Some hypotheses regarding mapping the earth:
1. If the earth is flat it will be possible and easy to create an accurate flat representation map of the world. All distances of a flat earth will correctly line up on a flat representation.
2. If the earth is spherical it will be impossible to create an accurate flat representation map of the world. All distances of a spherical earth will incorrectly line up on a flat representation.
3. If the earth is flat it will be impossible to create an accurate spherical representation map of the world. All distances of a flat earth will incorrectly line up on a spherical representation.
4. If the earth is spherical it will be possible and easy to create an accurate spherical representation map of the world. All distances of a spherical earth will correctly line up on a spherical representation.
Ancillary hypotheses:
1a. Flat earth maps onto a flat representation would not require a projection, it's a simple 1:x ratio with no distortion.
2a. Round earth maps onto a flat representation require an inaccurate projection, there is no consistent 1:x ratio possible, all things are inconsistently distorted.
3a. Flat earth maps onto a globe representation require an inaccurate projection, there is no consistent 1:x ratio possible, all things are inconsistently distorted.
4a. Round earth maps onto a globe would not require a projection, it's a simple 1:x ratio with no distortion.
Some observations:
* All flat earth maps I have seen do not work in regard to distances. Land masses usually get distorted and distances always are incorrect somewhere. This does not match hypothesis #1.
* All round earth projections onto a flat map work generally for short distances and fail for long distances, land masses on the extremes of the map get distorted, for example, Greenland on the Mercator projection. This matches hypothesis #2.
* I have never seen anyone try to take a flat earth map and match it to a globe. This is probably because there is no accepted flat earth map and, understandably, no flat earth proponent would have a reason to attempt to place a flat earth map onto a globe. Nothing I have seen tests hypothesis #3.
* All round earth models I have seen match hypothesis #4. For example: GPS mapping, Google earth, airline flight times regardless of location on the earth, all globes I have seen, country boundary mapping, ancient long distance triangulation measurements, international shipping experiences, drive times across large distances.
My experiences are not exhaustive of course, there may be some observations that match hypothesis #1. Please include any observations that match hypothesis #1.