{ "what": "The entropy deciders' arithmetic checked against Arb (python-flint): instruments/bigfloat as instruments/sumdiff and instruments/optq/twolaw.js call it, and the Python standard library's decimal as tools/verify_sumdiff.py, tools/verify_twolaw.py and instruments/fei/fei.py call it, each run through its REAL functions on seeded random and adversarial inputs and on the registry's own certificates; every enclosure required to contain the exact value, decided in Arb at 256 bits and up from the exact inputs (exactly, in FLINT's rationals, when the value is rational); every malformed input required to be refused; planted wrong functions required to be caught. Written by tools/audit-entropy-arb.js with tools/audit-entropy-arb.py.", "generated": "2026-10-19T22:49:00+00:00", "seconds": 111.9, "secondsJs": 71.2, "secondsPython": 30.5, "seed": 20261019, "n": 20000, "node": "v24.14.1", "python": { "impl": "3.9.6", "libmpdec": "2.5.0", "arb": "3.12.13", "pythonFlint": "0.9.0" }, "arbPrec": [ 256, 1024, 4096, 16384 ], "implPrecision": { "bigfloat": "bits per case (u.b): 128, 256, 512, 1024 as sumdiff/twolaw escalate; ln2 also 24..4096", "verify_sumdiff/verify_twolaw": "120 digits", "fei": "60 digits" }, "modules": { "instruments/bigfloat/bigfloat.js": "106ead607bf310a02ecc1df8d23574a8c8a9aee017a48ec613cb5e7dececefb3", "instruments/bigfloat/functions.js": "b66409f2295613c4b8410736394fdf42007b8ea014e84757fd16cac87a3145ce", "instruments/sumdiff/sumdiff.js": "e5919271fff8d04171120437a6fc78e537489f9459826f58cefb3387fdeca487", "instruments/optq/twolaw.js": "ad15c46f2469dc9ca944ae0450d40d7045871a52c047abdab2d9d954f86391f5", "tools/verify_sumdiff.py": "850e901f46d6c4180138c60fee10cade3d59ded904a6409c7c379ea16c2fd3db", "tools/verify_twolaw.py": "2405d1c4b76595155aa4fe739c0fde0cebb03c048107d8b28cf29b37a2344304", "instruments/fei/fei.py": "271e0f4def7ccffb5113a3d166fe73d1ec0de5942035631c5a7d9c672e11d6de" }, "inputs": { "corpus/optimization-constants/mi2026/c3b_pr92/certificate_3b_13pt.json": "2efe99360a1ff5e3736e734ed06e5ab8e0cdcc33583ef22ece9e927e9e45365c", "corpus/optimization-constants/mi2026/c3c_pr93/certificate_3c_v3_95pt.json": "2ebd7ba7a32e776ff5d271510b63a0999869fecfb148087a3d0fdd57cc4ffc3e", "corpus/optimization-constants/l2026/certificate_3c_147pt.json": "b3dc9587ba7a733a3f68b66017d7b5936f38040f8e9ca31a057bb65a6f227999", "corpus/optconst-queue/sources/zenodo-23289305/certificate.json": "8b6612ef91342b900733f87b00a8bc0ceebf88727eeceb2e5b390e2ccbfb97a4", "corpus/optimization-constants/num2026/fei_c71_n18_artifact.json": "57f5a1ad2bf5bcdee873413e796bc868ccc6131521af799c5eb48d5ae13267ba", "certs/sumdiff-ledger.json": "bf8857db9f64b6508da042b5f0d4e4b30fe452f47fb14a64b701482e7ab94ac4", "certs/optq-ledger.json": "db1d13b12e10582c57a33c67cf04fb64ba47073770ea6330c1ad517c6ded8510", "certs/fei-ledger.json": "ad853dabf838029896747144ee765fd418830311dc7c4b133575119d08bcf59e" }, "counts": { "cases": 298746, "arithmetic": { "cases": 264031, "contained": 264015, "violations": 0, "undecided": 0, "refused": 16 }, "printed": { "cases": 2715, "contained": 2715, "violations": 0, "undecided": 0, "refused": 0 }, "beyondTheDoor": { "cases": 4000, "contained": 2000, "violations": 0, "undecided": 0, "refused": 2000 }, "malformed": 71, "malformedRefused": 71, "baselines": 5, "baselinesDecided": 5 }, "redControls": { "red:bf.entropy.lognearest": { "cases": 4000, "violationsCaught": 800 }, "red:bf.fromRatio.shrunk": { "cases": 4000, "violationsCaught": 4000 }, "red:bf.log.nearest": { "cases": 4000, "violationsCaught": 3472 }, "red:fei.ln_iv.nearest": { "cases": 4000, "violationsCaught": 4000 }, "red:py.entropy.lnnearest": 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width 0 left out", "widest": null }, "violationsByName": { "None": 4000 }, "casesByName": { "None": 4000 } }, "red:bf.log.nearest": { "cases": 4000, "contained": 528, "violations": 3472, "undecided": 0, "refused": 0, "retried": 163, "byKind": { "red": 4000 }, "refusedBy": {}, "class": "red", "impl": "planted", "widthUlps": { "median": null, "p99": null, "max": null, "n": 0, "note": "the enclosure width over |value| in units of the last place at the case's precision (2^-bits, or 10^-(digits-1)); point values only, ties of width 0 left out", "widest": null }, "violationsByName": { "None": 3472 }, "casesByName": { "None": 4000 } }, "red:fei.ln_iv.nearest": { "cases": 4000, "contained": 0, "violations": 4000, "undecided": 0, "refused": 0, "retried": 0, "byKind": { "red": 4000 }, "refusedBy": {}, "class": "red", "impl": "planted", "widthUlps": { "median": null, "p99": null, "max": null, "n": 0, "note": "the enclosure width over |value| in units of the last place at the case's precision (2^-bits, or 10^-(digits-1)); point values only, ties of width 0 left out", "widest": null }, "violationsByName": { "None": 4000 }, "casesByName": { "None": 4000 } }, "red:py.entropy.lnnearest": { "cases": 4000, "contained": 2803, "violations": 1197, "undecided": 0, "refused": 0, "retried": 2803, "byKind": { "red": 4000 }, "refusedBy": {}, "class": "red", "impl": "planted", "widthUlps": { "median": 0.328, "p99": 0.942, "max": 1.0, "n": 2803, "note": "the enclosure width over |value| in units of the last place at the case's precision (2^-bits, or 10^-(digits-1)); point values only, ties of width 0 left out", "widest": { "kind": "red", "truthSpec": { "t": "plnp", "q": "2796444458774269183706139420991555352052520629700050220493639941173916997012665/100000000000000000000000000000000000000000000000000000000000000000000000000000000" }, "u": { "d": 120 }, "res": [ "0.100023824172884134572073862643085569050959487644332664747551018620999753295866226087699818448997792665884967294445260112", "0.100023824172884134572073862643085569050959487644332664747551018620999753295866226087699818448997792665884967294445260113" ] } }, "violationsByName": { "None": 1197 }, "casesByName": { "None": 4000 } }, "red:py.ln_out.nearest": { "cases": 4000, "contained": 1, "violations": 3999, "undecided": 0, "refused": 0, "retried": 0, "byKind": { "red": 4000 }, "refusedBy": {}, "class": "red", "impl": "planted", "widthUlps": { "median": null, "p99": null, "max": null, "n": 0, "note": "the enclosure width over |value| in units of the last place at the case's precision (2^-bits, or 10^-(digits-1)); point values only, ties of width 0 left out", "widest": null }, "violationsByName": { "None": 3999 }, "casesByName": { "None": 4000 } }, "red:py.outward.shrunk": { "cases": 4000, "contained": 0, "violations": 4000, "undecided": 0, "refused": 0, "retried": 0, "byKind": { "red": 4000 }, "refusedBy": {}, "class": "red", "impl": "planted", "widthUlps": { "median": null, "p99": null, "max": null, "n": 0, "note": "the enclosure width over |value| in units of the last place at the case's precision (2^-bits, or 10^-(digits-1)); point values only, ties of width 0 left out", "widest": null }, "violationsByName": { "None": 4000 }, "casesByName": { "None": 4000 } } }, "malformed": { "cases": 71, "refused": 71, "baselines": 5, "baselinesDecided": 5, "byImpl": { "bigfloat": { "cases": 10, "refused": 10 }, "sumdiff": { "cases": 15, "refused": 15 }, "verify_sumdiff": { "cases": 15, "refused": 15 }, "twolaw": { "cases": 11, "refused": 11 }, "verify_twolaw": { "cases": 11, "refused": 11 }, "fei": { "cases": 6, "refused": 6 }, "fei.ln_iv": { "cases": 2, "refused": 2 }, "verify_sumdiff.ln_out": { "cases": 1, "refused": 1 } }, "list": [ { "impl": "bigfloat", "name": "bigfloat.fromRatio: denominator 0", "expectRefusal": true, "threw": "bigfloat: division by zero", "res": null }, { "impl": "bigfloat", "name": "bigfloat.fromRatio: NaN", "expectRefusal": true, "threw": "Cannot convert NaN to a BigInt", "res": null }, { "impl": "bigfloat", "name": "bigfloat.fromRatio: a non-integer numerator", "expectRefusal": true, "threw": "The number 1.5 cannot be converted to a BigInt because it is not an integer", "res": null }, { "impl": "bigfloat", "name": "bigfloat.log: p = 0", "expectRefusal": true, "threw": "bigfloat.log: need positive interval", "res": null }, { "impl": "bigfloat", "name": "bigfloat.log: p < 0", "expectRefusal": true, "threw": "bigfloat.log: need positive interval", "res": null }, { "impl": "bigfloat", "name": "bigfloat.log: an interval straddling 0", "expectRefusal": true, "threw": "bigfloat.log: need positive interval", "res": null }, { "impl": "bigfloat", "name": "bigfloat.log: an inverted interval", "expectRefusal": true, "threw": "bigfloat: inverted interval", "res": null }, { "impl": "bigfloat", "name": "bigfloat.logPoint: 0", "expectRefusal": true, "threw": "bigfloat.log: need positive value", "res": null }, { "impl": "bigfloat", "name": "bigfloat.div: by an interval containing 0", "expectRefusal": true, "threw": "bigfloat: interval division by an interval containing 0", "res": null }, { "impl": "bigfloat", "name": "bigfloat.div: by exactly 0", "expectRefusal": true, "threw": "bigfloat: interval division by an interval containing 0", "res": null }, { "impl": "sumdiff", "name": "sumdiff: baseline: a valid four-point law", "expectRefusal": false, "threw": null, "res": "verdict CERTIFIED, rho 1.067130275691..1.067130275692" }, { "impl": "verify_sumdiff", "name": "verify_sumdiff: baseline: a valid four-point law", "expectRefusal": false, "threw": null, "res": "verdict CERTIFIED, 1.06713027569165740086587964747775500913, 1.06713027569165740086587964747775500913" }, { "impl": "sumdiff", "name": "sumdiff: a weight is 0", "expectRefusal": true, "threw": "REFUSED: a weight is not positive", "res": null }, { "impl": "verify_sumdiff", "name": "verify_sumdiff: a weight is 0", "expectRefusal": true, "threw": "ValueError: REFUSED: a weight is not positive", "res": null }, { "impl": "sumdiff", "name": "sumdiff: a weight is negative (p > 1 elsewhere)", "expectRefusal": true, "threw": "REFUSED: a weight is not positive", "res": null }, { "impl": "verify_sumdiff", "name": "verify_sumdiff: a weight is negative (p > 1 elsewhere)", "expectRefusal": true, "threw": "ValueError: REFUSED: a weight is not positive", "res": null }, { "impl": "sumdiff", "name": "sumdiff: the weights sum to 1 + 10^-320", "expectRefusal": true, "threw": "REFUSED: the weights sum to 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", "res": null }, { "impl": "verify_sumdiff", "name": "verify_sumdiff: the weights sum to 1 + 10^-320", "expectRefusal": true, "threw": "ValueError: REFUSED: the weights are not a probability law (they sum to more than 1)", "res": null }, { "impl": "sumdiff", "name": "sumdiff: the weights sum to 1 - 10^-320", "expectRefusal": true, "threw": "REFUSED: the weights sum to 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999", "res": null }, { "impl": "verify_sumdiff", "name": "verify_sumdiff: the weights sum to 1 - 10^-320", "expectRefusal": true, "threw": "ValueError: REFUSED: the weights are not a probability law (they sum to less than 1)", "res": null }, { "impl": "sumdiff", "name": "sumdiff: one weight p = 3/2 alone (p > 1)", "expectRefusal": true, "threw": "REFUSED: the weights sum to 3/2, not 1", "res": null }, { "impl": "verify_sumdiff", "name": "verify_sumdiff: one weight p = 3/2 alone (p > 1)", "expectRefusal": true, "threw": "ValueError: REFUSED: the weights are not a probability law (they sum to more than 1)", "res": null }, { "impl": "sumdiff", "name": "sumdiff: a point listed twice", "expectRefusal": true, "threw": "REFUSED: a point is listed twice", "res": null }, { "impl": "verify_sumdiff", "name": "verify_sumdiff: a point listed twice", "expectRefusal": true, "threw": "ValueError: REFUSED: a point is listed twice: (0, 0)", "res": null }, { "impl": "sumdiff", "name": "sumdiff: a non-integer weight that truncates to a law (num 1.9/2 and 1.2/2)", "expectRefusal": true, "threw": "an unsafe integer reached the reader: 1.9", "res": null }, { "impl": "verify_sumdiff", "name": "verify_sumdiff: a non-integer weight that truncates to a law (num 1.9/2 and 1.2/2)", "expectRefusal": true, "threw": "ValueError: REFUSED: a weight is not an exact integer: 1.9", "res": null }, { "impl": "sumdiff", "name": "sumdiff: a boolean weight (true/2)", "expectRefusal": true, "threw": "not an integer: true", "res": null }, { "impl": "verify_sumdiff", "name": "verify_sumdiff: a boolean weight (true/2)", "expectRefusal": true, "threw": "ValueError: REFUSED: a weight is not an exact integer: True", "res": null }, { "impl": "sumdiff", "name": "sumdiff: a non-integer coordinate (x = 0.5)", "expectRefusal": true, "threw": "an unsafe integer reached the reader: 0.5", "res": null }, { "impl": "verify_sumdiff", "name": "verify_sumdiff: a non-integer coordinate (x = 0.5)", "expectRefusal": true, "threw": "ValueError: REFUSED: a coordinate is not an exact integer: 0.5", "res": null }, { "impl": "sumdiff", "name": "sumdiff: a NaN weight", "expectRefusal": true, "threw": "Unexpected token 'N', ...\"0, \"num\": NaN, \"den\"\"... is not valid JSON", "res": null }, { "impl": "verify_sumdiff", "name": "verify_sumdiff: a NaN weight", "expectRefusal": true, "threw": "ValueError: REFUSED: a weight is not an exact integer: nan", "res": null }, { "impl": "sumdiff", "name": "sumdiff: an unknown problem (3a)", "expectRefusal": true, "threw": "not a 3b or 3c certificate: 3a", "res": null }, { "impl": "verify_sumdiff", "name": "verify_sumdiff: an unknown problem (3a)", "expectRefusal": true, "threw": "ValueError: REFUSED: not a 3b or 3c certificate", "res": null }, { "impl": "sumdiff", "name": "sumdiff: a denominator not of the form 10^k", "expectRefusal": true, "threw": "denominator not of the form 10^k: 1000", "res": null }, { "impl": "verify_sumdiff", "name": "verify_sumdiff: a denominator not of the form 10^k", "expectRefusal": true, "threw": "ValueError: REFUSED: the denominator is not of the form 10^k: '1000'", "res": null }, { "impl": "sumdiff", "name": "sumdiff: an empty law", "expectRefusal": true, "threw": "REFUSED: the weights sum to 0/1, not 1", "res": null }, { "impl": "verify_sumdiff", "name": "verify_sumdiff: an empty law", "expectRefusal": true, "threw": "ValueError: REFUSED: the law has no points", "res": null }, { "impl": "sumdiff", "name": "sumdiff: a one-point law (p = 1: every entropy 0, the ratio 0/0)", "expectRefusal": true, "threw": "bigfloat: interval division by an interval containing 0", "res": null }, { "impl": "verify_sumdiff", "name": "verify_sumdiff: a one-point law (p = 1: every entropy 0, the ratio 0/0)", "expectRefusal": true, "threw": "ValueError: REFUSED: a one-point law: every entropy is 0, so the ratio is 0/0", "res": null }, { "impl": "sumdiff", "name": "sumdiff: a one-point law in 3c, unreduced (10^320/10^320)", "expectRefusal": true, "threw": "bigfloat: interval division by an interval containing 0", "res": null }, { "impl": "verify_sumdiff", "name": "verify_sumdiff: a one-point law in 3c, unreduced (10^320/10^320)", "expectRefusal": true, "threw": "ValueError: REFUSED: a one-point law: every entropy is 0, so the ratio is 0/0", "res": null }, { "impl": "twolaw", "name": "twolaw: baseline: a valid two-law certificate", "expectRefusal": false, "threw": null, "res": "verdict CERTIFIED, rho 0.494968961125..0.494968961126" }, { "impl": "verify_twolaw", "name": "verify_twolaw: baseline: a valid two-law certificate", "expectRefusal": false, "threw": null, "res": "verdict CERTIFIED, 0.49496896112507365590413242725352025639, 0.49496896112507365590413242725352025639" }, { "impl": "twolaw", "name": "twolaw: a count is 0", "expectRefusal": true, "threw": "REFUSED: counts and laws disagree or a count is not positive", "res": null }, { "impl": "verify_twolaw", "name": "verify_twolaw: a count is 0", "expectRefusal": true, "threw": "ValueError: REFUSED: one positive count per law is needed", "res": null }, { "impl": "twolaw", "name": "twolaw: a count is negative", "expectRefusal": true, "threw": "REFUSED: counts and laws disagree or a count is not positive", "res": null }, { "impl": "verify_twolaw", "name": "verify_twolaw: a count is negative", "expectRefusal": true, "threw": "ValueError: REFUSED: one positive count per law is needed", "res": null }, { "impl": "twolaw", "name": "twolaw: points and weights of unequal length", "expectRefusal": true, "threw": "REFUSED: points and weights of unequal length", "res": null }, { "impl": "verify_twolaw", "name": "verify_twolaw: points and weights of unequal length", "expectRefusal": true, "threw": "ValueError: REFUSED: a law lists 3 points and 2 weights", "res": null }, { "impl": "twolaw", "name": "twolaw: a law does not sum to 1", "expectRefusal": true, "threw": "REFUSED: the weights sum to 11/15, not 1", "res": null }, { "impl": "verify_twolaw", "name": "verify_twolaw: a law does not sum to 1", "expectRefusal": true, "threw": "ValueError: REFUSED: the weights are not a probability law (they sum to less than 1)", "res": null }, { "impl": "twolaw", "name": "twolaw: a point listed twice", "expectRefusal": true, "threw": "REFUSED: a point is listed twice", "res": null }, { "impl": "verify_twolaw", "name": "verify_twolaw: a point listed twice", "expectRefusal": true, "threw": "ValueError: REFUSED: a point is listed twice: (1, 0)", "res": null }, { "impl": "twolaw", "name": "twolaw: a weight is 0", "expectRefusal": true, "threw": "REFUSED: a weight is not positive", "res": null }, { "impl": "verify_twolaw", "name": "verify_twolaw: a weight is 0", "expectRefusal": true, "threw": "ValueError: REFUSED: a weight is not positive", "res": null }, { "impl": "twolaw", "name": "twolaw: the base too small for injectivity (base 2)", "expectRefusal": true, "threw": "verdict REFUSED: max |L| = 3 exceeds (base - 1)/2: the digit map is not shown injective, so the entropies do not add", "res": null }, { "impl": "verify_twolaw", "name": "verify_twolaw: the base too small for injectivity (base 2)", "expectRefusal": true, "threw": "ValueError: REFUSED: a form reaches 3 on a support, past (base-1)/2 = 0 \u2014 the digit map need not be injective", "res": null }, { "impl": "twolaw", "name": "twolaw: a non-integer weight that truncates to a law (1.9/2 and 1.2/2)", "expectRefusal": true, "threw": "Cannot convert 1.9 to a BigInt", "res": null }, { "impl": "verify_twolaw", "name": "verify_twolaw: a non-integer weight that truncates to a law (1.9/2 and 1.2/2)", "expectRefusal": true, "threw": "ValueError: REFUSED: a weight is not an exact integer: 1.9", "res": null }, { "impl": "twolaw", "name": "twolaw: a non-integer coordinate (0.5)", "expectRefusal": true, "threw": "Cannot convert 0.5 to a BigInt", "res": null }, { "impl": "verify_twolaw", "name": "verify_twolaw: a non-integer coordinate (0.5)", "expectRefusal": true, "threw": "ValueError: REFUSED: a coordinate is not an exact integer: 0.5", "res": null }, { "impl": "twolaw", "name": "twolaw: not a 3c certificate", "expectRefusal": true, "threw": "not a 3c certificate: C_3b", "res": null }, { "impl": "verify_twolaw", "name": "verify_twolaw: not a 3c certificate", "expectRefusal": true, "threw": "ValueError: REFUSED: not a 3c certificate: 'C_3b'", "res": null }, { "impl": "twolaw", "name": "twolaw: two one-point laws (every entropy 0, the ratio 0/0)", "expectRefusal": true, "threw": "bigfloat: interval division by an interval containing 0", "res": null }, { "impl": "verify_twolaw", "name": "verify_twolaw: two one-point laws (every entropy 0, the ratio 0/0)", "expectRefusal": true, "threw": "ValueError: REFUSED: every entropy is 0 (each law has one point), so the ratio is 0/0", "res": null }, { "impl": "fei", "name": "fei.decide: baseline: majority of 5, valid", "expectRefusal": false, "threw": null, "res": "verdict CERTIFIED, 3.80042946289491307862493203667206816167, 3.80042946289491307862493203667206816167" }, { "impl": "fei", "name": "fei.decide: the table is not the one its sha256 binds", "expectRefusal": true, "threw": "verdict REFUSED", "res": null }, { "impl": "fei", "name": "fei.decide: I misstated", "expectRefusal": true, "threw": "verdict REFUSED", "res": null }, { "impl": "fei", "name": "fei.decide: unbalanced (at-least-2-of-5, monotone)", "expectRefusal": true, "threw": "verdict REFUSED", "res": null }, { "impl": "fei", "name": "fei.decide: a dictator (I = 1: the rule does not apply)", "expectRefusal": true, "threw": "verdict REFUSED", "res": null }, { "impl": "fei", "name": "fei.decide: a parity (H = 0)", "expectRefusal": true, "threw": "verdict REFUTED", "res": null }, { "impl": "fei", "name": "fei.decide: a mask with bits beyond 2^n inputs", "expectRefusal": true, "threw": "verdict REFUSED", "res": null }, { "impl": "fei.ln_iv", "name": "fei.ln_iv: k = 0 (the docstring's domain is a positive integer)", "expectRefusal": true, "threw": "ValueError: ln_iv: k must be a positive integer, got 0", "res": null }, { "impl": "fei.ln_iv", "name": "fei.ln_iv: k = -3 (the docstring's domain is a positive integer)", "expectRefusal": true, "threw": "ValueError: ln_iv: k must be a positive integer, got -3", "res": null }, { "impl": "verify_sumdiff.ln_out", "name": "verify_sumdiff.ln_out: x < 0", "expectRefusal": true, "threw": "InvalidOperation: []", "res": null } ] }, "witnesses": { "py.plnp>1:refused": [ { "truth": { "t": "plnp", "q": "283298057813736/1000000000000" }, "threw": "ValueError: REFUSED: an atom of weight outside (0, 1]" }, { "truth": { "t": "plnp", "q": "24055626856285821734916066884810591764415237298028211408345406136276170472278593824215466883519340361767935297192863269795805929581052883402092281439366405080782598919576067218648443882397550553660811352258826467863690042403987130/47379092172262845921294540053176957863064148471578876072358765808900480733837185251796826441303908628860099203403658239934435426150964794660775368501012608007948455078761288621529091968737745508583465551566054344204493617364992" }, "threw": "ValueError: REFUSED: an atom of weight outside (0, 1]" }, { "truth": { "t": "plnp", "q": "39982991379102425881880117596625708142827924431611954648756817996453506117064182716880676960099097710234048858760342432226110551566957846371943549242188738588464387744265726288847893160914187904664427704815100549613756198195469160720616380297890023211723329886237606434716074320125291184319634202259974096776126030443910304350752149766740963426952657886961656625780651384166480329818044276765419231240937955599920554545596639162240990465104206055365188428435157267727155989670390097697492338650563083200/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000" 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"3053840887627167867809901856563976040617805774418512721772776873555879797639787851331001744336052718247356763/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000" }, "inputs": null, "u": { "d": 120 }, "res": [ "0.362242021425737811433222308844479527575387209053139347030256041768033428634460091832207805849971474077195307863318946185", "0.362242021425737811433222308844479527575387209053139347030256041768033428634460091832207805849971474077195307863318946186" ], "exact": [ "[0.3622420214257378114332223088444795275753872090531393470302560417680334286345 +/- 4.46e-77]", "[0.3622420214257378114332223088444795275753872090531393470302560417680334286345 +/- 4.46e-77]" ] } ] }, "beyondTheDoor": "bf.plnp>1 and py.plnp>1 feed the bare -p ln p routines a p > 1, which no door admits (sumdiff.door, verify_sumdiff.read and verify_twolaw.laws require a law summing to 1). They are a measurement of how load-bearing the door is, not a requirement: verify_sumdiff.entropy takes p_lo\u00b7(\u2212ln_hi) as its lower end, which holds only while \u2212ln p >= 0.", "foundAtPin": { "what": "The findings this audit made at its first full run, recorded in certs/entropy-arb-audit.json at the commit named, before any fix (the gaps recorded at the pin), with the digests of the modules audited then; each was fixed afterwards in the file named. The audit's declared list (KNOWN in tools/audit-entropy-arb.py) is now empty: any one of these reproducing again fails the run.", "commit": "470fe0b2c7fe83856eb2b4354dff890899588172", "date": "2026-10-19", "generated": "2026-10-19T21:24:08+00:00", "seed": 20261019, "n": 20000, "cases": 298710, "modules": { "instruments/bigfloat/bigfloat.js": "106ead607bf310a02ecc1df8d23574a8c8a9aee017a48ec613cb5e7dececefb3", "instruments/bigfloat/functions.js": "b66409f2295613c4b8410736394fdf42007b8ea014e84757fd16cac87a3145ce", "instruments/sumdiff/sumdiff.js": "e5919271fff8d04171120437a6fc78e537489f9459826f58cefb3387fdeca487", "instruments/optq/twolaw.js": "ad15c46f2469dc9ca944ae0450d40d7045871a52c047abdab2d9d954f86391f5", "tools/verify_sumdiff.py": "0e44946d156ba937f1cb1f96908d14b6c25d94002adc6a5a14a08c8835672384", "tools/verify_twolaw.py": "3714b0026bb21b39039b009c78d375dded77cc79dffbc3d298ffa08af0ddc142", "instruments/fei/fei.py": "5825dc475292bf4e02e995e276c1a3422de52796bbea89b758149448fcd78141" }, "findings": [ { "class": "printed", "op": "fei.printed", "name": "decide()[\"H\"]", "misses": 202, "of": 308, "why": "fei.decide returns its enclosures as strings cut from str(): ratio [str(r_lo)[:44], str(r_hi)[:44]] and H [str(H_lo)[:30], str(H_hi)[:30]]. Cutting a decimal string rounds toward zero \u2014 for a positive upper end, DOWN \u2014 so the returned upper end falls below the exact value whenever the cut drops a nonzero digit. The verdict is not affected: decide() compares the full 60-digit r_lo with the threshold, and the full-precision ends (fei.H, fei.ratio) contain the value.", "witness": { "truthSpec": { "t": "feiH", "f": "f2" }, "inputs": { "n": 8, "hex": "7b4389cd3423d9e2815e4dc2af123f5f0c5f8c24e15d6431ae1aeda59978bc2c", "shape": "random" }, "printed": [ "7.0063574474519147637191728366", "7.0063574474519147637191728366" ], "exact": [ "[7.006357447451914763719172836656825781433861850569686597978495674240193918239 +/- 3.35e-76]", "[7.006357447451914763719172836656825781433861850569686597978495674240193918240 +/- 2.55e-76]" ] }, "fixedIn": "instruments/fei/fei.py: outward(x, digits, up) prints the lower end rounded down and the upper end up (ROUND_FLOOR / ROUND_CEILING) at 43 (ratio) and 29 (H) significant digits" }, { "class": "printed", "op": "fei.printed", "name": "decide()[\"ratio\"]", "misses": 216, "of": 308, "why": "fei.decide returns its enclosures as strings cut from str(): ratio [str(r_lo)[:44], str(r_hi)[:44]] and H [str(H_lo)[:30], str(H_hi)[:30]]. Cutting a decimal string rounds toward zero \u2014 for a positive upper end, DOWN \u2014 so the returned upper end falls below the exact value whenever the cut drops a nonzero digit. The verdict is not affected: decide() compares the full 60-digit r_lo with the threshold, and the full-precision ends (fei.H, fei.ratio) contain the value.", "witness": { "truthSpec": { "t": "feiRho", "f": "f2" }, "inputs": { "n": 8, "hex": "7b4389cd3423d9e2815e4dc2af123f5f0c5f8c24e15d6431ae1aeda59978bc2c", "shape": "random" }, "printed": [ "2.372523156809113994063635246275327248739508", "2.372523156809113994063635246275327248739508" ], "exact": [ "[2.372523156809113994063635246275327248739508774796084350638220757414668840039 +/- 5.30e-76]", "[2.372523156809113994063635246275327248739508774796084350638220757414668840039 +/- 4.57e-77]" ] }, "fixedIn": "instruments/fei/fei.py: outward(x, digits, up) prints the lower end rounded down and the upper end up (ROUND_FLOOR / ROUND_CEILING) at 43 (ratio) and 29 (H) significant digits" }, { "class": "printed", "op": "py.printed", "name": "verify_sumdiff.py main(): \"ratio in [...]\"", "misses": 4, "of": 4, "why": "verify_sumdiff.py and verify_twolaw.py print \"ratio in [str(lo)[:44], str(hi)[:44]]\": the same cut toward zero, so the printed upper end is below the exact ratio. The verdicts compare the full 120-digit ends; tools/run-optq-ledger.py parses verify_twolaw's printed line into certs/optq-ledger.json.", "witness": { "truthSpec": { "t": "rho", "law": "real:3b-13" }, "inputs": { "problem": "3b", "file": "corpus/optimization-constants/mi2026/c3b_pr92/certificate_3b_13pt.json" }, "printed": [ "1.778988841420693549805193887477289434416066", "1.778988841420693549805193887477289434416066" ], "exact": [ "[1.778988841420693549805193887477289434416066785870268133201048179037478760510 +/- 5.40e-77]", "[1.778988841420693549805193887477289434416066785870268133201048179037478760511 +/- 2.21e-76]" ] }, "fixedIn": "tools/verify_sumdiff.py, tools/verify_twolaw.py: the printed \"ratio in [lo, hi]\" quantized outward (ROUND_FLOOR for lo, ROUND_CEILING for hi)" }, { "class": "printed", "op": "py.printed", "name": "verify_twolaw.py main(): \"ratio in [...]\"", "misses": 1, "of": 1, "why": "verify_sumdiff.py and verify_twolaw.py print \"ratio in [str(lo)[:44], str(hi)[:44]]\": the same cut toward zero, so the printed upper end is below the exact ratio. The verdicts compare the full 120-digit ends; tools/run-optq-ledger.py parses verify_twolaw's printed line into certs/optq-ledger.json.", "witness": { "truthSpec": { "t": "mixRho", "tl": "real:twolaw-23289305" }, "inputs": { "file": "corpus/optconst-queue/sources/zenodo-23289305/certificate.json" }, "printed": [ "1.674894211307116228764343737060957167779440", "1.674894211307116228764343737060957167779440" ], "exact": [ "[1.674894211307116228764343737060957167779440390811255252575175205962832958036 +/- 3.92e-76]", "[1.674894211307116228764343737060957167779440390811255252575175205962832958038 +/- 4.12e-76]" ] }, "fixedIn": "tools/verify_sumdiff.py, tools/verify_twolaw.py: the printed \"ratio in [lo, hi]\" quantized outward (ROUND_FLOOR for lo, ROUND_CEILING for hi)" }, { "class": "printed", "op": "rec.fei-ledger", "name": "C71 ratio", "misses": 1, "of": 1, "why": "certs/fei-ledger.json records fei.decide's strings: ratio = [6.521845710923046575658143972948578468366662, the same] \u2014 one point, below the exact ratio. The CERTIFIED verdict (C71 > 6.521845710923046575) is sound: the full enclosure lies above the threshold.", "witness": { "truthSpec": { "t": "feiRho", "f": "real:c71-n18" }, "inputs": { "file": "corpus/optimization-constants/num2026/fei_c71_n18_artifact.json" }, "printed": [ "6.521845710923046575658143972948578468366662", "6.521845710923046575658143972948578468366662" ], "exact": [ "[6.521845710923046575658143972948578468366662292229938598252894129325503563413 +/- 5.45e-76]", "[6.521845710923046575658143972948578468366662292229938598252894129325503563416 +/- 3.15e-76]" ] }, "fixedIn": "certs/fei-ledger.json regenerated by tools/run-fei-ledger.py from the fixed fei.py: the recorded ratio and H are fei.decide's outward strings" }, { "class": "printed", "op": "rec.fei-ledger", "name": "C71 H", "misses": 1, "of": 1, "why": "certs/fei-ledger.json records H = [6.7766053090059780825197902218, the same] \u2014 one point, below the exact H.", "witness": { "truthSpec": { "t": "feiH", "f": "real:c71-n18" }, "inputs": { "file": "corpus/optimization-constants/num2026/fei_c71_n18_artifact.json" }, "printed": [ "6.7766053090059780825197902218", "6.7766053090059780825197902218" ], "exact": [ "[6.776605309005978082519790221891882314787235038020170574747147806252281046359 +/- 4.47e-76]", "[6.776605309005978082519790221891882314787235038020170574747147806252281046362 +/- 4.13e-76]" ] }, "fixedIn": "certs/fei-ledger.json regenerated by tools/run-fei-ledger.py from the fixed fei.py: the recorded ratio and H are fei.decide's outward strings" }, { "class": "printed", "op": "rec.optq-ledger", "name": "optq-221-3c rhoPython (verify_twolaw.py)", "misses": 1, "of": 1, "why": "certs/optq-ledger.json records rhoPython = verify_twolaw's printed line, [1.674894211307116228764343737060957167779440, the same] \u2014 one point, below the exact ratio (and the row's \"how\" text cuts it again to 34 characters). The row's twolaw.js rho (outward, 30 places) contains it; the verdict is sound.", "witness": { "truthSpec": { "t": "mixRho", "tl": "real:twolaw-23289305" }, "inputs": { "file": "corpus/optconst-queue/sources/zenodo-23289305/certificate.json" }, "printed": [ "1.674894211307116228764343737060957167779440", "1.674894211307116228764343737060957167779440" ], "exact": [ "[1.674894211307116228764343737060957167779440390811255252575175205962832958036 +/- 3.92e-76]", "[1.674894211307116228764343737060957167779440390811255252575175205962832958038 +/- 4.12e-76]" ] }, "fixedIn": "tools/run-optq-ledger.py records verify_twolaw's outward printed line; certs/optq-ledger.json regenerated (every verdict unchanged)" }, { "class": "malformed", "impl": "verify_sumdiff", "name": "verify_sumdiff: a non-integer weight that truncates to a law (num 1.9/2 and 1.2/2)", "decided": "verdict CERTIFIED, 0.99999999999999999999999999999999999999, 1.00000000000000000000000000000000000000", "why": "read() applies int() to JSON numbers (1.9 -> 1, 1.2 -> 1), so a certificate whose weights are not integers is decided as the law {1/2, 1/2}; sumdiff.js refuses it.", "fixedIn": "tools/verify_sumdiff.py: law() refuses a coordinate or weight that is not an exact integer (a JSON float or a boolean) and a one-point law (every entropy 0, the ratio 0/0); ratio() refuses when the maximum entropy is not separated from 0; each with its reason" }, { "class": "malformed", "impl": "verify_sumdiff", "name": "verify_sumdiff: a boolean weight (true/2)", "decided": "verdict CERTIFIED, 0.99999999999999999999999999999999999999, 1.00000000000000000000000000000000000000", "why": "read(): int(True) = 1; sumdiff.js refuses it.", "fixedIn": "tools/verify_sumdiff.py: law() refuses a coordinate or weight that is not an exact integer (a JSON float or a boolean) and a one-point law (every entropy 0, the ratio 0/0); ratio() refuses when the maximum entropy is not separated from 0; each with its reason" }, { "class": "malformed", "impl": "verify_sumdiff", "name": "verify_sumdiff: a non-integer coordinate (x = 0.5)", "decided": "verdict CERTIFIED, 0.99999999999999999999999999999999999999, 1.00000000000000000000000000000000000000", "why": "read() never checks that a coordinate is an integer (the law is taken on R^2); sumdiff.js refuses it.", "fixedIn": "tools/verify_sumdiff.py: law() refuses a coordinate or weight that is not an exact integer (a JSON float or a boolean) and a one-point law (every entropy 0, the ratio 0/0); ratio() refuses when the maximum entropy is not separated from 0; each with its reason" }, { "class": "malformed", "impl": "verify_sumdiff", "name": "verify_sumdiff: a one-point law (p = 1: every entropy 0, the ratio 0/0)", "decided": "verdict REFUTED, -1, -1", "why": "every entropy is 0, so the ratio is 0/0; ratio() divides by the max's enclosure [-1E-1000118, 1E-1000118] without checking its sign and returns [-1, -1], and decide() says REFUTED. sumdiff.js refuses it (division by an interval containing 0).", "fixedIn": "tools/verify_sumdiff.py: law() refuses a coordinate or weight that is not an exact integer (a JSON float or a boolean) and a one-point law (every entropy 0, the ratio 0/0); ratio() refuses when the maximum entropy is not separated from 0; each with its reason" }, { "class": "malformed", "impl": "verify_sumdiff", "name": "verify_sumdiff: a one-point law in 3c, unreduced (10^320/10^320)", "decided": "verdict REFUTED, -1, -1", "why": "as the one-point law above.", "fixedIn": "tools/verify_sumdiff.py: law() refuses a coordinate or weight that is not an exact integer (a JSON float or a boolean) and a one-point law (every entropy 0, the ratio 0/0); ratio() refuses when the maximum entropy is not separated from 0; each with its reason" }, { "class": "malformed", "impl": "verify_twolaw", "name": "verify_twolaw: a non-integer weight that truncates to a law (1.9/2 and 1.2/2)", "decided": "verdict CERTIFIED, 0.47368421052631578947368421052631578947, 0.47368421052631578947368421052631578947", "why": "laws() applies int() to the weights; twolaw.js refuses it.", "fixedIn": "tools/verify_twolaw.py: decide() refuses a problem field that does not name 3c and two one-point laws (0/0); laws() reads each law through verify_sumdiff's law(), so a non-integer coordinate or weight is refused there; each with its reason" }, { "class": "malformed", "impl": "verify_twolaw", "name": "verify_twolaw: a non-integer coordinate (0.5)", "decided": "verdict CERTIFIED, 0.99999999999999999999999999999999999999, 1.00000000000000000000000000000000000000", "why": "laws() never checks that a point has integer coordinates (the injectivity bound is computed on the floats); twolaw.js refuses it.", "fixedIn": "tools/verify_twolaw.py: decide() refuses a problem field that does not name 3c and two one-point laws (0/0); laws() reads each law through verify_sumdiff's law(), so a non-integer coordinate or weight is refused there; each with its reason" }, { "class": "malformed", "impl": "verify_twolaw", "name": "verify_twolaw: not a 3c certificate", "decided": "verdict CERTIFIED, 0.49496896112507365590413242725352025639, 0.49496896112507365590413242725352025639", "why": "decide() never reads the certificate's problem field (twolaw.js requires 3c); main() decides only the pinned file.", "fixedIn": "tools/verify_twolaw.py: decide() refuses a problem field that does not name 3c and two one-point laws (0/0); laws() reads each law through verify_sumdiff's law(), so a non-integer coordinate or weight is refused there; each with its reason" }, { "class": "malformed", "impl": "verify_twolaw", "name": "verify_twolaw: two one-point laws (every entropy 0, the ratio 0/0)", "decided": "verdict REFUTED, -1, -1", "why": "as verify_sumdiff's one-point law: the ratio [-1, -1], REFUTED; twolaw.js refuses it.", "fixedIn": "tools/verify_twolaw.py: decide() refuses a problem field that does not name 3c and two one-point laws (0/0); laws() reads each law through verify_sumdiff's law(), so a non-integer coordinate or weight is refused there; each with its reason" }, { "class": "malformed", "impl": "fei.ln_iv", "name": "fei.ln_iv: k = 0 (the docstring's domain is a positive integer)", "decided": "verdict returned, -Infinity, -9.9999999999999999999999999999999999999", "why": "returns (-Infinity, -9.99...E+999999) rather than refusing; unreachable from fei.decide, which calls ln_iv only for a nonzero |a_S|.", "fixedIn": "instruments/fei/fei.py: ln_iv(k) refuses k <= 0" } ] }, "problems": [] }