The Fractal Asymmetry Kernel II (FAK-2): Tail Structure, Moments, and Energetic Regimes

This archive accompanies the paper: The Fractal Asymmetry Kernel II (FAK-2):Tail Structure, Moments, and Energetic Regimes Scientific context This paper presents FAK-2, a focused empirical dissection of theFractal Asymmetry Kernel (FAK) previously identified in FAK-1. While FAK-1 established the existence, universality, and robustness of adomain-independent kernel governing normalized multifractal asymmetry updates,the present study examines the internal statistical structure of that samekernel in detail. FAK-2 treats the Fractal Asymmetry Kernel as a fixed, measurement-level empiricalobject and characterizes its: global distributional form, deviations from Gaussianity, tail behaviour and tail-index stability, energetic contribution of rare update events, regime structure across update magnitudes, dataset-level variation in tail accessibility, robustness under alternative normalization protocols. No new kernel is introduced.All analyses refer to the same Fractal Asymmetry Kernel defined in FAK-1. Definition and methodological scope The Fractal Asymmetry Kernel is defined operationally as the pooled empiricaldistribution of normalized actualization increments ΔZ=Z(n+1)−Z(n),\Delta Z = Z(n+1) - Z(n),ΔZ=Z(n+1)−Z(n), derived from subset-level fractal-dimension outputs of theUnified Temporal–Measurement Framework (UTMF). FAK-2 operates entirely in post-processing on archived UTMF metadata.No raw experimental data are accessed, and no UTMF computations are rerun. Although incremental ensemble construction was originally developed forEmergence Depth (ED) and structural analyses, it is used here solely as anobservational scaffold to generate large numbers of comparable local updateevents.No temporal interpretation, causality, learning, or evolutionary dynamics areassumed or inferred. Empirical scope and findings Using the same heterogeneous ensemble of 114 datasets(81 empirical and 33 synthetic) spanning: gravitational-wave strain measurements, cosmological maps, large-scale structure surveys, particle-physics event streams, pulsar timing residuals, atomic spectra, astrometric catalogues, quantum random number generators, FAK-2 demonstrates that: the Fractal Asymmetry Kernel exhibits a sharply peaked central core withsystematic, symmetric heavy tails; deviations from Gaussian behaviour occur already at moderate quantiles andpersist across the full support; finite, stable tail indices are observed over wide threshold ranges,indicating constrained heavy-tailed behaviour rather than scale-freedivergence; rare intermediate and extreme updates, while infrequent, contribute adisproportionate fraction of total update energy; a clear regime structure separates frequent small updates from energeticallydominant rare events; tail accessibility varies systematically across datasets and empiricaldomains, even under identical measurement protocols; kernel structure is robust under alternative normalization choices andestimation diagnostics. The Fractal Asymmetry Kernel is shown to be decoupled from Emergence Depth,hierarchical position, stabilization timing, and relational geometry.It therefore constitutes a local, measurement-level statistical constraint,independent of higher-order structural organization. No generative, causal, or physical interpretation is imposed.All conclusions are strictly empirical and descriptive. Contents of this archive This Zenodo release contains: the final PDF of the paper“The Fractal Asymmetry Kernel II (FAK-2): Tail Structure, Moments, and Energetic Regimes”; a paper-grade Python analysis cell that reproduces all figures and numericalresults reported in the paper directly from archived metadata and theaccompanying CSV; the same derived CSV file used in FAK-1, containing per-dataset structuraldescriptors (final ED50_{50}50 position, rank stabilization timing, volatilitymeasures, insertion-impact statistics); the exact Python cell used to generate this CSV, included unchanged fortransparency and provenance; a README.md describing execution, dependencies, and full reproducibility. The CSV filename encodes the timestamp of the source JSON snapshot, ensuringexplicit traceability between metadata and analysis outputs. Data dependency All analyses rely on the following archived metadata snapshot: UTMF v5.2 FULL_DETAILS metadata archiveFilename: FULL_DETAILS_utmf_v5.2_20260112_011327 UTC.jsonDOI: https://doi.org/10.5281/zenodo.18226887 This JSON archive contains complete subset-level multifractal outputs for alldatasets processed under an identical UTMF v5.2 configuration. All Fractal Asymmetry Kernel quantities—including normalized update increments,distributional diagnostics, tail analyses, energetic decompositions, andconditioning statistics—are reconstructed deterministically in post-processingfrom this archive. The included CSV is a derived artifact produced exclusively from this snapshotusing the provided generator cell. Scope and reuse This release documents a fully reproducible, data-driven statisticalcharacterization of the internal structure of the Fractal Asymmetry Kernel. It is intended as a reference for: empirical analysis of heavy-tailed update statistics in multifractalmeasurements; cross-domain comparison of incremental emergence behaviour; methodological studies of tail estimation, regime structure, and energeticdominance; future work on kernel parametrization and constrained stochastic modelling. The archive does not include: raw experimental measurements; UTMF execution code; the FULL_DETAILS metadata itself (archived and cited separately). Relation to prior and subsequent work FAK-2 builds directly on: Fractal Asymmetry Theory (FAT), the Unified Temporal–Measurement Framework (UTMF v5.2), the original identification of the Fractal Asymmetry Kernel in FAK-1, earlier Emergence Depth (ED) and structural analyses. It is analytically distinct from work on hierarchical embedding, post-emergenttrajectories, confinement, or incremental growth dynamics. FAK-2 serves as the distributional dissection of the kernel identified inFAK-1 and provides the empirical foundation for future studies of constrainedupdate dynamics and kernel-level modelling. All upstream software, metadata archives, and datasets are archivedindependently and cited explicitly in the paper.