The Matrix of David: A Minimal Triangular Construction from the Repdigit 111

Abstract We present a construction of a finite fractal system derived from the repdigit 111 and realized as a finite Pascal's triangle, which we call the Matrix of David. Beginning from the unique prime factorization 111 = 3×37, we prove that the triangular number T(73) = 2701 serves as the minimal container for the nested system T(73)⊃T(37)⊃T(19), where all indices are prime numbers and all triangular values satisfy a digit-sum unity invariant. Using graph-theoretic methods, we demonstrate that the Matrix of David naturally embeds Fibonacci routing through diagonal summation, maintains golden ratio (φ) stability in interior regions, and exhibits Sierpiński fractal self-similarity under modulo-2 reduction. We formalize the problem of multiplicity of equivalent routes within the finite container and prove that the structure is completely described by ten independent parameters, but requires the introduction of an eleventh parameter—a parameter of distinction—for the transition from existence to interpretability. A concrete example reveals the existence of approximately 4.86×10¹² mathematically equivalent routes between two points in the structure. We introduce a hierarchical routing protocol that demonstrates how the choice of a realized path from among equivalent ones leads to structural invariants (97.5% weight concentration in five positions) not derivable from the ten-parameter description. This result establishes a fundamental boundary between existence and meaning in finite combinatorial systems. The Matrix of David is autonomous, requiring no appeal to infinity, and suggests deep connections between discrete geometry, combinatorics, and the problem of interpretation in mathematics. Keywords: Pascal's triangle, finite combinatorial structures, graph theory, Sierpiński fractal, golden ratio, Fibonacci sequence, interpretability, triangular numbers, repdigits, discrete geometry MSC (2020): 05A10 (Factorials, binomial coefficients, combinatorial functions), 05C20 (Directed graphs), 11B39 (Fibonacci and Lucas numbers), 28A80 (Fractals), 05A15 (Exact enumeration problems)