The clash over “Corollary 3.12” is not a simple disagreement over a calculation error. Instead, it is a profound conflict between two fundamentally different mathematical visions regarding what the true identity of a “number” actually is.
Here is a deeper look into the core of this dispute.
1. The Critics’ Stance (Prof. Scholze et al.): “A Logical Collapse”
After rigorously breaking down this specific section of the paper, Professor Peter Scholze and his colleagues arrived at a troubling conclusion:
“Mochizuki claims to move numbers across different mathematical universes to derive a brand-new inequality. However, when you translate that process of movement into strict, conventional mathematical language, he is actually identifying the numbers in the new universe as being completely identical to the ones in the original universe.”
Why is this a fatal flaw from their perspective? If you are simply sliding the original numbers into a new setting without changing their relationships, no new information or constraints are created.
In the critics’ words, “In the end, this simplifies to something like $a \le a$ (an obvious statement that $a$ is less than or equal to itself), just wrapped in incredibly complex machinery. This does not provide the powerful new inequality needed to solve the ABC Conjecture.” They referred to this as a total loss of meaningful content, or a “logical collapse.”
2. Prof. Mochizuki’s Counterargument: “Do Not Measure It by Old Rules”
In response, Professor Shinichi Mochizuki and his supporters argue that this criticism is a perfect example of trying to view IUT theory through the lens of traditional mathematics.
The core of Mochizuki’s argument is that when a number is sent to another universe, the core substance of the number itself (the “capsule”) remains unchanged, but the way addition and multiplication intertwine around it is intentionally blurred or altered to re-wrap the capsule.
- The Critics’ View: “Since the capsule (the content of the number) is identical, nothing has changed—therefore, it is mathematically meaningless.”
- Mochizuki’s View: “Even if the capsule is the same, the wrapping (the structure) was altered during the transfer. When we bring it back to the original universe, that journey leaves a measurable ‘distortion’ in the equation, which yields the proof.”
In short, from Mochizuki’s perspective, the critics are taking a newly constructed, multi-dimensional structure and forcing it down (projecting it) into a flat, traditional mathematical framework. As a result, they misinterpret it as being redundant.
3. Why It Remains a Deadlock
In normal mathematics, peer review progresses through a straightforward dialogue:
- Reviewer: “Does this step follow from established Theorem A?”
- Author: “Yes, because of the following logic…”
However, regarding Corollary 3.12, the dialogue breaks down completely:
- Critics: “According to standard definitions, this step forces these two things to be strictly equal, which makes the whole thing trivial.”
- Mochizuki’s Side: “No, you cannot use the traditional concept of ‘equality’ there. It must be viewed through this entirely new type of relationship that IUT defines.”
Because the very definitions of the words and the underlying logical foundations do not align, both sides—composed of world-class mathematical minds—sincerely believe that the other side is the one making a logical error. This is why the debate remains a complete parallel line, unresolved to this day.