Continuous pipe no fittings, off the top of my head using poly CST it will be 1-1/2" minimum, probably 2"

Every joint creates turbulence to flow and takes away some pressure when the water is moving.

There really are advantages to seamless for applications like this.

To run the 1400ft line with sched.40 PVC you would need 69 couplings and two male thread adapters(one on each end)

Fitting insertion losses are expressed as an equivalent length of straight pipe.
An 1-1/4 coupling = 1.25' x 69 couplings = 86.25'
An 1-1/4 male thread adapter = 2.75' x 2 MTA's = 5.5'
Total developed length = 1400'+86.25'+5.5'= 1491.25'
1-1/4" pipe @ 30gpm FHL (friction head loss) = 0.1185psi/ft

1-1/4psi @30gpm the friction head loss is 1491.25' x 0.1185psi = 176.31psi loss to pipe friction plus 4.33psi Vertical static head loss so the total loss is 180.64psi.

with 1-1/2" the loss is over 85psi
with 2" pipe the total loss is over 28psi
with 2-1/2" the loss is over 14psi

3" sched 40 couplings equal 3' x 69couplings = 207'
3" male thread adapter = 6.5' x 2 = 13'
TDL = 1400' + 207' + 13" = 1620'

3" sched 40 @ 30gpm FHL = 0.0023/ft x 1620' = 3.76psi loss

Total loss = 3.76psi FHL + 4.33 VSH = 7.52psi

Min Static head at the main water shutoff in the structure is 40psi therefore if the pressure on the municipal main is less than 47.52psi a boost pump would be required even with 3" pipe.

That is true, but the load is not 11gpm, its 30gpm and @ 30gpm the loss in PE pipe is 5.2psi/100ft. (0.052psi/ft)

1400ft x 0.052psi/ft = 72.8psi loss

Why would the load be 30 GPM?
When residences are piped with 3/4 and 1" services and 3/4 distribution systems?
You counting every fixture on? We know there isnt a house in TN that could do that.
A common well pump is 7-12 GPM, my whole street service isntover 6"

When designing a water supply & distribution system we begin by making a list of all fixtures in the structure and consult the code table to determine the demand load for each fixture.

In this example we have 2.5 bathrooms, a dishwasher and no irrigation so refering to table T 2903.6(1) we get the following load:

BATHROOM #1;
Lavatory............ 2
W.C. (tank type)... 3
Tub/shower.......... 4

BATHROOM #2;
Lavatory............ 2
W.C. (tank type)... 3
Tub/shower.......... 4

Kitchen sink........ 2.5
Dishwasher.......... 2.75

Laundry ............ 4

Hose Bibb........... 5
____________________________

TOTAL LOAD......... 32.25gpm

For single family residential applications code requires we design the water supply line at 100% load(rounded off to the nearest 5)

The load was 32.5 so I rounded to the nearest 5 and ended up with 30gpm.

For multi-family (up to 4 living units) & commercial multi-family (five or more living units code allows us to de-rate the load,(typically 80% for multi-family & 60% for commercial multi-family.)

Next we must determine the TDL (total developed length) of the line. TDL = length of the pipe + fitting allowances.

FITTING ALLOWANCE-The additional friction introduced by a fitting expressed as an equivalent length of straight pipe.

When you consult a Friction loss table you will note that is expresses both the friction loss in psi/100ft & the velocity of flow.

To prevent pipe wall erosion Code limits the Velocity of flow in a copper pipe to not more than 8ft/sec and 12ft/sec in plastic pipes. (Generally the friction loss tables only list the numbers that are within the velocity of flow range, or they may list all pipe sizes, but those that are within the velocity flow range are color highlighted.)

Next we need to determine the VSH (Vertical static head) loss. If your municipal water main is 10' below grade but the main water shutoff valve is 1' above grade on a slab house the vertical offset is 11' @ 0.434psi/ft vertical so the VSHL= 11 x 0.434= 4.77psi loss.

Code requires a minimum static head pressure of 40psi at the main water shutoff valve so we add the minimum supply pressure & VSHL, then deduct that from the municipal main static head pressure and use the friction head loss tables to determine the size of line that can supply minimum pressure at the structure.

Typically on a municipal main the physical setback from the main to the structure would be in the order of 50'-100'

For sched 40 PVC pipe @ 30gpm FHL (friction head loss) =
1" pipe..... 46.08psi/100ft
1.25" pipe.. 11.85psi/100ft
1.5" pipe... 5.53psi/100ft

For illustration, let us assume a typical urban house on slab with an 11ft VSH from the main to the water valve and a 50ft setback from the main to the house.

We then add the code minimum pressure (40psi) plus the VSH (4.77psi)and deduct that from the municipal main pressure.
Assuming a main pressure of 60psi we then get 60-44.7= 15.3psi.

We have a differential of 15.3psi and the 1.25" pipe is only 11.85psi loss so if you use 1.25 you still have a margin of 3.45psi.

In fact, if the municipal main pressure was excessively high we could opt to slightly undersize the line thereby using the FHL to reduce the pressure thus negating the need of a pressure reducing valve.

On the other hand, the original problem here is that the supply line is over a 1/4 mile long so we have to increase the diameter of the line to reduce the FHL.

In every case they require that you submit the complete specifications including the math in order to pull the permit.

I am only illustrating the manner inwhich a real plumber would work out the details in accordance with current code standards.

I personally could care less how the self appointed experts derive their solutions.