FA-69871 / Map projection transforms / Open access

Spherical web mercator inverse projection: exponent sign · case 01

Latitudes come back with the wrong hemisphere.

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

ROOT CAUSE

The exponent is negated, reflecting all northings through the equator.

VERIFIED REPAIR

At the exponent sign step restore `math.exp(py / R)`, leaving the rest of the model unchanged.

Unsuccessful approach: Using abs(y) makes all southern positions report northern latitudes.

Case contract

Input [x, y] metres (sphere radius 6378137). Eastings outside the world extent E = 20037508.342789244 are wrapped into [-E, E) first. Return [lon, lat] in degrees rounded to 7 decimals.

Why this case matters

Converting clicked map positions or tile corners back to geographic coordinates uses this inverse; faults misplace every reverse-geocoded point.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    px, py = x
    R = 6378137.0
    E = 20037508.342789244
    px = ((px + E) % (2 * E)) - E
    lon = math.degrees(px / R)
    lat = math.degrees(2 * math.atan(math.exp(-py / R)) - math.pi / 2)
    return [round(lon, 7), round(lat, 7)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression #0', [25000000.0, 1000000.0], [-135.421179, 8.9465739]), ('regression #1', [-30000000.0, -2000000.0], [90.5054148, -17.6789142]), ('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])], [('regression #1', [-30000000.0, -2000000.0], [90.5054148, -17.6789142]), ('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])], [('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('control #9', [500000.0, -500000.0], [4.4915764, -4.486983]), ('control #10', [-19000000.0, 12000000.0], [-170.679904, 72.6726763]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])], [('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('control #9', [500000.0, -500000.0], [4.4915764, -4.486983]), ('control #11', [3000000.0, 9000000.0], [26.9494585, 62.5882773]), ('regression #12', [45000000.0, 3000000.0], [44.2418779, 26.0074202]), ('control #13', [-1234567.0, 7654321.0], [-11.0903041, 56.4787668]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])], [('regression #0', [25000000.0, 1000000.0], [-135.421179, 8.9465739]), ('regression #1', [-30000000.0, -2000000.0], [90.5054148, -17.6789142]), ('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('regression #14', [-21000000.0, 500000.0], [171.3537903, 4.486983]), ('regression #15', [-25000000.0, -4000000.0], [135.421179, -33.7852301]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])]]
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
regression #0	`[-135.421179, -8.9465739]`	`[-135.421179, 8.9465739]`	Failed
regression #1	`[90.5054148, 17.6789142]`	`[90.5054148, -17.6789142]`	Failed
boundary #2	`[-180.0, 0.0]`	`[-180.0, 0.0]`	Passed
control #3	`[0.0, 0.0]`	`[0.0, 0.0]`	Passed
control #4	`[9.9999999, -10.0]`	`[9.9999999, 10.0]`	Failed
control #5	`[-73.9999998, -40.7139556]`	`[-73.9999998, 40.7139556]`	Failed
control #6	`[134.7472926, 47.3537047]`	`[134.7472926, -47.3537047]`	Failed
regression #16	`[6.2e-06, 0.0]`	`[6.2e-06, 0.0]`	Passed

SHA-256 / 1452dde37b83b68e6d525ac143db42cd163f514e0eafac3ef53839bcec8dd1fc

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    px, py = x
    R = 6378137.0
    E = 20037508.342789244
    px = ((px + E) % (2 * E)) - E
    lon = math.degrees(px / R)
    lat = math.degrees(2 * math.atan(math.exp(abs(py) / R)) - math.pi / 2)
    return [round(lon, 7), round(lat, 7)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression #0', [25000000.0, 1000000.0], [-135.421179, 8.9465739]), ('regression #1', [-30000000.0, -2000000.0], [90.5054148, -17.6789142]), ('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])], [('regression #1', [-30000000.0, -2000000.0], [90.5054148, -17.6789142]), ('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])], [('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('control #9', [500000.0, -500000.0], [4.4915764, -4.486983]), ('control #10', [-19000000.0, 12000000.0], [-170.679904, 72.6726763]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])], [('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('control #9', [500000.0, -500000.0], [4.4915764, -4.486983]), ('control #11', [3000000.0, 9000000.0], [26.9494585, 62.5882773]), ('regression #12', [45000000.0, 3000000.0], [44.2418779, 26.0074202]), ('control #13', [-1234567.0, 7654321.0], [-11.0903041, 56.4787668]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])], [('regression #0', [25000000.0, 1000000.0], [-135.421179, 8.9465739]), ('regression #1', [-30000000.0, -2000000.0], [90.5054148, -17.6789142]), ('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('regression #14', [-21000000.0, 500000.0], [171.3537903, 4.486983]), ('regression #15', [-25000000.0, -4000000.0], [135.421179, -33.7852301]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])]]
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
regression #0	`[-135.421179, 8.9465739]`	`[-135.421179, 8.9465739]`	Passed
regression #1	`[90.5054148, 17.6789142]`	`[90.5054148, -17.6789142]`	Failed
boundary #2	`[-180.0, 0.0]`	`[-180.0, 0.0]`	Passed
control #3	`[0.0, 0.0]`	`[0.0, 0.0]`	Passed
control #4	`[9.9999999, 10.0]`	`[9.9999999, 10.0]`	Passed
control #5	`[-73.9999998, 40.7139556]`	`[-73.9999998, 40.7139556]`	Passed
control #6	`[134.7472926, 47.3537047]`	`[134.7472926, -47.3537047]`	Failed
regression #16	`[6.2e-06, 0.0]`	`[6.2e-06, 0.0]`	Passed

SHA-256 / df65d08b498065ca0f4a1b350b80c1705a114c818090dd1b48c245ac946f1b34

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    px, py = x
    R = 6378137.0
    E = 20037508.342789244
    px = ((px + E) % (2 * E)) - E
    lon = math.degrees(px / R)
    lat = math.degrees(2 * math.atan(math.exp(py / R)) - math.pi / 2)
    return [round(lon, 7), round(lat, 7)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('regression #0', [25000000.0, 1000000.0], [-135.421179, 8.9465739]), ('regression #1', [-30000000.0, -2000000.0], [90.5054148, -17.6789142]), ('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])], [('regression #1', [-30000000.0, -2000000.0], [90.5054148, -17.6789142]), ('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])], [('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #4', [1113194.9, 1118889.97], [9.9999999, 10.0]), ('boundary #7', [0.0, 20037508.342789244], [0.0, 85.0511288]), ('boundary #8', [0.0, -20037508.342789244], [0.0, -85.0511288]), ('control #9', [500000.0, -500000.0], [4.4915764, -4.486983]), ('control #10', [-19000000.0, 12000000.0], [-170.679904, 72.6726763]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])], [('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #5', [-8237642.3, 4970241.3], [-73.9999998, 40.7139556]), ('control #9', [500000.0, -500000.0], [4.4915764, -4.486983]), ('control #11', [3000000.0, 9000000.0], [26.9494585, 62.5882773]), ('regression #12', [45000000.0, 3000000.0], [44.2418779, 26.0074202]), ('control #13', [-1234567.0, 7654321.0], [-11.0903041, 56.4787668]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])], [('regression #0', [25000000.0, 1000000.0], [-135.421179, 8.9465739]), ('regression #1', [-30000000.0, -2000000.0], [90.5054148, -17.6789142]), ('boundary #2', [20037508.342789244, 0.0], [-180.0, 0.0]), ('control #3', [0.0, 0.0], [0.0, 0.0]), ('control #6', [15000000.0, -6000000.0], [134.7472926, -47.3537047]), ('regression #14', [-21000000.0, 500000.0], [171.3537903, 4.486983]), ('regression #15', [-25000000.0, -4000000.0], [135.421179, -33.7852301]), ('regression #16', [-40075016.0, 0.0], [6.2e-06, 0.0])]]
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
regression #0	`[-135.421179, 8.9465739]`	`[-135.421179, 8.9465739]`	Passed
regression #1	`[90.5054148, -17.6789142]`	`[90.5054148, -17.6789142]`	Passed
boundary #2	`[-180.0, 0.0]`	`[-180.0, 0.0]`	Passed
control #3	`[0.0, 0.0]`	`[0.0, 0.0]`	Passed
control #4	`[9.9999999, 10.0]`	`[9.9999999, 10.0]`	Passed
control #5	`[-73.9999998, 40.7139556]`	`[-73.9999998, 40.7139556]`	Passed
control #6	`[134.7472926, -47.3537047]`	`[134.7472926, -47.3537047]`	Passed
regression #16	`[6.2e-06, 0.0]`	`[6.2e-06, 0.0]`	Passed

SHA-256 / dda471717214331ba1674be3d715f5c127147f549e94ee0b0ecb52673eb20be7

Verification & scope

Stipulated deterministic toy contract on a bounded input domain; results are rounded as stated and no conformance with any published standard or library is claimed. 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 2026-09-29T14:48:15.404360+00:00.

Case digest / 3b0e41bec299f3d1c4d16ea5b14f45089a7660d257698506bd6084e1cd934ab6