Projective Trajectory Branching and the de Sitter Limit

The de Sitter solution shows that accelerated expansion can be a property of empty geometry rather than a consequence of matter content. This note identifies the projective analogue of that statement inside the Heisenberg spectral admissibility cascade. The result does not claim that Heisenberg balls grow exponentially. They do not: by the Bass–Guivarc'h formula their growth is polynomial of degree four. The exponential component survives only after replacing endpoint counting by projectively distinguishable trajectory counting. We define a hierarchy of trajectory distinguishability notions and show that any endpoint-based definition collapses to polynomial growth. The canonical O12-compatible residual channel is $b$-only and therefore bounded by the non-backtracking abelian-shadow entropy. For a generic probe, the residual profile separates the $b$-sequences on the pre-saturation window. Consequently, the distinguishable-history count is the Pell count \[ {N_b}(n)=2{N_b}(n-1)+{N_b}(n-2), \qquad {h_b}=\lim_{n\to\infty}\tfrac{1}{n}\log {N_b}(n)=\log(1+\sqrt{2}). \] The equality is witnessed exactly on the sampled histories of a multi-prime campaign ($q\in\{29,61,101,211\}$) and the genericity condition on the probe is proved to fail only on a finite union of proper real-algebraic hypersurfaces. The full-history Gabor channel is strictly compressive: a geodesic-death mechanism confines nonzero symbols to trajectories whose abelian shadow is an outward geodesic, and for a generic probe its distinguishable-history count is exactly $2^{n+2}-4n-1$ on the pre-saturation window, so the channel has positive and exactly pinned symbolic entropy $\log 2$, strictly below the canonical rate. The last separation step behind the exact count — mirror pairs at equal $|b|$ — is closed by proving the generic non-vanishing of a mixed two-anchor coefficient, through a Hermitian jet at the orthogonal point of an anchor moment domain. Its finite-horizon compression profile is retained as the candidate input for an evolving effective equation of state, now gated by a conditional no-go: under the minimal rank–time dictionary and count-to-density mappings its observable content is confined to the early low-rank window. Structural reading (interpretation, not a derived result): accelerated expansion, if inherited by the cosmological branch, is not driven by exponential growth of Heisenberg states but by exponential branching of projectively distinguishable histories, and the branching skeleton of that history space is now an exactly enumerated combinatorial object.