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 $\mathbb{Z}_2^3$, 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 $L=0,\ldots,6$ form a second copy of $PG(2,2)$ under bitwise XOR: orbits $\mathcal{O}_L$, $\mathcal{O}_M$, $\mathcal{O}_K$ satisfy $\mathcal{O}_L \oplus \mathcal{O}_M \subseteq \mathcal{O}_K$ iff $L \oplus M = K$ in $\mathbb{Z}_2^3$. This is the XOR-Fano composition rule governing magic state combination. Level 2: Each orbit $\mathcal{O}_L$ splits into two disjoint Lagrangian 4-cliques $\mathcal{O}_L^A \sqcup \mathcal{O}_L^B$, 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 $\mathcal{N}(\rho) = \sum_L \mathcal{N}_L(\rho)$, where $\mathcal{N}_L$ is the negativity on orbit $\mathcal{O}_L$. 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 $PSL(2,7) \to S_4$ symmetry reduction of the 6-7 architecture is the discrete finite-group analogue of $SU(2) \times U(1) \to U(1)$ 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 $\mathcal{O}_L$ plays the role of the Goldstone mode; $\eta > 0$ is the order parameter. The 7 choices of broken line are the 7 discrete vacua. The FMO complex is the biological realisation: the symmetric ($\eta = 0$) 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. 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. 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 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
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.
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. Nonlocal games provide an operational framework for distinguishing quantum from classical resources: quantum players sharing entanglement can win certain games with higher probability than any classical strategy. This paper introduces magic-valence nonlocal games — a family of games whose winning probability is determined not by entanglement alone but by the Fano orbit valence of the quantum state shared between players. The central result is that the seven Fano orbit valences {L=0,...,6} of the 3-qubit symplectic polar space W(5,2) correspond to seven distinct winning-probability tiers in the Fano Line Verification Game. Stabiliser states (L=0) win at the classical bound 1/7. Each non-classical orbit valence L≥1 achieves a strictly higher winning probability ω_L ∈ (1/7, 1], with the maximum achieved by states with full secondary-orbit support {4,5,6}. The magic valence mismatch theorem is proved: a player who injects magic of the wrong orbit valence — primary-orbit magic in a game requiring secondary-orbit magic — scores 0.472, strictly below the 0.500 classical bound. Wrong-valence magic is operationally worse than no magic for the intended game. This gives a sharp operational definition of magic valence: it is the resource that determines game-winning ability, not just total Wigner negativity. Five Clifford-equivalence classes of CS-family magic states are identified and their game distinguishability characterised. The maximum game distinguishability of 7/8 is proved: no single game can distinguish all seven orbit valences simultaneously, but 7 out of 8 can be separated by the Fano Line Verification Game family. The paper establishes the connection between magic-valence nonlocal games and the ORBIT opcode: the 7 stabiliser measurements of ORBIT are precisely the measurements needed to determine which game tier a state belongs to, at zero additional circuit cost beyond syndrome extraction. Keywords Nonlocal Games, Magic Valence, Fano Orbit, Orbit Valence, Quantum Advantage, Fano Line Verification Game, W(5,2), Symplectic Polar Space, Wigner Negativity, Magic State, Valence Mismatch, ORBIT Opcode, CS Gate, Clifford Equivalence, Game Distinguishability, Stabiliser States, Quantum Resource Theory, Bell Inequality, Fano Plane, PG(2,2), Origami ISA, Magic-ISA, TriQ, Classical Simulation Bound
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 Gottesman-Knill (GK) theorem establishes that Clifford circuits on stabiliser states are efficiently classically simulable. The simulability boundary is at zero magic: any state with positive Wigner negativity N(ρ) > 0 is beyond GK. This paper establishes that this boundary is not sharp enough. Two states can both lie beyond GK — both have N > 0 — yet differ in computational power in a precise, measurable way. The distinguishing invariant is the Fano orbit valence label {p_L}: the decomposition of the state's Wigner characteristic function over the seven orbit valences of the symplectic polar space W(5,2). The orbit valence provides a strictly finer simulability classification than total negativity N. The central new result is the magic valence mismatch theorem: injecting a magic state of the wrong orbit valence into a circuit that expects a different valence yields a winning probability of 0.472 in the Fano Line Verification Game — strictly below the 0.500 classical bound. Wrong-valence magic is not merely suboptimal; it is worse than no magic at all for the intended computation. Critically, this failure mode is invisible to syndrome-only decoders: CS₀₁, CS₀₂, and CS₁₂ states all produce identical stabiliser syndromes (+1)^7 but distinct orbit valence labels, detectable only by the ORBIT opcode. A valence-aware magic state factory using the ORBIT opcode accepts states by orbit valence, achieving distillation overhead O(1/p) per accepted state rather than the O(1/p²) of standard T-gate distillation — a ~1000× overhead reduction at physical error rate p = 10⁻³ for CS-family computations. The island graph picture of quantum circuit computation is introduced: entangled registers form islands connected by magic-carrying channels, and the computational power of the circuit is determined not by total magic N but by the magic current profile {I_L}. The ORBIT opcode serves as an ammeter for magic current: 7 Pauli measurements identify which orbit valences are flowing at each node. Keywords Gottesman-Knill Theorem, Classical Simulation, Magic State, Wigner Negativity, Fano Orbit, Magic Valence, Orbit Valence Label, W(5,2), Symplectic Polar Space, ORBIT Opcode, Magic State Distillation, CS Gate, Controlled-S Gate, Valence Mismatch, Island Graph, Magic Current, Fault-Tolerant Quantum Computation, Stabiliser Syndrome, Coherent Error Detection, Origami ISA, Magic-ISA, TriQ, SQU, Fano Plane, PG(2,2), Quantum Resource Theory, Simulability Boundary
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. WITHDRAWN BY THE AUTHOR (2026-08-02). The central claim of this record isincorrect and it should not be cited. The abstract states that two CZ-type stabiliser states with "identicalfidelity, identical stabiliser Renyi entropy, and identical Schmidt spectrum"can be distinguished only by a Pauli measurement named by Fano orbitmembership. The premise is false. For the paper's own witness statespsi_1 = H_0.CZ_01|+++> and psi_3 = H_2.CZ_12|+++>, the single-qubit Schmidteigenvalues are: cut 0: psi_1 = {0.5, 0.5} psi_3 = {1, 0} cut 1: psi_1 = {0.5, 0.5} psi_3 = {0.5, 0.5} cut 2: psi_1 = {1, 0} psi_3 = {0.5, 0.5} The Schmidt rank vectors are (2,2,1) and (1,2,2). These states are alreadyseparated by the Schmidt rank vector, the standard invariant the paper's owncomparison table invokes. Only the unordered multiset of spectra over all cutscoincides, which is a relabelling artefact rather than an invariant collision.The stabiliser Renyi entropies agree only because both are zero, as they mustbe for stabiliser states, so that agreement carries no information. The residual true content is textbook: a stabiliser state is determined by itsstabiliser group, and two distinct groups are separated by any element of thesymmetric difference. Here ZZI suffices. No finite geometry is needed toidentify that measurement. The structural background IS correct but is prior art not cited here. The 7+56decomposition of the 63 non-identity three-qubit Pauli operators, withcommutation as collinearity and maximal commuting sets as Fano planes, is thesymplectic polar space W(5,2). See M. Saniga and M. Planat, "Multiple Qubits asSymplectic Polar Spaces of Order Two", Adv. Stud. Theor. Phys. 1 (2007) 1-4,arXiv:quant-ph/0612179, and the subsequent literature by Saniga, Planat, Levay,Havlicek and Thas. Two further internal inconsistencies: the paper's GHZ/W comparison table namesSchmidt rank as the analogous invariant while the abstract claims Schmidtspectra are identical; and Theorem 4.1(v) states a single Pauli suffices when|<X|Y>|^2 = 0, while the witness pair has |<X|Y>|^2 = 1/4.