Discrete Algebraic Geometry in High-Dimensional State Spaces: From Combinatorial Optimization to Post-Quantum Cryptography

This monograph unifies three distinct mathematical breakthroughs under a single foundational architecture: the application of discrete algebraic geometry—specifically, the GF(2^7) Möbius inversion map (the fractional linear group)—to high-dimensional state spaces. Traditional optimization and cryptographic engines bleed computational energy through continuous drift or iterative multi-round hashing. The architectures presented in this unified thesis abandon these approximations in favor of a single-evaluation, discrete algebraic bijection, mapping perfectly into L1 cache memory to achieve zero-friction, constant-time execution (SIMT). This archive contains three strictly compartmentalized proofs: Part I (H-CAM v23.0): Applies the GF(2^7) map to Kinetic Routing, solving 100-dimensional NP-hard combinatorial landscapes by dynamically injecting spatial momentum to escape localized minimum gravity wells. Part II (NTT Shuffle): Applies the GF(2^7) map to Cryptographic Side-Channel Shielding, formally verified via pq-verify as a signed permutation group (B_128) that masks execution order in lattice-based Key Encapsulation Mechanisms (ML-KEM) with <1% overhead. Part III (Provenance Concentration): Operates in the domain of Ideal Lattices and SVP Oracles, utilizing exact Bézout invariants and LLL transformation matrices to exponentially reduce the search space of shortest vectors. Rooted in the foundational mandate of "iamweare"—to learn more to displace less computational and universal energy—this architecture proves that strict cryptographic avalanche diffusion translates seamlessly into global algorithmic optimization. Version 2 adds: the APN Optimality Theorem (the GF(2^7) shuffle achieves differential uniformity Δ=2, the theoretical minimum, strictly stronger than the AES S-box at Δ=4); large-prime validation of the Provenance Concentration Theorem at dimension 82 via production fpylll; complete NTT benchmarks (6.4μs per protected transform, 1.35× algorithmic overhead); and Ali 2024 prior art citation.