Casimir effect for a gapped scalar field: Exact crossover at the Compton scale

Version 4.0 Update Notes:
This version incorporates significant refinements, terminological clarifications, and the addition of cross-references to the unified theoretical framework established in the companion paper (Pillar 6: The Hard Gap Criterion).

Key updates applied across the series:

1. Unified Framework Integration: Added explicit cross-references linking this specific physical regime (dephasing, Casimir, Unruh, or Bekenstein bounds) to the universal thermodynamic scaling and universality classes of gapped baths rigorously derived in Pillar 6.

2. Self-Sufficiency: Ensured that the manuscript is fully self-contained and mathematically independent, making it ready for direct submission to peer-reviewed journals without requiring the reader to consult prior papers.

3. Terminological and Physical Precision: Corrected minor conjectures (e.g., the exact $1/T$ thermal scaling of the plateau threshold), refined physical terminology (e.g., specifying intermediate-time windows for gapped baths), and updated pedagogical framing to align perfectly with standard literature.

4. Bibliography: Updated the references to include the latest Zenodo DOIs for the complete six-paper series.