{
  "index": "1960-B-5",
  "type": "ANA",
  "tag": [
    "ANA",
    "ALG"
  ],
  "difficulty": "",
  "question": "5. Define a sequence as follows:\n\\[\n\\begin{array}{l}\na_{0}=0 \\\\\na_{1}=1+\\sin (-1) \\\\\n\\vdots \\\\\na_{n}=1+\\sin \\left(a_{n-1}-1\\right) \\\\\n\\vdots\n\\end{array}\n\\]\n\nEvaluate\n\\[\n\\lim _{n \\rightarrow \\infty} \\frac{1}{n} \\sum_{k=1}^{n} a_{k} .\n\\]",
  "solution": "Solution. Let \\( b_{n}=a_{n}-1 \\). Then \\( b_{0}=-1 \\) and\n\\[\nb_{n}=\\sin b_{n-1}, \\quad n=1,2,3, \\ldots .\n\\]\n\nThe polygonal representation of this recursion (see page 223) suggests that \\( b_{n} \\) increases to 0 as \\( n \\rightarrow \\infty \\). Analytically, we note that for \\( -\\pi<x \\) \\( <0, x<\\sin x<0 \\). So from \\( -1 \\leq b_{n-1}<0 \\) there follows \\( b_{n-1}<b_{n}<0 \\). Hence\n\\[\n-1=b_{0}<b_{1}<b_{2}<\\cdots<0 .\n\\]\n\nTherefore \\( \\lim b_{n}=c \\) exists. Passing to the limit in (1) we obtain \\( \\sin c=c \\), so \\( c=0 \\).\n\nWe next prove that\n\\[\n\\lim _{n \\rightarrow \\infty} \\frac{1}{n} \\sum_{k=1}^{n} b_{k}=0 .\n\\]\n\nLet \\( \\epsilon>0 \\) be given and choose \\( p \\) so that \\( b_{k}>-\\epsilon \\) for \\( k>p \\). Choose \\( m \\) so that \\( m \\epsilon>p \\) (and \\( m>p \\) ). Then if \\( n>m \\), we have\n\\[\n\\sum_{k=1}^{n}\\left(-b_{k}\\right)=\\sum_{k=1}^{n}\\left(-b_{k}\\right)+\\sum_{k=p+1}^{n}\\left(-b_{k}\\right)<p+n \\epsilon<2 n \\epsilon,\n\\]\nand hence\n\\[\n0>\\frac{1}{n} \\sum_{k=1}^{n} b_{k}>-2 \\epsilon .\n\\]\n\nSince \\( \\epsilon \\) was arbitrary, (2) follows.\nFinally, we have\n\\[\n\\lim _{n \\rightarrow \\infty} \\frac{1}{n} \\sum_{k=1}^{n} a_{k}=\\lim _{n-\\infty}\\left(1+\\frac{1}{n} \\sum_{k=1}^{n} b_{k}\\right)=1 .\n\\]",
  "vars": [
    "a_0",
    "a_1",
    "a_n",
    "a_k",
    "b_n",
    "b_0",
    "b_n-1",
    "b_1",
    "b_2",
    "b_k",
    "n",
    "k",
    "x"
  ],
  "params": [
    "c",
    "\\\\epsilon",
    "p",
    "m"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "a_0": "zeroterm",
        "a_1": "firstterm",
        "a_n": "nthterm",
        "a_k": "kthterm",
        "b_n": "iterbval",
        "b_0": "bzeroterm",
        "b_n-1": "prevbterm",
        "b_1": "bfirstterm",
        "b_2": "bsecond",
        "b_k": "bkthterm",
        "n": "indexn",
        "k": "indexk",
        "x": "varxvalue",
        "c": "limitval",
        "\\epsilon": "tolerance",
        "p": "thresholdp",
        "m": "boundm"
      },
      "question": "5. Define a sequence as follows:\n\\[\n\\begin{array}{l}\nzeroterm=0 \\\\\nfirstterm=1+\\sin (-1) \\\\\n\\vdots \\\\\nnthterm=1+\\sin \\left(a_{indexn-1}-1\\right) \\\\\n\\vdots\n\\end{array}\n\\]\n\nEvaluate\n\\[\n\\lim _{indexn \\rightarrow \\infty} \\frac{1}{indexn} \\sum_{indexk=1}^{indexn} kthterm .\n\\]",
      "solution": "Solution. Let \\( iterbval = nthterm-1 \\). Then \\( bzeroterm=-1 \\) and\n\\[\niterbval = \\sin prevbterm, \\quad indexn = 1,2,3, \\ldots .\n\\]\n\nThe polygonal representation of this recursion (see page 223) suggests that iterbval increases to 0 as \\( indexn \\rightarrow \\infty \\). Analytically, we note that for \\( -\\pi<varxvalue<0,\\; varxvalue<\\sin varxvalue<0 \\). So from \\( -1 \\leq prevbterm<0 \\) there follows \\( prevbterm<iterbval<0 \\). Hence\n\\[\n-1=bzeroterm<bfirstterm<bsecond<\\cdots<0 .\n\\]\n\nTherefore \\( \\lim iterbval = limitval \\) exists. Passing to the limit in (1) we obtain \\( \\sin limitval = limitval \\), so \\( limitval = 0 \\).\n\nWe next prove that\n\\[\n\\lim _{indexn \\rightarrow \\infty} \\frac{1}{indexn} \\sum_{indexk=1}^{indexn} bkthterm = 0 .\n\\]\n\nLet tolerance>0 be given and choose thresholdp so that bkthterm>\\!-tolerance for indexk>thresholdp. Choose boundm so that boundm\\,tolerance>thresholdp (and boundm>thresholdp). Then if indexn>boundm, we have\n\\[\n\\sum_{indexk=1}^{indexn}\\left(-bkthterm\\right)=\\sum_{indexk=1}^{indexn}\\left(-bkthterm\\right)+\\sum_{indexk=thresholdp+1}^{indexn}\\left(-bkthterm\\right)<thresholdp+indexn\\,tolerance<2\\,indexn\\,tolerance,\n\\]\nand hence\n\\[\n0>\\frac{1}{indexn} \\sum_{indexk=1}^{indexn} bkthterm>-2\\,tolerance .\n\\]\n\nSince tolerance was arbitrary, (2) follows.\nFinally, we have\n\\[\n\\lim _{indexn \\rightarrow \\infty} \\frac{1}{indexn} \\sum_{indexk=1}^{indexn} kthterm = \\lim _{indexn\\to\\infty}\\left(1+\\frac{1}{indexn} \\sum_{indexk=1}^{indexn} bkthterm\\right)=1 .\n\\]"
    },
    "descriptive_long_confusing": {
      "map": {
        "a_0": "violetleaf",
        "a_1": "copperring",
        "a_n": "marblestone",
        "a_k": "silktracer",
        "b_n": "amberglint",
        "b_0": "opalshadow",
        "b_n-1": "hazelribbon",
        "b_1": "ivorygrove",
        "b_2": "pearlforge",
        "b_k": "onyxlantern",
        "n": "driftwood",
        "k": "lilystem",
        "x": "sageblossom",
        "c": "graniteveil",
        "\\\\epsilon": "zephyrwave",
        "p": "thornquill",
        "m": "emberstride"
      },
      "question": "5. Define a sequence as follows:\n\\[\n\\begin{array}{l}\nvioletleaf=0 \\\\\ncopperring=1+\\sin (-1) \\\\\n\\vdots \\\\\nmarblestone=1+\\sin \\left(a_{n-1}-1\\right) \\\\\n\\vdots\n\\end{array}\n\\]\n\nEvaluate\n\\[\n\\lim _{driftwood \\rightarrow \\infty} \\frac{1}{driftwood} \\sum_{lilystem=1}^{driftwood} silktracer .\n\\]",
      "solution": "Solution. Let \\( amberglint=marblestone-1 \\). Then \\( opalshadow=-1 \\) and\n\\[\namberglint=\\sin hazelribbon, \\quad driftwood=1,2,3, \\ldots .\n\\]\n\nThe polygonal representation of this recursion (see page 223) suggests that amberglint increases to 0 as driftwood \\rightarrow \\infty. Analytically, we note that for \\( -\\pi<sageblossom \\) \\( <0, sageblossom<\\sin sageblossom<0 \\). So from \\( -1 \\leq hazelribbon<0 \\) there follows \\( hazelribbon<amberglint<0 \\). Hence\n\\[\n-1=opalshadow<ivorygrove<pearlforge<\\cdots<0 .\n\\]\n\nTherefore \\( \\lim amberglint=graniteveil \\) exists. Passing to the limit in (1) we obtain \\( \\sin graniteveil=graniteveil \\), so graniteveil=0.\n\nWe next prove that\n\\[\n\\lim _{driftwood \\rightarrow \\infty} \\frac{1}{driftwood} \\sum_{lilystem=1}^{driftwood} onyxlantern=0 .\n\\]\n\nLet zephyrwave>0 be given and choose thornquill so that onyxlantern>-zephyrwave for lilystem>thornquill. Choose emberstride so that emberstride zephyrwave>thornquill (and emberstride>thornquill ). Then if driftwood>emberstride, we have\n\\[\n\\sum_{lilystem=1}^{driftwood}\\left(-onyxlantern\\right)=\\sum_{lilystem=1}^{driftwood}\\left(-onyxlantern\\right)+\\sum_{lilystem=thornquill+1}^{driftwood}\\left(-onyxlantern\\right)<thornquill+driftwood \\, zephyrwave<2 driftwood \\, zephyrwave,\n\\]\nand hence\n\\[\n0>\\frac{1}{driftwood} \\sum_{lilystem=1}^{driftwood} onyxlantern>-2 zephyrwave .\n\\]\n\nSince zephyrwave was arbitrary, (2) follows.\nFinally, we have\n\\[\n\\lim _{driftwood \\rightarrow \\infty} \\frac{1}{driftwood} \\sum_{lilystem=1}^{driftwood} silktracer=\\lim _{driftwood-\\infty}\\left(1+\\frac{1}{driftwood} \\sum_{lilystem=1}^{driftwood} onyxlantern\\right)=1 .\n\\]"
    },
    "descriptive_long_misleading": {
      "map": {
        "a_0": "finalterm",
        "a_1": "latenumber",
        "a_n": "earlyterm",
        "a_k": "fixedpiece",
        "b_n": "staticvalue",
        "b_0": "peakvalue",
        "b_n-1": "nextvalue",
        "b_1": "apexvalue",
        "b_2": "nadirvalue",
        "b_k": "constantpiece",
        "n": "startindex",
        "k": "endcount",
        "x": "constantval",
        "c": "changingvar",
        "\\\\epsilon": "bigdelta",
        "p": "endlessval",
        "m": "tinyindex"
      },
      "question": "5. Define a sequence as follows:\n\\[\n\\begin{array}{l}\nfinalterm=0 \\\\\nlatenumber=1+\\sin (-1) \\\\\n\\vdots \\\\\nearlyterm=1+\\sin \\left(earlyterm-1\\right) \\\\\n\\vdots\n\\end{array}\n\\]\n\nEvaluate\n\\[\n\\lim _{startindex \\rightarrow \\infty} \\frac{1}{startindex} \\sum_{endcount=1}^{startindex} fixedpiece .\n\\]",
      "solution": "Solution. Let \\( staticvalue=earlyterm-1 \\). Then \\( peakvalue=-1 \\) and\n\\[\nstaticvalue=\\sin nextvalue, \\quad startindex=1,2,3, \\ldots .\n\\]\n\nThe polygonal representation of this recursion (see page 223) suggests that \\( staticvalue \\) increases to 0 as \\( startindex \\rightarrow \\infty \\). Analytically, we note that for \\( -\\pi<constantval<0, constantval<\\sin constantval<0 \\). So from \\( -1 \\leq nextvalue<0 \\) there follows \\( nextvalue<staticvalue<0 \\). Hence\n\\[\n-1=peakvalue<apexvalue<nadirvalue<\\cdots<0 .\n\\]\n\nTherefore \\( \\lim staticvalue=changingvar \\) exists. Passing to the limit in (1) we obtain \\( \\sin changingvar=changingvar \\), so \\( changingvar=0 \\).\n\nWe next prove that\n\\[\n\\lim _{startindex \\rightarrow \\infty} \\frac{1}{startindex} \\sum_{endcount=1}^{startindex} staticvalue=0 .\n\\]\n\nLet \\( bigdelta>0 \\) be given and choose \\( endlessval \\) so that \\( staticvalue>-bigdelta \\) for \\( endcount>endlessval \\). Choose \\( tinyindex \\) so that \\( tinyindex bigdelta>endlessval \\) (and \\( tinyindex>endlessval \\) ). Then if \\( startindex>tinyindex \\), we have\n\\[\n\\sum_{endcount=1}^{startindex}\\left(-staticvalue\\right)=\\sum_{endcount=1}^{endlessval}\\left(-staticvalue\\right)+\\sum_{endcount=endlessval+1}^{startindex}\\left(-staticvalue\\right)<endlessval+startindex bigdelta<2 startindex bigdelta,\n\\]\nand hence\n\\[\n0>\\frac{1}{startindex} \\sum_{endcount=1}^{startindex} staticvalue>-2 bigdelta .\n\\]\n\nSince \\( bigdelta \\) was arbitrary, (2) follows.\nFinally, we have\n\\[\n\\lim _{startindex \\rightarrow \\infty} \\frac{1}{startindex} \\sum_{endcount=1}^{startindex} earlyterm=\\lim _{startindex \\rightarrow \\infty}\\left(1+\\frac{1}{startindex} \\sum_{endcount=1}^{startindex} staticvalue\\right)=1 .\n\\]"
    },
    "garbled_string": {
      "map": {
        "a_0": "qpcnamdz",
        "a_1": "yzgfhrso",
        "a_n": "kldhtmva",
        "a_k": "mhvcnals",
        "b_n": "jvsrqwpe",
        "b_0": "lrhxbmco",
        "b_n-1": "gztfsqwa",
        "b_1": "vdhrqmso",
        "b_2": "nfsqplkm",
        "b_k": "cbgtrmow",
        "n": "dlbqsfma",
        "k": "wprvztgh",
        "x": "tzbqnwle",
        "c": "vkshdmea",
        "\\\\epsilon": "qbznlfyu",
        "p": "hslkranv",
        "m": "dfqvbnza"
      },
      "question": "5. Define a sequence as follows:\n\\[\n\\begin{array}{l}\nqpcnamdz=0 \\\\\nyzgfhrso=1+\\sin (-1) \\\\\n\\vdots \\\\\nkldhtmva=1+\\sin \\left(a_{n-1}-1\\right) \\\\\n\\vdots\n\\end{array}\n\\]\n\nEvaluate\n\\[\n\\lim _{dlbqsfma \\rightarrow \\infty} \\frac{1}{dlbqsfma} \\sum_{wprvztgh=1}^{dlbqsfma} mhvcnals .\n\\]",
      "solution": "Solution. Let \\( jvsrqwpe = kldhtmva - 1 \\). Then \\( lrhxbmco = -1 \\) and\n\\[\njvsrqwpe = \\sin gztfsqwa, \\quad dlbqsfma = 1,2,3, \\ldots .\n\\]\n\nThe polygonal representation of this recursion (see page 223) suggests that \\( jvsrqwpe \\) increases to 0 as \\( dlbqsfma \\rightarrow \\infty \\). Analytically, we note that for \\( -\\pi< tzbqnwle <0, tzbqnwle<\\sin tzbqnwle<0 \\). So from \\( -1 \\leq jvsrqwpe <0 \\) there follows \\( gztfsqwa < jvsrqwpe <0 \\). Hence\n\\[\n-1=lrhxbmco<vdhrqmso<nfsqplkm<\\cdots<0 .\n\\]\n\nTherefore \\( \\lim jvsrqwpe = vkshdmea \\) exists. Passing to the limit in (1) we obtain \\( \\sin vkshdmea = vkshdmea \\), so \\( vkshdmea = 0 \\).\n\nWe next prove that\n\\[\n\\lim _{dlbqsfma \\rightarrow \\infty} \\frac{1}{dlbqsfma} \\sum_{wprvztgh=1}^{dlbqsfma} cbgtrmow = 0 .\n\\]\n\nLet \\( qbznlfyu > 0 \\) be given and choose \\( hslkranv \\) so that \\( cbgtrmow > -qbznlfyu \\) for \\( wprvztgh > hslkranv \\). Choose \\( dfqvbnza \\) so that \\( dfqvbnza\\, qbznlfyu > hslkranv \\) (and \\( dfqvbnza > hslkranv \\) ). Then if \\( dlbqsfma > dfqvbnza \\), we have\n\\[\n\\sum_{wprvztgh=1}^{dlbqsfma}\\left(-cbgtrmow\\right)=\\sum_{wprvztgh=1}^{dlbqsfma}\\left(-cbgtrmow\\right)+\\sum_{wprvztgh=hslkranv+1}^{dlbqsfma}\\left(-cbgtrmow\\right)<hslkranv+dlbqsfma\\, qbznlfyu<2\\, dlbqsfma\\, qbznlfyu,\n\\]\nand hence\n\\[\n0>\\frac{1}{dlbqsfma} \\sum_{wprvztgh=1}^{dlbqsfma} cbgtrmow > -2\\, qbznlfyu .\n\\]\n\nSince \\( qbznlfyu \\) was arbitrary, (2) follows.\nFinally, we have\n\\[\n\\lim _{dlbqsfma \\rightarrow \\infty} \\frac{1}{dlbqsfma} \\sum_{wprvztgh=1}^{dlbqsfma} mhvcnals = \\lim _{dlbqsfma-\\infty}\\left(1+\\frac{1}{dlbqsfma} \\sum_{wprvztgh=1}^{dlbqsfma} cbgtrmow \\right)=1 .\n\\]"
    },
    "kernel_variant": {
      "question": "Let  \n\n\\[\nd:=2024 ,\\qquad \\mathbf 1\\in\\mathbb R^{d}\n\\]\n\nbe the all–ones column vector.\n\n1.\\;(\\emph{Row–stochastic circulant kernel})  \n   Put  \n\n   \\[\n      \\beta:=1-2^{-d},\\qquad \\alpha:=\\beta^{-1},\n   \\]\n   and define the circulant matrix  \n\n   \\[\n     C:=\\operatorname{circ}\\bigl(\\alpha 2^{-1},\\alpha 2^{-2},\\dots ,\\alpha 2^{-d}\\bigr)\\tag{$\\dagger$}\n   \\]\n   whose first row is the displayed vector.  \n   Since $\\sum_{k=1}^{d}\\alpha 2^{-k}=\\alpha\\beta=1$, each row of $C$\n   sums to $1$; hence $C$ is row–stochastic.  \n   Its smallest entry equals  \n\n   \\[\n     \\varepsilon:=\\alpha 2^{-d}=2^{-d}/(1-2^{-d})>0 .\n   \\]\n\n2.\\;(\\emph{Block kernels})  \n   On $\\mathbb R^{d^{2}}=\\mathbb R^{d}\\otimes\\mathbb R^{d}$ (Kronecker basis) set  \n\n   \\[\n     K_{1}:=C\\otimes C,\\qquad \n     K_{2}:=C\\otimes I_{d}.\n   \\]\n   Both $K_{1}$ and $K_{2}$ are row–stochastic.  \n   – All entries of $K_{1}$ are positive and its minimal entry equals\n     $\\varepsilon^{2}$.  \n   – $K_{2}$ contains many zeros, so its minimal entry is $0$, while the\n     smallest \\emph{positive} entry equals $\\varepsilon$.\n\n3.\\;(\\emph{Non–linearities})  \n   Fix  \n\n   \\[\n      \\lambda:=\\tfrac16\n   \\]\n   and introduce the component–wise nonlinearities  \n\n   \\[\n     g(x)=\\lambda\\bigl(\\sin x+\\tfrac12\\sin^{2}x\\bigr),\\qquad\n     h(x)=\\lambda\\bigl(x-\\sin x\\bigr),\\qquad x\\in\\mathbb R.\\tag{$\\ddagger$}\n   \\]\n\n4.\\;(\\emph{Initial data and second–order scheme})  \n   Prescribe  \n\n   \\[\n     A_{0}(i,j)=(-1)^{\\,i+j},\\qquad A_{1}=5\\mathbf 1_{d^{2}} .\n   \\]\n   For $n\\ge 2$ define  \n\n   \\[\n      A_{n}=5\\mathbf 1_{d^{2}}\n             +K_{1}g\\!\\bigl(A_{n-1}-5\\mathbf 1_{d^{2}}\\bigr)\n             +K_{2}h\\!\\bigl(A_{n-2}-5\\mathbf 1_{d^{2}}\\bigr).\\tag{$\\star$}\n   \\]\n\n5.\\;(\\emph{Required limit})  \n   Let  \n\n   \\[\n     \\bar a_{n}:=\\frac1{d^{2}}\\mathbf 1_{d^{2}}^{\\!\\top}A_{n}\\qquad (n\\ge 0)\n   \\]\n   be the coordinate averages.  Show that the Cesàro limit  \n\n   \\[\n     L:=\\lim_{N\\to\\infty}\\frac1N\\sum_{n=1}^{N}\\bar a_{n}\\tag{1}\n   \\]\n   exists and determine its value.",
      "solution": "Throughout $\\|\\cdot\\|_{\\infty}$ denotes the supremum norm on\n$\\mathbb R^{d^{2}}$, and  \n\n\\[\n   \\operatorname{diam}(v):=\\max_{k}v_{k}-\\min_{k}v_{k}.\n\\]\n\nStep 1.\\;Uniform boundedness of $A_{n}$.  \nWrite  \n\n\\[\n   B_{n}:=A_{n}-5\\mathbf 1_{d^{2}}\\qquad(n\\ge 0).\n\\]\n\nEquation $(\\star)$ becomes  \n\n\\[\n   B_{n}=K_{1}g(B_{n-1})+K_{2}h(B_{n-2}).\\tag{2}\n\\]\n\nFrom $(\\ddagger)$ we have  \n\n\\[\n   g'(x)=\\lambda\\cos x\\,(1+\\sin x),\\qquad \n   h'(x)=\\lambda\\,(1-\\cos x).\n\\]\nHence  \n\n\\[\n   |g'(x)|\\le 2\\lambda=\\tfrac13,\\qquad\n   |h'(x)|\\le 2\\lambda=\\tfrac13\n   \\qquad\\forall x\\in\\mathbb R.\n\\]\n\nFix  \n\n\\[\n   L:=\\tfrac13,\n\\]\nso that the Lipschitz estimates  \n\n\\[\n   |g(x)-g(y)|,\\;|h(x)-h(y)|\\le L|x-y|\\tag{3}\n\\]\nhold.  Because $K_{1},K_{2}$ are row–stochastic,\n\n\\[\n   \\|K_{i}x\\|_{\\infty}\\le\\|x\\|_{\\infty}\\qquad(i=1,2).\\tag{4}\n\\]\n\nCombining (2)–(4) and using $g(0)=h(0)=0$ gives  \n\n\\[\n   \\|B_{n}\\|_{\\infty}\\le\n   L\\bigl(\\|B_{n-1}\\|_{\\infty}+\\|B_{n-2}\\|_{\\infty}\\bigr)\\qquad(n\\ge 2).\n\\]\n\nDenote  \n\n\\[\n   m_{n}:=\\|B_{n}\\|_{\\infty}\\qquad(n\\ge 0).\n\\]\n\nInitial values: $B_{0}$ takes the two values $-4$ and $-6$, so\n$m_{0}=6$, while $B_{1}=0$ gives $m_{1}=0$.\n\nIntroduce an auxiliary sequence  \n\n\\[\n   u_{0}=6,\\quad u_{1}=6,\\quad\n   u_{n}=L\\bigl(u_{n-1}+u_{n-2}\\bigr)\\quad(n\\ge 2).\\tag{5}\n\\]\n\nA simple induction shows $m_{n}\\le u_{n}$ for all $n$.  Sequence\n$(u_{n})$ solves  \n\n\\[\n   u_{n}-Lu_{n-1}-Lu_{n-2}=0.\n\\]\n\nIts characteristic polynomial $t^{2}-Lt-L=0$ has roots  \n\n\\[\n   r_{\\pm}=\\frac{L\\pm\\sqrt{L^{2}+4L}}{2}\n   \\quad\\Longrightarrow\\quad\n   0<r_{-}<r_{+}<1.\n\\]\n\nFor $L=\\tfrac13$, $r_{+}\\approx 0.767140$.  Consequently  \n\n\\[\n   u_{n}=c_{+}r_{+}^{\\,n}+c_{-}r_{-}^{\\,n}=O(r_{+}^{\\,n}),\\tag{6}\n\\]\nand therefore  \n\n\\[\n   \\|B_{n}\\|_{\\infty}=O(r_{+}^{\\,n}).\\tag{7}\n\\]\n\nThus $(B_{n})$ remains in $[-6,6]^{d^{2}}$ and tends to $0$ in the\nsup–norm.\n\nStep 2.\\;Exponential flattening of $B_{n}$.  \nLet $P$ be a row–stochastic matrix with entries $P_{ij}\\ge\\eta>0$.\nDobrushin’s contraction theorem (see, e.g.\\ P.\\ Diaconis,\n\\emph{Group Representations in Probability and Statistics}, \\S\\,3)\nstates  \n\n\\[\n   \\operatorname{diam}(Px)\\le(1-2\\eta)\\operatorname{diam}(x)\n   \\qquad\\forall x\\in\\mathbb R^{m}. \\tag{8}\n\\]\n\nAll entries of $K_{1}$ satisfy $K_{1}^{(k,\\ell)}\\ge\\varepsilon^{2}$, so\n(8) with $P=K_{1}$ yields  \n\n\\[\n   \\operatorname{diam}(K_{1}v)\\le(1-2\\varepsilon^{2})\\,\n   \\operatorname{diam}(v)\\qquad\\forall v.\\tag{9}\n\\]\n\nNo uniform positivity is available for $K_{2}$.  Put  \n\n\\[\n   D_{n}:=\\operatorname{diam}(B_{n}).\n\\]\n\nUsing (2), (3) and (9):\n\n\\[\n   D_{n}\n   \\le L\\bigl[(1-2\\varepsilon^{2})D_{\\,n-1}+D_{\\,n-2}\\bigr]\n   \\qquad(n\\ge 2). \\tag{10}\n\\]\n\nDefine a comparison sequence $(E_{n})$ by  \n\n\\[\n   E_{0}=D_{0},\\quad E_{1}=D_{1},\\quad\n   E_{n}=aE_{\\,n-1}+bE_{\\,n-2}\\quad(n\\ge 2),\\tag{11}\n\\]\nwith  \n\n\\[\n   a:=(1-2\\varepsilon^{2})L,\\qquad b:=L.\n\\]\n\nBecause $0<\\varepsilon\\ll1$ and $L=\\tfrac13$, we have $0<a<\\tfrac13$ and\n$b=\\tfrac13$.  Inequality (10) implies $D_{n}\\le E_{n}$ for all $n$.\n\nThe characteristic equation $t^{2}-at-b=0$ has largest root  \n\n\\[\n   \\rho:=\\frac{a+\\sqrt{a^{2}+4b}}{2}<1\n   \\quad(\\rho\\approx 0.768).\\tag{12}\n\\]\n\nHence  \n\n\\[\n   D_{n}=O(\\rho^{\\,n}).\\tag{13}\n\\]\n\nStep 3.\\;Reduction to a scalar recursion.  \nSet  \n\n\\[\n   \\bar a_{n}:=\\frac1{d^{2}}\\mathbf 1^{\\top}A_{n},\\qquad\n   y_{n}:=\\bar a_{n}-5=\\frac1{d^{2}}\\mathbf 1^{\\top}B_{n}.\\tag{14}\n\\]\n\nBecause $K_{1},K_{2}$ are row–stochastic, (2) gives  \n\n\\[\n   y_{n}\n     =\\frac1{d^{2}}\\mathbf 1^{\\top}g(B_{n-1})\n      +\\frac1{d^{2}}\\mathbf 1^{\\top}h(B_{n-2}).\\tag{15}\n\\]\n\nFor $n\\ge 2$ decompose  \n\n\\[\n   B_{n-1}=y_{n-1}\\mathbf 1+e_{n-1},\\qquad\n   B_{n-2}=y_{n-2}\\mathbf 1+e_{n-2},\n\\]\nwhere, by construction,  \n\n\\[\n   \\frac1{d^{2}}\\mathbf 1^{\\top}e_{n-1}=0,\\qquad\n   \\frac1{d^{2}}\\mathbf 1^{\\top}e_{n-2}=0,\n\\]\nand, by (13),\n\n\\[\n   \\|e_{n-1}\\|_{\\infty},\\;\\|e_{n-2}\\|_{\\infty}=O(\\rho^{\\,n}).\\tag{16}\n\\]\n\nUsing the mean–value theorem together with (3) we obtain, for every\ncomponent $k$,  \n\n\\[\n   |g(B_{n-1}^{(k)})-g(y_{n-1})|\n      \\le L|e_{n-1}^{(k)}|\n      \\le LC\\rho^{\\,n},\n\\]\nand similarly for $h$.  Summing over $k$ and exploiting the vanishing\naverages of $e_{n-1},e_{n-2}$ eliminates any first–order contribution,\nwhence  \n\n\\[\n   \\frac1{d^{2}}\\mathbf 1^{\\top}g(B_{n-1})\n      =g(y_{n-1})+O(\\rho^{\\,n}),\\qquad\n   \\frac1{d^{2}}\\mathbf 1^{\\top}h(B_{n-2})\n      =h(y_{n-2})+O(\\rho^{\\,n}).\\tag{17}\n\\]\n\nConsequently  \n\n\\[\n   y_{n}=g(y_{n-1})+h(y_{n-2})+r_{n},\\qquad\n   |r_{n}|\\le c\\rho^{\\,n}\\quad(n\\ge 2).\\tag{18}\n\\]\n\nStep 4.\\;Stability of the perturbed scalar recursion.  \nPut  \n\n\\[\n   s_{n}:=|y_{n}|\\qquad(n\\ge 0).\n\\]\n\nBy (3) and (18),  \n\n\\[\n   s_{n}\\le L\\bigl(s_{n-1}+s_{n-2}\\bigr)+c\\rho^{\\,n}\\qquad(n\\ge 2).\\tag{19}\n\\]\n\nConsider the inhomogeneous linear recurrence  \n\n\\[\n   T_{n}=L\\bigl(T_{n-1}+T_{n-2}\\bigr)+c\\rho^{\\,n},\\qquad\n   T_{0}=5,\\;T_{1}=5.\\tag{20}\n\\]\n\nAs before, let $r_{\\pm}$ be the roots of $t^{2}-Lt-L=0$.  Seek a\nparticular solution $T_{n}^{\\text{p}}=K\\rho^{\\,n}$.  Substituting into\n(20) gives  \n\n\\[\n   K\\rho^{\\,2}-LK\\rho-LK=c\\rho^{\\,2}\n   \\Longrightarrow\n   K=\\frac{c\\rho^{\\,2}}{\\rho^{\\,2}-L(\\rho+1)}\n   =\\frac{c\\rho}{\\rho-L}\\,.\\tag{21}\n\\]\n\nBecause $\\rho\\in(r_{+},1)$ and $L<\\rho$, the denominator is strictly\npositive but extremely small ($\\mathcal O(\\varepsilon^{2})$); hence $K$\nis very large.  Nevertheless the factor $\\rho^{\\,n}$ in\n$T_{n}^{\\text{p}}$ guarantees exponential decay, so boundedness is\npreserved.  The general solution is  \n\n\\[\n   T_{n}=A_{+}r_{+}^{\\,n}+A_{-}r_{-}^{\\,n}+K\\rho^{\\,n}.\\tag{22}\n\\]\nSince $|r_{+}|,|r_{-}|,\\rho<1$,  \n\n\\[\n   T_{n}=O(\\rho_{*}^{\\,n}),\\qquad\n   \\rho_{*}:=\\max\\{r_{+},\\rho\\}<1. \n\\]\n\nBecause $T_{0}\\ge s_{0}$ and $T_{1}\\ge s_{1}$, induction based on\n(19)–(20) yields  \n\n\\[\n   s_{n}\\le T_{n}\\quad\\forall n\\quad\\Longrightarrow\\quad\n   s_{n}=O(\\rho_{*}^{\\,n}).\\tag{23}\n\\]\n\nHence  \n\n\\[\n   \\lim_{n\\to\\infty}y_{n}=0\\quad\\text{exponentially fast}.\n\\]\n\nStep 5.\\;Cesàro limit of the averages.  \nSince $\\bar a_{n}=5+y_{n}$ and $y_{n}\\to0$,  \n\n\\[\n   L=\\lim_{N\\to\\infty}\\frac1N\\sum_{n=1}^{N}\\bar a_{n}\n     =\\lim_{N\\to\\infty}\\Bigl(\n       5+\\frac1N\\sum_{n=1}^{N}y_{n}\\Bigr)=5.\\tag{24}\n\\]\n\nStep 6.\\;Conclusion for the full vectors.  \nBecause $\\bar a_{n}=\\tfrac1{d^{2}}\\mathbf 1^{\\!\\top}A_{n}$, equation\n(24) is exactly the value required in (1):\n\n\\[\n   \\boxed{L=5}.\n\\]\n\n\\hfill$\\square$",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.524727",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher dimensionality: the problem moves from 2024 variables to d² = 2024² ≈ 4 million variables via Kronecker products, greatly enlarging the state space.\n2. Second-order non-linear recursion: unlike the first-order scheme of the kernel variant, (★) depends on the two previous iterates, forcing analysis of a dynamical system on ℝ^{d²}×ℝ^{d²}.\n3. Multiple interacting stochastic operators (K₁, K₂) require recognition of invariance properties and tensor-product structure; simple eigenvalue observations for a single circulant matrix no longer suffice.\n4. Two different non-linearities (g and h) act simultaneously, and both enter with full matrix coupling, so one must control combined Lipschitz constants and fixed-point behaviour.\n5. Proving convergence entails a mix of techniques: identification of an invariant sub-space, reduction to a coupled scalar system, fixed-point analysis with trigonometric equations, and contraction/Lyapunov arguments for a two-step map.\n6. The final Cesàro evaluation is easy only after all previous difficulties are resolved; naïve summation or pattern spotting is impossible.  \n\nCollectively, these additions render the enhanced variant substantially more intricate than both the original single-scalar problem and the current kernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Let  \n\n\\[\nd:=2024 ,\\qquad \\mathbf 1\\in\\mathbb R^{d}\n\\]\n\nbe the all–ones column vector.\n\n1.\\;(\\emph{Row–stochastic circulant kernel})  \n   Put  \n\n   \\[\n      \\beta:=1-2^{-d},\\qquad \\alpha:=\\beta^{-1},\n   \\]\n   and define the circulant matrix  \n\n   \\[\n     C:=\\operatorname{circ}\\bigl(\\alpha 2^{-1},\\alpha 2^{-2},\\dots ,\\alpha 2^{-d}\\bigr)\\tag{$\\dagger$}\n   \\]\n   whose first row is the displayed vector.  \n   Since $\\sum_{k=1}^{d}\\alpha 2^{-k}=\\alpha\\beta=1$, each row of $C$\n   sums to $1$; hence $C$ is row–stochastic.  \n   Its smallest entry equals  \n\n   \\[\n     \\varepsilon:=\\alpha 2^{-d}=2^{-d}/(1-2^{-d})>0 .\n   \\]\n\n2.\\;(\\emph{Block kernels})  \n   On $\\mathbb R^{d^{2}}=\\mathbb R^{d}\\otimes\\mathbb R^{d}$ (Kronecker basis) set  \n\n   \\[\n     K_{1}:=C\\otimes C,\\qquad \n     K_{2}:=C\\otimes I_{d}.\n   \\]\n   Both $K_{1}$ and $K_{2}$ are row–stochastic.  \n   – All entries of $K_{1}$ are positive and its minimal entry equals\n     $\\varepsilon^{2}$.  \n   – $K_{2}$ contains many zeros, so its minimal entry is $0$, while the\n     smallest \\emph{positive} entry equals $\\varepsilon$.\n\n3.\\;(\\emph{Non–linearities})  \n   Fix  \n\n   \\[\n      \\lambda:=\\tfrac16\n   \\]\n   and introduce the component–wise nonlinearities  \n\n   \\[\n     g(x)=\\lambda\\bigl(\\sin x+\\tfrac12\\sin^{2}x\\bigr),\\qquad\n     h(x)=\\lambda\\bigl(x-\\sin x\\bigr),\\qquad x\\in\\mathbb R.\\tag{$\\ddagger$}\n   \\]\n\n4.\\;(\\emph{Initial data and second–order scheme})  \n   Prescribe  \n\n   \\[\n     A_{0}(i,j)=(-1)^{\\,i+j},\\qquad A_{1}=5\\mathbf 1_{d^{2}} .\n   \\]\n   For $n\\ge 2$ define  \n\n   \\[\n      A_{n}=5\\mathbf 1_{d^{2}}\n             +K_{1}g\\!\\bigl(A_{n-1}-5\\mathbf 1_{d^{2}}\\bigr)\n             +K_{2}h\\!\\bigl(A_{n-2}-5\\mathbf 1_{d^{2}}\\bigr).\\tag{$\\star$}\n   \\]\n\n5.\\;(\\emph{Required limit})  \n   Let  \n\n   \\[\n     \\bar a_{n}:=\\frac1{d^{2}}\\mathbf 1_{d^{2}}^{\\!\\top}A_{n}\\qquad (n\\ge 0)\n   \\]\n   be the coordinate averages.  Show that the Cesàro limit  \n\n   \\[\n     L:=\\lim_{N\\to\\infty}\\frac1N\\sum_{n=1}^{N}\\bar a_{n}\\tag{1}\n   \\]\n   exists and determine its value.",
      "solution": "Throughout $\\|\\cdot\\|_{\\infty}$ denotes the supremum norm on\n$\\mathbb R^{d^{2}}$, and  \n\n\\[\n   \\operatorname{diam}(v):=\\max_{k}v_{k}-\\min_{k}v_{k}.\n\\]\n\nStep 1.\\;Uniform boundedness of $A_{n}$.  \nWrite  \n\n\\[\n   B_{n}:=A_{n}-5\\mathbf 1_{d^{2}}\\qquad(n\\ge 0).\n\\]\n\nEquation $(\\star)$ becomes  \n\n\\[\n   B_{n}=K_{1}g(B_{n-1})+K_{2}h(B_{n-2}).\\tag{2}\n\\]\n\nFrom $(\\ddagger)$ we have  \n\n\\[\n   g'(x)=\\lambda\\cos x\\,(1+\\sin x),\\qquad \n   h'(x)=\\lambda\\,(1-\\cos x).\n\\]\nHence  \n\n\\[\n   |g'(x)|\\le 2\\lambda=\\tfrac13,\\qquad\n   |h'(x)|\\le 2\\lambda=\\tfrac13\n   \\qquad\\forall x\\in\\mathbb R.\n\\]\n\nFix  \n\n\\[\n   L:=\\tfrac13,\n\\]\nso that the Lipschitz estimates  \n\n\\[\n   |g(x)-g(y)|,\\;|h(x)-h(y)|\\le L|x-y|\\tag{3}\n\\]\nhold.  Because $K_{1},K_{2}$ are row–stochastic,\n\n\\[\n   \\|K_{i}x\\|_{\\infty}\\le\\|x\\|_{\\infty}\\qquad(i=1,2).\\tag{4}\n\\]\n\nCombining (2)–(4) and using $g(0)=h(0)=0$ gives  \n\n\\[\n   \\|B_{n}\\|_{\\infty}\\le\n   L\\bigl(\\|B_{n-1}\\|_{\\infty}+\\|B_{n-2}\\|_{\\infty}\\bigr)\\qquad(n\\ge 2).\n\\]\n\nDenote  \n\n\\[\n   m_{n}:=\\|B_{n}\\|_{\\infty}\\qquad(n\\ge 0).\n\\]\n\nInitial values: $B_{0}$ takes the two values $-4$ and $-6$, so\n$m_{0}=6$, while $B_{1}=0$ gives $m_{1}=0$.\n\nIntroduce an auxiliary sequence  \n\n\\[\n   u_{0}=6,\\quad u_{1}=6,\\quad\n   u_{n}=L\\bigl(u_{n-1}+u_{n-2}\\bigr)\\quad(n\\ge 2).\\tag{5}\n\\]\n\nA simple induction shows $m_{n}\\le u_{n}$ for all $n$.  Sequence\n$(u_{n})$ solves  \n\n\\[\n   u_{n}-Lu_{n-1}-Lu_{n-2}=0.\n\\]\n\nIts characteristic polynomial $t^{2}-Lt-L=0$ has roots  \n\n\\[\n   r_{\\pm}=\\frac{L\\pm\\sqrt{L^{2}+4L}}{2}\n   \\quad\\Longrightarrow\\quad\n   0<r_{-}<r_{+}<1.\n\\]\n\nFor $L=\\tfrac13$, $r_{+}\\approx 0.767140$.  Consequently  \n\n\\[\n   u_{n}=c_{+}r_{+}^{\\,n}+c_{-}r_{-}^{\\,n}=O(r_{+}^{\\,n}),\\tag{6}\n\\]\nand therefore  \n\n\\[\n   \\|B_{n}\\|_{\\infty}=O(r_{+}^{\\,n}).\\tag{7}\n\\]\n\nThus $(B_{n})$ remains in $[-6,6]^{d^{2}}$ and tends to $0$ in the\nsup–norm.\n\nStep 2.\\;Exponential flattening of $B_{n}$.  \nLet $P$ be a row–stochastic matrix with entries $P_{ij}\\ge\\eta>0$.\nDobrushin’s contraction theorem (see, e.g.\\ P.\\ Diaconis,\n\\emph{Group Representations in Probability and Statistics}, \\S\\,3)\nstates  \n\n\\[\n   \\operatorname{diam}(Px)\\le(1-2\\eta)\\operatorname{diam}(x)\n   \\qquad\\forall x\\in\\mathbb R^{m}. \\tag{8}\n\\]\n\nAll entries of $K_{1}$ satisfy $K_{1}^{(k,\\ell)}\\ge\\varepsilon^{2}$, so\n(8) with $P=K_{1}$ yields  \n\n\\[\n   \\operatorname{diam}(K_{1}v)\\le(1-2\\varepsilon^{2})\\,\n   \\operatorname{diam}(v)\\qquad\\forall v.\\tag{9}\n\\]\n\nNo uniform positivity is available for $K_{2}$.  Put  \n\n\\[\n   D_{n}:=\\operatorname{diam}(B_{n}).\n\\]\n\nUsing (2), (3) and (9):\n\n\\[\n   D_{n}\n   \\le L\\bigl[(1-2\\varepsilon^{2})D_{\\,n-1}+D_{\\,n-2}\\bigr]\n   \\qquad(n\\ge 2). \\tag{10}\n\\]\n\nDefine a comparison sequence $(E_{n})$ by  \n\n\\[\n   E_{0}=D_{0},\\quad E_{1}=D_{1},\\quad\n   E_{n}=aE_{\\,n-1}+bE_{\\,n-2}\\quad(n\\ge 2),\\tag{11}\n\\]\nwith  \n\n\\[\n   a:=(1-2\\varepsilon^{2})L,\\qquad b:=L.\n\\]\n\nBecause $0<\\varepsilon\\ll1$ and $L=\\tfrac13$, we have $0<a<\\tfrac13$ and\n$b=\\tfrac13$.  Inequality (10) implies $D_{n}\\le E_{n}$ for all $n$.\n\nThe characteristic equation $t^{2}-at-b=0$ has largest root  \n\n\\[\n   \\rho:=\\frac{a+\\sqrt{a^{2}+4b}}{2}<1\n   \\quad(\\rho\\approx 0.768).\\tag{12}\n\\]\n\nHence  \n\n\\[\n   D_{n}=O(\\rho^{\\,n}).\\tag{13}\n\\]\n\nStep 3.\\;Reduction to a scalar recursion.  \nSet  \n\n\\[\n   \\bar a_{n}:=\\frac1{d^{2}}\\mathbf 1^{\\top}A_{n},\\qquad\n   y_{n}:=\\bar a_{n}-5=\\frac1{d^{2}}\\mathbf 1^{\\top}B_{n}.\\tag{14}\n\\]\n\nBecause $K_{1},K_{2}$ are row–stochastic, (2) gives  \n\n\\[\n   y_{n}\n     =\\frac1{d^{2}}\\mathbf 1^{\\top}g(B_{n-1})\n      +\\frac1{d^{2}}\\mathbf 1^{\\top}h(B_{n-2}).\\tag{15}\n\\]\n\nFor $n\\ge 2$ decompose  \n\n\\[\n   B_{n-1}=y_{n-1}\\mathbf 1+e_{n-1},\\qquad\n   B_{n-2}=y_{n-2}\\mathbf 1+e_{n-2},\n\\]\nwhere, by construction,  \n\n\\[\n   \\frac1{d^{2}}\\mathbf 1^{\\top}e_{n-1}=0,\\qquad\n   \\frac1{d^{2}}\\mathbf 1^{\\top}e_{n-2}=0,\n\\]\nand, by (13),\n\n\\[\n   \\|e_{n-1}\\|_{\\infty},\\;\\|e_{n-2}\\|_{\\infty}=O(\\rho^{\\,n}).\\tag{16}\n\\]\n\nUsing the mean–value theorem together with (3) we obtain, for every\ncomponent $k$,  \n\n\\[\n   |g(B_{n-1}^{(k)})-g(y_{n-1})|\n      \\le L|e_{n-1}^{(k)}|\n      \\le LC\\rho^{\\,n},\n\\]\nand similarly for $h$.  Summing over $k$ and exploiting the vanishing\naverages of $e_{n-1},e_{n-2}$ eliminates any first–order contribution,\nwhence  \n\n\\[\n   \\frac1{d^{2}}\\mathbf 1^{\\top}g(B_{n-1})\n      =g(y_{n-1})+O(\\rho^{\\,n}),\\qquad\n   \\frac1{d^{2}}\\mathbf 1^{\\top}h(B_{n-2})\n      =h(y_{n-2})+O(\\rho^{\\,n}).\\tag{17}\n\\]\n\nConsequently  \n\n\\[\n   y_{n}=g(y_{n-1})+h(y_{n-2})+r_{n},\\qquad\n   |r_{n}|\\le c\\rho^{\\,n}\\quad(n\\ge 2).\\tag{18}\n\\]\n\nStep 4.\\;Stability of the perturbed scalar recursion.  \nPut  \n\n\\[\n   s_{n}:=|y_{n}|\\qquad(n\\ge 0).\n\\]\n\nBy (3) and (18),  \n\n\\[\n   s_{n}\\le L\\bigl(s_{n-1}+s_{n-2}\\bigr)+c\\rho^{\\,n}\\qquad(n\\ge 2).\\tag{19}\n\\]\n\nConsider the inhomogeneous linear recurrence  \n\n\\[\n   T_{n}=L\\bigl(T_{n-1}+T_{n-2}\\bigr)+c\\rho^{\\,n},\\qquad\n   T_{0}=5,\\;T_{1}=5.\\tag{20}\n\\]\n\nAs before, let $r_{\\pm}$ be the roots of $t^{2}-Lt-L=0$.  Seek a\nparticular solution $T_{n}^{\\text{p}}=K\\rho^{\\,n}$.  Substituting into\n(20) gives  \n\n\\[\n   K\\rho^{\\,2}-LK\\rho-LK=c\\rho^{\\,2}\n   \\Longrightarrow\n   K=\\frac{c\\rho^{\\,2}}{\\rho^{\\,2}-L(\\rho+1)}\n   =\\frac{c\\rho}{\\rho-L}\\,.\\tag{21}\n\\]\n\nBecause $\\rho\\in(r_{+},1)$ and $L<\\rho$, the denominator is strictly\npositive but extremely small ($\\mathcal O(\\varepsilon^{2})$); hence $K$\nis very large.  Nevertheless the factor $\\rho^{\\,n}$ in\n$T_{n}^{\\text{p}}$ guarantees exponential decay, so boundedness is\npreserved.  The general solution is  \n\n\\[\n   T_{n}=A_{+}r_{+}^{\\,n}+A_{-}r_{-}^{\\,n}+K\\rho^{\\,n}.\\tag{22}\n\\]\nSince $|r_{+}|,|r_{-}|,\\rho<1$,  \n\n\\[\n   T_{n}=O(\\rho_{*}^{\\,n}),\\qquad\n   \\rho_{*}:=\\max\\{r_{+},\\rho\\}<1. \n\\]\n\nBecause $T_{0}\\ge s_{0}$ and $T_{1}\\ge s_{1}$, induction based on\n(19)–(20) yields  \n\n\\[\n   s_{n}\\le T_{n}\\quad\\forall n\\quad\\Longrightarrow\\quad\n   s_{n}=O(\\rho_{*}^{\\,n}).\\tag{23}\n\\]\n\nHence  \n\n\\[\n   \\lim_{n\\to\\infty}y_{n}=0\\quad\\text{exponentially fast}.\n\\]\n\nStep 5.\\;Cesàro limit of the averages.  \nSince $\\bar a_{n}=5+y_{n}$ and $y_{n}\\to0$,  \n\n\\[\n   L=\\lim_{N\\to\\infty}\\frac1N\\sum_{n=1}^{N}\\bar a_{n}\n     =\\lim_{N\\to\\infty}\\Bigl(\n       5+\\frac1N\\sum_{n=1}^{N}y_{n}\\Bigr)=5.\\tag{24}\n\\]\n\nStep 6.\\;Conclusion for the full vectors.  \nBecause $\\bar a_{n}=\\tfrac1{d^{2}}\\mathbf 1^{\\!\\top}A_{n}$, equation\n(24) is exactly the value required in (1):\n\n\\[\n   \\boxed{L=5}.\n\\]\n\n\\hfill$\\square$",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.438691",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher dimensionality: the problem moves from 2024 variables to d² = 2024² ≈ 4 million variables via Kronecker products, greatly enlarging the state space.\n2. Second-order non-linear recursion: unlike the first-order scheme of the kernel variant, (★) depends on the two previous iterates, forcing analysis of a dynamical system on ℝ^{d²}×ℝ^{d²}.\n3. Multiple interacting stochastic operators (K₁, K₂) require recognition of invariance properties and tensor-product structure; simple eigenvalue observations for a single circulant matrix no longer suffice.\n4. Two different non-linearities (g and h) act simultaneously, and both enter with full matrix coupling, so one must control combined Lipschitz constants and fixed-point behaviour.\n5. Proving convergence entails a mix of techniques: identification of an invariant sub-space, reduction to a coupled scalar system, fixed-point analysis with trigonometric equations, and contraction/Lyapunov arguments for a two-step map.\n6. The final Cesàro evaluation is easy only after all previous difficulties are resolved; naïve summation or pattern spotting is impossible.  \n\nCollectively, these additions render the enhanced variant substantially more intricate than both the original single-scalar problem and the current kernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}