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Sketches of the different taxonomy rules. Figure \subref{sf.sketch1} shows the tower vectors of a KK tower and a string tower, while Figure \subref{sf.sketch2} represents two different KK towers decompactifying either $n$ or $m$ extra dimensions. Both the lengths and angles between the vectors are fixed by the nature of the tower. Figure \subref{sf.sketch3} depicts an example of a polytope that is consistent with the taxonomy rules. In the three figures the lower bound of $\frac{1}{\sqrt{d-2}}$ for the exponential mass decay rate is depicted in gray.

Sketches of the different taxonomy rules. Figure \subref{sf.sketch1} shows the tower vectors of a KK tower and a string tower, while Figure \subref{sf.sketch2} represents two different KK towers decompactifying either $n$ or $m$ extra dimensions. Both the lengths and angles between the vectors are fixed by the nature of the tower. Figure \subref{sf.sketch3} depicts an example of a polytope that is consistent with the taxonomy rules. In the three figures the lower bound of $\frac{1}{\sqrt{d-2}}$ for the exponential mass decay rate is depicted in gray.

Sketches of the different taxonomy rules. Figure \subref{sf.sketch1} shows the tower vectors of a KK tower and a string tower, while Figure \subref{sf.sketch2} represents two different KK towers decompactifying either $n$ or $m$ extra dimensions. Both the lengths and angles between the vectors are fixed by the nature of the tower. Figure \subref{sf.sketch3} depicts an example of a polytope that is consistent with the taxonomy rules. In the three figures the lower bound of $\frac{1}{\sqrt{d-2}}$ for the exponential mass decay rate is depicted in gray.

: Tower polytope example.

: Species polytope example

: Planckian phase.

: Stringy phase

: Plankian phase.

: Stringy phase

: Two duality frames with the same $\vec{\mathcal{Z}}_{\rm QG}$.

: Two duality frames with different $\vec{\mathcal{Z}}_{\rm QG}$.

A ``good'' projection of a rank 3 frame simplex down to a rank 2 ``effective'' frame simplex. To perform the projection, the tower vectors $\{\vec\zeta_1,\,\vec\zeta_2,\,\vec\zeta_3\}$ are partitioned into disjoint faces $\mathcal{F}_1 = \{ \vec\zeta_1 \}$, $\mathcal{F}_2 = \{ \vec\zeta_2, \vec\zeta_3 \}$ and the frame simplex is projected onto the plane generated by the face pericenters $\vec\zeta_{\mathcal{F}_1}$, $\vec\zeta_{\mathcal{F}_2}$, which are the tower vectors of the effective frame simplex. Note that the intersection of the frame simplex with this plane is equal to its projection onto the plane, which is an equivalent way to define a ``good'' projection.

: Rank-2 tower polytope in $d=7$.

: Rank-3 tower polytope in $d=4$.

: Rank-2 species polytope in $d=7$.

: Rank-3 species polytope in $d=4$.

Tower polytope for Type IIB string theory compactified on $S^1$ to $d=9$, with the different KK (in blue) and string oscillator towers depicted, as well as the different duality frame descriptions. The moduli correspond to the canonically normalized 10d dilaton and the $S^1$ radius. The sphere of radius $\frac{1}{\sqrt{d-2}}=\frac{1}{\sqrt{7}}$ is shown in gray and the Type IIB self-dual line in red. See \cite{Etheredge:2023odp} for more details on the limits of the duality frames and the expression of the different towers tower vectors.

Example of a direction $\hat{t}$ over which which several towers (in this case those associated to $\vec{\zeta}_1$ and $\vec{\zeta}_2$) become degenerate in a way that we can ``ignore''. The rank 3 Planckian phase is the same as depicted in Figure \ref{f.3dFRAMEch1}.

Frame simplices in different asymptotic limits of the I$'_{SO(32)}$ regions in $SO(32)$ heterotic string theory compactified on $S^1$. In Type I$'$ infinite-distance limits parallel to the self-duality line (red), the taxonomy rules are violated, as the tower vector for the Type I$'$ KK modes takes values along a segment (green arrows) orthogonal to this line. Such limits are irregular due to the effects of warping. In all other Type I$'$ infinite-distance limits, however, the KK, I$'^{\rm(warp)}$ vector approaches either KK, I$'$ or KK, I$'^{\rm (dual)}$. Such limits are regular, as the effects of warping become negligible in the asymptotic limit, so the frame simplices of these limits satisfy the taxonomy rules. Note that the heterotic towers (along the thin black segment) remain fixed in any limit.

: 2d tower polygons for 9, 8, 7, and 6 dimensional theories, with 11d as the maximum decompactification dimension. String oscillators depicted in red \fcolorbox{black}{red}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, while one, two, three, four and five dimensional KK modes appear in blue \fcolorbox{black}{blue}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, green \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, yellow \fcolorbox{black}{yellow}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, orange \fcolorbox{black}{orange}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and pink \fcolorbox{black}{Magenta}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, respectively. The point closest to the origin of edges associated to decompactification of several dimensions is highlighted. The disk of radius $\frac{1}{\sqrt{d-2}}$ is depicted in gray. The polygons of which no string embedding is known are highlighted in red.

: 2d species polygons for 9, 8, 7, and 6 dimensional theories, with 11d as the maximum decompactification dimension. Species scales given by the string scale are depicted in red \fcolorbox{black}{red}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, while Planck masses in 1, 2, 3, 4 and 5 dimensions more appear in blue \fcolorbox{black}{blue}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, green \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, yellow \fcolorbox{black}{yellow}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, orange \fcolorbox{black}{orange}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and pink \fcolorbox{black}{Magenta}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, respectively. The inner and outer disks have of radii $\frac{1}{\sqrt{d-2}}$ and $\frac{1}{\sqrt{(d-1)(d-2)}}$. The polygons of which no string embedding is known are highlighted in red.

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: $\mathcal{P}_{(10,A)}$: IIA

: $\mathcal{P}_{(10,B)}$: IIB, I, heterotic.

: $\mathcal{P}_{(10,C)}$: No known example.

: $\mathcal{P}_{(10,A)}^\circ$: IIA

: $\mathcal{P}_{(10,B)}^\circ$: IIB, I, heterotic.

: $\mathcal{P}_{(10,C)}^\circ$: No known example.

Maximal tower polytope for 9d theories. Red and blue points respectively correspond to string oscillators and KK towers associated with decompactifying 1 dimension. The edge colored in green, with its closest point highlighted, is associated to decompactification of 2 dimensions.

Maximal tower polytope of the 8d theory, $\mathcal{P}_{(8)}$. The string and $\vec{\zeta}_{\rm KK_1}$ towers are depicted in red \fcolorbox{black}{red}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and blue blue \fcolorbox{black}{blue}{\rule{0pt}{6pt}\rule{6pt}{0pt}}. The edges and facets associated to decompactification of two and three internal dimensions are depicted in green \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and yellow \fcolorbox{black}{yellow}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, with their closest point to the origin highlighted. The ball of radius $\frac{1}{\sqrt{d-2}}=\frac{1}{\sqrt{6}}$ is presented in gray and the triangulation of the polytope into frame simplices is depicted with blue lines.

Maximal species polytope for 8d theories. Points associated to the string scale \fcolorbox{black}{red}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, while 9, 10 and 11 dimensional Planck mass appear in blue \fcolorbox{black}{blue}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, green \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and yellow \fcolorbox{black}{yellow}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, respectively. The sphere of radius $\frac{1}{\sqrt{(d-1)(d-2)}}=\frac{1}{\sqrt{42}}$ is depicted in red and the triangulation of the species polytope in blue lines.

Shaded in black, the $\mathcal{P}_{(10,A)}$ (lower) and $\mathcal{P}_{(10,B)}$ (upper) tower polytopes are obtained from $\mathcal{P}_{(9)}$ after decompactifying along the two inequivalent $\vec{\zeta}_{\rm KK_1}$ vertices. The disk of radius $\frac{1}{\sqrt{(d-2)}}=\frac{1}{\sqrt{7}}$ is depicted in gray.

: $\mathcal{P}_{(9)}$ embedded in $\mathcal{P}_{(8)}$

: $\mathcal{P}_{(9)}^\circ$ embedded in $\mathcal{P}_{(8)}^\circ$

: $\mathcal{P}_{(10,A)}$ embedded in $\mathcal{P}_{(8)}$

: $\mathcal{P}_{(10,B)}$ embedded in $\mathcal{P}_{(8)}$

: $\mathcal{P}_{(8)}$ embedded in $\mathcal{P}_{(7)}$

: $\mathcal{P}_{(7)}$ embedded in $\mathcal{P}_{(6)}$

Tentative steps in trying to consistently compactify the polygon $\Pi_{\rm (6,V)}$ from Table \ref{tab:BIG} (in dashed gray lines) from $d=6$ to 5 dimensions. The depicted sphere has radius $\frac{1}{\sqrt{5-2}}$, while the black circumference has radius $\frac{1}{\sqrt{6-2}}$. The string towers are depicted in \fcolorbox{black}{red}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, whilst KK towers associated to decompactification of one, two, three and four dimensions are in colors \fcolorbox{black}{blue}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, \fcolorbox{black}{yellow}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and \fcolorbox{black}{orange}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, respectively.

Tentative steps in trying to consistently compactify the polygon $\Pi_{\rm (6,V)}$ from Table \ref{tab:BIG} (in dashed gray lines) from $d=6$ to 5 dimensions. The depicted sphere has radius $\frac{1}{\sqrt{5-2}}$, while the black circumference has radius $\frac{1}{\sqrt{6-2}}$. The string towers are depicted in \fcolorbox{black}{red}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, whilst KK towers associated to decompactification of one, two, three and four dimensions are in colors \fcolorbox{black}{blue}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, \fcolorbox{black}{yellow}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and \fcolorbox{black}{orange}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, respectively.

Tentative steps in trying to consistently compactify the polygon $\Pi_{\rm (6,V)}$ from Table \ref{tab:BIG} (in dashed gray lines) from $d=6$ to 5 dimensions. The depicted sphere has radius $\frac{1}{\sqrt{5-2}}$, while the black circumference has radius $\frac{1}{\sqrt{6-2}}$. The string towers are depicted in \fcolorbox{black}{red}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, whilst KK towers associated to decompactification of one, two, three and four dimensions are in colors \fcolorbox{black}{blue}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, \fcolorbox{black}{yellow}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and \fcolorbox{black}{orange}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, respectively.

Tentative steps in trying to consistently compactify the polygon $\Pi_{\rm (6,V)}$ from Table \ref{tab:BIG} (in dashed gray lines) from $d=6$ to 5 dimensions. The depicted sphere has radius $\frac{1}{\sqrt{5-2}}$, while the black circumference has radius $\frac{1}{\sqrt{6-2}}$. The string towers are depicted in \fcolorbox{black}{red}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, whilst KK towers associated to decompactification of one, two, three and four dimensions are in colors \fcolorbox{black}{blue}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, \fcolorbox{black}{yellow}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and \fcolorbox{black}{orange}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, respectively.

Consistent 3d convex hull in $d=5$ from which the $d=6$ polygon $\Pi_{\rm (6,I)}$ depicted in Table \ref{tab:BIG} (in dashed gray lines) can be recovered from decompactifying one dimension. The depicted sphere has radius $\frac{1}{\sqrt{5-2}}$, while the black circumference has radius $\frac{1}{\sqrt{6-2}}$. The string towers are depicted in red \fcolorbox{black}{red}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, while KK towers and edges/facets (in these cases with the points closest to the origin highlighted) associated to decompactification of one, two, three, four, five and six dimensions appear in blue \fcolorbox{black}{blue}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, green \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, yellow \fcolorbox{black}{yellow}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, orange \fcolorbox{black}{orange}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, pink \fcolorbox{black}{Magenta}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and brown \fcolorbox{black}{Brown}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, respectively.

Consistent 3d convex hull in $d=4$ from which the $d=6$ polygon $\Pi_{\rm (6,V)}$ depicted in Table \ref{tab:BIG} (in dashed gray lines) can be recovered from decompactifying \emph{two} dimensions. The depicted sphere has radius $\frac{1}{\sqrt{4-2}}$, while the black circumference has radius $\frac{1}{\sqrt{6-2}}$. The string towers are depicted in red \fcolorbox{black}{red}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, whilst KK towers and edges/facets (for these the closest point to the origin is highlighted) associated to decompactification of two, four and six dimensions appear in green \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, orange \fcolorbox{black}{orange}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and brown \fcolorbox{black}{Brown}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, respectively. Note that in this slice of the moduli space there would not be decompacitification limits to an odd number of dimensions.

Recovery of the tower and species polytopes $\mathcal{P}_{(10,C)}$ and $\mathcal{P}_{(10,C)}^\circ$ of the 10d theory from decompactification of a 4d theory with tower and species polygons $\Pi_{\rm (6,XI)}$ (in solid lines) and $\Pi^\circ_{\rm (6,XI)}$ (dashed), see Tables \ref{tab:BIG} and \ref{tab:BIGsp}. These are located at the same height as the species scale $\vec{\mathcal{Z}}_{\rm Pl_{10}}$.

Tower polytope $\mathcal{P}_{(8)}$ and the action on it by the group $\mathsf{G}_{8}=S_3\times S_2=\left \langle y,\,a: y^2=a^3=e,\, yay=a^{-1} \right\rangle\times\left\langle x: x^2=e\right\rangle$. The fixed loci correspond to those invariant under $x$ ($\Pi_{\rm (8,I)}$ in Table {\ref{tab:BIG}}) and $y$ ($\Pi_{\rm (8,II)}$).

Unique three-dimensional polytope obtained by applying the taxonomy rules for $d=7$. The string, $\vec{\zeta}_{\rm KK_1}$ and $\vec{\zeta}_{\rm KK_2}$ towers are depicted in red \fcolorbox{black}{red}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, blue \fcolorbox{black}{blue}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and green \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}}. Edges and facets associated to decompactifications of two, three and four dimensions appear in green \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, yellow \fcolorbox{black}{yellow}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and orange \fcolorbox{black}{orange}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, with their closest point to origin highlighted. Note the sphere with radius $\frac{1}{\sqrt{d-2}}=\frac{1}{\sqrt{5}}$ is contained inside it.

Species polytope in $d=7$ dual to that pictured in Figure \ref{f.7d3}. The $\mathcal{Z}$-vectors associated to the string scale are depicted in red \fcolorbox{black}{red}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, while those to the 8, 9, 10 and 11-dimensional Planck mass appear in blue \fcolorbox{black}{blue}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, green \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, yellow \fcolorbox{black}{yellow}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and orange \fcolorbox{black}{orange}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, respectively. The sphere with radius $\frac{1}{\sqrt{(d-2)(d-1)}}=\frac{1}{\sqrt{30}}$ is depicted. The facets recovered from rank 2 species polytopes for $(D=7+n)$-dimensional theories are outlined in blue ($n=1$, corresponding to $\Pi^\circ_{\rm (8,I)}$ and $\Pi^\circ_{\rm (8,II)}$ in Table \ref{tab:BIGsp}) and green ($n=2$, $P^\circ_{(9)}=\Pi^\circ_{\rm (9,I)}$). The facets not outlined are congruent to those that are. The $n$ associated to each facet must be subtracted from each vertex to recover the appropriate one in the $D$-dimensional theory.

Representation of the three rank 3 tower polytopes in $d=6$ obtained as slices of the maximal $\mathcal{P}_{(6)}$ tower polytope. The string, $\vec{\zeta}_{\rm KK_1}$, $\vec{\zeta}_{\rm KK_2}$ and $\vec{\zeta}_{\rm KK_3}$ towers are depicted in red \fcolorbox{black}{red}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, blue \fcolorbox{black}{blue}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, green \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and yellow \fcolorbox{black}{yellow}{\rule{0pt}{6pt}\rule{6pt}{0pt}}. Edges and facets associated to decompactifications of two, three, four and five dimensions appear in green \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, yellow \fcolorbox{black}{yellow}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, orange \fcolorbox{black}{orange}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and pink \fcolorbox{black}{Magenta}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, with their closest point to origin highlighted. In gray the sphere with radius $\frac{1}{\sqrt{d-2}}=\frac{1}{2}$ is depicted. Note that the cube depicted in Figure \ref{sf.3d-6d1} is nothing but that from Figure \ref{f.cube4dTOW} under the rescaling described before Section \ref{s.fullpolytope}.

Representation of the three rank 3 tower polytopes in $d=6$ obtained as slices of the maximal $\mathcal{P}_{(6)}$ tower polytope. The string, $\vec{\zeta}_{\rm KK_1}$, $\vec{\zeta}_{\rm KK_2}$ and $\vec{\zeta}_{\rm KK_3}$ towers are depicted in red \fcolorbox{black}{red}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, blue \fcolorbox{black}{blue}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, green \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and yellow \fcolorbox{black}{yellow}{\rule{0pt}{6pt}\rule{6pt}{0pt}}. Edges and facets associated to decompactifications of two, three, four and five dimensions appear in green \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, yellow \fcolorbox{black}{yellow}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, orange \fcolorbox{black}{orange}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and pink \fcolorbox{black}{Magenta}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, with their closest point to origin highlighted. In gray the sphere with radius $\frac{1}{\sqrt{d-2}}=\frac{1}{2}$ is depicted. Note that the cube depicted in Figure \ref{sf.3d-6d1} is nothing but that from Figure \ref{f.cube4dTOW} under the rescaling described before Section \ref{s.fullpolytope}.

Representation of the three rank 3 tower polytopes in $d=6$ obtained as slices of the maximal $\mathcal{P}_{(6)}$ tower polytope. The string, $\vec{\zeta}_{\rm KK_1}$, $\vec{\zeta}_{\rm KK_2}$ and $\vec{\zeta}_{\rm KK_3}$ towers are depicted in red \fcolorbox{black}{red}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, blue \fcolorbox{black}{blue}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, green \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and yellow \fcolorbox{black}{yellow}{\rule{0pt}{6pt}\rule{6pt}{0pt}}. Edges and facets associated to decompactifications of two, three, four and five dimensions appear in green \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, yellow \fcolorbox{black}{yellow}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, orange \fcolorbox{black}{orange}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and pink \fcolorbox{black}{Magenta}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, with their closest point to origin highlighted. In gray the sphere with radius $\frac{1}{\sqrt{d-2}}=\frac{1}{2}$ is depicted. Note that the cube depicted in Figure \ref{sf.3d-6d1} is nothing but that from Figure \ref{f.cube4dTOW} under the rescaling described before Section \ref{s.fullpolytope}.

An illustration of the unique four-dimensional tower polytope in $d=6$ obtained as a slice of the maximal tower polytope $\mathcal{P}_{(6)}$ of M-theory compactified on $T^5$ (note that other rank 4 tower polytopes following the taxonomy rules could exist, associated to non-toroidal compactifications). For simplicity, the only depicted towers are those corresponding to vertices generating the polytope, associated to decompactifications of either one or two dimensions, respectively in blue \fcolorbox{black}{blue}{\rule{0pt}{6pt}\rule{6pt}{0pt}} or green \fcolorbox{black}{dark-green}{\rule{0pt}{6pt}\rule{6pt}{0pt}}.

Scalar charge-to-mass ratio vectors for the different towers (outer polygon) and cut-offs (inner polygon), for M-theory compactified on ${\rm K3}$, in the moduli space slice spanned by the canonically normalized $\hat{\mathcal{V}}_{{\rm K3}}\in\mathbb{R}$ and $\hat{\tilde{t}}_0>0$, \cite{Castellano:2023jjt, Castellano:2023stg}. Note that they correspond to half of the polygon $\Pi_{\rm(7,II)}$ depicted in Table \ref{tab:BIG} and its dual polygon ($\Pi^\circ_{\rm(7,II)}$ in Table \ref{tab:BIGsp}). The different theories resulting from each limit are also depicted.