SIPA: Quantifying Physical Integrity and the Sim-to-Real Gap in 7-DoF Trajectories

[zc_Liu]

Introduction:

SIPA (Spatial Intelligence Physical Audit) is a trajectory-level physical consistency diagnostic. It does not require source code access or internal simulator states and directly audits 7-DoF CSV trajectories. By design, SIPA is compatible with any system that produces spatial motion data. Its principle is based on the Non-Associative Residual Hypothesis (NARH).

The Logic: Non-Associative Residual Hypothesis (NARH)

NARH posits that physical inconsistency stems from discrete solver ordering rather than just algebraic error.

(1)Setting

Consider a rigid-body simulation system defined by:

• State space S \subset \mathbb{R}^n

• Associative update operator \Phi \Delta t : S \to S

• Parallel constraint resolution composed of sub-operators `\{\Psi_i\}_{i=1}^k`

  ​The simulator implements a discrete update:

s_{t+1} = \Psi_{\sigma(k)} \circ \cdots \circ \Psi_{\sigma(1)} (s_t)

where 𝜎 is an execution order induced by:

• constraint partitioning

• thread scheduling

• contact batching

• solver splitting

Each \Psi_i is individually well-defined, but their composition order may vary.

(2) Order Sensitivity

Although each operator Ψi belongs to an associative algebra (e.g., matrix multiplication, quaternion composition), the composition of numerically approximated operators may satisfy:

(\Psi_a \circ \Psi_b) \circ \Psi_c \neq \Psi_a \circ (\Psi_b \circ \Psi_c)

due to:

• finite precision arithmetic

• projection steps

• iterative convergence truncation

• asynchronous execution

Define the discrete associator:

A(a,b,c;s) = \bigl( (\Psi_a \circ \Psi_b) \circ \Psi_c \bigr)(s) - \bigl( \Psi_a \circ (\Psi_b \circ \Psi_c) \bigr)(s)

(3) Definition: Non-Associative Residual

We define the Non-Associative Residual (NAR) at state s_t as:

R_t = \lVert A(a,b,c; s_t) \rVert

for a chosen triple of sub-operators representative of contact or constraint updates.

This residual measures path-dependence induced by discrete solver ordering, not algebraic non-associativity of the state representation.

(4) Hypothesis (NARH)

In high-interaction-density regimes (e.g., contact-rich robotics, high-speed manipulation), the Non-Associative Residual R_t becomes non-negligible relative to scalar stability metrics, and accumulates over time as a structured drift term.

Formally, there exists a regime such that:

\sum_{t=0}^{T} R_t \not\approx 0

even when:

\Vert s_{t+1} - s_t \Vert remains bounded.

(5) Interpretation

This hypothesis does not claim:

• that simulators are mathematically invalid,

• that associative algebras are incorrect,

• or that hardware tiling causes topological inconsistency.

Instead, it asserts:

Discrete parallel constraint resolution introduces a measurable order-dependent residual that is not explicitly encoded in the state space.

This residual may contribute to:

• sim-to-real divergence,

• policy brittleness,

• instability under reordering of equivalent control inputs.

(6) Falsifiability

NARH is falsified if:

1. s_t remains within numerical noise across interaction densities.

2. Reordering constraint application yields statistically indistinguishable trajectories.

3. Scalar metrics (e.g., kinetic energy norm, velocity norm) detect instability earlier or equally compared to any associator-derived signal.

(7) Research Implication

If validated, NARH suggests that:

• Order sensitivity is a structural property of discrete solvers.

• Additional diagnostic signals (e.g., associator magnitude) may serve as early-warning indicators.

• Embodied AI training in simulation may implicitly depend on hidden order-stability assumptions.

If invalidated, the experiment establishes an empirically order-invariant regime — a valuable boundary characterization of solver behavior.

GitHub Repository: https://github.com/ZC502/SIPA.git