The Informational Constraint Framework: Topology, Phase Transitions, and the Tensor-Rank Hierarchy

Classical thermodynamics successfully describes phase transitions and entropy as macroscopic functions of kinetic energy and temperature. However, it treats these physical states as fundamental, leaving the underlying ontological mechanism of matter and dimensional interaction unaddressed. This paper introduces the Informational Constraint Framework (ICF), positing that classical spacetime and physical matter are not fundamental objects, but rather lower-dimensional projections emergent from a deeper informational substrate. We redefine physical phase transitions as purely geometric phenomena occurring on a multidimensional constraint surface, driven by the topological collapse of available degrees of freedom (N) against the steepness of local informational cost gradients (Z). To formalize this projection, we introduce the Tensor-Rank Constraint Hierarchy, a unifying geometric structure revealing how all physical forces, material responses, and spacetime curvature arise from a single generative principle. By mapping the thermodynamic collapse to its corresponding tensor rank, we demonstrate that physical states are simply the geometric restriction of an informational pattern. Bridging Shannon entropy, Landauer's Principle, and Richard Feynman’s intuition of tensors as rules of response, we propose the Relaxation Principle: the axiom that all physical matter, and ultimately the rank-6 closure of gravity, are simply the substrate resting at the lowest informational cost permitted by local spacetime constraints. Furthermore, we demonstrate the strict dimensional consistency of this framework—proving that informational constraint density equates directly to classical thermodynamic work—and computationally ground the phase transition in the binary collapse of continuous macroscopic degrees of freedom. Version 2. 0 Major Updates and Additions: Unified Theoretical Architecture: Integrated the Tensor-Rank Constraint Hierarchy directly into the core thermodynamic equations of the ICF. The Feynman-ICF Tensor Matrix: Added a formal table (Table 1) mapping classical physical states (from scalar mass to the rank-6 closure of gravity) to their corresponding geometric derivatives on the constraint surface. Computational Falsification: Incorporated the "Benzene Snowflake" experiment directly into the manuscript as the Rank 4 falsification of substrate-emergent geometry. Dimensional & Microscopic Proofs (Appendices C & D): Appended rigorous proofs demonstrating the dimensional consistency of the constraint field (mapping abstract density to Joules via Landauer's Principle) and the discrete binary collapse of the 6-bit capacity limit. Visual Topology (Appendices A & B): Added full-scale topological diagrams illustrating the N-value and Z-value dynamics during phase collapse, alongside the full Tensor-Rank continuum graph. Version 2. 1 Update Notes: This revised manuscript (v2. 1) incorporates new empirical validation derived from high-performance computational sweeps. Following the initial theoretical establishment of the Rank 4 topological override (The Benzene Snowflake), the Informational Constraint Framework (ICF) has now been rigorously tested at the macroscopic limit. Volumetric Validation (New Section 4. 1): Integrates the recent 256³ supercomputer execution, proving that the topological collapse of the constraint surface remains mathematically stable across 16. 7 million simultaneous spatial coordinates. Updated Bibliography: Includes direct citations and links to the newly published computational data vaults and reproducible JAX tensor scripts that empirically verify the volumetric phase boundary.