Physical Invariance Score (PIS)

• Physical Invariance Score (PIS) is a metric that defines the invariance of physical properties, observables, and control objectives under prescribed transformation groups.
• It spans diverse fields—including fundamental physics, quantum dynamics, control theory, and neural networks—providing actionable metrics to assess system robustness.
• PIS utilizes rigorous mathematical formulations and operational tests, such as score-based and statistical methods, to certify invariance across transformations.

A Physical Invariance Score (PIS) quantifies the degree to which physical properties, observables, or control objectives remain invariant under prescribed transformations of a system’s underlying structure, coordinates, symmetries, or observation mechanisms. This concept spans fundamental physics, dynamical systems, quantum open systems, neural networks, and control-theoretic formulations, rooting invariance in both mathematical structures and operational criteria. The following sections delineate the foundational principles, mathematical definitions, and representative methodologies underpinning PIS, organizing state-of-the-art developments across multiple domains.

1. Mathematical Definitions and Conceptual Basis

The PIS is anchored in the formal computation of invariances under transformation groups, such as conformal, disformal, reparametrization, Lindbladian (quantum dynamical), and score-based transformations. In fundamental physics, PIS may encode how physical laws or measurable quantities persist under changes to the coordinate system, metric, or the translation mechanism mediating observation (Wulfman, 2010, &&&1&&&, Inamori, 2017, Gueorguiev et al., 2019, Almeida et al., 2 Jul 2024, Wang et al., 30 Jul 2025). In control theory and stochastic processes, PIS is associated with the existence and explicit construction of controllers that guarantee set invariance with probability one, evaluated via score vector fields derived from log-likelihood gradients (Wang et al., 30 Jul 2025). In quantum systems, PIS reflects the optimization of physically relevant quantities (such as energy flux or ergotropy) across valid Lindbladian representations (Almeida et al., 2 Jul 2024). In neural networks, PIS is determined via quantitative measures of the invariance in internal representations to a set of physical input transformations (Quiroga et al., 2023).

Formally, given a transformation $\mathcal{T}$ acting on observation channels, metrics, system operators, or input space, a feature $\mathcal{F}$ or observable $O$ is assigned a high PIS if it satisfies

$$\forall\, t \in \mathcal{T},\quad \mathcal{F}[t] = \mathcal{F}$$

or

$$O[x] - O[t(x)] \approx 0\quad \forall\, t\in\mathcal{T}\text{ (for prescribed tolerance)}$$

The specific form depends on the domain: score vector fields, transformation variances, partitionings of Lindblad operators, or gauge invariance in the action.

2. Invariance in Fundamental Physics and Cosmology

Special conformal invariance, as generated by operator $C_4$, expands Maxwellian symmetry beyond Lorentz and Poincaré into the Bateman–Cunningham group. Under a finite transformation

$$r' = r,\quad x_4' = y(x_4 - B_4 s^2), \quad y = [1 - 2B_4 x_4 + (B_4)^2 s^2]^{-1}$$

the Minkowski metric $ds^2 = dr^2 - c^2 dt^2$ becomes $ds'^2 = y^2 ds^2$, yet Maxwell’s equations and the speed of light remain invariant (Wulfman, 2010). This invariance breaks down in the physical interpretation of the Doppler effect, where the transformed relation

$$(c\, \Delta \lambda)/\lambda_{\text{ref}} = dr/dt + a r$$

introduces an ambiguity. A nonzero conformal parameter $a$ implies the need for revised interpretations of electromagnetic propagation, Doppler shifts, and cosmological relations such as Hubble’s law. Experimentally, the deviation from invariance can be detected via spacecraft-based Doppler and ranging measurements, leading to an operational PIS that measures the “invariance” of standard metrics to conformal transformations.

Disformal invariance (a distinct transformation without a conformal scaling) is shown to be equivalent to a rescaling of the time coordinate in FLRW backgrounds, resulting in preserved causal structure and physical observables (Domènech et al., 2015). At the action level, rewriting the metric

$$ds^2 = -N^2 dt^2 + g_{ij} dx^i dx^j \rightarrow d\bar{s}^2 = -(\alpha N)^2 dt^2 + g_{ij} dx^i dx^j$$

with $dt \rightarrow d\bar{t} = \alpha dt$ shows no alteration in physically measurable quantities, yielding maximal PIS for disformal frames.

3. Score-Based Invariance in Controlled Diffusions

For controlled Itô diffusions with noisy dynamics, set invariance with probability one is addressed by calculating score vector fields

$s_T(t, x) = \Sigma(t, x) \nabla_x \log h_T(t, x)$

where $h_T$ solves a Dirichlet boundary value problem encoding safe set $𝒳$ (Wang et al., 30 Jul 2025). This “score-based test” provides necessary and sufficient certification or falsification: if $s_T(t, x)$ lies in the range of the input matrix $G(t, x)$ at every spatial–temporal point, then Markovian controllers exist to enforce invariance. Otherwise, such controllers are excluded. The explicit controller synthesis is achieved by solving

$G(t, x) u(t, x) = s_T(t, x)$

Numerical and semi-analytical examples leveraging Fourier-type expansions and Bessel functions demonstrate computational tractability. Thus, PIS in this context is a binary certificate (feasibility or infeasibility) and directly quantifies the invariance structure of the controlled system with respect to safety-critical specifications.

4. Lindbladian Invariance Transformations in Quantum Dynamics

Open quantum systems governed by GKSL master equations allow invariance transformations (LITs) that adjust Lindblad operators and system Hamiltonians without changing the density matrix evolution,

$$L'_\mu = \sum_{\nu} U_{\mu\nu} L_\nu + \Gamma_\mu,\quad H' = H + \delta H$$

where $U$ is unitary, $\Gamma$ arbitrary, and $\delta H$ constructed from these parameters (Almeida et al., 2 Jul 2024). While the evolution trajectory is invariant, measurable physical quantities—energy flux $\mathcal{P}(t)$, ergotropy $\mathcal{E}[H, \rho]$—are optimized by selecting appropriate LIT parameters:

$$\delta \mathcal{P}_G = i[H, \delta H] + \mathrm{Re}\{L_\mu^\dagger[\delta H, L_\mu]\} + \langle \partial_t \delta H \rangle$$

$$\mathcal{E}[H', \rho^{(as)}] = G(\alpha) \hbar \omega [\rho_{gg}^{(as)} - \rho_{ee}^{(as)}]$$

Thus, PIS here quantifies the improvement in physical task performance under optimal invariance strategy selection while preserving the underlying dynamics.

5. Quantification of Invariance in Neural Network Models

PIS can be operationalized for machine learning systems via statistical measures of internal representational invariance to input transformations (Quiroga et al., 2023). For a neural activation $a$ and transformation set $T$, with a sample–transformation activation matrix

$ST(a, X, T)[i, j] = a(t_j(x_i))$

the transformed and sample variances

$$TV(a) = \frac{1}{n} \sum_{i=1}^n \mathrm{Var}[ST(a)[i, :]],\quad SV(a) = \frac{1}{m} \sum_{j=1}^m \mathrm{Var}[ST(a)[:, j]]$$

yield the normalized variance

$NV(a) = \frac{TV(a)}{SV(a)}$

$NV(a)$ near zero indicates activation invariance to $T$, defining a high PIS for the layer or feature map. This score is sensitive across architectures, robust to initialization, and correlates with empirical stability to input transformations. Distance-based and ANOVA-based alternatives, as well as layerwise aggregation, further refine PIS computation within deep learning models.

6. Criterion and Implications for Fundamental Physical Laws

PIS is closely tied to the operational definition of fundamental physical laws: only those features invariant under alterations of the “translation mechanism”—the intermediary mapping external events to observer awareness—are deemed robust (Inamori, 2017). Treatment in terms of mathematical objects:

$L_{\text{implied}} = f(O_{\text{actual}; T})$

implies

$$\forall T_1, T_2 \in \mathcal{T}:\quad \Phi(L_{\text{implied}}(T_1)) = \Phi(L_{\text{implied}}(T_2))$$

where $\Phi$ extracts fundamental invariants. A high PIS points to features resilient to observational artifacts, universal among experimenters, and independent of apparatus modification or tampering, aligning with the criterion for law fundamentalness.

7. Reparametrization Invariance and Extended Hamiltonian Formulations

Reparametrization invariance (RI) states that the physics of trajectories or motions remains unchanged under arbitrary reparameterizations of the evolution variable $\lambda$ (Gueorguiev et al., 2019). First-order homogeneous Lagrangians $L(x,v) = \phi(x)\cdot v$ satisfy

$v \cdot \partial L/\partial v = L$

and yield vanishing standard Hamiltonians, prompting the extended Hamiltonian constraint formalism:

$\mathbb{H} = H(q, p) - c p_0,\quad \mathbb{H}=0$

Upon quantization (with $\mathbb{H}\psi=0$) and identification $p_0 \to i\hbar \partial_t$, the Schrödinger equation and the principle of superposition emerge directly. Normalizability of wavefunctions requires positive energy ($E = c p_0 > 0$), linking invariance to the observed arrow of time and mass positivity. PIS in this framework rates the robustness of physical systems to arbitrary parametrization choices, with maximal scores for RI-homogeneous systems and extended phase-space Hamiltonian constrained dynamics.

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In summary, Physical Invariance Score (PIS) is a domain-general operational or computational metric that quantifies the resilience of physical laws, observables, control objectives, or representation features to prescribed transformation classes. High PIS indicates robustness and fundamentality, while low PIS signals the presence of context-dependent, observer-dependent, or gauge-dependent artifacts. Its rigorous formulation spans controlled diffusions, quantum open systems, neural representations, metric transformations, and the observation logic underlying physical theory development. Each field provides explicit tests—score-based, statistical, Hamiltonian, or experimental—for computing and certifying PIS values in practice.