At the admissible pair \[ (n,k)=(1000000,499985), \] the coefficient has 53,478 distinct prime factors. Its exponent histogram is \[ (1:53413),(2:56),(3:3),(4:2),(5:2),(8:1),(12:1). \] Consequently \[ \max_{1\leq k<n\leq10^6}\tau\binom nk \geq 2^{53413}3^{56}4^3 5^2 6^2\cdot9\cdot13. \] This exact integer has 16,113 decimal digits. Its first 64 digits are `2901061995181429015403180177031159054152063659198892515558624106`, its final 64 digits are `0248962087137382083803874069801496392268550388351507391473254400`, and its SHA-256 digest is `37e6b0aec8c146fa82e6e8d0eb776dbb1504fb2fff80e5fa74bff8eaddfee951`.

For complete factorization data, the artifact computes \[ v_p\binom nk=\sum_{j\geq1}\left(\left\lfloor\frac n{p^j}\right\rfloor-\left\lfloor\frac k{p^j}\right\rfloor-\left\lfloor\frac{n-k}{p^j}\right\rfloor\right) \] for every prime \(p\leq10^6\). Joining the 53,478 nonzero pairs as ascending `p^e` terms gives SHA-256 digest `fb3db7328c0c940f68d88e873c6554c9ec65616c825f5b933fd2e55e492e1be2`. This certifies the lower bound without asserting that this pair is globally optimal.

Using \(k\leq n/2\) by symmetry, the exact prefix record is attained at \[ (n,k)=(19971,9949). \] Its divisor count is \[ \boxed{2348387885204790879830610540244537680880212204266092283665817947799112331315722388739095226210492168628520272301982570612855049692984247653163656175812378814497351725872902200308583530162859385427897865865873620878587583646796243101440659894852088456115074260792754959770155740574399754249752554204207499681748634739352025413649254209837693841360334377369629570834866128375706364256515169543130470863765460450748221403831366558257535620632378709575168022539059180077056}. \] The prime-exponent histogram of \(\binom{19971}{9949}\) is \[ \#\{p:v_p=1\}=1527,\quad \#\{p:v_p=2\}=9,\quad \#\{p:v_p=3\}=4, \] with one exponent 8 and one exponent 10. The factors having exponent above one are \[ 2^{10}3^8 5^3 7^2 11^3 13^3 19^2 23^3 41^2 59^2 79^2 107^2 109^2 127^2 131^2. \] Thus the displayed value also equals \(2^{1527}3^9 4^4 9\cdot11\). The ordered full factorization has SHA-256 digest `11d4dc8a9b2dc54225e841546f236a3a9cd5c2f61fe0db6ab2455f28749b9c6c` under the `p^e` convention documented in the certificate artifact.

OEIS A048784 tabulates \(\tau\binom{2m}{m}\) for central binomial coefficients and cites Fedorov's asymptotic formula for that sequence. Fedorov's separate Mathematical Notes paper studies upper and lower orders of ratios between divisor counts of adjacent binomial coefficients. Erdős and Kolesnik study prime-power divisibility near the middle. These sources concern central values, asymptotics, or individual prime powers.

Searches by the exact objective, divisor-record terminology, and the pair of bounds \(k<n\leq10^6\) found no published table or complete two-parameter sweep. The exact value at the requested cutoff therefore remains unresolved in this record. The prefix theorem and cutoff-scale lower bound above are the certified results supplied here.

For each fixed \(n\), the program starts at \(\binom n0=1\) and applies \[ \binom nk=\binom n{k-1}\frac{n-k+1}{k}. \] A smallest-prime-factor sieve decomposes both update terms. When an exponent changes from \(e\) to \(e'\), the program replaces the divisor-count factor \(e+1\) by \(e'+1\). Its base-\(10^9\) integer class performs every division exactly and aborts on a remainder or a negative binomial exponent.

The sweep visits exactly 100,000,000 states and performs 490,041,974 prime-exponent updates. It hashes each row maximum, its location, and its decimal length in increasing row order. The final FNV-1a value is `15944218793165208384`. The one-line standard output has SHA-256 digest `acd2be4af4c40571118fca0c72418c8785bc595949f3d5cecc44bcd02f8f7c69`.

The verifier sieves every prime through each top parameter. Legendre's factorial valuation gives every binomial exponent as an integer. The code joins the nonzero pairs in ascending prime order using `p^e` and hashes the resulting factorization. It then multiplies every \(e+1\), hashes the exact decimal divisor count, and reports each exponent histogram.

The prefix factorization contains 1,542 terms. The cutoff-scale factorization contains 53,478 terms. Their report payload has SHA-256 digest `fea7d94ea1b9b59e253076f13b8caf82c47db0d875c7e0b71d351554ee7eb723`.