Research Article
Prime Incidence Analysis Under Bounded Integer-Line Isometries
Ioannis Papadakis*
Issue:
Volume 11, Issue 6, December 2026
Pages:
118-136
Received:
4 October 2026
Accepted:
19 October 2026
Published:
14 November 2026
DOI:
10.11648/j.mcs.20261106.11
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Abstract: For N≥3, let ℒ(N) denote the primes in [3,2N-3]. The binary Goldbach condition at 2N is equivalent to positive prime incidence under reflection about the midpoint N. This paper studies prime incidence under two bounded integer-line isometries: (1) displaced reflection (reflected branch) and (2) translation (translated branch). In both cases, the prime and its image must remain in ℒ(N). For either branch, the Maximal Displacement Envelope (MDE) is the largest even displacement through which incidence remains positive at every even displacement, including zero; if incidence fails at zero, that branch has no MDE. When both branch envelopes exist, their minimum defines the combined MDE(N). The Prime Incidence Conjecture (PIC) asserts that MDE(N) exists for every N≥3 and that MDE(N)/N→2 as N→∞. At zero displacement, reflected incidence is precisely the binary Goldbach condition. Both branches admit single-parameter arithmetic formulations through the least prime realizing a prescribed even prime difference and the least symmetric prime-pair displacement about a midpoint. Exhaustive computation through N=3×107 determines the branch-specific and combined envelopes over the full admissible displacement range and shows that the normalized first-failure displacements trend toward the normalized endpoint 2-6/N over the computed range. Established analytic estimates further imply that, for every fixed 0<ε<2 and A>0, the number of even displacements s≤(2-ε)N at which at least one branch fails is OA,ε(N(log N)-A) for all sufficiently large N.
Abstract: For N≥3, let ℒ(N) denote the primes in [3,2N-3]. The binary Goldbach condition at 2N is equivalent to positive prime incidence under reflection about the midpoint N. This paper studies prime incidence under two bounded integer-line isometries: (1) displaced reflection (reflected branch) and (2) translation (translated branch). In both cases,...
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