DOI 10.5281/zenodo.20410167 carries no abstract — its description contains only notes-to-self ("translate 'computational entropy' into the standard unit, add the baseline anchor"). This note attempts the requested baseline anchor and reports why it cannot be supplied as asked, together with a structural finding that supersedes the question. The central object encodes nothing The deposited decoder assigns self.n, self.k, self.d = 96, 6, 8 as literal constants. Computed from its own parity-check matrices over GF(2): rank(Hz)=23, rank(Hx)=73, 23+73=96=n, hence k=0. Equivalently rowspace(Hx) = ker(Hz) exactly, both of dimension 73 = the cycle-space dimension E−V+1: every cycle is a triangle sum and no logical operator exists. This is the expected topological answer — the 24-cell's boundary is a triangulated 3-sphere, simply connected, so it encodes nothing. Confirmed three independent ways. The CSS condition itself is satisfied exactly: the geometry is right, the label is wrong. The claimed d=8 is undefined rather than merely unverified. Consequence for the requested benchmark "Iso-accuracy against an off-the-shelf decoder" is undefinable when there are no logical qubits. Measured on the metric that survives (2000 trials per rate, versus PyMatching 2.4.0 exact MWPM on the same Hz): both always return to the codespace, but the deposit's decode_mwpm — named MWPM, in fact greedy nearest-pair — returns corrections 5.9% to 12.8% heavier across p=0.01–0.10, the gap widening with noise, never better. What could not be measured, and is therefore not reported Energy. RAPL exists on this machine but energy_uj is root-only and perf_event_paranoid blocks the power PMU. No FLOPs/joule figure is given; wall-clock, peak RSS (77,592 KB), matrix bytes and operation counts are reported instead. Separately, "computational entropy" has no operational definition anywhere in the corpus — every occurrence is unsourced marketing copy with mutually inconsistent percentages. The quantity the note-to-self asks to convert cannot be converted. Propagation This defect propagates to DOI 10.5281/zenodo.20407370, whose published claim is "composing six [[96,6,8]] 24-cell honeycomb codes": compose_24d_prototype.py is byte-identical in both deposits and imports the same k=0 class.
DOI 10.5281/zenodo.20410167 carries no abstract — its description contains only notes-to-self ("translate 'computational entropy' into the standard unit, add the baseline anchor"). This note attempts the requested baseline anchor and reports why it cannot be supplied as asked, together with a structural finding that supersedes the question. The central object encodes nothing The deposited decoder assigns self.n, self.k, self.d = 96, 6, 8 as literal constants. Computed from its own parity-check matrices over GF(2): rank(Hz)=23, rank(Hx)=73, 23+73=96=n, hence k=0. Equivalently rowspace(Hx) = ker(Hz) exactly, both of dimension 73 = the cycle-space dimension E−V+1: every cycle is a triangle sum and no logical operator exists. This is the expected topological answer — the 24-cell's boundary is a triangulated 3-sphere, simply connected, so it encodes nothing. Confirmed three independent ways. The CSS condition itself is satisfied exactly: the geometry is right, the label is wrong. The claimed d=8 is undefined rather than merely unverified. Consequence for the requested benchmark "Iso-accuracy against an off-the-shelf decoder" is undefinable when there are no logical qubits. Measured on the metric that survives (2000 trials per rate, versus PyMatching 2.4.0 exact MWPM on the same Hz): both always return to the codespace, but the deposit's decode_mwpm — named MWPM, in fact greedy nearest-pair — returns corrections 5.9% to 12.8% heavier across p=0.01–0.10, the gap widening with noise, never better. What could not be measured, and is therefore not reported Energy. RAPL exists on this machine but energy_uj is root-only and perf_event_paranoid blocks the power PMU. No FLOPs/joule figure is given; wall-clock, peak RSS (77,592 KB), matrix bytes and operation counts are reported instead. Separately, "computational entropy" has no operational definition anywhere in the corpus — every occurrence is unsourced marketing copy with mutually inconsistent percentages. The quantity the note-to-self asks to convert cannot be converted. Propagation This defect propagates to DOI 10.5281/zenodo.20407370, whose published claim is "composing six [[96,6,8]] 24-cell honeycomb codes": compose_24d_prototype.py is byte-identical in both deposits and imports the same k=0 class.
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.
A CSS quantum error-correcting code is, on the nose, a length-three chain complex over F2 : the commutation law HZ H⊤ X = 0 that makes the code valid is exactly the condition ∂ 2 = 0 that makes a chain complex. We take this identification literally and show that the entire working theory of a quantum code is one homological object the same engine computes, at scale, and certifies. The number of logical qubits is the first Betti number, k = dim H1 = n − rank HX − rank HZ = b1 ; the code distance is the systole, the shortest non-contractible cycle. We reproduce the published parameters of state-of-the-art codes exactly—the toric and hypergraph-product families and the IBM bivariate-bicycle codes [[72, 12, 6]], [[144, 12, 12]], [[90, 8, 10]]—cross-validated four independent ways and kernelcertified for the small cases in Cubical Agda, on CPU and GPU. The two scale certificates—the Betti count b1 and the rank-derived logical count k—are themselves kernel-proved to be a single first-homology certificate (a DescentCertificate both readings factor through), closed terminally through a fixed-point engine. We then run the code, not just measure it. (i) The Union-Find decoder is the engine's connected-components reduction, and "a logical error occurred" is the engine's coboundary decision (the residual cycle has nonzero winding); decoding the toric code at growing distance recovers the accuracy threshold near 9.5%, and the optimal (maximum-likelihood) threshold near 11% is the engine's two homology sectors competing as the domain-wall free energy of a random-bond Ising model on the Nishimori line. (ii) A code's distance scaling is its Tanner-graph spectral expansion: the second eigenvalue λ2 versus the Ramanujan bound is a computable certificate (Sipser–Spielman), exact on the topological tier where the spectral gap obeys γ d2 → 4π 2 . (iii) The full chain complex carries metachecks—redundancies among the parity checks—and we compute them: the bivariate-bicycle codes have six, the engines of single-shot error correction; the 4-dimensional toric code's homology (1, 4, 6, 4, 1), with k = b2 = 6, is the tensor of four circle complexes. (iv) The logical error-rate prefactor is the multiplicity of the systole, A = 2L for the toric code, which we compute exactly. (v) Finally the engine measures a topological phase transition: as random noise thickens, torus-wrapping "giant" cycles appear in a sharp homological-percolation transition at pc ≈ 0.5 that sharpens with system size—the same homology machinery now detecting when noise spontaneously writes a logical operator. (vi) The engine runs over any coefficients: qudit (Fp ) codes are chain complexes over Fp , and the integral homology, computed by the Smith normal form, exposes torsion—so a code's logical-qudit count can depend on the qudit dimension, an extra logical appearing exactly when p divides a torsion coefficient (reproducing the known homology of the torus, the Moore spaces, the Klein bottle, and RP2 ). We are careful to fence what is not claimed: minimum distance is NP-hard in general and we compute it only on the systolic tier; the giant-cycle transition is the random-cluster one, distinct from the maximum-likelihood decoding threshold of the random-bond Ising model. The thesis is singular and operational: a quantum code's parameters, its decoder, its quality, its higher-dimensional structure, its performance prefactor, and its phase transition are not six theories but one—homology you can run. Published by The Institute for Applied Ontological Mathematics.
This note corrects and improves the choice of code in the 4D cubic toric code family studied in Zenodo 20304690, 20301574, 20322507, and 20336622. At fixed k = 6 logical qubits (k = b2(T^4) = 6, a topological invariant), the torus shape is a free parameter. Minimizing physical-qubit count n = 6*prod(sides) at target distance d = minimum pairwise side product, the efficient frontier is the ISOTROPIC family (s,s,s,s) at n/d^2 = 6. The previously published anisotropic (2,4,4,4) -> [[768,6,8]] sits at n/d^2 = 12 (a 2x physical-qubit overhead) and is Pareto-dominated by the isotropic (3,3,3,3) -> [[486,6,9]]. [[486,6,9]] improves on [[768,6,8]] simultaneously on three axes:- 37% fewer physical qubits (486 vs 768);- higher distance (9 vs 8);- 2-6x lower logical failure rate under identical BP+OSD-CS-7 decoding and i.i.d. X noise across p = 0.04-0.08 (non-overlapping 95% confidence intervals, N = 4000 trials/rate). All parameters are verified by explicit construction (k by GF(2) rank; distance as minimum logical-operator weight, matching the geometric minimum non-contractible 2-surface). As a cross-check, the [[768,6,8]] failure rates measured here reproduce the previously published values to within statistical agreement, corroborating the original simulation. Method: direct code construction and Monte-Carlo decoder simulation (GF(2) linear algebra + ldpc 2.3.9 BP+OSD). No collapse/octonion engine is involved at any stage; every result is reproducible from the attached scripts (torus_code_4d.py, gate9_magnification.py, hh_threshold.py, burst_recheck_v2.py). Scope: results are for one decoder (BP+OSD-CS-7) and one noise model (i.i.d. X errors, perfect syndrome) -- the same setting as the prior deposits, so the comparison is direct. This is a code-parameter and decoder-performance result, not a hardware-viability claim; 4D toric codes require non-local qubit connectivity, which is not addressed here. Correction (2026-07-29): The Pareto-dominance claim in this record's title and abstract compares exactly two points — one isotropic (s,s,s,s) member against the single anisotropic [[768,6,8]] construction — and no frontier over the k=6 side-tuple space was computed, so dominance over the anisotropic family is not established by the evidence presented. What the comparison supports is the narrower statement that this isotropic code improves on [[768,6,8]] in n/d². A sweep of all k=6 side-tuples to n ≈ 2000 with the real n/d² frontier plotted is required before any Pareto claim can be restated. No erratum record exists for this item as of 2026-07-29; this is a self-correction pending that sweep. Addendum, 2026-07-31 — the efficient frontier, swept. The description above states that “the efficient frontier is the ISOTROPIC family (s,s,s,s) at n/d² = 6”. Sweeping every k=6 side-tuple with sides 2–6 and keeping the non-dominated points — using gate9_magnification.py, already attached to this record — shows that statement is too strong, and it is corrected here. Isotropic shapes uniquely attain the optimal ratio n/d² = 6, and every isotropic shape lies on the frontier. But they do not constitute the frontier: 4 of its 9 points are anisotropic. The reason is that isotropic shapes realise only square distances (4, 9, 16, 25, 36), so anisotropic shapes remain non-dominated wherever they supply an intermediate distance (6, 12, 20, 30) that no isotropic code of comparable size reaches. sidescoden/d²shape (2,2,2,2)[[96,6,4]]6.00isotropic (2,3,3,3)[[324,6,6]]9.00anisotropic (3,3,3,3)[[486,6,9]]6.00isotropic (3,4,4,4)[[1152,6,12]]8.00anisotropic (4,4,4,4)[[1536,6,16]]6.00isotropic (4,5,5,5)[[3000,6,20]]7.50anisotropic (5,5,5,5)[[3750,6,25]]6.00isotropic (5,6,6,6)[[6480,6,30]]7.20anisotropic (6,6,6,6)[[7776,6,36]]6.00isotropic Corrected statement. Isotropic tori uniquely minimise n/d², attaining 6 with equality; anisotropic tori are strictly worse on that ratio but still occupy the frontier at the distances isotropic shapes skip. The record's headline two-point comparison is unaffected — [[486,6,9]] has both fewer qubits than [[768,6,8]] (486 < 768) and greater distance (9 > 8), so that domination holds exactly as stated, as does the 37% qubit reduction. The sweep is bounded at side length 6 (n ≤ 7776). The alternating pattern continues beyond it by the same argument, but the frontier above n = 7776 was not enumerated.
The isotropic 4D cubic toric code on T^4 with sides (3,3,3,3) — the [[486,6,9]] code — has a transversal CNOT and a transversal Hadamard. Its non-Clifford completion had not previously been costed. This note re-verifies the code from an independent construction (n=486, rank(Hx)=rank(Hz)=240, k=6, logical weight 9), classifies its transversal gates, and derives the distillation cost. Two no-gos, both computed rather than assumed No transversal phase gate exists at all. Over all 2^486 subsets R of the qubits, the only product of S gates preserving the stabiliser group is the empty product. This is a complete classification, not a sample: the admissible subsets form a GF(2) nullspace, computed to have dimension 0. No transversal T. The matrix [Hx ; L_X] is not triorthogonal (3132 of 54285 pairs and 2268 of 5935160 triples have odd overlap), verified against a positive control on [[15,1,3]] which passes the identical test. Distillation cost The k=6 structure turns an apparent 6x qubit penalty into six 15-to-1 factories running in lockstep for the price of one, because every operation in the protocol is a block-transversal CNOT between distinct blocks. With single-shot error correction (a 4D-specific property), the space-time cost per output magic state is 7.7x lower than a rotated-surface-code factory at the same distance, despite using 7.04x more physical qubits. Output error coefficient is exactly 35 (the A_3 coefficient of the [15,11,3] Hamming code); the break-even fixed point is 0.141480. Code switching, quantified and rejected The only natural same-lattice partner has d=L, so matching d=9 costs n=26244 qubits for k=4 — 54x the qubits for fewer logical qubits. Distillation wins decisively. Relation to prior work This supersedes the branch note describing S and T as "unresolved": S was already resolved by |Y⟩-state injection in the 2026-06-23 session records. The open question was the cost, which this supplies. Two results here independently reproduce earlier corpus numbers from a different derivation (exact Hamming weight enumerator rather than 2^15 enumeration plus fit). Erratum (2026-07-29): The rail-parallelism design result in §5.2 — six 15-to-1 factories running in lockstep on the six rails of this code, driven by block-transversal CNOT, for a 7.67x space-time advantage — is prior art and is withdrawn as a novel claim. Xu et al., arXiv:2407.18490 (2024) §V.C.2, state that k magic states can be distilled in parallel in [[n,k,d]] patches, that this “mostly requires inter-block Clifford operations”, and explicitly that “the same protocol can be applied to distilling |T⟩ states in parallel using, e.g. a 15-to-1 distillation protocol, with the same scaling of the space and the time overhead”; their Appendix A additionally constructs the parallel magic-state injection that §7 of this record identified as its own unconstructed assumption. Separately, this record omits a cost the lockstep layout necessarily incurs: six rails sharing one block cannot post-select independently without rail-level addressability, so a round yields six magic states with probability p^6 or none, a space-time multiplier of 1/p^5 (1.08 at eps_in = 1e-3, 2.12 at 1e-2). The exact distillation results — coefficient 35, eps* = 0.141480, the depth-14 Koenig schedule — and both transversal no-gos are unaffected and are re-verified in the erratum. See 10.5281/zenodo.21697731.
DOI 10.5281/zenodo.20301574 reports 2,694 trials at zero logical failures. That is a floor, not a curve — the failure boundary was never located. This note finds it on every axis, on the isotropic [[486,6,9]] rather than the superseded anisotropic [[768,6,8]]. The boundary axislargest clean burstfirst failure L1 ball radius (exhaustive, all 81 centres)r=2, weight 342 — 0/81r=3, weight 462 — 81/81 localized burst, exact weightw=30 — 0/20,000w=31 — 2/20,000 uniform random support (control)w=12 — 0/20,000w=13 — 1/20,000 Three findings the original could not have seen The original result was true and uninformative. All 81 weight-24 vertex bursts decode to exact zero residual, matching the record's 176/176 — but the record tested only weights 6 and 24, both inside the protected region and 7 qubits below the boundary. Structural tolerance quantified for the first time: 2.5x. At equal resolution, unstructured errors begin failing at w=13, geometrically localized bursts at w=30. Code distance is 9, so a distance-only guarantee covers only weight 4 — the remainder is degeneracy. The curve is strongly non-monotone, with protected spikes at complete balls. Failure rate reaches 0.802 at w=143, collapses to 0/5,000 at w=144 (=|B1|) and w=145, then returns to 0.293 at w=146. A wider protected window spans w≈320–352 around |B2|=342. No single-number burst-tolerance figure is meaningful for this code. The r=3 failure is intrinsic, not decoder-attributable The all-ones vector is a nontrivial logical operator with zero syndrome, and B3 ⊕ shell4 = all-ones with shell4 of weight 24 — so any maximum-likelihood decoder must prefer the weight-24 explanation. Failures reach all 63 non-trivial logical classes. Decoder: ldpc 2.4.1 BP+OSD-CS-7, min-sum, settings copied verbatim from the record's own burst_recheck_v2.py. 3,374,789 decodes. No zero is reported as "0%" anywhere — every zero carries Wilson and Clopper-Pearson upper bounds. Re-run at a different worker count (6→11): every count, rate, CI and seed byte-identical. Version 1.1 (2026-09-22): exact coset-weight audit Nothing in version 1.0 is withdrawn, and no count, rate or interval changes. This version adds an exact audit that changes how two of v1.0's findings should be read. Method. For a burst e with logical class l, the coset weight A(e) is the weight of the lightest error with the same syndrome and the same class. W(e) is the lightest with the same syndrome and any other class. Both are solved exactly as integer programs (scipy.optimize.milp / HiGHS), and every value quoted is a proven optimum. A minimum-weight (MW) decoder succeeds if and only if W > A. 1. The boundary is a coset-weight boundary. Random supports are not reduced at all (A = w). Localized bursts are reduced to about half their weight: the weight-24 vertex burst has A = 8, and w = 30 has A = 12–14. Every one of the 15 localized BP+OSD failures at w = 31–35 has W = A − 1, so an MW decoder fails on them too. Of the 11 random-support failures at w = 13–16, 7 have W = A − 1 and 4 are decoder shortfalls. Both families begin to fail at coset weight about 13–15. The 2.5× "structural tolerance" in v1.0 is real when counted in flipped qubits, but it is almost entirely the factor-of-two stabilizer reduction of a compact error. 2. Complete balls (A, W): B0 (8, 15); B1 (32, 33); B2 (6, 13); B3 (30, 24). The 342-qubit radius-2 ball is exactly equivalent to a weight-6 error with margin 7. The wide window around |B2| is genuine protection. 3. The radius-1 spike is a decoder near-tie, not protection built into the code. Throughout w = 143–146 the margin W − A is ±1. Over 40 sampled bursts per weight: wMW failuresBP+OSD failures1439/4031/401440/400/401450/400/401460/4010/40 At w = 146 every BP+OSD failure is a burst an MW decoder corrects. The v1.0 reading of the w = 144 drop as a protected spike of the code should be taken as a property of BP+OSD on a one-qubit near-tie. 4. Decoder strength. Re-decoding every boundary failure at OSD-CS order 12 instead of 7 rescued none, which is consistent with point 1. Files added: spike_exact.py / spike_exact2.py (MW audit; .log, .json), coset_weight_scan.py (A vs w; .log, .json), boundary_check.py (OSD-12 re-decode; .log), mechanism_check.py (syndrome and correction weights; .log), make_fig.py and fig_failure_curve.pdf, and VOODOO_SCREEN.md (an instrument screen only: a non-discriminative negative, and no result rests on it). All scripts run against the v1.0 code (code486.py, burst_sweep.py, same seeds). A manuscript based on this analysis is in preparation for Physical Review A.
We construct the 4D cubic toric code on a parameterized 4-torus T^4 = (Z/N0) x (Z/N1) x (Z/N2) x (Z/N3) with qubits on 2-cells, Z-stabilizers on 3-cells, and X-stabilizers on 1-cells, yielding [[6 * prod Ni, 6, min Ni*Nj]]. Three concrete sizes are benchmarked: [[96,6,4]] (cubic 2^4 torus), [[192,6,4]] (sides 2,2,2,4), and [[768,6,8]] (sides 2,4,4,4). Across 2,694 trials spanning three localized error families (weight-24 vertex bursts, weight-6 edge bursts, weight-{4,8,16} logical seeds) and three decoders (permissive GF(2) inversion, greedy minimum-syndrome, BP+OSD-CS-7 via the ldpc 2.3.9 package), we find: (1) all vertex and edge bursts at all three sizes are stabilizer-equivalent under the permissive decoder, with zero logical failures across 880 burst trials per decoder; (2) BP+OSD-CS-7 decodes every vertex burst to exact-zero residual at all three sizes (176/176 total) -- the decoder identifies the burst pattern itself as the minimum-weight correction; (3) logical seed errors are correctly classified into six distinct sectors, confirming the k=6 logical space is faithfully resolved. This empirically realizes the burst-protection consequence of the palindromic preservation theorem (Zenodo 10.5281/zenodo.20040776) at three independent code sizes, and closes the decoder gap left open by greedy methods at higher weights. A side finding: the cubic 2^4 torus yields distance d=4, not d=8 as sometimes quoted for the [[96,6,*]] toric code on the 24-cell; reaching d=8 requires sides (2,4,4,4) and n=768 qubits. This refines the parameter claim without affecting the palindromic theorem itself. The bundle contains the source (torus_code_4d.py, burst_recheck_v2.py), the full per-burst log (burst_recheck_all_sizes.log), parsed numeric results (results.json), and reproducibility instructions. Total compute: about 78 seconds on a single x86 VPS core for all three sizes. Entry 1 - Identifier: 10.5281/zenodo.20040776 - Relation: is supplement to - Resource type: Publication / Preprint Entry 2 - Identifier: 10.5281/zenodo.18815664 - Relation: is part of - Resource type: Publication / Other Erratum (2026-07-29): The 2,694 trials reported here at zero logical failures are a floor, not a failure curve — the record tested only burst weights 6 and 24, both inside the protected region and roughly 7 qubits below the boundary, so the failure boundary was never located. It has since been located on the isotropic [[486,6,9]]: localized bursts stay clean to weight 30 (0/20,000) and first fail at weight 31 (2/20,000), against weight 13 for unstructured random support, giving 2.5× structural tolerance at equal resolution. The curve is also strongly non-monotone, with protected spikes at complete balls — failure rate 0.802 at w=143, 0/5,000 at w=144 and w=145, back to 0.293 at w=146 — so no single-number burst-tolerance figure is meaningful for this code. See 10.5281/zenodo.21684236.