Depth-Mode Spectrum and Stability

This record presents TA17 (Depth-Mode Spectrum and Stability), part of the Q5 Transport Architecture Series developed under the Zero-Point Hypothesis framework. TA16 established qualitatively that the continuum depth evolution dynamically suppresses high-roughness depth modes. TA17 makes this quantitative by computing the depth roughness/curvature functional explicitly for the natural mode bases on the depth interval d in 0, 1. For periodic modes, the roughness functional evaluates to (2pin) ⁴. For fixed-endpoint sine modes, it evaluates to \ ( (n*pi) ⁴ / 2 \). In both cases, the scaling law is the same: the roughness cost grows as the fourth power of the mode number n. This result is exact within the stated setup and follows directly from the structure of the second derivative operator; differentiating twice introduces a factor of n², and squaring and integrating gives n⁴. The stability hierarchy is therefore explicit. Modes n=0 and n=1 carry low roughness cost and are comparatively long-lived under TA15 evolution. Modes with large n carry roughness penalties scaling as n⁴ and are strongly and increasingly suppressed. Each doubling of spatial frequency increases the roughness penalty by a factor of 16. The roughness hierarchy is quartic even though the immediate diffusive damping rate from TA15 scales quadratically in frequency; these two scaling laws are distinct, and the corollary separates them explicitly. Three lemmas establish the result: the roughness of periodic modes (Lemma 1), the roughness of fixed-endpoint sine modes (Lemma 2), and the fourth-power suppression law common to both bases (Lemma 3). Three corollaries follow: the quantitative jolt cost (n⁴ per mode), low-mode dominance at late times, and the bridge to TA18, where the roughness spectrum is connected to K-channel leakage amplitude. The fourth-power scaling is robust across both boundary condition choices, differing only in the numerical prefactor. The mode expansions are continuum approximations to the finite transport spectrum of the underlying 320-slot depth graph; at the discrete level, the spectrum is indexed from n=0 to n=319 with a natural cutoff at n=160 imposed by the finite graph structure. No physical identification of mode number n with any observable quantum number, energy level, or particle property is claimed. The connection between high roughness cost and physical leakage amplitude is the subject of TA18. The theorem chain progressively derives the structure of the effective transport generator \ Gₑff = PiY G PiY + K†BK \, from which observable phase, leakage, decoherence, and residual correction emerge as structural consequences of projected transport closure on Q5.