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

The capacity exponent encodes the sub-linear energy-rank relation predicted by the Cosmochrony spectral framework. Prior work established a k=3 permutation-path fingerprint for measuring 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 (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 (q^1/2) BFS steps. Exact BFS on primes q \11, , 29\ confirms that the Heisenberg diameter grows as (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 D 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 in O10.