Let \(R_{n,r}\) count symmetric \(n\times n\) matrices over \(\mathbb F_2\) of rank \(r\), with unrestricted diagonal. Put \(Q_{n,r}=R_{n,r+1}/R_{n,r}\) for \(0\leq r<n\). Substitution in the MacWilliams formula gives \[ Q_{n,2s}=2^{n-2s}-1 \] and \[ Q_{n,2s+1}=\frac{2^{2s+2}}{2^{2s+2}-1}\left(2^{n-2s-1}-1\right). \] These quotients decrease strictly. At an even internal rank \(2s\), the preceding quotient has both a larger power-of-two factor and a multiplier greater than one: \[ Q_{n,2s-1}=\frac{2^{2s}}{2^{2s}-1}\left(2^{n-2s+1}-1\right)>2^{n-2s}-1=Q_{n,2s}. \] At an odd internal rank \(2s+1\), write \(a=n-2s-1\geq1\). Since \(2^{2s+2}/(2^{2s+2}-1)\leq4/3\), \[ Q_{n,2s+1}\leq\frac43(2^a-1)<2^{a+1}-1=Q_{n,2s}. \] Thus \(Q_{n,r-1}>Q_{n,r}\), which is equivalent to \[ R_{n,r}^2>R_{n,r-1}R_{n,r+1} \] for every \(n\geq2\) and \(1\leq r<n\). In particular, all 1,225 inequalities requested for \(1\leq n\leq50\) hold strictly.

MacWilliams's Theorem 2 counts symmetric matrices of each rank over a finite field. Lewis, Liu, Morales, Panova, Sam, and Zhang reproduce it as Equation (4.5), using \(\operatorname{sym}(n,r)\) for symmetric matrices with no diagonal restriction. At \(q=2\), the result is \[ R_{n,r}= \left(\prod_{i=1}^{\lfloor r/2\rfloor}\frac{2^{2i}}{2^{2i}-1}\right) \left(\prod_{i=0}^{r-1}(2^{n-i}-1)\right), \] with empty products equal to one. This formula supplies every exact rank vector through order 50. The executable artifact evaluates all 1,325 entries as integers, checks that each row sums to \(2^{n(n+1)/2}\), and records an unambiguous SHA-256 digest of the complete coefficient stream.

The diagonal convention was checked in two ways. The later paper explicitly distinguishes \(\operatorname{sym}(n,r)\), with unrestricted diagonal, from \(\operatorname{sym}_0(n,r)\), whose diagonal is zero. Independent enumeration of all \(2^{n(n+1)/2}\) upper-triangular bit assignments for every \(1\leq n\leq6\) agrees with the product formula. The first six rows are \[ \begin{aligned} &(1,1),\\ &(1,3,4),\\ &(1,7,28,28),\\ &(1,15,140,420,448),\\ &(1,31,620,4340,13888,13888),\\ &(1,63,2604,39060,291648,874944,888832). \end{aligned} \]

MacWilliams's Theorem 2 is the source of the fixed-rank count. The 2011 exposition reproduces the formula as Equation (4.5) and defines \(\operatorname{sym}(n,r)\) as the unrestricted symmetric count. Its separate notation \(\operatorname{sym}_0(n,r)\) imposes a zero diagonal, so the two families cannot be confused in the displayed formula.

At \(q=2\), exact evaluation gives the six rank vectors stated in the candidate record. Their totals are \(2^1,2^3,2^6,2^{10},2^{15},2^{21}\), as required when the upper triangle, including every diagonal entry, is free. The independent enumeration in the artifact checks every one of the 2,131,018 matrices in these six orders. This resolves the characteristic-two convention before the formula is used through order 50.

The program evaluates the MacWilliams product with exact rational arithmetic and asserts integrality. It checks each rank row against the total number \(2^{n(n+1)/2}\) of symmetric binary matrices. For orders at most six, a separate Gray-code enumeration visits every upper-triangular bit assignment, forms the symmetric matrix, computes its rank over \(\mathbb F_2\), and compares the complete row with the formula.

The program then checks the two adjacent-quotient formulas and all 1,225 strict log-concavity inequalities. The smallest margin is 5 at \((n,r)=(2,1)\), with triple \((1,3,4)\).

For the coefficient digest, each row begins with two-byte big-endian fields for its order and length. Each coefficient is encoded by a two-byte byte-length followed by its minimal unsigned big-endian representation. Concatenating the 50 rows in increasing order gives SHA-256 digest `298e2b5d779b4b3c344d57e3d9c47bd7d35dfe27d42cbc4229be2ffd86519470`.