Differential Geometric and Physical Invariants for Distinguishing AI-Generated and Real Images: A Comprehensive Mathematical Framework

The proliferation of photorealistic AI-generated images poses existential challenges to digital forensics, journalism, and societal trust in visual media. Current detection methods predominantly deep learning approaches exhibit catastrophic failure when encountering novel generative architectures, with accuracy plummeting from 99% on training distributions to below 65% on unseen generators. This brittleness stems from reliance on learned statistical patterns rather than fundamental constraints imposed by physical reality. We present a radically different paradigm: a mathematically rigorous framework grounded in differential geometry, information theory, and physical optics that distinguishes real from synthetic images by testing violations of physical laws. Real photographs arise from a well-understood physical process 3D scene geometry projected through camera optics with quantum sensor noise—that imposes stringent mathematical structure on the resulting 2D intensity fields. AI synthesis, operating in pixel or latent space without explicit 3D geometric or physical rendering, systematically violates these constraints. We derive three fundamental invariants with complete mathematical proofs: (1) Gradient Covariance Anisotropy—physical surfaces project to highly oriented gradient fields with eigenvalue ratios λ₁/λ₂ ≫ 1, violated by localized convolutional synthesis; (2) Cross-Scale Consistency Gaussian smoothing preserves principal gradient directions across scales for real structures, violated by multi-resolution generative architectures; (3) Noise-Signal Decoupling—quantum sensor noise is statistically independent of scene geometry, violated by learned upsampling that couples residuals with gradients. Comprehensive experiments on 12,000 images spanning six state-of-the-art generators (StyleGAN2/3, Stable Diffusion, Midjourney, DALL-E 2, BigGAN) demonstrate 97.6% average accuracy with remarkable cross-generator stability (variance 0.9%), compared to 91.0% (variance 8.7%) for ResNet-50. Under severe JPEG compression (Q=50), our method maintains 94.8% accuracy while deep learning degrades to 65.7%—a 25.3% catastrophic drop versus our 2.8% graceful degradation. The framework requires zero training on target generators, executes in O(N log N) time (150ms for 1024² images on CPU), and provides interpretable spatial localization achieving 0.883 pixel-level AUROC on forensic benchmarks. Crucially, we prove that simultaneously satisfying all three invariants requires generative models to solve the inverse graphics problem—reconstructing 3D geometry, physical materials, lighting, and camera parameters from 2D training images—a computationally intractable barrier (NP-hard). This theoretical result establishes that our framework identifies fundamental limitations of current AI synthesis paradigms rather than exploitable artifacts, providing detection guarantees that transcend specific architectures.