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Unified Formalism of the Lawson Doubled Mirror Bidirectional Framework: Algebraic, Geometric, and Topolo

Amanda Lawson

Published Jun 1, 2026• 0 citations
Publisher

Abstract

This paper presents the unified mathematical formalism of the Lawson Doubled Mirror Bidirectional Framework, a branch-coupled bidirectional quantum evolution architecture constructed on a quadrupled Hilbert space with Clifford algebra $\mathrm{Cl}(1,3)$ structure.

DOI
10.5281/zenodo.20481531

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This paper presents the unified mathematical formalism of the Lawson Doubled Mirror Bidirectional Framework, a branch-coupled bidirectional quantum evolution architecture constructed on a quadrupled Hilbert space with Clifford algebra Cl(1,3)\mathrm{Cl}(1,3)Cl(1,3) structure.
I am not a physicist, but I see systems fully formed in 3 and 4D mentally in a synesthetic neurodivergence. I have been using AI to help translate what I see into formal language. This started with a tweak to the TSVF - The LANL "Arrow of Time" paper from March 2026. I saw a way to double space and "bank shot" photons in the double slit experiment in order to create symmetry. I am an outsider to the field and do not pretend to know more than I do - it is just that what I see without experience ends up matching established math and I feel like it is important. I am sorry that I do not know who to credit because I do not know well the research or context for what I am doing. If this ends up being nonsense so be it - I don't mind being another casualty to this transformative era of technology and of mind and mystery.This work presents the Lawson Doubled Mirror Recoherence Framework, in which the quantum state space is formulated as a four-dimensional Klein bottle (S3/Z2 S^3/\mathbb{Z}_2 S3/Z2) realized as a Möbius torus. The geometry is constructed from two conjugate implementations of the Clifford algebra Cl(1,3) \mathrm{Cl}(1,3) Cl(1,3), distinguished by opposite ±45∘ \pm 45^\circ ±45∘ Hadamard tilts and joined along their single edges by inter-mirror entanglement CAB C_{AB} CAB. The conventional Bloch sphere arises only as a three-dimensional projection of this structure, measurement corresponds to irreversible loss of information about which chiral implementation is realized, and time appears as a bidirectional, branch-coupled parameter.
This paper presents a unified algebraic framework for nnn-qubit quantum computation rooted in the complex Clifford algebra Cl(2n,C)Cl(2n, \mathbb{C})Cl(2n,C). We leverage the fundamental isomorphism Cl(2n,C)≅M(2n,C)Cl(2n, \mathbb{C}) \cong M(2^n, \mathbb{C})Cl(2n,C)≅M(2n,C) to directly identify the algebra of all nnn-qubit quantum operators (including states, gates, and observables) with elements of Cl(2n,C)Cl(2n, \mathbb{C})Cl(2n,C). This operator-centric perspective treats quantum states, whether pure or mixed, as density multivectors ρ∈Cl(2n,C)\rho \in Cl(2n, \mathbb{C})ρ∈Cl(2n,C) satisfying standard physical constraints (ρ†=ρ\rho^\dagger = \rhoρ†=ρ, 2n⟨ρ⟩0=12^n \langle \rho \rangle_0 = 12n⟨ρ⟩0​=1, positivity), and unitary gates UUU as multivectors obeying U†U=IU^\dagger U = IU†U=I. Quantum evolution ρ′=UρU†\rho' = U\rho U^\daggerρ′=UρU† and measurements p(m)=2n⟨Pmρ⟩0p(m) = 2^n \langle P_m \rho \rangle_0p(m)=2n⟨Pm​ρ⟩0​ are thus described by intrinsic Clifford algebraic operations. This contrasts with, yet encompasses, specific realizations like spinor representations based on Witt ideals (S=Cl(2n,C)∏jfjfj†S = Cl(2n, \mathbb{C})\prod_j f_j f_j^\daggerS=Cl(2n,C)∏j​fj​fj†​), which are shown to be particular constructions within this universal algebra. By analyzing Deutsch's algorithm through these different lenses, we demonstrate the generality and unifying power of the operator-centric multivector approach. This formulation not only provides a comprehensive algebraic language that naturally handles mixed states and general quantum processes but also opens avenues for applying the rich geometric and structural tools of Clifford algebras—such as grade decomposition and invariant theory—to gain deeper insights into quantum information, entanglement, and dynamics.
The earlier approach is used for description of qubits and geometric phase parameters, the things critical in the area of topological quantum computing. The used tool, Geometric (Clifford) Algebra is the most convenient formalism for that case. Generalization of formal complex plane to an an arbitrary plane in three dimensions and of usual Hopf fibration to the map generated by an arbitrary unit value element of even sub-algebra of the three-dimensional Geometric Algebra are resulting in more profound description of qubits compared to quantum mechanical Hilbert space formalism.