{ "what": "mu(5) <= 1 + pi/m, certified per rung: component (i) — the reduction of the non-gcd-bounded tuples to a finite list — is CONSUMED from Mercer, INTEGERS 19 (2019) #A4 §6 (his Lemma 6.2; the general-m statement is his p. 16), the way Krawczyk's theorem is consumed elsewhere in this lab. Components (ii) the finite search and (iii) the case checks are certified here in exact rational arithmetic. mu is indexed by NUMBER OF TERMS. m = 5, 6 are calibrations against Mercer's printed tables (and, at 6, the source lab's record); m >= 7 are new.", "sourcePdf": { "file": "corpus/sources/mercer2019_integers.pdf", "sha256": "1f648ff3aff79991fdf57213a0a0a270b83db64eefe1f4477739cb9ea7f10749" }, "rows": { "5": { "m": 5, "statement": "mu(5) <= 1 + pi/5 = 1.6283185307179586 — component (i), the reduction to this finite list, is Mercer 2019 §6 (general m stated p. 16; his Lemma 6.2), CONSUMED; components (ii) the finite search and (iii) the case checks are certified here in exact arithmetic.", "bar": { "piLoExact": "62225653328057771307630486155/19807040628566084398385987584", "onePlusExact": "32252171294177638659912084815/19807040628566084398385987584", "onePlusFloat": 1.6283185307179586, "barExact": "1040202553188976099411873896361567962548693624879753584225/392318858461667547739736838950479151006397215279002157056", "barFloat": 2.6514212374794917, "provenance": "instruments/bigfloat/functions.pi(96), Machin, outward-rounded; piLo = the exact rational of the dyadic lower end; pi is irrational and piLo dyadic, so piLo < pi STRICTLY" }, "search": { "poolSize": 5, "examined": 250, "rows": [ "1/4 1/4 1/3 2/3" ] }, "discarded": [], "cases": [ { "tuple": [ 1, 2, 3, 4 ], "fromRows": [ { "r": "1/4", "s": "1/4", "u": "1/3", "v": "2/3", "t": "1/2", "w": "1" } ], "caseCheck": { "A": [ 0, 1, 2, 3, 4 ], "gcdOfDiffs": 1, "verdict": "CERTIFIED", "how": "exact-point", "at": { "y": "-1", "label": "y=-1 (z=-1)" }, "gExact": "1", "gFloat": 1 }, "minEnclosure": { "modSq": [ -1.5397067145083465e-26, 1.606022177801467e-29 ], "modulus": [ 0, 4.0075206522255975e-15 ] }, "crossCheck": { "ok": true, "sampledGMin": 2.465190328815662e-32, "K": 200000 } } ], "conservation": { "rows": 1, "derived": 1, "discarded": 0, "distinctTuples": 1, "certified": 1, "failed": 0, "refused": 0 }, "verdict": "CERTIFIED", "elapsedMs": 24, "calibration": { "printed": "Mercer Table 5 conclusion", "matches": true } }, "6": { "m": 6, "statement": "mu(5) <= 1 + pi/6 = 1.5235987755982987 — component (i), the reduction to this finite list, is Mercer 2019 §6 (general m stated p. 16; his Lemma 6.2), CONSUMED; components (ii) the finite search and (iii) the case checks are certified here in exact arithmetic.", "bar": { "piLoExact": "62225653328057771307630486155/19807040628566084398385987584", "onePlusExact": "181067897099454277697946411659/118842243771396506390315925504", "onePlusFloat": 1.523598775598299, "barExact": "32785583360018562830733469308003063245709624633674291132281/14123478904620031718630526202217249436230299750044077654016", "barFloat": 2.3213532290046355, "provenance": "instruments/bigfloat/functions.pi(96), Machin, outward-rounded; piLo = the exact rational of the dyadic lower end; pi is irrational and piLo dyadic, so piLo < pi STRICTLY" }, "search": { "poolSize": 9, "examined": 2916, "rows": [ "1/5 1/5 1/4 1/2", "1/5 2/5 1/4 3/4", "1/4 1/4 1/3 2/3", "1/3 1/3 2/5 3/5", "2/5 1/5 1/2 1/2", "3/5 1/5 2/3 1/3" ] }, "discarded": [], "cases": [ { "tuple": [ 2, 3, 4, 5 ], "fromRows": [ { "r": "1/5", "s": "1/5", "u": "1/4", "v": "1/2", "t": "2/5", "w": "3/4" } ], "caseCheck": { "A": [ 0, 2, 3, 4, 5 ], "gcdOfDiffs": 1, "verdict": "CERTIFIED", "how": "exact-point", "at": { "y": "-1", "label": "y=-1 (z=-1)" }, "gExact": "1", "gFloat": 1 }, "minEnclosure": { "modSq": [ 0.1360692973743835, 0.13606929737438359 ], "modulus": [ 0.36887572077107955, 0.3688757207710798 ] }, "crossCheck": { "ok": true, "sampledGMin": 0.13606929755748512, "K": 200000 } }, { "tuple": [ 1, 2, 4, 5 ], "fromRows": [ { "r": "1/5", "s": "2/5", "u": "1/4", "v": "3/4", "t": "3/5", "w": "1" } ], "caseCheck": { "A": [ 0, 1, 2, 4, 5 ], "gcdOfDiffs": 1, "verdict": "CERTIFIED", "how": "exact-point", "at": { "y": "-1", 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"y": "-1", "label": "y=-1 (z=-1)" }, "gExact": "1", "gFloat": 1 }, "minEnclosure": { "modSq": [ 0.032219090073830794, 0.0322190900738308 ], "modulus": [ 0.17949676897880582, 0.1794967689788059 ] }, "crossCheck": { "ok": true, "sampledGMin": 0.03221909052006608, "K": 200000 } } ], "conservation": { "rows": 6, "derived": 6, "discarded": 0, "distinctTuples": 6, "certified": 6, "failed": 0, "refused": 0 }, "verdict": "CERTIFIED", "elapsedMs": 129, "calibration": { "printed": "Mercer Tables 6 and 7", "matches": true } }, "7": { "m": 7, "statement": "mu(5) <= 1 + pi/7 = 1.4487989505128276 — component (i), the reduction to this finite list, is Mercer 2019 §6 (general m stated p. 16; his Lemma 6.2), CONSUMED; components (ii) the finite search and (iii) the case checks are certified here in exact arithmetic.", "bar": { "piLoExact": "62225653328057771307630486155/19807040628566084398385987584", "onePlusExact": "200874937728020362096332399243/138649284399962590788701913088", "onePlusFloat": 1.4487989505128276, "barExact": "40350740607236058271649564884867885729714703075912746973049/19223624064621709839247105108573478399313463548671105695744", "barFloat": 2.0990183990070705, "provenance": "instruments/bigfloat/functions.pi(96), Machin, outward-rounded; piLo = the exact rational of the dyadic lower end; pi is irrational and piLo dyadic, so piLo < pi STRICTLY" }, "search": { "poolSize": 11, "examined": 6655, "rows": [ "1/6 1/6 1/5 2/5", "1/6 1/3 1/5 3/5", "1/6 1/2 1/5 4/5", "1/6 1/6 1/4 3/4", "1/5 1/5 1/4 1/2", "1/5 2/5 1/4 3/4", "1/4 1/4 1/3 2/3", "1/3 1/6 2/5 2/5", "1/3 1/3 2/5 3/5", "2/5 1/5 1/2 1/2", "1/2 1/6 3/5 2/5", "3/5 1/5 2/3 1/3" ] }, "discarded": [], "cases": [ { "tuple": [ 3, 4, 5, 6 ], "fromRows": [ { "r": "1/6", "s": "1/6", "u": "1/5", "v": "2/5", "t": "1/3", "w": "3/5" } ], "caseCheck": { "A": [ 0, 3, 4, 5, 6 ], "gcdOfDiffs": 1, "verdict": "CERTIFIED", "how": "exact-point", "at": { "y": "-1", "label": "y=-1 (z=-1)" }, "gExact": "1", "gFloat": 1 }, 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case checks are certified here in exact arithmetic.", "bar": { "piLoExact": "62225653328057771307630486155/19807040628566084398385987584", "onePlusExact": "71866252551309807456684059983/59421121885698253195157962752", "onePlusFloat": 1.2094395102393196, "barExact": "5164758255768643409449575276168838578968637204556341960289/3530869726155007929657631550554312359057574937511019413504", "barFloat": 1.4627439289279252, "provenance": "instruments/bigfloat/functions.pi(96), Machin, outward-rounded; piLo = the exact rational of the dyadic lower end; pi is irrational and piLo dyadic, so piLo < pi STRICTLY" }, "search": { "poolSize": 63, "examined": 7751457, "rows": [ "1/14 1/14 1/13 2/13", "1/14 1/7 1/13 3/13", "1/14 3/14 1/13 4/13", "1/14 2/7 1/13 5/13", "1/14 5/14 1/13 6/13", "1/14 3/7 1/13 7/13", "1/14 1/2 1/13 8/13", "1/14 4/7 1/13 9/13", "1/14 9/14 1/13 10/13", "1/14 5/7 1/13 11/13", "1/14 11/14 1/13 12/13", "1/14 1/14 1/12 1/4", "1/14 1/7 1/12 1/3", "1/14 3/14 1/12 5/12", "1/14 2/7 1/12 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0.008026929631858177 ], "modulus": [ 0.08959313384327044, 0.08959313384327049 ] }, "crossCheck": { "ok": true, "sampledGMin": 0.008026995317479056, "K": 200000 } }, { "tuple": [ 1, 25, 27, 28 ], "fromRows": [ { "r": "6/7", "s": "1/14", "u": "8/9", "v": "1/9", "t": "13/14", "w": "1" } ], "caseCheck": { "A": [ 0, 1, 25, 27, 28 ], "gcdOfDiffs": 1, "verdict": "CERTIFIED", "how": "exact-point", "at": { "y": "-1", "label": "y=-1 (z=-1)" }, "gExact": "1", "gFloat": 1 }, "minEnclosure": { "modSq": [ 0.0007839123982922222, 0.0007839123982922223 ], "modulus": [ 0.02799843564008929, 0.027998435640089296 ] }, "crossCheck": { "ok": true, "sampledGMin": 0.0007839602979420098, "K": 200000 } } ], "conservation": { "rows": 749, "derived": 749, "discarded": 0, "distinctTuples": 749, "certified": 749, "failed": 0, "refused": 0 }, "verdict": "CERTIFIED", "elapsedMs": 44131 }, "16": { "m": 16, "statement": "mu(5) <= 1 + pi/16 = 1.196349540849362 — component (i), the reduction to this finite list, is Mercer 2019 §6 (general m stated p. 16; his Lemma 6.2), CONSUMED; components (ii) the finite search and (iii) the case checks are certified here in exact arithmetic.", "bar": { "piLoExact": "62225653328057771307630486155/19807040628566084398385987584", "onePlusExact": "379138303385115121681806287499/316912650057057350374175801344", "onePlusFloat": 1.196349540849362, "barExact": "143745853093743596536470740582194411676336158431169043675001/100433627766186892221372630771322662657637687111424552206336", "barFloat": 1.4312522238904795, "provenance": "instruments/bigfloat/functions.pi(96), Machin, outward-rounded; piLo = the exact rational of the dyadic lower end; pi is irrational and piLo dyadic, so piLo < pi STRICTLY" }, "search": { "poolSize": 71, "examined": 12526885, "rows": [ "1/15 1/15 1/14 1/7", "1/15 2/15 1/14 3/14", "1/15 1/5 1/14 2/7", "1/15 4/15 1/14 5/14", "1/15 1/3 1/14 3/7", "1/15 2/5 1/14 1/2", "1/15 7/15 1/14 4/7", "1/15 8/15 1/14 9/14", "1/15 3/5 1/14 5/7", "1/15 2/3 1/14 11/14", 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"distinctTuples": 1289, "certified": 1289, "failed": 0, "refused": 0 }, "verdict": "CERTIFIED", "elapsedMs": 95710 }, "18": { "m": 18, "statement": "mu(5) <= 1 + pi/18 = 1.174532925199433 — component (i), the reduction to this finite list, is Mercer 2019 §6 (general m stated p. 16; his Lemma 6.2), CONSUMED; components (ii) the finite search and (iii) the case checks are certified here in exact arithmetic.", "bar": { "piLoExact": "62225653328057771307630486155/19807040628566084398385987584", "onePlusExact": "418752384642247290478578262667/356526731314189519170947776512", "onePlusFloat": 1.174532925199433, "barExact": "175353559643568624423371878971844180986614998357364045952889/127111310141580285467674735819955244926072697750396698886144", "barFloat": 1.3795275923775367, "provenance": "instruments/bigfloat/functions.pi(96), Machin, outward-rounded; piLo = the exact rational of the dyadic lower end; pi is irrational and piLo dyadic, so piLo < pi STRICTLY" }, "search": { "poolSize": 95, 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true, "sampledGMin": 0.01615167240941147, "K": 200000 } } ], "conservation": { "rows": 1850, "derived": 1850, "discarded": 0, "distinctTuples": 1850, "certified": 1850, "failed": 0, "refused": 0 }, "verdict": "CERTIFIED", "elapsedMs": 193022 }, "19": { "m": 19, "statement": "mu(5) <= 1 + pi/19 = 1.1653469817678839 — component (i), the reduction to this finite list, is Mercer 2019 §6 (general m stated p. 16; his Lemma 6.2), CONSUMED; components (ii) the finite search and (iii) the case checks are certified here in exact arithmetic.", "bar": { "piLoExact": "62225653328057771307630486155/19807040628566084398385987584", "onePlusExact": "438559425270813374876964250251/376333771942755603569333764096", "onePlusFloat": 1.1653469817678839, "barExact": "192334369493866141010041658683520503094773609966298553563001/141627107904661984734044998861122973513309394715719778697216", "barFloat": 1.3580335879155165, "provenance": "instruments/bigfloat/functions.pi(96), Machin, outward-rounded; piLo = the 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