FA-63646 / Actuarial life tables / Open access

Gompertz-Makeham survival: Hazard integral starts at the wrong age · case 01

Survival from age x is computed as if from age x+t.

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

ROOT CAUSE

The age factor uses c^(x+t).

THE FAILURE

The age factor uses c^(x+t).

Unsuccessful approach: Dropping c^x computes survival from age zero.

Case contract

Input A, B, c, age, t (floats). B<0, c<1 or A<=-B -> 'invalid parameters'. Cumulative hazard H = A t + B c^x (c^t-1)/ln c (c=1: (A+B) t). Return [exp(-H), mu at x+t = A + B c^(x+t)] rounded to 8.

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):
    A=x['A']; B=x['B']; c=x['c']; a=x['age']; t=x['t']
    if B<0 or c<1 or A<=-B: return 'invalid parameters'
    if c==1: H=(A+B)*t
    else: H=A*t+B*c**(a+t)*(c**t-1)/math.log(c)
    mu=A+B*c**(a+t)
    return [round(math.exp(-H),8),round(mu,8)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['makeham 0', {'A': 0.001, 'B': 3e-05, 'c': 1.12, 'age': 76, 't': 2.5}, [0.61903389, 0.22014725]], ['makeham 1', {'A': 0.001, 'B': 5e-05, 'c': 1.08, 'age': 44, 't': 10}, [0.96826103, 0.00419046]], ['makeham 2', {'A': 0.001, 'B': 0.0001, 'c': 1.12, 'age': 42, 't': 10}, [0.7970066, 0.03725243]], ['makeham 3', {'A': 0.0005, 'B': 5e-05, 'c': 1.12, 'age': 61, 't': 10}, [0.39101048, 0.15661679]], ['gompertz only', {'A': 0.0, 'B': 4e-05, 'c': 1.1, 'age': 51, 't': 3}, [0.98222186, 0.00687488]], ['constant force degenerate', {'A': 0.002, 'B': 0.001, 'c': 1.0, 'age': 40, 't': 4}, [0.98807171, 0.003]], ['c below one rejected', {'A': 0.001, 'B': 0.0001, 'c': 0.95, 'age': 40, 't': 1}, 'invalid parameters'], ['negative total force rejected', {'A': -0.01, 'B': 0.001, 'c': 1.1, 'age': 40, 't': 1}, 'invalid parameters']], [['makeham 0', {'A': 0.001, 'B': 3e-05, 'c': 1.08, 'age': 35, 't': 2.5}, [0.99628416, 0.00153767]], ['makeham 1', {'A': 0.001, 'B': 3e-05, 'c': 1.1, 'age': 78, 't': 10}, [0.42348454, 0.13272783]], ['makeham 2', {'A': 0.0007, 'B': 0.0001, 'c': 1.08, 'age': 34, 't': 5}, [0.98822133, 0.00271153]], ['makeham 3', {'A': 0.0005, 'B': 3e-05, 'c': 1.08, 'age': 32, 't': 1}, [0.99913436, 0.00088028]], ['gompertz only', {'A': 0.0, 'B': 4e-05, 'c': 1.1, 'age': 52, 't': 3}, [0.98046153, 0.00756237]], ['constant force degenerate', {'A': 0.002, 'B': 0.002, 'c': 1.0, 'age': 40, 't': 4}, [0.98412732, 0.004]], ['c below one rejected', {'A': 0.001, 'B': 0.0001, 'c': 0.95, 'age': 40, 't': 1}, 'invalid parameters'], ['negative total force rejected', {'A': -0.01, 'B': 0.001, 'c': 1.1, 'age': 40, 't': 1}, 'invalid parameters']], [['makeham 0', {'A': 0.0005, 'B': 0.0001, 'c': 1.12, 'age': 44, 't': 2.5}, [0.95736913, 0.01993739]], ['makeham 1', {'A': 0.0005, 'B': 5e-05, 'c': 1.08, 'age': 40, 't': 5}, [0.99091741, 0.00209602]], ['makeham 2', {'A': 0.0007, 'B': 3e-05, 'c': 1.08, 'age': 42, 't': 2.5}, [0.99616179, 0.00162146]], ['makeham 3', {'A': 0.0005, 'B': 5e-05, 'c': 1.1, 'age': 58, 't': 5}, [0.92026523, 0.02076325]], ['gompertz only', {'A': 0.0, 'B': 4e-05, 'c': 1.1, 'age': 53, 't': 3}, [0.9785288, 0.0083186]], ['constant force degenerate', {'A': 0.002, 'B': 0.003, 'c': 1.0, 'age': 40, 't': 4}, [0.98019867, 0.005]], ['c below one rejected', {'A': 0.001, 'B': 0.0001, 'c': 0.95, 'age': 40, 't': 1}, 'invalid parameters'], ['negative total force rejected', {'A': -0.01, 'B': 0.001, 'c': 1.1, 'age': 40, 't': 1}, 'invalid parameters']], [['makeham 0', {'A': 0.0007, 'B': 3e-05, 'c': 1.1, 'age': 36, 't': 1}, [0.99832839, 0.00172012]], ['makeham 1', {'A': 0.0007, 'B': 5e-05, 'c': 1.1, 'age': 45, 't': 10}, [0.93431427, 0.01015296]], ['makeham 2', {'A': 0.0005, 'B': 0.0001, 'c': 1.1, 'age': 76, 't': 10}, [0.0958952, 0.36338659]], ['makeham 3', {'A': 0.0005, 'B': 5e-05, 'c': 1.1, 'age': 53, 't': 5}, [0.94881434, 0.01308189]], ['gompertz only', {'A': 0.0, 'B': 4e-05, 'c': 1.1, 'age': 54, 't': 3}, [0.9764072, 0.00915046]], ['constant force degenerate', {'A': 0.002, 'B': 0.004, 'c': 1.0, 'age': 40, 't': 4}, [0.97628571, 0.006]], ['c below one rejected', {'A': 0.001, 'B': 0.0001, 'c': 0.95, 'age': 40, 't': 1}, 'invalid parameters'], ['negative total force rejected', {'A': -0.01, 'B': 0.001, 'c': 1.1, 'age': 40, 't': 1}, 'invalid parameters']], [['makeham 0', {'A': 0.0005, 'B': 3e-05, 'c': 1.08, 'age': 65, 't': 10}, [0.93033322, 0.01013614]], ['makeham 1', {'A': 0.0005, 'B': 5e-05, 'c': 1.08, 'age': 49, 't': 2.5}, [0.9927903, 0.00313205]], ['makeham 2', {'A': 0.0005, 'B': 5e-05, 'c': 1.12, 'age': 76, 't': 10}, [0.00599143, 0.85501552]], ['makeham 3', {'A': 0.0005, 'B': 3e-05, 'c': 1.12, 'age': 54, 't': 10}, [0.77220735, 0.04287158]], ['gompertz only', {'A': 0.0, 'B': 4e-05, 'c': 1.1, 'age': 55, 't': 3}, [0.97407875, 0.01006551]], ['constant force degenerate', {'A': 0.002, 'B': 0.005, 'c': 1.0, 'age': 40, 't': 4}, [0.97238837, 0.007]], ['c below one rejected', {'A': 0.001, 'B': 0.0001, 'c': 0.95, 'age': 40, 't': 1}, 'invalid parameters'], ['negative total force rejected', {'A': -0.01, 'B': 0.001, 'c': 1.1, 'age': 40, 't': 1}, 'invalid parameters']]]
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
makeham 0	`[0.52948031, 0.22014725]`	`[0.61903389, 0.22014725]`	Failed
makeham 1	`[0.94360863, 0.00419046]`	`[0.96826103, 0.00419046]`	Failed
makeham 2	`[0.50477898, 0.03725243]`	`[0.7970066, 0.03725243]`	Failed
makeham 3	`[0.05469805, 0.15661679]`	`[0.39101048, 0.15661679]`	Failed
gompertz only	`[0.9764072, 0.00687488]`	`[0.98222186, 0.00687488]`	Failed
constant force degenerate	`[0.98807171, 0.003]`	`[0.98807171, 0.003]`	Passed
c below one rejected	`invalid parameters`	`invalid parameters`	Passed
negative total force rejected	`invalid parameters`	`invalid parameters`	Passed

SHA-256 / aa7225a5ca1b2d72fc70b23300afcdd3e746ad4e8007e8e14cfb912989eea014

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):
    A=x['A']; B=x['B']; c=x['c']; a=x['age']; t=x['t']
    if B<0 or c<1 or A<=-B: return 'invalid parameters'
    if c==1: H=(A+B)*t
    else: H=A*t+B*(c**t-1)/math.log(c)
    mu=A+B*c**(a+t)
    return [round(math.exp(-H),8),round(mu,8)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['makeham 0', {'A': 0.001, 'B': 3e-05, 'c': 1.12, 'age': 76, 't': 2.5}, [0.61903389, 0.22014725]], ['makeham 1', {'A': 0.001, 'B': 5e-05, 'c': 1.08, 'age': 44, 't': 10}, [0.96826103, 0.00419046]], ['makeham 2', {'A': 0.001, 'B': 0.0001, 'c': 1.12, 'age': 42, 't': 10}, [0.7970066, 0.03725243]], ['makeham 3', {'A': 0.0005, 'B': 5e-05, 'c': 1.12, 'age': 61, 't': 10}, [0.39101048, 0.15661679]], ['gompertz only', {'A': 0.0, 'B': 4e-05, 'c': 1.1, 'age': 51, 't': 3}, [0.98222186, 0.00687488]], ['constant force degenerate', {'A': 0.002, 'B': 0.001, 'c': 1.0, 'age': 40, 't': 4}, [0.98807171, 0.003]], ['c below one rejected', {'A': 0.001, 'B': 0.0001, 'c': 0.95, 'age': 40, 't': 1}, 'invalid parameters'], ['negative total force rejected', {'A': -0.01, 'B': 0.001, 'c': 1.1, 'age': 40, 't': 1}, 'invalid parameters']], [['makeham 0', {'A': 0.001, 'B': 3e-05, 'c': 1.08, 'age': 35, 't': 2.5}, [0.99628416, 0.00153767]], ['makeham 1', {'A': 0.001, 'B': 3e-05, 'c': 1.1, 'age': 78, 't': 10}, [0.42348454, 0.13272783]], ['makeham 2', {'A': 0.0007, 'B': 0.0001, 'c': 1.08, 'age': 34, 't': 5}, [0.98822133, 0.00271153]], ['makeham 3', {'A': 0.0005, 'B': 3e-05, 'c': 1.08, 'age': 32, 't': 1}, [0.99913436, 0.00088028]], ['gompertz only', {'A': 0.0, 'B': 4e-05, 'c': 1.1, 'age': 52, 't': 3}, [0.98046153, 0.00756237]], ['constant force degenerate', {'A': 0.002, 'B': 0.002, 'c': 1.0, 'age': 40, 't': 4}, [0.98412732, 0.004]], ['c below one rejected', {'A': 0.001, 'B': 0.0001, 'c': 0.95, 'age': 40, 't': 1}, 'invalid parameters'], ['negative total force rejected', {'A': -0.01, 'B': 0.001, 'c': 1.1, 'age': 40, 't': 1}, 'invalid parameters']], [['makeham 0', {'A': 0.0005, 'B': 0.0001, 'c': 1.12, 'age': 44, 't': 2.5}, [0.95736913, 0.01993739]], ['makeham 1', {'A': 0.0005, 'B': 5e-05, 'c': 1.08, 'age': 40, 't': 5}, [0.99091741, 0.00209602]], ['makeham 2', {'A': 0.0007, 'B': 3e-05, 'c': 1.08, 'age': 42, 't': 2.5}, [0.99616179, 0.00162146]], ['makeham 3', {'A': 0.0005, 'B': 5e-05, 'c': 1.1, 'age': 58, 't': 5}, [0.92026523, 0.02076325]], ['gompertz only', {'A': 0.0, 'B': 4e-05, 'c': 1.1, 'age': 53, 't': 3}, [0.9785288, 0.0083186]], ['constant force degenerate', {'A': 0.002, 'B': 0.003, 'c': 1.0, 'age': 40, 't': 4}, [0.98019867, 0.005]], ['c below one rejected', {'A': 0.001, 'B': 0.0001, 'c': 0.95, 'age': 40, 't': 1}, 'invalid parameters'], ['negative total force rejected', {'A': -0.01, 'B': 0.001, 'c': 1.1, 'age': 40, 't': 1}, 'invalid parameters']], [['makeham 0', {'A': 0.0007, 'B': 3e-05, 'c': 1.1, 'age': 36, 't': 1}, [0.99832839, 0.00172012]], ['makeham 1', {'A': 0.0007, 'B': 5e-05, 'c': 1.1, 'age': 45, 't': 10}, [0.93431427, 0.01015296]], ['makeham 2', {'A': 0.0005, 'B': 0.0001, 'c': 1.1, 'age': 76, 't': 10}, [0.0958952, 0.36338659]], ['makeham 3', {'A': 0.0005, 'B': 5e-05, 'c': 1.1, 'age': 53, 't': 5}, [0.94881434, 0.01308189]], ['gompertz only', {'A': 0.0, 'B': 4e-05, 'c': 1.1, 'age': 54, 't': 3}, [0.9764072, 0.00915046]], ['constant force degenerate', {'A': 0.002, 'B': 0.004, 'c': 1.0, 'age': 40, 't': 4}, [0.97628571, 0.006]], ['c below one rejected', {'A': 0.001, 'B': 0.0001, 'c': 0.95, 'age': 40, 't': 1}, 'invalid parameters'], ['negative total force rejected', {'A': -0.01, 'B': 0.001, 'c': 1.1, 'age': 40, 't': 1}, 'invalid parameters']], [['makeham 0', {'A': 0.0005, 'B': 3e-05, 'c': 1.08, 'age': 65, 't': 10}, [0.93033322, 0.01013614]], ['makeham 1', {'A': 0.0005, 'B': 5e-05, 'c': 1.08, 'age': 49, 't': 2.5}, [0.9927903, 0.00313205]], ['makeham 2', {'A': 0.0005, 'B': 5e-05, 'c': 1.12, 'age': 76, 't': 10}, [0.00599143, 0.85501552]], ['makeham 3', {'A': 0.0005, 'B': 3e-05, 'c': 1.12, 'age': 54, 't': 10}, [0.77220735, 0.04287158]], ['gompertz only', {'A': 0.0, 'B': 4e-05, 'c': 1.1, 'age': 55, 't': 3}, [0.97407875, 0.01006551]], ['constant force degenerate', {'A': 0.002, 'B': 0.005, 'c': 1.0, 'age': 40, 't': 4}, [0.97238837, 0.007]], ['c below one rejected', {'A': 0.001, 'B': 0.0001, 'c': 0.95, 'age': 40, 't': 1}, 'invalid parameters'], ['negative total force rejected', {'A': -0.01, 'B': 0.001, 'c': 1.1, 'age': 40, 't': 1}, 'invalid parameters']]]
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
makeham 0	`[0.99741664, 0.22014725]`	`[0.61903389, 0.22014725]`	Failed
makeham 1	`[0.98930468, 0.00419046]`	`[0.96826103, 0.00419046]`	Failed
makeham 2	`[0.98821185, 0.03725243]`	`[0.7970066, 0.03725243]`	Failed
makeham 3	`[0.99408845, 0.15661679]`	`[0.39101048, 0.15661679]`	Failed
gompertz only	`[0.99986109, 0.00687488]`	`[0.98222186, 0.00687488]`	Failed
constant force degenerate	`[0.98807171, 0.003]`	`[0.98807171, 0.003]`	Passed
c below one rejected	`invalid parameters`	`invalid parameters`	Passed
negative total force rejected	`invalid parameters`	`invalid parameters`	Passed

SHA-256 / b9a920fc905ca77e47d8c430e0f643badbeaa92c17c1aaaf0fddbbb833700c88

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 2026-09-29T14:47:16.478538+00:00.

Case digest / 76e4746b9abb4c5e20182db1da67fbff10c608748cf15b988d20fa8f88986992