THE QUANTUM INTELLIGENCER

A new quantum algorithm demonstrates sub-quadratic speed for computing modular factorials under a divisor condition, marking a theoretical advance with no immediate operational impact but contributing to the slow erosion of classical cryptographic assumptions. The algorithm, as presented, operates within a narrow algebraic promise: given a prime $p$, a divisor $q$ of $p-1$, and an integer $n < p$, it computes $n! \bmod p$ in expected time $\widetilde{O}(q^c + \sqrt{p/q})$. This breaks the exponent $1/2$ barrier when $q \approx p^{1/(2c+1)}$, a condition that depends on the structure of $p-1$. The result is bounded-error quantum, not deterministic, and its efficiency is asymptotic. It does not claim constant factors suitable for near-term hardware. The method relies on reconstructing Jacobi sums in compact algebraic form, a technique with polynomial dependence on $q$ and $\log p$. This is the core innovation, and it is presented as exact, not approximate. Extension to $n! \bmod p^2$ follows uniformly for $n < p^2$, and at $n = p-1$, this yields the Wilson quotient $\frac{(p-1)!+1}{p} \bmod p$. The quotient has number-theoretic significance, particularly in primality testing and the study of Wilson primes, but it does not feature in any deployed cryptographic standard. Its computation, even efficiently, does not correspond to a known attack vector on RSA, Diffie-Hellman, or elliptic curve systems. The authors conjecture that the divisor condition is not fundamental, suggesting a uniform algorithm for all primes may exist. This remains unproven. No implementation is reported. No benchmarking data appears. The work resides entirely in the domain of theoretical computer science. It does not describe a cryptanalytic event, nor does it indicate that any system is currently vulnerable. The advance is incremental, not disruptive. Yet it belongs to the broader pattern: quantum methods continue to chip away at problems once thought to require classical superpolynomial time. The pace is measured, the gains are narrow, but the direction is consistent. The gap between stated readiness and observed readiness remains instructive. Many migration plans assume that quantum advantage will arrive as a singular event, a sudden decryption of stored traffic. The reality is more diffuse: a series of small algorithmic gains, each insufficient alone, but collectively altering the cost surface of computation. This paper is one such gain. It does not force an immediate rotation. It does not trigger a patch cycle. It does not expose a flaw in a standard. But it is logged. Another data point for the timeline no one requested. The archive grows. Its conclusions remain undrawn. —Inspector Grey Dispatch from The Prepared E0