Transport Papers T03. The Tower and the CAR Limit: Iterated Transport, the Frozen Clock, and Second Quantisation

T3 of the Transport Papers. Version 1.0. 17 pages. Epistemic status: Tier A. The transport principle of this series completes Cl(p,q) to Cl(p+1,q+1); the completed algebra carries a null cone of its own, so the completion invites iteration. This paper settles what the iterated tower is. Rigidity first: Cl(p+k, q+k) ≅ Mat_{2^k}(Cl(p,q)) canonically, so the transport charge p - q is constant along the tower and the mod-8 clock never ticks. Bott periodicity manifests not as eventual return but as immediate freeze: every floor is Morita equivalent to the ground floor, and the eight clock positions classify eight parallel towers rather than stages of one. Each rung adjoins one hyperbolic plane, and the adjoined ladder pairs satisfy the canonical anticommutation relations exactly, as vectors: the tower is a fermionic register, one mode per rung, with Fock modules doubling at every floor and the Witt index counting capacity. The limit theorem settles the second-quantisation question: for even transport charge the grade twist in the connecting embeddings is inner by Skolem and Noether, the algebraic direct limit is Cl(p,q) ⊗ Mat_{2^∞}, and the completed complexified limit is the ground algebra tensored with the CAR algebra of countably many fermionic modes. Iterated transport is second quantisation. The odd-charge case, where the twist acts on the centre, is delimited precisely and posed. The paper is pure mathematics: complete proofs throughout, with the handful of imported classical results stated as facts carrying their provenance. Every computational identity is verified in exact arithmetic in an accompanying script. Position in the series: Within the series it relies on T1, T4. The series is surveyed in T17, The Transport Programme, whose status table records every paper's tier and check count. Verification: verify_T03.py checks every computational identity in the paper in exact rational arithmetic (31 checks, all passing at deposit). It imports GA.py, the series' shared exact-arithmetic library, deposited alongside so the record is self-contained. Run with a bare Python 3: python verify_T03.py. Files: T03_tower_car.pdf; T03_tower_car.tex; verify_T03.py; GA.py; LICENSE-CODE.txt. Licence: the paper (PDF and LaTeX source) is CC BY 4.0. The code files in this record are additionally released under the MIT licence; see LICENSE-CODE.txt. Authorship and disclosure. The founding idea of the series is the author's: that division by zero in a Clifford algebra should be read not as a prohibition but as a transport into a higher-dimensional algebra, and that the transported division need not be accompanied by an inverse operation. The direction of investigation and the critical review of the results are likewise the author's. The mathematical development and the drafting of the text were carried out in extended collaboration with Claude Fable 5 (Anthropic), an AI system.