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

A The Matrix of David: A Minimal Triangular Construction from the Repdigit 111 We construct a finite fractal system from the repdigit 111 = 3 × 37 (OEIS A010785), realized as a finite Pascal's triangle called the Matrix of David. The triangular number T(73) = 2701 (OEIS A000217) is the minimal container for the nested system T(73) ⊃ T(37) ⊃ T(19), where all indices are prime and all triangular values satisfy digit-sum unity. Graph-theoretic analysis shows the Matrix of David embeds Fibonacci routing (OEIS A000045) via diagonal summation, maintains golden ratio stability, and exhibits Sierpi'nski fractal self-similarity (OEIS A047999) under modulo-2 reduction. Ten independent parameters completely describe the structure. An eleventh parameter---a distinction function---is necessary and sufficient for interpretability. Approximately 9.69 × 10^(13) mathematically equivalent routes connect vertices (0,0) and (54,18), revealing a fundamental structural feature: the existence of a vast equivalence class of mathematically indistinguishable 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)