A Deterministic Information-Geometric Holographic Boundary Framework: Irreversible Dynamics, Compact Projection, and Emergent Temporal Ordering

This version presents a deterministic, information-geometric holographic boundary framework in which irreversible temporal ordering arises from the structural properties of a finite-capacity, expanding boundary. The framework is architectural in scope: it specifies how irreversible ordering can emerge from boundary-localized transformation rules without introducing new physical interactions, stochastic dynamics, or fundamental time variables.

Bulk fields are mapped to the boundary through a sequence of classical operations, including diffusion, compact projection, geometric expansion, and nonlinear saturation. These operations suppress fine-scale structure, stabilize coarse modes, and generate an irreversible sequence of boundary configurations that can be interpreted as coarse records of bulk evolution. Temporal ordering is therefore associated with the non-invertibility of the boundary update process rather than with an externally imposed physical time parameter.

In this framework, “holographic” refers to the conceptual analogy of boundary-local encoding and retention of information. It does not imply physical holography, AdS/CFT duality, string theory, quantum-gravitational bulk-boundary duality, or a physical holographic principle. The construction is theoretical and architectural; it does not rely on quantum, statistical, gravitational, or thermodynamic assumptions as primitive components, although such domains may provide later contexts for comparison or interpretation.

The present version replaces the earlier multi-supplement exploratory structure with one shortened combined supplement. This combined supplement consolidates the formal and interpretive background required to follow the revised HBF construction, including the boundary-operator structure, compact projection, saturation, expansion, irreversible ordering, record formation, and the interpretation of boundary-local spatial symptoms. Earlier separate exploratory supplements are not carried forward in this version, because the present release adopts a more compact and conservative structure.

The associated Python implementations provide illustrative computational realisations of the boundary architecture. They are included for transparency, reproducibility, and internal inspection of the proposed model behaviour. The scripts should not be understood as empirical simulations, predictive physical models, or numerical validation of a physical theory. In the release version, the illustrative scripts are deterministic by default and generate static PNG/PDF outputs without requiring manual interaction. Optional dynamic preview modes are included only for internal inspection.

The computational examples should be read as illustrative model realisations of boundary-local record formation. Part 1 provides time-series and boundary-archive diagnostics. Part 2 contains four complementary visual regimes: an asymmetric boundary patch, saturation rings, a fragmented boundary pattern, and a frozen relic region. These regimes are not successive versions of a single preferred model, but alternative diagnostic examples showing different structural symptoms of irreversible boundary-local record accumulation.

Together, the main manuscript, the shortened combined supplement, and the associated Python scripts form a complete and self-contained description of the framework. Terminology is used structurally: capacity refers to the finite number of distinguishable configurations that the boundary can store; record or trace refers to persistent boundary configurations and their irreversible ordering; boundary symptoms refer to stable spatial patterns arising from the update rules; and compact projection refers to the loss of fine-scale bulk information during boundary encoding.

No external datasets were generated or analyzed. The work is theoretical and conceptual in scope and does not rely on empirical input, numerical calibration, or data-based validation.

A later and distinct development of related ideas is Alysis, introduced in “Structural Irreversibility and Alysis: A Diagnostic Complement to Entropy in Macrostate Physics” (DOI: 10.5281/zenodo.18165132). In that later framework, structural irreversibility is treated diagnostically through the assessment of whether macrostates remain structurally admissible after relational decay. The present work should therefore be read as a conceptual precursor to later Alysis developments, not as an empirical application of Alysis itself.