The E$\Delta$-MHC-Geo Transformer: Adaptive Geodesic Operations with Guaranteed Orthogonality

Abstract

We present the E$\Delta$-MHC-Geo Transformer, a novel architecture that unifies Manifold-Constrained Hyper-Connections (mHC), Deep Delta Learning (DDL), and the Cayley transform to obtain input-adaptive, unconditionally orthogonal residual connections. Unlike DDL, whose Householder operator is orthogonal only at $\beta \in \{0,2\}$, our Data-Dependent Cayley rotation $Q(x)=(I+(\beta/2)A(x))^{-1}(I-(\beta/2)A(x))$ preserves orthogonality for all $\beta$ and all inputs. To handle negation, an eigenvalue $-1$ case that Cayley provably excludes, we introduce the E$\Delta$-MHC-Geo Hybrid, which combines Cayley rotation with Householder reflection via a learned operator-selection gate $X'=\gamma(X)Q(X)X+(1-\gamma(X))H_2(X)X$. A midpoint-collapse regularizer, $4\gamma(1-\gamma)$, encourages boundary gate decisions, where each selected component is orthogonal. In matched-parameter comparisons, with approximately 1.79M parameters per model and mean +/- standard deviation over 3 seeds, against four baselines including the concurrent JPmHC, E$\Delta$-MHC-Geo achieves the best long-horizon stability, 1.9x over JPmHC and 3.8x over GPT; the best near-$\pi$ rotation loss, 4.5x over JPmHC on single-plane; strong norm preservation, with 0.001 mean deviation; and 0.96 negation cosine alignment in a diagnostic reflection probe, all with 33% fewer layers. While JPmHC's wider representation excels on pure rotation, its finite Cayley residual mixer excludes an exact $\lambda=-1$ operator and has no reflection branch, motivating our hybrid approach for accessing both connected components of $O(n)$.