)$ with $x=(a_jt)^2$ is nondecreasing in $t$.\n\\end{proof}","\\begin{corollary}[Power envelope from one frequency]\n\\label{cor:power-envelope}\nFor $t\\ge t_0>0$,\n\\begin{equation}\n|\\phi_{\\theta}(t)|\n\\le\n|\\phi_{\\theta}(t_0)|\n\\left(\\frac{t}{t_0}\\right)^{-\\beta(t_0)}.\n\\label{eq:power-envelope}\n\\end{equation}\n\\end{corollary}","\\begin{proof}\nBy Lemma \\ref{lem:modulus},\n\\[\n\\frac{d\\log|\\phi_{\\theta}(t)|}{d\\log t}=-\\beta(t)\\le-\\beta(t_0)\n\\]\nfor $t\\ge t_0$. Integrating from $t_0$ to $t$ proves the claim.\n\\end{proof}","For $k\\ge1$, the exact integral \\eqref{eq:J-formula} implies\n\\begin{equation}\n|J_k|\n\\le \\frac{2}{u_k}.\n\\label{eq:J-bound}\n\\end{equation}","\\begin{theorem}[Omitted COS-mode certificate]\n\\label{thm:cos-tail}\nLet $N\\ge1$ and $b>q\\ge0$. Set $u_N=N\\pi/b$ and\n$\\beta_N=\\beta(u_N)$. If $\\beta_N>0$, then the contribution of all exact tilted-COS modes $k\\ge N$ to the weighted tail functional is bounded by\n\\begin{equation}\n\\eps_{\\mathrm{COS}}\n\\le\n e^{\\kappa(\\theta)-\\theta q}\n\\frac{4|\\phi_{\\theta}(u_N)|}{\\pi}\n\\left(\\frac1N+\\frac1{\\beta_N}\\right).\n\\label{eq:eps-cos}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nUsing the cosine coefficient magnitude $2|\\phi_{\\theta}(u_k)|/b$ and \\eqref{eq:J-bound}, the omitted series is at most\n\\[\ne^{\\kappa(\\theta)-\\theta q}\n\\frac{4}{\\pi}\n\\sum_{k=N}^{\\infty}\\frac{|\\phi_{\\theta}(u_k)|}{k}.\n\\]\nBy Corollary \\ref{cor:power-envelope},\n\\[\n|\\phi_{\\theta}(u_k)|\n\\le\n|\\phi_{\\theta}(u_N)|\\left(\\frac{k}{N}\\right)^{-\\beta_N}.\n\\]\nTherefore\n\\[\n\\sum_{k=N}^{\\infty}\\frac{|\\phi_{\\theta}(u_k)|}{k}\n\\le\n|\\phi_{\\theta}(u_N)|N^{\\beta_N}\n\\sum_{k=N}^{\\infty}k^{-\\beta_N-1}.\n\\]\nFinally,\n\\[\n\\sum_{k=N}^{\\infty}k^{-\\beta_N-1}\n\\le N^{-\\beta_N-1}+\\int_N^{\\infty}x^{-\\beta_N-1}\\,dx\n=N^{-\\beta_N-1}+\\frac{N^{-\\beta_N}}{\\beta_N},\n\\]\nwhich yields \\eqref{eq:eps-cos}.\n\\end{proof}","omitted series is at most\n\\[\ne^{\\kappa(\\theta)-\\theta q}\n\\frac{4}{\\pi}\n\\sum_{k=N}^{\\infty}\\frac{|\\phi_{\\theta}(u_k)|}{k}.\n\\]\nBy Corollary \\ref{cor:power-envelope},\n\\[\n|\\phi_{\\theta}(u_k)|\n\\le\n|\\phi_{\\theta}(u_N)|\\left(\\frac{k}{N}\\right)^{-\\beta_N}.\n\\]\nTherefore\n\\[\n\\sum_{k=N}^{\\infty}\\frac{|\\phi_{\\theta}(u_k)|}{k}\n\\le\n|\\phi_{\\theta}(u_N)|N^{\\beta_N}\n\\sum_{k=N}^{\\infty}k^{-\\beta_N-1}.\n\\]\nFinally,\n\\[\n\\sum_{k=N}^{\\infty}k^{-\\beta_N-1}\n\\le N^{-\\beta_N-1}+\\int_N^{\\infty}x^{-\\beta_N-1}\\,dx\n=N^{-\\beta_N-1}+\\frac{N^{-\\beta_N}}{\\beta_N},\n\\]\nwhich yields \\eqref{eq:eps-cos}.\n\\end{proof}","\\begin{theorem}[K-COS analytic error certificate]\n\\label{thm:combined}\nLet $p=\\Pp(Q\\ge q)$, and let $\\widehat p$ be the finite K-COS approximation formed with\n\\begin{enumerate}[label=(\\roman*)]\n\\item an upper interval endpoint $b>q$,\n\\item an integer $N\\ge1$ of cosine modes,\n\\item a packet approximation satisfying \\eqref{eq:Bk} on the finite grid.\n\\end{enumerate}\nUnder the positive-spectrum and parameter hypotheses of Proposition \\ref{prop:support},\n\\begin{equation}\n|p-\\widehat p|\n\\le\n\\eps_{\\mathrm{an}}\n:=\\eps_{\\mathrm{pkt}}+\\eps_{\\mathrm{sup}}+\\eps_{\\mathrm{COS}}.\n\\label{eq:combined}\n\\end{equation}\nIf $\\eps_{\\mathrm{an}}<|\\widehat p|$, then\n\\begin{equation}\n\\frac{|p-\\widehat p|}{|p|}\n\\le\n\\frac{\\eps_{\\mathrm{an}}}\n{|\\widehat p|-\\eps_{\\mathrm{an}}}.\n\\label{eq:relative-certificate","\\begin{theorem}[K-COS analytic error certificate]\n\\label{thm:combined}\nLet $p=\\Pp(Q\\ge q)$, and let $\\widehat p$ be the finite K-COS approximation formed with\n\\begin{enumerate}[label=(\\roman*)]\n\\item an upper interval endpoint $b>q$,\n\\item an integer $N\\ge1$ of cosine modes,\n\\item a packet approximation satisfying \\eqref{eq:Bk} on the finite grid.\n\\end{enumerate}\nUnder the positive-spectrum and parameter hypotheses of Proposition \\ref{prop:support},\n\\begin{equation}\n|p-\\widehat p|\n\\le\n\\eps_{\\mathrm{an}}\n:=\\eps_{\\mathrm{pkt}}+\\eps_{\\mathrm{sup}}+\\eps_{\\mathrm{COS}}.\n\\label{eq:combined}\n\\end{equation}\nIf $\\eps_{\\mathrm{an}}<|\\widehat p|$, then\n\\begin{equation}\n\\frac{|p-\\widehat p|}{|p|}\n\\le\n\\frac{\\eps_{\\mathrm{an}}}\n{|\\widehat p|-\\eps_{\\mathrm{an}}}.\n\\label{eq:relative-certificate}\n\\end{equation}\n\\end{","\\begin{theorem}[K-COS analytic error certificate]\n\\label{thm:combined}\nLet $p=\\Pp(Q\\ge q)$, and let $\\widehat p$ be the finite K-COS approximation formed with\n\\begin{enumerate}[label=(\\roman*)]\n\\item an upper interval endpoint $b>q$,\n\\item an integer $N\\ge1$ of cosine modes,\n\\item a packet approximation satisfying \\eqref{eq:Bk} on the finite grid.\n\\end{enumerate}\nUnder the positive-spectrum and parameter hypotheses of Proposition \\ref{prop:support},\n\\begin{equation}\n|p-\\widehat p|\n\\le\n\\eps_{\\mathrm{an}}\n:=\\eps_{\\mathrm{pkt}}+\\eps_{\\mathrm{sup}}+\\eps_{\\mathrm{COS}}.\n\\label{eq:combined}\n\\end{equation}\nIf $\\eps_{\\mathrm{an}}<|\\widehat p|$, then\n\\begin{equation}\n\\frac{|p-\\widehat p|}{|p|}\n\\le\n\\frac{\\eps_{\\mathrm{an}}}\n{|\\widehat p|-\\eps_{\\mathrm{an}}}.\n\\label{eq:relative-certificate}\n\\end{equation}\n\\end{theorem}","Let $p=\\Pp(Q\\ge q)$, and let $\\widehat p$ be the finite K-COS approximation formed with\n\\begin{enumerate}[label=(\\roman*)]\n\\item an upper interval endpoint $b>q$,\n\\item an integer $N\\ge1$ of cosine modes,\n\\item a packet approximation satisfying \\eqref{eq:Bk} on the finite grid.\n\\end{enumerate}\nUnder the positive-spectrum and parameter hypotheses of Proposition \\ref{prop:support},\n\\begin{equation}\n|p-\\widehat p|\n\\le\n\\eps_{\\mathrm{an}}\n:=\\eps_{\\mathrm{pkt}}+\\eps_{\\mathrm{sup}}+\\eps_{\\mathrm{COS}}.\n\\label{eq:combined}\n\\end{equation}\nIf $\\eps_{\\mathrm{an}}<|\\widehat p|$, then\n\\begin{equation}\n\\frac{|p-\\widehat p|}{|p|}\n\\le\n\\frac{\\eps_{\\mathrm{an}}}\n{|\\widehat p|-\\eps_{\\mathrm{an}}}.\n\\label{eq:relative-certificate}\n\\end{equation}\n\\end{theorem}","terval endpoint $b>q$,\n\\item an integer $N\\ge1$ of cosine modes,\n\\item a packet approximation satisfying \\eqref{eq:Bk} on the finite grid.\n\\end{enumerate}\nUnder the positive-spectrum and parameter hypotheses of Proposition \\ref{prop:support},\n\\begin{equation}\n|p-\\widehat p|\n\\le\n\\eps_{\\mathrm{an}}\n:=\\eps_{\\mathrm{pkt}}+\\eps_{\\mathrm{sup}}+\\eps_{\\mathrm{COS}}.\n\\label{eq:combined}\n\\end{equation}\nIf $\\eps_{\\mathrm{an}}<|\\widehat p|$, then\n\\begin{equation}\n\\frac{|p-\\widehat p|}{|p|}\n\\le\n\\frac{\\eps_{\\mathrm{an}}}\n{|\\widehat p|-\\eps_{\\mathrm{an}}}.\n\\label{eq:relative-certificate}\n\\end{equation}\n\\end{theorem}","ne modes,\n\\item a packet approximation satisfying \\eqref{eq:Bk} on the finite grid.\n\\end{enumerate}\nUnder the positive-spectrum and parameter hypotheses of Proposition \\ref{prop:support},\n\\begin{equation}\n|p-\\widehat p|\n\\le\n\\eps_{\\mathrm{an}}\n:=\\eps_{\\mathrm{pkt}}+\\eps_{\\mathrm{sup}}+\\eps_{\\mathrm{COS}}.\n\\label{eq:combined}\n\\end{equation}\nIf $\\eps_{\\mathrm{an}}<|\\widehat p|$, then\n\\begin{equation}\n\\frac{|p-\\widehat p|}{|p|}\n\\le\n\\frac{\\eps_{\\mathrm{an}}}\n{|\\widehat p|-\\eps_{\\mathrm{an}}}.\n\\label{eq:relative-certificate}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nDecompose the error into packet error on the retained grid, support and periodization error, and the exact omitted cosine series. The first is bounded by Theorem \\ref{thm:packet-tail}, the second by Proposition \\ref{prop:support}, and the third by Theorem \\ref{thm:cos-tail}. This proves \\eqref{eq:combined}. Also\n\\[\n|p|\\ge |\\widehat p|-|p-\\widehat p|\n\\ge |\\widehat p|-\\eps_{\\mathrm{an}},\n\\]\nwhich gives \\eqref{eq:relative-certificate}.\n\\end{proof}","The combined error terms correspond to three distinct obstructions. Packet error is reduced by splitting spectral packets or raising packet order. Support error is reduced by increasing $b$. Cosine-series error is reduced by increasing $N$.","\\begin{proposition}[Existence of a finite analytic refinement]\n\\label{prop:finite-refinement}\nFix a finite positive spectrum $\\{\\lambda_j\\}_{j=1}^{r}$, an upper-tail query $q>\\E Q$, and $\\eps>0$. There exist a finite interval endpoint $b>q$, a finite number of cosine modes $N$, and a finite packet partition (allowing singleton packets) for which the analytic error bound in \\eqref{eq:combined} is below $\\eps$.\n\\end{proposition}","{proposition}[Existence of a finite analytic refinement]\n\\label{prop:finite-refinement}\nFix a finite positive spectrum $\\{\\lambda_j\\}_{j=1}^{r}$, an upper-tail query $q>\\E Q$, and $\\eps>0$. There exist a finite interval endpoint $b>q$, a finite number of cosine modes $N$, and a finite packet partition (allowing singleton packets) for which the analytic error bound in \\eqref{eq:combined} is below $\\eps$.\n\\end{proposition}","\\begin{proof}\nFirst choose $b$ sufficiently large that \\eqref{eq:eps-support} is below $\\eps/3$. This is possible because the tilted law has a moment generating function in a nontrivial right neighborhood of zero and hence its Chernoff bound tends to zero exponentially as $b\\to\\infty$. Hold this $b$ fixed. By \\eqref{eq:phi-modulus}, $|\\phi_{\\theta}(t)|\\to0$ polynomially as $t\\to\\infty$, while $\\beta(t)\\to r/2>0$. Therefore \\eqref{eq:eps-cos} tends to zero as $N\\to\\infty$, so a finite $N$ makes it below $\\eps/3$. Finally, split every packet to singletons if necessary. A singleton has $\\delta_j=0$, so its packet remainder is exactly zero. Hence a finite partition makes $\\eps_{\\mathrm{pkt}}<\\eps/3$. Summing the three terms","\\begin{proof}\nFirst choose $b$ sufficiently large that \\eqref{eq:eps-support} is below $\\eps/3$. This is possible because the tilted law has a moment generating function in a nontrivial right neighborhood of zero and hence its Chernoff bound tends to zero exponentially as $b\\to\\infty$. Hold this $b$ fixed. By \\eqref{eq:phi-modulus}, $|\\phi_{\\theta}(t)|\\to0$ polynomially as $t\\to\\infty$, while $\\beta(t)\\to r/2>0$. Therefore \\eqref{eq:eps-cos} tends to zero as $N\\to\\infty$, so a finite $N$ makes it below $\\eps/3$. Finally, split every packet to singletons if necessary. A singleton has $\\delta_j=0$, so its packet remainder is exactly zero. Hence a finite partition makes $\\eps_{\\mathrm{pkt}}<\\eps/3$. Summing the three terms proves the claim.\n\\end{proof}",":eps-support} is below $\\eps/3$. This is possible because the tilted law has a moment generating function in a nontrivial right neighborhood of zero and hence its Chernoff bound tends to zero exponentially as $b\\to\\infty$. Hold this $b$ fixed. By \\eqref{eq:phi-modulus}, $|\\phi_{\\theta}(t)|\\to0$ polynomially as $t\\to\\infty$, while $\\beta(t)\\to r/2>0$. Therefore \\eqref{eq:eps-cos} tends to zero as $N\\to\\infty$, so a finite $N$ makes it below $\\eps/3$. Finally, split every packet to singletons if necessary. A singleton has $\\delta_j=0$, so its packet remainder is exactly zero. Hence a finite partition makes $\\eps_{\\mathrm{pkt}}<\\eps/3$. Summing the three terms proves the claim.\n\\end{proof}","ps/3$. This is possible because the tilted law has a moment generating function in a nontrivial right neighborhood of zero and hence its Chernoff bound tends to zero exponentially as $b\\to\\infty$. Hold this $b$ fixed. By \\eqref{eq:phi-modulus}, $|\\phi_{\\theta}(t)|\\to0$ polynomially as $t\\to\\infty$, while $\\beta(t)\\to r/2>0$. Therefore \\eqref{eq:eps-cos} tends to zero as $N\\to\\infty$, so a finite $N$ makes it below $\\eps/3$. Finally, split every packet to singletons if necessary. A singleton has $\\delta_j=0$, so its packet remainder is exactly zero. Hence a finite partition makes $\\eps_{\\mathrm{pkt}}<\\eps/3$. Summing the three terms proves the claim.\n\\end{proof}","ible because the tilted law has a moment generating function in a nontrivial right neighborhood of zero and hence its Chernoff bound tends to zero exponentially as $b\\to\\infty$. Hold this $b$ fixed. By \\eqref{eq:phi-modulus}, $|\\phi_{\\theta}(t)|\\to0$ polynomially as $t\\to\\infty$, while $\\beta(t)\\to r/2>0$. Therefore \\eqref{eq:eps-cos} tends to zero as $N\\to\\infty$, so a finite $N$ makes it below $\\eps/3$. Finally, split every packet to singletons if necessary. A singleton has $\\delta_j=0$, so its packet remainder is exactly zero. Hence a finite partition makes $\\eps_{\\mathrm{pkt}}<\\eps/3$. Summing the three terms proves the claim.\n\\end{proof}","neighborhood of zero and hence its Chernoff bound tends to zero exponentially as $b\\to\\infty$. Hold this $b$ fixed. By \\eqref{eq:phi-modulus}, $|\\phi_{\\theta}(t)|\\to0$ polynomially as $t\\to\\infty$, while $\\beta(t)\\to r/2>0$. Therefore \\eqref{eq:eps-cos} tends to zero as $N\\to\\infty$, so a finite $N$ makes it below $\\eps/3$. Finally, split every packet to singletons if necessary. A singleton has $\\delta_j=0$, so its packet remainder is exactly zero. Hence a finite partition makes $\\eps_{\\mathrm{pkt}}<\\eps/3$. Summing the three terms proves the claim.\n\\end{proof}","o\\infty$. Hold this $b$ fixed. By \\eqref{eq:phi-modulus}, $|\\phi_{\\theta}(t)|\\to0$ polynomially as $t\\to\\infty$, while $\\beta(t)\\to r/2>0$. Therefore \\eqref{eq:eps-cos} tends to zero as $N\\to\\infty$, so a finite $N$ makes it below $\\eps/3$. Finally, split every packet to singletons if necessary. A singleton has $\\delta_j=0$, so its packet remainder is exactly zero. Hence a finite partition makes $\\eps_{\\mathrm{pkt}}<\\eps/3$. Summing the three terms proves the claim.\n\\end{proof}","The packet planner accepts a finite positive spectrum, an upper-tail query\n$q>\\sum_j\\lambda_j$, and a requested relative tolerance $\\tau>0$.\nIts initial support parameter satisfies $L>0$, and its mode count is a\npositive integer.","\\begin{enumerate}[leftmargin=2em]\n\\item Solve $\\kappa'(\\theta_*)=q$.\n\\item Initialize an upper interval $b=q+L\\sqrt{\\kappa''(\\theta_*)+\\sqrt{\\kappa^{(4)}(\\theta_*)}}$ and a mode count $N$.\n\\item Retain the largest spectral spikes exactly and place the remaining sorted eigenvalues into coarse packets.\n\\item Evaluate $\\rho_P(\\theta_*+\\ii u_k)$ and the packet remainder bound on the actual COS grid. Split packets until the propagated packet contribution is within its budget.\n\\item Evaluate the support/alias certificate from \\eqref{eq:eps-support}--\\eqref{eq:eta}.\n\\item Evaluate the omitted-mode certificate \\eqref{eq:eps-cos}.\n\\item If the combined relative certificate exceeds $\\tau$, refine whichever analytic error source dominates: split packets, increase $b$","\\begin{enumerate}[leftmargin=2em]\n\\item Solve $\\kappa'(\\theta_*)=q$.\n\\item Initialize an upper interval $b=q+L\\sqrt{\\kappa''(\\theta_*)+\\sqrt{\\kappa^{(4)}(\\theta_*)}}$ and a mode count $N$.\n\\item Retain the largest spectral spikes exactly and place the remaining sorted eigenvalues into coarse packets.\n\\item Evaluate $\\rho_P(\\theta_*+\\ii u_k)$ and the packet remainder bound on the actual COS grid. Split packets until the propagated packet contribution is within its budget.\n\\item Evaluate the support/alias certificate from \\eqref{eq:eps-support}--\\eqref{eq:eta}.\n\\item Evaluate the omitted-mode certificate \\eqref{eq:eps-cos}.\n\\item If the combined relative certificate exceeds $\\tau$, refine whichever analytic error source dominates: split packets, increase $b$, or increase $N$.\n\\item Return a certified analytic result only after the combined bound meets\n$\\t","\\begin{enumerate}[leftmargin=2em]\n\\item Solve $\\kappa'(\\theta_*)=q$.\n\\item Initialize an upper interval $b=q+L\\sqrt{\\kappa''(\\theta_*)+\\sqrt{\\kappa^{(4)}(\\theta_*)}}$ and a mode count $N$.\n\\item Retain the largest spectral spikes exactly and place the remaining sorted eigenvalues into coarse packets.\n\\item Evaluate $\\rho_P(\\theta_*+\\ii u_k)$ and the packet remainder bound on the actual COS grid. Split packets until the propagated packet contribution is within its budget.\n\\item Evaluate the support/alias certificate from \\eqref{eq:eps-support}--\\eqref{eq:eta}.\n\\item Evaluate the omitted-mode certificate \\eqref{eq:eps-cos}.\n\\item If the combined relative certificate exceeds $\\tau$, refine whichever analytic error source dominates: split packets, increase $b$, or increase $N$.\n\\item Return a certified analytic result only after the combined bound meets\n$\\tau$. Return the tail","\\begin{enumerate}[leftmargin=2em]\n\\item Solve $\\kappa'(\\theta_*)=q$.\n\\item Initialize an upper interval $b=q+L\\sqrt{\\kappa''(\\theta_*)+\\sqrt{\\kappa^{(4)}(\\theta_*)}}$ and a mode count $N$.\n\\item Retain the largest spectral spikes exactly and place the remaining sorted eigenvalues into coarse packets.\n\\item Evaluate $\\rho_P(\\theta_*+\\ii u_k)$ and the packet remainder bound on the actual COS grid. Split packets until the propagated packet contribution is within its budget.\n\\item Evaluate the support/alias certificate from \\eqref{eq:eps-support}--\\eqref{eq:eta}.\n\\item Evaluate the omitted-mode certificate \\eqref{eq:eps-cos}.\n\\item If the combined relative certificate exceeds $\\tau$, refine whichever analytic error source dominates: split packets, increase $b$, or increase $N$.\n\\item Return a certified analytic result only after the combined bound meets\n$\\tau$. Return the tail estimate, the compiled spectral representation,\nand its error ledger together. If resource limits are reached first,\nreport failure to certify rather than a","nto coarse packets.\n\\item Evaluate $\\rho_P(\\theta_*+\\ii u_k)$ and the packet remainder bound on the actual COS grid. Split packets until the propagated packet contribution is within its budget.\n\\item Evaluate the support/alias certificate from \\eqref{eq:eps-support}--\\eqref{eq:eta}.\n\\item Evaluate the omitted-mode certificate \\eqref{eq:eps-cos}.\n\\item If the combined relative certificate exceeds $\\tau$, refine whichever analytic error source dominates: split packets, increase $b$, or increase $N$.\n\\item Return a certified analytic result only after the combined bound meets\n$\\tau$. Return the tail estimate, the compiled spectral representation,\nand its error ledger together. If resource limits are reached first,\nreport failure to certify rather than accepting the estimate.\n\\end{enumerate}","nder bound on the actual COS grid. Split packets until the propagated packet contribution is within its budget.\n\\item Evaluate the support/alias certificate from \\eqref{eq:eps-support}--\\eqref{eq:eta}.\n\\item Evaluate the omitted-mode certificate \\eqref{eq:eps-cos}.\n\\item If the combined relative certificate exceeds $\\tau$, refine whichever analytic error source dominates: split packets, increase $b$, or increase $N$.\n\\item Return a certified analytic result only after the combined bound meets\n$\\tau$. Return the tail estimate, the compiled spectral representation,\nand its error ledger together. If resource limits are reached first,\nreport failure to certify rather than accepting the estimate.\n\\end{enumerate}","actual COS grid. Split packets until the propagated packet contribution is within its budget.\n\\item Evaluate the support/alias certificate from \\eqref{eq:eps-support}--\\eqref{eq:eta}.\n\\item Evaluate the omitted-mode certificate \\eqref{eq:eps-cos}.\n\\item If the combined relative certificate exceeds $\\tau$, refine whichever analytic error source dominates: split packets, increase $b$, or increase $N$.\n\\item Return a certified analytic result only after the combined bound meets\n$\\tau$. Return the tail estimate, the compiled spectral representation,\nand its error ledger together. If resource limits are reached first,\nreport failure to certify rather than accepting the estimate.\n\\end{enumerate}","its budget.\n\\item Evaluate the support/alias certificate from \\eqref{eq:eps-support}--\\eqref{eq:eta}.\n\\item Evaluate the omitted-mode certificate \\eqref{eq:eps-cos}.\n\\item If the combined relative certificate exceeds $\\tau$, refine whichever analytic error source dominates: split packets, increase $b$, or increase $N$.\n\\item Return a certified analytic result only after the combined bound meets\n$\\tau$. Return the tail estimate, the compiled spectral representation,\nand its error ledger together. If resource limits are reached first,\nreport failure to certify rather than accepting the estimate.\n\\end{enumerate}","The resulting object is query dependent. Schematically,\n\\begin{equation}\n(\\Lambda,q,\\tau)\n\\longmapsto\n\\mathcal P_{\\Lambda,q,\\tau}\n\\longmapsto\n(\\widehat p,\\text{certificate}),\n\\label{eq:compiler-map}\n\\end{equation}\nwhere $\\mathcal P_{\\Lambda,q,\\tau}$ stores exact spikes, packet centers, centered moments, packet radii, and the selected COS parameters.","\\begin{itemize}\n\\item an explicit nonnegative spectrum.\n\\item a positive semidefinite matrix.\n\\item a dense factor \\(A\\).\n\\item a replayable streamed column-factor source.\n\\item or a covariance/quadratic-matrix pair\n\\((\\Sigma,M)\\).\n\\end{itemize}","A signed central engine accepts a signed spectrum, a symmetric operator, or\n\\[\n(\\Sigma,M),\\qquad M=M^\\top,\n\\]\nand performs the exact central-indefinite reduction proved in Chapter~\\ref{ch:gaussian}.  A\nseparate noncentral positive engine accepts\n\\[\n(\\mu,\\Sigma,M),\\qquad M\\succeq0,\n\\]\nand performs the corresponding noncentral reduction.","After separating the likelihood-ratio prefactor, the packet contribution\ncontains sums of the form\n\\[\n\\frac{2}{b}\n\\sum_k\n|J_k|\n\\left(\ne^{\\varepsilon_k}-1\n\\right).\n\\]\nDirect evaluation can overflow even when the candidate is destined to be\nrejected by the adaptive planner.  The implementation instead\ncomputes\n\\[\n\\log\\left(\n\\frac{2}{b}\n\\sum_k\n|J_k|\n\\left(\ne^{\\varepsilon_k}-1\n\\right)\n\\right)\n\\]\nby log-sum-exp arithmetic, and compares this quantity with the logarithm of\nthe requested tolerance before converting a final accepted bound back to\nfloat64.","An independently evaluated reference is useful only if its own numerical\nresolution has converged.  In a rank-350 deep-tail audit, an initial fixed-grid\nreference appeared to disagree with the adaptive answer.  Increasing the reference COS grid gave absolute discrepancies of approximately\n\\[\n1.88\\times10^{-2},\n\\quad\n2.6\\times10^{-3}\\text{--}4.0\\times10^{-3},\n\\quad\n1.39\\times10^{-4},\n\\quad\n2.60\\times10^{-8}\n\\]\ndecades as the grid was refined through\n\\[\nN=4096,\\ 8192,\\ 16384,\\ 32768.\n\\]","its own numerical\nresolution has converged.  In a rank-350 deep-tail audit, an initial fixed-grid\nreference appeared to disagree with the adaptive answer.  Increasing the reference COS grid gave absolute discrepancies of approximately\n\\[\n1.88\\times10^{-2},\n\\quad\n2.6\\times10^{-3}\\text{--}4.0\\times10^{-3},\n\\quad\n1.39\\times10^{-4},\n\\quad\n2.60\\times10^{-8}\n\\]\ndecades as the grid was refined through\n\\[\nN=4096,\\ 8192,\\ 16384,\\ 32768.\n\\]","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\linewidth]{figures_production/figure_21_reference_convergence.pdf}\n\\caption{The adaptive result is compared with progressively refined full-spectrum references. The reference discrepancy is not monotone across the intermediate grids. Its final recorded value is approximately $-2.60\\times10^{-8}$ decades, illustrating why a reference needs its own convergence check.}\n\\label{fig:reference-convergence}\n\\end{figure}","The notation follows the analytic development: $\\kappa$ is the cumulant generating function, $b$ the support endpoint, $N$ the retained mode count, and $J_k$ the payoff coefficient. The implementation-name correspondence is given in Appendix~\\ref{app:reproduction}.","For a positive central quadratic form\n\\[\nQ=\\sum_{j=1}^{r}\\lambda_jZ_j^2,\n\\qquad\n\\lambda_j>0,\n\\]\nthe output is not merely a point estimate.  It is\n\\[\n\\boxed{\n\\log_{10}P(Q\\ge q)\\in[d_-,d_+].\n}\n\\]\nHence the corresponding multiplicative uncertainty satisfies\n\\[\n\\boxed{\n\\frac{p_+}{p_-}\\le 10^{d_+-d_-}.\n}\n\\]","Choose any\n\\[\n0<\\theta<\\frac{1}{2\\lambda_{\\max}}.\n\\]\nExponential tilting gives\n\\[\nP(Q\\ge q)\n=\ne^{\\kappa(\\theta)-\\theta q}\nS_\\theta(q),\n\\]\nwhere\n\\[\nS_\\theta(q)\n=\nE_\\theta\\!\\left[\ne^{-\\theta(Q-q)}\n1_{\\{Q\\ge q\\}}\n\\right].\n\\]","The interval proof does not require \\(\\theta\\) to be the exact saddlepoint.\nA high-precision saddlepoint is used only as an efficient design choice.\nOnce serialized as an exact decimal input, it is simply a valid tilt inside\nthe MGF domain.","The same is true of the support endpoint \\(b\\) and the auxiliary Chernoff\nparameter \\(\\eta\\).  Correctness requires only\n\\[\nb>q\\ge0,\\qquad b>0,\n\\qquad\n\\theta<\\eta<\\frac{1}{2\\lambda_{\\max}}.\n\\]","Under the tilted measure,\n\\[\nP_\\theta(Q>b)\n\\le\n\\exp\\left\\{\n\\kappa(\\eta)-\\kappa(\\theta)-(\\eta-\\theta)b\n\\right\\}.\n\\]","The full-characteristic-function cosine expansion periodizes the tilted density\non \\([0,b]\\).  Because the original positive quadratic form is supported on\n\\([0,\\infty)\\), the total positive-tail mass aliased back into one fundamental\ninterval is at most \\(P_\\theta(Q>b)\\).","The actual payoff omitted above \\(b\\) is at most\n\\[\ne^{-\\theta(b-q)}P_\\theta(Q>b).\n\\]","Therefore\n\\[\n\\boxed{\nE_{\\rm support}\n\\le\nP_\\theta(Q>b)\n\\left(\n1+e^{-\\theta(b-q)}\n\\right).\n}\n\\]","Define\n\\[\na_j=\n\\frac{2\\lambda_j}{1-2\\theta\\lambda_j}.\n\\]\nThe normalized tilted characteristic function satisfies\n\\[\n|\\phi_\\theta(t)|\n=\n\\prod_{j=1}^r\n\\left(\n1+(a_jt)^2\n\\right)^{-1/4}.\n\\]","Since\n\\[\n1+(a_jt)^2\\ge(a_jt)^2,\n\\]\nwe have\n\\[\n|\\phi_\\theta(t)|\n\\le\nC_\\theta t^{-r/2},\n\\qquad\nC_\\theta=\\prod_j a_j^{-1/2}.\n\\]","For the payoff coefficient\n\\[\nJ_k=\n\\int_q^b\ne^{-\\theta(x-q)}\n\\cos(k\\pi x/b)\\,dx,\n\\]\nthe elementary bound\n\\[\n|J_k|\\le\\frac1\\theta\n\\]\nholds.","Let\n\\[\n\\nu=\\frac r2>1,\\qquad N\\ge1.\n\\]\nFor a finite transform retaining modes \\(0,\\ldots,N-1\\),\n\\[\n\\boxed{\nE_{\\rm series}\n\\le\n\\frac{2C_\\theta}{b\\theta}\n\\left(\\frac{b}{\\pi}\\right)^\\nu\n\\left[\nN^{-\\nu}\n+\n\\frac{N^{1-\\nu}}{\\nu-1}\n\\right].\n}\n\\]","This implemented interval backend therefore requires \\(r>2\\).  Lower-rank positive\nforms can use a different tail bound and are not claimed by this theorem.","Let \\(\\boldsymbol S_N\\) be the directed-rounding interval containing the finite\n\\(N\\)-mode tilted COS sum, and set\n\\[\nE=E_{\\rm support}+E_{\\rm series}.\n\\]\nThen\n\\[\n\\boxed{\nS_\\theta(q)\n\\in\n\\boldsymbol S_N+[-E,E].\n}\n\\]","If the lower endpoint of this interval is positive, directed rounding gives\n\\[\n\\boxed{\n\\log P(Q\\ge q)\n\\in\n\\kappa(\\theta)-\\theta q\n+\n\\log\\left(\n\\boldsymbol S_N+[-E,E]\n\\right).\n}\n\\]","\\section{Completing the running calculation}\\label{sec:worked-enclosure}\nThe factor, spectrum, and tilt of Sections~\\ref{sec:worked-law}--\\ref{sec:worked-tilt} now determine a complete positive-spectrum calculation at $q=40$. The exact diagonalization supplies the coefficients $(1,1,2,2)$. The implementation receives those coefficients as exact decimal literals, together with a serialized decimal approximation to the admissible tilt. Correctness does not require that decimal to equal the radical saddlepoint exactly.","\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=\\linewidth]{figures_production/worked_enclosure_refinement.pdf}\n\\caption{The running example under a requested multiplicative width of $2\\%$. Increasing the mode count narrows the directed interval from a failing width of approximately $2.461\\%$ to a passing width of approximately $1.373\\%$. Diamonds mark the independent closed-form reference. The reference-centered axis is rounded for display only.}\n\\label{fig:worked-enclosure-refinement}\n\\end{figure}","\\section{Independent exact interval control through \\texorpdfstring{\\(10^{-5000}\\)}{1e-5000}}","For the control\n\\[\nQ=0.02\\chi^2_{80},\n\\]\nthe survival probability has the finite exact formula\n\\[\nP(Q\\ge q)\n=\ne^{-x}\n\\sum_{j=0}^{39}\n\\frac{x^j}{j!},\n\\qquad\nx=\\frac{q}{0.04}.\n\\]","This finite expression was evaluated independently by directed-rounding\ninterval arithmetic, without using K-COS.  The K-COS enclosure and this exact\ncontrol enclosure are compared at thresholds targeting\n\\[\np\\approx10^{-100},\n\\qquad\n10^{-1000},\n\\qquad\n10^{-5000}.\n\\]","With $N=128$ and 90 interval decimal digits, the resulting K-COS widths\nare listed below. Widths and ratios are computed from the full outward\nendpoints before rounding the displayed values. At all three stored inputs,\nthe independent exact-law interval is fully contained in the K-COS interval.","\\begin{center}\n\\begin{tabular}{@{}rrr@{}}\n\\toprule\ntarget & width in $\\log_{10}p$ & multiplicative ratio minus one","\\midrule\n$10^{-100}$ &\n$8.93\\times10^{-13}$ &\n$2.06\\times10^{-12}$","$10^{-1000}$ &\n$7.01\\times10^{-12}$ &\n$1.62\\times10^{-11}$","$10^{-5000}$ &\n$3.34\\times10^{-11}$ &\n$7.70\\times10^{-11}$","Thus the \\(10^{-5000}\\) result is not merely a point estimate agreeing with a\nreference.  For the exact supplied spectrum, it comes with a directed-rounding\nlogarithmic enclosure whose probability ratio satisfies\n\\[\n\\boxed{\n\\frac{p_+}{p_-}\n\\le\n1+7.70\\times10^{-11}\n}\n\\]\nThe displayed upper bound is rounded upward from the directed ratio bound.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\linewidth]{figures_production/figure_24_interval_deep_tail_width.pdf}\n\\caption{Directed log-probability widths on exact scaled-chi-square controls at fixed $N=128$ and 90 interval digits. Values come from the corrected full decimal endpoint records. Subtracting rounded binary64 endpoints would give a different diagnostic.}\n\\label{fig:interval-tail-width}\n\\end{figure}","At the \\(10^{-5000}\\) control, the interval precision was swept from 40 to 120\ndecimal digits.  The total certified width remained essentially\n\\[\n3.34247\\times10^{-11}\n\\]\ndecades.","Over the same sweep, the arithmetic-only width contracted from approximately\n\\[\n7.5\\times10^{-37}\n\\]\nto\n\\[\n6.3\\times10^{-117}\n\\]\ndecades.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\linewidth]{figures_production/figure_25_interval_arithmetic_vs_analytic.pdf}\n\\caption{Increasing interval precision suppresses finite-arithmetic width but leaves the analytic remainder essentially unchanged at fixed support and mode count. Both quantities are shown separately for the control near $10^{-5000}$. More digits alone cannot remove a truncation error.}\n\\label{fig:interval-arithmetic}\n\\end{figure}","For an exact positive spectrum and threshold \\(q\\), the solver accepts a\nrequested relative multiplicative accuracy\n\\[\n\\varepsilon_{\\rm req}>0\n\\]\nand returns a certified interval\n\\[\n[p_-,p_+]\n\\]\nonly when\n\\[\n\\boxed{\n\\frac{p_+}{p_-}-1\n\\le\n\\varepsilon_{\\rm req}.\n}\n\\]","In logarithmic coordinates, if\n\\[\n\\log_{10}p\\in[d_-,d_+],\n\\]\nthe acceptance rule is\n\\[\n\\boxed{\n10^{d_+-d_-}-1\n\\le\n\\varepsilon_{\\rm req}.\n}\n\\]","No value of \\(N\\), support width, or arithmetic precision appears in the\nexternal contract.","\\begin{theorem}[Accuracy contract for the positive exact-spectrum backend]\\label{claim:final-5-1}\nAssume an exact supplied positive decimal spectrum of rank greater than two,\na threshold \\(q\\ge0\\) with valid interval-design parameters, and requested multiplicative tolerance\n\\(\\varepsilon_{\\rm req}>0\\).  Suppose the directed-rounding interval backend\nreturns an enclosure\n\\[\n\\log_{10}p\\in[d_-,d_+]\n\\]\nthat includes the proved support/alias and omitted-mode remainders.  If the\naccuracy-driven controller returns \\texttt{certified=True}, then\n\\[\n\\boxed{\nP(Q\\ge q)\\in[10^{d_-},10^{d_+}]\n}\n\\]\nand\n\\[\n\\boxed{\n\\frac{10^{d_+}}{10^{d_-}}-1\n\\le\n\\varepsilon_{\\rm req}.\n}\n\\]\nIf the resource caps are reached before this inequality is proved, the\ncontroller returns \\texttt{certified=False} rather than promoting the interval\nmidpoint to a certified answer.\n\\end{theorem}","rum of rank greater than two,\na threshold \\(q\\ge0\\) with valid interval-design parameters, and requested multiplicative tolerance\n\\(\\varepsilon_{\\rm req}>0\\).  Suppose the directed-rounding interval backend\nreturns an enclosure\n\\[\n\\log_{10}p\\in[d_-,d_+]\n\\]\nthat includes the proved support/alias and omitted-mode remainders.  If the\naccuracy-driven controller returns \\texttt{certified=True}, then\n\\[\n\\boxed{\nP(Q\\ge q)\\in[10^{d_-},10^{d_+}]\n}\n\\]\nand\n\\[\n\\boxed{\n\\frac{10^{d_+}}{10^{d_-}}-1\n\\le\n\\varepsilon_{\\rm req}.\n}\n\\]\nIf the resource caps are reached before this inequality is proved, the\ncontroller returns \\texttt{certified=False} rather than promoting the interval\nmidpoint to a certified answer.\n\\end{theorem}","with valid interval-design parameters, and requested multiplicative tolerance\n\\(\\varepsilon_{\\rm req}>0\\).  Suppose the directed-rounding interval backend\nreturns an enclosure\n\\[\n\\log_{10}p\\in[d_-,d_+]\n\\]\nthat includes the proved support/alias and omitted-mode remainders.  If the\naccuracy-driven controller returns \\texttt{certified=True}, then\n\\[\n\\boxed{\nP(Q\\ge q)\\in[10^{d_-},10^{d_+}]\n}\n\\]\nand\n\\[\n\\boxed{\n\\frac{10^{d_+}}{10^{d_-}}-1\n\\le\n\\varepsilon_{\\rm req}.\n}\n\\]\nIf the resource caps are reached before this inequality is proved, the\ncontroller returns \\texttt{certified=False} rather than promoting the interval\nmidpoint to a certified answer.\n\\end{theorem}","\\begin{proof}\nThe directed-rounding backend encloses the finite transform and exponential\nprefactor and then enlarges that enclosure by the proved support/alias and\ninfinite-mode error bounds.  Therefore the true tail probability lies inside\nthe returned interval.  The controller encloses\n\\(\\operatorname{expm1}(\\log(10)(d_+-d_-))\\) by directed rounding and\nsets \\texttt{certified=True} only when the upper endpoint of that ratio\nenclosure is no larger than the lower endpoint of the exact requested\ntolerance enclosure.  High precision alone is not an enclosure proof.  The fail-closed branch follows directly from\nthe controller definition.\n\\end{proof}","\\begin{enumerate}\n\\item the COS mode count \\(N\\).\n\\item the tilted support scale \\(L\\), which changes \\(b\\) and its associated\nChernoff parameter.\n\\item the interval arithmetic precision.\n\\end{enumerate}","At each stage the full directed-rounding interval is recomputed.  If the\nrounding component is material relative to the requested budget, arithmetic\nprecision is increased.  Otherwise the dominant analytic contribution\ndetermines the next refinement: mode error increases \\(N\\), while\nsupport/alias error increases \\(L\\).","Instead it returns\n\\[\n\\boxed{\\texttt{certified = false}}\n\\]\ntogether with the unresolved interval and a failure reason.","\\section{Accuracy ladder through \\texorpdfstring{\\(10^{-5000}\\)}{1e-5000}}","The exact control\n\\[\nQ=0.02\\chi^2_{80}\n\\]\nwas evaluated at tail depths\n\\[\n10^{-100},\n\\qquad\n10^{-1000},\n\\qquad\n10^{-5000}.\n\\]","For each depth the controller was asked to prove relative multiplicative\naccuracies\n\\[\n10^{-8},\n\\quad\n10^{-10},\n\\quad\n10^{-12},\n\\quad\n10^{-14}.\n\\]\nAll twelve contracts were satisfied.","At the \\(10^{-5000}\\) point, the controller produced:","\\begin{center}\n\\begin{tabular}{@{}rrrr@{}}\n\\toprule\nrequested accuracy & achieved ratio minus one & \\(N\\) & \\(L\\)","\\midrule\n\\(10^{-8}\\)  & \\(3.19\\times10^{-10}\\) & 64  & 10","\\(10^{-10}\\) & \\(7.70\\times10^{-11}\\) & 128 & 10","\\(10^{-12}\\) & \\(3.48\\times10^{-15}\\) & 128 & 12","\\(10^{-14}\\) & \\(3.48\\times10^{-15}\\) & 128 & 12","A further stress test requested\n\\[\n10^{-16}\n\\quad\\text{and}\\quad\n10^{-18}\n\\]\nrelative accuracy at the same \\(10^{-5000}\\) tail.  The solver obtained\napproximately\n\\[\n3.05\\times10^{-18}\n\\]\nand\n\\[\n9.20\\times10^{-20}\n\\]\nrespectively.","Converting a tiny log-width \\(w\\) using\n\\[\n10^w-1\n\\]\nunder an insufficient point-precision context can lose the width. At approximately \\(10^{-15}\\)\nrelative scales, the ratio can round to zero.","Even when the probability interval remains valid, that conversion cannot\nsupport a certification decision. The implementation encloses\n\\[\n\\boxed{\n\\operatorname{expm1}(\\log(10)\\,w)\n}\n\\]\nwith directed interval arithmetic before deciding whether the requested\naccuracy has been met.  Acceptance compares the upper endpoint of that ratio\nwith the lower endpoint of the requested exact tolerance, so a rounded point\nestimate cannot promote an unresolved boundary case to success.","\\section{Input semantics at \\texorpdfstring{\\(p\\approx10^{-16}\\)}{p about 1e-16}}","At this accuracy scale, the representation of the threshold \\(q\\) itself is\nmeasurable.  In the \\(0.02\\chi^2_{80}\\) control targeted at \\(10^{-16}\\),\nrounding the high-precision threshold to binary float64 shifts the exact tail\nby approximately\n\\[\n3.83\\times10^{-15}\n\\]\ndecades.  The binary64 solver evaluated at that same rounded input\nthen differs from the exact tail of the rounded input by approximately\n\\[\n5.17\\times10^{-15}\n\\]\ndecades.","curacy scale, the representation of the threshold \\(q\\) itself is\nmeasurable.  In the \\(0.02\\chi^2_{80}\\) control targeted at \\(10^{-16}\\),\nrounding the high-precision threshold to binary float64 shifts the exact tail\nby approximately\n\\[\n3.83\\times10^{-15}\n\\]\ndecades.  The binary64 solver evaluated at that same rounded input\nthen differs from the exact tail of the rounded input by approximately\n\\[\n5.17\\times10^{-15}\n\\]\ndecades.","The method is designed for a regime in which direct survival probabilities can\nunderflow long before the mathematical probability becomes numerically\nunresolvable.  The engine therefore treats \\(\\log p\\) as the primary numerical\nquantity and evaluates its error bounds on that logarithmic scale.","\\begin{center}\n\\small\n\\begin{tabular}{@{}P{0.27\\linewidth}P{0.13\\linewidth}P{0.22\\linewidth}P{0.16\\linewidth}P{0.12\\linewidth}@{}}\n\\toprule\nvalidation regime & deepest $\\log_{10}p$ & observed error & certificate & spectral size","\\midrule\ndirected-rounding $0.02\\chi^2_{80}$ control\n& $-5000$ & enclosure width $3.34\\times10^{-11}$ decades\n& multiplicative ratio $\\le 1+7.70\\times10^{-11}$ & 80 df","large-scale positive spectrum\n& $-922.031162$ & below stored reference resolution & $5.71\\times10^{-9}$ & rank 1997","strong-dependence kernel\n& $-457.973916$ & below stored reference resolution & $5.20\\times10^{-9}$ & rank 996","heterogeneous inversion benchmark near $10^{-16}$\n& $-16$ & $2.88$--$6.52\\times10^{-15}$ decades & independent integral converged & ranks 80--120","signed finite-endpoint exact control\n& $-200$ & $\\le 2.84\\times10^{-14}$ decades & $7.23\\times10^{-13}$ & rank 80","signed mixed multiscale\n& $-81.891937$ & $1.42\\times10^{-12}$ decades vs full grid & $5.78\\times10^{-9}$ & rank 340","noncentral exact controls\n& $-40$ & $\\le 9.26\\times10^{-15}$ decades & passing & up to 50 df","The analytic scaled-chi-square ladder also includes\n\\[\n\\log_{10}p=-100,-300,-1000,-5000,\n\\]\nwith respective reported absolute log-probability discrepancies\n\\[\n2.84\\times10^{-14},\\quad\n<10^{-15},\\quad\n3.41\\times10^{-13},\\quad\n<10^{-15}\n\\]\nin the stored benchmark output.  The corresponding analytic relative certificates\nremain finite throughout, reaching only\n\\[\n3.85\\times10^{-11}\n\\]\nat \\(10^{-5000}\\).","A large-scale run with \\(n=2000\\), \\(m=50{,}000\\), 49,951 retained\nfeatures, and quadratic-form rank 1997 produced\n\\[\n\\boxed{\\log_{10}p=-922.0311624669905}\n\\]\nusing 79 spectral packets and \\(N=128\\).  Its independently evaluated\nfull-spectrum tilted reference agreed within the stored reporting precision, while the\nreported relative certificate was\n\\[\n5.71\\times10^{-9}.\n\\]","The corrected strong-dependence case study independently reached\n\\[\n\\boxed{\\log_{10}p=-457.9739160722001}\n\\]\nwith a finite \\(5.20\\times10^{-9}\\) certificate.  The signed finite-endpoint\ncontrol passed its analytic ledger through \\(10^{-200}\\), while a rank-340 mixed-sign\nmultiscale spectrum passed its analytic ledger to\n\\[\n\\boxed{\\log_{10}p=-81.89193686056221}.\n\\]\nFinally, the noncentral engine matched exact noncentral-chi-square controls\nfrom \\(10^{-4}\\) through \\(10^{-40}\\), with maximum observed discrepancy\n\\[\n9.26\\times10^{-15}\n\\]\ndecades.","dently reached\n\\[\n\\boxed{\\log_{10}p=-457.9739160722001}\n\\]\nwith a finite \\(5.20\\times10^{-9}\\) certificate.  The signed finite-endpoint\ncontrol passed its analytic ledger through \\(10^{-200}\\), while a rank-340 mixed-sign\nmultiscale spectrum passed its analytic ledger to\n\\[\n\\boxed{\\log_{10}p=-81.89193686056221}.\n\\]\nFinally, the noncentral engine matched exact noncentral-chi-square controls\nfrom \\(10^{-4}\\) through \\(10^{-40}\\), with maximum observed discrepancy\n\\[\n9.26\\times10^{-15}\n\\]\ndecades.","oxed{\\log_{10}p=-457.9739160722001}\n\\]\nwith a finite \\(5.20\\times10^{-9}\\) certificate.  The signed finite-endpoint\ncontrol passed its analytic ledger through \\(10^{-200}\\), while a rank-340 mixed-sign\nmultiscale spectrum passed its analytic ledger to\n\\[\n\\boxed{\\log_{10}p=-81.89193686056221}.\n\\]\nFinally, the noncentral engine matched exact noncentral-chi-square controls\nfrom \\(10^{-4}\\) through \\(10^{-40}\\), with maximum observed discrepancy\n\\[\n9.26\\times10^{-15}\n\\]\ndecades.","nite \\(5.20\\times10^{-9}\\) certificate.  The signed finite-endpoint\ncontrol passed its analytic ledger through \\(10^{-200}\\), while a rank-340 mixed-sign\nmultiscale spectrum passed its analytic ledger to\n\\[\n\\boxed{\\log_{10}p=-81.89193686056221}.\n\\]\nFinally, the noncentral engine matched exact noncentral-chi-square controls\nfrom \\(10^{-4}\\) through \\(10^{-40}\\), with maximum observed discrepancy\n\\[\n9.26\\times10^{-15}\n\\]\ndecades.","\\subsection{Independent characteristic-function inversion near \\texorpdfstring{\\(10^{-16}\\)}{1e-16}}","Closed-form controls are important, but they can hide numerical difficulties\nthat appear only for heterogeneous spectra.  An independent inversion test was\ntherefore run on four generalized chi-square laws near\n\\[\nP(Q\\ge q)\\approx10^{-16}.\n\\]","The reference evaluates the original characteristic function directly through\nthe central Imhof inversion identity\n\\[\nP(Q\\ge q)\n=\n\\frac12\n+\n\\frac1\\pi\n\\int_0^\\infty\n\\frac{\\sin\\Theta(u)}{u\\,R(u)}\\,du,\n\\]\nwhere, for distinct positive coefficients \\(\\lambda_j\\) with multiplicities\n\\(h_j\\),\n\\[\n\\Theta(u)\n=\n\\frac12\\sum_j h_j\\arctan(\\lambda_j u)\n-\n\\frac12qu,\n\\qquad\nR(u)\n=\n\\prod_j\n\\left(1+\\lambda_j^2u^2\\right)^{h_j/4}.\n\\]\nThis route does not use exponential tilting, the COS expansion, spectral\npackets, or the K-COS error controller.","The four spectra include balanced, skewed, dominant-spike, and higher-rank\ncases.  Each reference integral was evaluated at two truncation limits.  The\nomitted integral after the frequency cutoff \\(T>0\\) was bounded using\n\\[\n\\frac1\\pi\n\\int_T^\\infty\\frac{du}{uR(u)}\n\\le\n\\frac{C}{\\pi\\nu}T^{-\\nu},\n\\qquad\n\\nu=\\frac12\\sum_jh_j,\n\\qquad\nC=\\prod_j\\lambda_j^{-h_j/2}.\n\\]","\\begin{center}\n\\small\n\\begin{tabular}{@{}lrrrr@{}}\n\\toprule\nspectrum & rank & \\(|\\Delta\\log_{10}p|\\) & relative difference & reference tail bound / \\(p\\)","\\midrule\nbalanced & 80 & $6.52\\times10^{-15}$ & $1.50\\times10^{-14}$ & $6.84\\times10^{-20}$","skewed & 80 & $4.13\\times10^{-15}$ & $9.50\\times10^{-15}$ & $4.44\\times10^{-18}$","spiky & 80 & $5.19\\times10^{-15}$ & $1.20\\times10^{-14}$ & $7.00\\times10^{-17}$","balanced & 120 & $2.88\\times10^{-15}$ & $6.64\\times10^{-15}$ & $4.19\\times10^{-29}$","The two inversion truncations agree to between approximately\n\\[\n1.2\\times10^{-21}\n\\quad\\text{and}\\quad\n3.7\\times10^{-17}\n\\]\ndecades across the four cases.  Thus the slow reference is numerically stable\nat a scale well below the observed K-COS-versus-inversion difference.\nAgreement between truncations is convergence evidence. The omitted-integral\nbound alone does not enclose quadrature discretization or numerical rounding,\nso this comparison is not assigned the directed-enclosure guarantee.","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\linewidth]{figures_production/figure_27_imhof_p1e16_validation.pdf}\n\\caption{Independent direct-inversion checks near $p=10^{-16}$ on four heterogeneous spectra. Circles show relative K-COS/reference differences and diamonds show the reference integral-tail bound. The latter is one error component, not a complete enclosure of all reference arithmetic.}\n\\label{fig:imhof-p1e16}\n\\end{figure}","\\section{Signed and noncentral controls}\n\\paragraph{Exact finite-endpoint control.}\nFor\n\\[\nQ=-0.02\\,\\chi^2_{80},\n\\]\nthe right tail is a lower chi-square tail with finite upper endpoint zero.  The\nsigned engine agrees with the exact distribution through\n\\[\n\\boxed{p=10^{-200}},\n\\]\nwith maximum observed log-probability error about $2.84\\times10^{-14}$ decades\nand certificates around $10^{-12}$.","\\paragraph{Mixed-sign high-rank analytic-ledger control.}\nA deterministic rank-340 multiscale spectrum with 180 positive and 160\nnegative coefficients was tested from approximately $10^{-14}$ to\n\\[\n\\boxed{p\\approx10^{-81.891937}}.\n\\]\nAt the deepest point the packetized solver used 35 packets, $N=4096$, and\nreported a relative certificate\n\\[\n5.78\\times10^{-9}.\n\\]\nIts log-probability differed from the unpacketized full-spectrum signed solver\nby about $1.42\\times10^{-12}$ decades.  Independent exponential-tilt\nimportance sampling placed the discrepancy at about $1.41$ Monte Carlo\nstandard errors.","l.}\nA deterministic rank-340 multiscale spectrum with 180 positive and 160\nnegative coefficients was tested from approximately $10^{-14}$ to\n\\[\n\\boxed{p\\approx10^{-81.891937}}.\n\\]\nAt the deepest point the packetized solver used 35 packets, $N=4096$, and\nreported a relative certificate\n\\[\n5.78\\times10^{-9}.\n\\]\nIts log-probability differed from the unpacketized full-spectrum signed solver\nby about $1.42\\times10^{-12}$ decades.  Independent exponential-tilt\nimportance sampling placed the discrepancy at about $1.41$ Monte Carlo\nstandard errors.","approximately $10^{-14}$ to\n\\[\n\\boxed{p\\approx10^{-81.891937}}.\n\\]\nAt the deepest point the packetized solver used 35 packets, $N=4096$, and\nreported a relative certificate\n\\[\n5.78\\times10^{-9}.\n\\]\nIts log-probability differed from the unpacketized full-spectrum signed solver\nby about $1.42\\times10^{-12}$ decades.  Independent exponential-tilt\nimportance sampling placed the discrepancy at about $1.41$ Monte Carlo\nstandard errors.","\\paragraph{Retained low-rank limitation.}\nFor independent $Y_1,Y_2\\sim\\chi^2_2$, the quadratic form\n\\[\nQ=0.07(Y_1-Y_2)\n\\]\nis Laplace distributed with scale $0.14$, so for $q\\ge0$,\n\\[\nP(Q\\ge q)=\\frac12e^{-q/0.14}.\n\\]\nThe computed values agree with this exact law through $10^{-200}$ to roughly\n$10^{-9}$ decades or better.  However, the implemented analytic COS-series\ncertificate rejects these low-rank mixed cases because the tilted\ncharacteristic function decays only as $|t|^{-2}$.  The implementation reports\nthis as a certificate failure rather than silently accepting the numerical\nvalue.","^2_2$, the quadratic form\n\\[\nQ=0.07(Y_1-Y_2)\n\\]\nis Laplace distributed with scale $0.14$, so for $q\\ge0$,\n\\[\nP(Q\\ge q)=\\frac12e^{-q/0.14}.\n\\]\nThe computed values agree with this exact law through $10^{-200}$ to roughly\n$10^{-9}$ decades or better.  However, the implemented analytic COS-series\ncertificate rejects these low-rank mixed cases because the tilted\ncharacteristic function decays only as $|t|^{-2}$.  The implementation reports\nthis as a certificate failure rather than silently accepting the numerical\nvalue.","\\begin{table}[htbp]\n\\centering\n\\caption{Synthetic rare-variant simulation scenarios.  Adjacent $r^2$ is\ncomputed within the simulated LD blocks after variant standardization.}\n\\begin{tabular}{lrrrrr}\n\\toprule\nscenario & $n$ & $m$ & rank & median adjacent $r^2$ & effective rank","Effective rank here is the participation ratio\n\\[\nr_{\\rm eff}=\\frac{(\\sum_j\\lambda_j)^2}{\\sum_j\\lambda_j^2}.\n\\]\nIt measures spectral concentration, not an integer count of nonzero directions.\nThe corresponding spectra therefore span a substantial range of concentration:\nstronger LD reduces effective rank even when the nominal quadratic-form rank\nremains close to the sample residual dimension.","\\[\n\\begin{aligned}\nG_b&\\longrightarrow G_b-\\bar G_b\n     \\longrightarrow \\widetilde G_b,\\\\\n\\widetilde G_b&\\longrightarrow M_X\\widetilde G_b\n     \\longrightarrow A_b=M_X\\widetilde G_bW_b^{1/2}\n     \\longrightarrow A_bA_b^\\top,\n\\end{aligned}\n\\]","where $\\widetilde G_b$ denotes the centered and scaled block. The full\nweighted factor is never materialized.","\\[\n\\boxed{6.61\\times10^{-16}}.\n\\]","For each scenario, 100 Gaussian null residual draws were evaluated in three\nways. Under the standardized Gaussian null, write $e=M_Xz$ with\n$z\\sim N(0,I)$. Because $M_XA=A$, the statistic used here is\n$Q=\\frac12 z^\\top Kz$, so its nonzero mixture coefficients are\n$\\frac12\\lambda_j(K)$. The factor $\\frac12$ belongs to this case-study\nnormalization, not to the general quadratic-form definition.","\\begin{enumerate}\n\\item directly through the factor,\n\\[\nQ=\\frac12\\|A^\\top e\\|_2^2.\n\\]\n\\item through the sample kernel,\n\\[\nQ=\\frac12 e^\\top Ke.\n\\]\n\\item through the eigencoordinates of $K$.\n\\end{enumerate}","\\[\n\\boxed{8.32\\times10^{-16}}.\n\\]","Thus the simulated phenotype path, the streamed kernel, and the generalized\n$\\chi^2$ spectral representation agree to floating-point precision.","\\[\n\\boxed{\\log_{10}p=-457.973916}.\n\\]","\\[\n\\boxed{3.21\\times10^{-12}\\ \\text{decades}}.\n\\]","\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\linewidth]{figures_production/figure_20_realistic_deep_tail_accuracy.pdf}\n\\caption{Absolute differences between K-COS and refined full-spectrum tilted references on the synthetic dependence scenarios. The corrected large copy-LD controls extend through approximately $10^{-458}$. Zeros are plotted at zero rather than assigned an artificial positive floor.}\n\\label{fig:realistic-deep-tail}\n\\end{figure}","For the $n=1000$, $m=5000$ corrected copy-LD spectrum, selected results are","\\begin{center}\n\\begin{tabular}{rrrr}\n\\toprule\ntilt fraction & $\\log_{10}p$ & COS modes $N$ & absolute error (decades)","\\midrule\n0.9950 & -65.950669 & 2048 & $4.69\\times10^{-13}$","0.9970 & -95.305034 & 2048 & $3.70\\times10^{-14}$","0.9990 & -240.602047 & 8192 & $4.94\\times10^{-14}$","To isolate the effect of exponential tilting, we first used\n\\[\nQ=0.02\\chi^2_{80},\n\\]\nwhose survival probabilities are available directly from the chi-square distribution. Table \\ref{tab:chi} compares a global COS calculation with query-centered tilted COS.","\\begin{table}[htbp]\n\\centering\n\\caption{Relative error in an exactly solvable chi-square family.}\n\\label{tab:chi}\n\\begin{tabular}{rrrr}\n\\toprule\nTarget tail $p$ & $N$ & Global COS rel. error & Tilted COS rel. error","\\midrule\n$10^{-6}$  & 64  & $3.56\\times10^{-8}$ & $2.71\\times10^{-14}$","$10^{-10}$ & 64  & $1.64\\times10^{-4}$ & $6.59\\times10^{-15}$","$10^{-10}$ & 128 & $2.03\\times10^{-6}$ & $9.05\\times10^{-15}$","$10^{-14}$ & 64  & $1.43$              & $3.53\\times10^{-14}$","$10^{-14}$ & 128 & $6.15\\times10^{-2}$ & $3.87\\times10^{-14}$","The second experiment used the existing Komorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were $w_j=(1-\\mathrm{MAF}_j)^{24}$, where MAF denotes minor-allele frequency. The factor used column scales $\\sqrt{w_j}$. Its positive quadratic-form spectrum was\n\\[\n\\lambda=\\tfrac12\\operatorname{eig}(AA^{\\mathsf T}).\n\\]\nThree independent $n=400$, $m=1200$ genotype sets yielded numerical rank $397$. Four upper-tail queries per spectrum were generated by choosing saddlepoint locations at fractions $0.70,0.80,0.85,0.90$ of the transform singularity. The comparison reference used exact-spectrum tilted COS with $N=2048$ and a large support parameter $L=20$. The adaptive method was asked for a relative analytic tolerance of $10^{-8}$.","xisting Komorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were $w_j=(1-\\mathrm{MAF}_j)^{24}$, where MAF denotes minor-allele frequency. The factor used column scales $\\sqrt{w_j}$. Its positive quadratic-form spectrum was\n\\[\n\\lambda=\\tfrac12\\operatorname{eig}(AA^{\\mathsf T}).\n\\]\nThree independent $n=400$, $m=1200$ genotype sets yielded numerical rank $397$. Four upper-tail queries per spectrum were generated by choosing saddlepoint locations at fractions $0.70,0.80,0.85,0.90$ of the transform singularity. The comparison reference used exact-spectrum tilted COS with $N=2048$ and a large support parameter $L=20$. The adaptive method was asked for a relative analytic tolerance of $10^{-8}$.","omorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were $w_j=(1-\\mathrm{MAF}_j)^{24}$, where MAF denotes minor-allele frequency. The factor used column scales $\\sqrt{w_j}$. Its positive quadratic-form spectrum was\n\\[\n\\lambda=\\tfrac12\\operatorname{eig}(AA^{\\mathsf T}).\n\\]\nThree independent $n=400$, $m=1200$ genotype sets yielded numerical rank $397$. Four upper-tail queries per spectrum were generated by choosing saddlepoint locations at fractions $0.70,0.80,0.85,0.90$ of the transform singularity. The comparison reference used exact-spectrum tilted COS with $N=2048$ and a large support parameter $L=20$. The adaptive method was asked for a relative analytic tolerance of $10^{-8}$.","notype matrices were centered and scaled, then residualized against three covariates. The kernel weights were $w_j=(1-\\mathrm{MAF}_j)^{24}$, where MAF denotes minor-allele frequency. The factor used column scales $\\sqrt{w_j}$. Its positive quadratic-form spectrum was\n\\[\n\\lambda=\\tfrac12\\operatorname{eig}(AA^{\\mathsf T}).\n\\]\nThree independent $n=400$, $m=1200$ genotype sets yielded numerical rank $397$. Four upper-tail queries per spectrum were generated by choosing saddlepoint locations at fractions $0.70,0.80,0.85,0.90$ of the transform singularity. The comparison reference used exact-spectrum tilted COS with $N=2048$ and a large support parameter $L=20$. The adaptive method was asked for a relative analytic tolerance of $10^{-8}$.","ere $w_j=(1-\\mathrm{MAF}_j)^{24}$, where MAF denotes minor-allele frequency. The factor used column scales $\\sqrt{w_j}$. Its positive quadratic-form spectrum was\n\\[\n\\lambda=\\tfrac12\\operatorname{eig}(AA^{\\mathsf T}).\n\\]\nThree independent $n=400$, $m=1200$ genotype sets yielded numerical rank $397$. Four upper-tail queries per spectrum were generated by choosing saddlepoint locations at fractions $0.70,0.80,0.85,0.90$ of the transform singularity. The comparison reference used exact-spectrum tilted COS with $N=2048$ and a large support parameter $L=20$. The adaptive method was asked for a relative analytic tolerance of $10^{-8}$.","{w_j}$. Its positive quadratic-form spectrum was\n\\[\n\\lambda=\\tfrac12\\operatorname{eig}(AA^{\\mathsf T}).\n\\]\nThree independent $n=400$, $m=1200$ genotype sets yielded numerical rank $397$. Four upper-tail queries per spectrum were generated by choosing saddlepoint locations at fractions $0.70,0.80,0.85,0.90$ of the transform singularity. The comparison reference used exact-spectrum tilted COS with $N=2048$ and a large support parameter $L=20$. The adaptive method was asked for a relative analytic tolerance of $10^{-8}$.","ctrum was\n\\[\n\\lambda=\\tfrac12\\operatorname{eig}(AA^{\\mathsf T}).\n\\]\nThree independent $n=400$, $m=1200$ genotype sets yielded numerical rank $397$. Four upper-tail queries per spectrum were generated by choosing saddlepoint locations at fractions $0.70,0.80,0.85,0.90$ of the transform singularity. The comparison reference used exact-spectrum tilted COS with $N=2048$ and a large support parameter $L=20$. The adaptive method was asked for a relative analytic tolerance of $10^{-8}$.","ndependent $n=400$, $m=1200$ genotype sets yielded numerical rank $397$. Four upper-tail queries per spectrum were generated by choosing saddlepoint locations at fractions $0.70,0.80,0.85,0.90$ of the transform singularity. The comparison reference used exact-spectrum tilted COS with $N=2048$ and a large support parameter $L=20$. The adaptive method was asked for a relative analytic tolerance of $10^{-8}$.","Across the twelve queries:\n\\begin{itemize}\n\\item every requested $10^{-8}$ analytic certificate passed.\n\\item the worst observed relative error was $6.148608\\times10^{-11}$.\n\\item the largest returned analytic relative bound was $7.917203\\times10^{-9}$.\n\\item the planner selected $N\\in\\{64,128\\}$, $L\\in\\{10,11,12,13\\}$, and 11--16 packets.\n\\end{itemize}","At a reference tail probability near $8.99\\times10^{-15}$, the same engine was requested to meet three tolerances. Table \\ref{tab:ladder} shows how the representation refined automatically.","\\begin{table}[htbp]\n\\centering\n\\caption{Adaptive refinement at an extreme-tail query.}\n\\label{tab:ladder}\n\\begin{tabular}{rrrrrr}\n\\toprule\nRequested $\\tau$ & $N$ & $L$ & Packets & Analytic bound & Observed error","\\midrule\n$10^{-6}$  & 128 & 10 & 10 & $3.56\\times10^{-7}$  & $3.63\\times10^{-12}$","$10^{-8}$  & 128 & 12 & 15 & $4.87\\times10^{-9}$  & $9.14\\times10^{-13}$","$10^{-10}$ & 256 & 14 & 24 & $8.21\\times10^{-11}$ & $5.65\\times10^{-14}$","\\midrule\n397  & 13 & 6.66 ms & 10.45 ms & 0.549 ms & $12.1\\times$","797  & 16 & 13.32 ms & 11.49 ms & 0.604 ms & $22.0\\times$","1197 & 17 & 21.52 ms & 11.24 ms & 0.651 ms & $33.1\\times$","\\midrule\ncoarse 4 & 4 & \\(2.91\\times10^{-16}\\) & \\(5.99\\times10^{-16}\\)","fine 64 & 32 & \\(3.39\\times10^{-16}\\) & \\(6.00\\times10^{-16}\\)","irregular & 9 & \\(3.34\\times10^{-16}\\) & \\(5.87\\times10^{-16}\\)","singleton-front & 21 & \\(2.43\\times10^{-16}\\) & \\(5.82\\times10^{-16}\\)","permuted, block 73 & 29 & \\(3.52\\times10^{-16}\\) & \\(5.89\\times10^{-16}\\)","Hierarchical refinement gave relative discrepancy $3.17\\times10^{-16}$.\nFor $H=L^\\top L$ with block-bidiagonal $L$, the smallest eigenvalue was\napproximately $2.86$ and the largest norm outside the exact light cone\nwas zero. The QR-core singular-spectrum discrepancy was\n$5.89\\times10^{-16}$ in relative $\\ell_2$ norm. These are finite controls,\nnot universal floating-point error bounds.","The solver is organized around four independent decomposition axes:\n\\[\n\\boxed{\n\\mathcal D\n=\n\\mathcal D_{\\rm storage}\n\\circ\n\\mathcal D_{\\rm phase}\n\\circ\n\\mathcal D_{\\rm rank}\n\\circ\n\\mathcal D_{\\rm spectrum}.\n}\n\\]\nThis display describes the pipeline schematically, not four endomorphisms\nproved to commute pairwise.  The precise commutation theorem below is for\nexact storage refinement and a fixed structural phase partition.  Full range\ncapture or residual completion preserves the nonzero spectrum exactly. A\ntruncated packet representation instead preserves the requested tail within\nits explicit analytic error bound.","Fix a structural phase partition\n\\[\n\\{1,\\ldots,m\\}=\\mathcal P_1\\sqcup\\cdots\\sqcup\\mathcal P_H\n\\]\nand let each phase be subdivided into arbitrary storage blocks\n\\[\n\\mathcal P_h\n=\nJ_{h1}\\sqcup\\cdots\\sqcup J_{hB_h}.\n\\]\nLet $S$ denote this physical storage subdivision.  The phase projectors\n$P_h$ are defined by the structural phases, not by $S$.","\\begin{theorem}[Storage-phase commutation theorem]\\label{claim:final-29-1}\nLet\n\\[\nA=M_XGW^{1/2},\n\\qquad H=A^\\top A,\n\\]\nand let $S$ and $S'$ be any two exact storage refinements that respect the\nsame structural phase partition.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item the exact operator $H$ is identical under $S$ and $S'$.\n\\item every phase coupling $P_hHP_g$ is identical under $S$ and $S'$.\n\\item the exact phase graph and every $k$-step polyphase light cone are\nidentical under $S$ and $S'$.\n\\item any exact matrix-free Krylov method applied through streamed products\nproduces the same Krylov subspaces in exact arithmetic under $S$ and $S'$.\n\\end{enumerate}\n\\end{theorem}","n theorem]\\label{claim:final-29-1}\nLet\n\\[\nA=M_XGW^{1/2},\n\\qquad H=A^\\top A,\n\\]\nand let $S$ and $S'$ be any two exact storage refinements that respect the\nsame structural phase partition.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item the exact operator $H$ is identical under $S$ and $S'$.\n\\item every phase coupling $P_hHP_g$ is identical under $S$ and $S'$.\n\\item the exact phase graph and every $k$-step polyphase light cone are\nidentical under $S$ and $S'$.\n\\item any exact matrix-free Krylov method applied through streamed products\nproduces the same Krylov subspaces in exact arithmetic under $S$ and $S'$.\n\\end{enumerate}\n\\end{theorem}","{1/2},\n\\qquad H=A^\\top A,\n\\]\nand let $S$ and $S'$ be any two exact storage refinements that respect the\nsame structural phase partition.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item the exact operator $H$ is identical under $S$ and $S'$.\n\\item every phase coupling $P_hHP_g$ is identical under $S$ and $S'$.\n\\item the exact phase graph and every $k$-step polyphase light cone are\nidentical under $S$ and $S'$.\n\\item any exact matrix-free Krylov method applied through streamed products\nproduces the same Krylov subspaces in exact arithmetic under $S$ and $S'$.\n\\end{enumerate}\n\\end{theorem}","qquad H=A^\\top A,\n\\]\nand let $S$ and $S'$ be any two exact storage refinements that respect the\nsame structural phase partition.  Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item the exact operator $H$ is identical under $S$ and $S'$.\n\\item every phase coupling $P_hHP_g$ is identical under $S$ and $S'$.\n\\item the exact phase graph and every $k$-step polyphase light cone are\nidentical under $S$ and $S'$.\n\\item any exact matrix-free Krylov method applied through streamed products\nproduces the same Krylov subspaces in exact arithmetic under $S$ and $S'$.\n\\end{enumerate}\n\\end{theorem}","enumerate}[label=(\\roman*)]\n\\item the exact operator $H$ is identical under $S$ and $S'$.\n\\item every phase coupling $P_hHP_g$ is identical under $S$ and $S'$.\n\\item the exact phase graph and every $k$-step polyphase light cone are\nidentical under $S$ and $S'$.\n\\item any exact matrix-free Krylov method applied through streamed products\nproduces the same Krylov subspaces in exact arithmetic under $S$ and $S'$.\n\\end{enumerate}\n\\end{theorem}","e}[label=(\\roman*)]\n\\item the exact operator $H$ is identical under $S$ and $S'$.\n\\item every phase coupling $P_hHP_g$ is identical under $S$ and $S'$.\n\\item the exact phase graph and every $k$-step polyphase light cone are\nidentical under $S$ and $S'$.\n\\item any exact matrix-free Krylov method applied through streamed products\nproduces the same Krylov subspaces in exact arithmetic under $S$ and $S'$.\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nStorage refinement changes only the order and grouping of finite sums.\nBy partition refinement invariance, the streamed products reconstruct the\nsame $A$, $A^\\top$, and therefore the same $H=A^\\top A$.  Since each\n$P_h$ is fixed by the structural phase partition, every block\n$P_hHP_g$ is unchanged.  The phase graph is defined only by these blocks, so\nits path structure and light cones are unchanged.  Finally, an exact\nmatrix-free Krylov method depends only on repeated applications of the same\noperator to the same starting block. Hence its generated subspaces are\nidentical.\n\\end{proof}","\\begin{theorem}[Exact frontier-to-I/O reduction]\\label{claim:final-29-3}\nSuppose a validated exact phase graph of the fixed operator $H=A^\\top A$\nis already supplied and the initial Krylov block is supported on a phase set\n$F_0$.  Keep covariate projections, weights, and structural phase metadata\nfixed, and assume all factor-action blocks required for the retained phase\noperator are available.  At depth $k$, let $F_k$ be the graph\n$k$-neighborhood of $F_0$.  Then no storage block lying wholly in a phase\noutside $F_k$ can affect the $k$-step Krylov vector.  Such a block need not be\nloaded when evaluating the requested power through the restricted factor\n$A_{F_k}$.  Storage blocks mixing active and inactive phases require a\nseparate refinement before this omission is valid.\n\\end{theorem}","heorem}[Exact frontier-to-I/O reduction]\\label{claim:final-29-3}\nSuppose a validated exact phase graph of the fixed operator $H=A^\\top A$\nis already supplied and the initial Krylov block is supported on a phase set\n$F_0$.  Keep covariate projections, weights, and structural phase metadata\nfixed, and assume all factor-action blocks required for the retained phase\noperator are available.  At depth $k$, let $F_k$ be the graph\n$k$-neighborhood of $F_0$.  Then no storage block lying wholly in a phase\noutside $F_k$ can affect the $k$-step Krylov vector.  Such a block need not be\nloaded when evaluating the requested power through the restricted factor\n$A_{F_k}$.  Storage blocks mixing active and inactive phases require a\nseparate refinement before this omission is valid.\n\\end{theorem}","xact frontier-to-I/O reduction]\\label{claim:final-29-3}\nSuppose a validated exact phase graph of the fixed operator $H=A^\\top A$\nis already supplied and the initial Krylov block is supported on a phase set\n$F_0$.  Keep covariate projections, weights, and structural phase metadata\nfixed, and assume all factor-action blocks required for the retained phase\noperator are available.  At depth $k$, let $F_k$ be the graph\n$k$-neighborhood of $F_0$.  Then no storage block lying wholly in a phase\noutside $F_k$ can affect the $k$-step Krylov vector.  Such a block need not be\nloaded when evaluating the requested power through the restricted factor\n$A_{F_k}$.  Storage blocks mixing active and inactive phases require a\nseparate refinement before this omission is valid.\n\\end{theorem}","O reduction]\\label{claim:final-29-3}\nSuppose a validated exact phase graph of the fixed operator $H=A^\\top A$\nis already supplied and the initial Krylov block is supported on a phase set\n$F_0$.  Keep covariate projections, weights, and structural phase metadata\nfixed, and assume all factor-action blocks required for the retained phase\noperator are available.  At depth $k$, let $F_k$ be the graph\n$k$-neighborhood of $F_0$.  Then no storage block lying wholly in a phase\noutside $F_k$ can affect the $k$-step Krylov vector.  Such a block need not be\nloaded when evaluating the requested power through the restricted factor\n$A_{F_k}$.  Storage blocks mixing active and inactive phases require a\nseparate refinement before this omission is valid.\n\\end{theorem}","laim:final-29-3}\nSuppose a validated exact phase graph of the fixed operator $H=A^\\top A$\nis already supplied and the initial Krylov block is supported on a phase set\n$F_0$.  Keep covariate projections, weights, and structural phase metadata\nfixed, and assume all factor-action blocks required for the retained phase\noperator are available.  At depth $k$, let $F_k$ be the graph\n$k$-neighborhood of $F_0$.  Then no storage block lying wholly in a phase\noutside $F_k$ can affect the $k$-step Krylov vector.  Such a block need not be\nloaded when evaluating the requested power through the restricted factor\n$A_{F_k}$.  Storage blocks mixing active and inactive phases require a\nseparate refinement before this omission is valid.\n\\end{theorem}","fixed operator $H=A^\\top A$\nis already supplied and the initial Krylov block is supported on a phase set\n$F_0$.  Keep covariate projections, weights, and structural phase metadata\nfixed, and assume all factor-action blocks required for the retained phase\noperator are available.  At depth $k$, let $F_k$ be the graph\n$k$-neighborhood of $F_0$.  Then no storage block lying wholly in a phase\noutside $F_k$ can affect the $k$-step Krylov vector.  Such a block need not be\nloaded when evaluating the requested power through the restricted factor\n$A_{F_k}$.  Storage blocks mixing active and inactive phases require a\nseparate refinement before this omission is valid.\n\\end{theorem}","top A$\nis already supplied and the initial Krylov block is supported on a phase set\n$F_0$.  Keep covariate projections, weights, and structural phase metadata\nfixed, and assume all factor-action blocks required for the retained phase\noperator are available.  At depth $k$, let $F_k$ be the graph\n$k$-neighborhood of $F_0$.  Then no storage block lying wholly in a phase\noutside $F_k$ can affect the $k$-step Krylov vector.  Such a block need not be\nloaded when evaluating the requested power through the restricted factor\n$A_{F_k}$.  Storage blocks mixing active and inactive phases require a\nseparate refinement before this omission is valid.\n\\end{theorem}","ghts, and structural phase metadata\nfixed, and assume all factor-action blocks required for the retained phase\noperator are available.  At depth $k$, let $F_k$ be the graph\n$k$-neighborhood of $F_0$.  Then no storage block lying wholly in a phase\noutside $F_k$ can affect the $k$-step Krylov vector.  Such a block need not be\nloaded when evaluating the requested power through the restricted factor\n$A_{F_k}$.  Storage blocks mixing active and inactive phases require a\nseparate refinement before this omission is valid.\n\\end{theorem}","\\begin{proof}\nFor every phase $h\\notin F_k$ and every starting phase $g\\in F_0$, the\npolyphase light-cone theorem gives $P_hH^kP_g=0$.  Storage-phase commutation\nallows each active phase operator to be evaluated through arbitrary exact\nstorage blocks.  Every path of length at most $k$ beginning in $F_0$ stays in $F_k$.\nLet $J_F$ be the coordinate inclusion from the retained coordinates\n$F=F_k$ into the full variant space. Thus $P_F=J_FJ_F^\\top$ and\n$A_F=AJ_F$. The compressed operator and its ambient embedding satisfy\n\\[\nH_F=J_F^\\top HJ_F=A_F^\\top A_F,\n\\qquad P_FHP_F=J_FH_FJ_F^\\top.\n\\]\nThese identities distinguish the compressed matrix from the larger ambient\nmatrix. Since every path of length at most $k$ from $F_0$ stays in $F$,\n\\[","\\begin{proof}\nFor every phase $h\\notin F_k$ and every starting phase $g\\in F_0$, the\npolyphase light-cone theorem gives $P_hH^kP_g=0$.  Storage-phase commutation\nallows each active phase operator to be evaluated through arbitrary exact\nstorage blocks.  Every path of length at most $k$ beginning in $F_0$ stays in $F_k$.\nLet $J_F$ be the coordinate inclusion from the retained coordinates\n$F=F_k$ into the full variant space. Thus $P_F=J_FJ_F^\\top$ and\n$A_F=AJ_F$. The compressed operator and its ambient embedding satisfy\n\\[\nH_F=J_F^\\top HJ_F=A_F^\\top A_F,\n\\qquad P_FHP_F=J_FH_FJ_F^\\top.\n\\]\nThese identities distinguish the compressed matrix from the larger ambient\nmatrix. Since every path of length at most $k$ from $F_0$ stays in $F$,\n\\[\nH^kX=J_F H_F^k J_F^\\top X.\n\\]\nThis i","\\begin{proof}\nFor every phase $h\\notin F_k$ and every starting phase $g\\in F_0$, the\npolyphase light-cone theorem gives $P_hH^kP_g=0$.  Storage-phase commutation\nallows each active phase operator to be evaluated through arbitrary exact\nstorage blocks.  Every path of length at most $k$ beginning in $F_0$ stays in $F_k$.\nLet $J_F$ be the coordinate inclusion from the retained coordinates\n$F=F_k$ into the full variant space. Thus $P_F=J_FJ_F^\\top$ and\n$A_F=AJ_F$. The compressed operator and its ambient embedding satisfy\n\\[\nH_F=J_F^\\top HJ_F=A_F^\\top A_F,\n\\qquad P_FHP_F=J_FH_FJ_F^\\top.\n\\]\nThese identities distinguish the compressed matrix from the larger ambient\nmatrix. Since every path of length at most $k$ from $F_0$ stays in $F$,\n\\[\nH^kX=J_F H_F^k J_F^\\top X.\n\\]\nThis includes the zero output outside $F$. Blocks wholly out","\\begin{proof}\nFor every phase $h\\notin F_k$ and every starting phase $g\\in F_0$, the\npolyphase light-cone theorem gives $P_hH^kP_g=0$.  Storage-phase commutation\nallows each active phase operator to be evaluated through arbitrary exact\nstorage blocks.  Every path of length at most $k$ beginning in $F_0$ stays in $F_k$.\nLet $J_F$ be the coordinate inclusion from the retained coordinates\n$F=F_k$ into the full variant space. Thus $P_F=J_FJ_F^\\top$ and\n$A_F=AJ_F$. The compressed operator and its ambient embedding satisfy\n\\[\nH_F=J_F^\\top HJ_F=A_F^\\top A_F,\n\\qquad P_FHP_F=J_FH_FJ_F^\\top.\n\\]\nThese identities distinguish the compressed matrix from the larger ambient\nmatrix. Since every path of length at most $k$ from $F_0$ stays in $F$,\n\\[\nH^kX=J_F H_F^k J_F^\\top X.\n\\]\nThis includes the zero output outside $F$. Blocks wholly outside the retained\nphase set may therefore be omitted from the restricted factor evaluation.\n\\end{proof}","\\notin F_k$ and every starting phase $g\\in F_0$, the\npolyphase light-cone theorem gives $P_hH^kP_g=0$.  Storage-phase commutation\nallows each active phase operator to be evaluated through arbitrary exact\nstorage blocks.  Every path of length at most $k$ beginning in $F_0$ stays in $F_k$.\nLet $J_F$ be the coordinate inclusion from the retained coordinates\n$F=F_k$ into the full variant space. Thus $P_F=J_FJ_F^\\top$ and\n$A_F=AJ_F$. The compressed operator and its ambient embedding satisfy\n\\[\nH_F=J_F^\\top HJ_F=A_F^\\top A_F,\n\\qquad P_FHP_F=J_FH_FJ_F^\\top.\n\\]\nThese identities distinguish the compressed matrix from the larger ambient\nmatrix. Since every path of length at most $k$ from $F_0$ stays in $F$,\n\\[\nH^kX=J_F H_F^k J_F^\\top X.\n\\]\nThis includes the zero output outside $F$. Blocks wholly outside the retained\nphase set may therefore be omitted from the restricted factor evaluation.\n\\end{proof}","starting phase $g\\in F_0$, the\npolyphase light-cone theorem gives $P_hH^kP_g=0$.  Storage-phase commutation\nallows each active phase operator to be evaluated through arbitrary exact\nstorage blocks.  Every path of length at most $k$ beginning in $F_0$ stays in $F_k$.\nLet $J_F$ be the coordinate inclusion from the retained coordinates\n$F=F_k$ into the full variant space. Thus $P_F=J_FJ_F^\\top$ and\n$A_F=AJ_F$. The compressed operator and its ambient embedding satisfy\n\\[\nH_F=J_F^\\top HJ_F=A_F^\\top A_F,\n\\qquad P_FHP_F=J_FH_FJ_F^\\top.\n\\]\nThese identities distinguish the compressed matrix from the larger ambient\nmatrix. Since every path of length at most $k$ from $F_0$ stays in $F$,\n\\[\nH^kX=J_F H_F^k J_F^\\top X.\n\\]\nThis includes the zero output outside $F$. Blocks wholly outside the retained\nphase set may therefore be omitted from the restricted factor evaluation.\n\\end{proof}","^kP_g=0$.  Storage-phase commutation\nallows each active phase operator to be evaluated through arbitrary exact\nstorage blocks.  Every path of length at most $k$ beginning in $F_0$ stays in $F_k$.\nLet $J_F$ be the coordinate inclusion from the retained coordinates\n$F=F_k$ into the full variant space. Thus $P_F=J_FJ_F^\\top$ and\n$A_F=AJ_F$. The compressed operator and its ambient embedding satisfy\n\\[\nH_F=J_F^\\top HJ_F=A_F^\\top A_F,\n\\qquad P_FHP_F=J_FH_FJ_F^\\top.\n\\]\nThese identities distinguish the compressed matrix from the larger ambient\nmatrix. Since every path of length at most $k$ from $F_0$ stays in $F$,\n\\[\nH^kX=J_F H_F^k J_F^\\top X.\n\\]\nThis includes the zero output outside $F$. Blocks wholly outside the retained\nphase set may therefore be omitted from the restricted factor evaluation.\n\\end{proof}","in $F_k$.\nLet $J_F$ be the coordinate inclusion from the retained coordinates\n$F=F_k$ into the full variant space. Thus $P_F=J_FJ_F^\\top$ and\n$A_F=AJ_F$. The compressed operator and its ambient embedding satisfy\n\\[\nH_F=J_F^\\top HJ_F=A_F^\\top A_F,\n\\qquad P_FHP_F=J_FH_FJ_F^\\top.\n\\]\nThese identities distinguish the compressed matrix from the larger ambient\nmatrix. Since every path of length at most $k$ from $F_0$ stays in $F$,\n\\[\nH^kX=J_F H_F^k J_F^\\top X.\n\\]\nThis includes the zero output outside $F$. Blocks wholly outside the retained\nphase set may therefore be omitted from the restricted factor evaluation.\n\\end{proof}","Let $J_F$ be the coordinate inclusion from the retained coordinates\n$F=F_k$ into the full variant space. Thus $P_F=J_FJ_F^\\top$ and\n$A_F=AJ_F$. The compressed operator and its ambient embedding satisfy\n\\[\nH_F=J_F^\\top HJ_F=A_F^\\top A_F,\n\\qquad P_FHP_F=J_FH_FJ_F^\\top.\n\\]\nThese identities distinguish the compressed matrix from the larger ambient\nmatrix. Since every path of length at most $k$ from $F_0$ stays in $F$,\n\\[\nH^kX=J_F H_F^k J_F^\\top X.\n\\]\nThis includes the zero output outside $F$. Blocks wholly outside the retained\nphase set may therefore be omitted from the restricted factor evaluation.\n\\end{proof}","he coordinate inclusion from the retained coordinates\n$F=F_k$ into the full variant space. Thus $P_F=J_FJ_F^\\top$ and\n$A_F=AJ_F$. The compressed operator and its ambient embedding satisfy\n\\[\nH_F=J_F^\\top HJ_F=A_F^\\top A_F,\n\\qquad P_FHP_F=J_FH_FJ_F^\\top.\n\\]\nThese identities distinguish the compressed matrix from the larger ambient\nmatrix. Since every path of length at most $k$ from $F_0$ stays in $F$,\n\\[\nH^kX=J_F H_F^k J_F^\\top X.\n\\]\nThis includes the zero output outside $F$. Blocks wholly outside the retained\nphase set may therefore be omitted from the restricted factor evaluation.\n\\end{proof}","rdinate inclusion from the retained coordinates\n$F=F_k$ into the full variant space. Thus $P_F=J_FJ_F^\\top$ and\n$A_F=AJ_F$. The compressed operator and its ambient embedding satisfy\n\\[\nH_F=J_F^\\top HJ_F=A_F^\\top A_F,\n\\qquad P_FHP_F=J_FH_FJ_F^\\top.\n\\]\nThese identities distinguish the compressed matrix from the larger ambient\nmatrix. Since every path of length at most $k$ from $F_0$ stays in $F$,\n\\[\nH^kX=J_F H_F^k J_F^\\top X.\n\\]\nThis includes the zero output outside $F$. Blocks wholly outside the retained\nphase set may therefore be omitted from the restricted factor evaluation.\n\\end{proof}","variant space. Thus $P_F=J_FJ_F^\\top$ and\n$A_F=AJ_F$. The compressed operator and its ambient embedding satisfy\n\\[\nH_F=J_F^\\top HJ_F=A_F^\\top A_F,\n\\qquad P_FHP_F=J_FH_FJ_F^\\top.\n\\]\nThese identities distinguish the compressed matrix from the larger ambient\nmatrix. Since every path of length at most $k$ from $F_0$ stays in $F$,\n\\[\nH^kX=J_F H_F^k J_F^\\top X.\n\\]\nThis includes the zero output outside $F$. Blocks wholly outside the retained\nphase set may therefore be omitted from the restricted factor evaluation.\n\\end{proof}","The theorem turns an algebraic support statement into an execution law:\n\\[\n\\boxed{\n\\text{Krylov depth}\n\\longrightarrow\n\\text{reachable phases}\n\\longrightarrow\n\\text{storage blocks that must be transferred}.\n}\n\\]\nDiscovery or verification of the phase graph may itself require all inputs.\nThe theorem does not claim minimum total I/O, omission of mixed-phase blocks,\nor correctness for an unverified approximate graph.  The scheduler is constrained\nby the exact operator graph rather than solely by available memory.","The representation pipeline may therefore be written as\n\\[\n\\boxed{\\begin{aligned}\nA\n&\\longrightarrow \\text{resident storage blocks}\\\\\n&\\longrightarrow \\text{structural phase frontier}\\\\\n&\\longrightarrow \\text{phase-local Krylov/range core}\\\\\n&\\longrightarrow \\text{residual moment certificate or exact completion}\\\\\n&\\longrightarrow \\text{spectrum or verified spectral enclosure}.\n\\end{aligned}}\n\\]","The resolved synthesis is\n\\[\n\\boxed{\\begin{aligned}\n&\\text{exact streamed factor actions}\n+\\text{ structural light cones}\n+\\text{ certified pruning}\\\\\n&+\\text{ phase-local rank}\n+\\text{ residual moment certificates}\n+\\text{ exact residual completion}\\\\\n&+\\text{ spectral packets}\n+\\text{ log-domain extreme-tail evaluation}.\n\\end{aligned}}\n\\]","Positive exact-spectrum tail enclosure &\ndirected rounding + proved remainders &\nFor rank \\(>2\\), the finite transform is enclosed numerically and enlarged by\nproved support/alias and infinite-mode bounds.","Accuracy-driven positive tail solver &\ncertified / fail-closed &\nA result is accepted only after\n\\(p_+/p_- - 1\\le\\varepsilon_{\\rm requested}\\) is proved.","The deepest retained exact control reaches\n\\[\np\\approx10^{-5000}.\n\\]\nAt that point the recorded accuracy-driven run reported relative multiplicative\nuncertainty below \\(10^{-18}\\).  This establishes that extreme-tail depth by\nitself is not the limiting numerical variable. The meaningful limit is whether\nthe complete error ledger can be made smaller than the requested tolerance.","\\[X=\\Sigma^{1/2}Z,\\quad Z\\sim N(0,I),\\quad H=\\Sigma^{1/2}M\\Sigma^{1/2}.\\]\n\\[X^{\\mathsf T}MX\\overset d=\\sum_{j=1}^{r}\\lambda_jZ_j^2,\\qquad \\lambda_j\\in\\operatorname{spec}(H)\\setminus\\{0\\}.\\]\nFor a central Gaussian vector with covariance $\\Sigma\\succeq0$ and symmetric $M$, the quadratic form becomes a weighted sum of independent standard-normal squares after whitening and orthogonal diagonalization. The operator may originate in a kernel, a signal model, or another symmetric construction. Once its spectrum has been correctly obtained, the tail layer sees those coefficients.","\\[A=[A_1\\ \\cdots\\ A_B],\\qquad AA^{\\mathsf T}=\\sum_{b=1}^B A_bA_b^{\\mathsf T}.\\]\n\\[KV=\\sum_b A_b(A_b^{\\mathsf T}V),\\qquad A_b\\in\\mathbb R^{n\\times c_b},\\quad V\\in\\mathbb R^{n\\times s}.\\]\nPartition a factor by columns and keep every block. Horizontal concatenation turns the sample-space Gram matrix into a sum of block Gram matrices. Changing the block widths or the order in which blocks are read preserves the represented operator.","The figure splits the declared $6\\times9$ integer factor into three blocks of width three. Every matrix entry in the displayed identity is an integer, and the regeneration script checks equality exactly. Block storage introduces no statistical approximation.","footnotesize\\color{navy}\\textbf{Reading connection.} Chapter~\\ref{ch:streaming}.}\n\\clearpage\n\\atlasheading{03}{Algebraic sizing}{Memory depends on the resident width}\n\\vspace{3pt}\n\\begin{center}\\includegraphics[width=\\linewidth]{figures_production/03_memory.pdf}\\end{center}\n{\\small\\color{navy}An explicitly counted binary64 payload with fixed resident width. Total-factor storage grows linearly in $m$.}","\\[\\text{working storage}=O(nc+ns+cs+np).\\]\n\\[\\text{illustrated payload}=8\\,(nc+2ns+cs+np)\\ \\text{bytes}.\\]\nA full factor holds $nm$ scalar entries. A streamed action with resident width $c$ and block-vector width $s$ keeps a factor block and a small number of action buffers. Residualization against a covariate basis of rank $p$ adds $np$ entries. At fixed $c,s,p$, this algebraic working set does not grow with total column count $m$.","cs+np).\\]\n\\[\\text{illustrated payload}=8\\,(nc+2ns+cs+np)\\ \\text{bytes}.\\]\nA full factor holds $nm$ scalar entries. A streamed action with resident width $c$ and block-vector width $s$ keeps a factor block and a small number of action buffers. Residualization against a covariate basis of rank $p$ adds $np$ entries. At fixed $c,s,p$, this algebraic working set does not grow with total column count $m$.","The sizing curve is an illustration, not a measured memory profile. It uses $n=2000$, $c=512$, $s=32$, $p=3$, and binary64 entries. The plotted streamed payload counts one resident factor, two sample buffers, one block intermediate, and the covariate basis.","\\[Q_X^{\\mathsf T}Q_X=I,\\quad M_X=I-Q_XQ_X^{\\mathsf T},\\quad M_X^2=M_X=M_X^{\\mathsf T}.\\]\n\\[A=M_XGW^{1/2},\\qquad K=AA^{\\mathsf T}.\\]\nFor a covariate range basis $Q_X$ with orthonormal columns, the residualizer $M_X=I-Q_XQ_X^{\\mathsf T}$ removes the component in the covariate span. It is symmetric and idempotent. Applying it to the weighted feature factor restricts the kernel to variation left after the null-model covariates have been removed.","The two-coordinate control draws the exact split $(4,3)=(4,0)+(0,3)$. The horizontal axis represents the covariate span and the vertical axis its orthogonal complement. This picture explains the geometry. The general projector acts in the full sample space.","The factor $A=M_XGW^{1/2}$ is one specialization of the domain-independent engine. A correct range basis must represent the actual covariate span. Rank deficiency cannot be handled by silently treating extra QR columns as covariate directions.","\\[\\sum_hP_h=I,\\qquad g\\to h\\iff P_hHP_g\\ne0.\\]\n\\[\\operatorname{dist}(g,h)>k\\quad\\Longrightarrow\\quad P_hH^kP_g=0.\\]\nChoose a complete orthogonal family of phase projectors on feature space and form $H=A^{\\mathsf T}A$. An edge from phase $g$ to phase $h$ exists when the transfer block $P_hHP_g$ is nonzero. Each operator application follows one graph edge. An iterate of depth $k$ cannot reach a phase farther than $k$ edges away.","quad g\\to h\\iff P_hHP_g\\ne0.\\]\n\\[\\operatorname{dist}(g,h)>k\\quad\\Longrightarrow\\quad P_hH^kP_g=0.\\]\nChoose a complete orthogonal family of phase projectors on feature space and form $H=A^{\\mathsf T}A$. An edge from phase $g$ to phase $h$ exists when the transfer block $P_hHP_g$ is nonzero. Each operator application follows one graph edge. An iterate of depth $k$ cannot reach a phase farther than $k$ edges away.","The finite control is a nine-phase chain with $H=2I+S+S^{\\mathsf T}$ and a unit start in phase five. The printed entries are exact integers in $H^kx$, and the shaded support expands by one phase at each step. The script checks every zero outside the cone.","\\[\\rho(X)_h=\\|P_hX\\|_F,\\qquad \\mathcal B_{hg}=\\|P_hHP_g\\|_2.\\]\n\\[\\rho(H^kX)\\le\\mathcal B^k\\rho(X),\\qquad\\rho((H^k-H_0^k)X)\\le(\\mathcal B^k-\\mathcal B_0^k)\\rho(X).\\]\nA weak nonzero coupling destroys the exact-zero conclusion. Record the spectral norm of every phase block in a nonnegative transfer matrix $\\mathcal B$. The phase-norm profile of any block vector is dominated componentwise by multiplication with $\\mathcal B$.","\\|P_hX\\|_F,\\qquad \\mathcal B_{hg}=\\|P_hHP_g\\|_2.\\]\n\\[\\rho(H^kX)\\le\\mathcal B^k\\rho(X),\\qquad\\rho((H^k-H_0^k)X)\\le(\\mathcal B^k-\\mathcal B_0^k)\\rho(X).\\]\nA weak nonzero coupling destroys the exact-zero conclusion. Record the spectral norm of every phase block in a nonnegative transfer matrix $\\mathcal B$. The phase-norm profile of any block vector is dominated componentwise by multiplication with $\\mathcal B$.","For a retained operator $H_0$, let $\\mathcal B_0$ keep the corresponding entries. Telescoping the two operator powers gives a phasewise bound on the omitted action. Symmetric block retention preserves self-adjointness. The power estimate itself does not require the retained operator to be positive.","The toy chain receives a signed weak edge of weight $-0.035$ between phases one and nine. Starting in phase five, that edge has no effect through depth four. At later depths the figure compares floating observed error with the transfer-envelope evaluation. The theorem supplies the general bound. The plot checks this finite instance.","\\[A=UB+F,\\quad U^{\\mathsf T}F=0,\\quad A^{\\mathsf T}A=B^{\\mathsf T}B+F^{\\mathsf T}F.\\]\n\\[t=\\operatorname{tr}R=\\|F\\|_F^2,\\quad v=\\operatorname{tr}(R^2),\\quad \\rho_R=\\|R\\|_2.\\]\nLet $U$ contain $d$ orthonormal sample-space directions. Project the factor into its captured range coordinates $B=U^{\\mathsf T}A$ and residual $F=(I-UU^{\\mathsf T})A$. Orthogonality eliminates the cross terms in the feature-space Gram matrix.","The missing term is the positive operator $R=F^{\\mathsf T}F$. Its trace is the missing mean of the Gaussian quadratic form. Its square trace measures concentration, and its spectral norm caps any unresolved coefficient. These invariants convey more information than one generic operator-norm error.","\\[U^{\\mathsf T}W=0,\\qquad A=[U\\ W]\\begin{bmatrix}B\\\\D\\end{bmatrix}.\\]\n\\[\\left\\lceil\\max\\{t/\\rho_R,t^2/v\\}\\right\\rceil\\le\\operatorname{rank}F\\le n-d\\quad(F\\ne0).\\]\nIf the residual admits $F=WD$ with orthonormal columns of $W$, its range lies orthogonal to $U$. Appending $W$ therefore creates an orthonormal synthesis factor, and stacking the two coordinate cores recovers the full factor.","D\\end{bmatrix}.\\]\n\\[\\left\\lceil\\max\\{t/\\rho_R,t^2/v\\}\\right\\rceil\\le\\operatorname{rank}F\\le n-d\\quad(F\\ne0).\\]\nIf the residual admits $F=WD$ with orthonormal columns of $W$, its range lies orthogonal to $U$. Appending $W$ therefore creates an orthonormal synthesis factor, and stacking the two coordinate cores recovers the full factor.","\\[\\kappa(s)=-\\frac12\\sum_j\\log(1-2s\\lambda_j),\\qquad 0<\\theta<\\frac1{2\\lambda_{\\max}}.\\]\n\\[P(Q\\ge q)=e^{\\kappa(\\theta)-\\theta q}\\,\\mathbb E_\\theta[e^{-\\theta(Q-q)}\\mathbf1_{\\{Q\\ge q\\}}].\\]\nFor a positive spectrum, the cumulant generating function is a sum of logarithms. A real tilt inside its domain defines a normalized reweighted law. The original upper tail becomes an exponential prefactor times a bounded weighted expectation under that law.","For an upper-tail planner query $q>\\mathbb E Q$, the saddlepoint solves $\\kappa'(\\theta)=q$. It centers the tilted law at the threshold and often improves numerical efficiency. The enclosure identity holds for any admissible positive tilt. Certification does not require solving the saddlepoint exactly.","The figure uses $Q=0.02\\chi^2_{80}$ and $q=4$. Its original mean is $1.6$, while the tilted mean is $4$ at $\\theta=15$. The plotted densities follow the same exact gamma family with different scales. No approximation to the probability law is introduced by tilting.","\\[M_{P,m}=\\sum_{j\\in P}(\\lambda_j-\\mu_P)^m,\\quad\\alpha_P(s)=\\frac{2s}{1-2s\\mu_P},\\quad\\rho_P(s)=|\\alpha_P(s)|\\max_{j\\in P}|\\lambda_j-\\mu_P|.\\]\n\\[\\kappa_P(s)=-\\frac{n_P}{2}\\log(1-2s\\mu_P)+\\frac12\\sum_{m=1}^{p}\\frac{\\alpha_P(s)^m M_{P,m}}m+R_{P,p}(s),\\]\n\\[|R_{P,p}(s)|\\le\\frac{n_P}{2}\\frac{\\rho_P(s)^{p+1}}{(p+1)(1-\\rho_P(s))}.\\]\nThe packet stores nearby eigenvalues through their mean $\\mu_P$ and centered power sums $M_{P,m}$, with $M_{P,1}=0$. It expands the log transform rather than substituting a surrogate distribution.","\\in P}(\\lambda_j-\\mu_P)^m,\\quad\\alpha_P(s)=\\frac{2s}{1-2s\\mu_P},\\quad\\rho_P(s)=|\\alpha_P(s)|\\max_{j\\in P}|\\lambda_j-\\mu_P|.\\]\n\\[\\kappa_P(s)=-\\frac{n_P}{2}\\log(1-2s\\mu_P)+\\frac12\\sum_{m=1}^{p}\\frac{\\alpha_P(s)^m M_{P,m}}m+R_{P,p}(s),\\]\n\\[|R_{P,p}(s)|\\le\\frac{n_P}{2}\\frac{\\rho_P(s)^{p+1}}{(p+1)(1-\\rho_P(s))}.\\]\nThe packet stores nearby eigenvalues through their mean $\\mu_P$ and centered power sums $M_{P,m}$, with $M_{P,1}=0$. It expands the log transform rather than substituting a surrogate distribution.","-\\mu_P)^m,\\quad\\alpha_P(s)=\\frac{2s}{1-2s\\mu_P},\\quad\\rho_P(s)=|\\alpha_P(s)|\\max_{j\\in P}|\\lambda_j-\\mu_P|.\\]\n\\[\\kappa_P(s)=-\\frac{n_P}{2}\\log(1-2s\\mu_P)+\\frac12\\sum_{m=1}^{p}\\frac{\\alpha_P(s)^m M_{P,m}}m+R_{P,p}(s),\\]\n\\[|R_{P,p}(s)|\\le\\frac{n_P}{2}\\frac{\\rho_P(s)^{p+1}}{(p+1)(1-\\rho_P(s))}.\\]\nThe packet stores nearby eigenvalues through their mean $\\mu_P$ and centered power sums $M_{P,m}$, with $M_{P,1}=0$. It expands the log transform rather than substituting a surrogate distribution.","Use a nonempty positive packet, integer order $p\\ge0$, and $\\Re s<1/(2\\lambda_{\\max})$. The factor logarithms use their principal branches and sum to the analytic $\\kappa$ normalized at zero. For $\\rho_P(s)<1$, the series has the stated remainder, which decreases geometrically when the radius ratio stays away from one.","The five-eigenvalue control uses $0.09,0.095,0.10,0.105,0.11$, a tilt of one, and 220 frequencies from zero to eighteen. The figure compares floating transform errors with the analytic remainder at orders one through eight. These computations illustrate the bound. They do not enclose library rounding.","\\[s_k=\\theta+i\\frac{k\\pi}{b},\\qquad\\rho_P(s_k)<1\\quad\\text{on every retained frequency}.\\]\n\\[(\\Lambda,q,\\varepsilon)\\longmapsto\\mathcal P_{\\Lambda,q,\\varepsilon}.\\]\nA distance between eigenvalues is insufficient to decide whether a packet is useful. The expansion must be controlled at the actual complex arguments $s_k=\\theta+i k\\pi/b$ selected by the tail query. The tilt, support endpoint, and retained mode count all change that grid.","\\[J_k=\\int_q^be^{-\\theta(x-q)}\\cos(u_kx)\\,dx=\\Re\\!\\left[e^{iu_kq}\\frac{1-e^{(-\\theta+iu_k)(b-q)}}{\\theta-iu_k}\\right].\\]\n\\[S_N=\\frac2b\\sum_{k=0}^{N-1}{}^{\\prime}\\, J_k\\,\\Re\\,\\phi_\\theta(u_k),\\qquad\\phi_\\theta(t)=e^{\\kappa(\\theta+it)-\\kappa(\\theta)}.\\]\nOn a positive-support interval $[0,b]$, the cosine frequencies are $u_k=k\\pi/b$. The payoff is zero below the threshold and decays exponentially above it. Integrating each cosine mode against that payoff gives an explicit coefficient.","The picture uses $q=4$, $b=7$, and $\\theta=15$. The payoff and three cosine modes share one coordinate system, with distinct line styles and an explicit payoff discontinuity. Correctness of the positive-support backend requires $q\\ge0$ and $b>q$. The saddlepoint planner additionally selects upper-tail queries above the original mean.","\\[a_j=\\frac{2\\lambda_j}{1-2\\theta\\lambda_j},\\quad|\\phi_\\theta(t)|=\\prod_j(1+(a_jt)^2)^{-1/4}\\le C_\\theta t^{-r/2}.\\]\n\\[E_{\\rm series}\\le\\frac{2C_\\theta}{b\\theta}\\left(\\frac b\\pi\\right)^\\nu\\left[N^{-\\nu}+\\frac{N^{1-\\nu}}{\\nu-1}\\right],\\quad\\nu=r/2>1.\\]\nThe modulus of the normalized tilted characteristic function has a positive product formula. Its decay is controlled by the actual transformed coefficients. More active positive coefficients increase the asymptotic decay exponent.","The rigorous interval backend uses a simple envelope $C_\\theta t^{-r/2}$ and the payoff bound $|J_k|\\le1/\\theta$. Its resulting series bound requires $r>2$. The packetized analytic path has a sharper frequency-dependent decay-rate bound. Those are distinct formulas with distinct implementation scopes.","\\[|p-\\widehat p|\\le E_{\\rm packet}+E_{\\rm support}+E_{\\rm modes}\\quad\\text{(analytic packet path)}.\\]\n\\[S_\\theta(q)\\in\\boldsymbol S_N+[-E,E],\\qquad E=E_{\\rm support}+E_{\\rm series}\\quad\\text{(exact-spectrum interval path)}.\\]\nPacket approximation changes the retained-grid transform. Support truncation and cosine periodization change the finite-window functional. Mode truncation removes infinitely many terms. These three mathematical approximation errors have different refinement controls and must be accounted for separately.","scalar tolerance can hide different unresolved causes.}\n\\vfill{\\footnotesize\\color{navy}\\textbf{Reading connection.} Chapter~\\ref{ch:inversion}.}\n\\clearpage\n\\atlasheading{15}{Requested accuracy}{Acceptance is a logarithmic interval test}\n\\vspace{3pt}\n\\begin{center}\\includegraphics[width=\\linewidth]{figures_production/15_acceptance.pdf}\\end{center}\n{\\small\\color{navy}An illustrative tolerance of $10^{-8}$ determines the permitted log width. The plotted curve is numerical.}","\\[\\log_{10}p\\in[d_-,d_+],\\quad w=d_+-d_-,\\quad\\frac{p_+}{p_-}-1=\\operatorname{expm1}(\\log(10)w).\\]\n\\[\\operatorname{expm1}(\\log(10)w)\\le\\varepsilon\\quad\\Longrightarrow\\quad\\text{requested enclosure width is met}.\\]\nA positive probability enclosure can be stored on a logarithmic scale even when its endpoints are below ordinary floating-point range. If $d_-$ and $d_+$ enclose $\\log_{10}p$, the width determines the maximum multiplicative uncertainty.","For validated acceptance, compute the width and $\\operatorname{expm1}(\\log(10)w)$ with directed interval arithmetic. Compare the ratio interval\\textquotesingle s upper endpoint to the exact tolerance enclosure\\textquotesingle s lower endpoint. Preserve the literal outer endpoints throughout parsing and conversion. High precision alone does not prove this decision.","\\[z=q/0.04,\\qquad P(Q\\ge q)=e^{-z}\\sum_{k=0}^{39}\\frac{z^k}{k!}.\\]\n\\[\\log_{10}P(Q\\ge q)=\\frac{-z+\\log\\!\\left(\\sum_{k=0}^{39}z^k/k!\\right)}{\\log(10)}.\\]\nFor an even-degree chi-square variable, the upper tail has a finite positive-sum formula. The control $Q=0.02\\chi^2_{80}$ is a gamma law of integer shape forty and scale $0.04$. Its survival probability can therefore be evaluated independently of cosine inversion.","The curve computes the logarithm first. At thresholds from $1.6$ through $500$, it evaluates the finite sum with ninety decimal digits and plots its logarithm. The exact law supplies the formula. The numeric points are high-precision evaluations rather than directed-rounding enclosures.","\\bottomrule\\end{tabular}\\end{center}\n\\[\\text{matrix input}\\ \\longrightarrow\\ \\text{spectral enclosure}\\ \\longrightarrow\\ \\text{tail enclosure}\\ \\longrightarrow\\ \\text{accepted width}.\\]\nThe main chapters explain the operator identities, propagation bounds, residual structure, packet construction, and probability enclosures. The atlas makes those mechanisms inspectable through small declared cases. The formalization appendix identifies exactly which finite-algebra or order statements are represented by its source files and what the checked build establishes.","The factor has a finite sample index type $I$ and an arbitrary variant index\n type $V$. A variant block is a finite set $s\\subseteq V$. The factor entries\nare exact real numbers, $A:I\\times V\\to\\R$. The Lean type is\n\\decl{Row -> Variant -> Real}. The sample coordinates carry a\n\\decl{Fintype} instance. The variants carry a \\decl{DecidableEq} instance to\nsupport finite-set unions, singleton blocks, and reindexing.","The residualized weighted factor in the main text is\n\\[\n A=M_X G W^{1/2}.\n\\]\nFor this companion, $A$ is supplied. Proving that a particular floating-point\nresidualization procedure computes that mathematical factor would require\nadditional implementation and numerical-error reasoning. These modules do\nnot assume a covariance model or a distribution for the vector $v$.","There are five central finite-algebra definitions:\n\\begin{align*}\n \\operatorname{gramEntry}(s,A,i,j)&=\\sum_{k\\in s}A_{ik}A_{jk},\\\\[6pt]\n \\operatorname{matrixAction}(K,v,i)&=\\sum_{j\\in I}K_{ij}v_j,\\\\[6pt]\n \\operatorname{streamedAction}(s,A,v,i)&=\\sum_{k\\in s}A_{ik}\\sum_{j\\in I}A_{jk}v_j,\\\\[6pt]\n \\operatorname{quadraticForm}(K,v)&=\\sum_{i\\in I}v_i\\operatorname{matrixAction}(K,v,i),\\\\[6pt]\n \\operatorname{factorEnergy}(s,A,v)&=\\sum_{k\\in s}\\left(\\sum_{i\\in I}A_{ik}v_i\\right)^2.\n\\end{align*}\nThe source uses ordinary finite sums in $\\R$. The factor convention is the\nsample Gram $AA^\\top$, matching \\decl{stream_sample_gram}. It is not the dual\nGram $A^\\top A$. The case-study statistic $Q=\\tfrac12 v^\\top Kv$ requires the\nadditional scalar factor $\\tfrac12$ when applying these identities.","[6pt]\n \\operatorname{factorEnergy}(s,A,v)&=\\sum_{k\\in s}\\left(\\sum_{i\\in I}A_{ik}v_i\\right)^2.\n\\end{align*}\nThe source uses ordinary finite sums in $\\R$. The factor convention is the\nsample Gram $AA^\\top$, matching \\decl{stream_sample_gram}. It is not the dual\nGram $A^\\top A$. The case-study statistic $Q=\\tfrac12 v^\\top Kv$ requires the\nadditional scalar factor $\\tfrac12$ when applying these identities.","The principal refinement theorem is \\decl{gramEntry_union}. Its hypotheses\nare finite sets $s,t\\subseteq V$ and $\\operatorname{Disjoint}(s,t)$. It proves,\nfor all sample coordinates $i,j$,\n\\[\n \\operatorname{gramEntry}(s\\cup t,A,i,j)\n =\\operatorname{gramEntry}(s,A,i,j)+\\operatorname{gramEntry}(t,A,i,j).\n\\]\nDisjointness prevents repeated columns from being counted twice. It is a\nmathematical condition, not a performance assumption. The theorem applies to\nempty blocks and zero factors. It does not require positive weights, since\n$A$ is already supplied as a real factor.","nt}(s,t)$. It proves,\nfor all sample coordinates $i,j$,\n\\[\n \\operatorname{gramEntry}(s\\cup t,A,i,j)\n =\\operatorname{gramEntry}(s,A,i,j)+\\operatorname{gramEntry}(t,A,i,j).\n\\]\nDisjointness prevents repeated columns from being counted twice. It is a\nmathematical condition, not a performance assumption. The theorem applies to\nempty blocks and zero factors. It does not require positive weights, since\n$A$ is already supplied as a real factor.","The same refinement has formal consequences for the operator action and the\nquadratic statistic:\n\\begin{align*}\n (K_{s\\cup t}v)_i&=(K_s v)_i+(K_t v)_i,\\\\\n v^\\top K_{s\\cup t}v&=v^\\top K_s v+v^\\top K_t v.\n\\end{align*}\nThese are \\decl{matrixAction_gram_union} and\n\\decl{quadraticForm_gram_union}. The independent\n\\decl{streamedAction_union} proves the corresponding streamed accumulation\nidentity directly.","The reindexing theorem \\decl{gramEntry_reindex} handles a bijection\n$e:V\\simeq V$. It moves the block to $e(s)$ and transforms the factor to\n$A'_{ik}=A_{i,e^{-1}(k)}$. The resulting Gram entry is unchanged. Permuting\nonly the coordinates without moving their factor columns falls outside this\nstatement.","\\decl{matrixAction_gram_eq_streamed} proves\n\\[\n \\sum_{j\\in I}\\left(\\sum_{k\\in s}A_{ik}A_{jk}\\right)v_j\n =\\sum_{k\\in s}A_{ik}\\left(\\sum_{j\\in I}A_{jk}v_j\\right).\n\\]\nThis is the exact algebra behind multiplication by a sample Gram matrix\nwithout constructing that matrix. Lean checks the finite-sum interchange and\nassociativity of real multiplication. It does not assert that a given\nsequence of device operations has exact real semantics.","The next theorem, \\decl{quadraticForm_gram_eq_energy}, proves\n\\[\n v^\\top K_s v=\\sum_{k\\in s}\\left(\\sum_{i\\in I}A_{ik}v_i\\right)^2.\n\\]\nIt expands the action, interchanges the finite sums, factors each coordinate\nsum, and identifies the product of a projection with itself. The resulting\n\\decl{quadraticForm_gram_nonneg} states $0\\le v^\\top K_s v$ for every real\nvector $v$. This is positive semidefiniteness of the supplied Gram form.","Consider one sample coordinate and two variant columns with\n\\[\n A=(3\\quad4),\\qquad v=2.\n\\]\nThe two singleton Gram contributions are $9$ and $16$. Their disjoint union\nhas Gram entry $25$. Matrix multiplication gives $Kv=50$ and\n$v^\\top Kv=100$. The streamed action instead computes\n\\[\n 3(3\\cdot2)+4(4\\cdot2)=18+32=50.\n\\]\nThe energy expression gives\n\\[\n (3\\cdot2)^2+(4\\cdot2)^2=36+64=100.\n\\]\nThus the engine statistic with the stated half normalization is $Q=50$.\nAll paths are exact real identities. This example explains the type and\nnormalization choices. The universal theorems establish them for arbitrary\nfinite sample coordinates and variant blocks.","ith\n\\[\n A=(3\\quad4),\\qquad v=2.\n\\]\nThe two singleton Gram contributions are $9$ and $16$. Their disjoint union\nhas Gram entry $25$. Matrix multiplication gives $Kv=50$ and\n$v^\\top Kv=100$. The streamed action instead computes\n\\[\n 3(3\\cdot2)+4(4\\cdot2)=18+32=50.\n\\]\nThe energy expression gives\n\\[\n (3\\cdot2)^2+(4\\cdot2)^2=36+64=100.\n\\]\nThus the engine statistic with the stated half normalization is $Q=50$.\nAll paths are exact real identities. This example explains the type and\nnormalization choices. The universal theorems establish them for arbitrary\nfinite sample coordinates and variant blocks.","If the two blocks overlap in one column, their sum repeats that column's\nouter product. For $s=t=\\{1\\}$ and $A_{i1}=3$, the union has contribution $9$\nwhile the sum of the two blocks is $18$. This is why\n\\decl{gramEntry_union} requires disjointness.","The separate definition\n\\[\n \\operatorname{weightedSquares}(s,c,z)=\\sum_{k\\in s}c_k z_k^2\n\\]\nmodels a supplied diagonal quadratic form. Its coefficients need not be\neigenvalues obtained by a proved diagonalization. It has three theorems:\n\\begin{itemize}\n \\item \\decl{weightedSquares_mono}: if $l_k\\le u_k$ for every $k\\in s$, then\n $\\sum l_kz_k^2\\le\\sum u_kz_k^2$.\n \\item \\decl{weightedSquares_nonneg}: if $c_k\\ge0$ for every $k\\in s$, then\n $\\sum c_kz_k^2\\ge0$.\n \\item \\decl{weightedSquares_tail_imp}: coefficientwise order gives the\n pointwise implication $q\\le\\sum l_kz_k^2\\Rightarrow q\\le\\sum u_kz_k^2$.\n\\end{itemize}\nThe first and third allow signed coefficients because squares remain\nnonnegative. The second has an explicit nonnegativity hypothesis.","parate definition\n\\[\n \\operatorname{weightedSquares}(s,c,z)=\\sum_{k\\in s}c_k z_k^2\n\\]\nmodels a supplied diagonal quadratic form. Its coefficients need not be\neigenvalues obtained by a proved diagonalization. It has three theorems:\n\\begin{itemize}\n \\item \\decl{weightedSquares_mono}: if $l_k\\le u_k$ for every $k\\in s$, then\n $\\sum l_kz_k^2\\le\\sum u_kz_k^2$.\n \\item \\decl{weightedSquares_nonneg}: if $c_k\\ge0$ for every $k\\in s$, then\n $\\sum c_kz_k^2\\ge0$.\n \\item \\decl{weightedSquares_tail_imp}: coefficientwise order gives the\n pointwise implication $q\\le\\sum l_kz_k^2\\Rightarrow q\\le\\sum u_kz_k^2$.\n\\end{itemize}\nThe first and third allow signed coefficients because squares remain\nnonnegative. The second has an explicit nonnegativity hypothesis.","\\operatorname{weightedSquares}(s,c,z)=\\sum_{k\\in s}c_k z_k^2\n\\]\nmodels a supplied diagonal quadratic form. Its coefficients need not be\neigenvalues obtained by a proved diagonalization. It has three theorems:\n\\begin{itemize}\n \\item \\decl{weightedSquares_mono}: if $l_k\\le u_k$ for every $k\\in s$, then\n $\\sum l_kz_k^2\\le\\sum u_kz_k^2$.\n \\item \\decl{weightedSquares_nonneg}: if $c_k\\ge0$ for every $k\\in s$, then\n $\\sum c_kz_k^2\\ge0$.\n \\item \\decl{weightedSquares_tail_imp}: coefficientwise order gives the\n pointwise implication $q\\le\\sum l_kz_k^2\\Rightarrow q\\le\\sum u_kz_k^2$.\n\\end{itemize}\nThe first and third allow signed coefficients because squares remain\nnonnegative. The second has an explicit nonnegativity hypothesis.","weightedSquares}(s,c,z)=\\sum_{k\\in s}c_k z_k^2\n\\]\nmodels a supplied diagonal quadratic form. Its coefficients need not be\neigenvalues obtained by a proved diagonalization. It has three theorems:\n\\begin{itemize}\n \\item \\decl{weightedSquares_mono}: if $l_k\\le u_k$ for every $k\\in s$, then\n $\\sum l_kz_k^2\\le\\sum u_kz_k^2$.\n \\item \\decl{weightedSquares_nonneg}: if $c_k\\ge0$ for every $k\\in s$, then\n $\\sum c_kz_k^2\\ge0$.\n \\item \\decl{weightedSquares_tail_imp}: coefficientwise order gives the\n pointwise implication $q\\le\\sum l_kz_k^2\\Rightarrow q\\le\\sum u_kz_k^2$.\n\\end{itemize}\nThe first and third allow signed coefficients because squares remain\nnonnegative. The second has an explicit nonnegativity hypothesis.","ot be\neigenvalues obtained by a proved diagonalization. It has three theorems:\n\\begin{itemize}\n \\item \\decl{weightedSquares_mono}: if $l_k\\le u_k$ for every $k\\in s$, then\n $\\sum l_kz_k^2\\le\\sum u_kz_k^2$.\n \\item \\decl{weightedSquares_nonneg}: if $c_k\\ge0$ for every $k\\in s$, then\n $\\sum c_kz_k^2\\ge0$.\n \\item \\decl{weightedSquares_tail_imp}: coefficientwise order gives the\n pointwise implication $q\\le\\sum l_kz_k^2\\Rightarrow q\\le\\sum u_kz_k^2$.\n\\end{itemize}\nThe first and third allow signed coefficients because squares remain\nnonnegative. The second has an explicit nonnegativity hypothesis.","The interval module uses two elementary predicates:\n\\begin{align*}\n \\operatorname{Enclosed}(x,L,U)&\\;:\\Longleftrightarrow\\;L\\le x\\le U,\\\\\n \\operatorname{ErrorBound}(x,\\widehat x,r)&\\;:\\Longleftrightarrow\\;\n |x-\\widehat x|\\le r.\n\\end{align*}\nThe equivalence \\decl{errorBound_iff_enclosed} proves\n\\[\n |x-\\widehat x|\\le r\n \\quad\\Longleftrightarrow\\quad\n \\widehat x-r\\le x\\le\\widehat x+r.\n\\]\nThere is no extra nonnegativity hypothesis: if this predicate holds, then the\nradius is necessarily nonnegative. The definition accepts arbitrary real\ninputs, and the theorem states the exact logical equivalence.","\\operatorname{ErrorBound}(x,\\widehat x,r)&\\;:\\Longleftrightarrow\\;\n |x-\\widehat x|\\le r.\n\\end{align*}\nThe equivalence \\decl{errorBound_iff_enclosed} proves\n\\[\n |x-\\widehat x|\\le r\n \\quad\\Longleftrightarrow\\quad\n \\widehat x-r\\le x\\le\\widehat x+r.\n\\]\nThere is no extra nonnegativity hypothesis: if this predicate holds, then the\nradius is necessarily nonnegative. The definition accepts arbitrary real\ninputs, and the theorem states the exact logical equivalence.","For the K-COS error ledger, distinguish the ideal analytic quantity $x$, an\nexact finite transform $m$, and the computed estimate $\\widehat x$. If\n\\[\n |x-m|\\le a,\\qquad |m-\\widehat x|\\le b,\n\\]\nthen \\decl{errorBound_compose} proves\n\\[\n |x-\\widehat x|\\le a+b.\n\\]\nThe proof is the triangle inequality. The hypotheses must be proved\nindependently. In particular, $a$ can contain a support/alias bound and an\nomitted-mode bound. This theorem only composes those supplied facts.","\\decl{errorBound_accept} adds the hypothesis $a+b\\le\\varepsilon$ and concludes\n$|x-\\widehat x|\\le\\varepsilon$. This is the arithmetic core of acceptance by a\nproved budget. A sequence of numerically stable estimates, without valid\nbounds, does not satisfy its hypotheses.","The remaining interval statements are reusable exact-real rules.\n\\decl{enclosed_add} adds two enclosed quantities.\n\\decl{enclosed_scale} scales the interval by a nonnegative real factor.\n\\decl{enclosed_widen} permits an explicitly wider interval.\n\\decl{enclosed_intersect} combines two independently valid enclosures of the\nsame quantity using $\\max$ on the lower endpoints and $\\min$ on the upper\nendpoints.","Finally, \\decl{enclosed_monotone} transports an interval through a supplied\nmonotone function. Its hypothesis is global monotonicity on $\\R$. The source\ndoes not directly instantiate this rule with the real logarithm, which needs\na positive-domain formulation. A log-tail certificate therefore still needs\nboth positivity and a separate domain-correct monotonicity argument. The\ninterval-library implementation also requires its own correctness contract.","For weighted blocks before residualization, \\texttt{project} applies the\nfixed projector $M_X$. For already residualized factor blocks, use the\nidentity default. The listing is a binary64 reference implementation of the\nstreamed identity, not an interval-arithmetic proof of that identity. The\ntail import is available when \\path{engine/} is on the Python import path.","\\section{Names used by the implementation}\nThe analysis uses $\\kappa(s)$, $b$, $J_k$, and $N$. The numerical source retains the names \\texttt{K(s)} for the cumulant, \\texttt{U} for the support endpoint, \\texttt{I\\_k} for the payoff integral, and \\texttt{K} for the integer mode count. These names do not denote additional mathematical objects. The exact inputs and serialized output intervals determine the calculation.","CDF & Cumulative distribution function, $F(q)=\\Pr(Q\\le q)$.","Effective rank & The participation ratio $(\\sum_j\\lambda_j)^2/\\sum_j\\lambda_j^2$ for a nonzero positive spectrum. It need not be an integer.","Gram representation & The operator $AA^\\top$ or its dual $A^\\top A$. Their positive eigenvalues agree with multiplicity.","MiB & Mebibyte, equal to $2^{20}$ bytes. Memory tables report modeled payload, not total device allocation.","Noncentrality & The squared mean displacement in a standardized Gaussian direction. The symbol $\\delta_i$ in a noncentral chi-square component differs from a centered packet deviation.","Saddlepoint tilt & An admissible exponential tilt satisfying $\\kappa'(\\theta_*)=q$. It improves centering, but exact solution of that equation is not required for the directed enclosure theorem.","Survival probability & The upper-tail probability $\\Pr(Q\\ge q)$. For the nondegenerate continuous laws considered here, strict and non-strict thresholds agree.","\\midrule\n\\endhead\n$Q$ & A quadratic random variable. In the candidate-eigensystem theorem, the local symbol $Q$ instead denotes a candidate basis matrix.","$A$, $A_b$ & A real factor and its resident column blocks.","$H$, $K$ & Symmetric operators, with sample Gram $K=AA^\\top$ and feature Gram $H=A^\\top A$ in the streaming construction.","$\\Sigma$, $M$, $\\mu$ & Covariance, quadratic weight, and Gaussian mean.","$\\lambda_j$ & Spectral coefficients. Positivity is imposed whenever a positive-support theorem requires it.","$q$ & The tail threshold.","$\\kappa$ & Cumulant generating function. The implementation name \\texttt{K(s)} is mapped in Appendix~\\ref{app:reproduction}.","$\\theta$, $\\eta$ & Admissible exponential tilts, with $\\eta>\\theta$ when used in a Chernoff support bound.","$b$ & Upper endpoint of the positive COS support window. The implementation calls this \\texttt{U}.","$N$ & Retained COS mode count. The implementation argument is \\texttt{K}.","$P_h$ & Orthogonal structural phase projectors.","$M_{P,m}$ & The $m$th centered power sum of a spectral packet $P$.","$\\rho_P(s)$ & The packet's complex-grid convergence radius parameter.","$\\varepsilon_{\\rm pkt}$, $\\varepsilon_{\\rm sup}$, $\\varepsilon_{\\rm COS}$ & Packet, support/folding, and omitted-mode contributions to the analytic budget.","$[p_-,p_+]$ & A probability enclosure. Acceptance at relative tolerance $\\varepsilon$ requires $p_->0$ and $p_+/p_- -1\\le\\varepsilon$.","$[d_-,d_+]$ & A base-ten logarithmic probability enclosure.","A recurring example begins in Section~\\ref{sec:worked-law} with an exact dyadic factor and spectrum $(1,1,2,2)$. Section~\\ref{sec:worked-stream} changes its storage partition. Section~\\ref{sec:worked-tilt} derives the tilt and packets. Section~\\ref{sec:worked-enclosure} completes the enclosure and compares it with a closed-form probability.","\\section*{Interpreting the answer}\nA directed calculation returns an outward interval $[p_-,p_+]$. Requested multiplicative tolerance $\\varepsilon>0$ is accepted only if $p_->0$ and\n\\[\n\\frac{p_+}{p_-}-1\\le\\varepsilon.\n\\]\nA plausible point estimate can still fail this test. Failure means that the requested accuracy was not certified. Agreement with an independent value is useful evidence, but cannot replace the interval inequality.","midrule\\endhead\n$A$ & A factor. In the streaming construction, $A\\in\\mathbb R^{n\\times m}$.","$K=AA^\\top$ & The sample-space Gram matrix.","$H=A^\\top A$ & The feature-space Gram matrix in the phase construction. Elsewhere $H$ denotes the explicitly defined symmetric coefficient matrix.","$Q$ & The scalar Gaussian quadratic form whose law is being evaluated.","$\\lambda_j$ & Its nonzero spectral coefficients, including multiplicity.","$q$ & The threshold in $P(Q\\ge q)$.","$\\kappa(s)$ & The cumulant generating function, $\\log\\mathbb E e^{sQ}$.","$\\theta$ & A positive tilt in the moment-generating-function domain.","$b$, $N$ & The upper support endpoint and the number of retained COS modes.","$u_k$, $J_k$ & Frequency $k\\pi/b$ and its exponentially weighted payoff integral.","$\\|\\cdot\\|_2$ & Euclidean vector norm or induced matrix norm, as appropriate.","$\\|\\cdot\\|_F$ & Frobenius norm.","$M\\succeq0$ & Positive semidefiniteness.","\\bottomrule\n\\end{longtable}\nThe symbols $U$, $B$, and $R$ have locally defined linear-algebra roles. A core or basis is never assumed to span the exact range unless that hypothesis is stated. Nonzero spectra are compared with multiplicity. Additional zero eigenvalues in an oversized core do not change the quadratic-form law.","y}{\\chapter*{\\bibname}}{\\section*{\\bibname}}{}{}\n\\begin{thebibliography}{9}\n\\bibitem{Imhof1961} J.~P. Imhof, Computing the distribution of quadratic forms\nin normal variables, \\emph{Biometrika} 48(3--4) (1961), 419--426.\n\\href{https://doi.org/10.1093/biomet/48.3-4.419}{doi:10.1093/biomet/48.3-4.419}.\n\\bibitem{Davies1980} R.~B. Davies, Algorithm AS~155: The distribution of a\nlinear combination of $\\chi^2$ random variables,\n\\emph{Applied Statistics} 29(3) (1980), 323--333.\n\\href{https://doi.org/10.2307/2346911}{doi:10.2307/2346911}.\n\\bibitem{FangOosterlee2008} F. Fang and C.~W. Oosterlee, A novel pricing\nmethod for European options based on Fourier-cosine series expansions,\n\\emph{SIAM Journal on Scientific Computing} 31(2) (2008--2009), 826--848.\nPublished online in 2008.\n\\href{https://doi.org/10.1137/080718061}{doi:10.1137/080718061}.\n\\bibitem{JunikePankrashkin2022} G. Junike and K. Pankrashkin","The exact-spectrum control $0.02\\chi^2_{80}$ near $10^{-5000}$ met a\nrequested multiplicative tolerance of $10^{-18}$. Its directed ratio upper\nbound was less than $9.202\\times10^{-20}$, and an independently evaluated\nfinite Erlang log interval was fully contained in the returned interval.","The dense $64\\times64$ exact-binary64 Hadamard-factor control near\n$10^{-16}$ met a requested tolerance of $10^{-12}$. All 64 known exact\neigenvalues were enclosed. The final directed probability ratio upper bound\nwas less than $2.783\\times10^{-15}$. These displayed bounds are rounded\nupward. The full outward endpoints and literal inputs are in the accompanying\ncontrol record.","With $b=512$, $N=16384$, auxiliary tilt $\\eta=0.245$, and 60 interval decimal digits, the supplied backend returns a logarithmic interval. Rounding its lower endpoint downward and its upper endpoint upward gives the readable enclosure\n\\[\n\\log_{10}P(Q\\ge40)\\in\\bigl[-4.04489575,-4.03897381\\bigr].\n\\]\nAt $N=8192$, the directed ratio upper bound is approximately $0.0246131$, which fails the requested tolerance $0.02$. Doubling the retained mode count, rather than relaxing the tolerance, gives the interval above. The full serialized interval has a directed multiplicative ratio upper bound below $0.013731$. The displayed, coarser interval also meets the requested tolerance $0.02$. An independently evaluated i","With $b=512$, $N=16384$, auxiliary tilt $\\eta=0.245$, and 60 interval decimal digits, the supplied backend returns a logarithmic interval. Rounding its lower endpoint downward and its upper endpoint upward gives the readable enclosure\n\\[\n\\log_{10}P(Q\\ge40)\\in\\bigl[-4.04489575,-4.03897381\\bigr].\n\\]\nAt $N=8192$, the directed ratio upper bound is approximately $0.0246131$, which fails the requested tolerance $0.02$. Doubling the retained mode count, rather than relaxing the tolerance, gives the interval above. The full serialized interval has a directed multiplicative ratio upper bound below $0.013731$. The displayed, coarser interval also meets the requested tolerance $0.02$. An independently evaluated interval for","With $b=512$, $N=16384$, auxiliary tilt $\\eta=0.245$, and 60 interval decimal digits, the supplied backend returns a logarithmic interval. Rounding its lower endpoint downward and its upper endpoint upward gives the readable enclosure\n\\[\n\\log_{10}P(Q\\ge40)\\in\\bigl[-4.04489575,-4.03897381\\bigr].\n\\]\nAt $N=8192$, the directed ratio upper bound is approximately $0.0246131$, which fails the requested tolerance $0.02$. Doubling the retained mode count, rather than relaxing the tolerance, gives the interval above. The full serialized interval has a directed multiplicative ratio upper bound below $0.013731$. The displayed, coarser interval also meets the requested tolerance $0.02$. An independently evaluated interval for the exact expression $2e^{-1","With $b=512$, $N=16384$, auxiliary tilt $\\eta=0.245$, and 60 interval decimal digits, the supplied backend returns a logarithmic interval. Rounding its lower endpoint downward and its upper endpoint upward gives the readable enclosure\n\\[\n\\log_{10}P(Q\\ge40)\\in\\bigl[-4.04489575,-4.03897381\\bigr].\n\\]\nAt $N=8192$, the directed ratio upper bound is approximately $0.0246131$, which fails the requested tolerance $0.02$. Doubling the retained mode count, rather than relaxing the tolerance, gives the interval above. The full serialized interval has a directed multiplicative ratio upper bound below $0.013731$. The displayed, coarser interval also meets the requested tolerance $0.02$. An independently evaluated interval for the exact expression $2e^{-10}-e^{-20}$ lies entirely inside the COS enclosure.","2$, $N=16384$, auxiliary tilt $\\eta=0.245$, and 60 interval decimal digits, the supplied backend returns a logarithmic interval. Rounding its lower endpoint downward and its upper endpoint upward gives the readable enclosure\n\\[\n\\log_{10}P(Q\\ge40)\\in\\bigl[-4.04489575,-4.03897381\\bigr].\n\\]\nAt $N=8192$, the directed ratio upper bound is approximately $0.0246131$, which fails the requested tolerance $0.02$. Doubling the retained mode count, rather than relaxing the tolerance, gives the interval above. The full serialized interval has a directed multiplicative ratio upper bound below $0.013731$. The displayed, coarser interval also meets the requested tolerance $0.02$. An independently evaluated interval for the exact expression $2e^{-10}-e^{-20}$ lies entirely inside the COS enclosure.","t upward gives the readable enclosure\n\\[\n\\log_{10}P(Q\\ge40)\\in\\bigl[-4.04489575,-4.03897381\\bigr].\n\\]\nAt $N=8192$, the directed ratio upper bound is approximately $0.0246131$, which fails the requested tolerance $0.02$. Doubling the retained mode count, rather than relaxing the tolerance, gives the interval above. The full serialized interval has a directed multiplicative ratio upper bound below $0.013731$. The displayed, coarser interval also meets the requested tolerance $0.02$. An independently evaluated interval for the exact expression $2e^{-10}-e^{-20}$ lies entirely inside the COS enclosure.",",-4.03897381\\bigr].\n\\]\nAt $N=8192$, the directed ratio upper bound is approximately $0.0246131$, which fails the requested tolerance $0.02$. Doubling the retained mode count, rather than relaxing the tolerance, gives the interval above. The full serialized interval has a directed multiplicative ratio upper bound below $0.013731$. The displayed, coarser interval also meets the requested tolerance $0.02$. An independently evaluated interval for the exact expression $2e^{-10}-e^{-20}$ lies entirely inside the COS enclosure.","approximately $0.0246131$, which fails the requested tolerance $0.02$. Doubling the retained mode count, rather than relaxing the tolerance, gives the interval above. The full serialized interval has a directed multiplicative ratio upper bound below $0.013731$. The displayed, coarser interval also meets the requested tolerance $0.02$. An independently evaluated interval for the exact expression $2e^{-10}-e^{-20}$ lies entirely inside the COS enclosure.","This same fixed enclosure does not meet tolerance $0.001$. The acceptance routine rejects that tighter request. This is an intended distinction between an accurate point estimate and an established enclosure width. The rank-four example has slower Fourier decay than the rank-eighty control developed later, so its mode count should not be interpreted as a general cost prediction.","midrule\n0 & $2.3289\\times10^{-4}$ & $3.9366\\times10^{-2}$","2 & $3.6881\\times10^{-8}$ & $4.8893\\times10^{-6}$","4 & $8.7572\\times10^{-12}$ & $1.0931\\times10^{-9}$","6 & $2.419\\times10^{-15}$ & $2.9091\\times10^{-13}$","8 & $7.1896\\times10^{-19}$ & $8.4306\\times10^{-17}$","10 & $2.2307\\times10^{-22}$ & $2.5701\\times10^{-20}$"]}