FA-63551 / Actuarial life tables / Open access

Joint-life and last-survivor status: Expectation stops at the shorter table · case 01

Last-survivor expectation ignores years when only the longer-lived table survives.

Verified by executionVariant 1 · 7 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The summation horizon uses the shorter table.

THE FAILURE

The summation horizon uses the shorter table.

Unsuccessful approach: Using the first life's table is still too short when the second is longer.

Case contract

Input lx, ly (independent lives, survival zero beyond each table), status joint|last (else 'unknown status'), t. Joint: tp_x*tp_y; last survivor: tp_x+tp_y-tp_x*tp_y. Return [status survival at t, curtate expectation = sum over k from 1 to the longer table length - 1], rounded to 6.

Why this case matters

Life-table arithmetic compounds across ages, so one misplaced index or assumption shifts every downstream value.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
from fractions import Fraction
N = 1
observations = []
def solve(x):
    lx=x['lx']; ly=x['ly']
    def p(l,t): return Fraction(l[t],l[0]) if t<len(l) else Fraction(0)
    def status(t):
        a=p(lx,t); b=p(ly,t)
        if x['status']=='joint': return a*b
        if x['status']=='last': return a+b-a*b
        return None
    if status(0) is None: return 'unknown status'
    T=x['t']
    e=sum(status(k) for k in range(1,min(len(lx),len(ly))))
    return [round(float(status(T)),6), round(float(e),6)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['joint life', {'lx': [100000, 99760, 99486, 99134, 98731, 98239, 97663, 0], 'ly': [100000, 99760, 99459, 99109, 98669, 98170, 97567, 96868, 96074, 95178], 'status': 'joint', 't': 1}, [0.995206, 5.85864]], ['last survivor unequal tables', {'lx': [100000, 99626, 99174, 98609, 97946, 0], 'ly': [100000, 99746, 99485, 99140, 98725, 98253, 97716, 97067, 96305, 95471], 'status': 'last', 't': 1}, [0.999991, 8.847686]], ['last survivor beyond shorter table', {'lx': [100000, 99547, 99068, 98510, 0], 'ly': [100000, 99650, 99240, 98743, 98172, 97496, 96738, 95876, 94860], 'status': 'last', 't': 7}, [0.95876, 7.831146]], ['last survivor open tables', {'lx': [100000, 99439, 98772, 97997, 97122, 96107, 94953], 'ly': [100000, 99716, 99346, 98921, 98456, 97885, 97234], 'status': 'last', 't': 3}, [0.999784, 5.997024]], ['joint life open tables', {'lx': [100000, 99684, 99310, 98868, 98342, 97727, 97000, 96166, 95225], 'ly': [100000, 99674, 99301, 98855, 98291, 97654], 'status': 'joint', 't': 2}, [0.986158, 4.878065]], ['joint beyond table', {'lx': [100000, 99570, 99088, 98538, 0], 'ly': [100000, 99747, 99450, 99136, 98733, 98245, 97704, 97074], 'status': 'joint', 't': 9}, [0.0, 2.955477]], ['unknown status', {'lx': [100000, 99617, 99173, 98642, 0], 'ly': [100000, 99511, 98916, 98270, 97551], 'status': 'reversionary', 't': 1}, 'unknown status']], [['joint life', {'lx': [100000, 99448, 98811, 98046, 97202, 96224, 95108, 0], 'ly': [100000, 99673, 99302, 98874, 98346, 97733, 97042, 96243, 95322, 94293], 'status': 'joint', 't': 4}, [0.955943, 5.761177]], ['last survivor unequal tables', {'lx': [100000, 99718, 99414, 99081, 98674, 0], 'ly': [100000, 99715, 99364, 98938, 98447, 97838, 97148, 96371, 95513, 94519], 'status': 'last', 't': 1}, [0.999992, 8.813541]], ['last survivor beyond shorter table', {'lx': [100000, 99428, 98762, 98025, 0], 'ly': [100000, 99693, 99289, 98823, 98276, 97648, 96919, 96073, 95105], 'status': 'last', 't': 7}, [0.96073, 7.839872]], ['last survivor open tables', {'lx': [100000, 99783, 99509, 99191, 98791, 98327, 97812], 'ly': [100000, 99761, 99460, 99128, 98713, 98242, 97687], 'status': 'last', 't': 3}, [0.999929, 5.998942]], ['joint life open tables', {'lx': [100000, 99667, 99266, 98772, 98175, 97495, 96689, 95778, 94734], 'ly': [100000, 99563, 99077, 98505, 97818, 97009], 'status': 'joint', 't': 2}, [0.983498, 4.854883]], ['joint beyond table', {'lx': [100000, 99653, 99229, 98732, 0], 'ly': [100000, 99509, 98964, 98317, 97578, 96754, 95832, 94804], 'status': 'joint', 't': 9}, [0.0, 2.94435]], ['unknown status', {'lx': [100000, 99557, 99059, 98473, 0], 'ly': [100000, 99384, 98656, 97854, 96902], 'status': 'reversionary', 't': 1}, 'unknown status']], [['joint life', {'lx': [100000, 99420, 98745, 98004, 97134, 96149, 95028, 0], 'ly': [100000, 99781, 99500, 99136, 98731, 98273, 97761, 97153, 96455, 95674], 'status': 'joint', 't': 1}, [0.992023, 5.77901]], ['last survivor unequal tables', {'lx': [100000, 99469, 98862, 98156, 97349, 0], 'ly': [100000, 99528, 98974, 98327, 97564, 96691, 95712, 94591, 93350, 91946], 'status': 'last', 't': 1}, [0.999975, 8.721804]], ['last survivor beyond shorter table', {'lx': [100000, 99476, 98855, 98104, 0], 'ly': [100000, 99661, 99248, 98756, 98176, 97498, 96731, 95865, 94884], 'status': 'last', 't': 7}, [0.95865, 7.8312]], ['last survivor open tables', {'lx': [100000, 99450, 98799, 98048, 97168, 96153, 95008], 'ly': [100000, 99755, 99464, 99076, 98648, 98129, 97541], 'status': 'last', 't': 3}, [0.99982, 5.997412]], ['joint life open tables', {'lx': [100000, 99685, 99309, 98892, 98407, 97822, 97160, 96399, 95545], 'ly': [100000, 99396, 98724, 97967, 97081, 96062], 'status': 'joint', 't': 2}, [0.980418, 4.835105]], ['joint beyond table', {'lx': [100000, 99548, 99025, 98421, 0], 'ly': [100000, 99505, 98968, 98338, 97593, 96743, 95786, 94672], 'status': 'joint', 't': 9}, [0.0, 2.938435]], ['unknown status', {'lx': [100000, 99455, 98849, 98120, 0], 'ly': [100000, 99494, 98921, 98296, 97615], 'status': 'reversionary', 't': 1}, 'unknown status']], [['joint life', {'lx': [100000, 99524, 98933, 98258, 97497, 96646, 95677, 0], 'ly': [100000, 99594, 99118, 98553, 97900, 97141, 96258, 95231, 94074, 92762], 'status': 'joint', 't': 3}, [0.968362, 5.754458]], ['last survivor unequal tables', {'lx': [100000, 99732, 99425, 99030, 98600, 0], 'ly': [100000, 99770, 99497, 99164, 98735, 98241, 97675, 97023, 96270, 95436], 'status': 'last', 't': 3}, [0.999919, 8.846157]], ['last survivor beyond shorter table', {'lx': [100000, 99698, 99335, 98894, 0], 'ly': [100000, 99669, 99278, 98818, 98308, 97710, 97016, 96215, 95327], 'status': 'last', 't': 7}, [0.96215, 7.845571]], ['last survivor open tables', {'lx': [100000, 99784, 99547, 99245, 98874, 98425, 97879], 'ly': [100000, 99504, 98929, 98302, 97565, 96734, 95811], 'status': 'last', 't': 3}, [0.999872, 5.998136]], ['joint life open tables', {'lx': [100000, 99606, 99145, 98568, 97907, 97123, 96202, 95190, 94059], 'ly': [100000, 99527, 98969, 98322, 97575, 96743], 'status': 'joint', 't': 2}, [0.981228, 4.836642]], ['joint beyond table', {'lx': [100000, 99565, 99039, 98435, 0], 'ly': [100000, 99674, 99279, 98817, 98295, 97694, 96998, 96233], 'status': 'joint', 't': 9}, [0.0, 2.948359]], ['unknown status', {'lx': [100000, 99508, 98911, 98195, 0], 'ly': [100000, 99742, 99447, 99070, 98653], 'status': 'reversionary', 't': 1}, 'unknown status']], [['joint life', {'lx': [100000, 99789, 99523, 99217, 98831, 98388, 97851, 0], 'ly': [100000, 99688, 99282, 98825, 98282, 97645, 96945, 96125, 95188, 94126], 'status': 'joint', 't': 3}, [0.980512, 5.84403]], ['last survivor unequal tables', {'lx': [100000, 99741, 99448, 99116, 98714, 0], 'ly': [100000, 99521, 98971, 98345, 97624, 96787, 95853, 94770, 93551, 92159], 'status': 'last', 't': 1}, [0.999988, 8.730679]], ['last survivor beyond shorter table', {'lx': [100000, 99668, 99245, 98758, 0], 'ly': [100000, 99563, 99025, 98427, 97706, 96930, 96030, 94987, 93814], 'status': 'last', 't': 7}, [0.94987, 7.794387]], ['last survivor open tables', {'lx': [100000, 99694, 99322, 98864, 98352, 97783, 97128], 'ly': [100000, 99535, 98960, 98318, 97580, 96708, 95703], 'status': 'last', 't': 3}, [0.999809, 5.997361]], ['joint life open tables', {'lx': [100000, 99664, 99263, 98780, 98192, 97509, 96759, 95881, 94872], 'ly': [100000, 99636, 99196, 98689, 98098, 97408], 'status': 'joint', 't': 2}, [0.984649, 4.865571]], ['joint beyond table', {'lx': [100000, 99667, 99263, 98779, 0], 'ly': [100000, 99685, 99331, 98920, 98454, 97919, 97309, 96596], 'status': 'joint', 't': 9}, [0.0, 2.956642]], ['unknown status', {'lx': [100000, 99732, 99419, 99061, 0], 'ly': [100000, 99758, 99459, 99121, 98740], 'status': 'reversionary', 't': 1}, 'unknown status']]]
for label, args, expected in fixtures[N-1]:
    check(label, 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 fixture	Actual	Expected	Outcome
joint life	`[0.995206, 5.85864]`	`[0.995206, 5.85864]`	Passed
last survivor unequal tables	`[0.999991, 4.982096]`	`[0.999991, 8.847686]`	Failed
last survivor beyond shorter table	`[0.95876, 3.981446]`	`[0.95876, 7.831146]`	Failed
last survivor open tables	`[0.999784, 5.997024]`	`[0.999784, 5.997024]`	Passed
joint life open tables	`[0.986158, 4.878065]`	`[0.986158, 4.878065]`	Passed
joint beyond table	`[0.0, 2.955477]`	`[0.0, 2.955477]`	Passed
unknown status	`unknown status`	`unknown status`	Passed

SHA-256 / a44d010b5528b48ae7c039c21be5a40d5c9c7a8ad19370786555a80f9165dfda

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
from fractions import Fraction
N = 1
observations = []
def solve(x):
    lx=x['lx']; ly=x['ly']
    def p(l,t): return Fraction(l[t],l[0]) if t<len(l) else Fraction(0)
    def status(t):
        a=p(lx,t); b=p(ly,t)
        if x['status']=='joint': return a*b
        if x['status']=='last': return a+b-a*b
        return None
    if status(0) is None: return 'unknown status'
    T=x['t']
    e=sum(status(k) for k in range(1,len(lx)))
    return [round(float(status(T)),6), round(float(e),6)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['joint life', {'lx': [100000, 99760, 99486, 99134, 98731, 98239, 97663, 0], 'ly': [100000, 99760, 99459, 99109, 98669, 98170, 97567, 96868, 96074, 95178], 'status': 'joint', 't': 1}, [0.995206, 5.85864]], ['last survivor unequal tables', {'lx': [100000, 99626, 99174, 98609, 97946, 0], 'ly': [100000, 99746, 99485, 99140, 98725, 98253, 97716, 97067, 96305, 95471], 'status': 'last', 't': 1}, [0.999991, 8.847686]], ['last survivor beyond shorter table', {'lx': [100000, 99547, 99068, 98510, 0], 'ly': [100000, 99650, 99240, 98743, 98172, 97496, 96738, 95876, 94860], 'status': 'last', 't': 7}, [0.95876, 7.831146]], ['last survivor open tables', {'lx': [100000, 99439, 98772, 97997, 97122, 96107, 94953], 'ly': [100000, 99716, 99346, 98921, 98456, 97885, 97234], 'status': 'last', 't': 3}, [0.999784, 5.997024]], ['joint life open tables', {'lx': [100000, 99684, 99310, 98868, 98342, 97727, 97000, 96166, 95225], 'ly': [100000, 99674, 99301, 98855, 98291, 97654], 'status': 'joint', 't': 2}, [0.986158, 4.878065]], ['joint beyond table', {'lx': [100000, 99570, 99088, 98538, 0], 'ly': [100000, 99747, 99450, 99136, 98733, 98245, 97704, 97074], 'status': 'joint', 't': 9}, [0.0, 2.955477]], ['unknown status', {'lx': [100000, 99617, 99173, 98642, 0], 'ly': [100000, 99511, 98916, 98270, 97551], 'status': 'reversionary', 't': 1}, 'unknown status']], [['joint life', {'lx': [100000, 99448, 98811, 98046, 97202, 96224, 95108, 0], 'ly': [100000, 99673, 99302, 98874, 98346, 97733, 97042, 96243, 95322, 94293], 'status': 'joint', 't': 4}, [0.955943, 5.761177]], ['last survivor unequal tables', {'lx': [100000, 99718, 99414, 99081, 98674, 0], 'ly': [100000, 99715, 99364, 98938, 98447, 97838, 97148, 96371, 95513, 94519], 'status': 'last', 't': 1}, [0.999992, 8.813541]], ['last survivor beyond shorter table', {'lx': [100000, 99428, 98762, 98025, 0], 'ly': [100000, 99693, 99289, 98823, 98276, 97648, 96919, 96073, 95105], 'status': 'last', 't': 7}, [0.96073, 7.839872]], ['last survivor open tables', {'lx': [100000, 99783, 99509, 99191, 98791, 98327, 97812], 'ly': [100000, 99761, 99460, 99128, 98713, 98242, 97687], 'status': 'last', 't': 3}, [0.999929, 5.998942]], ['joint life open tables', {'lx': [100000, 99667, 99266, 98772, 98175, 97495, 96689, 95778, 94734], 'ly': [100000, 99563, 99077, 98505, 97818, 97009], 'status': 'joint', 't': 2}, [0.983498, 4.854883]], ['joint beyond table', {'lx': [100000, 99653, 99229, 98732, 0], 'ly': [100000, 99509, 98964, 98317, 97578, 96754, 95832, 94804], 'status': 'joint', 't': 9}, [0.0, 2.94435]], ['unknown status', {'lx': [100000, 99557, 99059, 98473, 0], 'ly': [100000, 99384, 98656, 97854, 96902], 'status': 'reversionary', 't': 1}, 'unknown status']], [['joint life', {'lx': [100000, 99420, 98745, 98004, 97134, 96149, 95028, 0], 'ly': [100000, 99781, 99500, 99136, 98731, 98273, 97761, 97153, 96455, 95674], 'status': 'joint', 't': 1}, [0.992023, 5.77901]], ['last survivor unequal tables', {'lx': [100000, 99469, 98862, 98156, 97349, 0], 'ly': [100000, 99528, 98974, 98327, 97564, 96691, 95712, 94591, 93350, 91946], 'status': 'last', 't': 1}, [0.999975, 8.721804]], ['last survivor beyond shorter table', {'lx': [100000, 99476, 98855, 98104, 0], 'ly': [100000, 99661, 99248, 98756, 98176, 97498, 96731, 95865, 94884], 'status': 'last', 't': 7}, [0.95865, 7.8312]], ['last survivor open tables', {'lx': [100000, 99450, 98799, 98048, 97168, 96153, 95008], 'ly': [100000, 99755, 99464, 99076, 98648, 98129, 97541], 'status': 'last', 't': 3}, [0.99982, 5.997412]], ['joint life open tables', {'lx': [100000, 99685, 99309, 98892, 98407, 97822, 97160, 96399, 95545], 'ly': [100000, 99396, 98724, 97967, 97081, 96062], 'status': 'joint', 't': 2}, [0.980418, 4.835105]], ['joint beyond table', {'lx': [100000, 99548, 99025, 98421, 0], 'ly': [100000, 99505, 98968, 98338, 97593, 96743, 95786, 94672], 'status': 'joint', 't': 9}, [0.0, 2.938435]], ['unknown status', {'lx': [100000, 99455, 98849, 98120, 0], 'ly': [100000, 99494, 98921, 98296, 97615], 'status': 'reversionary', 't': 1}, 'unknown status']], [['joint life', {'lx': [100000, 99524, 98933, 98258, 97497, 96646, 95677, 0], 'ly': [100000, 99594, 99118, 98553, 97900, 97141, 96258, 95231, 94074, 92762], 'status': 'joint', 't': 3}, [0.968362, 5.754458]], ['last survivor unequal tables', {'lx': [100000, 99732, 99425, 99030, 98600, 0], 'ly': [100000, 99770, 99497, 99164, 98735, 98241, 97675, 97023, 96270, 95436], 'status': 'last', 't': 3}, [0.999919, 8.846157]], ['last survivor beyond shorter table', {'lx': [100000, 99698, 99335, 98894, 0], 'ly': [100000, 99669, 99278, 98818, 98308, 97710, 97016, 96215, 95327], 'status': 'last', 't': 7}, [0.96215, 7.845571]], ['last survivor open tables', {'lx': [100000, 99784, 99547, 99245, 98874, 98425, 97879], 'ly': [100000, 99504, 98929, 98302, 97565, 96734, 95811], 'status': 'last', 't': 3}, [0.999872, 5.998136]], ['joint life open tables', {'lx': [100000, 99606, 99145, 98568, 97907, 97123, 96202, 95190, 94059], 'ly': [100000, 99527, 98969, 98322, 97575, 96743], 'status': 'joint', 't': 2}, [0.981228, 4.836642]], ['joint beyond table', {'lx': [100000, 99565, 99039, 98435, 0], 'ly': [100000, 99674, 99279, 98817, 98295, 97694, 96998, 96233], 'status': 'joint', 't': 9}, [0.0, 2.948359]], ['unknown status', {'lx': [100000, 99508, 98911, 98195, 0], 'ly': [100000, 99742, 99447, 99070, 98653], 'status': 'reversionary', 't': 1}, 'unknown status']], [['joint life', {'lx': [100000, 99789, 99523, 99217, 98831, 98388, 97851, 0], 'ly': [100000, 99688, 99282, 98825, 98282, 97645, 96945, 96125, 95188, 94126], 'status': 'joint', 't': 3}, [0.980512, 5.84403]], ['last survivor unequal tables', {'lx': [100000, 99741, 99448, 99116, 98714, 0], 'ly': [100000, 99521, 98971, 98345, 97624, 96787, 95853, 94770, 93551, 92159], 'status': 'last', 't': 1}, [0.999988, 8.730679]], ['last survivor beyond shorter table', {'lx': [100000, 99668, 99245, 98758, 0], 'ly': [100000, 99563, 99025, 98427, 97706, 96930, 96030, 94987, 93814], 'status': 'last', 't': 7}, [0.94987, 7.794387]], ['last survivor open tables', {'lx': [100000, 99694, 99322, 98864, 98352, 97783, 97128], 'ly': [100000, 99535, 98960, 98318, 97580, 96708, 95703], 'status': 'last', 't': 3}, [0.999809, 5.997361]], ['joint life open tables', {'lx': [100000, 99664, 99263, 98780, 98192, 97509, 96759, 95881, 94872], 'ly': [100000, 99636, 99196, 98689, 98098, 97408], 'status': 'joint', 't': 2}, [0.984649, 4.865571]], ['joint beyond table', {'lx': [100000, 99667, 99263, 98779, 0], 'ly': [100000, 99685, 99331, 98920, 98454, 97919, 97309, 96596], 'status': 'joint', 't': 9}, [0.0, 2.956642]], ['unknown status', {'lx': [100000, 99732, 99419, 99061, 0], 'ly': [100000, 99758, 99459, 99121, 98740], 'status': 'reversionary', 't': 1}, 'unknown status']]]
for label, args, expected in fixtures[N-1]:
    check(label, 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 fixture	Actual	Expected	Outcome
joint life	`[0.995206, 5.85864]`	`[0.995206, 5.85864]`	Passed
last survivor unequal tables	`[0.999991, 4.982096]`	`[0.999991, 8.847686]`	Failed
last survivor beyond shorter table	`[0.95876, 3.981446]`	`[0.95876, 7.831146]`	Failed
last survivor open tables	`[0.999784, 5.997024]`	`[0.999784, 5.997024]`	Passed
joint life open tables	`[0.986158, 4.878065]`	`[0.986158, 4.878065]`	Passed
joint beyond table	`[0.0, 2.955477]`	`[0.0, 2.955477]`	Passed
unknown status	`unknown status`	`unknown status`	Passed

SHA-256 / edd0ff933756e7f0083e32a3cba64bc3109710d19a322166d16b82d4cbf81825

HELD IN THE MEMBER ARCHIVE

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

This mechanism has 7 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 of a stipulated toy actuarial contract. Rates are small synthetic tables, exact rationals are used where practical and floats are rounded at the output; it makes no claim of conformance with any published table, standard of practice or regulation. 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 2026-09-29T14:47:15.676432+00:00.

Case digest / 576becfb0d82fd4b0ae673240350857d008795fa4a5126f2428c641cda03de2a