Remark. From real recurrences to the residue table [fcap-0016]

✍️sourceAGENTDRAFTED

The mixed \((1,1)\) recurrence removes matched axes but leaves a one-sided algebra \(\mathcal {C}\kern -2pt\ell _{r,0}\) or \(\mathcal {C}\kern -2pt\ell _{0,r}\), which it cannot classify. Together with the signature switch in Theorem [fcap-001X] and initial entries in Example [fcap-0015], the two recurrences below complete the calculation.

The two one-sided recurrences are \[\begin {aligned} \mathcal {C}\kern -2pt\ell _{p,q+2}&\simeq \mathcal {C}\kern -2pt\ell _{q,p}\otimes _{\mathbb R}\mathbb H,\\ \mathcal {C}\kern -2pt\ell _{p+8,q}&\simeq \mathcal {C}\kern -2pt\ell _{p,q}\otimes _{\mathbb R}M_{16}(\mathbb R). \end {aligned}\] The first line is the actual equivalence TauCeti.realCliffordQuaternionRecurrenceEquiv. For the second, the formal theorem supplies existence TauCeti.nonempty_realCliffordEightPeriodicityEquiv.

They yield the residue-indexed mod-eight classification TauCeti.realClifford_classification. The classical table is [lawson2016spin, I.4, Theorem 4.3 and Tables I--II, pp. 27--29].