Curvature-Induced Leakage Law

This record presents TA18 (Curvature-Induced Leakage Law), part of the Q5 Transport Architecture Series developed under the Zero-Point Hypothesis framework. TA17 established that depth modes carry roughness cost scaling as the fourth power of the mode number. TA18 connects this roughness spectrum to the leakage amplitude, where the leakage operator \ (K = I - PiY \) measures the component of a state that fails to align with the admissible gate mode. This is the first theorem in the depth architecture series to establish an explicit mechanistic chain: depth roughness drives gate misalignment, which drives leakage into the complement sector. The central structural assumption is that the admissible gate projection PiY acts effectively as a low-pass selector in the depth basis. This is motivated by two independent considerations: the Riemann-Lebesgue decay of Fourier coefficients for smooth functions (overlap with high-frequency modes decays as mode frequency increases), and the T122 structure of the gate mode, which selects a balanced slowly-varying phase configuration. Under this model, gate overlap scales approximately as \ (1/ (1 + c*n²) \) for depth mode number n, where c is a structural constant determined by the Q5 geometry of the gate mode. The leakage amplitude then satisfies the exact expression ellₙ = cn² / (1 + cn²). In the low-mode regime where \ (c*n² \) is small, this gives ellₙ approximately proportional to n². Combined with the TA17 result that roughness scales as n⁴, the two quantities are related by ellₙ proportional to the square root of the roughness functional in the low-mode regime. Leakage therefore grows monotonically with depth roughness but more slowly: quadratically rather than quartically. An important correction is stated explicitly: an earlier qualitative claim that leakage is proportional to curvature is too strong. The correct relationship is that leakage grows monotonically with roughness but sub-linearly in the roughness functional. Monotonicity and proportionality are distinct, and the distinction matters for the scaling hierarchy. The leakage saturates to ellₙ approaching 1 for high modes. This saturation is structurally significant: beyond a critical roughness scale, additional oscillation does not meaningfully increase misalignment because the state is already almost entirely outside the admissible sector. This behaviour is architecturally adjacent to the rupture regime identified in TA27 and the bear-clause destabilization in T88 and T89, though the full saturation-to-rupture transition is not derived here. The quadratic leakage scaling is model-dependent and should be verified against the explicit Q5 construction of PiY when available. The qualitative conclusion that leakage increases monotonically with mode frequency is robust across model variations. TA18 provides the leakage spectrum that feeds into TA19 (Observable Deviation from Leakage Spectrum), which connects the mode-dependent leakage amplitude to deviations in interference observables and probability outputs, closing the chain back to the T71 through T79 experimental prediction arc. 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.