FA-5901 / Planar geometry / Open access

Picks theorem interior points · case 01

Boundary points are subtracted with full instead of half weight.

Verified by executionVariant 1 · 4 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Boundary points are subtracted with full instead of half weight.

THE FAILURE

Boundary points are subtracted with full instead of half weight.

Unsuccessful approach: The boundary term is added with the wrong sign.

Case contract

Integer coordinates and lengths, with exact arithmetic except for indicated division results. Lengths are nonnegative. Rectangles are (xmin,ymin,xmax,ymax) with ordered bounds; coordinate vectors have matching dimensions. double_area is twice the unsigned area and boundary is the lattice boundary-point count of a simple nondegenerate lattice polygon. Exact operational definition: (double_area-boundary+2)//2

Why this case matters

Small exact fixtures expose this error without platform timing, external services, or probabilistic observations. Planar geometry results depend on the stated convention.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import calendar
import statistics
import itertools
from fractions import Fraction
from datetime import date, datetime, timedelta, timezone
from decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR

N = 1
observations = []
def solve(double_area, boundary):
    return double_area//2-boundary+1
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (8, 8)', solve(*(8, 8)), 1)
check('fixture 2: (1, 3)', solve(*(1, 3)), 0)
check('fixture 3: (18, 12)', solve(*(18, 12)), 4)
check('fixture 4: (9, 9)', solve(*(9, 9)), 1)
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
fixture 1: (8, 8)	`-3`	`1`	Failed
fixture 2: (1, 3)	`-2`	`0`	Failed
fixture 3: (18, 12)	`-2`	`4`	Failed
fixture 4: (9, 9)	`-4`	`1`	Failed

SHA-256 / b568b3fbb655e454aac5f49b06551bb538d377bc2a9a1b2a7e4187669a437a5c

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import calendar
import statistics
import itertools
from fractions import Fraction
from datetime import date, datetime, timedelta, timezone
from decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR

N = 1
observations = []
def solve(double_area, boundary):
    return (double_area+boundary-2)//2
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (8, 8)', solve(*(8, 8)), 1)
check('fixture 2: (1, 3)', solve(*(1, 3)), 0)
check('fixture 3: (18, 12)', solve(*(18, 12)), 4)
check('fixture 4: (9, 9)', solve(*(9, 9)), 1)
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
fixture 1: (8, 8)	`7`	`1`	Failed
fixture 2: (1, 3)	`1`	`0`	Failed
fixture 3: (18, 12)	`14`	`4`	Failed
fixture 4: (9, 9)	`8`	`1`	Failed

SHA-256 / 2c90df70e55df5a8a5c327c4ccbd81f978f4f2766d570c05d4d2bae864223284

HELD IN THE MEMBER ARCHIVE

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

This mechanism has 4 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

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 2026-09-29T14:37:55.265843+00:00.

Case digest / 9a9c400217a61ef0abf567e01ecb5a31879dc727fc701360d9120e5174ad0345