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 Erratum, 2026-07-31 — the novelty claim is withdrawn, and what replaces it.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.
Title:Homological Closure of the Planar Surface Code: A Fully Rigorous Relative Chain Complex Formulation Description: This paper presents a fully closed and homologically rigorous formulation of the planar open-boundary surface code. Unlike standard expositions that rely on heuristic parity arguments or generator counting, we derive every structural property directly from the relative chain complex C2→∂2C1→∂1C0C_2 \xrightarrow{\partial_2} C_1 \xrightarrow{\partial_1} C_0C2∂2C1∂1C0 over F2\mathbb{F}_2F2. The core results are: Abelian stabilizer structure derived solely from the exactness condition ∂1∘∂2=0\partial_1 \circ \partial_2 = 0∂1∘∂2=0. Logical qubit count identified with the relative homology group H1(Σ,∂rough)H_1(\Sigma,\partial_{\rm rough})H1(Σ,∂rough), computed via the long exact sequence. Anticommuting logical operators guaranteed by Poincaré–Lefschetz duality and nondegenerate pairing. Code distance d=Ld=Ld=L rigorously established using the min-cut/max-flow theorem (Menger’s theorem), eliminating geometric intuition gaps. All arguments are presented in quotient-space form, reducing the surface code to a purely homological stabilization mechanism. Structural Extension In the Appendix, we establish a structural isomorphism between: Homological stabilizer projection in quantum error correction, and Viability-preserving projection in Yang Jihoon Dynamics. Both systems are shown to be quotient constructions under constraint-preserving equivalence relations. This yields a unified structural perspective: Fault-tolerant homological protection≡Viability-preserving dynamical projection Significance This work provides: A mathematically sealed derivation of the planar surface code A homological closure free of informal parity reasoning A structural bridge between quantum error correction and viability dynamics A foundation for interpreting fault tolerance as constrained topological persistence The result is not an analogy, but a quotient-level structural correspondence.
This repository contains the complete research artifact and computational verification suite for scaling the discrete Lorentzian causal diamond into high-performance Quantum Low-Density Parity-Check (qLDPC) and finite-block codes. By systematically exploring algebraic amplification (Hypergraph Product), geometric tessellation (toric scaling), and lattice enrichment (
E8 Lorentzian substrates), this work identifies precise structural obstructions in existing approaches and introduces novel, Pareto-optimal CSS codes tailored for near-term hardware in the
N∼100–200 physical qubit regime. Scientific Contributions The Augmented-Seed Code Family: The central discovery that a specific weight-3 row (one of 64 in two
B4 symmetry orbits) appended to the primal causal diamond seed raises the classical distance from 4 to 6. This breaks the intrinsic distance ceiling of the basic geometry, unlocking a family of finite-block codes that significantly outperform planar codes (e.g., surface/toric codes) at small block sizes. Methodological Warning on Sampling Blindness: Demonstration that Belief Propagation (BP)-style random sampling fails catastrophically in high-dimensional kernels. Earlier sampled bounds of
d≥67 for the `[[193, 25]]` code are formally retracted; an exact Integer Linear Program (ILP) oracle is introduced, definitively proving
d=4. The
E8 Structural Obstruction (Theorem 4.5): Rigorous proof that the 28-qubit
E8 Lorentzian code cannot be punctured to yield a valid quantum code. Its 7-disconnected-4-cycle geometry guarantees
dZ=1 after any single row drop, pointing toward the Euclidean
E8 root lattice as the necessary next step. Naïve CSS Lift Theorem (Theorem 5.3): Algebraic proof that setting
HX=ker(HZ) definitionally forces
k=0, regardless of chain or torus topology, resolving anomalies in earlier geometric scaling attempts. Highlighted Code FamiliesThe augmented geometric construction yields a frontier of codes optimized for neutral-atom and trapped-ion architectures, including: [[112, 4, (6, 6)]]: Achieves an exceptional figure of merit
kd2/N≈1.29, making it a highly efficient logical block for near-term implementation. [[176, 32, (3, 6)]: Features an asymmetric distance profile naturally aligned with the
Z-biased noise characteristic of superconducting hardware. [[208, 16, 6]]: A self-hypergraph product code with
d=6 proven algebraically via the Tillich–Zémor theorem. Artifact Contents Verification Suite (`verification_modular_assembly.py`): A unified, self-contained Python script using `scipy.optimize.milp` that computationally verifies every theorem, constructs all augmented seeds, performs exact distance certifications via ILP in under 100ms per operator, and reproduces the geometric obstructions. Manuscript: Complete LaTeX source files and compiled PDF of the paper.
Three-dimensional (3D) quantum XYZ product can construct a class of non-CSS quantum codes by using three classical codes. However, there has been limited study on their error-correcting performance so far and whether this code construction can be generalized to higher dimension is an open question. In this paper, we first study the error-correcting performance of the 3D Chamon code, which is an instance of the 3D XYZ product of three repetition codes. Second, we show that the 3D XYZ product can be generalized to four dimension and propose four-dimensional (4D) XYZ product code construction, which constructs a class of non-CSS quantum codes by using either four classical codes or two CSS quantum codes. Compared with the 4D homological product, we show that the 4D XYZ product can construct non-CSS codes with higher code dimension or code distance. Third, we consider two instances of the 4D XYZ product, to which we refer as the 4D Chamon code and the 4D XYZ product concatenated code, respectively. Our simulation results show that, the 4D XYZ product can construct non-CSS codes with better error-correcting performance against Pauli-
Z-biased noise than CSS codes constructed by the 4D homological product. Finally, we present the geometric arrangement of the 4D Chamon code within a 4D cubic lattice, demonstrating that it possesses two key characteristics of fracton models, which strongly suggest that it is a novel 4D fracton model.