Travel Distance Errors?

In Chapter 1 of the Payer's Handbook (5.5), the section on travel lists various speeds and distances. The line for "Fast" pace lists a speed of about 4 miles per hour, but then states that this would result in 30 instead of 32 miles in one day (despite the DMG making it clear that the core assumption is for 8 hours of travel per day) - I think this may have been a hold-over from the identical 2014 table?
I'm not sure if there's some hidden assumption that Fast pace requires an additional 30 minutes of rest per day, or whether it was initially done to make the differences in paces all equal to 6 miles (which might be beneficial for traditional 24-mile and 6-mile hexes, but means that the maths is incorrect for more free-form travel). The miles per hour is actually, if anything, a slightly underestimate of total speed, as travelling at 400 feet per minute (which is identical to travelling 40 feet every "round") is a speed of over 4.5 mph, but I think it's reasonable to round down to 4mph as you might take the odd breather etc.
Similarly, the daily total for Slow pace is 18 miles, instead of 8 hours travelling at 2 miles per hour which should give 16 miles (however travelling exactly 20 feet per minute for a full 60 minutes, and then multiplying that distance by 8 *before* rounding, does give a total distance of just over 18 miles). Again, I suspect this was done to make all of the total distances vary by 6 miles, but it makes it look like either a) there are errors in the calculations or b) there's a whole extra series of hidden assumptions about what proportion of the "8 hours" are you actually moving depending on your pace.

The above Fast and Slow Travel daily distances also contradict the guidance set out in the DMG, which gives explicit formulas for working out speeds for any other reason, which otherwise agree with the whole table:
Miles per hour = Speed ÷ 10
Miles per day (Normal pace) = Miles per hour × number of hours traveled (typically 8)
Fast pace = Miles per day × 1⅓ (round down)
Slow pace = Miles per day × 2/3 (round down).

(I also think there's a typo in the DMG with "traveled")
Whilst I appreciate varying all of the daily distances by 6 miles does specifically make 6-mile hexes easier, it still seems like it's just messing up the otherwise fairly neat calculations that could otherwise be done?

(On a related note, I do think that sometimes there are hidden assumption with speeds - the Phantom Steed spell creates a mount which has a Speed of 100 feet but is later stated to travel at "13 miles in an hour" - initially I thought they'd done the above calculations wrong as it seems like it should be 10mph, but I believe there is a hidden assumption that the Phantom Steed over any non-battlefield distance will be able to sustain a "Fast" pace and therefore will travel at 13mph, not 10.)

I acknowledge that this is an unbelievably niche area to delve into, but I just thought I'd see if anyone can offer any further insight.

Quote from Gloomweaver>>

In Chapter 1 of the Payer's Handbook (5.5), the section on travel lists various speeds and distances. The line for "Fast" pace lists a speed of about 4 miles per hour, but then states that this would result in 30 instead of 32 miles in one day (despite the DMG making it clear that the core assumption is for 8 hours of travel per day) - I think this may have been a hold-over from the identical 2014 table?
I'm not sure if there's some hidden assumption that Fast pace requires an additional 30 minutes of rest per day, or whether it was initially done to make the differences in paces all equal to 6 miles (which might be beneficial for traditional 24-mile and 6-mile hexes, but means that the maths is incorrect for more free-form travel). The miles per hour is actually, if anything, a slightly underestimate of total speed, as travelling at 400 feet per minute (which is identical to travelling 40 feet every "round") is a speed of over 4.5 mph, but I think it's reasonable to round down to 4mph as you might take the odd breather etc.
Similarly, the daily total for Slow pace is 18 miles, instead of 8 hours travelling at 2 miles per hour which should give 16 miles (however travelling exactly 20 feet per minute for a full 60 minutes, and then multiplying that distance by 8 *before* rounding, does give a total distance of just over 18 miles). Again, I suspect this was done to make all of the total distances vary by 6 miles, but it makes it look like either a) there are errors in the calculations or b) there's a whole extra series of hidden assumptions about what proportion of the "8 hours" are you actually moving depending on your pace.

 I believe there are some assumptions, as you said. Being honest, I don't know why for a Slow Pace it's 18 instead of 16, but I could understand for Fast Pace it's 30 if you assume you can't keep those 4 miles / hour for a period of 8 hours, so it's not a constant speed.

Regarding hexes, we had also this useful table in the 5e DMG: Map Travel Pace

The above Fast and Slow Travel daily distances also contradict the guidance set out in the DMG, which gives explicit formulas for working out speeds for any other reason, which otherwise agree with the whole table:
Miles per hour = Speed ÷ 10
Miles per day (Normal pace) = Miles per hour × number of hours traveled (typically 8)
Fast pace = Miles per day × 1⅓ (round down)
Slow pace = Miles per day × 2/3 (round down).

(I also think there's a typo in the DMG with "traveled")
Whilst I appreciate varying all of the daily distances by 6 miles does specifically make 6-mile hexes easier, it still seems like it's just messing up the otherwise fairly neat calculations that could otherwise be done?

 In this case, isn't this the calculation for Special Movement? It was the same in the 5e DMG.

(On a related note, I do think that sometimes there are hidden assumption with speeds - the Phantom Steed spell creates a mount which has a Speed of 100 feet but is later stated to travel at "13 miles in an hour" - initially I thought they'd done the above calculations wrong as it seems like it should be 10mph, but I believe there is a hidden assumption that the Phantom Steed over any non-battlefield distance will be able to sustain a "Fast" pace and therefore will travel at 13mph, not 10.)

 Magic? :D This is from the 5e DMG:

Similarly, a phantom steed spell creates a magical mount with a speed of 100 feet that doesn’t tire like a real horse. A character on a phantom steed can cover 1,000 feet in 1 minute at a normal pace, 1,333 feet at a fast pace, or 666 feet at a slow pace. In 1 hour, the character can travel 7, 10, or 13 miles.

(if you check, the spell's description is not exactly the same between versions: Phantom Steed)

I acknowledge that this is an unbelievably niche area to delve into, but I just thought I'd see if anyone can offer any further insight.

 I like these kinds of discussions. Niche or not, it's interesting to think about :)

Quote from Gloomweaver >>

In Chapter 1 of the Payer's Handbook (5.5), the section on travel lists various speeds and distances. The line for "Fast" pace lists a speed of about 4 miles per hour, but then states that this would result in 30 instead of 32 miles in one day (despite the DMG making it clear that the core assumption is for 8 hours of travel per day) - I think this may have been a hold-over from the identical 2014 table?

It's not.

If it is wrong, it has been wrong throughout 5e. The table is the same in the 2014 and the 2024 PHBs, but the guideline you mentioned is also present in both the 2014 and 2024 DMGs. The only thing that has changed between editions is that in 2014, travelling at a Fast Pace imposed a -5 penalty to passive Wisdom (Perception) checks while in 2024, it imposes disadvantage on Wisdom (Perception) checks.

I'm not saying that the numbers are right, I am just saying that any issues are not because of the edition change.

Most likely, the issue that someone just wanted to round it to 30 to make it look nicer.

In all honesty, according to basic math principles (you perform any rounding as the last step of any calculation), some of the numbers are wrong. For a 30-foot speed:

• Fast Pace should be 390 feet per minute / 4 miles per hour / 32 miles per day.
• Normal Pace is correct.
• Slow Pace is correct.

Technically, they have simplified the math on Speed to Miles per hour, but it works out because of their rounding.

• Fast pace would be 4.43 miles per hour and 35.45 miles per day*. 
• Normal pace would 3.41 miles per hour and 27.27 miles per day*.
• Slow pace would be 2.27 miles per hour and 18.18 miles per day.

* 32 and 24 would appear correct if you multiply the rounded miles per hour instead of using the number before rounding. Their formula is to multiply the miles per hour by 8, but using the rounded number is poor math.

If you are travelling at nonstandard speeds, you may want to prepare your own charts with your own rounding standards. I recommend embracing fractional miles for a smoother transition between movement speeds.