Overview
The white paper attributed the CMB large-angle power deficit to a global projective coherence constraint, illustrated by a two-parameter attenuation envelope. This note replaces that ansatz by a conditional mechanism with no adjustable parameter: under the rank–time dictionary, the largest angular modes probe the first admissibility ranks, whose number of projectively distinguishable histories is fixed exactly by the Pell count of the trajectory-branching note — $N_{\mathrm{hist}}(n) = ((1+\sqrt2)^{n+1} + (1-\sqrt2)^{n+1})/2$, so the first ranks hold only $3$, $7$, $17$ histories. A mode supported by so few distinguishable histories cannot carry the full ensemble variance if the capacity ceiling is saturated. The large-angle deficit is therefore tested here as a small-$N$ saturation benchmark, not presented as a derived prediction.
Core results
- No-fit saturation grid against Planck. Conditional capacity ceilings are crossed with two angular dictionaries and anchored only by $n=1 \leftrightarrow \ell=2$. With Planck PR3 calibration and the real-$m=0$ correction applied to the TT Bessel cell, its diagnostic $\chi^2$ changes from $27.26$ for $\Lambda$CDM to $17.70$ — zero fitted parameters. This measures proximity to the saturation curve of a bound; it is not a derived prediction.
- Honest polarization status. The minimal TE/EE propagation and the alternative capacity-ratio cells remain uncorrected for real-$m=0$. A dedicated low-$\ell$ polarization transfer treatment is open.
- Discrete signature. The envelope is a staircase in $\ell$ (integer admissibility ranks) — a structure no smooth damping law reproduces.
- Re-audited $8/3$ witness. The $S^3$ fibre-to-base stiffness ratio is geometric (uniform $S^3$ moments times the Born–Infeld factor), not tied to any graph substrate: $R = 2.666993 \pm 0.001188$ at $10^6$ samples, with its own observable, the modulation ratio $\sqrt{8/3} \simeq 1.633$.
Status
Imported. The exact Pell count of distinguishable histories (trajectory-branching note). Conditional theorem chain. The kernel and capacity ceiling hold under their stated support, independence, locking, and dictionary assumptions. Numerical, no fit. The saturation-grid outcome; the $\chi^2$ values are diagnostics (symmetrised errors, no multipole covariance), not a likelihood analysis or a detection claim. Open. Saturation of the ceiling, the angular dictionary, the consumer-side support and independence assumptions, the TE/EE transfer treatment, and the $\ell_2/\ell_1 \simeq 1.633$ modulation search.
Relation to the Cosmochrony programme
The note builds directly on the trajectory-branching note (source of the count) and refines the large-angle discussion of the cosmology paper; it corrects the phenomenological low-$\ell$ appendix of the white paper, whose two-parameter envelope becomes a descriptive diagnostic rather than the core claim.
References
Jérôme Beau. Low-Multipole Suppression from Distinguishable-History Capacity. Working paper, 2026. 10.5281/zenodo.21204352