Finite Relational Geometry for Prime Boundary References

Abstract

This paper proposes a finite relational geometric framework for examining prime-related boundary structures without attempting to replace standard limits, trigonometric functions, irrational numbers, or established number theory. The framework begins with a Meta point M and boundary nodes arranged on a circle, where I(M) denotes a Meta-to-boundary reference and I(B) denotes a boundary-to-boundary reference. For a regular configuration, the boundary reference is expressed by I(B)_k = 2R sin(πk/n), so trigonometric quantities are interpreted first as exact geometric ratios rather than as numerical approximations. When the number of boundary nodes is prime, the available chord-reference classes increase with the prime point count and form an increasingly dense finite reference spectrum, while the special ratio k/n = 1/6 acts as a sign-transition boundary for the defect variables ε = I(M) − I(B), δ = I(M) − I(B′), and ω = I(B) − I(B′), which satisfy the exact closure relation ω = δ − ε. In the generalized configuration 1_M + 2q_B, the total Meta reference is T_M = 2qR and exactly matches the factor-2 diameter structure T_B(2) = 2qR, whereas a factor-q boundary structure has T_B(q) = 4qR sin(π/q). Before Meta removal this gives ε_Σ = 0 and δ_Σ = ω_Σ = 2qR[1 − 2 sin(π/q)]. After the Meta point is removed, the system reorganizes into boundary-only defects δ′_q(k) and ω′_q(k), satisfying δ′_q(k) + ω′_q(k) = T_B(2) − T_B(q), while the former Meta–Boundary defect is transferred into the new Boundary–Boundary defect through ω′_q(k) = ε_pre,q(k). The sign change remains controlled by the same geometric condition k/(2q) = 1/6, suggesting that the special angle π/6 is better interpreted as a finite reference-transition boundary than as an isolated trigonometric value. As the number of prime boundary nodes increases, the number of admissible reference branches grows, the branches become denser around this transition boundary, and the closure relations remain exact. The proposed contribution is therefore not a new trigonometric identity or a replacement for conventional analysis, but a finite relational representation in which prime point-count growth, geometric ratios, defect branching, closure, and Meta removal can be studied within a single visual and algebraic framework.