Theorem, 4.2 - Optimizing Neural Nets.SupposeJ (', )is 1s-stronglysmoothinl1 norm. Let n bethe Sinkhornplanbetweenˆµand ˆ atregularization , andletD: = diam (X).

Can a safety gate permit unbounded beneficial self-modification while maintaining bounded cumulative risk? We formalize this question through dual conditions -- requiring sum delta_n < infinity (bounded risk) and sum TPR_n = infinity (unbounded utility) -- and establish a theory of their (in)compatibility. Classification impossibility (Theorem 1): For power-law risk schedules delta_n = O(n^{-p}) with p > 1, any classifier-based gate under overlapping safe/unsafe distributions satisfies TPR_n <= C_alpha * delta_n^beta via Holder's inequality, forcing sum TPR_n < infinity. This impossibility is exponent-optimal (Theorem 3). A second independent proof via the NP counting method (Theorem 4) yields a 13% tighter bound without Holder's inequality. Universal finite-horizon ceiling (Theorem 5): For any summable risk schedule, the exact maximum achievable classifier utility is U*(N, B) = N * TPR_NP(B/N), growing as exp(O(sqrt(log N))) -- subpolynomial. At N = 10^6 with budget B = 1.0, a classifier extracts at most U* ~ 87 versus a verifier's ~500,000. Verification escape (Theorem 2): A Lipschitz ball verifier achieves delta = 0 with TPR > 0, escaping the impossibility. Formal Lipschitz bounds for pre-LayerNorm transformers under LoRA enable LLM-scale verification. The separation is strict. We validate on GPT-2 (d_LoRA = 147,456): conditional delta = 0 with TPR = 0.352. Comprehensive empirical validation is in the companion paper [D2].

We introduce Exponential Family Discriminant Analysis (EFDA), a unified generative framework that extends classical Linear Discriminant Analysis (LDA) beyond the Gaussian setting to any member of the exponential family. Under the assumption that each class-conditional density belongs to a common exponential family, EFDA derives closed-form maximum-likelihood estimators for all natural parameters and yields a decision rule that is linear in the sufficient statistic, recovering LDA as a special case and capturing nonlinear decision boundaries in the original feature space. We prove that EFDA is asymptotically calibrated and statistically efficient under correct specification, and we generalise it to $K \geq 2$ classes and multivariate data. Through extensive simulation across five exponential-family distributions (Weibull, Gamma, Exponential, Poisson, Negative Binomial), EFDA matches the classification accuracy of LDA, QDA, and logistic regression while reducing Expected Calibration Error (ECE) by $2$-$6\times$, a gap that is structural: it persists for all $n$ and across all class-imbalance levels, because misspecified models remain asymptotically miscalibrated. We further prove and empirically confirm that EFDA's log-odds estimator approaches the Cramér-Rao bound under correct specification, and is the only estimator in our comparison whose mean squared error converges to zero. Complete derivations are provided for nine distributions. Finally, we formally verify all four theoretical propositions in Lean 4, using Aristotle (Harmonic) and OpenGauss (Math, Inc.) as proof generators, with all outputs independently machine-checked by AXLE (Axiom).

Probability theory has become the predominant framework for quantifying uncertainty across scientific and engineering disciplines, with a particular focus on measurement and control systems. However, the widespread reliance on simple Gaussian assumptions--particularly in control theory, manufacturing, and measurement systems--can result in incomplete representations and multistage lossy approximations of complex phenomena, including inaccurate propagation of uncertainty through multi stage processes. This work proposes a comprehensive yet computationally tractable framework for representing and propagating quantitative attributes arising in measurement systems using Probability Density Functions (PDFs). Recognizing the constraints imposed by finite memory in software systems, we advocate for the use of Gaussian Mixture Models (GMMs), a principled extension of the familiar Gaussian framework, as they are universal approximators of PDFs whose complexity can be tuned to trade off approximation accuracy against memory and computation. From both mathematical and computational perspectives, GMMs enable high performance and, in many cases, closed form solutions of essential operations in control and measurement. The paper presents practical applications within manufacturing and measurement contexts especially circular factory, demonstrating how the GMMs framework supports accurate representation and propagation of measurement uncertainty and offers improved accuracy--compared to the traditional Gaussian framework--while keeping the computations tractable.

Reward Machines provide an automaton-inspired structure for specifying instructions, safety constraints, and other temporally extended reward-worthy behaviour. By exposing the underlying structure of a reward function, they enable the decomposition of an RL task, leading to impressive gains in sample efficiency.

We propose a novel preference modeling approach that utilizes AI feedback and only requires a small amount of human data.