FA-63831 / Actuarial life tables / Open access

Median and quantile future lifetime: Degenerate quantiles are accepted · case 01

p=0 or p=1000 returns a time instead of invalid.

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

ROOT CAUSE

The range check is inclusive.

THE FAILURE

The range check is inclusive.

Unsuccessful approach: Still accepts p=1000.

Case contract

Input l (from age x) and p (per mille surviving, 0<p<1000 else 'invalid'; also l_0<=0). Find the first t with l_t <= l_0 p/1000 and interpolate linearly: t-1 + (l_{t-1}-target)/(l_{t-1}-l_t). No crossing -> 'beyond table'. 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):
    l=x['l']
    if l[0]<=0 or not 0<=x['p']<=1000: return 'invalid'
    target=Fraction(l[0]*x['p'],1000)
    for t in range(1,len(l)):
        if l[t]<=target:
            frac=(l[t-1]-target)/(l[t-1]-l[t])
            return round(float(t-1+frac),6)
    return 'beyond table'
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['median', {'l': [100000, 99387, 98676, 97881, 96960, 95885, 94662, 93295, 91792, 90121, 88233, 0], 'p': 500}, 10.433319], ['upper quartile survival', {'l': [100000, 99475, 98885, 98208, 97456, 96585, 95576, 94404, 93122, 91705, 90114, 0], 'p': 750}, 10.167721], ['crossing in first year', {'l': [1000, 301, 100, 0], 'p': 500}, 0.715308], ['crossing at last age', {'l': [1000, 900, 800, 399], 'p': 500}, 2.74813], ['never crosses', {'l': [1000, 950, 900], 'p': 500}, 'beyond table'], ['p zero rejected', {'l': [1000, 500, 0], 'p': 0}, 'invalid'], ['p full rejected', {'l': [1000, 500, 0], 'p': 1000}, 'invalid'], ['low quantile', {'l': [100000, 99796, 99551, 99257, 98907, 98508, 98032, 97478, 96832, 96111, 95312, 0], 'p': 101}, 10.894032]], [['median', {'l': [100000, 99728, 99400, 99011, 98531, 97972, 97317, 96577, 95722, 94733, 93633, 0], 'p': 500}, 10.466], ['upper quartile survival', {'l': [100000, 99497, 98924, 98231, 97459, 96611, 95636, 94522, 93247, 91843, 90240, 0], 'p': 750}, 10.168883], ['crossing in first year', {'l': [1000, 302, 100, 0], 'p': 500}, 0.716332], ['crossing at last age', {'l': [1000, 900, 800, 398], 'p': 500}, 2.746269], ['never crosses', {'l': [1000, 950, 900], 'p': 500}, 'beyond table'], ['p zero rejected', {'l': [1000, 500, 0], 'p': 0}, 'invalid'], ['p full rejected', {'l': [1000, 500, 0], 'p': 1000}, 'invalid'], ['low quantile', {'l': [100000, 99549, 99016, 98408, 97709, 96926, 96052, 95042, 93885, 92604, 91159, 0], 'p': 102}, 10.888108]], [['median', {'l': [100000, 99519, 98953, 98284, 97516, 96640, 95632, 94490, 93212, 91778, 90214, 0], 'p': 500}, 10.445762], ['upper quartile survival', {'l': [100000, 99696, 99292, 98815, 98275, 97687, 97001, 96211, 95276, 94196, 92976, 0], 'p': 750}, 10.19334], ['crossing in first year', {'l': [1000, 303, 100, 0], 'p': 500}, 0.71736], ['crossing at last age', {'l': [1000, 900, 800, 397], 'p': 500}, 2.744417], ['never crosses', {'l': [1000, 950, 900], 'p': 500}, 'beyond table'], ['p zero rejected', {'l': [1000, 500, 0], 'p': 0}, 'invalid'], ['p full rejected', {'l': [1000, 500, 0], 'p': 1000}, 'invalid'], ['low quantile', {'l': [100000, 99674, 99256, 98791, 98244, 97618, 96907, 96078, 95119, 94025, 92792, 0], 'p': 103}, 10.888999]], [['median', {'l': [100000, 99629, 99193, 98668, 98069, 97392, 96584, 95658, 94604, 93390, 92018, 0], 'p': 500}, 10.456628], ['upper quartile survival', {'l': [100000, 99416, 98748, 97985, 97109, 96096, 94934, 93641, 92205, 90578, 88748, 0], 'p': 750}, 10.154911], ['crossing in first year', {'l': [1000, 304, 100, 0], 'p': 500}, 0.718391], ['crossing at last age', {'l': [1000, 900, 800, 396], 'p': 500}, 2.742574], ['never crosses', {'l': [1000, 950, 900], 'p': 500}, 'beyond table'], ['p zero rejected', {'l': [1000, 500, 0], 'p': 0}, 'invalid'], ['p full rejected', {'l': [1000, 500, 0], 'p': 1000}, 'invalid'], ['low quantile', {'l': [100000, 99428, 98766, 98052, 97212, 96267, 95157, 93939, 92567, 91036, 89325, 0], 'p': 104}, 10.883571]], [['median', {'l': [100000, 99481, 98876, 98212, 97434, 96547, 95571, 94472, 93234, 91812, 90252, 0], 'p': 500}, 10.445996], ['upper quartile survival', {'l': [100000, 99429, 98814, 98062, 97236, 96330, 95319, 94161, 92874, 91437, 89822, 0], 'p': 750}, 10.165015], ['crossing in first year', {'l': [1000, 305, 100, 0], 'p': 500}, 0.719424], ['crossing at last age', {'l': [1000, 900, 800, 395], 'p': 500}, 2.740741], ['never crosses', {'l': [1000, 950, 900], 'p': 500}, 'beyond table'], ['p zero rejected', {'l': [1000, 500, 0], 'p': 0}, 'invalid'], ['p full rejected', {'l': [1000, 500, 0], 'p': 1000}, 'invalid'], ['low quantile', {'l': [100000, 99395, 98716, 97924, 97019, 95986, 94803, 93476, 91996, 90309, 88450, 0], 'p': 105}, 10.881289]]]
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
median	`10.433319`	`10.433319`	Passed
upper quartile survival	`10.167721`	`10.167721`	Passed
crossing in first year	`0.715308`	`0.715308`	Passed
crossing at last age	`2.74813`	`2.74813`	Passed
never crosses	`beyond table`	`beyond table`	Passed
p zero rejected	`2.0`	`invalid`	Failed
p full rejected	`0.0`	`invalid`	Failed
low quantile	`10.894032`	`10.894032`	Passed

SHA-256 / 3338b970f1b8860b2c172f4b09a2637ebc01b15930d46cdb67da9fcfff3bc973

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):
    l=x['l']
    if l[0]<=0 or not 0<x['p']<=1000: return 'invalid'
    target=Fraction(l[0]*x['p'],1000)
    for t in range(1,len(l)):
        if l[t]<=target:
            frac=(l[t-1]-target)/(l[t-1]-l[t])
            return round(float(t-1+frac),6)
    return 'beyond table'
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['median', {'l': [100000, 99387, 98676, 97881, 96960, 95885, 94662, 93295, 91792, 90121, 88233, 0], 'p': 500}, 10.433319], ['upper quartile survival', {'l': [100000, 99475, 98885, 98208, 97456, 96585, 95576, 94404, 93122, 91705, 90114, 0], 'p': 750}, 10.167721], ['crossing in first year', {'l': [1000, 301, 100, 0], 'p': 500}, 0.715308], ['crossing at last age', {'l': [1000, 900, 800, 399], 'p': 500}, 2.74813], ['never crosses', {'l': [1000, 950, 900], 'p': 500}, 'beyond table'], ['p zero rejected', {'l': [1000, 500, 0], 'p': 0}, 'invalid'], ['p full rejected', {'l': [1000, 500, 0], 'p': 1000}, 'invalid'], ['low quantile', {'l': [100000, 99796, 99551, 99257, 98907, 98508, 98032, 97478, 96832, 96111, 95312, 0], 'p': 101}, 10.894032]], [['median', {'l': [100000, 99728, 99400, 99011, 98531, 97972, 97317, 96577, 95722, 94733, 93633, 0], 'p': 500}, 10.466], ['upper quartile survival', {'l': [100000, 99497, 98924, 98231, 97459, 96611, 95636, 94522, 93247, 91843, 90240, 0], 'p': 750}, 10.168883], ['crossing in first year', {'l': [1000, 302, 100, 0], 'p': 500}, 0.716332], ['crossing at last age', {'l': [1000, 900, 800, 398], 'p': 500}, 2.746269], ['never crosses', {'l': [1000, 950, 900], 'p': 500}, 'beyond table'], ['p zero rejected', {'l': [1000, 500, 0], 'p': 0}, 'invalid'], ['p full rejected', {'l': [1000, 500, 0], 'p': 1000}, 'invalid'], ['low quantile', {'l': [100000, 99549, 99016, 98408, 97709, 96926, 96052, 95042, 93885, 92604, 91159, 0], 'p': 102}, 10.888108]], [['median', {'l': [100000, 99519, 98953, 98284, 97516, 96640, 95632, 94490, 93212, 91778, 90214, 0], 'p': 500}, 10.445762], ['upper quartile survival', {'l': [100000, 99696, 99292, 98815, 98275, 97687, 97001, 96211, 95276, 94196, 92976, 0], 'p': 750}, 10.19334], ['crossing in first year', {'l': [1000, 303, 100, 0], 'p': 500}, 0.71736], ['crossing at last age', {'l': [1000, 900, 800, 397], 'p': 500}, 2.744417], ['never crosses', {'l': [1000, 950, 900], 'p': 500}, 'beyond table'], ['p zero rejected', {'l': [1000, 500, 0], 'p': 0}, 'invalid'], ['p full rejected', {'l': [1000, 500, 0], 'p': 1000}, 'invalid'], ['low quantile', {'l': [100000, 99674, 99256, 98791, 98244, 97618, 96907, 96078, 95119, 94025, 92792, 0], 'p': 103}, 10.888999]], [['median', {'l': [100000, 99629, 99193, 98668, 98069, 97392, 96584, 95658, 94604, 93390, 92018, 0], 'p': 500}, 10.456628], ['upper quartile survival', {'l': [100000, 99416, 98748, 97985, 97109, 96096, 94934, 93641, 92205, 90578, 88748, 0], 'p': 750}, 10.154911], ['crossing in first year', {'l': [1000, 304, 100, 0], 'p': 500}, 0.718391], ['crossing at last age', {'l': [1000, 900, 800, 396], 'p': 500}, 2.742574], ['never crosses', {'l': [1000, 950, 900], 'p': 500}, 'beyond table'], ['p zero rejected', {'l': [1000, 500, 0], 'p': 0}, 'invalid'], ['p full rejected', {'l': [1000, 500, 0], 'p': 1000}, 'invalid'], ['low quantile', {'l': [100000, 99428, 98766, 98052, 97212, 96267, 95157, 93939, 92567, 91036, 89325, 0], 'p': 104}, 10.883571]], [['median', {'l': [100000, 99481, 98876, 98212, 97434, 96547, 95571, 94472, 93234, 91812, 90252, 0], 'p': 500}, 10.445996], ['upper quartile survival', {'l': [100000, 99429, 98814, 98062, 97236, 96330, 95319, 94161, 92874, 91437, 89822, 0], 'p': 750}, 10.165015], ['crossing in first year', {'l': [1000, 305, 100, 0], 'p': 500}, 0.719424], ['crossing at last age', {'l': [1000, 900, 800, 395], 'p': 500}, 2.740741], ['never crosses', {'l': [1000, 950, 900], 'p': 500}, 'beyond table'], ['p zero rejected', {'l': [1000, 500, 0], 'p': 0}, 'invalid'], ['p full rejected', {'l': [1000, 500, 0], 'p': 1000}, 'invalid'], ['low quantile', {'l': [100000, 99395, 98716, 97924, 97019, 95986, 94803, 93476, 91996, 90309, 88450, 0], 'p': 105}, 10.881289]]]
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
median	`10.433319`	`10.433319`	Passed
upper quartile survival	`10.167721`	`10.167721`	Passed
crossing in first year	`0.715308`	`0.715308`	Passed
crossing at last age	`2.74813`	`2.74813`	Passed
never crosses	`beyond table`	`beyond table`	Passed
p zero rejected	`invalid`	`invalid`	Passed
p full rejected	`0.0`	`invalid`	Failed
low quantile	`10.894032`	`10.894032`	Passed

SHA-256 / b2bf0ed659a3f7e18db25bbb67b718b8df8b97aea37dc4f9a1d6f26f7235afcb

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 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 2026-09-29T14:47:18.418553+00:00.

Case digest / 00d2a450ce96ff7decf95f0edd6ee1becb52054cf4373d9910df1c1c19c78a0f