Projective Capacity Beyond Expanders: Polynomial-Growth Relaxation Graphs and the Capacity Exponent

The capacity exponent δ encodes the sub-linear energy-rank relation predicted by the Cosmochrony spectral framework. Prior work established a $k=3$ permutation-path fingerprint for measuring δ on Lubotzky--Phillips--Sarnak (LPS) expander graphs~{SpectralO8}, building on the discrete projective capacity state law of SpectralO7 paper. A geometric obstruction was identified: the exponential shell growth |Sₙ| ~ pⁿ on LPS compresses the entire pre-saturation window into only O(log q) BFS steps, preventing stable log-log slope estimation. We resolve this obstruction by replacing LPS with Cayley graphs of the discrete Heisenberg group Heis₃(Z/qZ), whose ball growth is polynomial: |Bₙ| ~ n⁴ (homogeneous dimension $D = 4$, Bass--Guivarc'h theorem). We transport the $k=3$ fingerprint and the discrete projective capacity framework of~{SpectralO7} to this new family, and prove a Window-Depth Theorem showing that a vertex window of O(q²) now spans Ω(q1/2) BFS steps. Exact BFS on primes q ∈ \11, …, 29\ confirms that the Heisenberg diameter grows as Θ(q), providing up to 5.4× more data points than LPS at $q=29$, with the ratio growing polynomially in q. The ball growth exponent D̂ converges monotonically toward $4$ as q increases, consistent with the Bass--Guivarc'h prediction. These results validate the geometric obstruction hypothesis of~{SpectralO8} and establish polynomial-growth Cayley graphs as the correct discrete setting for the subsequent extraction of δ in O10.