Let S_10 be the number in the question and let T=2^100=1,267,650,600,228,229,401,496,703,205,376. The certified interval is \[ 126{,}174{,}821{,}830{,}345{,}268{,}667{,}240{,}568{,}576 \leq S_{10}\leq 901{,}210{,}462{,}928{,}281{,}273{,}073{,}900{,}978{,}176. \] Equivalently, for a uniform random binary matrix, \[ \frac{492870397774786205731408471}{4951760157141521099596496896} \leq \Pr(\det A=0)\leq \frac{25613941912987493}{36028797018963968}. \] The decimal endpoints are approximately 0.09953438416518605 and 0.7109297015802510.

For the lower bound, first count every matrix having a zero row or two equal rows. Its complement consists of ten ordered, distinct, nonzero vectors chosen from 1,023 possibilities, so this first family has size T-(1023)_10, where (a)_k=a(a-1)\cdots(a-k+1). Among matrices outside that family, consider the 55 column events consisting of ten zero-column events and 45 equal-column events. Each single event leaves 511 possible nonzero row patterns, giving (511)_10 row-distinct matrices. Any two distinct column events impose two independent binary equations on a row, giving (255)_10 matrices. The first Bonferroni lower bound for their union is 55(511)_10-1485(255)_10. Adding the disjoint row-degenerate family gives the stated lower endpoint. Every counted matrix has determinant zero over the reals.

For the upper bound, every binary matrix invertible over F_2 has odd integer determinant and is invertible over the reals. There are \[ |\operatorname{GL}(10,2)|=\prod_{i=0}^{9}(2^{10}-2^i) =366{,}440{,}137{,}299{,}948{,}128{,}422{,}802{,}227{,}200 \] such matrices. Subtracting this from T gives the upper endpoint.

The interval leaves the exact value unresolved.

For a binary n by n matrix A, define the normalized sign matrix \[ H(A)=\begin{pmatrix}1&\mathbf1^{\mathsf T}\\ \mathbf1&J-2A\end{pmatrix}. \] Subtracting the first row from every other row gives \[ \det H(A)=(-2)^n\det A. \] Every sign matrix can be normalized to first row and first column equal to one by row and column sign changes. Each normalized matrix has the same number 2^(2n+1) of sign matrices in its normalization fiber. Consequently, the singular probability for n by n uniform binary matrices equals the singular probability for (n+1) by (n+1) uniform sign matrices.

Tikhomirov proved that the latter probability is (1/2+o(1))^(n+1). This asymptotic theorem explains the scale of the problem. Its constants do not give a numerical order-eleven bound, so the finite interval in rsbm10-claim-certified-count-interval uses exact elementary counts.

The current A046747 table gives \[ (S_1,\ldots,S_9)=(1,10,338,42976,21040112,39882864736, 292604283435872,8286284310367538176,902616230040448613437952). \] Its July 2026 extension credits Dominik Beck for the order-nine term. The table contains no value for order ten as checked on 2026-07-25.

Živković's published classification partitions all binary matrices through order eight under row and column permutations. It gives 10,160,459,763,342,013,440 nonsingular matrices at order eight, whose complement in 2^64 is S_8 above. The order-nine figure is newer than that classification, and the audit located no primary paper or replayable orbit certificate for it. The order-nine value is therefore recorded as sourced database evidence.

Metropolis and Stein counted a broad structural family of singular binary matrices using binomial coefficients and signed Stirling numbers. Živković classified row and column permutation classes through order eight. Bourgain, Vu, and Wood proved exponential singularity bounds for discrete random matrices. Tikhomirov later proved the sharp exponential scale for uniform sign matrices. These sources establish the finite counting framework and the asymptotic scale.

The targeted search found no primary source giving S_10. OEIS A046747 was updated in July 2026 with S_9 and still ends there. This table endpoint is evidence about the search, rather than proof of novelty.

An exact continuation should enumerate row and column permutation orbits of 10 by 10 bipartite adjacency matrices. For each canonical orbit representative, exact Bareiss elimination decides real singularity, while the reciprocal automorphism-group order supplies its labeled weight. The certificate should preserve every representative, its automorphism order, determinant or rank, and a running weighted checksum. Independent sums must recover 2^100 across all orbits and the stated singular total across the zero-determinant orbits.

The verifier enumerates 2^(n^2) labeled matrices for n=1,2,3,4 and evaluates each determinant with fraction-free Bareiss elimination. It reproduces the singular counts 1, 10, 338, and 42,976. At order four it checks the full determinant histogram \[ -3:60,\ -2:1200,\ -1:10020,\ 0:42976,\ 1:10020,\ 2:1200,\ 3:60. \] The canonical count line has SHA-256 digest cb8f2c764fb76fd45e8557a6cb27b6c670e3526d60721dfdc093293f64825ae0. The histogram line has digest ab7118ff1c833f08b01258733186ebe654157be07d1705ae3c387000c14888a3.

The same program evaluates the falling-factorial Bonferroni expression, the order of GL(10,2), and both reduced probability fractions. The order-ten part certifies arithmetic and the combinatorial formulas stated in the claim. It does not enumerate 2^100 matrices.