Beyond a Proven “Defeat”:How the Algorithmic Legitimacy Shift (ALS) and the Ghost Drift Theorem Redefine Responsibility and Legitimacy

In our previous article, Cognitive Legitimacy: A Minimax-Risk Definition of When Algorithms Are More Legitimate Than Humans, we established a stark mathematical fact.

Under structural information constraints (B<JB < JB<J), human judgment (the Human Channel) admits an unavoidable lower bound on minimax risk—no amount of effort, expertise, or care can push the error rate below this floor.

When this condition holds, the locus of legitimacy shifts from human evaluation to algorithmic evaluation.We refer to this phenomenon as the Algorithmic Legitimacy Shift (ALS).

Equations do not lie. Yet understanding what these equations imply for real social systems is more consequential than solving them. This is not a claim that “AI is smarter than humans.” Rather, it signals a structural redefinition of what responsibility, trust, and correctness mean in domains where ALS occurs.

In this article, we examine three (at least structurally) irreversible changes quietly declared by that inequality—and explain how the Ghost Drift Theorem exposes their deeper implications.

1. Not a “Limit of Ability” but a “Limit of Structure”:

The Invalidation of the Logic of Apology

Until now, when oversights or judgment errors occurred in organizations, we labeled them “human error” and relied on countermeasures like “we should have been more careful” or “we will implement thorough double-checks.”

However, our proof fundamentally denies this approach.

When the range a human can view at once ($B$) is smaller than the total volume of the target ($J$), an “oversight” is not a probabilistic accident, but a structural inevitability (Lower Bound).

There is no room for “carelessness” or “negligence” to enter the equation. The error is physically invisible. Therefore, for a human to bow their head post-incident and claim, “The cause was a lack of effort,” is a logic that does not hold mathematically. It is equivalent to saying, “Next time, I will transcend the laws of physics.”

The first implication Ghost Drift theory presents is this: “Against structural defeat, apologies based on ‘willpower’ are invalid.”

Note: What is denied here is the specific explanation that ‘more effort or attention could have broken the structural lower bound,’ not the act of apology or compensation to the victims itself.

2. Responsibility Engineering: From Post-Hoc to Ex-Ante

If there exists a domain humans inevitably cannot protect, and if it is proven that “an algorithm can approach zero risk (by allocating sufficient sample size and computational resources $m$),” what follows? (Provided, of course, that the definition of $J$ and the assumptions of the data generation process are valid.)

Here, the definition of “Responsibility” changes.

Traditional responsibility was about explaining “why you made a mistake” after the accident occurred (Post-Hoc). However, responsibility in the new definition resides in the selection made beforehand (Ex-Ante Selection).

In other words, at the moment you choose “Which channel to use: Human (High Risk) or Algorithm (Low Risk),” the bulk of the responsibility is already determined.

If a mathematically lower-risk channel (AI) exists, yet one persists in adopting human judgment and an accident ensues, it is not the “operator’s mistake.” It can potentially constitute “design negligence” on the part of the architect (or management) who selected that channel.

“Walking a dangerous route while knowing a safer one exists.” This fact is what will be questioned in the coming era. This is the core of “Responsibility Engineering.”

3. The Drift of Cognitive Legitimacy (Ghost Drift)

When we use a calculator, few would argue, “It would be more correct if I calculated it myself.” In complex calculations, we universally admit that the calculator’s answer is more “Legitimate” than the human’s.

This theorem demonstrates that the same phenomenon — this “trust in calculators” — is occurring even in the advanced domains of “Judgment” and “Cognition.”

In the region of $B < J$, the moment algorithmic risk falls below the human lower bound, social trust (Legitimacy) quietly but surely migrates from the biological human to the mathematical structure.

This is Ghost Drift.

This is not a Sci-Fi fear of “AI domination.” Just as water flows from high to low, it is a natural phenomenon where trust physically flows out from the “high-risk vessel (Human)” to the “low-risk vessel (Structure).”

Conclusion: What Should We Do?

This proof is not a “recommendation for retirement” to humanity. Rather, it presses humans for a “Redefinition of Roles.”

Algorithms excel at outputting the optimal solution within a given $J$ (the total set to be confirmed). However, they cannot decide “what should be confirmed in the first place (the definition of $J$).”

Admitting defeat in the battle of $B < J$ is not shameful. It is a necessary rite of passage for humans to entrust “full enumeration” and “pattern recognition” — tasks we are weak at — to structure, and to migrate to the upper layers where we truly belong: “Setting the Questions” and “Designing Risk Tolerance.”

That equation is not despair. It is the first clue for us to break free from the spell of “mentalism” and design a mathematically sincere social system.