FA-86091 / Game economy crafting balance / Open access

Stamina regeneration: Refills cannot exceed cap · case 01

Stamina potions used near full are wasted.

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

ROOT CAUSE

Refills are clamped to the regeneration cap instead of the overflow ceiling.

VERIFIED REPAIR

Restore `min(999, s + amount)` at the overflow ceiling step.

Unsuccessful approach: Clamping between cap and the ceiling refills a spend from overflow back up to cap.

Case contract

Stamina regenerates 1 per 300 s while below cap, anchored at last_tick. Events [time, amount] are applied in time order (then amount); before each event (and finally at now) regeneration catches up: if it reaches cap the value becomes cap and the anchor moves to the event time, otherwise the anchor advances by whole periods. While at or above cap the anchor follows the event time. Amounts may push stamina above cap up to 999; negative amounts are spends that are rejected if they would go below zero. Return [stamina, anchor].

Why this case matters

Game economies leak or destroy currency when one crafting or pricing rule is off by one boundary, rounding stage or state update; the defect is observable in exact integer outcomes.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(stamina, cap, last_tick, now, refills):
    events = sorted(refills) + [[now, 0]]
    t = last_tick
    s = stamina
    for at, amount in events:
        if s < cap:
            gained = (at - t) // 300
            if s + gained >= cap:
                s = cap
                t = at
            else:
                s += gained
                t += gained * 300
        else:
            t = at
        if s + amount >= 0:
            s = min(cap, s + amount)
    return [s, t]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[('overflow potion persists #1', [5, 5, 0, 1200, [[100, 10]]], [15, 1200]),
  ('spend from full restarts timer #1', [10, 10, 0, 1000, [[400, -4]]], [8, 1000]),
  ('regression overflow ceiling #1', [23, 20, 972, 1572, [[1185, -20], [1315, -20], [1066, 20]]], [3, 1315]),
  ('fault site overflow ceiling #1', [12, 13, 710, 1010, [[793, 7], [721, 13]]], [32, 1010]),
  ('partial repair boundary #1', [9, 9, 266, 266, [[266, -9], [266, -10]]], [0, 266]),
  ('partial repair boundary #2', [16, 17, 83, 1773, [[154, -6]]], [15, 1583]),
  ('exact refill to cap #1', [3, 5, 0, 600, []], [5, 600]),
  ('control #1', [0, 11, 224, 4427, []], [11, 4427])],
 [('overflow potion persists #1', [5, 5, 0, 1200, [[100, 10]]], [15, 1200]),
  ('spend from full restarts timer #1', [10, 10, 0, 1000, [[400, -4]]], [8, 1000]),
  ('fault site overflow ceiling #1', [12, 13, 710, 1010, [[793, 7], [721, 13]]], [32, 1010]),
  ('fault site overflow ceiling #2', [11, 12, 57, 163, [[92, 12], [139, 12]]], [35, 163]),
  ('regression overflow ceiling #1', [23, 20, 972, 1572, [[1185, -20], [1315, -20], [1066, 20]]], [3, 1315]),
  ('partial repair boundary #1', [16, 17, 83, 1773, [[154, -6]]], [15, 1583]),
  ('exact refill to cap #1', [3, 5, 0, 600, []], [5, 600]),
  ('control #1', [0, 11, 224, 4427, []], [11, 4427])],
 [('overflow potion persists #1', [5, 5, 0, 1200, [[100, 10]]], [15, 1200]),
  ('spend from full restarts timer #1', [10, 10, 0, 1000, [[400, -4]]], [8, 1000]),
  ('fault site overflow ceiling #1', [4, 13, 569, 868, [[776, 10]]], [14, 868]),
  ('regression overflow ceiling #1', [0, 15, 688, 1288, [[1231, 15], [743, 9], [808, 15]]], [39, 1288]),
  ('partial repair boundary #1', [17, 18, 353, 353, []], [17, 353]),
  ('partial repair boundary #2', [0, 18, 176, 176, []], [0, 176]),
  ('exact refill to cap #1', [3, 5, 0, 600, []], [5, 600]),
  ('control #1', [14, 15, 389, 2081, []], [15, 2081])],
 [('overflow potion persists #1', [5, 5, 0, 1200, [[100, 10]]], [15, 1200]),
  ('spend from full restarts timer #1', [10, 10, 0, 1000, [[400, -4]]], [8, 1000]),
  ('fault site overflow ceiling #1', [0, 7, 587, 587, [[587, 8], [587, 9]]], [17, 587]),
  ('fault site overflow ceiling #2', [17, 17, 382, 982, [[483, 17], [862, -6], [923, 17]]], [45, 982]),
  ('regression overflow ceiling #1', [0, 15, 688, 1288, [[1231, 15], [743, 9], [808, 15]]], [39, 1288]),
  ('partial repair boundary #1', [8, 9, 913, 1212, [[1053, -9], [963, -6], [1014, -5]]], [2, 913]),
  ('exact refill to cap #1', [3, 5, 0, 600, []], [5, 600]),
  ('control #1', [0, 9, 303, 303, [[303, 9]]], [9, 303])],
 [('overflow potion persists #1', [5, 5, 0, 1200, [[100, 10]]], [15, 1200]),
  ('spend from full restarts timer #1', [10, 10, 0, 1000, [[400, -4]]], [8, 1000]),
  ('regression overflow ceiling #1', [4, 11, 767, 2462, [[2177, 7], [1959, -11], [1673, 11]]], [14, 2462]),
  ('fault site overflow ceiling #1', [12, 12, 443, 1043, [[843, 12]]], [24, 1043]),
  ('partial repair boundary #1', [17, 17, 907, 907, [[907, -12], [907, -17], [907, -17]]], [0, 907]),
  ('partial repair boundary #2', [0, 14, 147, 147, []], [0, 147]),
  ('exact refill to cap #1', [3, 5, 0, 600, []], [5, 600]),
  ('control #1', [8, 8, 764, 764, [[764, 0]]], [8, 764])]]
for label, args, expected in cases[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
overflow potion persists #1	`[5, 1200]`	`[15, 1200]`	Failed
spend from full restarts timer #1	`[8, 1000]`	`[8, 1000]`	Passed
regression overflow ceiling #1	`[1, 1485]`	`[3, 1315]`	Failed
fault site overflow ceiling #1	`[13, 1010]`	`[32, 1010]`	Failed
partial repair boundary #1	`[0, 266]`	`[0, 266]`	Passed
partial repair boundary #2	`[15, 1583]`	`[15, 1583]`	Passed
exact refill to cap #1	`[5, 600]`	`[5, 600]`	Passed
control #1	`[11, 4427]`	`[11, 4427]`	Passed

SHA-256 / 14784042d6a747cbbb041c841225c1d06a2236e924eac38a568e7e0b6a295c5f

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(stamina, cap, last_tick, now, refills):
    events = sorted(refills) + [[now, 0]]
    t = last_tick
    s = stamina
    for at, amount in events:
        if s < cap:
            gained = (at - t) // 300
            if s + gained >= cap:
                s = cap
                t = at
            else:
                s += gained
                t += gained * 300
        else:
            t = at
        if s + amount >= 0:
            s = max(cap, min(999, s + amount))
    return [s, t]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[('overflow potion persists #1', [5, 5, 0, 1200, [[100, 10]]], [15, 1200]),
  ('spend from full restarts timer #1', [10, 10, 0, 1000, [[400, -4]]], [8, 1000]),
  ('regression overflow ceiling #1', [23, 20, 972, 1572, [[1185, -20], [1315, -20], [1066, 20]]], [3, 1315]),
  ('fault site overflow ceiling #1', [12, 13, 710, 1010, [[793, 7], [721, 13]]], [32, 1010]),
  ('partial repair boundary #1', [9, 9, 266, 266, [[266, -9], [266, -10]]], [0, 266]),
  ('partial repair boundary #2', [16, 17, 83, 1773, [[154, -6]]], [15, 1583]),
  ('exact refill to cap #1', [3, 5, 0, 600, []], [5, 600]),
  ('control #1', [0, 11, 224, 4427, []], [11, 4427])],
 [('overflow potion persists #1', [5, 5, 0, 1200, [[100, 10]]], [15, 1200]),
  ('spend from full restarts timer #1', [10, 10, 0, 1000, [[400, -4]]], [8, 1000]),
  ('fault site overflow ceiling #1', [12, 13, 710, 1010, [[793, 7], [721, 13]]], [32, 1010]),
  ('fault site overflow ceiling #2', [11, 12, 57, 163, [[92, 12], [139, 12]]], [35, 163]),
  ('regression overflow ceiling #1', [23, 20, 972, 1572, [[1185, -20], [1315, -20], [1066, 20]]], [3, 1315]),
  ('partial repair boundary #1', [16, 17, 83, 1773, [[154, -6]]], [15, 1583]),
  ('exact refill to cap #1', [3, 5, 0, 600, []], [5, 600]),
  ('control #1', [0, 11, 224, 4427, []], [11, 4427])],
 [('overflow potion persists #1', [5, 5, 0, 1200, [[100, 10]]], [15, 1200]),
  ('spend from full restarts timer #1', [10, 10, 0, 1000, [[400, -4]]], [8, 1000]),
  ('fault site overflow ceiling #1', [4, 13, 569, 868, [[776, 10]]], [14, 868]),
  ('regression overflow ceiling #1', [0, 15, 688, 1288, [[1231, 15], [743, 9], [808, 15]]], [39, 1288]),
  ('partial repair boundary #1', [17, 18, 353, 353, []], [17, 353]),
  ('partial repair boundary #2', [0, 18, 176, 176, []], [0, 176]),
  ('exact refill to cap #1', [3, 5, 0, 600, []], [5, 600]),
  ('control #1', [14, 15, 389, 2081, []], [15, 2081])],
 [('overflow potion persists #1', [5, 5, 0, 1200, [[100, 10]]], [15, 1200]),
  ('spend from full restarts timer #1', [10, 10, 0, 1000, [[400, -4]]], [8, 1000]),
  ('fault site overflow ceiling #1', [0, 7, 587, 587, [[587, 8], [587, 9]]], [17, 587]),
  ('fault site overflow ceiling #2', [17, 17, 382, 982, [[483, 17], [862, -6], [923, 17]]], [45, 982]),
  ('regression overflow ceiling #1', [0, 15, 688, 1288, [[1231, 15], [743, 9], [808, 15]]], [39, 1288]),
  ('partial repair boundary #1', [8, 9, 913, 1212, [[1053, -9], [963, -6], [1014, -5]]], [2, 913]),
  ('exact refill to cap #1', [3, 5, 0, 600, []], [5, 600]),
  ('control #1', [0, 9, 303, 303, [[303, 9]]], [9, 303])],
 [('overflow potion persists #1', [5, 5, 0, 1200, [[100, 10]]], [15, 1200]),
  ('spend from full restarts timer #1', [10, 10, 0, 1000, [[400, -4]]], [8, 1000]),
  ('regression overflow ceiling #1', [4, 11, 767, 2462, [[2177, 7], [1959, -11], [1673, 11]]], [14, 2462]),
  ('fault site overflow ceiling #1', [12, 12, 443, 1043, [[843, 12]]], [24, 1043]),
  ('partial repair boundary #1', [17, 17, 907, 907, [[907, -12], [907, -17], [907, -17]]], [0, 907]),
  ('partial repair boundary #2', [0, 14, 147, 147, []], [0, 147]),
  ('exact refill to cap #1', [3, 5, 0, 600, []], [5, 600]),
  ('control #1', [8, 8, 764, 764, [[764, 0]]], [8, 764])]]
for label, args, expected in cases[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
overflow potion persists #1	`[15, 1200]`	`[15, 1200]`	Passed
spend from full restarts timer #1	`[10, 1000]`	`[8, 1000]`	Failed
regression overflow ceiling #1	`[20, 1572]`	`[3, 1315]`	Failed
fault site overflow ceiling #1	`[32, 1010]`	`[32, 1010]`	Passed
partial repair boundary #1	`[9, 266]`	`[0, 266]`	Failed
partial repair boundary #2	`[17, 1773]`	`[15, 1583]`	Failed
exact refill to cap #1	`[5, 600]`	`[5, 600]`	Passed
control #1	`[11, 4427]`	`[11, 4427]`	Passed

SHA-256 / b2e7f469689f20663f5d4d32d937a44af7d0902969d33e31167c409b31122cae

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(stamina, cap, last_tick, now, refills):
    events = sorted(refills) + [[now, 0]]
    t = last_tick
    s = stamina
    for at, amount in events:
        if s < cap:
            gained = (at - t) // 300
            if s + gained >= cap:
                s = cap
                t = at
            else:
                s += gained
                t += gained * 300
        else:
            t = at
        if s + amount >= 0:
            s = min(999, s + amount)
    return [s, t]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[('overflow potion persists #1', [5, 5, 0, 1200, [[100, 10]]], [15, 1200]),
  ('spend from full restarts timer #1', [10, 10, 0, 1000, [[400, -4]]], [8, 1000]),
  ('regression overflow ceiling #1', [23, 20, 972, 1572, [[1185, -20], [1315, -20], [1066, 20]]], [3, 1315]),
  ('fault site overflow ceiling #1', [12, 13, 710, 1010, [[793, 7], [721, 13]]], [32, 1010]),
  ('partial repair boundary #1', [9, 9, 266, 266, [[266, -9], [266, -10]]], [0, 266]),
  ('partial repair boundary #2', [16, 17, 83, 1773, [[154, -6]]], [15, 1583]),
  ('exact refill to cap #1', [3, 5, 0, 600, []], [5, 600]),
  ('control #1', [0, 11, 224, 4427, []], [11, 4427])],
 [('overflow potion persists #1', [5, 5, 0, 1200, [[100, 10]]], [15, 1200]),
  ('spend from full restarts timer #1', [10, 10, 0, 1000, [[400, -4]]], [8, 1000]),
  ('fault site overflow ceiling #1', [12, 13, 710, 1010, [[793, 7], [721, 13]]], [32, 1010]),
  ('fault site overflow ceiling #2', [11, 12, 57, 163, [[92, 12], [139, 12]]], [35, 163]),
  ('regression overflow ceiling #1', [23, 20, 972, 1572, [[1185, -20], [1315, -20], [1066, 20]]], [3, 1315]),
  ('partial repair boundary #1', [16, 17, 83, 1773, [[154, -6]]], [15, 1583]),
  ('exact refill to cap #1', [3, 5, 0, 600, []], [5, 600]),
  ('control #1', [0, 11, 224, 4427, []], [11, 4427])],
 [('overflow potion persists #1', [5, 5, 0, 1200, [[100, 10]]], [15, 1200]),
  ('spend from full restarts timer #1', [10, 10, 0, 1000, [[400, -4]]], [8, 1000]),
  ('fault site overflow ceiling #1', [4, 13, 569, 868, [[776, 10]]], [14, 868]),
  ('regression overflow ceiling #1', [0, 15, 688, 1288, [[1231, 15], [743, 9], [808, 15]]], [39, 1288]),
  ('partial repair boundary #1', [17, 18, 353, 353, []], [17, 353]),
  ('partial repair boundary #2', [0, 18, 176, 176, []], [0, 176]),
  ('exact refill to cap #1', [3, 5, 0, 600, []], [5, 600]),
  ('control #1', [14, 15, 389, 2081, []], [15, 2081])],
 [('overflow potion persists #1', [5, 5, 0, 1200, [[100, 10]]], [15, 1200]),
  ('spend from full restarts timer #1', [10, 10, 0, 1000, [[400, -4]]], [8, 1000]),
  ('fault site overflow ceiling #1', [0, 7, 587, 587, [[587, 8], [587, 9]]], [17, 587]),
  ('fault site overflow ceiling #2', [17, 17, 382, 982, [[483, 17], [862, -6], [923, 17]]], [45, 982]),
  ('regression overflow ceiling #1', [0, 15, 688, 1288, [[1231, 15], [743, 9], [808, 15]]], [39, 1288]),
  ('partial repair boundary #1', [8, 9, 913, 1212, [[1053, -9], [963, -6], [1014, -5]]], [2, 913]),
  ('exact refill to cap #1', [3, 5, 0, 600, []], [5, 600]),
  ('control #1', [0, 9, 303, 303, [[303, 9]]], [9, 303])],
 [('overflow potion persists #1', [5, 5, 0, 1200, [[100, 10]]], [15, 1200]),
  ('spend from full restarts timer #1', [10, 10, 0, 1000, [[400, -4]]], [8, 1000]),
  ('regression overflow ceiling #1', [4, 11, 767, 2462, [[2177, 7], [1959, -11], [1673, 11]]], [14, 2462]),
  ('fault site overflow ceiling #1', [12, 12, 443, 1043, [[843, 12]]], [24, 1043]),
  ('partial repair boundary #1', [17, 17, 907, 907, [[907, -12], [907, -17], [907, -17]]], [0, 907]),
  ('partial repair boundary #2', [0, 14, 147, 147, []], [0, 147]),
  ('exact refill to cap #1', [3, 5, 0, 600, []], [5, 600]),
  ('control #1', [8, 8, 764, 764, [[764, 0]]], [8, 764])]]
for label, args, expected in cases[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
overflow potion persists #1	`[15, 1200]`	`[15, 1200]`	Passed
spend from full restarts timer #1	`[8, 1000]`	`[8, 1000]`	Passed
regression overflow ceiling #1	`[3, 1315]`	`[3, 1315]`	Passed
fault site overflow ceiling #1	`[32, 1010]`	`[32, 1010]`	Passed
partial repair boundary #1	`[0, 266]`	`[0, 266]`	Passed
partial repair boundary #2	`[15, 1583]`	`[15, 1583]`	Passed
exact refill to cap #1	`[5, 600]`	`[5, 600]`	Passed
control #1	`[11, 4427]`	`[11, 4427]`	Passed

SHA-256 / 4353d756bdfc5533076257122f7985aa228463cc3ff5abf7b443d543ed8d3d6b

Verification & scope

Deterministic toy contract stipulated for this model; integer or exact arithmetic only, not a reproduction of any specific game engine. 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 2026-09-29T14:50:46.336590+00:00.

Case digest / 705e797c6a7e522ac659903abffd37ddaf8f5c0d60f17eaf1a91ecdb38cdfd7a