Stateless Quantum-Resistant Zero-Knowledge Proofs via Topological Invariants of Finite-Field Sierpiński Gaskets

Modern zero-knowledge proof (ZKP) systems face critical bottlenecks: substantial proving-key memory footprints, computationally intensive polynomial commitments via pairings or FFTs, and persistent state-retention constraints. These limitations severely hinder deployment on resource-constrained embedded nodes, micro-aerial vehicles (MAVs), and air-gapped cryptographic environments. In this paper, we propose a stateless, quantum-resistant zero-knowledge proof framework founded on the topological invariants of discrete Sierpiński gaskets mapped onto prime finite fields F_q. By discretizing classical contractive iterated function systems (IFS) through modular contraction operators coupled with non-linear permutation foldings, our construction guarantees deterministic convergence toward an invariant fractal attractor A ⊂ F_q^d while inducing a topologically inaccessible, measure-zero complement subspace (void) F = F_q^d \ A. We construct an interactive Σ-protocol wherein the private witness is formulated as a non-linear spatiotemporal sequence of contraction choices, and validity is established via a geometric trapdoor: any forged execution path deflects into the void F with probability 1 - |A|/q^d. The proposed protocol operates strictly with O(1) RAM utilization, eliminates common reference strings (CRS), and executes in microsecond latency (4.18 μs on x86_64 CPUs, 2.84 ms on 168 MHz ARM Cortex-M4 microcontrollers). Furthermore, because the non-commutative fractal dynamics destroy cyclic subgroup periodicity, the protocol provides innate resilience against Shor-type period-finding and Quantum Fourier Transform (QFT) algorithms. The structural foundations and density theorems are formally mechanized and verified in Lean 4 without axioms or sorry stubs.