The Hybrid Group ⊕_γ over ℝ and 𝔽_p: Face Operations, Transport Rigidity, Gradient-free Learning, and a Fully-Resident Software Protection Loader

Abstract

This work develops a self-contained algebraic framework based on the hybrid group operator a ⊕_γ b = a + b + γ ab, defined simultaneously on real vector spaces and prime fields. We establish the face-operation calculus (M, A, D, E faces), prove the dual-derivative power theorem which gives an exact additive-to-multiplicative homomorphism across characteristics, and present two rigidity lemmas that characterize when a one-dimensional operation is forced to be ⊕_γ. A single-query recovery proposition shows that parameter hiding is ineffective against an operation oracle but can be effective against a decision-only interface, which forms the design principle of the accompanying software loader. The discrete logarithm problem on the hybrid group is shown to be exactly equivalent to the classical discrete logarithm in 𝔽_p^×, providing an honest security boundary. We then introduce Algebraic Residual Iteration (ARI), a gradient-free, closed-form learning algorithm that achieves millisecond convergence with multiplicative pyramid expansions; DeepTower is included as a specialisation using the σ_λ spectrum. The SAGA byte-level aggregation provides integrity folding for code protection. Finally, we specify HGP-Loader v5 (payload decryption mode) and v6 (fully-resident interpretation mode), where business logic never materialises as native instructions. Validation includes 16/16 tests (correctness, avalanche, RAW-face non-collapse, instance diversity, static residency) and performance measurements showing ~159% end-to-end overhead—dramatically lower than typical VM-based protectors. A reverse-engineering case study confirms that while algebraic parameters can be recovered, the key itself remains protected by the data-dependent walk and verdict-as-decryption architecture.