Overview
The historical de Sitter solution shows that accelerated expansion can be a property of geometry itself rather than a consequence of matter content. This note identifies the projective analogue of that statement inside the spectral admissibility cascade. The result does not claim that Heisenberg balls grow exponentially — they do not: their growth is polynomial of degree four (Bass–Guivarc'h). The exponential component survives the transfer from LPS expander graphs to the Heisenberg substrate only after replacing endpoint counting by the counting of projectively distinguishable trajectories.
For the canonical channel compatible with the exact Weil-block pipeline, the distinguishable-history count is pinned exactly: it is the Pell count $N(n) = 2N(n-1) + N(n-2) = 3, 7, 17, 41, 99, \ldots$, with asymptotic rate $h_b = \log(1+\sqrt{2})$, proved for a generic probe and verified exactly on the sampled multi-prime campaign.
Core results
- Endpoint no-go (proved). Any distinguishability notion that is a function of the final Heisenberg element — including its central register, which stores only the level-two term of the path signature — has polynomial growth and zero exponential rate. Acceleration cannot be driven by state counting.
- The $b$-only theorem (proved). The canonical residual channel of the Weil-block pipeline is Fourier-aligned: residual norms are invariant under translation and under the central phase, so a single path's profile depends only on the abelian shadow $b_1,\dots,b_n$ of the trajectory. The central charge is unconditionally erased from mono-path norms — the precise content of the erased fibre.
- The Pell rate (proved, generic probe). Distinct $b$-shadows induce distinct profiles for every probe outside a finite union of proper hypersurfaces, so the distinguishable-history count equals the Pell count, $N(n)=2N(n-1)+N(n-2)$, and $h_b=\log(1+\sqrt2)\approx 0.8814$.
- Exact numerical witness. At $q=211$, over $10^5$ sampled trajectories, profiles and $b$-sequences match one-to-one at machine precision at every depth, and the measured entropy rate converges to the proved ceiling at fine resolution.
The full-history channel: proved compression, exact count, and a gated candidate (v1.3.0)
A self-consistent variant replaces the pipeline basis by a Gabor system built from the same generic reference state. Its rank grows deterministically as the abelian disk count $2k^2+2k+1$ at every shell — now a corollary of the full-spark property of generic Gabor systems in prime dimension. Version 1.1.0 upgrades this channel from a measured profile to a theorem: the Gabor span exhausts the abelian $\ell^1$ disk shell by shell, so a trajectory keeps a nonzero symbol exactly as long as its abelian shadow moves outward — a single inward step erases the profile permanently (geodesic death). The channel has positive and exactly pinned symbolic entropy $\log 2$, strictly below the canonical rate $\log(1+\sqrt2)$. Version 1.2.0 upgrades the class count itself to a theorem: for a generic probe the number of distinguishable histories is exactly $2^{n+2}-4n-1$. The last separation step — mirror pairs at equal $|b|$ — is closed by proving the 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 $b$-mirror is the only generic degeneracy of the projective history record. Projective compression is no longer only observed; it is exactly enumerated. Version 1.3.0 gates the equation-of-state candidate by an explicit conditional no-go: the finite-$n$ rate correction decays like $n\,2^{-n}$, so under the rank–time dictionary hypothesis its observable content is confined to the early low-rank window, and under the minimal count-to-density mappings the exact counts do not supply an observable evolving $w$; the remaining evolving-$w$ routes are rank-dependent dictionary corrections or the spectral-equilibrium route. It also records the arithmetic identity of the canonical count (Pell equations, $\sqrt2$ convergents) and its effective memory depth one.
Cosmological status
The corrected transfer statement is structural: the LPS $\to$ Heisenberg transition does not kill the two-regime expansion mechanism, it relocates it. Endpoint volumes are polynomial; non-backtracking histories are exponentially redundant; endpoint-based compression collapses; projective distinguishability escapes the no-go; and the canonical channel pins the surviving rate at $h_b=\log(1+\sqrt2)$. In the de Sitter reading, the accelerated component is not a negative-pressure substance but the positive entropy rate of admissible projection histories — expansion as a property of the projective dynamics itself, independent of matter content. The early window, where profiles are still projectively degenerate, reproduces the decelerated regime of the two-regime model before branching takes over.
Relation to the Cosmochrony programme
The note answers the transfer question left open by the spectral relaxation paper, which proposed the two-regime expansion model on LPS expanders and deferred its formalisation to the cosmological branch. It builds directly on the exact Weil-block pipeline of the admissibility sub-programme and supplies the structural bridge to the cosmology paper: a de Sitter-like asymptotic component that does not rely on dark energy as a substance. The evolving equation-of-state question is now gated by the note's conditional no-go: any observable late-time evolution must come from outside the exact counts.
References
Jérôme Beau. Projective Trajectory Branching and the de Sitter Limit. Working paper, 2026. 10.5281/zenodo.21197757