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Inspired by many-body effects, we propose a novel design for Boltzmann machine (BM)-based invertible logic (IL) using probabilistic bits (p-bits). A CMOS-based XNOR gate is derived to serve as the hardware implementation of many-body interactions, and an IL family is built based on this design. Compared to the conventional two-body-based design framework, the many-body-based design enables compact configuration and provides the simplest binarized energy landscape for fundamental IL gates; furthermore, we demonstrate the composability of the many-body-based IL circuit by merging modular building blocks into large-scale integer factorizers (IFs). To optimize the energy landscape of large-scale combinatorial IL circuits, we introduce degeneracy in energy levels, which enlarges the probabilities for the lowest states. Circuit simulations of our IFs reveal a significant boost in factorization accuracy. An example of a 2- <inline-formula> <tex-math notation="LaTeX">$\times2$ </tex-math></inline-formula>-bit IF demonstrated an increment of factorization accuracy from 64.99&#x0025; to 91.44&#x0025; with a reduction in the number of energy levels from 32 to 9. Similarly, our 6- <inline-formula> <tex-math notation="LaTeX">$\times6$ </tex-math></inline-formula>-bit IF increases the accuracy from 4.430&#x0025; to 83.65&#x0025; with the many-body design. Overall, the many-body-based design scheme provides promising results for future IL circuit designs.