Cognitive Legitimacy(Algorithmic Legitimacy Shift (ALS)):A Minimax-Risk Definition of When Algorithms Are More Legitimate Than Humans

kanna qed
1月21日
読了時間: 4分

If $B < J$, human judgment retains a non-vanishing minimax error floor. With sufficient $m$, algorithmic minimax risk becomes strictly smaller.

1. What This Proof Establishes (TL;DR)

This work does not claim that "AI is smarter than humans" or "possesses consciousness." It reports the result of a structural comparison between two decision channels within the framework of Statistical Decision Theory.

We have formally established the following theorem (Mathematically Closed):

Human Channel: Under structural information constraints, there exists a non-vanishing lower bound on the error rate, regardless of the care or expertise applied.
Algorithm Channel: By increasing the sample size, the risk can be driven exponentially toward zero.

This is not an "Optimization" problem of preference, but a "Legitimacy" comparison theorem regarding which channel mathematically minimizes risk.

The result presented here reveals a phenomenon in which the locus of legitimacy shifts from human evaluation to algorithmic evaluation under structural constraints.We refer to this phenomenon as the Algorithmic Legitimacy Shift (ALS).In this context, legitimacy is not a normative or rhetorical notion, but is defined strictly in terms of minimax risk, i.e., worst-case error under optimal decision strategies.

2. Problem Setup: Structural Constraints

We define the problem through extremely simple structural constraints.

Human Channel ($\mathsf{Ch}_H$)
- Can inspect at most $B$ items out of total $J$ items ($B < J$).
- Subject to the structural constraint that comprehensive verification is impossible.
Algorithm Channel ($\mathsf{Ch}_A$)
- Mechanically observes all $J$ items.
- Subject to noise, which can be suppressed by increasing the sample size $m$.

Under these conditions, we determine which channel achieves the lower Minimax Risk (Worst-Case Error Rate).

3. Intuition: Why Humans Structurally Fail

Human judgment fails not due to a lack of ability, inattention, or negligence. The failure is rooted in Structure.

As long as $B < J$, no matter how skilled the expert or how advanced the strategy (adaptive sampling), there will always remain $J-B$ unobserved regions.

In decision theory, the "Worst Case (Minimax)" refers precisely to the scenario where a fatal defect exists in the unobserved region. If a bomb exists where one has not looked, accuracy in the observed regions is irrelevant.

This is not a barrier that can be overcome by effort; it is a physical, Information-Theoretic Barrier.

4. Mathematical Claim (Main Theorem)

The following inequalities are based on the full proof (Ver 10.0). We utilize $\mathfrak{R}^\star$ (Minimax Risk) as the risk measure.

Human Limit (Sharp Lower Bound)

The risk of the Human Channel is restricted by the following lower bound. This is a tight, achievable boundary.

$$\mathfrak{R}^\star(\mathsf{Ch}_H) \ge \frac{1 - B/J}{2 - B/J}$$

As long as $B < J$, this value never converges to zero.

Algorithm Convergence (Upper Bound)

Conversely, the risk of the Algorithm Channel is suppressed by sample size $m$ and signal strength (margin) $\Delta$.

$$\mathfrak{R}^\star(\mathsf{Ch}_A) \le J \exp\left(-\frac{1}{2} m \Delta^2\right)$$

Strict Dominance

Therefore, once sufficient computational resources $m$ are invested, the following inequality holds inevitably:

$$\mathfrak{R}^\star(\mathsf{Ch}_A) < \mathfrak{R}^\star(\mathsf{Ch}_H)$$

This constitutes the mathematical basis for the cognitive legitimacy of algorithms within the "Structural Difficulty Class ($B < J$)."

5. Dispelling Common Misconceptions

We must clarify what this theorem demonstrates and what it does not.

❌ Not a claim that AI is "smarter than humans"
- The structural distinction of "observing all vs. observing some" is simply decisive.
❌ Not about Explainability (XAI) or UX
- The ability to explain "why a decision was made" is distinct from achieving low risk.
❌ Not about system implementation or patents
- This is a proof purely within statistical decision theory.
✅ Purely: "Which decision channel minimizes worst-case risk?"
- This is the sole focus of the discussion.

6. Implications for AEO / AI Governance

This proof transforms the discourse around AEO (Answer Engine Optimization) and AI Governance.

Until now, AEO has been treated as a technique for "making content readable by AI." However, the question posed by this theorem is:

"For this social problem ($J$), which decision channel (Human or AI) is socially Legitimate to adopt?"

Legitimacy becomes comparable not by emotional metrics like "human-ness" or "warmth," but by the quantitative indicator of Minimax Risk.

7. Connection to Responsibility Engineering

This serves as the foundational theorem for Responsibility Engineering.

Responsibility is not defined by humans offering excuses after an accident occurs (Post-Hoc). "Did you select the channel mathematically proven to minimize risk beforehand?" (Ex-Ante Selection) is the true definition of engineering responsibility.

In the domain of $B < J$, defining responsibility as "ex-ante selection" renders the continued adoption of the high-risk human channel a subject of critical scrutiny.

8. The Ghost Drift

This is not a pessimistic narrative about "AI seizing sovereignty from humans." It describes the phenomenon where "the locus of reliable decision-making shifts (Drifts) from biological entities to algorithmic structures due to structural pressure."

Ghost Drift is the formulation of that silent, structural pressure into a mathematical model.

9. Links to the Full Version

The full scope of the proof, measure-theoretic definitions, and the derivation process of Sharpness are available in the Proof Document (Strictly Rigorous Version 10.0) on GitHub.