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. 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. This paper proves that the GHZ-Fano game — a nonlocal game on seven constraints defined by the Fano plane PG(2,2) acting as the constraint graph of the GHZ stabiliser group — is a complete self-test: any quantum strategy achieving the maximum winning score must be locally unitarily equivalent to the GHZ state (|000⟩ + |111⟩)/√2 with Pauli measurements {σ_z, σ_x} for each party. The proof proceeds in four exact steps: A sum-of-squares (SOS) certificate {K_i = √(2/7) P⁻_i} whose common kernel is span{|GHZ⟩} — the state self-test. Fano incidence forces anti-commutation {M^Z_P, M^X_P} = 0 on each party's local observables. Jordan's lemma decomposes each party's Hilbert space into 2-dimensional Pauli blocks. Irreducibility of the M_2(ℂ) representation identifies each block with (σ_z, σ_x). All four steps hold for arbitrarily large (possibly infinite-dimensional) Hilbert spaces. The state self-test achieves linear robustness 1 − F ≤ 0.875ε rather than the generic √ε scaling. This follows from the uniform spectrum of the Fano constraint operator on the complement of |GHZ⟩ — itself a consequence of the transitive action of PSL(2,7) on the seven Fano lines. As a corollary, the game provides a device-independent protocol for certifying the Steane [[7,1,3]] stabiliser code without trusting the measurement apparatus. Keywords Self-Test, Nonlocal Games, GHZ State, Fano Plane, PG(2,2), PSL(2,7), Sum-of-Squares Certificate, Linear Robustness, Device-Independent Certification, Steane Code, [[7,1,3]] Code, GHZ Stabiliser, Anti-Commutation, Jordan Lemma, Quantum Entanglement, Bell Inequality, Quantum Verification, Origami ISA, W(5,2), Symplectic Polar Space, Quantum Error Correction
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 standard resource theory of magic treats magic as a scalar quantity — the total Wigner negativity of a quantum state. This paper shows that for 3-qubit systems, magic has additional structure: a valence, encoded in a 7-component orbit label indexed by the seven lines of the Fano plane PG(2,2). The 63-point 3-qubit symplectic phase space W(5,2) decomposes into the Fano plane (7 classical points) plus 7 orbits of 8 points each under the GHZ stabiliser group; each orbit corresponds to a distinct Fano line and carries a distinct magic valence. Just as chemical valence determines what bonds an atom can form, the magic valence of a quantum state determines what gates it can natively implement and what other magic states it can combine with. The central result is the CS Magic Composition Theorem: for any subset of controlled-S (CS) gates applied to the all-plus state, the negative-chi support is determined exactly by a symmetric-difference rule on the Fano geometry. Single CS gates produce primary-orbit magic; pairs produce secondary-orbit magic via the XOR-Fano rule; all three CS gates together produce magic exclusively in the three secondary orbits. The valence composition rule is bitwise XOR in Z₂³ — the Clebsch-Gordan rule of the representation ring R(Z₂³). The orbit label is stated in terms of the Pauli characteristic function χ rather than the Gross-Wigner function — a necessity, not a limitation, as a theorem of Raussendorf, Okay, Zurel and Feldmann establishes cohomological obstructions to Clifford-covariant Wigner functions for qubits. The characteristic function χ is the canonical magic indicator forced by these obstructions. Operational consequences: the orbit valence label distinguishes coherent gate-type errors (wrong CS gate applied to the wrong qubit pair) that are invisible to total negativity. The 7 stabiliser measurements of the Fano self-test simultaneously certify the GHZ state and read the magic valence label at zero additional circuit cost. Note: Earlier records (v1.0) used the term "magic type." Version 1.1 adopts the preferred term "magic valence" throughout, and adds a footnote connecting the concept to Queen's A Kind of Magic (1986). The mathematics is identical.
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 63-point 3-qubit symplectic polar space W(5,2) contains the Fano plane PG(2,2) as the 7-point GHZ stabiliser subspace. This paper proves that the complementary 56 points decompose into exactly seven orbits of eight points each under the action of the GHZ stabiliser group Z23, with one orbit per Fano line. This decomposition has a three-level nested structure. Level 0: The Fano plane PG(2,2) itself — the 7 GHZ stabilisers, the Wigner support of the GHZ state, and the difference set for all orbits. Level 1: The 7 orbit labels L=0,…,6 form a second copy of PG(2,2) under bitwise XOR: orbits OL, OM, OK satisfy OL⊕OM⊆OK iff L⊕M=K in Z23. This is the XOR-Fano composition rule governing magic state combination. Level 2: Each orbit OL splits into two disjoint Lagrangian 4-cliques OLA⊔OLB, each a maximal abelian subspace of W(5,2) — the same type of structure as the Fano plane itself. The Wigner negativity of any 3-qubit state decomposes as N(ρ)=∑LNL(ρ), where NL is the negativity on orbit OL. The orbit index L simultaneously identifies which Fano line is broken (activated) in the resource-theory sense, which magic valence the state carries, and which line is asymmetric in the 6-7 directed-routing architecture. Magic is broken-line activation in phase space. A remark establishes the formal analogy with spontaneous symmetry breaking: the PSL(2,7)→S4 symmetry reduction of the 6-7 architecture is the discrete finite-group analogue of SU(2)×U(1)→U(1) in the electroweak sector. The 6 intact Fano lines play the role of the massless photon; the 1 broken line plays the role of the massive gauge bosons; the activated orbit OL plays the role of the Goldstone mode; η>0 is the order parameter. The 7 choices of broken line are the 7 discrete vacua. The FMO complex is the biological realisation: the symmetric (η=0) configuration is evolutionarily unstable, and the broken-symmetry configuration is the stable ground state selected over a billion years — the closest natural analogue to a thermodynamically driven symmetry-breaking transition.
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
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
Projective Geometry as the Mother Tongue of Quantum Mechanics: From Brody & Hughston's CPn to the Fano Orbit Decomposition
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. Two programmes have placed projective geometry at the centre of quantum mechanics, arriving from opposite directions. The Brody–Hughston geometric quantum mechanics programme identifies the space of pure states with the complex projective space CPn, equipped with the Fubini-Study symplectic form. Our companion programme identifies the 3-qubit discrete phase space with the finite projective space PG(5,2) over GF(2), equipped with a binary symplectic form. We show that both are specialisations of a single abstract structure: a projective space over a field F carrying a non-degenerate alternating bilinear form and a central extension of its symplectic group. The central result concerns the Wigner functor W, sending each quantum system to its Wigner function. Numerical experiment shows that W is covariant under the real symplectic subgroup of the Clifford group (CNOT-type gates) but fails for phase gates (CS, S). The obstruction is the metaplectic character χ:Sp(2n,GF(2))→U(1) satisfying a group 2-cocycle condition. The precise unification is: Brody–Hughston's prequantum U(1) line bundle on CPn (the Maslov bundle, required for geometric quantisation) and the metaplectic character χ on PG(2n−1,2) (required for discrete Wigner covariance) are the same construction over different base fields — a central extension of the symplectic group by a circle. Three additional connections are established. (i) The Fano orbit decomposition of PG(5,2) — seven orbits of eight points, one per Fano line — is a purely discrete phenomenon with no continuous analogue in CPn; magic valence is intrinsically a GF(2) structure invisible to characteristic-zero geometry. (ii) The 6j symbols governing both programmes — SU(2) spin networks in the Brody–Hughston setting, Z23 Fano symbols in ours — are evaluations of the same abstract Racah coefficient for different groups; the Origami ISA is the spin-21 discrete limit of Penrose's spin-network geometry; Racah (1942), Mac Lane (1963), Penrose (1971), and the present work all evaluate the same Pentagon identity for different groups. (iii) The finite Klein correspondence (PG(5,2) points ↔PG(3,2) lines) suggests a discrete twistor interpretation of the magic-valence structure, directly connecting to Hughston's twistor programme. The unifying category MetaProjSymp(F) is defined, the Wigner functor is shown to be natural on it, and the bridge theorem is proved: Kibble's geometrisation of quantum mechanics, the Brody–Hughston programme, and the discrete Fano programme are all full subcategories of the same abstract construction, distinguished only by the choice of field.