Transport Papers T01. The Transport Principle: Total Division in Clifford Algebras via Minimal Completion

T01 of the Transport Papers. Version 3.0. 51 pages. Epistemic status: Tier A. The usual response to a zero divisor is exclusion. This paper, the root of the programme, takes the opposite: read the failure as an instruction. Division fails in a real Clifford algebra of nondegenerate form in exactly three ways: at isotropic vectors, central idempotents, and non-central idempotents. We prove the trichotomy exhaustive and, for each failure, the minimal repair and the theorem that makes it forced. From dimension three, isotropic vectors admit no equivariant repair at the level of elements, in the Clifford algebra of any extension: the null rotations shear every candidate partner, and the obstruction is a tautological cohomology class vanishing only for the hyperbolic plane. The ascent repairs instead the space of points: the reciprocal extends totally, continuously and uniquely to the null quadric of Cl(p+1,q+1), and the increment (+1,+1) is forced twice over, by a sign argument on lifts and an embedding-independent rank computation. Central idempotents ascend by one generator, making the grade involution inner and fusing the two ideals into Mat_2(Cl^0(p,q)), minimal among the simple real algebras into which the split algebra embeds unitally; the sign of the adjoined direction is a bit the algebra forgets. Non-central idempotents demand nothing, because eee = e where vvv = 0. Under iteration, Rule 1 fixes the mod-8 clock, fusion occurs at most once, and no unital homomorphism returns to the base. A closing section states the principle and a six-step algorithm. Every computational identity is machine-verified in exact rational arithmetic, except one flagged floating-point Moore-Penrose check. Position in the series: Within the series it relies on T0. The series is surveyed in T17, The Transport Programme, whose status table records every paper's tier and check count. Verification: verify_T01.py checks every computational identity in the paper in exact rational arithmetic (115 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_T01.py. T01_kernel.lean formalises the paper's associative-ring and finite-group kernel in Lean 4 with Mathlib (76 theorems, no sorry, no admit, no axiom of its own); it cites T1 by label. Files: T01_transport_principle.pdf; T01_transport_principle.tex; verify_T01.py; GA.py; T01_kernel.lean; 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.