Holographic Information Resonance Networks (H-IRN) I. Foundations, Nonrelativistic Limit, and Tsallis Non-Extensivity

We develop the foundational structure of Holographic Information Resonance Networks (H-IRN) and derive their nonrelativistic gravitational limit. The central idea is to describe the quantum vacuum as a polarizable medium whose coarse-grained free-energy density F(s)F(s)F(s) depends only on the dimensionless invariant s=g2/a02s = g^2/a_0^2s=g2/a02, where g=∣∇Φ∣g = |\nabla \Phi|g=∣∇Φ∣ is the local gravitational field and a0a_0a0 the MOND acceleration scale. Starting from a minimal set of axioms, we derive a rigorous bridge between (i) the microscopic IRN free-energy density F(s)F(s)F(s), (ii) an effective k-essence–type Lagrangian ftot(s)f_{\mathrm{tot}}(s)ftot(s) for the Newtonian potential, and (iii) the MOND/AQUAL interpolation function μ(x)\mu(x)μ(x) with x=g/a0x = g/a_0x=g/a0.
We obtain explicit formulae for the forward and inverse maps F(s)↔ftot(s)↔μ(x)F(s) \leftrightarrow f_{\mathrm{tot}}(s) \leftrightarrow \mu(x)F(s)↔ftot(s)↔μ(x) and prove that any admissible μ(x)\mu(x)μ(x) can be represented by a suitable choice of F(s)F(s)F(s). A general exponent relation α=1+β/2\alpha = 1 + \beta/2α=1+β/2 connects the small-field behaviour μ(x)∼xβ\mu(x) \sim x^\betaμ(x)∼xβ to the microscopic scaling F(s)∼sαF(s) \sim s^\alphaF(s)∼sα.
A central result is the derivation of the effective dimension deff=3d_{\mathrm{eff}} = 3deff=3 from holographic consistency requirements in de Sitter spacetime (the detailed proof is given in Paper IV of this series). This effective dimension implies Tsallis non-extensive statistics with q=4/3q = 4/3q=4/3 which, via the proven identity α=1/(2−q)\alpha = 1/(2-q)α=1/(2−q), yields the exponent α=3/2\alpha = 3/2α=3/2. By the relation β=2(α−1)\beta = 2(\alpha - 1)β=2(α−1), this gives β=1\beta = 1β=1, precisely matching the deep-MOND behaviour μ(x)∝x\mu(x) \propto xμ(x)∝x observed in SPARC/RAR galaxy rotation curve data. The universal exponents q=4/3q = 4/3q=4/3, α=3/2\alpha = 3/2α=3/2, and β=1\beta = 1β=1 are thus derived from first principles, with no free parameters.
This paper provides the nonrelativistic mathematical backbone for the QVP-G / H-IRN framework; questions of relativistic completion and cosmology are addressed in subsequent papers of this series.

The authors gratefully acknowledges the use of a novel multi-agent artificial intelligence system based on iterative Socratic dialogue as a conceptual support tool in the development of this work.