For your reference, we have uploaded AGEP v3.14—which is scheduled for peer-review submission to JMLR next month—to Zenodo.

https://doi.org/10.5281/zenodo.22716189

Rapid Rigor Upgrade: Preprint v1.1 is Out! (Complete Proof of Theorem 4.1 & Scheme Correction)

Hi everyone,

In the spirit of rigorous, open-source AI safety research, I want to share a rapid and exciting upgrade to our AGEP preprint on Zenodo, moving from **v1 to v1.1**.

A sharp commenter recently pointed out that in our $2 \times 2$ toy model, the determinantal ideal $I = \langle xy - zw \rangle$ is prime (since $xy - zw$ is irreducible in a UFD), which mathematically makes the coordinate ring an integral domain. Thus, the rank-1 collapse subscheme $S_{\text{collapse}} = \text{Spec}(R/I)$ is technically a **reduced scheme** containing no non-zero nilpotents.

We have addressed this feedback immediately! Describing the PoC model as containing nilpotents in v1 was indeed a technical mischaracterization, and we have fully corrected the wording in **v1.1**.

However, this feedback has actually provided a beautiful mathematical bridge to our upcoming **Work Package 1 (WP1)**:
- In real-world deep learning loss landscapes, we *do* encounter true non-reduced structures (learning plateaus with severe flatness).
- To model these plateaus, we must extend our determinantal ideals to non-reduced forms, such as $I_{\text{non-reduced}} = \langle (xy - zw)^2 \rangle$ (representing quadratic flatness near the singularity) or $I_{\text{non-reduced}} = \langle xy - zw, x^2 \rangle$ (representing isolated nilpotent "spikes" when query weights completely die). This non-reduced framework will be the core of our v2.

#### 🎓 Presenting the Complete Proof of Theorem 4.1 (Appendix)
To satisfy the rigorous standards of both algebraic geometers and mathematical statisticians, we have added a comprehensive, **step-by-step mathematical proof of Theorem 4.1** in the Appendix of v1.1.

This 4-page mathematical appendice showcases the precise calculus behind how:
1. The monoidal blow-up at the origin of $\mathbb{A}^4$ pulls back and factors the singular cone into a smooth exceptional divisor $E$ and a resolved hypersurface.
2. The local Kullback-Leibler (KL) divergence and volume Jacobian are simultaneously monomialized into $K(u) = u_1^4 (u_2')^2$ and $|g'(u)| = u_1^3$.
3. The Real Log Canonical Threshold (RLCT) is algebraically derived as $\lambda = 0.5$, bounding the local bracketing entropy to $\log N_{[]} \le C \log(1/\varepsilon)$.
4. Dudley's bracketing entropy integral converges to a finite value as the scale parameter $\delta \to 0$ via a rigorous change of variables ($t = \sqrt{\log(1/\varepsilon)}$) and integration by parts (utilizing Mill's ratio bound for the Gaussian tail).

#### 🐎 Speed & Capital Efficiency
The transition from receiving technical feedback to compiling a revised 17-page mathematical draft with a complete proof took us **less than 24 hours**.

This is the power of the **AI-Co-PI paradigm**. By leveraging highly-synchronized interactive AI pipelines, we can bypass the slow administrative overhead of traditional academic research and deliver world-class mathematical safety foundations at lightning speed.

The revised preprint v1.1 is now available for download on Zenodo. We hope you enjoy reading the beauty of resolved singularities and uniform weak convergence.

As always, we are eager to hear your thoughts, comments, and questions. Let's keep the momentum going!

Best regards,
Hideki Ishiyama
PI, AGEP Lab

Quick Update: Correcting a Mathematical Term in v1 (Reduced vs. Non-reduced Schemes) & The Road to v2

I want to address a very sharp and helpful feedback I received regarding the algebraic properties of our $2 \times 2$ linear attention singularity model in the pre-print.

In the draft, I described the rank-1 collapse subscheme $S_{\text{collapse}} = \text{Spec}(\mathbb{R}[x, y, z, w] / \langle xy - z w \rangle)$ as a "non-reduced" scheme containing nilpotent elements.

As a math-rigor check, the commenter is 100% correct: since $xy - zw$ is an irreducible polynomial in a UFD, the generated ideal $I = \langle xy - zw \rangle$ is prime. Therefore, the coordinate ring is an integral domain, making the scheme reduced (meaning it contains no nilpotents). Describing the $2 \times 2$ toy model as non-reduced was a technical misnomer on my part, and I will correct this wording in the next revision (v2) on Zenodo.

Does this affect the core claims of the AGEP Theorem? Absolutely not. The core mathematical proof of Theorem 4.1 (the algebraic restoration of the Donsker property) and our PyTorch simulation results remain completely untouched and robust. The monoidal blow-up still successfully regularizes the KL divergence and the Jacobian into normal crossing monomials, and the Dudley entropy integral still beautifully converges on the exceptional divisor.

Why this feedback actually accelerates our roadmap: This correction does not weaken our theory; rather, it provides the perfect mathematical bridge to our next generalized phase.

In real-world deep learning, we do encounter true non-reduced structures. The severe "learning plateaus" (flat regions where gradients collapse) are best modeled as nilpotents. As suggested, by extending our ideal to non-reduced forms such as $I = \langle (xy - zw)^2 \rangle$ (representing quadratic flatness near the singularity) or $I = \langle xy - zw, x^2 \rangle$ (representing isolated nilpotent "spikes" when certain weights die), we can mathematically describe the dynamic of SGD getting stuck in plateaus.

I am extremely grateful for this high-level feedback. It has already sharpened our focus for Work Package 1 (WP1), and we are now formulating the "Non-Reduced AGEP Theorem" for v2 to directly capture these flat plateaus.

Thank you for watching and supporting our journey! I will keep you posted as we refine the draft.

Thank you for the flag! Yes, Pangram is 100% correct in detecting AI assistance in the prose, and this is actually by design. 

As an independent researcher without the backing of a major US academic institution or a native English-speaking PR team, I utilized my customized LLM pipeline (Gemini Notebook) as an interactive "Co-PI" to draft and polish the English prose of this proposal to meet international standards. This is explicitly disclosed and detailed in "Part 5: The 'AI-Co-PI' Research Paradigm" of this proposal.

I drive 100% of the mathematical vision, topological intuition, and hands-on PyTorch coding. However, delegating the heavy lifting of English translation and formatting to AI allowed me to produce a citable 14-page LaTeX pre-print and submit this grant in less than a week. 

This flag actually proves the exact efficiency of the human-AI co-working model that I am proposing to fund. I invite reviewers to look past the "perfect" AI-assisted prose and evaluate the core mathematical novelty of the AGEP Theorem and the verified PyTorch simulation data hosted on my Zenodo DOI.