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4946 bytes, as of the version from 2026-10-24 21:50 (current). Recorded here, not run.
# Sources and mathematical evidence for T469
The table's quantity is E||X-E[X]|| for the stated uniform distribution. It is the first distance moment about the centroid, not the root mean square radius and not the mean pair distance. The selection and normalizations were written down in convention.md before the first computation.
## Primary sources read
* H. Li and X. Qiu (2016), DOI 10.1155/2016/8371750. The publisher HTML names the quantity and the regular-polygon construction but omits mathematical formula content in text extraction. A public copy of the original open-access paper at https://pdfs.semanticscholar.org/74f1/54828d0193a818819985a99d61937f8cd8f9.pdf supplied the displayed formulas. Equation (23) gives h/3 [sec(alpha)+cot(alpha) log(sec(alpha)+tan(alpha))]. Example 1 gives the rectangle formula, Example 2 the unit-square formula, Example 3 the decomposition of every regular polygon and the disk limit. The paper's equilateral-triangle vertex distance in equation (25) is NOT the centroid distance; the generator uses equation (23) on the three center-to-edge triangles, with alpha=pi/3 and h=sqrt(3)/6.
* D. H. Bailey, J. M. Borwein and R. E. Crandall, Box integrals, DOI 10.1016/j.cam.2006.06.010, and the author manuscript at https://www.davidhbailey.com/dhbpapers/boxintegrals.pdf. Equation (4) relates center distance to half the corner distance; section 7 gives B2(1) and B3(1). The manuscript downloaded here is dated March 26, 2020; its internal cross-reference placeholders are incomplete. The actual formulas and defining integrals were read. Nothing relies on its incidental naming of constants or its asymptotic formula.
Both source PDFs and extracted text are retained here. The Wiley HTML download returned 403 in curl, but the browser tool read it and the independent PDF was accessible. PDF screenshots failed for the Li-Qiu mirror; the formula text and independent integral checks supply the evidence.
## Derivations behind the checks
For a centered regular n-gon of circumradius R and inradius h, the angular measure at distance r is 2*pi when r<=h and 2*pi-2*n*acos(h/r) when h<=r<=R. Multiplying by r and dividing by the polygon area gives the radial density used by checks.py. Its zeroth moment is 1; its second moment is (R^2+2h^2)/6. The generator instead uses the closed form from Li-Qiu equation (23). These computations share no implementation.
For a rectangle with half-sides A and B, first-quadrant rays reach the boundary at A/cos(theta) until theta=atan(B/A), and at B/sin(theta) afterwards. Integrating r^(p+1) along each ray gives the independently coded moment integral. Its second moment is (A^2+B^2)/3. Both 1-by-1 and 1-by-2 rectangles were checked.
For the centered unit cube, restrict by symmetry to the positive octant and split into the three regions where one coordinate is largest. Write the largest coordinate as t and the other two as tu and tv. The Jacobian t^2 gives E[r^p]=3/(2^p*(p+3))*int_0^1 int_0^1 (1+u^2+v^2)^(p/2) du dv. Polar coordinates in the u,v square yield the one-dimensional integral in checks.py. This computes mean distance from the uniform volume measure without using the generator's logarithmic closed form. Its known zeroth and second moments are 1 and 1/4.
For unit balls, the volume within radius r is r^d times the total volume; the radial density d*r^(d-1) gives d/(d+1) exactly. For the interval of length 1 this is evaluated with radius 1/2, giving 1/4. Circle and sphere points all have radius 1, so their distances are exactly 1. These exactness arguments do not use digit patterns.
If Y=aX+b for a>0, then E[Y]=aE[X]+b and ||Y-E[Y]||=a||X-E[X]|| pointwise. Thus M(aK+b)=aM(K). The quadrature code evaluates every region at scale 3/2 to check the size convention; translation invariance follows pointwise without numerical integration.
## Observed results
All 13 normalization, second-moment, scaling and mean checks passed at 130 and 170 decimal digits with absolute tolerance 1e-120. Eight irrational entries carry finite Arb balls; their maximum radius was 4.298304014e-124. Five entries are exact rationals. All 13 regions were also independently sampled with 300000 points each, fixed seed 46920261024. The largest centroid discrepancy was 2.943 standard errors, second-moment discrepancy 2.509 standard errors, and mean discrepancy 2.680 standard errors, below the predetermined six-standard-error limit. The circle and sphere radius checks used the exact constant-radius property.
The dry run found 13 entries, a longest working-precision ball representation of 142 characters, and a 3 KB entries block including comments. The actual written 100-digit entries have length at most 102 characters. The final stored intervals are checked by exact parsing of their decimal centers and last-place widths. Each full Arb ball must fit strictly inside the stored interval, and the independent quadrature must lie inside it as well.