The 7 nodes of the Fano plane — the combinatorial geometry underlying octo- nion multiplication — map exactly onto 7 orbital quantum dot (QD) modes in the Transdimensional Anomalous Hall Effect (TDAHE) ground state of rhombohedral graphene. The adjacency graph of the Fano plane is the complete graph K7: every QD couples to all 6 others, with the coupling sign and mediation structure deter- mined by the unique Fano line through each pair. The ternary gate F({i, j}) = |k⟩ (output is the unique third node on the Fano line through i and j) gives a complete 7-gate primitive set closed under G2. We prove that the Fano-Toffoli associator FT(i, j, k) = [(eiej)ek] −[ei(ejek)] is zero if and only if (i, j, k) lies on a Fano line — precision 1.000, recall 1.000, confirmed exhaustively over all 210 ordered triples. This makes FT a perfect hardware error detector: TDAHE ground states are Fano- line correlated (FT = 0 always); any perturbation exits the Fano line (FT ̸= 0 immediately). The chiral Bell measurement uses ABC/CBA-stacked interferometer pairs; fidelity per 3-step teleportation protocol is (1 −fbulk)3 = 89.3% from pion kinematics alone, versus 82±1% for Laneve et al. (2025) using engineered dissimilar quantum dots. Contents Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.
The 7 nodes of the Fano plane — the combinatorial geometry underlying octo- nion multiplication — map exactly onto 7 orbital quantum dot (QD) modes in the Transdimensional Anomalous Hall Effect (TDAHE) ground state of rhombohedral graphene. The adjacency graph of the Fano plane is the complete graph K7: every QD couples to all 6 others, with the coupling sign and mediation structure deter- mined by the unique Fano line through each pair. The ternary gate F({i, j}) = |k⟩ (output is the unique third node on the Fano line through i and j) gives a complete 7-gate primitive set closed under G2. We prove that the Fano-Toffoli associator FT(i, j, k) = [(eiej)ek] −[ei(ejek)] is zero if and only if (i, j, k) lies on a Fano line — precision 1.000, recall 1.000, confirmed exhaustively over all 210 ordered triples. This makes FT a perfect hardware error detector: TDAHE ground states are Fano- line correlated (FT = 0 always); any perturbation exits the Fano line (FT ̸= 0 immediately). The chiral Bell measurement uses ABC/CBA-stacked interferometer pairs; fidelity per 3-step teleportation protocol is (1 −fbulk)3 = 89.3% from pion kinematics alone, versus 82±1% for Laneve et al. (2025) using engineered dissimilar quantum dots. Contents Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.
The 7 nodes of the Fano plane — the combinatorial geometry underlying octo- nion multiplication — map exactly onto 7 orbital quantum dot (QD) modes in the Transdimensional Anomalous Hall Effect (TDAHE) ground state of rhombohedral graphene. The adjacency graph of the Fano plane is the complete graph K7: every QD couples to all 6 others, with the coupling sign and mediation structure deter- mined by the unique Fano line through each pair. The ternary gate F({i, j}) = |k⟩ (output is the unique third node on the Fano line through i and j) gives a complete 7-gate primitive set closed under G2. We prove that the Fano-Toffoli associator FT(i, j, k) = [(eiej)ek] −[ei(ejek)] is zero if and only if (i, j, k) lies on a Fano line — precision 1.000, recall 1.000, confirmed exhaustively over all 210 ordered triples. This makes FT a perfect hardware error detector: TDAHE ground states are Fano- line correlated (FT = 0 always); any perturbation exits the Fano line (FT ̸= 0 immediately). The chiral Bell measurement uses ABC/CBA-stacked interferometer pairs; fidelity per 3-step teleportation protocol is (1 −fbulk)3 = 89.3% from pion kinematics alone, versus 82±1% for Laneve et al. (2025) using engineered dissimilar quantum dots. Contents Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.
PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested. The Fano plane PG(2,2) — seven points, seven lines, three points per line — appears three times in quantum error correction: as the stabiliser geometry of the 3-qubit GHZ state, as the parity-check matrix of the 7-qubit Steane ⟦7,1,3⟧ code, and as the constraint graph of a device-independent self-testing protocol. We prove that these three appearances are facets of a single geometric object. The central result is a three-way identification: the non-trivial elements of the GHZ stabiliser group (Z₂)³\{0} are canonically bijective with the 7 points of PG(2,2) = GF(2)³\{0}, which are the columns of the [7,4,3] Hamming parity-check matrix. The 3-qubit GHZ stabiliser group generates the 7-qubit Steane code's parity-check matrix. We prove an orbit decomposition of 3-qubit phase space: PG(5,2)\PG(2,2) = ⊔_{L=0}^{6} O_L, seven orbits of eight phase-space points (one per Fano line), giving a new Clifford-invariant magic label {p_L} that refines Wigner negativity into seven independent channels. The Fano game — measuring the seven GHZ stabilisers — achieves quantum maximum w=1, classical bound w≤5/7, and self-testing robustness C=7/8. The value 7/8 = |PG(2,2)|/2³ is a Fano-geometric invariant. Three practical tools follow: a device-independent GHZ certification protocol, the orbit-label magic invariant for magic state distillation, and a hardware coupling specification for a Fano-structured quantum processor (6 of 7 Fano lines at full coupling, one at r≈0.18, matching the FMO efficiency η≈0.1825). Keywords: quantum error correction, Fano plane, Steane code, GHZ state, magic states, self-testing, Wigner function, orbit decomposition, stabiliser codes, device-independent, PSL(2,7), quantum computing
PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested. A practitioner's primer for quantum computing researchers. No knowledge of octonions, sheaf theory, or category theory is assumed. Knowledge of stabiliser states, the Clifford group, and T-gates is assumed. The central message: the structure of the 3-qubit Pauli group is secretly governed by the Fano plane PG(2,2), and the Fano plane is the geometry of the 731 instruction set architecture (731 ISA) — a unified framework for quantum computation that organises every opcode as a Pachner move on a simplicial complex. The framework reveals three levels of quantum computational resource:• Level 0 (Clifford): stabiliser states; Gottesman-Knill simulable; the Fano lines are the free sector.• Level 1 (Standard magic): T-gates; Wigner function negativity; non-Clifford but accessible to the Origami ISA (regime 2).• Level 2 (Associamancy): the SPIN opcode; the Schur boundary; genuinely complex irreps of PSL(2,7); not accessible to any associative hardware. The Origami ISA (five opcodes: SPLIT, SPLAT, TWIST, FLIP, FLOP) subsumes Bell sampling, qudit stabiliser learning, and the abelian hidden subgroup problem as special cases. The 731 ISA adds SPIN and BIND, unlocking Level 2. This primer is a companion to "The Topology of Risk" (doi:10.5281/zenodo.20642983), which covers the same H⁰/H¹/H² framework applied to financial risk. Keywords: quantum computing, Fano plane, Origami ISA, 731 ISA, stabiliser states, magic states, associamancy, Schur boundary, primer, tutorial, hidden subgroup problem, MBQC, photonics
FINDING: The Fano plane encodes the non-associative multiplication table of octonions, and this algebraic structure underlies the Peres-Mermin square proof of quantum contextuality. MATH: Octonion multiplication is defined by 7 imaginary units e1,…,e7 with ei2=−1 and eiej=ϵijkek where ϵijk is given by the 7 lines (each a triple) of the Fano plane. The Peres-Mermin square is a 3×3 array of observables whose row and column products yield ±1, demonstrating contextuality via the Kochen-Specker theorem. No new constants or ratios appear. CONNECTION: The Fano plane is the projective plane of order 2, with 7 points and 7 lines, each line containing 3 points. This is a finite geometry with symmetry group PSL(2,7) of order 168. No direct link to golden ratio, base-60, or crystallographic root systems. However, the octonion algebra is intimately tied to the exceptional Lie group G2 (automorphism group of octonions) and the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com