FA-69976 / Map projection transforms / Open access

Web mercator ground resolution and scale denominator: zoom exponent · case 01

Every zoom level reports half the correct resolution.

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

ROOT CAUSE

The zoom factor uses 2**(z+1), as though zoom levels started at 1.

VERIFIED REPAIR

At the zoom exponent step restore `2 ** z)`, leaving the rest of the model unchanged.

Unsuccessful approach: Clamping the exponent at 1 is correct except at zoom 0, which is reported at zoom 1 resolution.

Case contract

Input [lat, z, tile_size, dpi]. Ground resolution in metres per pixel is 2*pi*6378137*cos(lat)/(tile_size*2**z) with the equatorial circumference computed exactly. The map scale denominator is resolution*dpi/0.0254. Return [resolution rounded to 6 decimals, scale denominator rounded to the nearest integer] computed from the unrounded resolution.

Why this case matters

Scale bars, level-of-detail choice and print exports depend on these values.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    lat, z, tile, dpi = x
    circumference = 2 * math.pi * 6378137.0
    res = circumference * math.cos(math.radians(lat)) / (tile * 2 ** (z + 1))
    scale = res * dpi / 0.0254
    return [round(res, 6), round(scale)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-0.88, 17, 256, 72], [1.194188, 3385]), ('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #2', [-22.5, 12, 512, 72], [17.654651, 50045]), ('control #3', [75.86, 10, 256, 300], [37.345917, 441094]), ('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #5', [-75.78, 4, 256, 300], [2403.383393, 28386418]), ('control #6', [-15.49, 9, 512, 72], [147.321225, 417603]), ('control #7', [76.6, 19, 512, 300], [0.034598, 409])], [('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #5', [-75.78, 4, 256, 300], [2403.383393, 28386418]), ('control #6', [-15.49, 9, 512, 72], [147.321225, 417603]), ('control #7', [76.6, 19, 512, 300], [0.034598, 409]), ('control #8', [20.0, 19, 512, 72], [0.140288, 398]), ('control #9', [-14.77, 4, 256, 96], [9460.648974, 35756784]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #2', [-22.5, 12, 512, 72], [17.654651, 50045]), ('control #8', [20.0, 19, 512, 72], [0.140288, 398]), ('control #9', [-14.77, 4, 256, 96], [9460.648974, 35756784]), ('control #10', [6.58, 15, 256, 300], [4.745845, 56053]), ('control #11', [-18.37, 9, 256, 72], [290.167542, 822522]), ('control #12', [-51.53, 14, 512, 300], [2.97199, 35102]), ('control #13', [-69.8, 6, 256, 300], [844.594183, 9975522])], [('control #3', [75.86, 10, 256, 300], [37.345917, 441094]), ('control #10', [6.58, 15, 256, 300], [4.745845, 56053]), ('control #11', [-18.37, 9, 256, 72], [290.167542, 822522]), ('control #12', [-51.53, 14, 512, 300], [2.97199, 35102]), ('control #13', [-69.8, 6, 256, 300], [844.594183, 9975522]), ('control #14', [-23.41, 10, 256, 72], [140.290274, 397673]), ('control #15', [-37.12, 12, 256, 96], [30.474424, 115179]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #14', [-23.41, 10, 256, 72], [140.290274, 397673]), ('control #15', [-37.12, 12, 256, 96], [30.474424, 115179]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711]), ('boundary #17', [60.0, 1, 512, 96], [19567.879241, 73957339]), ('regression #18', [45.0, 10, 512, 90.7], [54.049141, 193002]), ('boundary #19', [-60.0, 3, 256, 300], [9783.939621, 115558342])]]
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
control #0	`[0.597094, 1693]`	`[1.194188, 3385]`	Failed
control #1	`[32795.943288, 92964879]`	`[65591.886576, 185929757]`	Failed
control #2	`[8.827326, 25022]`	`[17.654651, 50045]`	Failed
control #3	`[18.672958, 220547]`	`[37.345917, 441094]`	Failed
control #4	`[0.064508, 183]`	`[0.129015, 366]`	Failed
control #5	`[1201.691697, 14193209]`	`[2403.383393, 28386418]`	Failed
control #6	`[73.660612, 208802]`	`[147.321225, 417603]`	Failed
control #7	`[0.017299, 204]`	`[0.034598, 409]`	Failed

SHA-256 / b9d4431f3beca4dc8c94729a32e1b169a422645d1e264cd14c0889cea2d92c3d

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    lat, z, tile, dpi = x
    circumference = 2 * math.pi * 6378137.0
    res = circumference * math.cos(math.radians(lat)) / (tile * 2 ** max(z, 1))
    scale = res * dpi / 0.0254
    return [round(res, 6), round(scale)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-0.88, 17, 256, 72], [1.194188, 3385]), ('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #2', [-22.5, 12, 512, 72], [17.654651, 50045]), ('control #3', [75.86, 10, 256, 300], [37.345917, 441094]), ('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #5', [-75.78, 4, 256, 300], [2403.383393, 28386418]), ('control #6', [-15.49, 9, 512, 72], [147.321225, 417603]), ('control #7', [76.6, 19, 512, 300], [0.034598, 409])], [('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #5', [-75.78, 4, 256, 300], [2403.383393, 28386418]), ('control #6', [-15.49, 9, 512, 72], [147.321225, 417603]), ('control #7', [76.6, 19, 512, 300], [0.034598, 409]), ('control #8', [20.0, 19, 512, 72], [0.140288, 398]), ('control #9', [-14.77, 4, 256, 96], [9460.648974, 35756784]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #2', [-22.5, 12, 512, 72], [17.654651, 50045]), ('control #8', [20.0, 19, 512, 72], [0.140288, 398]), ('control #9', [-14.77, 4, 256, 96], [9460.648974, 35756784]), ('control #10', [6.58, 15, 256, 300], [4.745845, 56053]), ('control #11', [-18.37, 9, 256, 72], [290.167542, 822522]), ('control #12', [-51.53, 14, 512, 300], [2.97199, 35102]), ('control #13', [-69.8, 6, 256, 300], [844.594183, 9975522])], [('control #3', [75.86, 10, 256, 300], [37.345917, 441094]), ('control #10', [6.58, 15, 256, 300], [4.745845, 56053]), ('control #11', [-18.37, 9, 256, 72], [290.167542, 822522]), ('control #12', [-51.53, 14, 512, 300], [2.97199, 35102]), ('control #13', [-69.8, 6, 256, 300], [844.594183, 9975522]), ('control #14', [-23.41, 10, 256, 72], [140.290274, 397673]), ('control #15', [-37.12, 12, 256, 96], [30.474424, 115179]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #14', [-23.41, 10, 256, 72], [140.290274, 397673]), ('control #15', [-37.12, 12, 256, 96], [30.474424, 115179]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711]), ('boundary #17', [60.0, 1, 512, 96], [19567.879241, 73957339]), ('regression #18', [45.0, 10, 512, 90.7], [54.049141, 193002]), ('boundary #19', [-60.0, 3, 256, 300], [9783.939621, 115558342])]]
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
control #0	`[1.194188, 3385]`	`[1.194188, 3385]`	Passed
control #1	`[32795.943288, 92964879]`	`[65591.886576, 185929757]`	Failed
control #2	`[17.654651, 50045]`	`[17.654651, 50045]`	Passed
control #3	`[37.345917, 441094]`	`[37.345917, 441094]`	Passed
control #4	`[0.129015, 366]`	`[0.129015, 366]`	Passed
control #5	`[2403.383393, 28386418]`	`[2403.383393, 28386418]`	Passed
control #6	`[147.321225, 417603]`	`[147.321225, 417603]`	Passed
control #7	`[0.034598, 409]`	`[0.034598, 409]`	Passed

SHA-256 / c585587757ff04d7f544aebb69e7ebfbc241c98e3691b930a5bf6be3a6a5584f

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    lat, z, tile, dpi = x
    circumference = 2 * math.pi * 6378137.0
    res = circumference * math.cos(math.radians(lat)) / (tile * 2 ** z)
    scale = res * dpi / 0.0254
    return [round(res, 6), round(scale)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('control #0', [-0.88, 17, 256, 72], [1.194188, 3385]), ('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #2', [-22.5, 12, 512, 72], [17.654651, 50045]), ('control #3', [75.86, 10, 256, 300], [37.345917, 441094]), ('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #5', [-75.78, 4, 256, 300], [2403.383393, 28386418]), ('control #6', [-15.49, 9, 512, 72], [147.321225, 417603]), ('control #7', [76.6, 19, 512, 300], [0.034598, 409])], [('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #5', [-75.78, 4, 256, 300], [2403.383393, 28386418]), ('control #6', [-15.49, 9, 512, 72], [147.321225, 417603]), ('control #7', [76.6, 19, 512, 300], [0.034598, 409]), ('control #8', [20.0, 19, 512, 72], [0.140288, 398]), ('control #9', [-14.77, 4, 256, 96], [9460.648974, 35756784]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #2', [-22.5, 12, 512, 72], [17.654651, 50045]), ('control #8', [20.0, 19, 512, 72], [0.140288, 398]), ('control #9', [-14.77, 4, 256, 96], [9460.648974, 35756784]), ('control #10', [6.58, 15, 256, 300], [4.745845, 56053]), ('control #11', [-18.37, 9, 256, 72], [290.167542, 822522]), ('control #12', [-51.53, 14, 512, 300], [2.97199, 35102]), ('control #13', [-69.8, 6, 256, 300], [844.594183, 9975522])], [('control #3', [75.86, 10, 256, 300], [37.345917, 441094]), ('control #10', [6.58, 15, 256, 300], [4.745845, 56053]), ('control #11', [-18.37, 9, 256, 72], [290.167542, 822522]), ('control #12', [-51.53, 14, 512, 300], [2.97199, 35102]), ('control #13', [-69.8, 6, 256, 300], [844.594183, 9975522]), ('control #14', [-23.41, 10, 256, 72], [140.290274, 397673]), ('control #15', [-37.12, 12, 256, 96], [30.474424, 115179]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711])], [('control #1', [-33.07, 0, 512, 72], [65591.886576, 185929757]), ('control #4', [30.21, 20, 256, 72], [0.129015, 366]), ('control #14', [-23.41, 10, 256, 72], [140.290274, 397673]), ('control #15', [-37.12, 12, 256, 96], [30.474424, 115179]), ('boundary #16', [0.0, 0, 256, 96], [156543.033928, 591658711]), ('boundary #17', [60.0, 1, 512, 96], [19567.879241, 73957339]), ('regression #18', [45.0, 10, 512, 90.7], [54.049141, 193002]), ('boundary #19', [-60.0, 3, 256, 300], [9783.939621, 115558342])]]
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
control #0	`[1.194188, 3385]`	`[1.194188, 3385]`	Passed
control #1	`[65591.886576, 185929757]`	`[65591.886576, 185929757]`	Passed
control #2	`[17.654651, 50045]`	`[17.654651, 50045]`	Passed
control #3	`[37.345917, 441094]`	`[37.345917, 441094]`	Passed
control #4	`[0.129015, 366]`	`[0.129015, 366]`	Passed
control #5	`[2403.383393, 28386418]`	`[2403.383393, 28386418]`	Passed
control #6	`[147.321225, 417603]`	`[147.321225, 417603]`	Passed
control #7	`[0.034598, 409]`	`[0.034598, 409]`	Passed

SHA-256 / c35c61a0af55be5a9e2eaa8afb63fccc44415c8b13dda42d2d0a8fd1e2dc5b0a

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 2026-09-29T14:48:16.407849+00:00.

Case digest / bee108c8127d425b55ec2c75ae282f7659bb65bcab992cbf860de03e3382080c