What is hierarchical depth, physically? This document provides the constitutional answer within the Energy-Efficiency Theory framework: hierarchical depth L is the number of nested constraint boundaries that maintain a given subnetwork as a Being. It is the constitutional measure of how many times basal difference (L2) has been recursively selected, bounded, and maintained through the Encapsulate operation --- the Formation of a new constraint boundary around an existing subnetwork, with the subnetwork itself serving as the template. Version 3. 2 is a comprehensive constitutional alignment and structural deepening upgrade from v3. 1 (May 2026). It implants the full v6. 1 constitutional architecture --- the L0 Sole Meta-Axiom, the L1 Constitutional Presuppositions, the Five-Pillar Methodology, the Dual-Identity registration of Causality and Contradiction, the Bifurcation Duality Theorem, and the global terminological convention. It conditionalizes Barrier Asymmetry per Inertia v3. 4 Theorem I, corrects ``Constraint + Difference'' to ``Constraint + Being'' throughout per Difference v3. 3 Being face registration and CND v4. 0 third fatal correction, and anchors hierarchical depth in the full Existence--Difference--Distinction--Being--non-Being constitutional chain. The document establishes three irreducible constitutional foundations absent from v3. 1. First, the operational definition of hierarchical depth: Encapsulation is not a sixth independent constraint operation but Formation applied to a subnetwork, with the subnetwork serving as the template. Decapsulation is Meltdown applied to an encapsulating boundary. The triple L distinction (L / L₄₅₅ / Lₒ₄₂ₓₑ₀₋) is upgraded to CLOSED-in-EET governance, and Lₒ₄₂ₓₑ₀₋ 1/₁ is established as the only directly measurable depth measure. Second, the vibrational basis of temporal stratification: the constraint-network wave equation (Graph-Theoretic v3. 0, Part IV-bis) provides an independent vibrational foundation for the time stratification theorem (L) (L^/kB T₄₅₅), complementing the Arrhenius derivation. Deeply encapsulated constraints vibrate more slowly but persist longer --- they have longer natural periods (T₁ = 2/₁ L) and higher quality factors (Qₖ 1/ₖ). This explains the observed longevity gradient across hierarchical scales without requiring thermal activation. Third, the five-channel collapse framework: hierarchy collapse is not a single process but the joint effect of five independently irreversible entropy-production channels (formation, meltdown, maintenance, transient, and capture), each contributing distinctively to the loss of hierarchical depth. The hierarchy collapse theory is substantially deepened. Collapse is established as the LIFO trace of encapsulation history --- the outermost (most recently formed) encapsulation layer dissolves first because its meltdown barrier Eb^melt (L) L^ is lowest among all active layers. The depth-lifetime trade-off t^* (L) (- L) reveals that deeper networks have shorter lifetimes, all else being equal, because their exponentially larger maintenance burden consumes their finite power budget faster. The shadow hierarchy --- C (t) as structural memory --- is established: the fossil record of past meltdown events encodes the topological template of the network's historically deepest structure, enabling faster rebuilding up to the shadow depth Lₒ₇₀₃₎ₖ. The robustness illusion theorem (anchored in Inertia v3. 4 Theorem V) demonstrates that encapsulation can temporarily maintain functional robustness while inertial degradation accelerates --- systems crash abruptly, not gradually. A completely new part (Part VI) establishes the spectral and vibrational diagnostics of hierarchical depth. The ringing test enables layer-resolved measurement of quality factors Qₖ (L) across encapsulation levels, detecting deep-layer overdamping before global spectral gap decline. A fluctuation-dissipation theorem for hierarchical depth connects microscopic transient encapsulation fluctuations to macroscopic L relaxation. An inverse problem methodology enables inference of L from observable data (relaxation time spectra, 1/f noise exponents, event size distributions) when full constraint-network topology is unavailable. Graph entropy S (G) is established as a complementary diagnostic --- it can detect mesoscale structural complexity loss before ₁ signals global connectivity decline. The bridge system expands from approximately 13 to 46 bidirectional bridges, spanning the complete companion ontology network through June 2026. An Academic Priority Registration appendix establishes sixteen priority claims across four constitutional tiers, following the format precedent of Gravitation v3. 1 Appendix G. The document concludes with ten directly testable predictions (five original upgraded to PM-9 statistical compliance, five new) and thirty-six Deep Insights (twelve new). Hierarchical depth is the constitutional measure of how deeply a Being is nested. From the proton (L=1) to the universe (L), every L is a Being; every Being is matter; every matter is the substrate maintained at a specific depth of encapsulation. This document provides the complete constitutional foundation. Keywords: Hierarchical depth; encapsulation; scaling laws; hierarchy collapse; five-channel thermodynamics; vibrational diagnostics; spectral gap; complexity; Energy-Efficiency Theory
Hongpu Yang (Sat,) studied this question.