Algebra-Relative Branch Multiplicity in Decoherent Histories: Orbit Reduction, Record Compression, and Logical Recovery

Everettian branch multiplicity is often discussed as though it were a count of wavefunction terms or decoherent alternatives. This paper argues that the relevant object is an algebra-relative packing number on physically admissible history classes. Candidate histories, already assumed to belong to a decoherent or approximately decoherent family, are first quotiented by exact physical symmetries and then compared only through distinctions visible in a specified operator system of admissible effects. The resulting effective branch packing number \( N_{\mathrm{eff}}(t;{\mathcal A},\varepsilon) \) is the maximal number of mutually \( \varepsilon \)-distinguishable symmetry classes of histories relative to \( \mathcal A \). The paper proves a sequence of structural results: an orbit-packing reduction theorem; monotonicity under refinement of the admissible operator system; metric packing bounds; polynomial and, under the natural affine-rank condition, sharp asymptotic record-compression theorems for collective linear records of identical bosonic qudits; contraction of channel distinguishability under recovery; and a logical cluster theorem for recovered channels. In particular, \( r_0 \) independent collective record directions give \( \Theta(N^{r_0}) \) record growth rather than the labeled \( d^N \) growth; and separated clusters of recovered logical channels, rather than microscopic histories, determine logical branch multiplicity. Applications include the three-qubit bit-flip code, bosonic qudit occupation sectors, and a formal orbit-record model for dipositronium annihilation histories. The framework relates decoherent histories, Quantum Darwinism, exchange symmetry, and quantum error correction without asserting literal merger of globally orthogonal branches.