Spectral Gap and Finite-Level Cusp-Depth Diagnostics of the Weighted Farey Laplacian

Let F N be the Farey fractions of level N, and let L N be the unnormalized weighted graph Laplacian with edge weights w {uv}=(q uq v)⁻². We prove three results. First, the weights admit the exact Ford-circle representation w {uv}=exp(-(d(B infinity,B u)+d(B infinity,B v))), where d is signed hyperbolic distance between horoballs. The Ford geometry is classical; the weighting identity is the observation used here, and no uniqueness characterization is claimed. Second, a golden-convergent test vector gives lambda 2(L N) = c/(N^4 log N) for an unspecified absolute c 0. The selected parent edges form a spanning tree, denominators at least double down its descendant direction, and the resulting congestion is O(mN^4 log N). Thus the gap is bracketed to one logarithmic factor; neither a matching lower bound nor a numerical lower constant is proved. The BC-010 gap values through N=128 remain inherited reports. In contrast, the governed reproduction bundle replays the single level N=80 with Python 3.12.10, NumPy 2.4.6, one-thread parity-block diagonalization, generated JSON/CSV tables, and an exact-scope SHA-256 manifest. It gives m=1967, lambda 2 approximately 1.95557e-7 in the odd sector and lambda 3 approximately 1.95679e-7 in the even sector, with cross-sector relative separation approximately 6.221e-4. The inherited value 1.2e-3 is not reproduced under the declared 10 percent comparison tolerance and is replaced. The bottom-five projector has mean continued-fraction depth 22.9993 against graph mean 15.9985, while its mean Ford-horoball depth 2 log q is 8.75875 against graph mean 7.76982. These are distinct single-level observables. Reflection supplies invariant parity sectors but no pairing theorem, and no cross-level localization law follows. We close by separating volume and counting-measure isoperimetry. A separate same-system diagnostic at N=20,40,60,80 is also reported with its prior feasibility probe disclosed; it is descriptive, not preregistered, independent, or an asymptotic theorem.