The theoretical foundation rests on the Trit-Plane Decomposition used in Post-Training Quantization (PTQTP). A full-precision weight matrix W is decomposed into a linear superposition of dual ternary trit-planes modulated by row-wise scalar columns: W ≈ ∑_k s_k ⊙ P_k. Here, P_k ∈ {-1, 0, 1} represents the discrete topology, while s_k ∈ ℝ^m represents the continuous magnitude. The inference equation becomes y = x W^T = ∑_k s_k ⊙ (x P_k^T). In standard linear-algebraic terms, the "trit-plane decomposition" is PTQTP's naming for a structured rank-D approximation of W as a sum of D ternary planes, each modulated by a row-wise scale vector — i.e., a low-rank factorization whose geometric "plane" language denotes exactly this factorization, not a novel algebraic structure. This formulation enables multiplication-free additive inference. The term (x P_k^T) is computed via add/subtract/skip trees, and the scaling s_k is applied via bitwise shifts if s_k is a power of two. A distinct mechanism operates below the ternary entropy floor: Tiled Bit Networks (TBN) exploit structural sharing to achieve sub-1-bit *effective storage*, which is a compression-ratio metric rather than a representation in the ternary {-1, 0, +1} domain. TBN's weights are strictly binary: a single small learnable tile t ∈ {±1}^k×m is trained per layer (1 bit per stored entry) and reused R times to tile the full layer, W_l = tile_expand(t_l), where the logical matrix spans R×k×m entries. Each stored bit is thus amortized across R logical weights, yielding 1/R bits per weight — 8x compression below binary networks corresponds to a reuse factor R = 8, i.e., 0.125 effective bits per weight. Because the ternary domain carries an entropy floor of log_2(3) ≈ 1.585 bits per unique value, sub-1-bit storage is information-theoretically impossible without such redundancy; TBN's inference datapath is consequently a binary expansion/broadcast path, not a ternary add/subtract/skip path. 3/6