The Theory of Entropicity (ToE) Living Review Letters Series - Letter IA: The Entropic Rosetta Stone: How John Haller's Action-as-Entropy Anticipates and Validates the Theory of Entropicity (ToE) — A Deep Comparative Analysis of John Haller (2015), the Entropy–Action Equivalence Tradition, and Obidi's Theory of Entropicity (ToE)

This Letter presents a deep comparative analysis demonstrating that John L. Haller Jr.’s 2015 result — the exact identification of a diffusing particle’s self‑information with the classical action, — constitutes a precise conceptual precursor to Obidi’s Theory of Entropicity (ToE). Haller’s derivation, grounded in Shannon entropy, Hirshman’s Fourier‑linked entropies, Bernoulli diffusion, and Gaussian channel mutual information, shows that the principle of least action is a direct consequence of the second law of thermodynamics. This entropy–action equivalence aligns with and independently validates the foundational inversion at the heart of ToE: entropy is not a derivative thermodynamic quantity but the primary ontological substrate from which geometry, fields, and physical law emerge. By situating Haller’s work within the broader entropy‑as‑generator tradition — including Bekenstein, Jacobson, Verlinde, Padmanabhan, Frieden, and Jaynes — this Letter demonstrates that ToE provides the unifying variational field theory that completes this lineage. Here, we distinguish between the Haller–Obidi Actions: Haller’s particle‑level entropy–action identity and the Obidi entropic field action from which it emerges as a limiting case. The Obidi Action, the Master Entropic Equation, and the entropic α‑connection generalize Haller’s particle‑level correspondence into a universal entropic dynamics, establishing the Theory of Entropicity (ToE) as the natural culmination of six decades of converging insights linking information, entropy, action, and the structure of physical law. A central contribution of this Letter is the formal identification of the Obidi–Haller Correspondence, which reveals that Haller’s particle‑level entropy–action identity is the specific, calculable instance of the broader structural principle that the Theory of Entropicity (ToE) elevates to a universal law. Haller’s result establishes the Entropy–Action Equivalence in its most explicit form, showing that the classical action is an informational quantity arising from conditional entropy and mutual information. The Theory of Entropicity generalizes this insight by promoting entropy to a fundamental field whose variational dynamics generate geometry, fields, and physical law. The Obidi–Haller Correspondence therefore demonstrates that the Theory of Entropicity (ToE) is not an isolated theoretical construction but the natural completion of a deep, convergent pattern in modern physics: the recognition that action is entropic in origin, and that the entropic field is the true dynamical substrate of the universe. Thus, while the Obidi–Haller Correspondence expresses how Haller’s entropy–action identity emerges as the single‑particle limit of the Obidi entropic field theory (ToE), the Haller–Obidi Actions refer to the distinct variational structures at the particle and field levels. This Letter also situates Haller’s result within a deeper historical arc that begins with Louis de Broglie’s proposal of a hidden thermodynamic mechanism underlying quantum behaviour. Haller’s entropy–action identity provides the first precise informational signature of the process de Broglie intuited, while the Theory of Entropicity (ToE) elevates this structure to a universal field‑theoretic principle. Together they form what may be called the de Broglie–Haller–Obidi Entropic Lineage: a conceptual progression in which de Broglie envisioned a hidden entropic substrate, Haller revealed its mathematical imprint, and ToE constructs the full entropic dynamics from which geometry, fields, and physical law emerge.