Admissible History Deformations and Central 2-Torsion Residues: The 𝘈₅/2𝘐 Case

<jats:p>A history-sensitive observable can be lost even when a chosen coarse endpoint or quotient is unchanged, so the relevant conservation question is not merely whether two states agree but which admissible deformations preserve the observable. We formulate this question algebraically using a deformation setup \(\mathscr{S} = {(F,G,Q,\phi,\text{proj},\text{lab},R_{p},\text{Adm})}\), where the kernel \(K = {\ker{(\text{proj})}}\) is central and has exponent at most two, while \(R_{p}\) records declared relation-preserving deformations. For \({D{(R_{p})}} = {R_{p}^{2}{\lbrack F,R_{p}\rbrack}}\), the residue is annihilated on \(D{(R_{p})}\), and its restriction to \(R_{p}\) therefore descends to a canonical homomorphism \(\bar\rho:R_p/D(R_p)\longrightarrow K.\) The quotient carrier is commutative of exponent two. Independently of Case A, we prove the exact criterion \({{\rho{({Hw})}} = {\rho{(H)}}}\Leftrightarrow{{\rho{(w)}} = 1}\), and hence \(\mathrm{ConservationOnAllowed}(A)\iff A\subseteq\ker\phi.\) The exact kernel \(\ker\phi\) is setup-specific, whereas \(D{(R_{p})}\) depends only on \((F,R_{p})\). A canonical quotient setup realizes equality of its residue kernel with \(D{(R_{p})}\), showing that Case A is the largest setup-independent, residue-blind subgroup guaranteed safe from \((F,R_{p})\) alone. On the bit domain \(R_{p} \cap \text{Adm}\), the residue-changing sector is exactly the nontrivial-residue part of Case B. We then specialize to \({SL{(2,5)}}\rightarrow{PSL{(2,5)}} \cong A_{5}\). The original torus witness is local to an embedded \(V_{4}/Q_{8}\) carrier, but a third primitive label globalizes the history alphabet: its labels generate all of \({PSL{(2,5)}} \cong A_{5}\) while the same commutator retains residue \(- I\) and remains an explicit Case-B witness. We further prove that, for commuting pairs in the concrete \(A_{5}\) base, failure of simultaneous commuting liftability is equivalent to Klein-four type. The associated central-cover commutator pairing is independent of lift choice, binary-valued on the commuting locus, and has nontrivial support exactly on the Klein-four locus. The paper separates these algebraic results from downstream physical realization: it does not claim a physical history bit, a cosmological initial bit, or a topological \(H^{2}{(T^{2};{\mathbb{Z}}_{2})}\) identification.</jats:p>