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Large-Scale Autonomous Discovery of Kissing Number Constructions

Shuxing Yang, Rui Zhao, Junyao Wu, Yize Wang, Fujia Chen, Kaihao Zhu, Wenhao Li, Zichen Li, Yaqi Li, Shenzhan Hong, Yuang Pan, Junjie Yang, Taowen Deng, Jincheng Mi, Hongsheng Chen, Yihao Yang

arXiv:2609.35051v1cs.ITmath.IT

TL;DR

For highly constrained configurations, improvements may require reorganizing points, while fixed search representations can obscure the degrees of freedom needed to find and certify new constructions. Qiushi Engine carries out construction searches, mathematical analysis, and computational verification, using distinct structural mechanisms such as layer motion, labelled lifting, exchanges, signed-code replacements, lattice constructions, and moment arguments. The paper reports new kissing-number lower bounds in nineteen dimensions; for every pair (d, Nd) in Table 1, a kissing configuration of Nd points in Rd gives K(d) ≥ Nd.

  • Problem

    For highly constrained configurations, improvements may require reorganizing points, while fixed search representations can obscure the degrees of freedom needed to find and certify new constructions.

    Source: §1 · p. 2

  • Method

    Qiushi Engine carries out construction searches, mathematical analysis, and computational verification, using distinct structural mechanisms such as layer motion, labelled lifting, exchanges, signed-code replacements, lattice constructions, and moment arguments.

    Source: §1 · pp. 2–3

  • Results

    The paper reports new kissing-number lower bounds in nineteen dimensions; for every pair (d, Nd) in Table 1, a kissing configuration of Nd points in Rd gives K(d) ≥ Nd.

    Source: §1 · p. 2, §1.1 · p. 3

  • Takeaways & Limitations

    The results illustrate how autonomous research systems can contribute by discovering mathematical representations and constructions beyond optimization within fixed formulations.

    Source: §1 · p. 3

    The restricted signed-code capacity results and the dimension-36 isolated construction do not establish optimality for arbitrary twelve-coordinate signed codes.

    Source: §6.4 · p. 25

Abstract

from arXiv · show

The kissing-number problem is a classical problem in discrete geometry whose exact solution is known in only a few dimensions. Recent artificial-intelligence approaches have begun to discover improved configurations through large-scale numerical and combinatorial search, but converting such searches into general mathematical constructions and rigorous proofs remains challenging. Here we use Qiushi Engine, an autonomous multi-agent research system, to investigate kissing numbers and obtain new lower bounds in nineteen dimensions: $25$, $27$, $32$--$39$, $43$, $45$, and $49$--$55$. The resulting constructions arise from distinct structural mechanisms, including coordinated motions of contact layers, labelled direction reuse, joint support exchanges, signed-code replacements, cross-shell lattice constructions, low-overlap lattice isometries, and spherical-design moment certificates. They yield sharp capacities for parameterized signed-code models, deterministic image-union guarantees, and exact section and projection counts controlled by embedded root systems and anchor-graph statistics. These methods yield, among others, $K(25)\ge197580$, $K(27)\ge201567$, $K(38)\ge591900$, $K(43)\ge2553792$, $K(45)\ge7380090$, and $K(55)\ge53301140$. The autonomous system carried out the construction searches, mathematical analysis and computational verification, while each final result was reduced to explicit mathematical arguments and independently checkable finite certificates. Our results illustrate how autonomous research systems can move beyond optimization within fixed formulations to discover new mathematical representations and constructions at scale.

1. Introduction

The paper presents an autonomous investigation that establishes new kissing-number lower bounds in nineteen dimensions through distinct structural constructions. It situates these results against the difficulty of changing constrained representations, summarizes the construction families and their section-count consequences, and relates them to earlier methods and the paper’s notation and organization.

1. Introduction

  • The paper frames the central difficulty as finding new degrees of freedom when constrained configurations and fixed search representations obscure possible improvements.It asks whether an autonomous research system can move beyond optimizing a fixed formulation to discover new representations, constructions, and proofs.Source: §1 · p. 2

1.1. Main results and structural statements.

  • New lower bounds are established in dimensions 25, 27, 32–39, 43, 45, and 49–55, with a finite configuration witness for every listed pair in Table 1.The paper states K(d) ≥ N_d for each pair in the table.Source: §1 · p. 2, §1.1 · p. 3

1.2. Structural consequences.

  • The nineteen constructions use layer motion and labelled lifting, shared-deletion exchanges and signed designs, shell completion and image unions, and moment-certified section statistics.The section-count arguments use an embedded parent geometry for dimension 43 and an anchor-graph common-neighbour count for dimension 45; the latter gives 7,380,090 points.Source: §1.2 · p. 4

1.3. Relation to earlier constructions.

  • Code-based constructions use layers of Edel–Rains–Sloane, dense codes of Cheng–Sloane, and constant-weight constructions and tables [17, 18, 24]; new supports, transverse choices, signed replacements and boundary embeddings supply improvements.Local replacement has a direct precursor in Takhanov–Yun [25, Section V]; the constructions specify added shell lines, an enlarged line class and image families, and section statistics and witnesses used in stated bounds.Source: §1.3 · p. 5

1.4. Notation and organization.

  • The introduction defines code and kissing-condition notation, separates products between different shells, and outlines the proof sections, appendices, and independent-checking materials.It defines A(n, δ, w) as the largest binary constant-weight code with the stated length, weight, and minimum distance.Source: §1.4 · p. 5

2. Lifting and the cost of reusing a direction

The lifting framework adds points in an enlarged space by replacing selected base directions with labelled copies and retaining the unused directions at the equator. Its compatibility criteria separate constraints on blocks from constraints on repeated directions, yielding sharp capacity bounds and explicit simplex-based lifts.

2. Lifting and the cost of reusing a direction

  • A labelled lifting is a kissing configuration exactly when its mixed-point inner products satisfy the stated pairwise compatibility conditions.The construction retains base points outside the selected sets at the equator and can include pure-tail points.Source: §2 · pp. 5–6

2.1. The incidence-capacity inequality.

  • The weighted incidence bound controls how many base directions can be reused across label sets, with equality requiring tight label cliques and full positive-weight capacities.Source: §2.1 · pp. 6–7
  • A direction used at equal tail norm h^2 < 1/2 has multiplicity at most the bound in Proposition 2.3, and that multiplicity bound is sharp.A regular simplex realizes the bound for one direction at a prescribed height, although block and pure-tail conditions need not be simultaneously saturated.Source: §2.1 · p. 7
  • For r ≥ 2, an equal-latitude simplex lift exists exactly when k ≥ r − 1 and distinct directions in B have inner products at most 1/(r + 1).It replaces each direction in B by r copies and yields |C| + (r − 1)|B| points without pure-tail points.Source: §2.1 · p. 7
  • Disjoint blocks lifted to antipodal tails produce a kissing configuration, adding one point per selected direction to the orthogonal union of the base and tail configurations.The construction uses two copies of each selected direction, one at each antipodal tail.Source: §2.1 · p. 8

3. Coordinated shell motion in dimension 25

A coordinated chord motion preserves the internal products of a 552-point contact layer, allowing one additional point while two cap points are replaced. The finite boundary test and pair checks establish a 197580-point kissing configuration in dimension 25.

3. Coordinated shell motion in dimension 25

  • The construction moves 552 retained equatorial points while preserving every internal product of the moved layer.It adjoins one point and replaces two cap points.Source: §3 · p. 9
  • For rational half-plane data and rational h^2, the boundary test checks at most 2m + 1 candidates to minimize deletions.The existence of replacements is a separate requirement.Source: §3 · p. 9
  • At least two original cap points must change when the new point is fixed, and the specified construction attains this minimum.Source: §3 · p. 10
  • Outward-rounded rational upper bounds verify the remaining pair groups, while the construction’s count changes only through the one added point.The four reported upper bounds are 1.999372, 1.965926, 1.994722, and 1.987522.Source: §3 · p. 11
  • 197580 points form a kissing configuration in dimension 25, improving the supplied 197579-point configuration by one point.Source: §3 · pp. 9, 11

4. Reuse geometry and two-layer Leech lifting

The section develops direction-reuse lifts and applies them to dimension-25 and dimension-27 Leech-shell constructions, culminating in a 201567-point configuration. It also gives an equality criterion for prescribed label-system capacity and identifies compatible 496-point blocks for lifting.

4. Reuse geometry and two-layer Leech lifting

  • Four pairwise disjoint compatible Leech-shell blocks of 496 vectors satisfy the antipodal lift’s 1/3 condition with the stronger maximum normalized inner product 1/4.Source: §4 · pp. 11–12

4.1. The endpoint construction in dimension 25.

  • The dimension-25 endpoint construction gives the historical bound K(25) ≥ 197058, with the endpoint choice permitting both poles without a larger Leech block.Source: §4.1 · p. 12

4.2. The tail geometry in dimension 27.

  • The dimension-27 four-triangle allocation gives 200540 points; compared with the stated alternative, it uses 496 fewer original directions and retains 496 more equatorial points.Source: §4.2 · p. 13

4.3. Capacity and its equality case.

  • Equality in the prescribed label-system capacity holds exactly when each used row occurs at three labels and every label contains α rows; the four-triangle construction attains equality for α = 496.Source: §4.3 · pp. 13–14

4.4. A two-layer configuration with 201567 points.

  • 201567 points form a kissing configuration in dimension 27 using the retained first-layer heads and a replacement second layer.The construction uses a 311-owner witness and includes the retained equator and twelve pure-tail points.Source: §4.4 · pp. 14–15, §4.3 · p. 14

5. Layered binary codes and joint support exchanges

This section develops layered binary-code constructions and joint exchanges that account for conflicts shared across inserted points. It gives exact replacement criteria and finite witnesses for the resulting support and code families.

5.1. Compatible replacements and shared deletion costs.

  • For an inserted set I, retaining X\F(I) gives net gain |I|−|F(I)|, charging each old conflict only once.The replacement is a kissing configuration when I is independent and all conflicts F(I) are deleted.Source: §5.1 · p. 15
  • For an internally compatible candidate pool, maximum-matching deficiency identifies the best subexchange; a matching covering every candidate rules out positive subexchanges.The certificate applies to the specified candidate pool, not to every possible replacement.Source: §5.1 · p. 16

5.2. The three-layer construction.

  • The three-layer construction combines a binary-code layer, a middle support-and-sign layer, and normalized D_n roots.Each additional support contributes 128 points, while each additional sign word contributes one.Source: §5.2 · pp. 16–17

5.3. The support families.

  • Finite verification finds pairwise intersections at most four and distinct five-subset keys, establishing A(n, 8, 8) ≥ 1803, 1960, 2843, 3077, and 3383 for n=33, 34, 37, 38, and 39, respectively.Source: §5.3 · p. 17
  • Within the supplied pools, maximum matchings certify exchange gains of 1, 1, 3, 1, and 1 for dimensions 33, 34, 37, 38, and 39.The displayed exchanges attain those gains; the certificate does not claim optimality over all possible replacements.Source: §5.3 · p. 18

5.4. Dense layers from coset unions.

  • The five code-based constructions combine support families with sign codes; the distance checks verify the required code conditions, including coset unions at lengths 38 and 39.The sign-code verification establishes minimum distances of at least ⌈m/4⌉ for Proposition 5.3.Source: §5.4 · pp. 18–19
  • At the first three lengths, the increase over the cited comparison is 128 times the support-size increase: 15, 24, and 11 supports, respectively.For the last two lengths, both larger support families and explicit coset unions contribute to the final count.Source: §5.4 · p. 19

5.5. A new length-32 support family.

  • The same construction applies to a verified family of 1,676 eight-subsets of, whose 1,403,650 unordered support pairs have an intersection histogram in orders zero through four.Source: §5.5 · p. 19

6. Local signed replacements

The section develops signed weight-eight replacements whose compatibility and gain can be characterized exactly within specified models. Hadamard transfers and boundary matching constraints yield explicit capacities and dimension-specific constructions.

6.1. A compatible replacement region.

  • A replacement preserves the kissing condition exactly when the new supports satisfy the stated intersection criterion, with gain |Q| − 128|D|.For prescribed Q, deleting all bundles whose blocks intersect a new support in more than four coordinates is the least whole-bundle deletion set.Source: §6.1 · p. 20

6.2. Hadamard transfer.

  • 576 signed weight-eight vectors in twelve coordinates are attainable in the complete-sign transverse 2 + 2 model, which is sharp within that model.Compatibility requires both left and right fibres to be matchings; equality occurs when every fibre is a perfect matching.Source: §6.2 · p. 21

6.3. Boundary-constrained capacity.

  • For boundary-restricted transfers, compatible source blocks require matching fibres, and a vertex cover supplies an explicit upper bound on capacity.In the twelve-coordinate model, the exact capacity is controlled by the matching number of a derived double-cover graph.Source: §6.3 · p. 22
  • 32 source blocks, or 512 signed points, are the maximum when a twelve-coordinate boundary forbids at least one pair type; forbidding exactly one still attains the bound.For fixed coordinate pairing, division into halves, and b prescribed deletions, the model gives an exact best gain.Source: §6.3 · p. 23

6.1. A compatible replacement region.

  • 409,676 points in dimension 35 and 484,760 in dimension 36 follow from compatible signed weight-eight replacements of complete middle-layer bundles.The isolated-region rule gives gain |Q| − 128b when an internally compatible code replaces b contained bundles and every other block intersects the region in at most four coordinates.Source: §6.1 · p. 20, §6.3 · p. 24

6.3. Boundary-constrained capacity.

  • 409,676 points in dimension 35 and 484,760 in dimension 36 are verified using archived support families and compatible replacements.The support, region, and transverse-block hypotheses are checked against the archived data.Source: §6.3 · p. 24

6.4. A 512-point replacement in dimension 37.

  • A 512-point signed replacement adds 128 points after deleting three bundles, and the full dimension-37 construction gains 1,792 points over its comparison family.The gain combines thirteen extra supports contributing 1,664 points with the signed replacement’s net 128 points.Source: §6.4 · p. 24
  • 512 is the exact capacity among complete-sign 2 + 2 transfers across the ten divisions of the six fixed coordinate pairs, but not for arbitrary twelve-coordinate signed codes.The division capacities are four at 512, four at 448, and two at 384; the supplied 32-block witness falls outside the fixed-division model.Source: §6.4 · p. 25

7. Second-shell completion in dimension 38

The dimension-38 construction completes a layered configuration with equatorial directions from a second Leech-lattice shell. Its certified 144 added lines raise the configuration to 591,900 points; the automatic shell window does not permit the next shell.

7.1. The base configuration.

  • The base configuration combines 98,280 antipodal lines from the norm-32 Golay–Leech shell with 1,932 pure-tail points arranged in 644 zero-sum triples.The shell lines are partitioned into classes with absolute inner products at most eight; the rotation checks use exact integer arithmetic.Source: §7.1 · pp. 25–26

7.3. Multiple shells and the next-shell obstruction.

  • The automatic shell window allows shell norms 32 < b ≤ 48 against minimal heads, but it controls new–old pairs only.Compatibility between added lines from different shells requires a separate pairwise separation condition.Source: §7.2 · p. 26, §7.3 · p. 27

7.2. The 144 added lines.

  • 144 norm-48 Leech lines satisfy the cross-shell compatibility condition and each has exactly 1,656 contacts with the old cap layer.A minimal head is in contact precisely when its difference from the added line is a minimal lattice vector.Source: §7.2 · p. 26
  • 591,900 points form a kissing configuration in dimension 38, adding 288 points from 144 equatorial lines to the 591,612-point base.The base has an unoccupied equator, enabling directions from a different Leech shell.Source: §7 · p. 25, §7.2 · p. 27

7.3. Multiple shells and the next-shell obstruction.

  • Norm-64 Leech vectors cannot be added at the equator of the unchanged base: each has 46 minimal-vector contacts at each of inner products 32 and −32.Source: §7.3 · p. 27

8. Seven lifts from a compatible line class

This section lifts compatible antipodal line classes using root-system tail configurations and controls the usable lines through image unions. The resulting bounds depend on certified unions, while class enlargement and image selection contribute separately to improvements.

8.1. The tail configurations.

  • The tail configurations use normalized antipodal root systems A1, A2, D3, D4, D5, E6, and E7, with 2, 6, 12, 24, 40, 72, and 126 points.Their kissing conditions follow from coordinate overlaps; the final two codes come from the integer E8 model.Source: §8.1 · p. 28

8.2. Rigidity of antipodal lifts with pure tails.

  • Paired lifts with pure tails are kissing exactly when every height is 1/2, the selected head sets are pairwise disjoint, and distinct heads within each set have inner product at most 1/3.For an antipodal base, the model’s maximum is achieved by assigning each line, with both signs, to its first selected compatible class.Source: §8.2 · p. 28

8.3. An orbitwise image-union guarantee.

  • For a finite group action, deterministic image selection guarantees a union at least as large as the orbitwise expected-union bound.At each step, choose an image maximizing the conditional expected union size.Source: §8.3 · p. 29

8.4. Selection from a finite image pool.

  • For a finite pool of overlapping line classes, some t classes have a guaranteed union size, and a sequential integer-score rule can find such a selection.The selection guarantee follows by choosing at each step to maximize the conditional expectation of the final union.Source: §8.4 · p. 29

8.5. The 7077-line class and its automorphism images.

  • The 7,077-line mother class and its isometric images supply seven certified bounds; the union, not the sum of image sizes, determines available lines.Disjoint assigned line sets are lifted with both signs and assigned injectively to antipodal tail lines.Source: §8.5 · p. 30, Table 6 · p. 30
  • K(48 + k) ≥ 52,416,000 + 2S + τ_k, with τ_k = 2t, for the selected line-class images and root-system tail code.The certificate supports seven bounds by verifying the 7,077-line class, isometries, and image unions.Source: §8.5 · p. 30

8.6. Class growth and image selection.

  • In dimension 55, class enlargement contributes 498 distinct lines rather than 504, while image selection contributes another 207 lines.The selected images overlap, so an eight-line enlargement need not provide eight new lines per image.Source: §8.6 · p. 31, Table 7 · p. 31

9. Embedded root systems and moment-certified sections

The section uses embedded root systems and spherical-design moments to count lattice-shell vectors perpendicular to specified sections. It derives exact counts for fixed sections and a nested coordinate family, with the dimension-43 improvement attributed to the parent embedding.

9.1. Signature domains and target statistics.

  • Spherical-design moments constrain all admissible projection signatures, while finite domain inequalities make the signature set finite and separate exceptional singleton fibres.The proof does not require a primitive section; antipodality removes odd total-degree moment rows, not monomials with odd individual exponents.Source: §9.1 · p. 32

9.2. A realization in P48n.

  • 2,553,792 norm-six vectors are perpendicular to the specified D5 section, giving K(43) ≥ 2,553,792.The count follows from an exact moment certificate and applies to every W(E8)-conjugate of the specified D5 subsystem.Source: §9 · p. 31, §9.2 · p. 33

9.3. The complete projection domain. √

  • The complete projection domain has 26,641 signatures, including 240 exceptions, and the minimal shell is an 11-design.The design supplies moments through degree ten for the projection count.Source: §9.3 · pp. 33–34

9.4. Moments and the averaging explanation.

  • Weyl averaging determines an average contact count, but orbit-average multiplicities need not be integers and do not alone determine a fixed child’s count.A target-preserving subgroup and rational certificate then establish the exact count for the specified child.Source: §9.4 · p. 34, §9.5 · p. 34

9.5. Exact count for the specified child.

  • The fixed-child certificate uses 43 moment or fixed-root rows and a rational multiplier to prove the exact count 2,553,792.Its inputs depend only on the root system and spherical moments; conjugate cases follow by precomposing the embedding with a Weyl element.Source: §9.5 · p. 34

9.6. A nested family of fixed coordinate sections. √

  • A nested family of eight fixed coordinate sections has exact counts established using full-rank moment systems and target-signature counts.The proof uses full column rank, not radial moments alone, to establish counts for the fixed cuts.Source: §9.6 · pp. 34–35

9.7. What accounts for the improvement.

  • The dimension-43 gain comes from the embedded E8 parent: identical intrinsic section data yield 2,553,792 contacts here versus 2,545,056 in the comparison construction.Both sections have determinant 972, singleton cluster, β = 0, and permutation-isometric Grams.Source: §9.7 · p. 35

10. A moment-certified projection in dimension 45

The section projects fibres of a norm-six shell orthogonally to a three-dimensional section and certifies the resulting count with design moments and an anchor graph. For the specified QR-neighbour construction, it gives a dimension-45 kissing configuration of 7,380,090 points.

10.1. The specified quadratic-residue neighbour.

  • The specified quadratic-residue-neighbour lattice has minimum squared norm six, and its minimal shell has 52,416,000 vectors and is an 11-design.The minimum is established using code-weight and parity arguments.Source: §10.1 · pp. 36–37

10.2. A complete signature domain and its moments.

  • The moment system reduces the projection count to an exact identity after excluding signatures with positive dual slack.The complete domain is refined from 239 to 201 signatures; restricting to the refined domain gives the exact identity.Source: §10.2 · p. 37

10.3. Witness amplification.

  • Three fibres project injectively to a kissing configuration with 7,377,408 + 9f(−2, −2, 3) points.Each distinct norm-six vector with signature (−2, −2, 3) supplies one witness counted by f.Source: §10.3 · p. 38

10.4. A regular anchor graph.

  • For any norm-eight anchor, the graph has 1,128 vertices and degree 368; each nonadjacent pair represents a section with the specified Gram matrix.The corresponding fibre size is determined by the common-neighbour count of that pair.Source: §10.4 · p. 38

10.5. Signed spectrum and lattice restrictions.

  • The signed adjacency matrix has spectrum {92 (47), −4 (1081)}; lattice restrictions also force even codegrees and forbid four mutually negative signed products.The unsigned adjacency matrix satisfies A + 16I ⪰ 0.Source: §10.5 · p. 39

10.6. The certified anchor and its optimum.

  • The value 7,380,090 is optimal for the specified anchor and section Gram, not across other anchors, lattices, or Grams.The complete anchor graph enumeration finds a unique nonadjacent pair with codegree 70 among 428,076 pairs.Source: §10.6 · p. 39
  • 7,380,090 points form a dimension-45 kissing configuration, so K(45) ≥ 7,380,090.The count combines a base projection of 7,377,408 points with 9 times the 298 witnesses in the specified fibre.Source: §10.6 · p. 40

11. Finite search and mathematical structure

The constructions exploit different structural freedoms in constrained configurations, and the paper states explicit limits for several mechanisms. Section counts use moment reductions, with finite witnesses supplying the hypotheses for the nineteen lower bounds.

11. Finite search and mathematical structure

  • The constructions exploit layer motion, signed-support changes, another lattice shell, and low-overlap images, while stating limits for specific insertions and lifts.Two cap changes are necessary for the fixed dimension-25 insertion, and the next Leech shell cannot augment the unchanged dimension-38 base.Source: §11 · p. 40
  • Moment reductions identify section statistics without determining every fibre; together with finite witnesses, they yield the nineteen stated lower bounds.The embedded parent determines the dimension-43 count, while dimension-45 section vectors remove all positive dual-slack signatures.Source: §11 · p. 40

12. Finite verification

The verification section describes finite witnesses and checks supporting the construction proofs. It also reports that Qiushi Engine conducted the searches, analysis, and computational verification, with final results reduced to explicit arguments and checkable certificates.

12. Finite verification

  • Each lower bound combines a mathematical reduction covering point-pair types with a finite witness satisfying its hypotheses.The supplement contains the witnesses and programs used to check the required identities and inequalities.Source: §12 · p. 40

12.1. Autonomous mathematical research.

  • Qiushi Engine carried out construction searches, mathematical analysis, and computational verification across the nineteen constructions.The research connected representation changes with layer motion, joint label selection, signed-design transfer, and moment functionals for section counts.Source: §12.1 · p. 41

Appendix A. The integer Leech shell and tail conventions

The appendix specifies a Golay-code model for three families of integer vectors and verifies their sizes, norms, and pairwise inner-product bounds. It also fixes labels and coordinate changes used for the tail conventions.

Appendix A. The integer Leech shell and tail conventions

  • The Golay code has dimension 12, is self-orthogonal, and has weight enumerator 1 + 759z^8 + 2576z^12 + 759z^16 + z^24.An octad is the support of a weight-eight codeword; distinct octads intersect in at most four coordinates.Source: Appendix A · p. 42
  • The three families contain 1104, 97152, and 98304 vectors, respectively, totaling 196560 vectors of squared norm 32.The family cardinalities follow from the parameter-recovery argument, and each family has squared norm 32.Source: Appendix A · p. 42
  • All six pair types among the three families satisfy the stated inner-product bound of 16.The checks cover A–A, A–B, A–C, B–B, B–C, and C–C pairs.Source: Appendix A · p. 42
  • The triangle labels are {0, 7, 10}, {1, 6, 9}, {2, 3, 11}, and {4, 5, 8}, and exact coordinate-change tests check every label.The common rotation is checked to preserve within-set and cross-set inner products.Source: Appendix A · p. 43

Appendix B. Binary generators and coset representatives

Appendix B prints binary generators and coset representatives for the dense sign codes, with finite enumeration certifying their distance properties. Appendix C describes an explicit even-neighbour model and its minimal shell.

Appendix B. Binary generators and coset representatives

  • The printed data specify the generators and affine representatives for the length-32, length-38, and length-39 codes.The length-32 generator is used unchanged in dimensions 32–37.Source: Appendix B · p. 43, Table 11 · p. 44, Table 12 · p. 44
  • Finite enumeration establishes the kernel weight distributions and minimum weight ten between every distinct pair of affine representatives.The length-32 code uses only the zero representative; the length-38 and length-39 cases use 11 and 10 cosets.Source: Appendix B · p. 43

Appendix C. An explicit Pless neighbour and its minimal shell

  • The explicit integer model establishes the minimum and shell size for every extremal even unimodular lattice of rank 48.The stated scope includes the catalogue lattice in Section 8 and the QR neighbour in Section 10.Source: Appendix C · p. 44

C.1. The specified even neighbour.

The specified even neighbour is an even unimodular lattice with minimum squared norm 6 and a norm-six shell yielding a kissing configuration. The appendix also distinguishes this construction from other layered-code observations.

C.1. The specified even neighbour.

  • The lattice L′ is even unimodular, has minimum squared norm 6, and contains exactly 52,416,000 vectors of squared norm 6.The displayed characteristic congruence establishes evenness, and the construction has covolume one.Source: §C.1 · p. 45
  • The code-theoretic constraints exclude vectors of squared norm 2 or 4, while a displayed lattice vector has squared norm 6.The Pless symmetry code has minimum Hamming weight 15, and the complete-weight-enumerator consequence narrows the possible short vectors.Source: §C.1 · pp. 45–46
  • The norm-six shell, divided by 6, gives a kissing configuration of size 52,416,000.For distinct shell vectors, their difference has squared norm at least six, so their inner product is at most 3.Source: §C.1 · p. 46

Appendix D. Flexibility of auxiliary configurations

  • The dimension-38 flexibility observation concerns the layered-code construction, whereas the 591900-point construction uses the Leech lift.The appendix also identifies X25 as the 197058-point endpoint model and XT, XM, and XR as ERS layers.Source: Appendix D · p. 46

D.1. Paired motion in dimension 25.

In dimension 25, paired points can move together while preserving the kissing condition and specified contacts. In dimensions 38 and 39, dense-layer points can move independently without changing cardinality or creating new contacts involving them.

D.1. Paired motion in dimension 25.

  • In dimension 25, each paired direction can move independently by less than 1/24 while preserving the kissing condition and the specified contacts.With the equator, poles, and labels fixed, each pair's common direction remains free in an open subset of S23.Source: §D.1 · p. 47

D.2. Independent motion of the dense layers.

  • In dimensions 38 and 39, each point of XT can move independently within its displacement radius without changing cardinality or creating new contacts involving dense points.The respective radii are 1/76 and 1/156.Source: §D.2 · p. 47
  • The dimension-25 motion moves contacting pairs collectively, while the dimension-38 and -39 motions move strictly separated points independently.The dense-code starting configurations need not retain a binary description throughout their neighbourhoods.Source: §D.2 · p. 47
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