FAILURE MAP
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FA-14926 / Numerics / Open access

Pell positive solutions: quadratic unit real product · case 01

The exact pell positive solutions result violates the stated contract at quadratic unit real product.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The quadratic unit real product step uses a*u+b*v instead of a*u+D*b*v.

THE FAILURE

The quadratic unit real product step uses a*u+b*v instead of a*u+D*b*v.

Unsuccessful approach: The partial repair a*u-D*b*v still violates the quadratic unit real product invariant.

Case contract

Input [D,u,v,k], known positive fundamental solution u^2-D*v^2=1, k>=0; return first k positive solutions.

Why this case matters

Exact discrete arithmetic with observable algorithmic state; no floating point approximation is used.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
    D,u,v,k=x
    a,b=1,0
    out=[]
    for _ in range(k):
     a,b=a*u+b*v,a*v+b*u
     out.append([a,b])
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([2, 3, 2, 2], [[3, 2], [17, 12]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([2, 3, 2, 1], [[3, 2]]), ([2, 3, 2, 3], [[3, 2], [17, 12], [99, 70]]), ([2, 3, 2, 4], [[3, 2], [17, 12], [99, 70], [577, 408]]), ([2, 3, 2, 5], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]]), ([2, 3, 2, 6], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860]])], [([2, 3, 2, 3], [[3, 2], [17, 12], [99, 70]]), ([2, 3, 2, 5], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([3, 2, 1, 8], [[2, 1], [7, 4], [26, 15], [97, 56], [362, 209], [1351, 780], [5042, 2911], [18817, 10864]]), ([5, 9, 4, 0], []), ([5, 9, 4, 1], [[9, 4]]), ([5, 9, 4, 2], [[9, 4], [161, 72]])], [([2, 3, 2, 4], [[3, 2], [17, 12], [99, 70], [577, 408]]), ([2, 3, 2, 8], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860], [114243, 80782], [665857, 470832]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([6, 5, 2, 7], [[5, 2], [49, 20], [485, 198], [4801, 1960], [47525, 19402], [470449, 192060], [4656965, 1901198]]), ([6, 5, 2, 8], [[5, 2], [49, 20], [485, 198], [4801, 1960], [47525, 19402], [470449, 192060], [4656965, 1901198], [46099201, 18819920]]), ([7, 8, 3, 0], []), ([7, 8, 3, 1], [[8, 3]])], [([2, 3, 2, 5], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]]), ([3, 2, 1, 4], [[2, 1], [7, 4], [26, 15], [97, 56]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([10, 19, 6, 6], [[19, 6], [721, 228], [27379, 8658], [1039681, 328776], [39480499, 12484830], [1499219281, 474094764]]), ([10, 19, 6, 7], [[19, 6], [721, 228], [27379, 8658], [1039681, 328776], [39480499, 12484830], [1499219281, 474094764], [56930852179, 18003116202]]), ([10, 19, 6, 8], [[19, 6], [721, 228], [27379, 8658], [1039681, 328776], [39480499, 12484830], [1499219281, 474094764], [56930852179, 18003116202], [2161873163521, 683644320912]]), ([11, 10, 3, 0], [])], [([2, 3, 2, 6], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860]]), ([3, 2, 1, 7], [[2, 1], [7, 4], [26, 15], [97, 56], [362, 209], [1351, 780], [5042, 2911]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([12, 7, 2, 5], [[7, 2], [97, 28], [1351, 390], [18817, 5432], [262087, 75658]]), ([12, 7, 2, 6], [[7, 2], [97, 28], [1351, 390], [18817, 5432], [262087, 75658], [3650401, 1053780]]), ([12, 7, 2, 7], [[7, 2], [97, 28], [1351, 390], [18817, 5432], [262087, 75658], [3650401, 1053780], [50843527, 14677262]]), ([12, 7, 2, 8], [[7, 2], [97, 28], [1351, 390], [18817, 5432], [262087, 75658], [3650401, 1053780], [50843527, 14677262], [708158977, 204427888]])]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("explicit oracle %d" % i, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
explicit oracle 0[[3, 2], [13, 12]][[3, 2], [17, 12]]Failed
explicit oracle 1[][]Passed
explicit oracle 2[[649, 180], [453601, 233640], [336442249, 233280540], [260341516801, 211958675280], [207114205954249, 184422653280900], [167613197254869601, 156970859051068920], [137035719647602776649, 132044463030020257260], [112704185396697848352001, 110363286043051646758560]][[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]Failed
explicit oracle 3[[3, 2]][[3, 2]]Passed
explicit oracle 4[[3, 2], [13, 12], [63, 62]][[3, 2], [17, 12], [99, 70]]Failed
explicit oracle 5[[3, 2], [13, 12], [63, 62], [313, 312]][[3, 2], [17, 12], [99, 70], [577, 408]]Failed
explicit oracle 6[[3, 2], [13, 12], [63, 62], [313, 312], [1563, 1562]][[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]]Failed
explicit oracle 7[[3, 2], [13, 12], [63, 62], [313, 312], [1563, 1562], [7813, 7812]][[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860]]Failed

SHA-256 / 1047fd6e2358e7714fd1b23d0a3acad59071510258cf4aab9cee87d7619f9094

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
    D,u,v,k=x
    a,b=1,0
    out=[]
    for _ in range(k):
     a,b=a*u-D*b*v,a*v+b*u
     out.append([a,b])
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([2, 3, 2, 2], [[3, 2], [17, 12]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([2, 3, 2, 1], [[3, 2]]), ([2, 3, 2, 3], [[3, 2], [17, 12], [99, 70]]), ([2, 3, 2, 4], [[3, 2], [17, 12], [99, 70], [577, 408]]), ([2, 3, 2, 5], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]]), ([2, 3, 2, 6], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860]])], [([2, 3, 2, 3], [[3, 2], [17, 12], [99, 70]]), ([2, 3, 2, 5], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([3, 2, 1, 8], [[2, 1], [7, 4], [26, 15], [97, 56], [362, 209], [1351, 780], [5042, 2911], [18817, 10864]]), ([5, 9, 4, 0], []), ([5, 9, 4, 1], [[9, 4]]), ([5, 9, 4, 2], [[9, 4], [161, 72]])], [([2, 3, 2, 4], [[3, 2], [17, 12], [99, 70], [577, 408]]), ([2, 3, 2, 8], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860], [114243, 80782], [665857, 470832]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([6, 5, 2, 7], [[5, 2], [49, 20], [485, 198], [4801, 1960], [47525, 19402], [470449, 192060], [4656965, 1901198]]), ([6, 5, 2, 8], [[5, 2], [49, 20], [485, 198], [4801, 1960], [47525, 19402], [470449, 192060], [4656965, 1901198], [46099201, 18819920]]), ([7, 8, 3, 0], []), ([7, 8, 3, 1], [[8, 3]])], [([2, 3, 2, 5], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]]), ([3, 2, 1, 4], [[2, 1], [7, 4], [26, 15], [97, 56]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([10, 19, 6, 6], [[19, 6], [721, 228], [27379, 8658], [1039681, 328776], [39480499, 12484830], [1499219281, 474094764]]), ([10, 19, 6, 7], [[19, 6], [721, 228], [27379, 8658], [1039681, 328776], [39480499, 12484830], [1499219281, 474094764], [56930852179, 18003116202]]), ([10, 19, 6, 8], [[19, 6], [721, 228], [27379, 8658], [1039681, 328776], [39480499, 12484830], [1499219281, 474094764], [56930852179, 18003116202], [2161873163521, 683644320912]]), ([11, 10, 3, 0], [])], [([2, 3, 2, 6], [[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860]]), ([3, 2, 1, 7], [[2, 1], [7, 4], [26, 15], [97, 56], [362, 209], [1351, 780], [5042, 2911]]), ([2, 3, 2, 0], []), ([13, 649, 180, 8], [[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]), ([12, 7, 2, 5], [[7, 2], [97, 28], [1351, 390], [18817, 5432], [262087, 75658]]), ([12, 7, 2, 6], [[7, 2], [97, 28], [1351, 390], [18817, 5432], [262087, 75658], [3650401, 1053780]]), ([12, 7, 2, 7], [[7, 2], [97, 28], [1351, 390], [18817, 5432], [262087, 75658], [3650401, 1053780], [50843527, 14677262]]), ([12, 7, 2, 8], [[7, 2], [97, 28], [1351, 390], [18817, 5432], [262087, 75658], [3650401, 1053780], [50843527, 14677262], [708158977, 204427888]])]]
for i, (args, expected) in enumerate(fixtures[N-1]):
    check("explicit oracle %d" % i, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
explicit oracle 0[[3, 2], [1, 12]][[3, 2], [17, 12]]Failed
explicit oracle 1[][]Passed
explicit oracle 2[[649, 180], [1, 233640], [-546716951, 151632540], [-709639444799, 467280], [-460557093109751, -127734796799100], [-2128918334399, -165800159882371080], [387970992456749302249, -107604686968959022740], [503588141611794410371201, -663200639531353440]][[649, 180], [842401, 233640], [1093435849, 303264540], [1419278889601, 393637139280], [1842222905266249, 510940703520900], [2391203911756701601, 663200639532988920], [3103780835237293411849, 860833919173116097260], [4028705132934095091878401, 1117361763886065161254560]]Failed
explicit oracle 3[[3, 2]][[3, 2]]Passed
explicit oracle 4[[3, 2], [1, 12], [-45, 38]][[3, 2], [17, 12], [99, 70]]Failed
explicit oracle 5[[3, 2], [1, 12], [-45, 38], [-287, 24]][[3, 2], [17, 12], [99, 70], [577, 408]]Failed
explicit oracle 6[[3, 2], [1, 12], [-45, 38], [-287, 24], [-957, -502]][[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378]]Failed
explicit oracle 7[[3, 2], [1, 12], [-45, 38], [-287, 24], [-957, -502], [-863, -3420]][[3, 2], [17, 12], [99, 70], [577, 408], [3363, 2378], [19601, 13860]]Failed

SHA-256 / 7465f0b3c1ac4ba2357e80daa22a14db1653b6d8fc511b4ea2f4d234261fd42d

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 8 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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Verification & scope

A deterministic bounded teaching model. Inputs are restricted to the explicit contract; this is not a production algebra library. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:39:21.795919+00:00.

Case digest / 9383d6e2e2a7b34ff3281bcfa1ffcdcb2504bfb0a018c8ea1038a046b0ff7c10