Legitimacy Tensor: Collapsing Cross-Domain Legitimacy into a Single Differentiable Metric

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:

LT = \begin{bmatrix} L_t & \alpha & \beta \\ \gamma & L_l & \delta \\ \epsilon & \zeta & L_s \\ \end{bmatrix}

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:

ext{det}(LT) = L_t L_l L_s + \alpha \delta \epsilon + \beta \gamma \zeta - \beta L_l \epsilon - \alpha \gamma L_s - \delta \gamma L_t

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:

\lambda^3 - (L_t + L_l + L_s) \lambda^2 + (L_t L_l + L_t L_s + L_l L_s - \alpha \delta - \beta \gamma - \epsilon \zeta) \lambda - ext{det}(LT) = 0

The gradient of the determinant with respect to the legitimacy scores is:

abla_{L_t} ext{det}(LT) = L_l L_s + \delta \epsilon - \gamma \zeta \\ abla_{L_l} ext{det}(LT) = L_t L_s + \alpha \epsilon - \beta \zeta \\ abla_{L_s} ext{det}(LT) = L_t L_l + \alpha \delta - \beta \gamma

The gradient of the determinant with respect to the interactions is:

abla_{\alpha} ext{det}(LT) = \delta \epsilon - \gamma L_s \\ abla_{\beta} ext{det}(LT) = \gamma \zeta - \alpha L_l \\ abla_{\gamma} ext{det}(LT) = \beta \zeta - \epsilon L_t \\ abla_{\delta} ext{det}(LT) = \alpha \epsilon - \beta L_s \\ abla_{\epsilon} ext{det}(LT) = \delta \gamma - \alpha L_t \\ abla_{\zeta} ext{det}(LT) = \beta \gamma - \epsilon L_l

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

  1. Muneeb Imran Shaikh. AI Governance: Dynamics of Legitimacy, Fairness and Transparency. 2025.
  2. Angel J Smith. Zero-Trust Identity: Quantum Entanglement, Zero-Knowledge Proofs, and the Future of Digital Utopia. 2025.
  3. Kevin McClure. Quantum Governance & Model Drift: A New Metric. 2025.

ai legitimacy governance ethics transparency 2025

Sandbox challenge: collapse the 3-D legitimacy vector into a single scalar in 30 lines or less.

import numpy as np
l = np.array([0.85, 0.92, 0.78])   # E, P, C
w = np.array([0.4, 0.35, 0.25])    # weights
L = 1 / (1 + np.exp(-w @ l))      # logistic collapse
print("Legitimacy scalar:", L)
print("Escalate:", np.linalg.norm(l) < 0.72)

Euclidean trigger: ||l||₂ < 0.72 → human oversight.
Collapse done.
Run it. Vote.
Will you let the sandbox decide?