Paper 7 of this series constructed an explicit Clifford representation ρ: ZD*(4) → Cliff(0,7) and observed, via exhaustive computation, that the triple-product identity ρ̃(e_p)ρ̃(e_r)ρ̃(e_t) = φ · Q_{p,r,t} defines a bijection from the 7 Fano lines to the 7 non-zero elements of {I,Z}^⊗3, coinciding with the projective duality of PG(2,𝔽₂). That paper left open whether this duality is the X/Z duality of the Steane [[7,1,3]] code. This paper closes the gap and proves the following. Main Theorem (CSS Bridge, n=4). For every Fano line {p,r,t}, the Z-string Q_{p,r,t} equals the projective orthogonal complement of {p,r,t} in 𝔽₂³, and coincides with the unique Z-stabiliser of the Steane [[7,1,3]] code whose support is the complement {1,...,7} \ {p,r,t}. The correspondence is equivariant for the contragredient action of GL(3,𝔽₂): lines transform covariantly, their dual Z-stabilisers transform contravariantly via g^{-T}. Theorem (n=5 CSS Bridge). The analogue holds for n=5: every hyperplane W of PG(3,𝔽₂) maps to the unique Z-stabiliser of the [[15,7,3]] Reed–Muller code with support equal to the complement of W. Here M = I₄ (no bit-reversal needed). Proposition (M = I_{n-1} for all n ≥ 2). Under the MSB-first generator convention, the CSS bridge S_Z(v_W) = {1,...,2^{n-1}-1} \ W holds analytically for all n ≥ 2, with M = I_{n-1}. As a corollary, the sign-parity constraint π_A π_B π_C = −1 (proved in the series anchor paper) acquires a unified three-way interpretation: algebraically as a sign-parity discriminant, quantum-error-correcting as the phase of a balanced ZD triple, and projective-geometrically as the orthogonality relation between a Fano line and its dual point in PG(2,𝔽₂). A second corollary makes precise why no Clifford representation can lift the full octonion multiplication to a group algebra: the 28 non-Fano triples do not satisfy the cocycle condition, and no Z-type projective dual can be assigned to them. Part of a series on zero-divisors in Cayley–Dickson algebras. Series anchor: https://doi.org/10.5281/zenodo.20315751
Paper 7 of this series constructed an explicit Clifford representation ρ: ZD*(4) → Cliff(0,7) and observed, via exhaustive computation, that the triple-product identity ρ̃(e_p)ρ̃(e_r)ρ̃(e_t) = φ · Q_{p,r,t} defines a bijection from the 7 Fano lines to the 7 non-zero elements of {I,Z}^⊗3, coinciding with the projective duality of PG(2,𝔽₂). That paper left open whether this duality is the X/Z duality of the Steane [[7,1,3]] code. This paper closes the gap and proves the following. Main Theorem (CSS Bridge, n=4). For every Fano line {p,r,t}, the Z-string Q_{p,r,t} equals the projective orthogonal complement of {p,r,t} in 𝔽₂³, and coincides with the unique Z-stabiliser of the Steane [[7,1,3]] code whose support is the complement {1,...,7} \ {p,r,t}. The correspondence is equivariant for the contragredient action of GL(3,𝔽₂): lines transform covariantly, their dual Z-stabilisers transform contravariantly via g^{-T}. Theorem (n=5 CSS Bridge). The analogue holds for n=5: every hyperplane W of PG(3,𝔽₂) maps to the unique Z-stabiliser of the [[15,7,3]] Reed–Muller code with support equal to the complement of W. Here M = I₄ (no bit-reversal needed). Proposition (M = I_{n-1} for all n ≥ 2). Under the MSB-first generator convention, the CSS bridge S_Z(v_W) = {1,...,2^{n-1}-1} \ W holds analytically for all n ≥ 2, with M = I_{n-1}. As a corollary, the sign-parity constraint π_A π_B π_C = −1 (proved in the series anchor paper) acquires a unified three-way interpretation: algebraically as a sign-parity discriminant, quantum-error-correcting as the phase of a balanced ZD triple, and projective-geometrically as the orthogonality relation between a Fano line and its dual point in PG(2,𝔽₂). A second corollary makes precise why no Clifford representation can lift the full octonion multiplication to a group algebra: the 28 non-Fano triples do not satisfy the cocycle condition, and no Z-type projective dual can be assigned to them. Part of a series on zero-divisors in Cayley–Dickson algebras. Series anchor: https://doi.org/10.5281/zenodo.20315751
We establish three results connecting the zero-divisor structure of the Cayley–Dickson algebras Aₙ to the stabilizer formalism of quantum error-correcting codes. Step 1 (Bijection, n=4). The 7 lo-flats of ZD(4) — weight-4 subsets {a,b,c,d} ⊂ {1,...,7} with a⊕b⊕c⊕d=0 — are in explicit bijection with the 7 non-identity elements of the X-stabilizer group of the Steane [[7,1,3]] code. All 7 lo-flats carry identical sign-parity pattern: π(EE|FF)=−1 (inactive), π(EF₁)=π(EF₂)=+1 (active). Step 2 (Operator translation). For every mixed flat F={i,j,8+k,8+l} in A₄, the algebraic identity π_A·π_B·π_C=−1 translates into the Clifford-group identity (Z_αZ_β)(X_αX_γ)(X_βX_γ) = −Y_αY_β. The global phase −1 is the precise Clifford avatar of π_Aπ_Bπ_C=−1. The operator −Y_αY_β is a detectable error with full-group syndrome weight 2^(n−2), proved for all n≥4 and verified on QPU. Step 3 (Extension to n=5, verified on QPU). The 15 Fano planes of PG(3,F₂) are in explicit bijection with the 15 non-identity elements of the X-stabilizer group S_X ≅ (Z/2Z)⁴ of the [[15,7,3]] code (4 generators, 2⁴−1=15 non-identity elements). The phase identity and syndrome formula extend to all n≥4 (Universal Phase Lemma). Syndrome structure verified on ibm_kingston (156-qubit QPU, job d8bm67ijki0s73aq6idg, 27 May 2026). The Type-A/Type-B criterion for Fano planes in PG(3,F₂) is resolved: a plane W is Type-B if and only if 8∈W or W={1,...,7}, where 8=(1,0,0,0)∈F₂⁴ is the bridge element between A₃ and A₄ in the Cayley–Dickson tower. All algebraic results are cited from the companion papers (doi:10.5281/zenodo.20315751, doi:10.5281/zenodo.20417859).Every claim is labelled [VERIFIED], [THEOREM], or [OPEN] throughout.
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. We articulate a structural parallel between the MIP*=RE theorem and the Origami Instruction Set Architecture (ISA). The algebraic rigidity of entangled-prover verification — exact-or-fail Pauli basis tests, low-degree tests, Mermin Magic Square constraints — corresponds precisely to the Frobenius axiom SPLIT∘SPLAT=1, the Pentagon identity, and the seven-point Fano plane PG(2,2) in the ISA framework. Six numerical experiments confirm the parallels: Pentagon unitarity holds exactly for all j ∈ {1/2,...,15/2}; Frobenius fidelity degrades as (1-p)^{2(2j+1)} under depolarising noise; the GHZ stabiliser group forms a perfect quantum Fano strategy satisfying all seven constraints simultaneously while no classical strategy can; G₂ non-associativity produces a sharp excluded-volume step function above j=7/2. This version adds a synthesis remark showing that the numbers 3 and 7 — the two dimensions where the cross product exists, gyroscopes are stable, and choreographic orbits can close — are not two separate coincidences but the same theorem: the Fano plane PG(2,F₂) has 7 points and 3 points per line because these are the imaginary unit counts of the octonions O and quaternions H respectively, forced by the unique projective plane of order q=2 over the field F₂. The broken Fano (6/7 architecture) and the magic/non-associativity conjecture are restated in this unified language. The practical implication is a conjecture that fault-tolerant quantum computing overhead is governed by the regime of the code's F-move amplitudes, with MSD-free computation possible at rung 1 and higher. Keywords MIP* Equals RE, Interactive Proof Systems, Quantum Entanglement, Pentagon Identity, Frobenius Axiom, Fano Plane, PG(2,2), GHZ Stabiliser, Origami ISA, Wigner 6j Symbol, Fault-Tolerant Quantum Computing, Magic State Distillation, Regime Ladder, Complexity Theory, Quantum Prover, Local Pauli Test, Low-Degree Test, G₂ Non-Associativity, Excluded Volume, Depolarising Noise, Rung-1, Quantum Verification, Adelic Simplicial Architecture
We characterize four groups of Clifford circuits for Calderbank--Shor--Steane (CSS) codes that are relevant to fault-tolerant logical operations. First, we show that every code-preserving Clifford circuit is a product of Z-diagonal circuits, composed of S and CZ gates, and their X-basis analogues. Second, we define the two-fold transversal group, generated by depth-one two-local code-preserving circuits, and show that each of its elements can be expressed as a product of layers consisting of either Z-diagonal, X-diagonal, or CNOT gates. As a corollary, every transversal gate is a product of three transversal diagonal circuits; for connected non-self-dual codes, two such circuits suffice.
We further show that every code-preserving automorphism circuit, consisting of single-qubit Clifford gates and permutations, has a normal form comprising a Hadamard layer, a permutation, and two diagonal circuits. We also define a two-fold automorphism group, in which a depth-one two-local circuit may be code-preserving up to a permutation, and show that its logical image can be larger than that of the two-fold transversal group.
For 231 CSS codes, we provide explicit generators and determine the logical image of the two-fold transversal group. We tabulate 173 codes whose full logical Clifford group is generated by two-fold-transversal circuits, including codes of distances 3, 4, 5, 6, 8, 10, 12, and 13, with respective best rates 2/5, 3/4, 34/77, 17/28, 14/25, 1/27, 3/56, and 1/33. We construct families of CSS codes from bipartite grids, cut-complements, and quadrics, and census the self-dual codes invariant under rank-3 permutation groups. Many of these codes realize the full logical Clifford group in this way.
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.