A Lagged Cross-Polytope Ladder for Kissing Numbers: Recursive Calibration and Falsifiable Bounds
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2026
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| author | He, Mingdong |
| author_facet | He, Mingdong |
| contents | <p><strong>Background and Motivation:</strong></p> <p>This work documents a heuristic observation (inspired by a visualization in a short video) outside the author's primary research field. While the study of kissing numbers is not the main focus of the author's academic work, the identified patterns are recorded here as a speculative project to facilitate further exploration and discussion.</p> <p><strong>Core Findings:</strong></p> <p>The manuscript proposes a structural framework relating kissing numbers k(n) to the face numbers of lagged cross-polytopes. We define the quantity B_d(n) := N_d(n-d+1) = 2^{d+1}\binom{n-d+1}{d+1}, suggesting a potential link between structural constraints and d-face counts.</p> <p><strong>The Lagged-Ladder Mechanism:</strong></p> <p>We identify a "sliding ladder" progression where the dominant structural hierarchy d^*(n) climbs through discrete geometric steps. The framework employs a recursive calibration rule: the system remains at level d as long as known lower bounds satisfy L(n) \le B_d(n), shifting to d+1 upon a forced transition.</p> <p><strong>Falsifiable Predictions:</strong></p> <p>The framework remains consistent with established data, including the Leech lattice at n=24. It provides sharp, non-trivial upper bound predictions for dimensions 12 through 15:</p> <p>k12≤1320 K13≤1760 K14≤2288 K15≤2912</p> <p>These predictions serve as immediate targets for computational verification or falsification.</p> <p> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19933251 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Lagged Cross-Polytope Ladder for Kissing Numbers: Recursive Calibration and Falsifiable Bounds He, Mingdong <p><strong>Background and Motivation:</strong></p> <p>This work documents a heuristic observation (inspired by a visualization in a short video) outside the author's primary research field. While the study of kissing numbers is not the main focus of the author's academic work, the identified patterns are recorded here as a speculative project to facilitate further exploration and discussion.</p> <p><strong>Core Findings:</strong></p> <p>The manuscript proposes a structural framework relating kissing numbers k(n) to the face numbers of lagged cross-polytopes. We define the quantity B_d(n) := N_d(n-d+1) = 2^{d+1}\binom{n-d+1}{d+1}, suggesting a potential link between structural constraints and d-face counts.</p> <p><strong>The Lagged-Ladder Mechanism:</strong></p> <p>We identify a "sliding ladder" progression where the dominant structural hierarchy d^*(n) climbs through discrete geometric steps. The framework employs a recursive calibration rule: the system remains at level d as long as known lower bounds satisfy L(n) \le B_d(n), shifting to d+1 upon a forced transition.</p> <p><strong>Falsifiable Predictions:</strong></p> <p>The framework remains consistent with established data, including the Leech lattice at n=24. It provides sharp, non-trivial upper bound predictions for dimensions 12 through 15:</p> <p>k12≤1320 K13≤1760 K14≤2288 K15≤2912</p> <p>These predictions serve as immediate targets for computational verification or falsification.</p> <p> </p> |
| title | A Lagged Cross-Polytope Ladder for Kissing Numbers: Recursive Calibration and Falsifiable Bounds |
| url | https://doi.org/10.5281/zenodo.19933251 |