Short result note of the Cosmochrony programme (bib key Beau2026tb).
The transfer of the spectral admissibility cascade from LPS expanders to the Heisenberg graph Heis3(Z/qZ) destroys exponential growth at the level of endpoint volumes (Bass-Guivarc'h degree 4), but not at the level of projectively distinguishable histories. The note defines a hierarchy of trajectory distinguishability notions (D1-D4), proves the endpoint no-go, proves that the canonical O12-compatible residual channel is b-only (the central charge is unconditionally erased from mono-path norms), and pins the canonical branching rate exactly: the distinguishable-history count is the Pell count N(n) = 2N(n-1) + N(n-2), with asymptotic rate h_b = log(1+sqrt(2)). Profile separation is proved for a generic probe (finite union of proper hypersurfaces) and witnessed exactly on the sampled multi-prime campaign (q in {29, 61, 101, 211}). The full-history Gabor channel (V3) is itself a theorem since v1.1.0: the Gabor span exhausts the abelian l1 disk shell by shell, so nonzero symbols are confined to l1-outward geodesics (geodesic death: one inward step erases the profile permanently), and for a generic probe (Lawrence-Pfander-Walnut full spark plus a Chebotarev-based separation lemma) the distinguishable-history count is exact: since v1.2.0, N(n) = (N_class+2)/2 = 2^(n+2) - 4n - 1 is a theorem on the window q > (4n-1)(2n+3), n >= 2. The last separation step (a-mirror pairs at equal |b|) is closed by proving generic non-vanishing of a mixed two-anchor coefficient -- a Hermitian jet at the orthogonal point of an anchor moment domain -- combined with a Klein rigidity property of outward geodesics; the channel has positive and exactly pinned symbolic entropy log 2, strictly below the canonical rate log(1+sqrt(2)). The count is witnessed exhaustively at q in {101, 211}. The finite-n compression profile is retained as the candidate input for an evolving effective equation of state, now gated by a conditional no-go (since v1.3.0): under the rank-time dictionary hypothesis and the minimal count-to-density mapping class, the exact counts do not supply an observable evolving equation of state -- their observable content is confined to the early low-rank window; the remaining evolving-w routes are rank-dependent dictionary corrections or the spectral-equilibrium route. Version 1.3.0 also adds two arithmetic remarks (Pell-equation identity of the canonical count; transfer-matrix formulation with effective memory depth one) and a labelled doubling-bracket reading.
Central result channel: V2 (generic probe, O12 basis), rate log(1+sqrt(2)). Full-history channel: V3 (Gabor basis), proved strictly compressive, rate log 2.
bash compile.sh
# Output: out/TrajectoryBranching.pdfScript and data: simulation/spectral/trajectory-entropy/trajectory_branching.py (same workspace),
outputs in traj_outputs/ (q{q}_traj.npz, trajectory_branching_h.pdf).
Exhaustive verification of the V3 theorem (all non-backtracking words, machine precision):
simulation/spectral/trajectory-entropy/v3_exact_check.py.
Certification of the mixed two-anchor coefficient (eps^2 identity, moment reduction, Hermitian jets):
simulation/spectral/trajectory-entropy/amirror_mixed_coeff.py and amirror_jet_certify.py.
Deposited on Zenodo, concept DOI 10.5281/zenodo.21197757 (latest version 1.3.0: conditional equation-of-state no-go, arithmetic and memory-depth remarks). Web page: https://cosmochrony.org/science/cosmology/trajectory-branching/