The EMCHS heuristic prime gap bound

Let pnp_n denote the nnth prime, and let δ\delta and ϵ\epsilon be parameters in the ranges considered by the Enhanced Multidimensional Chaotic Heuristic Sieve (EMCHS), including δ=0\delta=0 or 0.30.3 and small positive values such as ϵ=0.1\epsilon=0.1. Heuristic prime gap bound. Under the EMCHS model,

lim infn(pn+1pn)exp(2δϵ)ln ⁣(e2/δ).\liminf_{n\to\infty}(p_{n+1}-p_n)\leq \exp(2\delta-\epsilon)\,\ln\!\bigl(e^{2/\delta}\bigr).

For δ=0.3\delta=0.3 and ϵ=0.1\epsilon=0.1, the source gives the numerical consequence lim inf(pn+1pn)11\liminf(p_{n+1}-p_n)\leq 11 and suggests that a gap of 88 is attainable. These are explicitly heuristic estimates, not unconditional theorems, and the displayed formula is singular at δ=0\delta=0 unless an interpretation or limiting convention is supplied.

Sources & referencesView supporting material

Primary source

Milad Ghadimi, “Heuristic Bounded Prime Gaps via a Chaotic Multidimensional Sieve and Random Matrix Theory”, arXiv:2507.17986 (2025).

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