What question did this study set out to answer?

The study aims to explain the phenomenon of grokking in neural networks using classical nucleation theory.

July 8, 2026Open Access

The Grokking Nucleation Theorem: A Classical-Nucleation-Theory Account of the Grokking Transition and a Goldilocks Temperature

JHJustin HartUniversity of Providence A(Aristotle (Harmonic)Laboratoire Spécification et Vérification

Key Points

The study aims to explain the phenomenon of grokking in neural networks using classical nucleation theory.
Models generalization as a phase transition using classical nucleation theory.
Employs machine-checked proofs in Lean 4 to verify results related to nucleation barriers and critical nucleus sizes.
Identifies a unique Goldilocks temperature that minimizes expected grokking delay.
The nucleation barrier is confirmed to match classical forms.
Critical nucleus size identified as a point of equilibrium.
Goldilocks temperature exists that uniquely minimizes grokking delay, characterized by strict convexity.

Abstract

Series S6 (Cognition this record links to the spine via isDerivedFrom the concept DOI 10. 5281/zenodo. 19317982, and references the S6 series anchor (concept 10. 5281/zenodo. 21002783). “Grokking” — a neural network's sudden jump from memorization to generalization long after training loss has plateaued — has resisted a first-principles account of *why* the delay exists and *why* it is non-monotone in an effective temperature (learning rate / weight decay ratio, or similar control). The Grokking Nucleation Theorem (GNT) models generalization as a classical-nucleation-theory phase transition: a generalizing sub-network must cross a barrier set by a competition between a driving force Δ and a surface-tension-like cost σ, exactly the same mathematical structure already certified elsewhere in the Viridis canon for reforestation (the Forest Nucleation Theorem and Afforestation Stewardship Theorem). GNT proves the resulting nucleation barrier matches the classical form, identifies the critical nucleus size, and — the headline result — proves the existence of a unique interior Goldilocks temperature T* at which the expected grokking delay is minimized, with the delay strictly convex and non-monotone around that optimum. Five results are machine-checked in Lean 4 (zero sorry, axioms ⊆ propext, Classical. choice, Quot. sound), module GrokkingNucleation. lean: (i) gntbarrierₑqcnt — the nucleation barrier takes the classical CNT closed form; (ii) gntcriticalₚoint — the critical nucleus size is a stationary point of the driving/tension competition; (iii) taugrokₙonmonotoneᵤniqueₘin — existence and uniqueness of an interior Goldilocks temperature minimizing expected grokking delay, via strict convexity; (iv) nucleationᵢbfloor — a combined kinetic/information-generation floor on grokking delay, monotone in the information-generation term; (v) gntₙonvacuous — an explicit interior witness. GNT joins the S6 Cognition not peer-reviewed.

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