A Bures-type gradient flow with entropic regularisation: global existence in every dimension, convergence for d ≤ 5, and non-uniqueness of stationary states at d = 10,000

This paper studies a mathematical gradient flow on full-rank density matrices with an entropic term of weight γ. It proves that solutions exist for all time in every dimension d ≥ 2, and that every full-rank stationary state has a simple coaxial form. For d = 2, 3, 4 and 5 and every fixed γ it gives a proof that the stationary state is unique, strictly stable, and reached from every full-rank starting state. The one-variable inequalities behind that proof were settled by exact rational computation that two separate programs reproduced. New in this version: at d = 10,000 and γ = 0.299213 the coaxial stationary equation has at least three distinct solutions, certified by exact rational interval arithmetic and the intermediate value theorem. This disproves a uniqueness conjecture for larger d that was stated in an earlier, unposted draft, and the correction is recorded in the paper. What is not established: whether these are all the stationary states, whether they are stable in every direction, and where multiplicity first appears. The written proof chain for d ≤ 5 has not been checked line by line by anyone independent of the author and his AI collaborators. The flow is a mathematical model; no physical realisation is claimed. This record has not been peer reviewed. Files: the paper (PDF and LaTeX source), three verification scripts, their result files, and SHA256SUMS.txt. Each script prints its verdicts and writes a JSON file; see the Reproducibility section of the paper. The checksums show the files are the ones described; they do not show who wrote them. How AI was used in this work This research was led by one human, G. E. Nerik (Gerald Hopson). He proposed the hypothesis, set the rules for how it would be tested, chose what to include, and wrote or approved every statement of his own point of view. Several AI systems worked under his direction. Each is named below with what it did, because how the work was done is part of what a reader needs in order to judge it. • Claude (Anthropic) acted as the coordinating AI. It wrote and ran the registered computations, kept the numbered record of results (including failures), independently re-ran checks that other AIs reported, wrote one of the three verification scripts, and drafted the theorem paper and these summaries from the author's direction. • ChatGPT (OpenAI) acted as the independent reviewer. It reviewed selected proof arguments, audited and re-ran the exact-arithmetic certificates of the theorem paper, recomputed the local certificates of the registered study with separate code, and reviewed each draft. It wrote two of the three verification scripts, re-ran the scripts and obtained identical output, and wrote parts of the general-reader narrative. The complete written proof chain has not been checked line by line by anyone independent of the author and his AI collaborators, and the full registered study was not re-run. The review was done in more than one separate conversation. • Gemini (Google), in two forms (a Deep Think model, called Argon 4 in the lab, and a Pro model, called Silas), was used in the earliest research. Argon 4 wrote early test scripts, which the author ran on his own computer, since that chat could not run code. According to the author, Silas helped with breadth and transcription. Some of the earliest claims about the two- and three-dimensional cases came from this stage. They were not accepted as they stood: Claude and ChatGPT re-derived and re-checked them independently on 29 to 30 October. According to the author, Gemini was set aside because it took more checking than it saved. None of its work is in the final registered versions. What the AIs did not do. No AI is listed as an author. An AI review is not peer review, and no independent human has reviewed this work. The AIs can be wrong, and one of them was: an automated numerical checker, one of the AI-written tools, once reported success after mishandling invalid error information (a missing error value was treated as zero, so an invalid state was accepted). That result was rejected, the original numerical failure stayed on the record, and the mistake is recorded, not removed. The author is responsible for the content, including its limits. How we tried to keep it honest. Claims were registered before the main runs. Key checks were re-run by a second party where possible. Each finding states what was checked and what was not. Where the record shows a rejected or failed result, it stays in the record. The hardest tests were assigned to "subcontractors": temporary ChatGPT conversations with no memory of the project. These included the dimension-5 tests and the other script tests. By the documented procedure, each was given a written task specification intended to keep it from seeing the coordinating AI's conclusions. The full histories of those conversations, and their independence, have not been independently authenticated. The record also contains one unresolved item from this method: one result package was never received and the origin of one report copy is not settled, so the method is not claimed to be flawless.