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Oct 26 08:44
The selection and verification of research methods have been entrusted to Codex. At one point, a new exploration was paused to integrate the research contents of past materials that can be referenced, along with the background of important decisions, before resuming. A new decisive factor that demonstrates the Riemann Hypothesis as true has not yet been obtained.
It felt like I was doing something that had been published in the past again. Moving forward without sufficiently building on the history felt uncomfortable, like studying without a textbook.
After confirming the differences compared to past achievements, we will select the next candidates and verify them mathematically.
Oct 8 16:01
Although further research was conducted on the finite Weil, the two conditions necessary for proving the Riemann Hypothesis could not be resolved, and the matter was temporarily put on hold. Subsequently, I reconsidered fundamental issues, such as the structure of natural numbers, the nature of information, and what it would mean if the world were like a computer game. Ultimately, I arrived at the question of whether the Riemann Hypothesis is even a problem that can be proven with the current axioms of mathematics.
I had a feeling of going around in the same place for a long time. No matter how beautifully I can explain a structure, I cannot say for certain that there are absolutely no exceptions. It felt similar to how, even after observing a cat eating every day for years, I cannot prove it will eat tomorrow too. If the environment changes, the results will change, and perhaps there is no rule that says it must eat every day in the first place. Can we distinguish between "not finding a rule that prohibits exceptions" and "such a rule does not exist"? If there are premises unknown to us or things that cannot be captured within the current framework of mathematics, then no matter how much we continue to think within that framework, it seems unlikely that we will find an answer. From that sense of discomfort, I arrived at Gödel's Incompleteness Theorem.
Investigate the possibility that the Riemann Hypothesis is independent of the current axiomatic system, examining its relationship with known incompleteness theorems and computability, and determine whether there is a mathematical pathway for verification.
Oct 7 11:32
The structural theory of "displacement" did not reveal a direct bridge to the Riemann Hypothesis, leading us back to a frontal breakthrough with CMP and ES. Subsequently, I demonstrated a finite proposition that states if a strict even-odd order holds for actual finite Weil matrices, the simplicity of the lowest even state automatically follows. The main line toward the Riemann Hypothesis has been compressed to two conditions: "capture by the theta kernel" and "strict even-odd order" on the same co-finite sequence. The results have been reflected in the public research log, but the Riemann Hypothesis remains unresolved.
While searching for shortcuts, I had begun to focus on individual lemmas and error verification, so I returned to prioritizing genuinely reducing the conditions necessary for proving the Riemann Hypothesis, rather than the research subject itself.
Using the exchange relations that gave rise to CORE-S, we will verify with Codex whether we can compress the remaining conditions of "capture to the theta core" and "strict parity order" into a single mathematical condition.
Oct 2 22:51
In the previous investigation titled "Do the two candidates originate from the same blueprint?", we were unable to demonstrate the connection. Thus, we returned to the process of how zeta is created from integers and the history of research, reconsidering our questions through my papers and the metaphor of smoke and spirals, exploring the underlying mechanisms behind the visible results using AI. We also examined proposals for providing an appropriate "measuring method for length" in the mechanism of choosing zeros, but we could not construct the necessary measuring methods, and we have not reached a proof.
It seemed as if we repeatedly returned to the same explanation and were attempting to break through head-on with detailed calculations. Additionally, instead of increasing the confirmation items, I wanted to find the minimal principle that states, "With this method of construction, exceptions cannot be created."
Oct 1 08:10
The simplicity and evenness of the Weil minimum state, as well as its agreement with the prolate approximation, have not yet been proven. Therefore, I further abstracted the problem and examined whether the completed zeta structure has a "mirror" that reflects left and right, and whether there is a mechanism that does not allow anything outside the critical line. However, it became clear that deriving this mechanism unconditionally leads back to the same barrier as the known Riemann Hypothesis. Furthermore, it became evident that even if another complete picture with the correct properties could be created, it does not necessarily mean it is the same as the actual zeta.
Instead of cutting parts and approaching infinity, I believed that it was closer to the essence to view the structure from the completed form itself. However, upon verification, I realized that what remains in the end is not "the existence of a good object" but rather "the fact that it is the same object as the original." Therefore, I changed my perspective on comparison.
Directly compare whether the actual finite Weil minimum state and the prolate approximation stem from the same equations or generation rules, rather than how close they are.
Sep 30 23:37
Since the last time, instead of simply following the equations, I have abstracted the structure multiple times by substituting it with elements like the resonance in the acoustic room, the collision between bullets, and the retrieval of narrative foreshadowing. Within this process, I reexamined the idea that "primes, Gamma, and theta are not separate forces, but instead emerge from the same singular zeta structure as a result of trying to satisfy all consistencies." I also confirmed, through known characterization theorems, that when certain conditions are simultaneously met, the zeta function is almost uniquely determined. Furthermore, by reviewing existing research, I realized that the finite test functions and the operator theory with fixed ranges I was trying to create could rely on established theorems. Consequently, I narrowed my remaining research focus to the "simplicity and parity of finite ground states" and the "comparison between actual ground states and the prolate approximation."
Rather than continuing to pile up detailed lemmas, I believed it would be much more productive to revisit "why this mysterious structure must necessarily exist" from first principles. I did not adopt intuitions such as the golden ratio or resonance as answers; instead, I verified whether they could be derived from actual equations and adhered strictly to the principle of discarding anything without a solid basis.
The parts that are closed under existing theorems will not be recalculated. Instead, a direct comparison between the actual ground state and the prolate approximation, as well as two independent parallel verifications of the simplicity and parity of the ground state, will be conducted.
Sep 30 11:52
Continued autonomous exploration has not yet proven the Riemann Hypothesis itself, but has concretely specified the gaps necessary for proof, comparing "finite Weil minimum states with real zeros" and "prolate approximations converging to Xi." Furthermore, when cutting the candidate groups that correspond globally to the same points with finite support, the boundary value becomes the leading scoring item. It has been proven that in the ranges of k and k'', the minimal direction converges to k, while a unique selection for the entire system, including Fourier projections, has yet to be obtained.
Instead of directly connecting the two candidates, I shifted my perspective towards the possibility of viewing the same thing from different angles and the intuition that nature might leave behind rational forms through the simple repetition of rules, leading to a sequential focus on common optimization, speed and history, and hierarchical selection at the boundaries.
We will investigate whether the selection of k visible only at the boundary remains in the entire system that includes Fourier projections, focusing on the scale around N∼aY that resolves the boundary layer.
Sep 30 01:03
I allowed AI to autonomously search for a proof of the Riemann Hypothesis, repeatedly developing methods and refuting them. Although it has not yet reached a proof, through using the structure of prime numbers to reduce the problem, I have narrowed it down to one critical point: "Why do positive and negative elements cancel each other out so precisely?"
Many of the methods I have tried so far have merely transformed the form, ultimately returning to the same level of difficulty as the Riemann Hypothesis. Thus, I shifted my focus from re-expressing equations to searching for the very structure where cancellations occur.
We will verify whether the mechanism by which positive and negative structures nearly cancel each other out can be explained within the hierarchical structure created by prime numbers.
9 2026 30
The Riemann Hypothesis project was started.
Creators who are quickly testing what can be done with AI.
