Classical Factorization Barriers
Three independent barrier results for classical integer factorization of RSA semiprimes $N = pq$, developed as structured mathematical investigations.
Status and Transparency
These documents are public research notes / exploratory manuscript drafts. They are not peer-reviewed journal publications. They should be read as structured mathematical investigations and proposed barrier formulations, not as established theorems accepted by the research community. Some statements are explicitly model-relative or conjectural.
This project was developed through extended iterative dialogue with large language model systems, primarily Anthropic Claude and OpenAI ChatGPT. The repository owner acted as human orchestrator: setting the research direction, proposing and refining the problem framing, steering the investigation, and curating the resulting manuscripts. The mathematical content — theorem statements, proof sketches, counterexamples, barrier formulations, and manuscript language — was generated and refined through this human–AI collaboration. The AI systems are not listed as authors; they are tools used in the production of these notes.
Readers should independently verify all mathematical claims before citing or relying on them. Corrections, objections, and independent verification are welcome.
Paper A — The Joux–Buchmann Bridge Revisited
Revised account of why the Joux/BGJT quasi-polynomial strategy for discrete logarithms does not transfer to real quadratic regulators. Proves a rigorous Frobenius obstruction (Theorem T1: no public residue-level Frobenius substitute exists without factoring data) and a structural tower obstruction (Theorem T2: bounded-degree towers cannot create Joux-style recursive depth). Continued-fraction loophole analysis shows graph-only cycle selection and dynamic principality on the thin split family both collapse to static principality. No sub-$L[1/2]$ mechanism found.
Paper B — Compression Barrier: Pell Trace
Analyzes the regulator–factorization bridge via Pell trace residues $t_{mN} \bmod N$. Proves that computing the trace nontrivially is factoring-equivalent (Lemma 5.1). Classifies four natural routes (representation-theoretic, Kloosterman, multilevel, ray-class), each collapsing to the same CRT wall. States the OP4/H3 compression barrier conjecture.
Paper C — CRT Walls: Product-Functorial Isomorphism Encodings
Proves that semiprime factorization cannot be encoded as a small product-functorial group or matrix-space isomorphism problem suitable for Babai’s quasi-polynomial algorithm. Establishes a dichotomy: every such encoding is either CRT-symmetric or already generates a factoring certificate. Studies the genuine boundary in singular matrix spaces.
Authorship and Responsibility
These notes are published by the repository owner as the human orchestrator and curator of the project. Because the work was substantially AI-assisted and has not undergone independent expert verification, no claim is made that the results have the same status as peer-reviewed mathematical research. Corrections, objections, and independent verification are welcome.
Citation
If referring to this repository, please cite it as an AI-assisted exploratory research project rather than as a conventional peer-reviewed publication.
Suggested informal citation:
Faktorisierung 1: AI-assisted exploratory research notes on CRT barriers in classical factorization, public GitHub repository, 2026.
License
Released under CC BY 4.0.