Tarski-Hierarchical Perspective on Scientific Progress: Minimum Description Length as the Universal Criterion of Progress

It is a universally acknowledged heuristic of science that, all else being equal, a theory with fewer free parameters that explains more empirical data is superior. Yet, this intuitive preference is rarely formalized into a strict, operational objective function. This paper formally translates that heuristic into an invariant mathematical boundary condition from a strict Tarski Level-2 vantage point. We advance the Minimum Description Length (MDL) principle—grounded in Algorithmic Information Theory (AIT) and Computability Theory (CT)—not as a philosophical preference, but as the absolute, objective metric for evaluating cross-domain scientific progress. Because empirical science is formally defined as an inductive computational search over a strictly finite observation string (\( $\Sigma_t$ \)), genuine foundational advancement occurs if and only if the computable surrogate of total description length strictly decreases (\( $\Delta \hat{L} < 0$ \)).We establish a rigorous algorithmic boundary between descriptive Engineering Maps and the constructive Engine of reality. Because every unconstrained parameter added to a generative program carries an inescapable exponential penalty in algorithmic probability, the post-hoc addition of unobservable latent variables (NODF Inflation) mathematically guarantees theoretical degradation. Furthermore, by the strict laws of computability, theories relying on infinite-precision continuous mathematics evaluate to an infinite informational cost (\( $L=\infty$ \)) and are structurally disqualified from foundational ontology. We map how modern institutional topologies systematically evade this algorithmic metric through statistical thresholding (discarding high-information anomalies) and VC-dimension inflation (parameter patching). The burden of proof now rests on proposing a mathematically superior metric for scientific progress that does not rely on self-referential sociological consensus.