Time-varying Decay Uncertainty (TDU)

• Time-varying Decay Uncertainty (TDU) is defined as the modulation of decay processes over time, exhibiting periodic, stochastic, or event-driven fluctuations with typical amplitudes around 10⁻³.
• Mathematical models extend the classical exponential decay law by incorporating time-dependent modulations through harmonic functions and transient pulses, refining error budgets in experimental measurements.
• TDU impacts diverse fields—from nuclear metrology and solar physics to nanophotonic diagnostics and adaptive learning—necessitating enhanced uncertainty quantification and revised theoretical frameworks.

Time-varying Decay Uncertainty (TDU) refers to the explicit, quantifiable modulation or fluctuation of decay rates or analogous dynamical quantities as a function of time, in contrast to the assumption of strict exponential decay governed by a time-independent constant. TDU arises in a range of contexts—from nuclear and molecular decay processes sensitive to external fields, to dynamical systems where timedependent memory kernels, feedback delays, or model uncertainties introduce systematic, non-random temporal variation in decay, loss, or cost functions. Empirically, TDU is typically manifested as periodic, stochastic, or event-driven deviations on the scale of 10⁻⁴–10⁻² in the relevant rate or observable, often requiring reexamination of theoretical models and the structure of experimental uncertainty budgets.

1. Mathematical Models for TDU

The canonical exponential decay law, $N(t) = N_0 \exp(-\lambda_0 t)$, assumes a time-invariant decay constant $\lambda_0$. TDU generalizes this by introducing a time-dependent modulation,

$\lambda(t) = \lambda_0 [1 + \delta\lambda(t)],$

with $\delta\lambda(t)$ capturing deterministic (e.g., periodic) and stochastic (e.g., noise-driven) deviations. The survival law becomes

$$N(t) = N_0 \exp\left[-\int_0^t \lambda(\tau)\,d\tau\right],$$

and $\delta\lambda(t)$ may include harmonic expansions and pulses:

$$\delta\lambda(t) = A_1\cos\left(\frac{2\pi t}{T_\text{annual}} + \phi_1\right) + A_2\cos\left(\frac{2\pi t}{T_\text{rot}} + \phi_2\right) + \sum_j H_j(t-t_j)\Delta_j,$$

where $A_1 \sim 10^{-3}$ characterizes annual modulations (e.g., from Earth–Sun distance), $A_2$ captures solar rotational periodicities, and $H_j$ are window functions for transient events (e.g., solar flares) (Fischbach et al., 2011, Jenkins et al., 2011, Nistor et al., 2013). In optimization and control, TDU is recast as the temporal uncertainty in parameter estimates with explicitly decaying memory weights, e.g., via exponential or more general envelope functions (Aarnoudse et al., 27 Aug 2025, Wu et al., 11 Dec 2025).

2. Experimental Manifestations and Datasets

TDU was originally identified in high-precision nuclear decay measurements, where unexplained modulations persisted after accounting for standard environmental and instrumental effects. Table 1 summarizes representative experiments:

Experiment	Observable	TDU Amplitude	Periodicities/Events
BNL ³²Si	β⁻ decay	1–3×10⁻³	Annual, 32d, 173d
PTB ²²⁶Ra	γ emission	1.5×10⁻³	Annual, 32d, 173d
Purdue ⁵⁴Mn	γ after EC	~1×10⁻³ (flare)	Solar flares, annual

Multiple laboratories maintained temperature stability (ΔT < 1 °C), frequent electronic calibrations, and high count rates (typ. $10^6$ counts per hour), collectively controlling for Poisson and systematic uncertainties down to $10^{-4}$ (Fischbach et al., 2011, Jenkins et al., 2011). Beyond nuclear physics, engineered nanophotonic systems (phosphorescence near antennas) manifest TDU as a broadening and multimodality in the lifetime distribution $p(\tau)$, offering a probe of environmental fluctuations on ns–ms timescales (Kislov et al., 2019).

3. Quantitative Features, Statistical Analysis, and Uncertainty Propagation

TDU is operationally defined by the fractional variation in the decay rate,

$$TDU(t) \equiv \frac{\Delta\lambda(t)}{\lambda_0} = \sum_i \delta_i \cos(\omega t - \phi_i),$$

with root-mean-square amplitude $\text{rms}(TDU) = \alpha/\sqrt{2}$ ($\alpha$ the vector sum amplitude) (Nistor et al., 2013). Experimentally, typical amplitudes for annual or rotational periodicities are $10^{-3}$, and statistical significance routinely exceeds $5\sigma$ over multi-year baselines (Fischbach et al., 2011, Jenkins et al., 2011). The TDU uncertainty propagates to quantities such as half-life or branching ratios, imposing a systematic, time-dependent uncertainty at the $10^{-3}$ level—substantially larger than the precision achieved in some modern half-life measurements.

For time-dependent decay in nanostructures, the observed non-exponential decay curve $I(t)$ is interpreted as a Laplace mixture,

$I(t) = \int_0^\infty p(\tau) e^{-t/\tau} d\tau,$

with TDU characterized by the standard deviation $\sigma_\tau$ in $p(\tau)$. Environmental parameters (diffusion constant $D$, temperature $T$) map onto $p(\tau)$ and hence TDU, providing a quantitative diagnostic of dynamical environmental uncertainty (Kislov et al., 2019).

4. Physical Mechanisms and Theoretical Interpretations

In nuclear and atomic contexts, proposed sources of TDU include:

• Solar neutrino modulation: The flux $\phi_\nu$ at Earth modulates annually due to orbital eccentricity (Δ$\phi/\phi \approx$ 3.4%), and may undergo further rotational modulations if neutrino flavor transitions (e.g., RSFP with $\mu_\nu \sim 10^{-11}\,\mu_B$ and $B\sim10^5$ G) are active (Fischbach et al., 2011, Nistor et al., 2013, Jenkins et al., 2011).
• “Neutrello” sector: Introduced as generic weakly-interacting solar particles inducing TDU with cross-sections $\sim 4 \times 10^{-22}\,\text{cm}^2$—orders of magnitude above electroweak predictions—implying the need for new interaction mechanisms or collective enhancement (Fischbach et al., 2011).
• Cosmic neutrino background: Hypothesized to contribute with its own amplitude and characteristic phase ($\phi_{CNB}$), potentially explaining phase shifts in multi-component TDU analyses (Nistor et al., 2013).

In stochastic environments (e.g., molecular photonics), TDU is a direct physical measure of environmental heterogeneity, with diffusion, drift, and structural reconfiguration entering as drivers of the time-dependence of the effective decay kernel (Kislov et al., 2019).

5. Extensions in Dynamical Systems, Control, and Learning

TDU frameworks are generalized to uncertainty quantification in adaptive optimization and reinforcement learning. In time-varying or nonstationary problems, online estimates of performance, decay, or value functions employ exponentially decaying memory kernels,

$\text{weight} \propto \lambda^{2(k-j)},$

where $\lambda < 1$ discounts older data, embedding TDU directly in the uncertainty quantification for sequential model predictions (Aarnoudse et al., 27 Aug 2025). In ensemble reinforcement learning, TDU is embodied as a dynamic, iteration-indexed decay coefficient $\beta(n)$ multiplying the epistemic standard deviation across critics:

$Q_E(s,a) = \mu_Q(s,a) + \beta(n)\,\sigma_Q(s,a),$

with $\beta(n)$ exponentially decaying from an initial value $\beta_0$ to $\beta_{\min}$ over the training schedule. This captures the time-varying epistemic uncertainty—encouraging exploration under high uncertainty but enforcing contraction to mean-based estimates as uncertainty abates (Wu et al., 11 Dec 2025).

6. Implications and Applications

TDU imposes a non-negligible, structured uncertainty in any application relying on temporally stable decay laws or loss rates, including:

• Nuclear metrology: Precision sources and decay standards must incorporate TDU as an explicit systematic uncertainty (at the $10^{-3}$ level) (Fischbach et al., 2011, Jenkins et al., 2011, Nistor et al., 2013).
• Space-weather forecasting and solar physics: TDU’s correlation with solar activity motivates the concept of "helioradiology"—inferring solar interior dynamics via terrestrial decay monitoring (Fischbach et al., 2011).
• Nanoscale diagnostics: In nanophotonics and lab-on-a-chip devices, extraction of $p(\tau)$ and TDU enables contactless thermometry, flow, and diffusion measurements otherwise inaccessible to conventional approaches (Kislov et al., 2019).
• Adaptive and adversarial learning: TDU-based uncertainty modeling enables targeted, resource-efficient exploration–exploitation schedules, yielding gains in data efficiency and convergence robustness (Aarnoudse et al., 27 Aug 2025, Wu et al., 11 Dec 2025).

7. Open Questions and Future Directions

The origin of TDU in nuclear decay remains unsettled, with sizeable residual uncertainties between environmental controls, solar/astrophysical influences, and potential new physics. Key priorities include multi-isotope, multi-location campaigns with near-real-time environmental and neutrino-flux monitoring; space-based experiments to decouple gravitational and electromagnetic backgrounds; and laboratory exposure to controlled neutrino sources (Fischbach et al., 2011, Jenkins et al., 2011).

In engineered systems, systematic exploration of the sensitivity of TDU to environment geometry, dynamical noise, and feedback delays remains active. Further, formal integration of TDU-aware uncertainty models in learning and control algorithms offers a principled path toward robust performance in nonstationary environments (Aarnoudse et al., 27 Aug 2025, Wu et al., 11 Dec 2025).

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Key References: (Fischbach et al., 2011, Jenkins et al., 2011, Nistor et al., 2013, Kislov et al., 2019, Aarnoudse et al., 27 Aug 2025, Wu et al., 11 Dec 2025)