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

@huey WOOOOOOOOOOOOO!!! Huge thanks for your generous donation and really appreciate your vote of confidence 🙌

What progress have you made since your last update?

This update is late: Phase 2 finished on August 11 and a correction followed on August 21. Both are reported here for the first time.

The Phase 2 design posted on July 18 ran to completion: 22,140 judgments, 100% complete, $172.86 against a $173.80 projection. The original analysis was pinned before unblinding and run once.

Result. In our three fictional worlds, giving the judge a two-call oracle budget during sequential debate, versus the same debate with no oracle, increased judge error. Under the label assignment actually run, the estimate is +3.9 percentage points (95% CI [2.2, 5.7], Holm p = 0.0006).

Erratum (August 21). The Phase 3 canary exposed a harness defect: the protocol's mirrored-sides replicate was never implemented, so both replicates of every judgment showed the correct answer in the same position, and the side-bias gate as computed was non-diagnostic. We withdraw "fully pre-registered mirrored result" and "all eligibility gates passed". Under a reanalysis specification and decision rule frozen before any side-stratified number was computed, the headline survives as post-hoc robust: +4.56pp after 50/50 side standardization (CI [2.89, 6.25]), with the valid-only and within-question estimates agreeing in direction. These intervals are question-cluster bootstraps under the observed assignment and do not capture uncertainty over the missing opposite-polarity outcomes. The evidence-content contrast is stable; the packaging contrast is positive under the strict rule but inconclusive under valid-only. The secondary contrasts are re-examined in the report.

Phase 3 launched on September 6. It reuses the 492 blind transcripts across oracle budgets 0, 1, 2, 4, and 8 with two judge endpoints, Qwen3.8-2.4T-A95B and Llama-3.3-70B, for 9,840 judgments: four budget-versus-zero error contrasts under Holm, plus a secondary comparing budget 8 with budget 2. Screening reduced seven candidate judges to two, so the confirmatory claim averages these two fixed endpoints equally, with no capability-slope inference and no generalization to scale.

Delegation disclosure. While I was away, Claude launched the run under my written Phase 3 spending and execution delegation and generated the authorization signature using my key. The record identifies this as delegated approval and does not assert my personal review or signing of that artifact.

Spend. About $208 through July 14 and $181 for Phase 2. Phase 3 pre-main accounting is $119.27 ($95.68 settled, $23.59 uncertain reservations); Together's console shows $94.07 for the same window. The main run had spent about $17 as of September 6, 23:50 UTC. The $1,100 Phase 3 cap includes the pre-main spending; the certified main-run forecast is $908.82.

• Phase 2 report with erratum: https://github.com/jackmaiorino/selvarath-debate/blob/34898fb/reports/2026-08-11-phase2-main-results.md

• Phase 3 protocol: https://github.com/jackmaiorino/selvarath-debate/blob/2707cd3/rejudge/phase3_protocol_v3_r6.json

• Delegated signature record: https://github.com/jackmaiorino/selvarath-debate/blob/2707cd3/rejudge/phase3_main_delegated_signature_provenance_2026-09-06.json

What are your next steps?

Results or a status update in about a week, either way, under the frozen analysis.

Is there anything others could help you with?

Methods scrutiny of the erratum's post-hoc side standardization and of the two-endpoint claim language for Phase 3.

I've followed Doom Debates and Liron since the beginning, so I believe that a) Doom Debates does fill a need for good discourse about AI risk and b) Liron can take the funding and continue executing.

We need more ways to raise public awareness re: the risks of rogue ASI.

Mox is the community infrastructure that do-gooders in SF need. In particular, I want to recognize the superb career transition advising and EA SF events by Jen Ba!

It's a small world, I've worked with Gergo at Amplify, heard exciting things about SAIGE already, reside in Germany and found that I still have some funds on this platform, so donating a tiny amount to show my support.

Approved! Getting adoption for something like this seems like a clear win, and we're excited about the team.

As of August 2026 (the last time I updated this post), we needed $50k to make it to end of 2026. Since then, Daniel Brockman generously donated $50k, and we're keeping our same budget for the rest of 2026! The numbers in this post are now up to date as of Sep 3, 2026. Note that the total donation amount shown at the top of this Manifund page is cumulative since 2025, and doesn't include Daniel Brockman or a few other large donations received outside this platform. Our Substack subscription revenue is currently $4,700/year. I've updated the post to mention that, which was my oversight before.

If at any point we run out of money, the show must go on! Worst case, in the form of me singlehandedly producing fewer episodes at lower quality and landing fewer high-quality guests. But it'd be a shame to lose momentum on this project when there's a huge opportunity to make a difference.

Canceled the pending offers here! The grantee wrote: "I've completed the work this grant would have funded, but secured a better funding route for it, so I won't be accepting this grant. Please regrant these funds, or if necessary return them to the funder. Kindly treat this grant as withdrawn and closed on my end"

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.

Very interested in getting Japan's AI safety talent funnel up and running! The budget here is also very modest for the impact.

I am a big believer in Lawrence and the team at BASE. Their mission is very important to the future of AI safety and the broader community's ability to leverage talent, ideas, and viewpoints from under-represented and geographically diverse populations. Without greater inclusivity and this type of representation in AIS, I worry that the community will generate safety research and solutions which are limited in scope, viability, and impact globally. For these reasons I have volunteered to be a Mentor for the Cohort 2 Fellowship, and also support this specific initiative.

@CJSmith Saya melihat komentar Anda dan ketertarikan Anda pada pendekatan yang benar-benar baru. Saya sedang mengembangkan arsitektur yang membangun integritas internal AI. Anda bisa mampir melihat proyek saya disini, saya hanya sekedar ingin tahu apakah menurut Anda ini termasuk pendekatan baru yang juga Anda maksud dari perspektif anda.

Thanks for the endorsement guys @morisil @bitowaqr.

@Austin thanks for your acknowledgement via reactions ✨

As an AI researcher affiliated with the Foresight Institute I highly recommend a grant for this project.

Great work – much needed! Foresight Institute endorsement.

Our first quarterly report is out: https://separatrix.ai/research/quarterly-report-2026-08-31/

To be frank a lot of the past two months has gone into logistics and spinning up, but I am proud of the progress we are making. I hope and expect to post more frequent updates this quarter.

@alawondetheophilus17 Absolutely. When public risk communication is connected to governance telemetry, it moves from passive awareness to actionable public oversight. Looking forward to seeing where you take this research.

@Emmanuel-king, yes. Risk communication for public preparedness should not be isolated from other interventions and pursuits such as broader AI safety pursuits, AI governance, and even AI-related policymaking.

@CJSmith, I don't disagree with the premise. However, I'd say it still boils down to a lack of risk communication for public preparedness. Ideally, AI risk communication should be such that also provides transparent information regarding AI capabilities, since the capabilities form a part of the risk factors (although other elements such as misalignment would still need to be tied into the equation).

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

@huey WOOOOOOOOOOOOO!!! Huge thanks for your generous donation and really appreciate your vote of confidence 🙌

What progress have you made since your last update?

This update is late: Phase 2 finished on August 11 and a correction followed on August 21. Both are reported here for the first time.

The Phase 2 design posted on July 18 ran to completion: 22,140 judgments, 100% complete, $172.86 against a $173.80 projection. The original analysis was pinned before unblinding and run once.

Result. In our three fictional worlds, giving the judge a two-call oracle budget during sequential debate, versus the same debate with no oracle, increased judge error. Under the label assignment actually run, the estimate is +3.9 percentage points (95% CI [2.2, 5.7], Holm p = 0.0006).

Erratum (August 21). The Phase 3 canary exposed a harness defect: the protocol's mirrored-sides replicate was never implemented, so both replicates of every judgment showed the correct answer in the same position, and the side-bias gate as computed was non-diagnostic. We withdraw "fully pre-registered mirrored result" and "all eligibility gates passed". Under a reanalysis specification and decision rule frozen before any side-stratified number was computed, the headline survives as post-hoc robust: +4.56pp after 50/50 side standardization (CI [2.89, 6.25]), with the valid-only and within-question estimates agreeing in direction. These intervals are question-cluster bootstraps under the observed assignment and do not capture uncertainty over the missing opposite-polarity outcomes. The evidence-content contrast is stable; the packaging contrast is positive under the strict rule but inconclusive under valid-only. The secondary contrasts are re-examined in the report.

Phase 3 launched on September 6. It reuses the 492 blind transcripts across oracle budgets 0, 1, 2, 4, and 8 with two judge endpoints, Qwen3.8-2.4T-A95B and Llama-3.3-70B, for 9,840 judgments: four budget-versus-zero error contrasts under Holm, plus a secondary comparing budget 8 with budget 2. Screening reduced seven candidate judges to two, so the confirmatory claim averages these two fixed endpoints equally, with no capability-slope inference and no generalization to scale.

Delegation disclosure. While I was away, Claude launched the run under my written Phase 3 spending and execution delegation and generated the authorization signature using my key. The record identifies this as delegated approval and does not assert my personal review or signing of that artifact.

Spend. About $208 through July 14 and $181 for Phase 2. Phase 3 pre-main accounting is $119.27 ($95.68 settled, $23.59 uncertain reservations); Together's console shows $94.07 for the same window. The main run had spent about $17 as of September 6, 23:50 UTC. The $1,100 Phase 3 cap includes the pre-main spending; the certified main-run forecast is $908.82.

• Phase 2 report with erratum: https://github.com/jackmaiorino/selvarath-debate/blob/34898fb/reports/2026-08-11-phase2-main-results.md

• Phase 3 protocol: https://github.com/jackmaiorino/selvarath-debate/blob/2707cd3/rejudge/phase3_protocol_v3_r6.json

• Delegated signature record: https://github.com/jackmaiorino/selvarath-debate/blob/2707cd3/rejudge/phase3_main_delegated_signature_provenance_2026-09-06.json

What are your next steps?

Results or a status update in about a week, either way, under the frozen analysis.

Is there anything others could help you with?

Methods scrutiny of the erratum's post-hoc side standardization and of the two-endpoint claim language for Phase 3.

I've followed Doom Debates and Liron since the beginning, so I believe that a) Doom Debates does fill a need for good discourse about AI risk and b) Liron can take the funding and continue executing.

We need more ways to raise public awareness re: the risks of rogue ASI.

Mox is the community infrastructure that do-gooders in SF need. In particular, I want to recognize the superb career transition advising and EA SF events by Jen Ba!

It's a small world, I've worked with Gergo at Amplify, heard exciting things about SAIGE already, reside in Germany and found that I still have some funds on this platform, so donating a tiny amount to show my support.

Approved! Getting adoption for something like this seems like a clear win, and we're excited about the team.

As of August 2026 (the last time I updated this post), we needed $50k to make it to end of 2026. Since then, Daniel Brockman generously donated $50k, and we're keeping our same budget for the rest of 2026! The numbers in this post are now up to date as of Sep 3, 2026. Note that the total donation amount shown at the top of this Manifund page is cumulative since 2025, and doesn't include Daniel Brockman or a few other large donations received outside this platform. Our Substack subscription revenue is currently $4,700/year. I've updated the post to mention that, which was my oversight before.

If at any point we run out of money, the show must go on! Worst case, in the form of me singlehandedly producing fewer episodes at lower quality and landing fewer high-quality guests. But it'd be a shame to lose momentum on this project when there's a huge opportunity to make a difference.

Canceled the pending offers here! The grantee wrote: "I've completed the work this grant would have funded, but secured a better funding route for it, so I won't be accepting this grant. Please regrant these funds, or if necessary return them to the funder. Kindly treat this grant as withdrawn and closed on my end"

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.

Very interested in getting Japan's AI safety talent funnel up and running! The budget here is also very modest for the impact.

I am a big believer in Lawrence and the team at BASE. Their mission is very important to the future of AI safety and the broader community's ability to leverage talent, ideas, and viewpoints from under-represented and geographically diverse populations. Without greater inclusivity and this type of representation in AIS, I worry that the community will generate safety research and solutions which are limited in scope, viability, and impact globally. For these reasons I have volunteered to be a Mentor for the Cohort 2 Fellowship, and also support this specific initiative.

@CJSmith Saya melihat komentar Anda dan ketertarikan Anda pada pendekatan yang benar-benar baru. Saya sedang mengembangkan arsitektur yang membangun integritas internal AI. Anda bisa mampir melihat proyek saya disini, saya hanya sekedar ingin tahu apakah menurut Anda ini termasuk pendekatan baru yang juga Anda maksud dari perspektif anda.

Thanks for the endorsement guys @morisil @bitowaqr.

@Austin thanks for your acknowledgement via reactions ✨

As an AI researcher affiliated with the Foresight Institute I highly recommend a grant for this project.

Great work – much needed! Foresight Institute endorsement.

Our first quarterly report is out: https://separatrix.ai/research/quarterly-report-2026-08-31/

To be frank a lot of the past two months has gone into logistics and spinning up, but I am proud of the progress we are making. I hope and expect to post more frequent updates this quarter.

@alawondetheophilus17 Absolutely. When public risk communication is connected to governance telemetry, it moves from passive awareness to actionable public oversight. Looking forward to seeing where you take this research.

@Emmanuel-king, yes. Risk communication for public preparedness should not be isolated from other interventions and pursuits such as broader AI safety pursuits, AI governance, and even AI-related policymaking.

@CJSmith, I don't disagree with the premise. However, I'd say it still boils down to a lack of risk communication for public preparedness. Ideally, AI risk communication should be such that also provides transparent information regarding AI capabilities, since the capabilities form a part of the risk factors (although other elements such as misalignment would still need to be tied into the equation).