FA-59816 / Subscription proration billing / Open access
Package pricing with milli-unit usage: free units before packaging · case 01
The free allowance never reduces the bill.
ROOT CAUSE
Free units are not subtracted before packaging.
VERIFIED REPAIR
Restore the contract rule at the free units before packaging step: use `billable = max(0, units - x['free'])`.
Unsuccessful approach: The attempt treats the free allowance as packages instead of units.
Case contract
Input {events in milli-units, free units, div (units per package), round up|down, price per package}. Units = ceil(sum(events)/1000) once per invoice. Billable = max(0, units - free). Packages = ceil or floor(billable/div) by round. Return [units, packages, packages*price].
Why this case matters
Package pricing rounds usage twice, and doing either rounding at the wrong grain changes the bill.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
milli = sum(x['events'])
units = -(-milli // 1000)
billable = units
if x['round'] == 'up':
packs = -(-billable // x['div'])
else:
packs = billable // x['div']
return [units, packs, packs * x['price']]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression', {'events': [604, 2714, 1000, 776, 1000, 4405], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [11, 2, 500]), ('regression', {'events': [4863, 300, 2484, 893, 1000, 741], 'free': 10, 'div': 5, 'round': 'down', 'price': 999}, [11, 0, 0]), ('partial-repair probe', {'events': [391, 8, 2736, 4419], 'free': 5, 'div': 100, 'round': 'up', 'price': 250}, [8, 1, 250]), ('partial-repair probe', {'events': [4324, 534, 615, 25], 'free': 5, 'div': 100, 'round': 'up', 'price': 999}, [6, 1, 999]), ('normal control', {'events': [1332, 1000, 3819, 645, 1000, 333], 'free': 0, 'div': 10, 'round': 'up', 'price': 250}, [9, 1, 250]), ('normal control', {'events': [35, 1000, 2766, 1000, 937], 'free': 0, 'div': 10, 'round': 'up', 'price': 100}, [6, 1, 100]), ('normal control', {'events': [], 'free': 10, 'div': 100, 'round': 'up', 'price': 100}, [0, 0, 0]), ('normal control', {'events': [1000], 'free': 10, 'div': 100, 'round': 'down', 'price': 999}, [1, 0, 0])], [('regression', {'events': [673, 4210, 1000, 3643], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [10, 1, 250]), ('regression', {'events': [1000, 452], 'free': 5, 'div': 100, 'round': 'up', 'price': 999}, [2, 0, 0]), ('partial-repair probe', {'events': [604, 2714, 1000, 776, 1000, 4405], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [11, 2, 500]), ('partial-repair probe', {'events': [905, 3643, 1000, 755, 4581], 'free': 5, 'div': 100, 'round': 'up', 'price': 250}, [11, 1, 250]), ('normal control', {'events': [4706], 'free': 5, 'div': 10, 'round': 'down', 'price': 250}, [5, 0, 0]), ('normal control', {'events': [1000], 'free': 10, 'div': 10, 'round': 'down', 'price': 999}, [1, 0, 0]), ('normal control', {'events': [3902, 1000, 1000, 2138, 771, 1712], 'free': 0, 'div': 10, 'round': 'up', 'price': 100}, [11, 2, 200]), ('normal control', {'events': [576, 688, 449, 1000, 815], 'free': 0, 'div': 5, 'round': 'down', 'price': 999}, [4, 0, 0])], [('regression', {'events': [1000, 2653, 1000, 4233], 'free': 5, 'div': 5, 'round': 'up', 'price': 999}, [9, 1, 999]), ('regression', {'events': [1000, 1000, 577, 2375], 'free': 10, 'div': 1, 'round': 'down', 'price': 250}, [5, 0, 0]), ('partial-repair probe', {'events': [673, 4210, 1000, 3643], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [10, 1, 250]), ('partial-repair probe', {'events': [233, 824, 4771, 633, 598], 'free': 5, 'div': 10, 'round': 'up', 'price': 250}, [8, 1, 250]), ('normal control', {'events': [], 'free': 10, 'div': 1, 'round': 'up', 'price': 999}, [0, 0, 0]), ('normal control', {'events': [339, 1000, 1810, 2557, 4715], 'free': 10, 'div': 100, 'round': 'down', 'price': 100}, [11, 0, 0]), ('normal control', {'events': [959, 514, 670, 1000], 'free': 0, 'div': 1, 'round': 'down', 'price': 250}, [4, 4, 1000]), ('normal control', {'events': [1000, 3934, 1000, 1000], 'free': 10, 'div': 100, 'round': 'down', 'price': 100}, [7, 0, 0])], [('regression', {'events': [3797, 1000, 1000, 932], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [7, 1, 250]), ('regression', {'events': [460, 1000], 'free': 10, 'div': 100, 'round': 'up', 'price': 100}, [2, 0, 0]), ('partial-repair probe', {'events': [1000, 2653, 1000, 4233], 'free': 5, 'div': 5, 'round': 'up', 'price': 999}, [9, 1, 999]), ('partial-repair probe', {'events': [2261, 1497, 1000, 778, 722], 'free': 5, 'div': 100, 'round': 'up', 'price': 250}, [7, 1, 250]), ('normal control', {'events': [], 'free': 0, 'div': 10, 'round': 'down', 'price': 999}, [0, 0, 0]), ('normal control', {'events': [487, 2432, 2028, 558], 'free': 0, 'div': 10, 'round': 'up', 'price': 250}, [6, 1, 250]), ('normal control', {'events': [402], 'free': 0, 'div': 5, 'round': 'down', 'price': 100}, [1, 0, 0]), ('normal control', {'events': [], 'free': 0, 'div': 1, 'round': 'down', 'price': 999}, [0, 0, 0])], [('regression', {'events': [2133, 1366, 1000, 1863, 1000], 'free': 5, 'div': 5, 'round': 'up', 'price': 999}, [8, 1, 999]), ('regression', {'events': [716, 4530], 'free': 10, 'div': 10, 'round': 'up', 'price': 100}, [6, 0, 0]), ('partial-repair probe', {'events': [934, 3399, 485, 1083, 1187, 1000], 'free': 5, 'div': 10, 'round': 'up', 'price': 999}, [9, 1, 999]), ('partial-repair probe', {'events': [3797, 1000, 1000, 932], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [7, 1, 250]), ('normal control', {'events': [4797, 2089, 1000, 1954], 'free': 0, 'div': 10, 'round': 'down', 'price': 100}, [10, 1, 100]), ('normal control', {'events': [532, 140, 780, 1000], 'free': 0, 'div': 10, 'round': 'up', 'price': 250}, [3, 1, 250]), ('normal control', {'events': [], 'free': 0, 'div': 100, 'round': 'down', 'price': 999}, [0, 0, 0]), ('normal control', {'events': [1000, 2646], 'free': 0, 'div': 10, 'round': 'down', 'price': 100}, [4, 0, 0])]]
for i, (label, args, expected) in enumerate(fixtures[N-1]):
check("%s %d" % (label, 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 fixture | Actual | Expected | Outcome |
|---|---|---|---|
| regression 0 | [11, 3, 750] | [11, 2, 500] | Failed |
| regression 1 | [11, 2, 1998] | [11, 0, 0] | Failed |
| partial-repair probe 2 | [8, 1, 250] | [8, 1, 250] | Passed |
| partial-repair probe 3 | [6, 1, 999] | [6, 1, 999] | Passed |
| normal control 4 | [9, 1, 250] | [9, 1, 250] | Passed |
| normal control 5 | [6, 1, 100] | [6, 1, 100] | Passed |
| normal control 6 | [0, 0, 0] | [0, 0, 0] | Passed |
| normal control 7 | [1, 0, 0] | [1, 0, 0] | Passed |
SHA-256 / 02af113b0b670066c716a627316170f1c4a43444d9df1f20269c9cb78807f26e
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
milli = sum(x['events'])
units = -(-milli // 1000)
billable = max(0, units - x['free'] * x['div'])
if x['round'] == 'up':
packs = -(-billable // x['div'])
else:
packs = billable // x['div']
return [units, packs, packs * x['price']]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression', {'events': [604, 2714, 1000, 776, 1000, 4405], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [11, 2, 500]), ('regression', {'events': [4863, 300, 2484, 893, 1000, 741], 'free': 10, 'div': 5, 'round': 'down', 'price': 999}, [11, 0, 0]), ('partial-repair probe', {'events': [391, 8, 2736, 4419], 'free': 5, 'div': 100, 'round': 'up', 'price': 250}, [8, 1, 250]), ('partial-repair probe', {'events': [4324, 534, 615, 25], 'free': 5, 'div': 100, 'round': 'up', 'price': 999}, [6, 1, 999]), ('normal control', {'events': [1332, 1000, 3819, 645, 1000, 333], 'free': 0, 'div': 10, 'round': 'up', 'price': 250}, [9, 1, 250]), ('normal control', {'events': [35, 1000, 2766, 1000, 937], 'free': 0, 'div': 10, 'round': 'up', 'price': 100}, [6, 1, 100]), ('normal control', {'events': [], 'free': 10, 'div': 100, 'round': 'up', 'price': 100}, [0, 0, 0]), ('normal control', {'events': [1000], 'free': 10, 'div': 100, 'round': 'down', 'price': 999}, [1, 0, 0])], [('regression', {'events': [673, 4210, 1000, 3643], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [10, 1, 250]), ('regression', {'events': [1000, 452], 'free': 5, 'div': 100, 'round': 'up', 'price': 999}, [2, 0, 0]), ('partial-repair probe', {'events': [604, 2714, 1000, 776, 1000, 4405], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [11, 2, 500]), ('partial-repair probe', {'events': [905, 3643, 1000, 755, 4581], 'free': 5, 'div': 100, 'round': 'up', 'price': 250}, [11, 1, 250]), ('normal control', {'events': [4706], 'free': 5, 'div': 10, 'round': 'down', 'price': 250}, [5, 0, 0]), ('normal control', {'events': [1000], 'free': 10, 'div': 10, 'round': 'down', 'price': 999}, [1, 0, 0]), ('normal control', {'events': [3902, 1000, 1000, 2138, 771, 1712], 'free': 0, 'div': 10, 'round': 'up', 'price': 100}, [11, 2, 200]), ('normal control', {'events': [576, 688, 449, 1000, 815], 'free': 0, 'div': 5, 'round': 'down', 'price': 999}, [4, 0, 0])], [('regression', {'events': [1000, 2653, 1000, 4233], 'free': 5, 'div': 5, 'round': 'up', 'price': 999}, [9, 1, 999]), ('regression', {'events': [1000, 1000, 577, 2375], 'free': 10, 'div': 1, 'round': 'down', 'price': 250}, [5, 0, 0]), ('partial-repair probe', {'events': [673, 4210, 1000, 3643], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [10, 1, 250]), ('partial-repair probe', {'events': [233, 824, 4771, 633, 598], 'free': 5, 'div': 10, 'round': 'up', 'price': 250}, [8, 1, 250]), ('normal control', {'events': [], 'free': 10, 'div': 1, 'round': 'up', 'price': 999}, [0, 0, 0]), ('normal control', {'events': [339, 1000, 1810, 2557, 4715], 'free': 10, 'div': 100, 'round': 'down', 'price': 100}, [11, 0, 0]), ('normal control', {'events': [959, 514, 670, 1000], 'free': 0, 'div': 1, 'round': 'down', 'price': 250}, [4, 4, 1000]), ('normal control', {'events': [1000, 3934, 1000, 1000], 'free': 10, 'div': 100, 'round': 'down', 'price': 100}, [7, 0, 0])], [('regression', {'events': [3797, 1000, 1000, 932], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [7, 1, 250]), ('regression', {'events': [460, 1000], 'free': 10, 'div': 100, 'round': 'up', 'price': 100}, [2, 0, 0]), ('partial-repair probe', {'events': [1000, 2653, 1000, 4233], 'free': 5, 'div': 5, 'round': 'up', 'price': 999}, [9, 1, 999]), ('partial-repair probe', {'events': [2261, 1497, 1000, 778, 722], 'free': 5, 'div': 100, 'round': 'up', 'price': 250}, [7, 1, 250]), ('normal control', {'events': [], 'free': 0, 'div': 10, 'round': 'down', 'price': 999}, [0, 0, 0]), ('normal control', {'events': [487, 2432, 2028, 558], 'free': 0, 'div': 10, 'round': 'up', 'price': 250}, [6, 1, 250]), ('normal control', {'events': [402], 'free': 0, 'div': 5, 'round': 'down', 'price': 100}, [1, 0, 0]), ('normal control', {'events': [], 'free': 0, 'div': 1, 'round': 'down', 'price': 999}, [0, 0, 0])], [('regression', {'events': [2133, 1366, 1000, 1863, 1000], 'free': 5, 'div': 5, 'round': 'up', 'price': 999}, [8, 1, 999]), ('regression', {'events': [716, 4530], 'free': 10, 'div': 10, 'round': 'up', 'price': 100}, [6, 0, 0]), ('partial-repair probe', {'events': [934, 3399, 485, 1083, 1187, 1000], 'free': 5, 'div': 10, 'round': 'up', 'price': 999}, [9, 1, 999]), ('partial-repair probe', {'events': [3797, 1000, 1000, 932], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [7, 1, 250]), ('normal control', {'events': [4797, 2089, 1000, 1954], 'free': 0, 'div': 10, 'round': 'down', 'price': 100}, [10, 1, 100]), ('normal control', {'events': [532, 140, 780, 1000], 'free': 0, 'div': 10, 'round': 'up', 'price': 250}, [3, 1, 250]), ('normal control', {'events': [], 'free': 0, 'div': 100, 'round': 'down', 'price': 999}, [0, 0, 0]), ('normal control', {'events': [1000, 2646], 'free': 0, 'div': 10, 'round': 'down', 'price': 100}, [4, 0, 0])]]
for i, (label, args, expected) in enumerate(fixtures[N-1]):
check("%s %d" % (label, 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 fixture | Actual | Expected | Outcome |
|---|---|---|---|
| regression 0 | [11, 0, 0] | [11, 2, 500] | Failed |
| regression 1 | [11, 0, 0] | [11, 0, 0] | Passed |
| partial-repair probe 2 | [8, 0, 0] | [8, 1, 250] | Failed |
| partial-repair probe 3 | [6, 0, 0] | [6, 1, 999] | Failed |
| normal control 4 | [9, 1, 250] | [9, 1, 250] | Passed |
| normal control 5 | [6, 1, 100] | [6, 1, 100] | Passed |
| normal control 6 | [0, 0, 0] | [0, 0, 0] | Passed |
| normal control 7 | [1, 0, 0] | [1, 0, 0] | Passed |
SHA-256 / e6f6b79a1059eacd3425239737bbc423e91933ac81aefe19be8de92f82b0ea9c
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
milli = sum(x['events'])
units = -(-milli // 1000)
billable = max(0, units - x['free'])
if x['round'] == 'up':
packs = -(-billable // x['div'])
else:
packs = billable // x['div']
return [units, packs, packs * x['price']]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression', {'events': [604, 2714, 1000, 776, 1000, 4405], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [11, 2, 500]), ('regression', {'events': [4863, 300, 2484, 893, 1000, 741], 'free': 10, 'div': 5, 'round': 'down', 'price': 999}, [11, 0, 0]), ('partial-repair probe', {'events': [391, 8, 2736, 4419], 'free': 5, 'div': 100, 'round': 'up', 'price': 250}, [8, 1, 250]), ('partial-repair probe', {'events': [4324, 534, 615, 25], 'free': 5, 'div': 100, 'round': 'up', 'price': 999}, [6, 1, 999]), ('normal control', {'events': [1332, 1000, 3819, 645, 1000, 333], 'free': 0, 'div': 10, 'round': 'up', 'price': 250}, [9, 1, 250]), ('normal control', {'events': [35, 1000, 2766, 1000, 937], 'free': 0, 'div': 10, 'round': 'up', 'price': 100}, [6, 1, 100]), ('normal control', {'events': [], 'free': 10, 'div': 100, 'round': 'up', 'price': 100}, [0, 0, 0]), ('normal control', {'events': [1000], 'free': 10, 'div': 100, 'round': 'down', 'price': 999}, [1, 0, 0])], [('regression', {'events': [673, 4210, 1000, 3643], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [10, 1, 250]), ('regression', {'events': [1000, 452], 'free': 5, 'div': 100, 'round': 'up', 'price': 999}, [2, 0, 0]), ('partial-repair probe', {'events': [604, 2714, 1000, 776, 1000, 4405], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [11, 2, 500]), ('partial-repair probe', {'events': [905, 3643, 1000, 755, 4581], 'free': 5, 'div': 100, 'round': 'up', 'price': 250}, [11, 1, 250]), ('normal control', {'events': [4706], 'free': 5, 'div': 10, 'round': 'down', 'price': 250}, [5, 0, 0]), ('normal control', {'events': [1000], 'free': 10, 'div': 10, 'round': 'down', 'price': 999}, [1, 0, 0]), ('normal control', {'events': [3902, 1000, 1000, 2138, 771, 1712], 'free': 0, 'div': 10, 'round': 'up', 'price': 100}, [11, 2, 200]), ('normal control', {'events': [576, 688, 449, 1000, 815], 'free': 0, 'div': 5, 'round': 'down', 'price': 999}, [4, 0, 0])], [('regression', {'events': [1000, 2653, 1000, 4233], 'free': 5, 'div': 5, 'round': 'up', 'price': 999}, [9, 1, 999]), ('regression', {'events': [1000, 1000, 577, 2375], 'free': 10, 'div': 1, 'round': 'down', 'price': 250}, [5, 0, 0]), ('partial-repair probe', {'events': [673, 4210, 1000, 3643], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [10, 1, 250]), ('partial-repair probe', {'events': [233, 824, 4771, 633, 598], 'free': 5, 'div': 10, 'round': 'up', 'price': 250}, [8, 1, 250]), ('normal control', {'events': [], 'free': 10, 'div': 1, 'round': 'up', 'price': 999}, [0, 0, 0]), ('normal control', {'events': [339, 1000, 1810, 2557, 4715], 'free': 10, 'div': 100, 'round': 'down', 'price': 100}, [11, 0, 0]), ('normal control', {'events': [959, 514, 670, 1000], 'free': 0, 'div': 1, 'round': 'down', 'price': 250}, [4, 4, 1000]), ('normal control', {'events': [1000, 3934, 1000, 1000], 'free': 10, 'div': 100, 'round': 'down', 'price': 100}, [7, 0, 0])], [('regression', {'events': [3797, 1000, 1000, 932], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [7, 1, 250]), ('regression', {'events': [460, 1000], 'free': 10, 'div': 100, 'round': 'up', 'price': 100}, [2, 0, 0]), ('partial-repair probe', {'events': [1000, 2653, 1000, 4233], 'free': 5, 'div': 5, 'round': 'up', 'price': 999}, [9, 1, 999]), ('partial-repair probe', {'events': [2261, 1497, 1000, 778, 722], 'free': 5, 'div': 100, 'round': 'up', 'price': 250}, [7, 1, 250]), ('normal control', {'events': [], 'free': 0, 'div': 10, 'round': 'down', 'price': 999}, [0, 0, 0]), ('normal control', {'events': [487, 2432, 2028, 558], 'free': 0, 'div': 10, 'round': 'up', 'price': 250}, [6, 1, 250]), ('normal control', {'events': [402], 'free': 0, 'div': 5, 'round': 'down', 'price': 100}, [1, 0, 0]), ('normal control', {'events': [], 'free': 0, 'div': 1, 'round': 'down', 'price': 999}, [0, 0, 0])], [('regression', {'events': [2133, 1366, 1000, 1863, 1000], 'free': 5, 'div': 5, 'round': 'up', 'price': 999}, [8, 1, 999]), ('regression', {'events': [716, 4530], 'free': 10, 'div': 10, 'round': 'up', 'price': 100}, [6, 0, 0]), ('partial-repair probe', {'events': [934, 3399, 485, 1083, 1187, 1000], 'free': 5, 'div': 10, 'round': 'up', 'price': 999}, [9, 1, 999]), ('partial-repair probe', {'events': [3797, 1000, 1000, 932], 'free': 5, 'div': 5, 'round': 'up', 'price': 250}, [7, 1, 250]), ('normal control', {'events': [4797, 2089, 1000, 1954], 'free': 0, 'div': 10, 'round': 'down', 'price': 100}, [10, 1, 100]), ('normal control', {'events': [532, 140, 780, 1000], 'free': 0, 'div': 10, 'round': 'up', 'price': 250}, [3, 1, 250]), ('normal control', {'events': [], 'free': 0, 'div': 100, 'round': 'down', 'price': 999}, [0, 0, 0]), ('normal control', {'events': [1000, 2646], 'free': 0, 'div': 10, 'round': 'down', 'price': 100}, [4, 0, 0])]]
for i, (label, args, expected) in enumerate(fixtures[N-1]):
check("%s %d" % (label, 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 fixture | Actual | Expected | Outcome |
|---|---|---|---|
| regression 0 | [11, 2, 500] | [11, 2, 500] | Passed |
| regression 1 | [11, 0, 0] | [11, 0, 0] | Passed |
| partial-repair probe 2 | [8, 1, 250] | [8, 1, 250] | Passed |
| partial-repair probe 3 | [6, 1, 999] | [6, 1, 999] | Passed |
| normal control 4 | [9, 1, 250] | [9, 1, 250] | Passed |
| normal control 5 | [6, 1, 100] | [6, 1, 100] | Passed |
| normal control 6 | [0, 0, 0] | [0, 0, 0] | Passed |
| normal control 7 | [1, 0, 0] | [1, 0, 0] | Passed |
SHA-256 / 22dc179f263ac7df76aca5a793953ef3e8227346edfb9796154a05696e05e430
Verification & scope
A deterministic teaching model of a stipulated billing rule. It makes no claim to reproduce any billing provider's exact behaviour and is not billing software. 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 2026-09-29T14:46:39.791272+00:00.
Case digest / 697a1b0d62a5799fde6cd205ead9af9faceafbe8f685805f759dcc0cc0eda43d