Positroid-PQC: A Non-Lattice Cryptosystem based on Amplituhedron Geometry and 1RSB Thermodynamic Hardness

The ongoing standardization of Post-Quantum Cryptography (PQC) by NIST is heavily dominated by lattice-based schemes (e.g., ML-KEM, ML-DSA). This monoculture presents a systemic global risk: if the Learning With Errors (LWE) problem is broken, the entire cryptographic infrastructure collapses. We propose a fundamental departure from mathematical hardness conjectures toward physical thermodynamic security. Positroid-PQC is a non-lattice cryptosystem based on GF(2) cluster algebras and the Overlap Gap Property (OGP). Formal verification via Z3 SMT Solver guarantees a perfect Feistel involution with a 0.000% Bit Error Rate (BER). Empirically, legitimate decryption executes in 1.02 ms, whereas brute-force attackers are forced into the 1RSB (One-Step Replica Symmetry Breaking) regime, suffering a deterministic Out-of-Memory (OOM) hardware collapse in 681.21 ms. For strict artifact evaluation, the source code, Z3 proofs, and reproducibility harnesses are publicly available at https://github.com/MAJBS/Chronos-Dialeteia-Engine.