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 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
A quantum error-correcting code's logical operators, taken modulo its stabilizers, form a finite abelian group; that group, not any single number, is the code's logical identity. We identify the group as one mathematical object across a divide usually treated as fundamental: the cokernel of an integer matrix, read off its Smith normal form. For a Gottesman–Kitaev– Preskill bosonic code the matrix is an integral symplectic Gram matrix and the cokernel is Q 2 (Z/d i Z) ; for a quantum low-density-parity-check digital code the matrix is a boundary i map and the cokernel is its first homology; the reading is the same computation, and code concatenation is the cokernel of a direct sum, so the two Smith normal forms simply concatenate. Because the cokernel is the complete invariant of the two-sided unimodular action, a code's logical group is certified by verifying a witness rather than by trusting a table: a prover supplies (U, D, V ) and a checker re-derives U AV = D in pure integer arithmetic. Every result is checked in Lean 4, and its load-bearing arithmetic is reproved in a second, toolchain-disjoint Cubical Agda kernel from the Peano axioms up. We prove that the group order alone is an incomplete invariant—(Z/2Z)4 and (Z/4Z)2 share order sixteen but differ in two-torsion—give an analog-weighted decoder whose correctable region provably exceeds the hard-decision region because the soft metric widens the distance, and expose the entire certified construction inside a live PennyLane and Strawberry Fields workflow, the software and hardware setting of a photonic fault-tolerant program built on exactly this concatenation. Published by The Institute for Applied Ontological Mathematics.
Stabilizer codes are often constructed within the Calderbank--Shor--Steane (CSS) framework, where two mutually orthogonal binary classical codes define X and Z-type stabilizer generators. While this structure is algebraically convenient, additional non-CSS constraints may help suppress low-weight logical operators and improve decoding performance in the finite-length regime. We thus introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise orthogonal binary PCMs associated with X-, Y-, and Z-type stabilizer generators. A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS: the same stabilizer group may admit a CSS generating set. We characterize this collapse, obtaining algebraic and rank conditions for deciding when the Y-type checks are redundant and when they define genuinely non-CSS stabilizer constraints. We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators. The novel framework includes a known non-CSS topological code, namely the XYZ2 hexagonal code, and yields also sparse finite-length quantum low-density parity-check (qLDPC) constructions from intersecting-subset and quasi-dyadic code families. Simulations under depolarizing code-capacity noise and quaternary belief propagation decoding show that the proposed XYZ qLDPC instances can outperform representative CSS qLDPC instances with similar finite-length parameters.
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.
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.
This work explores the composition of Microsoft's 4D geometric error-correcting codes — built on the 24-cell honeycomb cellulation — into higher-dimensional logical structures targeting 24D stability. While 4D codes have demonstrated 1,000-fold error rate reduction and single-shot correction on neutral atom arrays, this work asks how these codes compose when stacked or tensored into 8D, 12D, or 24D logical architectures. We investigate the stabilizer structure under composition, whether single-shot correctability survives dimensional lifting, and the mapping of composed code geometries onto neutral atom hardware with all-to-all connectivity. This bridges the gap between demonstrated 4D fault-tolerant codes and the theoretical framework for topological codes beyond dimension 2. Update, 30 July 2026 — constructive resolution. The distance question raised here is now settled by direct construction in 10.5281/zenodo.21698294, which builds the 4D toric code from its cubical chain complex and obtains [[96,6,4]] at L=2: k=6 exactly by GF(2) rank, and distance 4 proven exhaustively over every error pattern of weight ≤4. Distance 8 is therefore not obtainable from the 4D toric construction. That bounds the 4D toric family only and does not establish that no [[96,6,8]] code exists in another family — the quantum Singleton bound permits it comfortably at these parameters. An earlier wording of this note, added and corrected on 30 July 2026, overstated the scope. That work was carried out independently, from first principles, and reaches the same conclusion. Erratum, 2026-07-31 — the novelty claim is withdrawn, and what replaces it. The deposited file states that "nobody has published on composing 4D geometric codes into higher-dimensional logical structures" and that the bridge paper "has not been written." That is false. Composing chain-complex CSS codes is the homological (tensor) product of chain complexes, and the cellular chain complex of a product complex is the tensor product of the factors', so the composition of two 4D cubic toric codes is the 8D cubic toric code — a published family, and the subject of this record's own reference #9. This was checked, not assumed: an explicitly composed 8D code and an independently built 8D cubic toric code agree at all 8 grades on shape, nonzero count, both check-weight spectra and exact GF(2) rank. The composition constructs no new code. What replaces the claim is a quantitative answer to the question the record was actually reaching for. For the balanced hypercubic toric code at middle grade, n/k = L^D exactly and independently of D — verified in 8 of 8 computed cases, with the qubit count a combinatorial count and the logical count a measured pair of GF(2) ranks, up to n = 459,270 where both ranks were measured directly. Composing therefore changes how many logical qubits sit in one block and changes nothing else. Head to head at the same distance 81, the 8D composite [[459270, 70, d ≤ 81]] and the plain 4D code [[39366, 6, d ≤ 81]] both spend 6,561 physical qubits per logical qubit, to the digit — with the 8D version paying weight-10 stabilizers where the 4D pays weight-6. Distance does not rescue it either: by Künneth the distance of a product is bounded by the minimum over grade splittings, not the product of the factors' distances, so each added dimension is a chance to lose distance — the 5D product of [[486,6,9]] at grade 1 is [[1215, 5, 3]], proved exactly. On single-shot correction: there is nothing to lift. D = 4 at q = 2 already has metachecks in both sectors (ranks 80 and 80 for [[486,6,9]]), so 8D, 12D and 24D add no sector that was not already covered — and lifting can lose it, the 5D product at grade 1 having no X-sector metachecks at all (rank d₀ = 0). On hardware: all-to-all connectivity is real but was never the binding constraint. The smallest possible balanced 24D hypercubic toric code needs 4.5 × 10¹³ physical qubits, 1.5 × 10¹⁰ times the 3,000-atom array cited above. What stands: the homological product is the correct composition operation and the framework is well-posed; composition genuinely does raise dimension and does buy more logical qubits per block (6 → 70 from 4D to 8D); the metacheck question was the right one to ask; and the reference library is accurate. Two limits on the erratum itself: all computation uses the cubic 4-torus code [[486,6,9]], not Microsoft's 24-cell honeycomb code, whose parity-check matrices were never obtained — so nothing here is a statement about that code; and no exact distance is claimed at n ≥ 2430, the 8D figures being certified upper bounds with no lower bound proved. The 1,000-fold error-rate reduction and the neutral-atom hardware results were not tested and are neither confirmed nor refuted. Full computation, code and logs: 10.5281/zenodo.20808710 and branch writeup at /home/voodoo/branch_work/18779369_composition/RESULT.md. Note: the source file QEC_Neutral_Atom_24D_Research.md itself still contains the withdrawn claim verbatim (a published file cannot be edited in place) — this erratum is the authoritative correction until a new version replaces the file text.