Reconstructing Cosmological Singularity Structures:

This preprint introduces a reconstruction framework for cosmological singularity structures. Rather than treating singularities as terminal failures of classical spacetime, the paper proposes that they may be understood as reconstructable transition boundaries whose geometric, differential, observer-dependent, and diagnostic information remains encoded. The main features of this work are as follows: 1. Singularity as a reconstructable transition boundary The paper reformulates cosmological singularities not as endpoints of physical theory, but as transition structures that can be reconstructed, encoded, projected, and tested. This shifts the interpretation of singularities from terminal breakdown to recoverable physical structure. 2. Geometric reconstruction through Hesse, Hironaka, and Mori frameworks The framework begins with Hesse-type local degeneracy diagnostics and uses Hironaka-type resolution and Mori-type birational flips as reconstruction templates. These tools are not applied literally to Lorentzian spacetime; instead, they provide a controlled model-space logic for replacing collapse-dominated singular loci with structured transition domains. 3. Minimal P = 5 regular-singular transition system The paper introduces a minimal P = 5 Fuchsian-type system as the smallest extension beyond the classical Heun P = 4 multi-singular structure. The fifth regular singular point represents one additional transition channel associated with dimensional or domain bridging, avoiding the excessive freedom of higher multi-singular systems. 4. D-module absorption of singular behavior Using the Kashiwara–Oshima perspective on regular singular differential systems, the singularity is not removed by hand. Instead, it is absorbed into the algebraic and microlocal structure of the differential system, where monodromy, connection data, boundary values, and solution-space structure carry the reconstructed information. 5. Promotion of the earlier Tij framework to a fourth-rank transition tensor The observer-dependent tensorial structure previously denoted by Tij is promoted to a fourth-rank transition tensor, denoted by a calligraphic T. This tensor acts as the source kernel for localized metric reconstruction, wave-operator deformation, and observer-dependent projection. 6. Observer-dependent projection into physical observables The reconstructed singularity is not assumed to be directly visible. It becomes physical only after projection into an observer’s measurement frame, where it may appear as phase residuals, frequency deviations, damping modifications, chromatic propagation effects, or waveform residuals. 7. Analytical origin of the f^ (3/2) gravitational-wave phase diagnostic A key result is the derivation of an f^ (3/2) -type phase response from the square-root branch behavior of the fifth regular singular point. This scaling is not introduced as a phenomenological fitting assumption; it arises from the minimal P = 5 transition channel under geometric-optics propagation, weak transition coupling, and regular-singular monodromy. 8. Connection to parameterized post-Einsteinian gravitational-wave tests The paper connects the predicted f^ (3/2) phase response to the type of power-law phase deviations often modeled phenomenologically in parameterized post-Einsteinian gravitational-wave analyses. In this framework, the exponent is not guessed from data; it is fixed by the micro-geometric structure of the transition channel. 9. Diagnostic matrix for gravitational-wave observables The work introduces a diagnostic matrix linking reconstruction axes—observer dependence, irreversibility, domain transition, and direct diagnostic accessibility—to gravitational-wave observables such as ringdown residuals, phase deviations, damping-rate shifts, frequency deviations, and quasinormal-mode connection anomalies. 10. Falsifiable route toward observational tests The framework provides a direct route from abstract singularity reconstruction to testable gravitational-wave diagnostics. It is designed to be constrained or supported by current ground-based detectors such as LIGO, Virgo, and KAGRA, as well as future high-precision programs including LISA, Einstein Telescope, and Cosmic Explorer. In summary, this work proposes that singularities are not merely failures of classical theory. They may be reconstructable transition structures whose hidden information can be carried by regular-singular differential systems, absorbed into D-module-type solution spaces, projected by an observer-dependent transition tensor, and tested through gravitational-wave observables. Version History: v1. 0 — Initial preprint release v2. 0 — Expanded and revised preprint release. This version substantially extends the initial manuscript by consolidating the singularity-reconstruction framework into a complete mathematical structure. The revision clarifies the roles of Hesse-type degeneracy diagnostics, Hironaka-type resolution, Mori-type flips, minimal P=5 transition systems, observer-dependent tensorial projection, and gravitational-wave diagnostic pathways. It also strengthens the logical connection between cosmological singularity structures, transition-domain reconstruction, and falsifiable observational signatures.