[Cryptanalysis] Theoretical reduction of logical depth (T-count) for Shor's algorithm via Z/6Z Topological Superselection (with exact MPDO simulation)

Hi everyone,

I am an independent researcher and I recently published a theoretical paper focusing on quantum cryptanalysis, specifically targeting the resource estimation and initialization phase of arithmetic search algorithms like Shor's.

Since this sub focuses heavily on modern cryptography and the ongoing transition to PQC, I wanted to share this work regarding the logical feasibility of FTQC attacks on current asymmetric protocols (RSA/ECC). I am looking for rigorous feedback from the cryptanalysis community.

**The Cryptanalytic Impact (T-count reduction):** The primary bottleneck for executing Shor's algorithm on early FTQC architectures is the massive non-Clifford gate volume (T-count) required for magic state distillation during the initialization of the uniform superposition.

My work proposes a topological superselection based on the modular ring **ℤ/6ℤ**. By confining the probability amplitude strictly to the resonant channels **1 (mod 6)** and **5 (mod 6)** (since all primes > 3 fall here), we analytically purge ≈ 66.6% of the sterile Hilbert space before the oracle even runs. This allows the adder architecture to inject Toffoli and T gates exclusively over the residual active entropy, massively contracting the logical depth and relaxing the Surface Code error correction thresholds required to break RSA-2048.

**The Physical Feasibility (Evading Thermalization):** To prove this state can actually survive in a physical register without melting into thermal noise, I formulated a Parent Lindbladian and simulated it using an exact Matrix Product Density Operator (MPDO) tensor network up to **N=60** qubits. The system enters a Non-Ergodic Extended (NEE) phase, strictly obeying an Area Law for bipartite Rényi entropy (saturating at ≈ 1.65 bits).

**Open Source Links:** I invite you to review the math and the simulation code:

* 📄 **Full Paper (Zenodo DOI):** [https://zenodo.org/records/19354011](https://zenodo.org/records/19354011)
* 💻 **GitHub (Python MPDO simulation code):** [https://github.com/NachoPeinador/Phase-Pi-Quantum-Prior](https://github.com/NachoPeinador/Phase-Pi-Quantum-Prior)

I would greatly appreciate any critiques, especially regarding the assumptions on T-count overhead and logical spacetime volume.

**Required AI / LLM Disclosure (Rule 8):** In accordance with subreddit rules and the declaration included in my manuscript: I used LLMs (specifically Gemini) as methodological assistants during this research. Their use was strictly limited to:

1. Adversarial review (red teaming) to stress-test the thermodynamic viability, which motivated the MPDO tensor network approach.
2. Code refactoring assistance for the Python simulation (memory management for the 1296x1296 transfer matrices).
3. Formatting and grammar refinement of the English text. The core mathematical breakthroughs, the ℤ/6ℤ topology, the analytic phase derivation, and the algorithmic conclusions are entirely my original work.