P3.14 Recursive Reality Project - A Complete Proof of the Interference Dimension Law

The Interference Dimension Law states: at Level k of recursive self-reference, the maximum number of independent geodesic predecessors of any state is k+1, equal to the spatial dimension. Previous papers verified this law computationally for k=1, 2, and 3, and the XY Predecessor Maximum paper proved the upper bound (max ≤ k+1) algebraically for all k via the hexagonal distance formula. The lower bound — that states actually achieving k+1 predecessors exist at every level — remained verified but unproved. The present paper closes this gap and delivers a complete proof of the law for all k. Throughout this paper, depth refers to Cayley graph distance from the origin (d=0 at the origin, d=1 for first-shell states). Paper B uses a shell index convention where the origin is shell n=1 and states at Cayley distance d are in shell n=d+1. The law’s statement ‘first achieved at shell n=k+2’ (Paper B) is equivalent to ‘first achieved at Cayley distance d=k+1’ (this paper). Two results establish the proof. First, the general distance formula for the Level k Cayley graph: d_k(a, b, z₁, …, zₖ₋₁) = max(|a|, |b|, |a−b|) + |z₁| + ⋯ + |zₖ₋₁| This is proved from the product structure of the Level k generator set (C₆ orbit in the XY plane plus independent translations in each Z_j direction) and verified computationally for k=1, 2, 3. Since XY generators do not affect any z_j coordinate, and each Z_j generator does not affect the a or b coordinates, interleaving XY and translational steps in any order cannot reduce the total below the sum of the component minima. Second, the constructive family: s_k(n) = (n, n−1, n−1, …, n−1) ∈ ℤ^(k+1) is proved to have exactly k+1 geodesic predecessors at every Cayley distance d ≥ k+1, for every k ≥ 1. This corresponds to shell index n ≥ 2 in Paper B’s convention. The proof uses the general distance formula to verify that each of the k+1 candidate predecessors is at Cayley distance k(n−1), exactly one less than the depth k(n−1)+1 of s_k(n), and that no further predecessors exist (by the established upper bound). The k+1 predecessors are mutually distinct: p₁ and p₂ differ in their b-coordinate (n−2 versus n−1); p₁ and p₂ both have first coordinate n−1 while each p_j has first coordinate n; and p_j differs from p_j’ in which z-coordinate holds the value n−2. Together with the XY Predecessor Maximum paper’s upper bound, this gives: the maximum number of independent geodesic predecessors at Level k is exactly k+1, first achieved at Cayley distance d=k+1 (equivalently, shell n=k+2 in Paper B). This upgrades Paper B’s main result from a verified conjecture to a proved theorem. The Interference Dimension Law is now a proved theorem.