CORRECTION NOTICE (2026-08). The central mechanism does not work. The quantum dimension sin(pπβ)/sin(πβ) vanishes at β = 1/p for every integer p, not only p = 3, 5, 7, so it cannot select those three codes. The conventional explanations (quantum Hamming bound, Fano-plane structure) are not superseded. A revised version is in preparation. Three quantum codes are foundational in fault-tolerant architecture: triangular colour codes, the [[5,1,3]] perfect code, and the Steane [[7,1,3]] code. Their canonical status is usually explained by counting arguments (quantum Hamming bound) or algebraic coincidence (Fano plane geometry). We give a unified structural explanation: all three sit at snap events of the beta-deformed G2 spider, where quantum dimensions vanish at roots of unity beta=1/p for p=3,5,7. At each snap, a continuous G2 symmetry breaks and a discrete symmetry crystallises — spontaneous symmetry making (SSM). The snap at beta=1/5 forces BIND self-annihilation, the algebraic condition for saturation of the quantum Hamming bound. The snap at beta=1/7 crystallises PSL(2,7) as the transversal T-gate group of the Steane code.
CORRECTION NOTICE (2026-08). The central mechanism does not work. The quantum dimension sin(pπβ)/sin(πβ) vanishes at β = 1/p for every integer p, not only p = 3, 5, 7, so it cannot select those three codes. The conventional explanations (quantum Hamming bound, Fano-plane structure) are not superseded. A revised version is in preparation. Three quantum codes are foundational in fault-tolerant architecture: triangular colour codes, the [[5,1,3]] perfect code, and the Steane [[7,1,3]] code. Their canonical status is usually explained by counting arguments (quantum Hamming bound) or algebraic coincidence (Fano plane geometry). We give a unified structural explanation: all three sit at snap events of the beta-deformed G2 spider, where quantum dimensions vanish at roots of unity beta=1/p for p=3,5,7. At each snap, a continuous G2 symmetry breaks and a discrete symmetry crystallises — spontaneous symmetry making (SSM). The snap at beta=1/5 forces BIND self-annihilation, the algebraic condition for saturation of the quantum Hamming bound. The snap at beta=1/7 crystallises PSL(2,7) as the transversal T-gate group of the Steane code.
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
This paper establishes that the logical space of the [[13,7,3]]_3 ternary CSS Hamming code — the quantum stabilizer code arising from the Topological Vortex Logic (TVL) framework and studied in "Quantum Error-Correcting Structure of Topological Vortex Logic: The [[13,7,3]]_3 CSS Family and its Algebraic Geometry" (Zenodo DOI: 10.5281/zenodo.19983025) — carries a hidden action of the exceptional finite simple group G₂(3). The logical space of the [[13,7,3]]_3 code is a 7-dimensional F₃-vector space L carrying a faithful action of the code's automorphism group PSL(3,3). As an SL(3,3)-module, L is the characteristic-3 adjoint module psl₃ (the short-root form of the Lie algebra of type A₂, which is irreducible in characteristic 3), and L carries a UNIQUE invariant non-degenerate symmetric bilinear form B and a UNIQUE invariant alternating trilinear form φ, the Cartan form φ(X,Y,Z) = tr([X,Y]Z). The joint stabilizer of the pair (B, φ) in GL(7,3) is exactly the exceptional finite simple group G₂(3), of order 4,245,696 = 2⁶ · 3⁶ · 7 · 13. The order is established directly by an orbit–stabilizer count inside O(7,3), using only the classical order formula and the Cartan–Dieudonné generation of the orthogonal group, so it does not depend on the characteristic-sensitive form-stabilizer theorem. The physical code automorphism group PSL(3,3) embeds in this G₂(3) at index 756. The G₂(3) is a symmetry of the LOGICAL space alone: the same prime-3 arithmetic that produces the group obstructs it physically, since ν₃(13!) = 5 < 6 = ν₃(|G₂(3)|) forbids any embedding of G₂(3) into S₁₃, hence any faithful monomial action of G₂(3) on the 13 physical qutrits. All structural claims are reproduced from scratch over F₃ by the accompanying verification script. This is a result in finite group theory and quantum coding. It is not a statement about G₂-holonomy geometry, with which the finite group G₂(3) shares only a name and a root system; no geometric or physical (octonionic / compactification) reading is claimed. Two Python verification scripts with an accompanying README are included as ancillary files, reproducing the computational results from scratch over F₃ (standard library and numpy only). Published in conjunction with the TVL framework: "T³ as a Closed Information-Processing Environment" (10.5281/zenodo.20806554, Concept DOI) and "Topological Vortex Logic: Generation and Colour Structure from T³/Z₃" (10.5281/zenodo.19682633, Concept DOI); and building directly on the code-theory paper "Quantum Error-Correcting Structure of Topological Vortex Logic" (10.5281/zenodo.19983025, Concept DOI). Concept DOIs always resolve to the latest version. Changelog v1.0.0 (June 2026) Initial release.
This paper establishes the algebraic and geometric structure of the [[13,7,3]]_3 ternary CSS quantum stabilizer code and its dimension-indexed family, arising from the Topological Vortex Logic (TVL) framework introduced in "T³ as a Closed Information-Processing Environment" (Zenodo DOI: 10.5281/zenodo.19682633). The 26 stable winding states of TVL on the three-torus T³ form 13 antipodal pairs {w, −w}. These pairs are in natural bijection with the 13 points of the projective plane P²(F₃), and the Z₃ charge structure of TVL induces a parity-check matrix making the antipodal pairs the physical qutrits of the [[13,7,3]]_3 ternary CSS quantum Hamming code (Ketkar et al. 2006). This paper studies the structural properties of this code and its dimension-indexed family [[(3^d−1)/2, (3^d−1)/2 − 2d, 3]]_3 for d ≥ 2. Ten structural results are established: (1) Family construction and self-orthogonality: The CSS construction is valid at every level d ≥ 2, with H·H^T = 0 over F₃ proved for all d. (2) Double distinction of d=3: The TVL code is the smallest family member with positive logical dimension and the unique member among d ≤ 6 with prime physical qutrit count (n=13; the next occurs at d=7, n=1093). (3) PSL irreducibility (iff d=3): PSL(d,F₃) acts absolutely irreducibly on the logical space L_d = C_d/C_d^⊥ if and only if d=3 — proved at d=3 via Burnside's criterion (the 7-dimensional logical module). For every d ≥ 4 the action is reducible: L_d is a uniserial F₃[SL(d,F₃)]-module with exactly d−2 composition factors, by the Bardoe–Sin submodule structure of the GL(n,F_q) permutation module on PG(n−1,q). The earlier converse-of-Schur argument fails in defining characteristic, where dim End_G = 1 does not imply irreducibility. The logical space exists only on the self-orthogonal range q ∈ {2,3}, where the same d−2-factor structure holds in both fields — the binary factors being the trivial module and the exterior powers Λ^k V of the natural module — so that L_d is irreducible exactly for (q,d) ∈ {(3,3),(2,3)}: the TVL code and the Steane code [[7,1,3]] (Proposition 7). (4) σ-structural properties of the family: The shift automorphism σ_d is an isometry of F₃^(n_d) for the weight form; its eigenspaces are mutually symplectically orthogonal; its maximum Jordan block size on the logical space L_d equals 3^(ν₃(d)) (the 3-adic valuation of d); the σ-fixed Gram matrix has rank 1; and the construction is self-orthogonal over F_q if and only if q ∈ {2,3} (the Q-Scope theorem, via a Gauss power-sum identity). (5) Stabilizer weight enumerator: The distribution {A₀=1, A₉=104, A₁₂=624} is derived entirely from the line-incidence geometry of P²(F₃), with each sub-case verified independently. (6) Closed-form weight enumerator: A(ω) = i^(d mod 2) · 3^(n_d/2) for all d ≥ 2, with |A(ω)|² = 3^(n_d) an exact integer. The formula is specific to q=3: the magnitude equality |1+(q−1)ω| = |1−ω| = √3 holds uniquely at q=3. (7) Quantum weight enumerators: The Shor–Laflamme stabilizer and normalizer weight enumerators of the family are given in closed form. The stabilizer enumerator A_d(z) generalizes the d=3 line-incidence count to hyperplanes in PG(d−1, F₃); the normalizer enumerator B_d(z) is its F₉ MacWilliams transform; and the minimum distance d_min = 3 is recovered directly. (8) Non-standard quadratic form: The weight-mod-3 function on C_d/C_d^⊥ defines a non-degenerate quadratic form of Witt class +1 for d ∈ {3,4,5}. At d=3 this differs from the standard sum-of-squares form on F₃^7 (Witt class −1); at d=4 both forms coincide at Witt class +1. (9) Shell stratification as level sets: TVL's face, edge, and corner shells (|w|²=1, 2, 3) correspond precisely to the level sets Q₃=1, 2, 0 of the standard quadratic form on the parity-check space F₃³. (10) Polar duality and parabolic nesting: The q₃=0 sector of TVL is the polar of the corner point (1,1,1) with respect to the absolute conic. The q₃=0 line is σ-invariant. The family nesting d→d+1 corresponds to maximal parabolic subgroup actions in PSL(d+1,F₃). The paper applies the automorphism-group-action framework of Grassl and Rötteler (2013) to the ternary projective Hamming family and identifies three structural features not studied in that context: the closed-form weight enumerator, the family-wide Witt-class result, and the q=3 exceptionality identity. Five Python verification scripts with accompanying README are included as ancillary files, reproducing the computational results cited in the paper. Published in conjunction with the TVL framework archived at: 10.5281/zenodo.19682633 (Concept DOI, always resolves to latest version). Note on this record. Every computational claim in this version is reproduced from first principles by the five ancillary scripts, and the central results have each been re-derived by an independent method. The revisions reflect that standard: open characterization questions were resolved one by one, and the result shown to be overstated — the PSL-irreducibility characterization — was corrected in the open, sharpening the paper's central claim that [[13,7,3]]₃ is the unique ternary code in its family with an irreducible logical space. Version 1.0.7 is the current, self-contained version. Changelog v1.0.7 (June 2026) Self-contained version: the version-history scaffolding is removed and one result is added to Section 4.1. Version-history scaffolding removed from the paper. The version note (which documented the v1.0.5 PSL-irreducibility correction and the v1.0.6 weight-enumerator additions) is removed, and the title page no longer carries a version marker or version-note pointer; the paper now reads as a standalone work. The full revision history is retained in this changelog (below) and in the Zenodo version record.2. Section 4 — composition structure across the self-orthogonal range (Proposition 7, new). The former "General q" remark — which had incorrectly stated that L_d = C_d/C_d^⊥ is defined for general q and that the ternary d=3 code is the unique irreducible instance — is replaced by Proposition 7: L_d exists only for q ∈ {2,3} (the self-orthogonal range); in both fields it has exactly d−2 composition factors (for q=3 of dimensions [x^j](1+x+x²)^d; for q=2 the trivial module together with the exterior powers Λ^k V of the natural module); and it is irreducible exactly for (q,d) ∈ {(3,3),(2,3)} — the TVL code [[13,7,3]]₃ and the Steane code [[7,1,3]]. The q=2 structure is classical (Bardoe–Sin for all q; the exterior filtration of the elementary-abelian group algebra) and is cited, not claimed as new. The dim End_G(L_d) = 1 remark — explaining why this does not force irreducibility in defining characteristic (3 | |SL(d,F₃)|, so L_d need not be semisimple; at d=4 reducible-but-indecomposable with End_G(L_d) = F₃) — is retained as a standalone caution. A previously unnumbered remark becomes a numbered proposition (Proposition 7), so the subsequent theorems, propositions, and corollaries shift up by one — for example the closed-form weight-enumerator theorem is now Theorem 14 and the family-wide Witt-class theorem is now Theorem 19 (it was Theorem 18 in v1.0.6). All cross-references are symbolic and update automatically; the ancillary verification scripts and README are updated to the new numbers. The paper remains 19 pages and compiles with no undefined references.3. The open-questions section is streamlined to the single genuinely open item, physical realisation; the questions resolved in earlier revisions (the closed-form classical and quantum weight enumerators, the family-wide d≥4 minimum distance, general PSL irreducibility, and now the general-q module structure via Proposition 7) are presented as results in the body.4. Conclusions item 3 is extended to record the two-field characterization and the Steane instance.5. Verification scripts: a fifth ancillary script, verify_logical_both_fields.py, is added; it reproduces the Proposition 7 q=2 composition factors, the well-posedness q ∈ {2,3}, and the irreducibility criterion by an independent method (exterior filtration and short exact sequences). The four earlier scripts are unchanged.6. Title page and date updated to v1.0.7. v1.0.6 (June 2026) Completes the weight-enumerator treatment and refines the open questions. This revision adds results: subsequent theorems and propositions are renumbered upward (for example the family-wide Witt-class theorem is now Theorem 18), with all cross-references symbolic and updating automatically.1. Classical weight enumerator (Theorem 13). The closed-form weight enumerator of the family is given as the complete polynomial W_d(z), with its value at omega = e^{2 pi i/3} — the Gauss-sum value i^{d mod 2} 3^{n_d/2} — recorded as Corollary 14. The d=3 expansion is given explicitly; its coefficient of z^11 is |PSL(3,3)| = 5616.2. Quantum weight enumerators (new Section 6.4). The Shor–Laflamme stabilizer and normalizer weight enumerators of the family are given in closed form (Propositions 15 and 16). The stabilizer enumerator A_d(z) generalizes the d=3 line-incidence count of the stabilizer-enumerator proposition (Section 6.1) to hyperplanes in PG(d−1,3); the normalizer enumerator B_d(z) is its F_9 MacWilliams transform; and the minimum distance d_min = 3 is recovered directly from the pair (Corollary 17).3. Open Question 2 closed. Items 1 and 2 together close the open question on the closed-form classical and quantum weight enumerators for the family.4. Open Question 1 narrowed (general q). The general q = p^t module structure is determined by Bardoe–Sin (already cited in the paper): Frobenius-twisted composition factors, irreducible only in special cases, the prime-field case uniserial; the genuinely remaining item is a self-contained description of the logical subquotient C_d/C_d^⊥ for general q.5. Open Question 5 closed (minimum distance). The family-wide d_min = 3 is established in Ketkar, Klappenecker, Kumar, and Sarvepalli (2006), and is now also exhibit
CORRECTION NOTICE (2026-08). The figure "4.3× directed magic-routing efficiency", quoted three times and attributed to a companion paper, does not appear in that paper. It is unsourced and should be disregarded. A revised version is in preparation. The Origami Instruction Set Architecture (O-ISA) was introduced alongside a conjectural exotic hardware substrate — the Resonance Processing Unit (RPU) — requiring non-associative materials to support octonion computation. This paper establishes that the base Origami ISA requires no exotic hardware whatsoever. It runs today on any quantum processor supporting three-qubit gates: superconducting transmons, trapped ions, neutral atoms, and photonic chips. Two complementary architectural primitives are specified. The TriQ (Origami Triple) is the fundamental three-qubit register: three physical qubits in Fano-plane incidence connectivity, corresponding to one logical qubit in the [[7,1,3]] Steane code. The SQU (Symplectic Quantum Unit) is the full processor architecture built from TriQ registers, with the Magic-ISA opcode set and inter-register typed routing. The SevenQ is the complete seven-qubit Fano register: all seven points of PG(2,2) simultaneously realised, corresponding to the full [[7,1,3]] Steane code with its six gauge degrees of freedom corresponding bijectively to the six non-classical Fano orbit valences. The key diagnostic opcode is ORBIT: seven simultaneous Pauli measurements returning the Fano orbit valence label of the register state, replacing the conjectured Freudenthal determinant ALU with seven standard readout operations. The SQU achieves fault-tolerant universal quantum computation with 7 physical qubits per logical qubit for CS-family gates via gauge-orbit selection, and 15 for CCZ gates via code switch — both at zero magic-state distillation overhead. A critical non-obvious result: SevenQ ORBIT is identical to standard Steane syndrome extraction. The seven syndrome measurements already performed at every error correction cycle are the ORBIT opcode, with the orbit valence label computed in classical post-processing at zero additional hardware cost. Three hardware tiers are defined: SQU (standard qubits, available today), SQU-diode (6-7 directed Fano coupling topology, near-term), and RPU (non-associative materials, long-term). The SevenQ-diode variant applies the 6-7 broken-symmetry topology to the full seven-qubit register for directed magic routing at 4× symmetric efficiency. The paper also establishes the exploration order for the projective tower: TriQ (3 qubits, minimal n=3) → SevenQ (7 qubits, complete n=3) → 1531-Anvil (15 qubits, first n=4). Keywords Origami ISA, TriQ, SevenQ, SQU, Symplectic Quantum Unit, Fano Plane, PG(2,2), Magic-ISA, ORBIT Opcode, Steane Code, [[7,1,3]] Code, Fault-Tolerant Quantum Computation, Magic State Distillation, CS Gate, Gauge Selection, Quantum Error Correction, Coherent Error Detection, Magic Valence, Fano Orbit, Superconducting Qubits, Trapped Ions, Neutral Atoms, Quantum Hardware, Instruction Set Architecture, 6-7 Topology, Directed Magic Routing, SevenQ-Diode, SQU-Diode, Adelic Simplicial Architecture, Resonance Processing Unit
We construct a two-parameter family of single-error-correcting seven-ququart codes with transversal A7~ symmetry, realizing the finite component of a two-qubit super-golden gate set. These ((7,4,3))4 codes encode two logical qubits and support non-Clifford operations by applying the same gate to each physical ququart.
Our method constructs invariant parent states rather than searching directly for a codespace. For n systems of local dimension q, suppose the local symmetry is a unitary t-group and the permutation symmetry is t-transitive. Maximal mixing of all marginals on at most t sites is then equivalent to linear equations in Schur--Weyl sector weights. Every feasible solution yields parents of pure distance at least t+1; puncturing one site produces a q-dimensional code of distance at least t with the prescribed local symmetry acting transversally.
Steane and the nonstabilizer Fake Steane code reveal why the construction leaves continuous freedom. They share the same prescribed transversal Clifford and permutation symmetries, and their eight-qubit purifications lie on a circle: error correction fixes two invariant-sector norms but leaves the relative phase free. Keeping the same permutation group and replacing Clifford symmetry by A7~≤SU(4) gives three fixed sector norms and two free relative phases, producing the new ququart family.