Piazzi Smyth was well aware that 1.001 inches was precise enough, but not exact, as the length of the Pyramid Inch is, as a matter of definition, the earth's polar radius divided by 250 million, such that the earth's polar radius also equals 10 million cubits of 25 Pyramid Inches.

Piazzi Smyth favoured Captain Clarke's computation on the Earth's polar radius:
250,261,452 inches = 250 million Pyramid Inches where Pyramid Inch = 1.00104.. British Inches as the best available estimate of the length of the Pyramid Inch in 1867.
This was rounded, rather sensibly, to 1.001 inches

For example, taking the hypothetical side-length of Corner Sockets as 365.242 x 25 Pyramid Inches:

1.001 inches = 9,140.2 British Inches

Clarke (1867) 1.00104.. = 9,140.6 British Inches

Barone (2011) 1.00106 = 9,140.7 British Inches

So any further refinement in the determination of the Earth's polar radius has little effect on the predicted side-length.

J.H. Cole determined the north side-length of the Corner Sockets as 231.863 metres (9,128.5 inches) and the east side-length of the Corner Sockets as 232.071 metres (9,136.7 inches), but was unable to determine the other two sides.

The average of the north side and the east side is 9,132.6 inches:

The as built dimensions of the sides of the true base square are within a range of 8 inches, and comparison with simple lengths in royal cubits within the pyramid indicates that the build standard is generally better than 1 part in 1000, especially on longer lengths.

If the plan was as Piazzi Smyth proposed then the side-length of the corner sockets should be within the range:

9,140.7 plus or minus 9.1 inches = 9,131.6 inches to 9,149.8 inches

The hypothesis just passes this test as a mean, but actually fails the test as all lengths should be in this range if intended.

Also, the foundation corner sockets did not define the true size of the pyramid.

This particular 'proof' of the hypothetical Pyramid Inch, as proposed by Piazzi Smyth, has to be rejected, and was never consistent with the facts:

The Royal Engineers had surveyed the sides as 9.127.5 inches, 9,129.5 inches, 9,121.0 inches and 9,140.0 inches so 3 out of 4 sides were too short.

Following Smyth's theory, Petrie disputed the position of the south-west corner socket, and in his judgement the south-side was 9,123.9 inches not 9,140.0 inches. Petrie's measurements of the other sides were similiar to those of the Royal Engineers.

Mark