Thermodynamic Probability Filter (TPF)

• TPF is a principled framework that leverages free energy and entropy to provide parameter-free, data-driven probability estimates.
• It unifies maximum-entropy and maximum-likelihood regimes through a minimum free energy principle with a closed-form, efficient update.
• In hardware applications, TPF enables early-abort classification in ASICs, achieving energy savings over 90% by halting unproductive computations.

The Thermodynamic Probability Filter (TPF) is a principled estimation and classification framework that leverages thermodynamic analogies—specifically free energy, entropy, and temperature—to unify probabilistic inference and physical computation. TPF robustly interpolates between maximum-entropy and maximum-likelihood regimes, providing parameter-free, data-driven probability estimates and enabling energy-efficient, real-time early-abort prediction in hardware systems such as Bitcoin mining ASICs. The method is formalized mathematically via a minimum free energy principle and is accompanied by machine-verified theorems that guarantee its information-theoretic and energy-saving properties (Isozaki, 2012, &&&1&&&).

1. Theoretical Foundation: Free Energy Functional

TPF is built on the formulation of a Helmholtz free energy functional, combining likelihood, entropy, and a sample-size-dependent temperature parameter. For a discrete probability mass function $p = (p_1, \ldots, p_K)$ over $K$ states and empirical distribution $\hat{q}$, the constituent components are:

• Shannon entropy: $H[p] = -\sum_{i=1}^K p_i \log p_i$
• Energy (cross-entropy): $U[p] = -\sum_{i=1}^K p_i \log \hat{q}_i$
• Temperature: $T>0$ (inverse: $\beta = 1/T$)

The free energy functional is defined as:

$$F[p] = U[p] - T H[p] = -\sum_{i=1}^K p_i \log \hat{q}_i - T \left(-\sum_{i=1}^K p_i \log p_i\right)$$

or equivalently,

$F[p] = U[p] - \frac{1}{\beta} H[p]$

The minimizer of $F[p]$ yields a probability estimate balancing fidelity to the data and entropy regularization (Isozaki, 2012).

2. Variational Solution and Algorithmic Implementation

Minimizing the free energy $F[p]$ with normalization constraints leads to a Gibbs (Boltzmann) distribution:

$$p_i^* = \frac{\exp(-\beta \varepsilon_i)}{Z(\beta)} = \frac{\hat{q}_i^\beta}{\sum_j \hat{q}_j^\beta}$$

where $\varepsilon_i = -\log \hat{q}_i$ and $Z(\beta)$ is the normalization constant.

The critical innovation of TPF is the data-adaptive temperature selection. For $n$ samples, define the geometric-mean mixture:

$$P^{(G)}_n = \left( \prod_{i=0}^n p^{(i)} \right)^{1/(n+1)}$$

Compute KL divergence $D(P^{(G)}_{n-1} \Vert \hat{q}^{(n)})$; set unnormalized inverse-temperature $\beta_0 = 1/D$, and normalized $\beta = \beta_0/(1+\beta_0)$. The probability estimate is then updated in closed form:

$$p_i^{(n)} = \frac{\left( \hat{q}_i^{(n)} \right)^\beta}{\sum_j \left( \hat{q}_j^{(n)} \right)^\beta}$$

No iterative inner loop is required; the update operates in $\mathcal{O}(K)$ per sample (Isozaki, 2012).

3. Extension to Hardware-Embedded Early-Abort Classification

In hardware applications, such as Bitcoin mining ASICs, TPF is employed as a real-time early-abort classifier to realize substantial energy savings (Lafuente et al., 17 Jan 2026). Here:

• Input features: Thermodynamic and timing signatures ($\Delta t$, temperature, voltage) from SHA-256 rounds $1$ to $k$.
• Classifier: A lightweight multilayer perceptron (MLP) is trained to approximate $\mathbb{P}(\mathrm{success} \mid X)$.
• Early-abort decision: If $\mathbb{P}(\mathrm{success} \mid X)<\tau$ at round $k$, computation is aborted, and energy is saved.

The theoretical energy savings from aborting at round $k$ of $n$ total is:

$\mathrm{Energy~Savings} = 1 - \frac{k}{n}$

For $k=5$, $n=64$, this yields $92.19\%$ energy reduction, as validated empirically and by formal proof (Lafuente et al., 17 Jan 2026).

4. Information-Theoretic Guarantees and Formal Verification

TPF's logical core is the detection of predictive dependence in early-round signatures:

• Accuracy Baseline: Maximum probability for a constant predictor, $\max_{y\in Y} P(Y=y)$.
• Achievable accuracy: $\text{accuracy}(P,g) = \sum_{(x,y):g(x)=y} P(x,y)$ for any function $g$.
• Key theorems (machine-checked in Lean 4/Mathlib):

1. Independence $\implies$ zero mutual information (leakage).
2. If a predictor $g$ beats baseline accuracy, the input and output are not independent.
3. Maximum provable energy savings: $1-k/n$ for given $k$, $n$.
4. Distinguishability of physically unclonable functions via concrete timing tests.

All proofs are mechanized and complete, with zero unproven "admits" (Lafuente et al., 17 Jan 2026).

5. Limiting Behavior, Empirical Evaluation, and Robustness

TPF's behavior interpolates smoothly between:

• Maximum entropy (ME) regime: As $n\to 0$, $\beta\to 0$, all $\hat{q}_i^\beta\to 1$, so $p_i\to 1/K$ (uniform).
• Maximum likelihood (ML) regime: As $n\to\infty$, $\beta\to 1$, so $p_i\to \hat{q}_i$.

Empirical evaluation (Isozaki, 2012):

• On small to moderate samples, TPF's minimum-free-energy estimates yield lower KL divergence to ground truth than ML, ME, or MAP-Dirichlet, except in lowest-uncertainty cases.
• TPF is stable against over- and under-fitting in finite-data conditions.

Experimental validation on ASIC hardware (Lafuente et al., 17 Jan 2026):

• Digital-twin simulation: $92.19\%$ energy reduction, $0\%$ false-abort rate.
• Physical ASICs (LV06): $88.50\%$ observed energy reduction (3.69\% gap is attributed to real-world noise, conservatism).

6. Extensions, Variations, and Contexts of Application

TPF is versatile, with theoretically justified adaptations:

• Conditional/joint distributions: Apply TPF to empirical conditional probabilities per context.
• Priors/bayesian posteriors: Use subjective or Bayesian priors as $\hat{q}$ for tempered posterior estimation.
• Continuous variables: Extend by replacing sums with integrals, using differential entropy and empirical densities.
• Alternative divergences: Free-energy minimization with other divergences (e.g., $\alpha$-divergence, Rényi cross-entropy).
• Hardware: Application to other cryptographic workloads and blockchains, adaptive abort strategies, and more granular measurements (Isozaki, 2012, Lafuente et al., 17 Jan 2026).

Typical assumptions include uniform per-round energy cost, sufficient signal in early-round measurements, fixed $k$ and $n$, and conservative threshold setting for zero false positives. TPF only reduces energy cost, not the stochastic variance of mining returns. Current models may omit network pipeline intricacies and sub-round timing effects.

7. Significance and Benchmark Contributions

TPF establishes a data-driven, physically grounded, and formally validated paradigm that unifies statistical estimation and energy-aware hardware control:

• It bridges maximum-entropy and maximum-likelihood inference, adapting automatically to data regime.
• In hardware, it transforms silicon substrates into interactive, energy-efficient computational reservoirs, offering a $>90\%$ reduction in waste compute on real ASICs.
• All information-theoretic and performance bounds are mechanized and proven via Lean 4/Mathlib (Lafuente et al., 17 Jan 2026).
• By fusing thermodynamics, information theory, reservoir computing, and formal methods, TPF exemplifies a uniquely rigorous approach to predictive filtering, resource allocation, and physical computation.

TPF thus defines a robust methodological benchmark for both statistical inference with limited data and for energy-aware computation in silicon devices.