Entropy and Closedness of the Inductive Chain

We propose a finite, operationally verifiable criterion for theexistence of a reliability basis for inductive chains, grounded in Shannonentropy. The key caveat is thermodynamic asymmetry: absolute zero (Kelvin) is agenuine lower bound dictated by the third law of thermodynamics, whereas infinitetemperature has no physical realisation. Carrying this oriented asymmetry intomathematical logic, we define the entropy of an inductive chain of lengthN and show that reliability reduces to the question of the degree of itsclosedness and self-consistency. The approach requires neither the axiom of infinity nor transfiniteordinals. The exposition is carried out within the framework of Errett Bishop'sconstructivism, Kohlenbach's proof-mining theory, and Gentzen's. Weestablish that the proposed criterion is strictly more finitist while remainingoperationally more informative. The logic is illustrated by dual proofs of Newton's binomial theorem andGauss's arithmetic-series formula. Building on the latter, we demonstrate thatBishop's modulus of convergence can be reduced to tabulatedvalues to assess the stability of a self-closed Ouroboros-type inductive-chainmodel for absolutely smooth functions in the sense of L.N. Bernstein. Keywords: inductive chains · Shannon entropy · constructive mathematics · proofmining · finitism · reliability criterion · thermodynamic analogyMSC 2020: 03F55 (Intuitionistic mathematics) · 03F60 (Constructive analysis) ·94A17 (Information measures)