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          "latex": "UD(x)\n=\n\\frac{3x+1}{4},"
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          "sha256": "bf91c2be3af65950a55e2596f63ec4ebe711a1b4ce6c01e18fcf45ef2a38dfcd",
          "latex": "DU(x)\n=\n\\frac{3x+2}{4}."
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          "latex": "\\boxed{\nDU(x)-UD(x)=\\frac14.\n}"
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          "latex": "\\boxed{\nw_{\\min}\n=\nU^uD^{k-u}.\n}"
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          "latex": "\\boxed{\nw_{\\max}\n=\nD^{k-u}U^u.\n}"
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          "latex": "b_{\\min}\n=\n\\sum_{t=1}^u\n2^{t-1}3^{u-t}."
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          "latex": "\\boxed{\nb_{\\min}\n=\n3^u-2^u.\n}"
        },
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          "latex": "b_{\\max}\n=\n\\sum_{t=1}^u\n2^{k-u+t-1}3^{u-t}."
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          "latex": "\\boxed{\nb_{\\max}\n=\n2^{k-u}(3^u-2^u).\n}"
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          "latex": "\\boxed{\n3^u-2^u\n\\le\nb_w\n\\le\n2^{k-u}(3^u-2^u).\n}"
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          "latex": "F_{UUU}(x)\n=\n\\frac{27x+19}{8}."
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          "latex": "W_{k,u}\n=\nb_{\\max}-b_{\\min}."
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          "latex": "\\boxed{\nW_{k,u}\n=\n(2^{k-u}-1)(3^u-2^u).\n}"
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          "latex": "\\log F_w(x)\n=\n\\log x\n+\nu\\log3\n-\nk\\log2\n+\n\\log\\left(\n1+\\frac{b_w}{3^ux}\n\\right)."
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          "latex": "\\boxed{\n\\Delta_w L\n=\nu\\log3-k\\log2\n+\nC_w(x),\n}"
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          "latex": "\\boxed{\nC_w(x)\n=\n\\log\\left(\n1+\\frac{b_w}{3^ux}\n\\right).\n}"
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          "latex": "T\\mu=\\nu T+C"
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          "latex": "\\boxed{\nu\\log3-k\\log2.\n}"
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          "latex": "\\boxed{\nC_w(x).\n}"
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          "latex": "x\\to\\infty."
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          "latex": "\\boxed{\n\\text{asymptotic drift is count-controlled}.\n}"
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          "latex": "F_w(n)\n=\n\\frac{3^un+b_w}{2^k}."
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          "latex": "\\boxed{\n\\text{temporal branch sequence}\n\\longrightarrow\n\\text{finite algebraic operator}.\n}"
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          "latex": "T(\\mu(x,y))\n=\n\\nu(Tx,Ty)+C_T(x,y)."
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          "latex": "\\text{Linear Core}+\\text{Correction}."
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          "latex": "\\text{Local Chart / Atlas}."
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          "latex": "\\boxed{\n\\text{finite itinerary}\n\\to\n\\text{affine core}\n+\n\\text{word-order correction}.\n}"
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          "latex": "\\boxed{\nF_w(x)\n=\n\\frac{3^ux+b_w}{2^k}.\n}"
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          "latex": "\\boxed{\n\\text{counts determine slope;}\n}"
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          "latex": "\\boxed{\n\\text{order determines offset.}\n}"
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          "latex": "\\boxed{\n\\text{parity word}\n\\longleftrightarrow\n\\text{unique residue cylinder modulo }2^k,\n}"
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          "latex": "\\boxed{\n\\psi_w\\circ T^k\\circ\\phi_w^{-1}\n=\n\\operatorname{id}.\n}"
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          "latex": "T(n)=\n\\begin{cases}\nn/2,&n\\equiv0\\pmod2,\\\\[2mm]\n(3n+1)/2,&n\\equiv1\\pmod2\n\\end{cases}"
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          "latex": "\\boxed{\nh(w)\n=\n\\min_{\nj\\le k,\\ \\Delta_j>0\n}\n\\left\\lfloor\n\\frac{b_j}{\\Delta_j}\n\\right\\rfloor\n}"
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          "latex": "\\boxed{\nH_w\n=\n\\Omega_w\n\\cap\n[1,h(w)].\n}"
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          "latex": "\\boxed{\n\\text{verify Collatz on }[2,N]\n}"
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          "sha256": "eba55640febc654f3bfa13d75d3e5736b476a31eeb4bd0a081b1fd5bcab2e8eb",
          "latex": "\\boxed{\n\\text{refine hard cylinders until }\\mathfrak F_k(N)=\\varnothing.\n}"
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          "latex": "\\gamma"
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          "latex": "\\boxed{\n\\text{all checks terminate in finite exact arithmetic}.\n}"
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          "latex": "T^j(n)\\in\\{1,2\\},"
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          "latex": "\\boxed{\n\\gamma_T(n,j)\n}"
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          "latex": "\\boxed{\n\\gamma_D(n,j)\n}"
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          "latex": "T^k(n)\n=\n\\frac{3^un+b_w}{2^k}."
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          "latex": "\\theta_w\n=\n\\left\\lfloor\n\\frac{b_w}{2^k-3^u}\n\\right\\rfloor+1."
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          "latex": "\\boxed{\nD_{\\gamma_w}\n=\n\\{\nn\\in\\Omega_w:n\\ge\\theta_w\n\\}\n}"
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          "latex": "\\boxed{\n(2^k-3^u)a\n>\nm_w-r_w.\n}"
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          "latex": "\\boxed{\na\n>\n\\frac{m_w-r_w}{2^k-3^u}.\n}"
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          "latex": "\\boxed{\nq_w\n=\n\\left\\lfloor\n\\frac{m_w-r_w}\n{2^k-3^u}\n\\right\\rfloor+1.\n}"
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          "latex": "\\boxed{\n\\Omega_w\n=\n\\{n>0:w\\text{ is the first }k\\text{-step parity word of }n\\}.\n}"
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          "latex": "\\Omega_D\n=\n2\\mathbb Z_{>0}\n=\n(0+2\\mathbb Z)\\cap\\mathbb Z_{>0},"
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          "latex": "T^k(n)\\pmod2\n=\nm_w+a\\pmod2."
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          "latex": "\\{r_{wD},r_{wU}\\}\n=\n\\{r_w,r_w+2^k\\}\n\\pmod{2^{k+1}}."
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          "latex": "\\gcd(3^u,2^k)=1,"
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          "latex": "\\boxed{\nn\n\\equiv\n-b_w3^{-u}\n\\pmod{2^k}.\n}"
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          "latex": "F_{UD}(n)\n=\n\\frac{3n+1}{4}."
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          "latex": "u=1,\n\\qquad\nb=1."
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          "latex": "3n+1\\equiv0\\pmod4."
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          "latex": "3^{-1}\\equiv3\\pmod4,"
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          "latex": "r_{UD}\n\\equiv\n-3\n\\equiv1\n\\pmod4."
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          "sha256": "ce3951ee49f315c58131077cd65e7e76b6edf6d7ef2572771e7a36176ea0e7d5",
          "latex": "\\Omega_{UD}\n=\n(1+4\\mathbb Z)\\cap\\mathbb Z_{>0}."
        },
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          "latex": "r=1."
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          "latex": "m=T^2(1)=1."
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          "latex": "\\boxed{\nT^2(1+4a)\n=\n1+3a.\n}"
        },
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          "latex": "w=DU"
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          "latex": "F_{DU}(n)\n=\n\\frac{3n+2}{4}."
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          "latex": "u=1,\n\\qquad\nb=2."
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          "latex": "3n+2\\equiv0\\pmod4."
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          "latex": "r_{DU}=2."
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          "latex": "m=T^2(2)=2."
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          "latex": "\\boxed{\nT^2(2+4a)\n=\n2+3a.\n}"
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          "latex": "F_w(n)\n=\n\\frac{9n+5}{16}."
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          "latex": "k=4,\\qquad u=2,\\qquad b=5."
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          "latex": "9n+5\\equiv0\\pmod{16}."
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          "latex": "9^{-1}\\equiv9\\pmod{16},"
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          "latex": "r_w\n\\equiv\n-45\n\\equiv3\n\\pmod{16}."
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          "latex": "3\\to5\\to8\\to4\\to2."
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          "latex": "m_w=2."
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          "latex": "\\boxed{\n3+16a\n\\longmapsto\n2+9a.\n}"
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          "latex": "\\phi_w(n)\n=\n\\frac{n-3}{16},"
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          "latex": "\\psi_w(y)\n=\n\\frac{y-2}{9}."
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          "latex": "\\boxed{\na\\mapsto a.\n}"
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          "latex": "\\boxed{\n\\mathcal A_w\n=\n(\n\\Omega_w,\n\\Gamma_w,\n\\phi_w,\n\\psi_w,\nT^k|_{\\Omega_w}\n).\n}"
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          "latex": "\\boxed{\n\\mathfrak A_k\n=\n\\{\n\\mathcal A_w:\nw\\in\\{D,U\\}^k\n\\}.\n}"
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          "latex": "\\boxed{\n\\mathbb Z_{>0}\n=\n\\bigsqcup_{w\\in\\{D,U\\}^k}\n\\Omega_w.\n}"
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          "latex": "\\boxed{\n\\mathfrak A_k\n\\text{ is source-complete}.\n}"
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          "latex": "\\boxed{\n\\text{source partition}\n\\neq\n\\text{target partition}.\n}"
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          "latex": "\\boxed{\n\\Omega_w\n=\n\\Omega_{wD}\n\\bigsqcup\n\\Omega_{wU}.\n}"
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          "latex": "\\boxed{\n\\text{one new itinerary symbol}\n\\leftrightarrow\n\\text{one new quotient bit}.\n}"
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          "latex": "r_{k+1}\n\\equiv\nr_k\n\\pmod{2^k}."
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          "latex": "\\boxed{\n\\text{舊的平均負漂移直覺}\n\\longrightarrow\n\\text{exact finite valuation-word drift}\n+\n\\text{separate statistical layer}.\n}"
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          "latex": "T(n)\n=\n\\begin{cases}\nn/2,&n\\text{ even},\\\\\n(3n+1)/2,&n\\text{ odd},\n\\end{cases}"
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          "latex": "\\mathcal O\n=\n\\{1,3,5,\\ldots\\}."
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          "latex": "\\boxed{\nk_{\\mathrm{parity}}=K,\n}"
        },
        {
          "kind": "display",
          "line_start": 680,
          "line_end": 684,
          "sha256": "345bcf1dea1a36193dabb2ff3e705858e2d60f7452400c1e8cc203f5abc9e40e",
          "latex": "\\boxed{\nu_{\\mathrm{parity}}=m.\n}"
        },
        {
          "kind": "display",
          "line_start": 694,
          "line_end": 698,
          "sha256": "d78acf05438eceb332b8b720547765ac71334848be650cadf43d96f1d1d3597c",
          "latex": "n_i\n=\n\\frac{3n_{i-1}+1}{2^{\\kappa_i}},"
        },
        {
          "kind": "display",
          "line_start": 702,
          "line_end": 708,
          "sha256": "4ee3616252fd9c061866f849805ebd65dc280d4199de7d39a2f4bc8b0134a50c",
          "latex": "\\boxed{\n2^{\\kappa_i}n_i\n=\n3n_{i-1}+1.\n}"
        },
        {
          "kind": "display",
          "line_start": 718,
          "line_end": 722,
          "sha256": "f3a7bce2457d691bc974769e73a8ea4f270b1219bbd44d0e1548d03dce9ce8f0",
          "latex": "n_1\n=\n\\frac{3n_0+1}{2^{\\kappa_1}},"
        },
        {
          "kind": "display",
          "line_start": 724,
          "line_end": 728,
          "sha256": "75ae5e750a088607b1033849c6180025ca805b105342306defcdc49abd6c0496",
          "latex": "n_2\n=\n\\frac{3n_1+1}{2^{\\kappa_2}}."
        },
        {
          "kind": "display",
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          "line_end": 740,
          "sha256": "356fa6dcf3795eb41498127c8c56e495314dd0c37cd5b8fb64cb7c6bb28d36a0",
          "latex": "n_2\n=\n\\frac{\n3\\frac{3n_0+1}{2^{\\kappa_1}}+1\n}{\n2^{\\kappa_2}\n}"
        },
        {
          "kind": "display",
          "line_start": 742,
          "line_end": 749,
          "sha256": "b13e32bd15a6830aa10c2e8b5aa100b38fb1bf6aed1495fac51e464cc85be7ca",
          "latex": "=\n\\frac{\n9n_0+3+2^{\\kappa_1}\n}{\n2^{\\kappa_1+\\kappa_2}\n}."
        },
        {
          "kind": "display",
          "line_start": 753,
          "line_end": 759,
          "sha256": "9d913361bf165904fae7c0b79d0b49e9a3354eb2e9ccb5b7467d4fe369e2e01d",
          "latex": "\\boxed{\nB_{(\\kappa_1,\\kappa_2)}\n=\n3+2^{\\kappa_1}.\n}"
        },
        {
          "kind": "display",
          "line_start": 771,
          "line_end": 775,
          "sha256": "2cc09de48ebf9344321c39463acb5481057ed66dedaf341747598be2c1d4c486",
          "latex": "\\boldsymbol\\kappa\n=\n(\\kappa_1,\\ldots,\\kappa_m),"
        },
        {
          "kind": "display",
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          "line_end": 789,
          "sha256": "d66e985030fa5c6886e0b21274c12e8c7040bb6b3fe94b641ba47a23719cbab5",
          "latex": "\\boxed{\nS^m(n_0)\n=\n\\frac{\n3^mn_0+B_{\\boldsymbol\\kappa}\n}{\n2^K\n},\n}"
        },
        {
          "kind": "display",
          "line_start": 793,
          "line_end": 800,
          "sha256": "a084027c595093eac87e362b7388fb6a3902c312d692e2369cefa39ab9a84ccb",
          "latex": "\\boxed{\nB_{\\boldsymbol\\kappa}\n=\n\\sum_{i=1}^{m}\n3^{m-i}2^{K_{i-1}}.\n}"
        },
        {
          "kind": "display",
          "line_start": 808,
          "line_end": 810,
          "sha256": "08e291e7f08c2d3fd1976b8a3ebf5c688b211a64c9cf32f0fe997087d18361ae",
          "latex": "B_0=0."
        },
        {
          "kind": "display",
          "line_start": 814,
          "line_end": 822,
          "sha256": "f39727468c701ed8086f47a30a1d11d3339c4bc1861b0d11dc5a2009b31f28c3",
          "latex": "n_{j-1}\n=\n\\frac{\n3^{j-1}n_0+B_{j-1}\n}{\n2^{K_{j-1}}\n}."
        },
        {
          "kind": "display",
          "line_start": 826,
          "line_end": 830,
          "sha256": "15213b2668b299f4ed9eb85b5a507f85908593ec8b8e9a3f4f95203c9bcee304",
          "latex": "n_j\n=\n\\frac{3n_{j-1}+1}{2^{\\kappa_j}}"
        },
        {
          "kind": "display",
          "line_start": 832,
          "line_end": 843,
          "sha256": "ffe55051017b42b07ffa380cabd9c542f48ac244f98000724299cc8c16deecef",
          "latex": "=\n\\frac{\n3^j n_0\n+\n3B_{j-1}\n+\n2^{K_{j-1}}\n}{\n2^{K_j}\n}."
        },
        {
          "kind": "display",
          "line_start": 847,
          "line_end": 855,
          "sha256": "f235bde0089104f4b0aea5d1d6ee891af01b110b5412d498fad69c7e52c0db22",
          "latex": "\\boxed{\nB_j\n=\n3B_{j-1}\n+\n2^{K_{j-1}}.\n}"
        },
        {
          "kind": "display",
          "line_start": 859,
          "line_end": 866,
          "sha256": "d58c749e2a7789b4d9eaa475cc6122b201cce02aedbefd18640367b626452ac6",
          "latex": "\\boxed{\nB_j\n=\n\\sum_{i=1}^{j}\n3^{j-i}2^{K_{i-1}}.\n}"
        },
        {
          "kind": "display",
          "line_start": 876,
          "line_end": 882,
          "sha256": "17ab25e7ceff242ec90b7c62de65a6307c6ca24bbc3f8d61cb84e81a3a9e46ff",
          "latex": "\\boxed{\n\\Sigma(\\boldsymbol\\kappa)\n=\n(m,K).\n}"
        },
        {
          "kind": "display",
          "line_start": 886,
          "line_end": 892,
          "sha256": "fd505d42e651f1a52977f5479ba650b815829538fc697d2e9aabcf401e31b533",
          "latex": "\\boxed{\n\\lambda_{\\boldsymbol\\kappa}\n=\n\\frac{3^m}{2^K}.\n}"
        },
        {
          "kind": "display",
          "line_start": 896,
          "line_end": 902,
          "sha256": "c5a7004c7858610f3085ae9d1bd0381275c952672b9235f97e5b6cc0aec7528a",
          "latex": "\\boxed{\nC(\\boldsymbol\\kappa)\n=\nB_{\\boldsymbol\\kappa}\n}"
        },
        {
          "kind": "display",
          "line_start": 908,
          "line_end": 916,
          "sha256": "8e79fca24299839ec998dd781b98497a8b4d55c5f3ca869adfeddd1fd0e05cb4",
          "latex": "\\boxed{\nS^m(n)\n=\n\\lambda_{\\boldsymbol\\kappa}n\n+\n\\frac{B_{\\boldsymbol\\kappa}}{2^K}.\n}"
        },
        {
          "kind": "display",
          "line_start": 924,
          "line_end": 926,
          "sha256": "62c66a7a5dd70c3146618063c344e531e6d4b59e379808443ce962b3abd63c5a",
          "latex": "m"
        },
        {
          "kind": "display",
          "line_start": 930,
          "line_end": 932,
          "sha256": "81af35d8b751ce79beab08d104b2132a00fe56cb49b547874903b3128a0e2ee3",
          "latex": "K,"
        },
        {
          "kind": "display",
          "line_start": 936,
          "line_end": 938,
          "sha256": "885b84309bfd5d21e4c8c6de6faa68ad5c3e69cdc3549e8b9fca4154b468d68f",
          "latex": "B_{\\boldsymbol\\kappa}."
        },
        {
          "kind": "display",
          "line_start": 942,
          "line_end": 944,
          "sha256": "239f55d6193ade085ecffd6e9227da596230105cf5d2e3786d234d4292684dd0",
          "latex": "(1,3)"
        },
        {
          "kind": "display",
          "line_start": 948,
          "line_end": 950,
          "sha256": "523743119dbfdfaa65f5f982c797b68aab0cd477f0cb0ea8957ebc7eee94d060",
          "latex": "(3,1)"
        },
        {
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          "sha256": "f870fe03725e4a2357ba3251d0f0968fd066af513f44b218d6a8b3815ce41207",
          "latex": "m=2,\n\\qquad\nK=4."
        },
        {
          "kind": "display",
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          "line_end": 968,
          "sha256": "2c7a48ab26eae4071f4f5423c31de1ae5f823446a79cc75605387f9cec063ed5",
          "latex": "B_{(1,3)}\n=\n3+2\n=\n5,"
        },
        {
          "kind": "display",
          "line_start": 970,
          "line_end": 976,
          "sha256": "45f94754d12568841b247284d59a50636e26d53a4d8f37050ac978b746142f2b",
          "latex": "B_{(3,1)}\n=\n3+8\n=\n11."
        },
        {
          "kind": "display",
          "line_start": 980,
          "line_end": 986,
          "sha256": "d2026eacac0ce57842a67bce650fee30d829405d291283e9fccc8fc92d0168fc",
          "latex": "\\boxed{\n\\text{same drift skeleton}\n\\not\\Rightarrow\n\\text{same finite operator}.\n}"
        },
        {
          "kind": "display",
          "line_start": 994,
          "line_end": 996,
          "sha256": "2a1ef2fd8b78d876e872ad0e47553be3db0321baf25f08dd4bc8f206bfff5550",
          "latex": "E(\\boldsymbol\\kappa)"
        },
        {
          "kind": "display",
          "line_start": 1000,
          "line_end": 1002,
          "sha256": "b9ea8e418748182c5ddbe2ab6c999b8601f964ed4acd040cd10b20f6b57f2c91",
          "latex": "k=K,"
        },
        {
          "kind": "display",
          "line_start": 1004,
          "line_end": 1006,
          "sha256": "2b9b551edfaefb1456f913aa789c85e67f116ad356c47f154255bc958db9ee47",
          "latex": "u=m."
        },
        {
          "kind": "display",
          "line_start": 1010,
          "line_end": 1014,
          "sha256": "dc869dd3e28ab8e67b676105c7f9600177c77526684a9ed25546b4aad9daa95b",
          "latex": "F_w(n)\n=\n\\frac{3^un+b_w}{2^k}"
        },
        {
          "kind": "display",
          "line_start": 1018,
          "line_end": 1028,
          "sha256": "e50eb28464a425d130cfba111b8eea396df491b29d7f337a7339ad1e7a52826a",
          "latex": "\\boxed{\nS^m(n)\n=\n\\frac{\n3^mn+B_{\\boldsymbol\\kappa}\n}{\n2^K\n}.\n}"
        },
        {
          "kind": "display",
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          "line_end": 1038,
          "sha256": "a3d3c5cc9b244c85cfeb8865b84c6c0a280caf93c3123a283aeb30bbca640db7",
          "latex": "\\boxed{\nB_{\\boldsymbol\\kappa}\n=\nb_{E(\\boldsymbol\\kappa)}\n}"
        },
        {
          "kind": "display",
          "line_start": 1052,
          "line_end": 1054,
          "sha256": "1a81f2631d06fa175eed514f64075588c5784de4a653eb1ae1261248a2770f75",
          "latex": "n_0>0,"
        },
        {
          "kind": "display",
          "line_start": 1058,
          "line_end": 1066,
          "sha256": "2ce3f7b3e22d8a780fdec9f7dadcfaa1fc860126b3d40f399e184addbb3f011d",
          "latex": "S^m(n_0)\n=\n\\frac{\n3^mn_0+B_{\\boldsymbol\\kappa}\n}{\n2^K\n}."
        },
        {
          "kind": "display",
          "line_start": 1070,
          "line_end": 1087,
          "sha256": "be61ab76ea79b0ed51f9cb604837b82b3376ea94053d68a04c258dc07189fc4b",
          "latex": "\\ln S^m(n_0)\n=\nm\\ln3\n-\nK\\ln2\n+\n\\ln n_0\n+\n\\ln\\left(\n1+\n\\frac{\nB_{\\boldsymbol\\kappa}\n}{\n3^m n_0\n}\n\\right)."
        },
        {
          "kind": "display",
          "line_start": 1091,
          "line_end": 1101,
          "sha256": "3bf90f483081aeeb91301d61b18e5587d71da3fcf89e3239dd311677ecd10dca",
          "latex": "\\boxed{\n\\Delta_{\\boldsymbol\\kappa}L\n=\nm\\ln3\n-\nK\\ln2\n+\nC_{\\boldsymbol\\kappa}(n_0),\n}"
        },
        {
          "kind": "display",
          "line_start": 1105,
          "line_end": 1117,
          "sha256": "106da8ca626c79dfee855ca96bd0e22d475c9c462407b89e220c68cee34d38fd",
          "latex": "\\boxed{\nC_{\\boldsymbol\\kappa}(n)\n=\n\\ln\\left(\n1+\\frac{\nB_{\\boldsymbol\\kappa}\n}{\n3^mn\n}\n\\right).\n}"
        },
        {
          "kind": "display",
          "line_start": 1125,
          "line_end": 1127,
          "sha256": "5b4093e6db6c519ae440d9886d115b00b6198e11e8ecdf1bc30990b8a8820ef3",
          "latex": "m\\ge1,"
        },
        {
          "kind": "display",
          "line_start": 1129,
          "line_end": 1131,
          "sha256": "185a3ed58345964ba25a91720ad405e1ec557ad7e4a4b9687ef17652ce4770db",
          "latex": "B_{\\boldsymbol\\kappa}>0."
        },
        {
          "kind": "display",
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          "line_end": 1137,
          "sha256": "cb1e57953684c3fb5a86b8db39139d7daa8c70607d43154d01e488efc383143c",
          "latex": "C_{\\boldsymbol\\kappa}(n)>0."
        },
        {
          "kind": "display",
          "line_start": 1141,
          "line_end": 1146,
          "sha256": "f01523b130213fe3ed84de2010ab0f1f14a965f18e9a919e889e5beb38ff4cae",
          "latex": "\\boxed{\nC_{\\boldsymbol\\kappa}(n)\\to0\n\\quad(n\\to\\infty)\n}"
        },
        {
          "kind": "display",
          "line_start": 1152,
          "line_end": 1156,
          "sha256": "a9872658e4442bb664ae57a7cf64cecf8a4549c14b1dc3986c3ea4b58553be3d",
          "latex": "\\boxed{\nm\\ln3-K\\ln2\n}"
        },
        {
          "kind": "display",
          "line_start": 1166,
          "line_end": 1168,
          "sha256": "e6aff33ba50f05da70c9bd0ad4ab54fe7aec07ac5f3ea45001c6f81a7abf6efa",
          "latex": "3^m<2^K."
        },
        {
          "kind": "display",
          "line_start": 1172,
          "line_end": 1174,
          "sha256": "077fb8ab0d82ca162ecf1e1773e19a1d4efe63770ffe364531165270c1c5659d",
          "latex": "\\log_2:"
        },
        {
          "kind": "display",
          "line_start": 1176,
          "line_end": 1178,
          "sha256": "2032fa6f55f2d2d1aa179fea0b4133d38544019ad04a63e12dd7c64e033d064c",
          "latex": "m\\log_2 3<K."
        },
        {
          "kind": "display",
          "line_start": 1182,
          "line_end": 1186,
          "sha256": "80d96236c4c4781ff932ef251bd5a722e8f108f332f1e45002e2681267cd1c9a",
          "latex": "\\boxed{\n\\frac Km>\\log_2 3.\n}"
        },
        {
          "kind": "display",
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          "line_end": 1195,
          "sha256": "296d5ea98a8e8b68dd46f36e9a84d7bee99207eca0fe5d45e45c2d2587c90786",
          "latex": "\\boxed{\n\\log_2 3\n\\approx1.5849625007.\n}"
        },
        {
          "kind": "display",
          "line_start": 1203,
          "line_end": 1207,
          "sha256": "555b9208b6ba75beff2e2e4991257c81df9e9f602d6f24c635c486cefd7d1d74",
          "latex": "\\frac uk\n<\n\\frac{\\ln2}{\\ln3}."
        },
        {
          "kind": "display",
          "line_start": 1211,
          "line_end": 1213,
          "sha256": "fd02d3e8a02c1a56dd3f4bfb2eacd32c169731f33900ad5550c2ca02a8f19ea7",
          "latex": "u=m,"
        },
        {
          "kind": "display",
          "line_start": 1215,
          "line_end": 1217,
          "sha256": "c8ce5123600da257a21d1b560d372b08d0fcb564d286a9eca1ef2d7eaf917bbf",
          "latex": "k=K."
        },
        {
          "kind": "display",
          "line_start": 1221,
          "line_end": 1225,
          "sha256": "43eee1de2818912cc996e34ad33f90a4d62ac4ccaf93e4c1a384af4fe4b3596c",
          "latex": "\\frac mK\n<\n\\frac{\\ln2}{\\ln3}."
        },
        {
          "kind": "display",
          "line_start": 1229,
          "line_end": 1237,
          "sha256": "2e02bf845e494a88e6e5ccb668ab7f188105b6a348f5512c0a492d0b5ba79688",
          "latex": "\\boxed{\n\\frac Km\n>\n\\frac{\\ln3}{\\ln2}\n=\n\\log_2 3.\n}"
        },
        {
          "kind": "display",
          "line_start": 1247,
          "line_end": 1251,
          "sha256": "b3fb08313571de31b4197b6c1e18f112f2fe4e64254cb7c5fbc3a9dcc2d13387",
          "latex": "S^m(n)\n=\n\\frac{3^mn+B}{2^K},"
        },
        {
          "kind": "display",
          "line_start": 1255,
          "line_end": 1257,
          "sha256": "55b9618ad8cc5367027a373a44fda03cb3f5388888a8f6b9b3ae622e998f3374",
          "latex": "S^m(n)<n"
        },
        {
          "kind": "display",
          "line_start": 1261,
          "line_end": 1267,
          "sha256": "361f44537a42201de68ea0282d388ce87f8b0938e143c821f90d67ceaeb01c5e",
          "latex": "\\boxed{\nB_{\\boldsymbol\\kappa}\n<\n(2^K-3^m)n.\n}"
        },
        {
          "kind": "display",
          "line_start": 1271,
          "line_end": 1273,
          "sha256": "8841d3048c0aa9d01889e344f71e317e61cd09523a9be74068cad943e882eaa0",
          "latex": "2^K>3^m,"
        },
        {
          "kind": "display",
          "line_start": 1277,
          "line_end": 1289,
          "sha256": "7ebf1899eec870f883506e166c74bcdd79dd3193e4bcba48648c459ce6cdcab4",
          "latex": "\\boxed{\n\\theta_{\\boldsymbol\\kappa}\n=\n\\left\\lfloor\n\\frac{\nB_{\\boldsymbol\\kappa}\n}{\n2^K-3^m\n}\n\\right\\rfloor+1.\n}"
        },
        {
          "kind": "display",
          "line_start": 1293,
          "line_end": 1295,
          "sha256": "8889a65a53fa4f4d21ed26044cbda7141b9e53e34a9879c68e0f50dc4e946164",
          "latex": "n\\ge\\theta_{\\boldsymbol\\kappa}"
        },
        {
          "kind": "display",
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          "line_end": 1305,
          "sha256": "663e37f0bf6ff94380e28f0538426e934f9a77e8e09ba9760c6240f518d4e22c",
          "latex": "\\boxed{\nS^m(n)<n.\n}"
        },
        {
          "kind": "display",
          "line_start": 1313,
          "line_end": 1315,
          "sha256": "64d42d0b6efa2e513b5cf17c84df4de170b3167aa477e3ad70b8997c78e03866",
          "latex": "2^K<3^m,"
        },
        {
          "kind": "display",
          "line_start": 1319,
          "line_end": 1321,
          "sha256": "93372114c66de14e19bde2f4ee45a9694a1895ea345911a14e5293f15b37a6ca",
          "latex": "(3^m-2^K)n>0,"
        },
        {
          "kind": "display",
          "line_start": 1325,
          "line_end": 1327,
          "sha256": "185a3ed58345964ba25a91720ad405e1ec557ad7e4a4b9687ef17652ce4770db",
          "latex": "B_{\\boldsymbol\\kappa}>0."
        },
        {
          "kind": "display",
          "line_start": 1331,
          "line_end": 1335,
          "sha256": "8290b7944f46a21aa90043abc9aa5f4483701021baee4e7271031f6ec8873869",
          "latex": "\\boxed{\nS^m(n)>n\n}"
        },
        {
          "kind": "inline",
          "line_start": 1337,
          "line_end": 1337,
          "sha256": "1b16b1df538ba12dc3f97edbb85caa7050d46c148134290feba80f8236c83db9",
          "latex": "n"
        },
        {
          "kind": "display",
          "line_start": 1347,
          "line_end": 1349,
          "sha256": "a982035f084466ec48c94f1a917549bd9b8bc944983242ca9a32e07d73ba550f",
          "latex": "\\kappa(n)=v_2(3n+1)"
        },
        {
          "kind": "display",
          "line_start": 1355,
          "line_end": 1357,
          "sha256": "6bd6f104110e76a60ac11544d06ad0460303d257ca3c8112b3ee5305b7c26084",
          "latex": "v_2(3n+1)=j"
        },
        {
          "kind": "display",
          "line_start": 1361,
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          "latex": "\\boxed{\n\\text{commutative leading multipliers}\n\\Rightarrow\n\\text{counts determine skeleton}.\n}"
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          "latex": "\\text{finite word}\n\\to\n\\text{same operator class}."
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          "latex": "\\text{counts}\\to\\text{leading skeleton},\n\\qquad\n\\text{order}\\to\\text{correction}."
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          "latex": "\\text{word}\n\\leftrightarrow\n\\text{one residue class}."
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          "latex": "\\text{target coordinate}\n\\to\n\\text{unique source}."
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          "latex": "\\|F(x)-F(y)\\|<\\|x-y\\|."
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          "latex": "A_wx+B_w\n\\equiv0\n\\pmod{D_w}."
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          "latex": "\\boxed{\nA_wx+B_w\n\\equiv0\n\\pmod I.\n}"
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          "latex": "[A_w][x]=-[B_w]."
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          "latex": "\\boxed{\n[A_w]\\in(R/I)^\\times,\n}"
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          "latex": "\\boxed{\n[x]\n=\n-[A_w]^{-1}[B_w]\n}"
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          "latex": "\\boxed{\n[A_w]\\text{ unit in }R/I\n\\Rightarrow\n\\text{unique residue chart}.\n}"
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          "latex": "\\gcd(3^u,2^k)=1,"
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          "latex": "M_{A_w}:R/I\\to R/I"
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          "latex": "A_wx=-B_w"
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          "latex": "\\boxed{\n\\text{one word}\n\\leftrightarrow\n\\text{one residue}\n}"
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          "sha256": "ee74c8d0e6994072abd7bab18d4f85549207f613db88c83aaf4ef97ac27c8098",
          "latex": "\\mathbb Z/6\\mathbb Z,"
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          "latex": "x\\equiv4."
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          "latex": "\\boxed{\n\\text{one equation has multiple residue charts}.\n}"
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          "latex": "2x\\equiv1\\pmod6"
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          "sha256": "207a538d708dd6f1fc3038197eaaa7fb62057a014f0cbb26be104b16ee830819",
          "latex": "\\boxed{\n\\text{zero / one / multiple charts}.\n}"
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          "sha256": "1e6eb42c125917912a936a4010b5935eac765b4c8611a55b588bae40552a9b61",
          "latex": "\\to"
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          "sha256": "e57a3253180637a7f75003ce7b83ac934e4bf6a827bc09a03d5f870a07eef8ee",
          "latex": "F_w(x)\n=\n\\frac{A_wx+B_w}{D_w}"
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          "sha256": "7bf52ac09b171bf1437aa9351e3312407a0dd371085defb31ccf7dfc66df6543",
          "latex": "\\boxed{\n\\text{operator closure survives}.\n}"
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          "latex": "\\boxed{\n\\text{residue uniqueness}.\n}"
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          "sha256": "92d42b78523b9cebde12911aaac2e3077256b4637ab9a0a021a94910e7044b28",
          "latex": "R=\\mathbb Z/6\\mathbb Z."
        },
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          "sha256": "aa85dd310d6e47064e196107ed2dba3500ac94e57003cf65eab1c8e6dec3c66c",
          "latex": "x\\mapsto2x."
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          "sha256": "bad26dc21b21d9a46b14d9760f3eba2a14a12d0c5e4228e2e74e71d59d0b5293",
          "latex": "2\\cdot1\n=\n2\n\\pmod6,"
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          "sha256": "9ade7031097f5215234837a2b4f7f0fe3eab0c997e8b612f33ae70b4ebcda5ce",
          "latex": "2\\cdot4\n=\n8\n\\equiv2\n\\pmod6."
        },
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          "sha256": "c4e591e45da54ab659b042f058e2ed6435eed94d6260d5e07edb33e800e6b3ae",
          "latex": "\\boxed{\n1\\neq4\n\\quad\\text{but}\\quad\n2\\cdot1=2\\cdot4.\n}"
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          "latex": "Ax=Ay,"
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          "sha256": "58df4b4e42bb1f8f2530ff05e0b29f9561538381f527de0f4f456a510797a628",
          "latex": "A(x-y)=0."
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          "sha256": "a775856cd2c2bf4b75fd762a121c77f4efff81ecd86efa2acf2603489a17f7af",
          "latex": "\\boxed{\nA\\text{ is regular / non-zero-divisor on the relevant module}.\n}"
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          "latex": "\\boxed{\nA\\text{ regular}\n\\Rightarrow\nx\\mapsto Ax+B\n\\text{ injective}.\n}"
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          "sha256": "02078dde2a68737246e04f6c470ddf7d076bb4922062364f401a9e41368a0493",
          "latex": "R=\\mathbb Z,"
        },
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          "sha256": "49f3f0ee23b584181d04de04b94d60bdae4f877581d587f8ccbbbcf94ff747a9",
          "latex": "2x=2y\n\\Rightarrow\nx=y."
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          "sha256": "267e7a6fa8045ebc3e0bf0e9f198ad20c5df909d8bddaa141a729304bd3c5b15",
          "latex": "\\boxed{\n\\text{non-unit}\n\\not\\Rightarrow\n\\text{non-injective}.\n}"
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          "sha256": "9cc88d30fb4d426967a9940b742dc45b31b40a2483357598ffbffd5106a30591",
          "latex": "\\operatorname{Frac}(R),"
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          "latex": "\\mathbb Q,\\mathbb R"
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          "latex": "F(n)<n."
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          "latex": "\\mathbb C"
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          "latex": "\\boxed{\nF(z)<z\n}"
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          "latex": "F(x)=\\lambda x+c,"
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          "sha256": "c99c80cb9fbf40d53a1aad4d32e4a0d22641ddc52bd732131b27a355eaf44948",
          "latex": "F(x)-F(y)\n=\n\\lambda(x-y)."
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          "sha256": "b8ee6882521887337ba167666ef7b0e31d865839d049283b01dc9daaaf6a9230",
          "latex": "|\\cdot|_v"
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          "sha256": "13507da446936331d01d4609968ef3c4c3e0e432084d218c12aa70aeaba779be",
          "latex": "\\boxed{\n|F(x)-F(y)|_v\n=\n|\\lambda|_v|x-y|_v.\n}"
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          "sha256": "925b2f75bdef6d75a458beab6dc21b9396a2540c81514168fd26f256f79eda5b",
          "latex": "\\boxed{\n|\\lambda|_v<1.\n}"
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          "sha256": "9d533de47f9cd31ee607fa62d83261667bf7d1b190ae4d45c486697661d78e06",
          "latex": "w=UUDD."
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          "sha256": "bee544e30c06294140ab0a0f6a21167e20a9c1c4c98080408ce5ae11cc687a86",
          "latex": "\\boxed{\nF_w(x)\n=\n\\frac{9x+5}{16}.\n}"
        },
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          "sha256": "46299f4d04bb1afb7c53d67e3309b49b7422ea1ccee29198322f2673461f2f6c",
          "latex": "\\lambda=\\frac9{16}."
        },
        {
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          "line_end": 1215,
          "sha256": "493c7f6be970f3dc52773bc2649908821cc455f981c400524e4b52c0814da60e",
          "latex": "\\boxed{\n\\left|\\frac9{16}\\right|_\\infty\n=\n\\frac9{16}<1.\n}"
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          "latex": "2"
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          "sha256": "5498263bae226400467ac4be2ba6389d62f16c9888f0db47be1ee53cdeb3f5ad",
          "latex": "|2|_2=\\frac12."
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          "sha256": "ddc2e412d6e58f4c993b7a1a2b4dae772be8f3a25cb1f48dea6c0c11d56e181e",
          "latex": "v_2(9)=0,\n\\qquad\nv_2(16)=4,"
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          "latex": "b_{\\min}\n=\nr\n\\sum_{t=1}^{u}\n2^{t-1}m^{u-t}."
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          "latex": "12\\alpha\n\\approx7.571."
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          "latex": "\\boxed{\nP_{12}\n=\n\\frac{3302}{4096}\n=\n80.615234375\\%.\n}"
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          "latex": "\\boxed{\nP_k\n\\text{ 對有限 }k\\text{ 不必單調。}\n}"
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          "latex": "\\boxed{\n\\frac{938413}{1048575}\n\\approx\n89.4941229\\%.\n}"
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          "latex": "P_{16}\n\\approx89.4943237\\%"
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          "latex": "2^{20}=16\\cdot2^{16},"
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          "latex": "2^{16}"
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          "latex": "n=0."
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          "latex": "938415."
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          "latex": "T^{16}(1)=1,"
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          "latex": "T^{16}(2)=2."
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          "latex": "938415-2\n=\n\\boxed{\n938413.\n}"
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          "latex": "\\boxed{\n\\text{benchmark count}\n=\n\\text{binomial class law}\n+\n\\text{finite-domain boundary correction}.\n}"
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          "latex": "20\\alpha\n\\approx12.6186."
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          "latex": "\\boxed{\nP_{20}\n=\n\\frac{910596}{1048576}\n\\approx86.8412\\%.\n}"
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          "latex": "\\boxed{\n\\kappa\\text{ 必須為奇數}.\n}"
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          "latex": "\\boxed{\n3\\mid t\n\\Rightarrow\nS^{-1}(t)=\\varnothing.\n}"
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          "latex": "\\boxed{\nS(n)\\not\\equiv0\\pmod3\n}"
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          "latex": "3n+1\\equiv1\\pmod3,"
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          "latex": "\\boxed{\n1,2\\pmod3.\n}"
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          "latex": "1\\equiv1\\pmod3,"
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          "latex": "R_{2j}(1)\n=\n\\frac{2^{2j}-1}{3}."
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          "latex": "\\boxed{\nR_{2j}(1)\n=\n\\frac{4^j-1}{3}.\n}"
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          "latex": "\\boxed{\nM_j.\n}"
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          "latex": "M_j\n=\n\\frac{4^j-1}{3}\n=\n1,5,21,85,341,\\ldots."
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          "latex": "\\boxed{\nv_2(3M_j+1)=2j.\n}"
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          "latex": "5\n=\n\\frac{4^2-1}{3}\n=\nR_4(1)."
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          "latex": "\\boxed{\n5\n\\text{ 只是 }t=1,\\kappa=4\\text{ 的 inverse-fiber member}.\n}"
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          "latex": "5\\to16\\to8\\to4\\to2\\to1."
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          "latex": "\\boxed{\n\\mathcal R(t)\n=\n\\left\\{\n\\frac{2^\\kappa t-1}{3}:\n2^\\kappa t\\equiv1\\pmod3\n\\right\\}\n}"
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          "latex": "t\\in\\mathbb Z_{>0}^{\\mathrm{odd}},\n\\qquad\n3\\nmid t"
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          "latex": "\\boxed{\n\\kappa=v_2(3n+1).\n}"
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          "latex": "\\boxed{\nn\n\\xrightarrow{\\;\\kappa\\;}\nt\n\\iff\nn=\\frac{2^\\kappa t-1}{3}.\n}"
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          "latex": "\\boxed{\nr_w+2^k a\n\\to\nm_w+3^u a.\n}"
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          "latex": "\\boxed{\nm_w+3^u a\n\\to\nr_w+2^k a.\n}"
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          "latex": "\\boxed{\nt\n\\leftarrow\n\\frac{2^\\kappa t-1}{3}.\n}"
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          "latex": "\\boxed{\n\\text{two compatible exact coordinate directions}.\n}"
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          "latex": "r_w+2^k\\mathbb Z\n\\leftrightarrow\nm_w+3^u\\mathbb Z."
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          "latex": "\\boxed{\n\\text{local bijection}\n\\neq\n\\text{global one-to-one dynamics}.\n}"
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          "latex": "\\boxed{\n\\text{source residue class}\n\\overset{F_w}{\\longleftrightarrow}\n\\text{target progression}\n}"
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          "latex": "\\text{$2$-adic integer}\n\\leftrightarrow\n\\text{infinite parity sequence}."
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          "sha256": "70a679bccadbc8a09255e0c527551969157735c5b212cfe9bb7df96d480fe079",
          "latex": "w\\in\\{D,U\\}^k"
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          "sha256": "e0d21dcb06366cc007ee4549879beb8779980e1ad1a0de28aee478c343915d2b",
          "latex": "r_w\\bmod2^k."
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