Random-Projector Quantum Diagnostics of Ramsey Numbers and A Prime-factor Heuristic for <i>R</i> (5, 5) = 45

Abstract We introduce a statistical framework for estimating Ramsey numbers by embedding two-color Ramsey instances into a ℤ 2 × ℤ 2 -graded Majorana algebra. This approach replaces brute-force enumeration with two randomized spectral diagnostics applied to operators of a given dimension d associated with Ramsey numbers: a linear projector P lin and an exponential map P exp ( α ), suitable for both classical and quantum computation. In the diagonal case, both diagnostics identify R (5, 5) at n = 45. The quantum realizations act on a reduced module and therefore require only five data qubits plus a few ancillas via block-encoding/qubitization for R (5, 5) = 45, in stark contrast to the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:mo>(</m:mo> <m:mtable> <m:mtr> <m:mtd> <m:mi>n</m:mi> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mn>2</m:mn> </m:mtd> </m:mtr> </m:mtable> <m:mo>)</m:mo> </m:mrow> <m:mo>≈</m:mo> <m:msup> <m:mn>10</m:mn> <m:mn>3</m:mn> </m:msup> </m:math> \left( \matrix{ n \cr 2 \cr} \right) \approx {10^3} logical qubits demanded by direct edge encodings. We also provide few-qubit estimates for R (6, 6) and R (7, 7), and propose a simple “prime-sequence” consistency heuristic that connects R (5, 5) = 45 to constrained diagonal growth. Our method echoes Erdős’s probabilistic paradigm, emphasizing randomized arguments rather than explicit colorings, and parallels the classical coin-flip approach to Ramsey bounds. Finally, we discuss potential applications of this framework to machine learning with a limited number of qubits.