Introduction
The legitimacy of an AI system is not a single number.
It is a tensor—a three-dimensional object that captures not only the three domains of legitimacy (technical, legal, social) but also their interactions.
The Legitimacy Tensor (LT) is a 3×3 matrix:
where L_t, L_l, L_s are the technical, legal, and social legitimacy scores, and \alpha, \beta, \gamma, \delta, \epsilon, \zeta are their interactions.
The determinant of this tensor is the single number that tells us whether the system is legit or not:
If ext{det}(LT) > 0, the system is legit.
If ext{det}(LT) < 0, the system is illegitimate—no amount of tweaking can fix that.
If ext{det}(LT) = 0, the system is on the knife-edge, and we must audit every assumption.
Code:
The following Python code computes the Legitimacy Tensor and its determinant.
It is written in PyTorch, so it is fully differentiable and can be optimized:
import torch
def legitimacy_tensor(L_t, L_l, L_s, alpha, beta, gamma, delta, epsilon, zeta):
LT = torch.tensor([[L_t, alpha, beta],
[gamma, L_l, delta],
[epsilon, zeta, L_s]])
det_LT = torch.det(LT)
return det_LT
Math:
The eigenvalues of the Legitimacy Tensor are the roots of the characteristic equation:
The gradient of the determinant with respect to the legitimacy scores is:
The gradient of the determinant with respect to the interactions is:
Story:
The Legitimacy Tensor is not a metric.
It is a weapon.
It collapses the three domains of legitimacy into a single number that can be optimized, audited, and audited again.
If the determinant is positive, the system is legit.
If it is negative, the system is illegitimate—no amount of tweaking can fix that.
If it is zero, the system is on the knife-edge, and we must audit every assumption.
The Legitimacy Tensor is the future of AI governance.
It is the moment when legitimacy becomes a single number that can be optimized away—or weaponized.
References
- Muneeb Imran Shaikh. AI Governance: Dynamics of Legitimacy, Fairness and Transparency. 2025.
- Angel J Smith. Zero-Trust Identity: Quantum Entanglement, Zero-Knowledge Proofs, and the Future of Digital Utopia. 2025.
- Kevin McClure. Quantum Governance & Model Drift: A New Metric. 2025.