The Circle Test: Transcendental Manifold Incompleteness, Thermodynamic Back-Reaction, and the Physical Boundaries of Discrete Turing Robotics

Abstract:Contemporary physical artificial intelligence (Physical AI) and vision-language-action (VLA) foundation models rest upon an unproven foundational axiom: that dynamic interaction within continuous physical space-time can be asymptotically resolved by scaling discrete floating-point operations (FLOPs) and temporal sampling rates under the Church-Turing thesis. This paper formally introduces The Circle Test for Robotics as a non-negotiable demarcation principle separating discrete statistical curve-fitting from genuine autonomous physical agency. Grounded in computational complexity, non-linear Hamiltonian mechanics, and non-equilibrium thermodynamics, we prove four foundational theorems:1. Symplectic Incompleteness Theorem: Operating under finite register bit-depth N and sampling period Δt, any discrete-time digital automaton degenerates the continuous Lie group SO(2)—characterized by the transcendental topological invariant π—into a discrete polygon P_M. Numerical round-off operators (fl(·)) violate symplectic form conservation (det R != 1), driving phase-volume drift along logarithmic spirals.2. Local Finite-Time Lyapunov Exponent (FTLE) Divergence: At quadrant velocity zero-crossing boundaries, non-linear Stribeck friction introduces instantaneous negative damping (∂F_f / ∂v < 0). Coupling with discrete control latency (τ > 0) causes the local FTLE to become strictly positive (λ_FTLE > 0), exponentially amplifying algorithmic truncation residuals ε_N into macroscopic limit-cycle chattering.3. Cascaded Thermodynamic Back-Reaction Law: In self-contained, integrated joint actuator modules, mitigating chattering via computational brute force (Δt -> 0, N -> ∞) triggers a fatal dissipation chain: microscopic non-ideal Landauer bit-erasure losses cascade into high-frequency motor coil pulse currents, exploding macroscopic Joule heating (I^2 R_m). The resulting structural thermal expansion (ΔL = α L ΔT) exceeds software-level precision optimizations by multiple orders of magnitude (e.g., 138 μm physical thermal expansion vs. 10^-9 m algorithmic optimization).4. The Bi-Directional Squeeze Fallacy: Hybrid control architectures—combining low-frequency foundation model planning (e.g., Google RT-2/VLA at 3–10 Hz) with high-frequency traditional joint interpolators (1–10 kHz)—do not resolve this crisis. Instead, they suffer simultaneously from open-loop latency at the macro scale and catastrophic thermal-dynamic chattering at the micro scale. Anchored in the Ye Qiang Human-AI Evolution/Synergy Theory (YQ-HAET), we formalize the triadic division of embodied agency: Human intellect establishes intentional low-dimensional topological manifolds (Conceptualization), continuous viscoelastic morphology resolves physical perturbations at zero latency via the Principle of Least Action (Actuation), and discrete deterministic micro-operators (e.g., zero-heap C99 estimators) remain strictly confined to hard real-time conservation verification (Computation). Keywords: The Circle Test, Embodied AI, Symplectic Invariance, Finite-Time Lyapunov Exponent (FTLE), Landauer Limit, Thermodynamic Back-Reaction, YQ-HAET, Morphological Computation.