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When \\(m\\) is large, the direct factor \\(A\\) may exceed device\nmemory even though the final statistic requires only a spectrum or repeated\noperator actions.  This manuscript formalizes a composable decomposition\ncalculus with four independent layers: storage blocks, structural\npolyphase packets, Krylov/range compression, and final spectral packets for\ndeep-tail evaluation.  The core exact result is that both \\(K\\) and every\nKrylov action can be evaluated by streaming variant blocks, so total variant\ncount controls arithmetic and I/O but need not control device working-set\nsize.  Storage refinement is proved inv","\\begin{abstract}\nSequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing \\cite{Wu2011}. Large rare-variant and kernel tests often begin with a weighted factor\n\\[\nA=M_XGW^{1/2},\n\\qquad\nK=AA^\\top,\n\\]\nwhere \\(G\\in\\mathbb R^{n\\times m}\\) is a genotype or feature matrix,\n\\(W\\) is diagonal and nonnegative, and \\(M_X\\) projects away null-model\ncovariates.  When \\(m\\) is large, the direct factor \\(A\\) may exceed device\nmemory even though the final statistic requires only a spectrum or repeated\noperator actions.  This manuscript formalizes a composable decomposition\ncalculus with four independent layers: storage blocks, structural\npolyphase packets, Krylov/range compression, and final spectral packets for\ndeep-tail evaluation.  The core exact result is that both \\(K\\) and every\nKrylov action can be evaluated by streaming variant blocks, so total variant\ncount controls arithmetic and I/O but need not control device working-set\nsize.  Storage refinement is proved invariant, two-dimensional sample/variant\nti","ernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing \\cite{Wu2011}. Large rare-variant and kernel tests often begin with a weighted factor\n\\[\nA=M_XGW^{1/2},\n\\qquad\nK=AA^\\top,\n\\]\nwhere \\(G\\in\\mathbb R^{n\\times m}\\) is a genotype or feature matrix,\n\\(W\\) is diagonal and nonnegative, and \\(M_X\\) projects away null-model\ncovariates.  When \\(m\\) is large, the direct factor \\(A\\) may exceed device\nmemory even though the final statistic requires only a spectrum or repeated\noperator actions.  This manuscript formalizes a composable decomposition\ncalculus with four independent layers: storage blocks, structural\npolyphase packets, Krylov/range compression, and final spectral packets for\ndeep-tail evaluation.  The core exact result is that both \\(K\\) and every\nKrylov action can be evaluated by streaming variant blocks, so total variant\ncount controls arithmetic and I/O but need not control device working-set\nsize.  Storage refinement is proved invariant, two-dimensional sample/variant\ntiling is exact, and the streaming layer composes …","s a variance-component framework for set-based rare-variant association testing \\cite{Wu2011}. Large rare-variant and kernel tests often begin with a weighted factor\n\\[\nA=M_XGW^{1/2},\n\\qquad\nK=AA^\\top,\n\\]\nwhere \\(G\\in\\mathbb R^{n\\times m}\\) is a genotype or feature matrix,\n\\(W\\) is diagonal and nonnegative, and \\(M_X\\) projects away null-model\ncovariates.  When \\(m\\) is large, the direct factor \\(A\\) may exceed device\nmemory even though the final statistic requires only a spectrum or repeated\noperator actions.  This manuscript formalizes a composable decomposition\ncalculus with four independent layers: storage blocks, structural\npolyphase packets, Krylov/range compression, and final spectral packets for\ndeep-tail evaluation.  The core exact result is that both \\(K\\) and every\nKrylov action can be evaluated by streaming variant blocks, so total variant\ncount controls arithmetic and I/O but need not control device working-set\nsize.  Storage refinement is proved invariant, two-dimensional sample/variant\ntiling is exact, and the streaming layer composes with an exact polyphase\nlight-cone","large, the direct factor \\(A\\) may exceed device\nmemory even though the final statistic requires only a spectrum or repeated\noperator actions.  This manuscript formalizes a composable decomposition\ncalculus with four independent layers: storage blocks, structural\npolyphase packets, Krylov/range compression, and final spectral packets for\ndeep-tail evaluation.  The core exact result is that both \\(K\\) and every\nKrylov action can be evaluated by streaming variant blocks, so total variant\ncount controls arithmetic and I/O but need not control device working-set\nsize.  Storage refinement is proved invariant, two-dimensional sample/variant\ntiling is exact, and the streaming layer composes with an exact polyphase\nlight-cone theorem, minimal phase-local rank allocation, low-rank interface\nmessages, and exact QR-core updates.  When phase couplings are weak rather\nthan zero, a nonnegative transfer matrix gives certified phasewise leakage\nbounds for operator powers and complete Krylov polynomials.  A dual\nquery/spectral budget propagates pruning uncertainty to generalized-\\(\\chi^2\\)\ntail brack…","er composes with an exact polyphase\nlight-cone theorem, minimal phase-local rank allocation, low-rank interface\nmessages, and exact QR-core updates.  When phase couplings are weak rather\nthan zero, a nonnegative transfer matrix gives certified phasewise leakage\nbounds for operator powers and complete Krylov polynomials.  A dual\nquery/spectral budget propagates pruning uncertainty to generalized-\\(\\chi^2\\)\ntail brackets, while residual trace, square-trace, and spectral-cap invariants\nsupply substantially sharper incomplete-range certificates and an exact\nmoment-triggered residual-completion rule.\nReference implementations reproduce\nexplicitly formed operators to roughly machine precision.  For\n\\(n=2000\\), Krylov width \\(16\\), covariate rank \\(4\\), and resident variant\nblock \\(1024\\), the algebraic streamed kernel-action working set is about\n\\(16.3\\) MiB and remains constant from \\(m=10^4\\) through \\(m=10^6\\), while\nthe full float64 factor grows from about \\(153\\) MiB to \\(15.3\\) GiB.\n\\end{abstract}","ylov polynomials.  A dual\nquery/spectral budget propagates pruning uncertainty to generalized-\\(\\chi^2\\)\ntail brackets, while residual trace, square-trace, and spectral-cap invariants\nsupply substantially sharper incomplete-range certificates and an exact\nmoment-triggered residual-completion rule.\nReference implementations reproduce\nexplicitly formed operators to roughly machine precision.  For\n\\(n=2000\\), Krylov width \\(16\\), covariate rank \\(4\\), and resident variant\nblock \\(1024\\), the algebraic streamed kernel-action working set is about\n\\(16.3\\) MiB and remains constant from \\(m=10^4\\) through \\(m=10^6\\), while\nthe full float64 factor grows from about \\(153\\) MiB to \\(15.3\\) GiB.\n\\end{abstract}","query/spectral budget propagates pruning uncertainty to generalized-\\(\\chi^2\\)\ntail brackets, while residual trace, square-trace, and spectral-cap invariants\nsupply substantially sharper incomplete-range certificates and an exact\nmoment-triggered residual-completion rule.\nReference implementations reproduce\nexplicitly formed operators to roughly machine precision.  For\n\\(n=2000\\), Krylov width \\(16\\), covariate rank \\(4\\), and resident variant\nblock \\(1024\\), the algebraic streamed kernel-action working set is about\n\\(16.3\\) MiB and remains constant from \\(m=10^4\\) through \\(m=10^6\\), while\nthe full float64 factor grows from about \\(153\\) MiB to \\(15.3\\) GiB.\n\\end{abstract}","propagates pruning uncertainty to generalized-\\(\\chi^2\\)\ntail brackets, while residual trace, square-trace, and spectral-cap invariants\nsupply substantially sharper incomplete-range certificates and an exact\nmoment-triggered residual-completion rule.\nReference implementations reproduce\nexplicitly formed operators to roughly machine precision.  For\n\\(n=2000\\), Krylov width \\(16\\), covariate rank \\(4\\), and resident variant\nblock \\(1024\\), the algebraic streamed kernel-action working set is about\n\\(16.3\\) MiB and remains constant from \\(m=10^4\\) through \\(m=10^6\\), while\nthe full float64 factor grows from about \\(153\\) MiB to \\(15.3\\) GiB.\n\\end{abstract}","generalized-\\(\\chi^2\\)\ntail brackets, while residual trace, square-trace, and spectral-cap invariants\nsupply substantially sharper incomplete-range certificates and an exact\nmoment-triggered residual-completion rule.\nReference implementations reproduce\nexplicitly formed operators to roughly machine precision.  For\n\\(n=2000\\), Krylov width \\(16\\), covariate rank \\(4\\), and resident variant\nblock \\(1024\\), the algebraic streamed kernel-action working set is about\n\\(16.3\\) MiB and remains constant from \\(m=10^4\\) through \\(m=10^6\\), while\nthe full float64 factor grows from about \\(153\\) MiB to \\(15.3\\) GiB.\n\\end{abstract}","race, and spectral-cap invariants\nsupply substantially sharper incomplete-range certificates and an exact\nmoment-triggered residual-completion rule.\nReference implementations reproduce\nexplicitly formed operators to roughly machine precision.  For\n\\(n=2000\\), Krylov width \\(16\\), covariate rank \\(4\\), and resident variant\nblock \\(1024\\), the algebraic streamed kernel-action working set is about\n\\(16.3\\) MiB and remains constant from \\(m=10^4\\) through \\(m=10^6\\), while\nthe full float64 factor grows from about \\(153\\) MiB to \\(15.3\\) GiB.\n\\end{abstract}","y substantially sharper incomplete-range certificates and an exact\nmoment-triggered residual-completion rule.\nReference implementations reproduce\nexplicitly formed operators to roughly machine precision.  For\n\\(n=2000\\), Krylov width \\(16\\), covariate rank \\(4\\), and resident variant\nblock \\(1024\\), the algebraic streamed kernel-action working set is about\n\\(16.3\\) MiB and remains constant from \\(m=10^4\\) through \\(m=10^6\\), while\nthe full float64 factor grows from about \\(153\\) MiB to \\(15.3\\) GiB.\n\\end{abstract}","rper incomplete-range certificates and an exact\nmoment-triggered residual-completion rule.\nReference implementations reproduce\nexplicitly formed operators to roughly machine precision.  For\n\\(n=2000\\), Krylov width \\(16\\), covariate rank \\(4\\), and resident variant\nblock \\(1024\\), the algebraic streamed kernel-action working set is about\n\\(16.3\\) MiB and remains constant from \\(m=10^4\\) through \\(m=10^6\\), while\nthe full float64 factor grows from about \\(153\\) MiB to \\(15.3\\) GiB.\n\\end{abstract}","gered residual-completion rule.\nReference implementations reproduce\nexplicitly formed operators to roughly machine precision.  For\n\\(n=2000\\), Krylov width \\(16\\), covariate rank \\(4\\), and resident variant\nblock \\(1024\\), the algebraic streamed kernel-action working set is about\n\\(16.3\\) MiB and remains constant from \\(m=10^4\\) through \\(m=10^6\\), while\nthe full float64 factor grows from about \\(153\\) MiB to \\(15.3\\) GiB.\n\\end{abstract}","completion rule.\nReference implementations reproduce\nexplicitly formed operators to roughly machine precision.  For\n\\(n=2000\\), Krylov width \\(16\\), covariate rank \\(4\\), and resident variant\nblock \\(1024\\), the algebraic streamed kernel-action working set is about\n\\(16.3\\) MiB and remains constant from \\(m=10^4\\) through \\(m=10^6\\), while\nthe full float64 factor grows from about \\(153\\) MiB to \\(15.3\\) GiB.\n\\end{abstract}","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 manuscript 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 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}[H]\n\\centering\n\\includegraphics[width=.72\\linewidth]{figure_01_residualization_geometry.png}\n\\caption{Residualization is the geometric starting point. Every sample-space vector splits into a component in the covariate span and an orthogonal residual component. The factor $A=M_XGW^{1/2}$ therefore acts only on variation remaining after the null-model covariates have been removed.}\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]\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}[H]\n\\centering\n\\includegraphics[width=.92\\linewidth]{figure_02_exact_block_kernel_assembly.png}\n\\caption{Exact block-kernel assembly. The sample interaction kernel carried by each resident variant block is $K_b=A_bA_b^\\top$, and the full kernel is their exact sum. Streaming changes the order of evaluation, not the operator.}\n\\label{fig:block-kernel-assembly}\n\\end{figure}","\\begin{theorem}[One-pass sample-space streaming action]\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\\)]\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]\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\\)]\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\\)]\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]\nAny exact-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]\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]\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 is\n\\[\nO(tc+pc+t^2).\n\\]","\\begin{theorem}[Partition-refinement invariance]\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]\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 \\(P_h\\) project onto variant phase \\(h\\), and 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]\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{figure}[H]\n\\centering\n\\includegraphics[width=.82\\linewidth]{figure_04_polyphase_lightcone.png}\n\\caption{The exact polyphase light cone. The support of the Krylov iterate expands by at most one graph edge per operator application. Phases outside graph distance $k$ remain identically inactive at depth $k$ and need not be loaded.}\n\\label{fig:polyphase-lightcone}\n\\end{figure}","\\begin{theorem}[Streaming-polyphase composition]\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]\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 polynomial theorem covers arbitrary vectors in a finite Krylov space,\nincluding polynomial filters and restarted combinations, rather than only the\nmonomial $H^kX$.","\\begin{corollary}[Certified phase pruning]\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.","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]\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]\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.  Under the additive certificate, a set of omitted 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 $r$ 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}\nAmong all subsets of cardinality $r+1$, the smallest possible sum of 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]\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.","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=.82\\linewidth]{figure_13_edge_risk_vs_block_norm.png}\n\\caption{Raw phase-block magnitude and finite-depth query risk are different quantities.  The edge score $\\beta_{hg}^{(5)}$ incorporates the start profile and every depth-five transfer path through the phase graph; the block norm $\\mathcal B_{hg}$ does not.}\n\\label{fig:edge-risk-vs-norm}\n\\end{figure}","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]\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.","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=.82\\linewidth]{figure_12_double_buffer_model.png}\n\\caption{Idealized effect of asynchronous double buffering over $64$ streamed blocks.  The largest modeled speedup occurs when transfer and computation times are comparable, because neither resource is left idle for most of the pipeline.  The additional cost is one extra resident block buffer.}\n\\label{fig:double-buffer-model}\n\\end{figure}","Let \\(R\\) be a newly generated Krylov residual and\n\\[\nR_h=P_hR.\n\\]","\\begin{theorem}[Minimal exact phase width]\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\\]","\\begin{theorem}[Exact QR-core update]\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 eigenvalues of the small core Gram.\n\\end{theorem}","0&R_\\perp\n\\end{pmatrix}.\n\\]\nTherefore the nonzero eigenvalues of\n\\([A,U][A,U]^\\top\\) are exactly the eigenvalues of the small core Gram.\n\\end{theorem}","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]\nThe nonzero eigenvalues of\n\\[\nK=AA^\\top\n\\]\nare exactly the eigenvalues of\n\\[\nBB^\\top.\n\\]\n\\end{theorem}","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\\]","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]\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]\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]\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}","A production 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\\]","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 true probability of \\(10^{-8}\\), the certified\ntail interval was\n\\[\n10^{-8.11143}\n\\le\np\n\\le\n10^{-7.89036},\n\\]\na width of about \\(0.221\\) decades.  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.","c 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 true probability of \\(10^{-8}\\), the certified\ntail interval was\n\\[\n10^{-8.11143}\n\\le\np\n\\le\n10^{-7.89036},\n\\]\na width of about \\(0.221\\) decades.  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]\nLet the singular values of \\(A\\) be \\(\\sigma_i(A)\\), and let\n\\(\\widetilde\\sigma_i(B)\\) denote the singular values of \\(B\\) padded by zeros\nto the ambient spectral dimension.  Then\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\\) are exactly those of \\(B\\).\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\\text{pruned operator for basis discovery}\n\\to\n\\text{one exact streamed factor check}\n\\to\n\\text{certified spectral core}.\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]\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 its\nlargest possible 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]\nFor the Gaussian quadratic form with coefficient matrix \\(H\\),\n\\[\nK_H(\\theta)\n=-\\frac12\\log\\det(I-2\\theta H)\n\\]\nsatisfies\n\\[\n\\boxed{\nK_H(\\theta)\n\\le\nK_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]\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 K_H(\\theta)-\\theta q\\},\n}\n\\]\nwhere \\(\\widehat K_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}[H]\n\\centering\n\\caption{Tail-bracket width at a true probability of \\(10^{-8}\\).  The\ntrace/moment certificate is still conservative, but it is orders of magnitude\nnarrower than coefficient-wise Weyl inflation before full range capture.}\n\\begin{tabular}{rrrr}\n\\toprule\nKrylov depth & basis dimension & v0.7 Weyl width & v0.8 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]\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 \\(W\\) has orthonormal columns.  Since \\(F\\) lies in the orthogonal\ncomplement of \\(U\\),\n\\[\nU^\\top W=0.\n\\]","\\begin{theorem}[Exact residual completion]\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 singular\nvalues 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 singular\nvalues 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 singular\nvalues of the stacked completion core\n\\[\n\\mathcal C_{\\rm complete}\n=\n\\begin{bmatrix}\nB\\\\D\n\\end{bmatrix}.\n\\]\n\\end{theorem}","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.","The resulting production logic is not ``approximate forever.''  It is a\ncertified handoff between two regimes:\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\\]","\\begin{table}[H]\n\\centering\n\\caption{Realistic 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","\\[\nG_b\n\\to\nG_b-\\bar G_b\n\\to\n\\text{variance scaling}\n\\to\nM_X G_b\n\\to\nW_b^{1/2}\n\\to\nA_bA_b^\\top,\n\\]","\\[\n\\boxed{6.61\\times10^{-16}}.\n\\]","\\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}[H]\n\\centering\n\\includegraphics[width=.84\\linewidth]{figure_20_realistic_deep_tail_accuracy.png}\n\\caption{K-COS versus the high-resolution full-spectrum tilted reference on\nrealistic LD spectra.  Accuracy remains at roughly machine-scale in log\nprobability through the deepest corrected copy-LD query, near $10^{-458}$.}\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 & $\\log_{10}p$ & K-COS $K$ & 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 & $4.94\\times10^{-14}$","The matrix packets above operate before eigendecomposition.  They should be\ndistinguished from the spectral packets used after the eigenvalues\n\\(\\lambda_j\\) are known.","The complete stack is\n\\[\n\\boxed{\n\\begin{aligned}\nG\n&\\to \\text{storage blocks}\\\\\n&\\to \\text{matrix phases}\\\\\n&\\to \\text{streamed }A,A^\\top,K\\\\\n&\\to \\text{polyphase/rank-adaptive Krylov}\\\\\n&\\to \\text{small spectral core}\\\\\n&\\to \\{\\lambda_j\\}\\\\\n&\\to \\text{centered spectral packets}\\\\\n&\\to \\text{deep-tail K-COS}.\n\\end{aligned}\n}\n\\]","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}[H]\n\\centering\n\\caption{Algebraic float64 memory model for \\(n=2000\\), \\(p=4\\),\nKrylov width \\(s=16\\), resident variant block \\(c=1024\\).}\n\\begin{tabular}{rrrr}\n\\toprule\n\\(m\\) & full factor (MiB) & streamed \\(KV\\) (MiB) & tiled Gram (MiB)","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=.84\\linewidth]{figure_03_memory_law.png}\n\\caption{The memory law. The complete factor grows linearly with total variant count $m$, whereas the streamed kernel action and the two-dimensional tiled construction remain bounded when resident block size, Krylov width, and covariate rank are fixed.}\n\\label{fig:memory-law}\n\\end{figure}","\\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}\\)","A hierarchical coarse-to-fine refinement gave relative discrepancy\n\\[\n3.17\\times10^{-16}.\n\\]","For an explicitly PSD polyphase operator\n\\[\nH=L^\\top L\n\\]\nwith block-bidiagonal \\(L\\), the smallest eigenvalue was approximately\n\\[\n2.86,\n\\]\nand the largest norm observed outside the exact \\(k\\)-step light cone was\n\\[\n\\boxed{0}.\n\\]","The QR-core update reproduced the full singular spectrum with relative\n\\(\\ell_2\\) discrepancy\n\\[\n5.89\\times10^{-16}.\n\\]","l complement. Exact block-Gram additivity then permits memory-bounded streaming. Partition invariance separates physical storage granularity from the operator itself, while the phase graph constrains propagation through an exact light cone. Phase-local rank removes redundant basis directions; QR-core reuse preserves spectra under small updates; and spectral packets compress the final generalized-$\\chi^2$ transform. The figures therefore represent a sequence of exact representation changes, each preserving the semantic object needed by the next layer.","factor exceeds device memory & streamed \\(AV\\), \\(A^\\top U\\), or \\(KV\\) & exact memory-bounded execution","Accordingly, the contribution claimed here is \\emph{not} the invention of\nblocking, streaming, matrix-free multiplication, or Krylov iteration.  The\ncandidate contribution is the exact composition of four independent\nrepresentations:\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\\]\nThe first controls device residency, the second controls structural\npropagation, the third controls spectral dimension, and the fourth controls\ndeep-tail transform cost.","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]\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}","Storage-phase commutation theorem]\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]\nSuppose the exact phase graph of $H$ is used and the current Krylov block is\nsupported on a phase set $F_0$.  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 during that step.\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.  Therefore blocks outside the active structural frontier\ncontribute identically zero to the requested iterate and may be omitted.\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\\]\nThis is stronger than ordinary chunking, because the scheduler is constrained\nby the exact operator graph rather than solely by available memory.","The candidate Komorebi contribution is the resolved synthesis\n\\[\n\\boxed{\\begin{aligned}\n&\\text{exact streamed factor actions}\n+\\text{ polyphase light cones}\n+\\text{ minimal phase rank}\\\\\n&+\\text{ low-rank messages}\n+\\text{ exact update cores}\n+\\text{ deep-tail spectral packets}.\n\\end{aligned}}\n\\]","streamed \\(KV\\) & proved & direct matrix identity","streamed \\(AV,A^\\top U\\) & proved & factor partition","- weighted phase-transfer matrix\n  \\(\\mathcal B_{hg}=\\|P_hHP_g\\|_2\\);\n- phasewise propagation envelope\n  \\(\\rho(H^kX)\\le \\mathcal B^k\\rho(X)\\);\n- thresholded-operator certificate\n  \\(\\rho((H^k-H_0^k)X)\\le(\\mathcal B^k-\\mathcal B_0^k)\\rho(X)\\);\n- certified approximate light cone for weak nonzero couplings;\n- polynomial/Krylov certificate for \\(p(H)-p(H_0)\\);\n- scalar depth-threshold law;\n- deterministic numerical validation for depths 1-8, a phase-threshold sweep,\n  and a signed degree-four polynomial;\n- two new explanatory figures integrated into the manuscript."]}