Essay IV of the Gradient Fractals suite executes the Informational layer of the ten-layer derivational chain. The preceding three essays have established the Gradient Fractal Field's ontological necessity (GF-I), algebraic-computational spine (GF-II), and geometric character D = 93/40 (GF-III). GF Essay IV now asks: what is the information-theoretic constitution of this field? What is the entropy rate of the single node, and how does that rate transform when Nₛat = 25 nodes operate in mutual Boundary contact? What information does each inter-node interface transfer, and how does the inter-node mutual information modify the total entropy budget of the fractal field at depth n? What is the information capacity of the Gradient Fractal Field, and what fraction of that capacity is irrecoverably committed to interface coordination rather than new registration? These questions are answered by derivation from the locked constants alone, with zero free parameters and full foreclosure of all alternatives. The derivation proceeds in seven movements. Part I re-derives the single-node entropy rate dS/dτ = log₂ (3) from the floor-function non-injectivity at d = 3 faces, establishing this as the foundational information-theoretic result (T. GF. ENT). Part II derives the face-weighted entropy: the logarithmic sensitivity analysis of G = (E×C) /F forces an exact three-way partition of log₂ (3), each face contributing log₂ (3) /3 = log₂ (∛3) bits per Chronon (T. GF. FWT). Part III derives the inter-node mutual information: the Boundary-Boundary contact forces a mutual information of (9/14) ×log₂ (3) bits per Chronon per shared interface (T. GF. IMI). Part IV derives the net Nₛat-node entropy budget: Hₙet (N) = (5N+9) /14 × log₂ (3) ; at N = 25: Hₙet = 67/7 × log₂ (3) ≈ 15. 17 bits/Chronon (T. GF. ENB). Part V derives the fractal entropy budget at depth n: Hfractal (n) = 25^ (n−1) × (67/7) × log₂ (3), establishing 25-fold amplification per depth level (T. GF. FEB). Part VI derives the information capacity, density, and the entropic clearance EC = 108/175: the fraction of potential information irrecoverably consumed by interface coordination (T. GF. ICP, T. GF. IDD, T. GF. ECL). Part VII establishes the co-constitutive synthesis. The profound finding of GF Essay IV: the fractal entropy budget Hfractal (n) = 25^ (n−1) × (67/7) × log₂ (3) and the fractal computational density Ωₜotal (n) = 25^ (n−1) × 134/9 are both forced by the same structural constants and exhibit the same 25-fold amplification per depth level. The information density per Ω-unit — (9/14) ×log₂ (3) bits per Ω per Chronon — is scale-invariant across all depths and equals exactly the inter-node mutual information per shared interface. The active fraction 67/175 ≈ 38. 3% governs both the computational density (T. GF. CPX) and the informational density (T. GF. ECL) — a forced structural identity between information and computation in the Gradient Fractal Field. Furthermore, the framework derives Shannon entropy log₂ (3) from first principles rather than assuming equal probabilities: the floor-function non-injectivity IS the equal-probability condition, making Shannon information theory a theorem of the Gradient Fractal Field rather than an independent axiomatic framework (R. I. 1).
Eugene Pretorius (Mon,) studied this question.