{ "what": "A chain of iterations of GNNW Theorem 14 (arXiv 2407.19026v2), each decided on all of (0, 1]: step 1 is the remark's G_AI in the region of the proved F_0.03; step k > 1 is a degree-9 correction proposed by this program's float optimiser, in the region of the bound step k-1 establishes (Lemma 15's hypotheses for that bound decided first). The conclusion is the paper's theorem applied six times: R(k, k) <= c^(k+o(k)) with c = e^F(1) of the last step. Re-run: python3 verify/verify_gnnw_gai.py certs/gnnw-chain-certificate.json", "generated": "2026-09-29", "source": { "paper": "arXiv:2407.19026v2", "sha256": "2d5be61d04424f096ab1e4f6d94aefaacda71189276f281807ee5e8650c2eae9" }, "steps": [ { "q": [ "-0.3864", "0.8347", "-2.0156", "2.7171", "-1.7541", "0.4522" ], "m": { "N": 100, "m0": "1.505717", "values": [ "1.458708", "1.414458", "1.372692", "1.333177", "1.295712", "1.260126", "1.226272", "1.194023", "1.163268", "1.133913", "1.105874", "1.079078", "1.053460", "1.028965", "1.005541", "0.983142", "0.961728", "0.941260", "0.921700", "0.903015", "0.885169", "0.868129", "0.851859", "0.836326", "0.821493", "0.807326", "0.799951", "0.796153", "0.792512", "0.789024", "0.785686", "0.782496", "0.779451", "0.776549", "0.772563", "0.759074", "0.746078", "0.733563", "0.721516", "0.709920", "0.698760", "0.688018", "0.677674", "0.667712", "0.658113", "0.648859", "0.639934", "0.631321", "0.623005", "0.614969", "0.607201", "0.599685", "0.592409", "0.585361", "0.578529", "0.571903", "0.565472", "0.559226", "0.553156", "0.547254", "0.541511", "0.535920", "0.530474", "0.525165", "0.519988", "0.514937", "0.510006", "0.505189", "0.500481", "0.495879", "0.491378", "0.486972", "0.482660", "0.478436", "0.474298", "0.470242", "0.466265", "0.462365", "0.458539", "0.454785", "0.451099", "0.447481", "0.443928", "0.440438", "0.437009", "0.433641", "0.430331", "0.427078", "0.423881", "0.420739", "0.417650", "0.414614", "0.411631", "0.408698", "0.405816", "0.402984", "0.400201", "0.397467", "0.394781", "0.392143" ] }, "note": "G_AI, the remark's polynomial, in the region of F_0.03" }, { "q": [ "-0.416545", "0.646462", "0.255568", "-4.938579", "10.000000", "-7.007958", "-1.007665", "3.557723", "-1.246039" ], "m": { "N": 100, "m0": "1.496675", "values": [ "1.434555", "1.380795", "1.334135", "1.293190", "1.256806", "1.223934", "1.193752", "1.165653", "1.139155", "1.113914", "1.089682", "1.066276", "1.043581", "1.021516", "1.000046", "0.979161", "0.958879", "0.939241", "0.920289", "0.902068", "0.884598", "0.867878", "0.851863", "0.836479", "0.821608", "0.807090", "0.801813", "0.798066", "0.794450", "0.790964", "0.787607", "0.784379", "0.781279", "0.778307", "0.775463", "0.763212", "0.749510", "0.736472", "0.723996", "0.712021", "0.700512", "0.689445", "0.678799", "0.668561", "0.658716", "0.649248", "0.640142", "0.631383", "0.622951", "0.614832", "0.607009", "0.599464", "0.592182", "0.585149", "0.578347", "0.571764", "0.565388", "0.559204", "0.553203", "0.547373", "0.541704", "0.536186", "0.530812", "0.525572", "0.520461", "0.515470", "0.510594", "0.505827", "0.501163", "0.496598", "0.492128", "0.487747", "0.483453", "0.479243", "0.475113", "0.471062", "0.467084", "0.463181", "0.459350", "0.455587", "0.451893", "0.448265", "0.444702", "0.441204", "0.437768", "0.434393", "0.431078", "0.427822", "0.424622", "0.421479", "0.418390", "0.415351", "0.412362", "0.409420", "0.406523", "0.403666", "0.400846", "0.398060", "0.395303", "0.392569" ] }, "note": "iteration 1: this program's float optimiser (degree 9, SLSQP, margin 5e-5 l on 215 points), in the region of step 1" }, { "q": [ "-0.434002", "0.835266", "-0.514657", "-3.672465", "10.000000", "-10.000000", "3.360082", "0.894786", "-0.628130" ], "m": { "N": 100, "m0": "1.491435", "values": [ "1.430945", "1.375686", "1.326144", "1.282480", "1.244353", "1.210978", "1.181371", "1.154593", "1.129845", "1.106503", "1.084119", "1.062365", "1.041025", "1.019977", "0.999178", "0.978658", "0.958522", "0.938923", "0.920016", "0.901909", "0.884617", "0.868069", "0.852136", "0.836691", "0.821714", "0.807455", "0.801800", "0.798016", "0.794366", "0.790850", "0.787468", "0.784218", "0.781100", "0.778115", "0.775261", "0.761900", "0.748306", "0.735300", "0.722905", "0.711062", "0.699711", "0.688801", "0.678293", "0.668167", "0.658406", "0.648999", "0.639936", "0.631209", "0.622806", "0.614713", "0.606917", "0.599405", "0.592160", "0.585167", "0.578410", "0.571875", "0.565547", "0.559415", "0.553463", "0.547684", "0.542062", "0.536591", "0.531261", "0.526063", "0.520989", "0.516033", "0.511189", "0.506450", "0.501812", "0.497269", "0.492818", "0.488454", "0.484173", "0.479974", "0.475851", "0.471804", "0.467831", "0.463927", "0.460093", "0.456327", "0.452626", "0.448989", "0.445415", "0.441904", "0.438453", "0.435061", "0.431728", "0.428450", "0.425228", "0.422061", "0.418945", "0.415881", "0.412865", "0.409895", "0.406969", "0.404085", "0.401238", "0.398426", "0.395646", "0.392892" ] }, "note": "iteration 2: this program's float optimiser (degree 9, SLSQP, margin 5e-5 l on 215 points), in the region of step 2" }, { "q": [ "-0.434013", "0.830597", "-0.499373", "-3.688645", "10.000000", "-10.000000", "3.384087", "0.866940", "-0.618928" ], "m": { "N": 100, "m0": "1.491432", "values": [ "1.428111", "1.371591", "1.321989", "1.279003", "1.241851", "1.209432", "1.180579", "1.154311", "1.129852", "1.106632", "1.084243", "1.062406", "1.040958", "1.019801", "0.998922", "0.978374", "0.958268", "0.938747", "0.919942", "0.901932", "0.884719", "0.868228", "0.852342", "0.836949", "0.822028", "0.807764", "0.801761", "0.797980", "0.794334", "0.790822", "0.787442", "0.784196", "0.781081", "0.778099", "0.775042", "0.761123", "0.747618", "0.734703", "0.722381", "0.710601", "0.699304", "0.688443", "0.677985", "0.667907", "0.658190", "0.648823", "0.639796", "0.631097", "0.622716", "0.614640", "0.606856", "0.599349", "0.592106", "0.585111", "0.578351", "0.571810", "0.565477", "0.559338", "0.553380", "0.547592", "0.541965", "0.536488", "0.531153", "0.525950", "0.520874", "0.515917", "0.511070", "0.506331", "0.501693", "0.497151", "0.492701", "0.488340", "0.484063", "0.479867", "0.475749", "0.471707", "0.467738", "0.463841", "0.460013", "0.456253", "0.452558", "0.448928", "0.445361", "0.441857", "0.438412", "0.435028", "0.431701", "0.428430", "0.425216", "0.422055", "0.418946", "0.415887", "0.412878", "0.409914", "0.406994", "0.404115", "0.401274", "0.398468", "0.395693", "0.392946" ] }, "note": "iteration 3: this program's float optimiser (degree 9, SLSQP, margin 5e-5 l on 215 points), in the region of step 3" }, { "q": [ "-0.434220", "0.831682", "-0.501864", "-3.686634", "10.000000", "-10.000000", "3.382308", "0.868797", "-0.619493" ], "m": { "N": 100, "m0": "1.491370", "values": [ "1.428171", "1.371631", "1.322037", "1.279033", "1.241857", "1.209398", "1.180510", "1.154219", "1.129738", "1.106504", "1.084105", "1.062266", "1.040815", "1.019662", "0.998789", "0.978250", "0.958153", "0.938641", "0.919842", "0.901837", "0.884629", "0.868142", "0.852262", "0.836877", "0.821967", "0.807714", "0.801762", "0.797981", "0.794334", "0.790821", "0.787442", "0.784195", "0.781080", "0.778097", "0.775052", "0.761137", "0.747637", "0.734725", "0.722403", "0.710621", "0.699322", "0.688458", "0.677997", "0.667915", "0.658197", "0.648829", "0.639800", "0.631100", "0.622719", "0.614642", "0.606856", "0.599349", "0.592106", "0.585111", "0.578350", "0.571809", "0.565475", "0.559334", "0.553376", "0.547588", "0.541960", "0.536482", "0.531146", "0.525943", "0.520866", "0.515906", "0.511060", "0.506320", "0.501681", "0.497139", "0.492688", "0.488326", "0.484048", "0.479850", "0.475733", "0.471690", "0.467720", "0.463822", "0.459994", "0.456233", "0.452538", "0.448907", "0.445340", "0.441835", "0.438390", "0.435006", "0.431678", "0.428408", "0.425193", "0.422031", "0.418922", "0.415863", "0.412853", "0.409889", "0.406970", "0.404090", "0.401249", "0.398443", "0.395669", "0.392921" ] }, "note": "iteration 4: this program's float optimiser (degree 9, SLSQP, margin 5e-5 l on 215 points), in the region of step 4" }, { "q": [ "-0.434241", "0.831665", "-0.501774", "-3.686715", "10.000000", "-10.000000", "3.382333", "0.868789", "-0.619496" ], "m": { "N": 100, "m0": "1.491363", "values": [ "1.428086", "1.371583", "1.321980", "1.278993", "1.241813", "1.209354", "1.180479", "1.154188", "1.129716", "1.106484", "1.084091", "1.062255", "1.040807", "1.019657", "0.998787", "0.978248", "0.958153", "0.938639", "0.919842", "0.901836", "0.884626", "0.868138", "0.852257", "0.836871", "0.821960", "0.807707", "0.801761", "0.797979", "0.794333", "0.790820", "0.787441", "0.784194", "0.781079", "0.778096", "0.775040", "0.761124", "0.747623", "0.734710", "0.722390", "0.710607", "0.699308", "0.688445", "0.677984", "0.667903", "0.658185", "0.648817", "0.639788", "0.631089", "0.622708", "0.614631", "0.606847", "0.599341", "0.592097", "0.585103", "0.578343", "0.571802", "0.565468", "0.559329", "0.553370", "0.547583", "0.541955", "0.536478", "0.531142", "0.525940", "0.520862", "0.515903", "0.511057", "0.506317", "0.501679", "0.497136", "0.492686", "0.488323", "0.484046", "0.479849", "0.475731", "0.471689", "0.467719", "0.463822", "0.459993", "0.456232", "0.452537", "0.448907", "0.445340", "0.441835", "0.438390", "0.435005", "0.431678", "0.428407", "0.425192", "0.422031", "0.418922", "0.415863", "0.412853", "0.409889", "0.406969", "0.404090", "0.401249", "0.398443", "0.395669", "0.392920" ] }, "note": "iteration 5: this program's float optimiser (degree 9, SLSQP, margin 5e-5 l on 215 points), in the region of step 5" } ], "verifier": { "file": "tools/verify_gnnw_gai.py", "sha256": "996c82c5500cde1b7ca72e55ee38b0ac27dd54be92387b4230117637e596a2f7" }, "second": { "file": "instruments/gnnw/second.js", "what": "JavaScript, instruments/bigfloat (dyadic intervals, its own exp and log), monotone endpoint bounds on [1/N, 1] instead of a written derivative, beta = f - t f' as written", "current": true, "agrees": true, "result": { "verdict": "CERTIFIED", "steps": [ { "tail": 79, "main": 11322, "minTailS": 4.714627115489143e-06, "minMainLB": 3.699262530843567e-08, "c": [ "3.7823287755373116217428004", "3.7823287755373116217428005" ] }, { "tail": 10, "main": 39820, "minTailS": 0.0003042932878499241, "minMainLB": 2.333032764449637e-10, "c": [ "3.7754710215368457179220446", "3.7754710215368457179220447" ] }, { "tail": 16, "main": 34758, "minTailS": 0.0004580344391602225, "minMainLB": 1.2534970345842026e-11, "c": [ "3.7725734617809080648710855", "3.7725734617809080648710856" ] }, { "tail": 12, "main": 31982, "minTailS": 0.0006724280866570116, "minMainLB": 5.351380827554986e-10, "c": [ "3.7722750853543349625183954", "3.7722750853543349625183955" ] }, { "tail": 12, "main": 31713, "minTailS": 0.0004343660924040097, "minMainLB": 2.222582973300355e-10, "c": [ "3.7721515782981542341194064", "3.7721515782981542341194065" ] }, { "tail": 12, "main": 32241, "minTailS": 0.00048575137953095737, "minMainLB": 3.7664359012154616e-11, "c": [ "3.7721307629003663892468671", "3.7721307629003663892468672" ] } ], "c": [ "3.7721307629003663892468671", "3.7721307629003663892468672" ], "seconds": 2099.275 }, "ran": "2026-09-29" }, "decided": { "verdict": "CERTIFIED", "c": [ "3.772130762900366389246867113974", "3.772130762900366389246867113974" ], "bases": [ "3.78232877553731", "3.77547102153684", "3.77257346178090", "3.77227508535433", "3.77215157829815", "3.77213076290036" ], "perStep": [ { "tail": 79, "main": 336, "minMainLower": 9.55470225858566e-07, "minTailS": 4.716462266515588e-06, "region": "strictly concave and increasing on (0, 1]; A < B" }, { "tail": 10, "main": 693, "minMainLower": 1.4124092925888733e-07, "minTailS": 0.00036069066423235314, "region": "strictly concave and increasing on (0, 1]; A < B" }, { "tail": 16, "main": 736, "minMainLower": 6.644697943638025e-08, "minTailS": 0.00048759773544233235, "region": "strictly concave and increasing on (0, 1]; A < B" }, { "tail": 12, "main": 718, "minMainLower": 1.0257094774408017e-07, "minTailS": 0.0007117294036585218, "region": "strictly concave and increasing on (0, 1]; A < B" }, { "tail": 12, "main": 717, "minMainLower": 1.6919464993415756e-07, "minTailS": 0.00047368557256367556, "region": "strictly concave and increasing on (0, 1]; A < B" }, { "tail": 12, "main": 718, "minMainLower": 1.0405172196340237e-08, "minTailS": 0.0005250751319474562, "region": "strictly concave and increasing on (0, 1]; A < B" } ], "seconds": 175.8 } }