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.
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.
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
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
Fault-tolerant quantum computation requires error-correcting codes that not only protect quantum information but also support useful logical operations with practical physical realizations. Here we develop a general framework in which the stabilizer syndromes of commuting Pauli rotations directly organize logical-gate design. We derive an exact codespace-preservation condition valid at arbitrary rotation angles, providing a common stabilizer-level description for CSS and non-CSS codes and for both Clifford and non-Clifford operations. A particularly useful equal-syndrome specialization reduces Clifford gate design to stabilizer factorization, yields a general generator weight bound of ⌈d/2⌉ for distance-d codes, and systematically extends to logical rotations higher in the Clifford hierarchy. Applied to bivariate-bicycle (BB) and non-CSS Mirror qLDPC codes, the framework gives explicit low-weight Clifford and controlled-phase gates. Our implementations yield new low-weight gates, including an addressed T gate on the [[16,4,4]] BB code, T-type rotations on the [[64,8,8]] BB code, and CCZ-, CS-, and T-type operations on the [[36,6,6]] Mirror code. We also develop analytic constructions using Clifford algebra bases, which turn the syndrome constraints into explicit algebraic relations and provide a systematic route to logical addressability. We use this approach to analytically construct addressed CNOT, SWAP, controlled, and CCZ-type gates on recursive hypercube codes. An independent construction on middle quantum Reed--Muller codes yields CrZ gates that saturate the equal-syndrome weight bound. These results establish stabilizer syndromes as a constructive link between the stabilizer structure of a quantum code, the logical computations it can directly support, and the fault-tolerance properties of their physical realizations.
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
Fibrational Tensor Codes (FTCs): Holonomic Quantum Computation, G2 Fiber Bundles, and the Cayley-Dickson Hardware Hierarchy
NOTICE (2026-08-04). A verification this record's framework relies upon has been shown to be vacuous, and the identity it claimed to establish does not hold in the naive form. The naive pentagon identity FAILS for octonionic labels. Over all 7^4 = 2401 imaginary labelings, the two independent rebracketing paths from ((ab)c)d to a(b(cd)) disagree in 1176 cases (49%). Explicit counterexample: (a,b,c,d) = (1,1,2,4) gives +e6 on one path and -e6 on the other. The verification previously relied upon (the Pachner prover of 10.5281/zenodo.19713350) is vacuous: it defines w5 := w1*w2*w3*w4 and then checks w1*w2*w3*w4*w5 = +1, i.e. x*x = 1 for x in {+1,-1}. The identical test passes 4096/4096 on RANDOM SIGNS with no octonions involved. That script's own comment concedes the equation is "tautologically true". What this does NOT mean. Octonions are non-associative by definition, so rebracketing must be path-dependent and the naive pentagon cannot hold. That is the content, not a defect. A coherent non-associative calculus must supply the associator as explicit data satisfying its own coherence condition (Mac Lane), which is what Kuperberg's spiders (Comm. Math. Phys. 1996) provide for rank-2 groups. The structural framing is defensible; the verification was measuring the wrong thing. Additional note. This record predates a change of vocabulary and, in places, of substance. Opcode names used here were subsequently revised (ORBIT to RESOLVE, LABEL to PROJECT, BIND to FUSE, MERGE to JOIN), so the terminology does not match the current reference. Readers wanting the current definitions should consult the maintained opcode reference rather than this record. This working paper introduces Fibrational Tensor Codes (FTCs), an exploratory quantum error correction paradigm that relocates the burden of error correction from macroscopic spatial topology — as in surface codes — to microscopic gauge topology. Rather than arranging qubits on 2D associative grids, the FTC architecture maps the Cayley-Dickson division algebra hierarchy directly onto hardware substrates: a Quaternionic routing layer (4 physical qubits, [[4,2,2]], PG(1,2)) and an Octonionic computational layer (7 physical qubits, PG(2,2), the Fano plane). The central conjecture proposes that the 7-qubit Hilbert space, isomorphic as a vector space to the Clifford algebra Cℓ(0,7), can be physically confined to the 8-dimensional minimal left ideal identified by C. Furey in her construction of the complex octonions. This confinement — achieved by appropriate physical coupling rather than discrete stabilizer constraints — would yield a topologically protected 8-dimensional logical qudit (the Logical Quoct) in which local errors are converted to heralded erasures via an Associator Penalty, enabling active feed-forward correction at approximately twice the rate of blind Pauli-flip correction. The key architectural principle is that non-associativity is confined strictly to the continuous Berry phase of the encoding map within a G2 fiber bundle; the logical gate group remains fully associative. The paper formally falsifies 7-qubit triorthogonality (Theorem 5.1), establishing that transversal CCZ requires scaling to the 15-qubit PG(3,2) Reed-Muller geometry, and identifies six concrete open problems required to advance this framework from architectural proposal to engineering specification.
We establish a precise mathematical correspondence between the classification of quantum CSS stabilizer codes and the Kodaira-Neron classification of elliptic curves over p-adic fields. The stabilizer lattice of a CSS code defines a filtered (phi,N)-module over Q_p whose reduction type determines the code's error-correcting properties. We verify this classification across 7 code families (100% agreement on KN Propositions 3.1-3.7), present the first explicit construction of a semistable non-crystalline code from the Shor 9-qubit stabilizer monodromy, and catalogue 26 conjectures spanning code distance estimates via Hasse-Weil zeta functions, ultrametric tree structures in error syndromes, and Yamada (1969) weight calibration.