This question is a copy of one I asked in the [Math StackExchange forum](https://math.stackexchange.com/questions/4623074/what-lattices-beyond-the-laminated-lattices-particularly-in-%E2%89%A4-24d-belong-to-a) a few days ago. I don't know if it qualifies as a research-level question, but it may be something beyond most people on the Math StackExchange forum.


Back in 2016, in his proposed answer to the question asked at <https://math.stackexchange.com/questions/1790672/packing-of-n-balls/1795044#1795044> , achille hui ( <https://math.stackexchange.com/users/59379/achille-hui> ) gave the following definition of laminated lattices, which I found to be very clearly worded and useful for what I will be asking further down.


"Laminated lattices $\Lambda\_n$ can be defined/constructed recursively.


* For $n = 1$, $\Lambda\_1$ is "the" lattice of even integers.
* For $n > 1$, $\Lambda\_n$ is "a" $n$-dim lattice [which] satisfies


1. the minimal spacing among lattice points is $2$.
2. contains "a" $\Lambda\_{n-1}$ as sub-lattice.
3. subject to 1) and 2), the volume of its fundamental cell is minimal."


He goes on to give the following textualization and note, which will also seem to work for my proposed generalization further down (although the last sentence of the second paragraph would no longer be correct):


"Geometrically, one can think of $\Lambda\_n$ as stacking copies of a lower dimensional laminated lattice $\Lambda\_{n-1}$ as tightly as possible without reducing the minimum lattice spacing.


With this definition, there is no guarantee $\Lambda\_n$ is unique for a given $n$. Indeed, it isn't unique in general. However, $\Lambda\_n$ is unique for $n \le 10$ and $14 \le n \le 24$."


I am envisioning a category of lattices (perhaps call them "psuedo-laminated lattices", if that isn't already defined to mean something else) where the definition is the same as achille hui's above apart from his #3 in the recursive definition:


"3. for at least one of the $\Upsilon\_{n-1}$'s it contains as a sub-lattice, among the $n$-dim lattices which satisfy 1) and which contain *that* $\Upsilon\_{n-1}$ as a sub-lattice, the volume of its fundamental cell is minimal."


(The Greek letter Upsilon doesn't seem to be used much in mathematics, and I like the way it looks, so I'm using $\Upsilon\_n$ here to denote a "pseudo-laminated" lattice in dimension $n$. I didn't think it necessary to repost the part of achille hui's definition of laminated lattices other than his condition #3 (where the one substantive change is) just to replace the symbols used, but each $\Lambda$ there would become "a" $\Upsilon$ (or maybe "an" $\Upsilon$) in my definition.)


This recursive definition would follow what I think of as an "any possible resolution of ties beginning with the earliest one" or "share of inheritance" (with inheritance shared in the case of any tie) model. If $\Lambda\_{13}^\mathsf{mid}$, the $\Lambda\_{13}$ with the mid-ranked kissing number of the three $\Lambda\_{13}$'s, was only infinitesimally more dense than $\Lambda\_{13}^\mathsf{max}$ or $\Lambda\_{13}^\mathsf{min}$, it wouldn't matter that the lattice known in actuality as $\Lambda\_{14}$ (which contains $\Lambda\_{13}^\mathsf{max}$ and $\Lambda\_{13}^\mathsf{min}$ as sub-lattices, but apparently not $\Lambda\_{13}^\mathsf{mid}$)... it wouldn't matter that that lattice is denser than any lattice formed by layers of $\Lambda\_{13}^\mathsf{mid}$ (which would in this case be the sole $\Lambda\_{13}$) and respecting the same minimum distance of $2$ between lattice points (or alternatively the same minimal norm of $4$). The densest lattice(s) with the same minimal norm which contained $\Lambda\_{13}^\mathsf{mid}$ (which would again be the sole $\Lambda\_{13}$ in this scenario) would be the laminated lattice(s) in dimension $14$. So why shouldn't that/those lattice(s) be considered as laminated lattices in actuality, given that tie in dimension $13$ and what the result of a resolution in favor of $\Lambda\_{13}^\mathsf{mid}$ would be?


Of course, a simple answer to that question is, "That's not how laminated lattices are defined." So, I'm basically asking, "But, what if they were defined that way?" Or rather I'm proposing a slightly generalized category of lattices using that definition as one worthy of note.


For all I know, the set of $\Upsilon\_n$'s could remain strictly larger than the set of $\Lambda\_n$'s for all $n \ge 14$ ($n = 14$ is the lowest $n$ for which there are any $\Upsilon\_n$'s that are not $\Lambda\_n$'s). Or it could be for all I know that the $\Upsilon\_{14}$'s other than $\Lambda\_{14}$ are sublattices of $\Lambda\_{15}$, and that $\Lambda\_{15}$ is the densest minimal norm $4$ lattice in dimension $15$ containing any of the $\Upsilon\_{14}$'s and is thus the sole $\Upsilon\_{15}$. And thus the set of pseudo-laminated lattices becomes equivalent to the set of laminated lattices from dimensions 15 through at least $25$ (if $\Lambda\_{24}$ (the Leech lattice) is the sole $\Upsilon\_{24}$, then the sets of $\Upsilon\_{25}$'s and $\Lambda\_{25}$'s must be equivalent), although the set of pseudo-laminated lattices could become larger again in higher dimensions. Or there could be (for all I know) a "collapsing" of the number of pseudo-laminated lattices to $1$ somewhere after dimension $15$ but before dimension $25$ where the number of laminated lattices itself expands again, seemingly not to collapse back to $1$ for a long while, if at all.


So my question is basically which of the scenarios I mentioned in the last paragraph is the case, and what additional pseudo-laminated lattices there are (or are presently known) in dimensions $14$ through $24$ (and to a lesser extent in dimensions beyond $24$). I don't know if this category of lattices has been studied at all, although I doubt I'm the first person who's thought of this.


Thanks to anyone who can help in answering my question here. Feel free to also comment on the utility and noteworthiness of this proposed category as compared to the category of laminated lattices as they are actually defined.



What lattices beyond the laminated lattices (particularly in $\le 24D$) belong to a slightly expanded category that includes "descendants" of Λ13_mid?
I finally found the answer to this question in a 1993 paper by W. Plesken and M. Pohst, *Constructing integral lattices with prescribed minimum. II* ( <https://www.ams.org/journals/mcom/1993-60-202/S0025-5718-1993-1176715-1/S0025-5718-1993-1176715-1.pdf> ). They are the lattices in Figure 1 of that article that are to the left or right of the main path of (strongly) laminated lattices, but to the right of the dashed dividing line, in $14$ through $18$ dimensions (where that "main path" forks beginning in dimension $11$ before merging at dimension $14$ are where the classic set of (strongly) laminated lattices itself has multiple non-isometric cases in a single dimension, with the number after the decimal point ascending with descending kissing number). To be specific, the lattices on the "lambda-lattices" side of that figure marked $14.2$, $14.4$, $14.3$, $15.2$, $15.3$, $15.4$, $16.2$, $16.3$, $16.4$, $17.2$, $17.3$, $17.4$, $18.2$ and $18.3$ are what I was thinking of as the "pseudo-laminated lattices" other than the laminated ones. Jacques Martinet in Chapter 3 of *Perfect lattices in Euclidean spaces*, which I bought earlier this year in search of an answer to this question, calls this group of lattices the *weakly laminated lattices above* $\Lambda0$ (or above $\Lambda1$ if I were to start the definition as with the one I quoted in the question), with what I had learned as being the laminated lattices being the *strongly laminated lattices*, or simply the *laminated lattices* (above $\Lambda0$ or $\Lambda1$). It was on the page after this definition in Martinet's work that I was directed to the two papers by Plesken and Pohst on the subject, where I finally found what I was looking for in the second of those two papers. The Leech lattice is indeed the sole weakly laminated lattice above $\Lambda0$ or $\Lambda1$ in $24$ dimensions, but it's not until dimension $19$ that all paths merge, and not until dimension $22$ (rather than $18$) that the Kappa lattices fully merge back into that path if you define allow for all weakly laminated lattices above $K7$, $K8$ aka $K8.1$ and $K9$ aka $K9.1$ are included (you have to use points other than deep holes of prior layers, presumably shallow holes but I don't know that for a fact, three times in a row in the Kappa family to avoid it quickly merging back into the Lambda family).


