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The projection from a 6D periodic lattice to 3D produces a quasicrystal in physical space and an orbifold $S^3/\mathrm{2I}$ — the Poincaré homology sphere — in the internal space. The constraint geometry and the spectral content of this orbifold are two outputs of that single projection. The constraint geometry provides the $\beta$-function. The spectral content provides the effective dimension $\mathcal{D}$ that enters the $\beta$-function’s dimensional correction. The full chain connecting these is developed in From Lattice Projection to Cosmic Expansion. This post computes the scalar Kaluza–Klein spectrum on $S^3/\mathrm{2I}$.

1. The Internal Space

The binary icosahedral group $\mathrm{2I} \cong \mathrm{SL}(2,5)$ is the double cover of the icosahedral rotation group $\mathrm{I} \cong A_5$. It has 120 elements — the largest finite subgroup of $\mathrm{SU}(2)$ — and acts freely on $S^3 \simeq \mathrm{SU}(2)$ by left multiplication. The quotient $\Sigma = S^3/\mathrm{2I}$ is a smooth, compact, oriented three-manifold with $\pi_1(\Sigma) = \mathrm{2I}$ and $H_1(\Sigma;\mathbb{Z}) = 0$. Poincaré constructed it in 1904 as a counterexample to the conjecture that homology determines the three-sphere¹. It is the unique spherical space form with icosahedral holonomy.

In a Kaluza–Klein compactification on $\Sigma$, the scalar field equation reduces to the eigenvalue problem for the Laplace–Beltrami operator $\Delta_\Sigma$. The eigenvalues of $\Delta_{S^3}$ on the unit three-sphere are

$$\lambda_k = k(k+2), \qquad k = 0, 1, 2, \ldots,$$

with degeneracy $(k+1)^2$². Each eigenspace carries the representation $D_{k/2} \otimes D_{k/2}$ of $\mathrm{SU}(2)_L \times \mathrm{SU}(2)_R$ under the Peter–Weyl decomposition of $L^2(\mathrm{SU}(2))$, where $D_j$ denotes the spin-$j$ irreducible representation of dimension $2j+1$³⁴.

On $\Sigma = S^3/\mathrm{2I}$, scalar wavefunctions must be $\mathrm{2I}$-invariant under the left action. Since $-I \in \mathrm{2I}$ acts as $(-1)^k$ on $D_{k/2}$, only even values $k = 2l$ contribute (integer spin $j = l$). For each surviving spin $l$, the eigenvalue is

$$\lambda_l = 4l(l+1),$$

and the full orbifold multiplicity is

$$m_l = (2l+1) \cdot n_l,$$

where the factor $(2l+1)$ comes from the unbroken $\mathrm{SU}(2)_R$ action on the opposite Peter–Weyl factor, and $n_l$ counts the $\mathrm{2I}$-invariant vectors in the spin-$l$ representation. The KK mass tower is therefore determined by the branching multiplicities $n_l$.

2. Branching Rules

The group $\mathrm{2I}$ has nine conjugacy classes. Each element $g \in \mathrm{2I} \subset \mathrm{SU}(2)$ has eigenvalues $e^{\pm i\alpha}$ in the fundamental representation, where $\alpha \in [0, \pi]$ is the half-angle parameter.

The total is $1 + 1 + 12 + 12 + 12 + 12 + 20 + 20 + 30 = 120$. The character of the spin-$l$ representation of $\mathrm{SU}(2)$ at half-angle $\alpha$ is

$$\chi_l(\alpha) = \frac{\sin((2l+1)\alpha)}{\sin(\alpha)},$$

with $\chi_l(0) = 2l+1$ and $\chi_l(\pi) = (-1)^{2l}(2l+1)$. The multiplicity of the trivial representation of $\mathrm{2I}$ in the restriction of spin-$l$ is

$$n_l = \frac{1}{120} \sum_{i} |C_i| \, \chi_l(\alpha_i),$$

where the sum runs over the nine conjugacy classes $C_i$ with half-angle parameters $\alpha_i$. This is an exact finite sum over nine terms.

3. The Spectrum

Evaluating $n_l$ for $l = 0$ through $l = 120$ partitions the spectrum into three regimes.

Below the Coxeter number ($l < 30$): The tower is sparse. Of the 30 spin values $l = 0, \ldots, 29$, exactly 15 survive with $n_l = 1$, and 15 are forbidden with $n_l = 0$. The surviving spin values are

$$l \in \{0, 6, 10, 12, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28\}.$$

The forbidden spin values are

$$l \in \{1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 17, 19, 23, 29\}.$$

At and above the Coxeter number ($30 \leq l \leq 59$): Every spin value is present. The invariant count begins to increase: $n_{30} = 2$, the first level carrying two independent $\mathrm{2I}$-invariants. Below $l = 30$, every surviving mode has $n_l = 1$; the transition at $l = 30$ marks both the onset of completeness and the first multiplicity enhancement.

Asymptotics ($l \gg 30$): The invariant count satisfies $n_l \to (2l+1)/120$ on average, and the full multiplicity $m_l = (2l+1) \cdot n_l \to (2l+1)^2/120$, recovering the expected $1/|\mathrm{2I}|$ factor relative to $S^3$.

The first massive mode occurs at $l = 6$, with eigenvalue $\lambda_6 = 4 \cdot 6 \cdot 7 = 168$. The five consecutive forbidden levels $l = 1, \ldots, 5$ create the largest gap in the tower. The eigenvalue gap between the zero mode ($\lambda = 0$) and the first excitation ($\lambda = 168$) is a factor of 56 above the generic $S^3$ first excitation at $k = 1$ ($\lambda = 3$). This protection factor — 56× — is the largest among all finite subgroup quotients of $S^3$.

4. The Numerical Semigroup

The forbidden spin values have algebraic structure. The generators $\{6, 10, 15\}$ are the degrees of the Klein invariant polynomials for $\mathrm{2I}$ acting on $\mathbb{C}^2$, divided by two. Klein showed that the ring of polynomial invariants $\mathbb{C}[x,y]^{\mathrm{2I}}$ is generated by three homogeneous polynomials⁵:

$$f_{12} \text{ (degree 12)}, \qquad H_{20} \text{ (degree 20)}, \qquad T_{30} \text{ (degree 30)},$$

satisfying the single relation $T_{30}^2 = H_{20}^3 - 1728 \, f_{12}^5$. This relation defines the $E_8$ simple singularity in the ADE classification of du Val singularities⁶. The factor of two between the Klein degrees $\{12, 20, 30\}$ and the semigroup generators $\{6, 10, 15\}$ arises because the Molien series counts invariants at polynomial degree $k$, while the spin quantum number is $l = k/2$.

Define the numerical semigroup

$$S = \langle 6, 10, 15 \rangle = \left\{ 6a + 10b + 15c \;\middle|\; a, b, c \in \mathbb{Z}_{\geq 0} \right\}.$$

Its gap set is

$$\mathbb{Z}_{\geq 0} \setminus S = \{1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 17, 19, 23, 29\}.$$

The semigroup has Frobenius number $F = 29$ (the largest gap) and genus $g = 15$ (the number of gaps). The identity $g = h/2$ — the genus equals half the Coxeter number — is specific to the $E_8$ case (§7).

Theorem (Semigroup selection rule). Let $\Gamma \subset \mathrm{SU}(2)$ be a finite subgroup and let $d_1, \ldots, d_r$ be the degrees of the homogeneous generators of the invariant ring $\mathbb{C}[x,y]^{\Gamma}$. Define the numerical semigroup $S = \langle d_1/2, \ldots, d_r/2 \rangle$. Then the spin-$l$ representation of $\mathrm{SU}(2)$ contains a $\Gamma$-invariant vector ($n_l \geq 1$) if and only if $l \in S$. Equivalently, the forbidden KK levels on $S^3/\Gamma$ are the gaps of $S$.

Proof. The Molien series for $\Gamma$ acting on $\mathrm{Sym}^{\bullet}(\mathbb{C}^2)$ is

$$M(t) = \sum_{k=0}^{\infty} \dim\!\left(\mathrm{Sym}^k(\mathbb{C}^2)^{\Gamma}\right) t^k = \frac{P(t)}{\prod_{i=1}^r (1 - t^{d_i})},$$

where $P(t)$ encodes the relations among the generators⁵⁷. Under the identification $\mathrm{Sym}^{2l}(\mathbb{C}^2) \cong D_l$, the coefficient of $t^{2l}$ in $M(t)$ equals $n_l$. Writing $M(t)$ in the variable $u = t^2$ gives $\widetilde{M}(u) = \sum_{l \geq 0} n_l \, u^l$, whose denominator is $\prod_i (1 - u^{d_i/2})$. The set of exponents $l$ with $n_l \geq 1$ is the numerical semigroup generated by the denominator exponents $d_i/2$⁸, and the set with $n_l = 0$ is its gap set. For $\Gamma = \mathrm{2I}$, the Molien series specializes to

$$M(t) = \frac{1 - t^{60}}{(1 - t^{12})(1 - t^{20})(1 - t^{30})},$$

and $S = \langle 6, 10, 15 \rangle$ with gap set identical to the forbidden KK levels. $\square$

The theorem reduces the determination of forbidden KK levels to a purely algebraic problem: given $\Gamma$, look up the Klein invariant degrees and compute the gap set of the resulting semigroup. The character sum is required only for multiplicities $n_l > 1$ above the Coxeter threshold.

5. The $E_8$ Exponents

The $E_8$ root system has eight exponents $m_i$ defined by the eigenvalues of the Coxeter element: $e^{2\pi i m_i/h}$ where $h = 30$. The exponents are

$$\{m_i\} = \{1, 7, 11, 13, 17, 19, 23, 29\}.$$

All eight $E_8$ exponents appear among the 15 forbidden KK levels. The remaining seven forbidden levels are $\{2, 3, 4, 5, 8, 9, 14\}$, which complete the gap set of the semigroup. The $E_8$ exponents are the integers $m \in [1, h-1]$ coprime to $h = 30$⁸: $\gcd(m, 30) = 1$. Since $30 = 2 \times 3 \times 5$, the exponents avoid multiples of 2, 3, and 5 — they are maximally incompatible with the generator structure $\{6, 10, 15\} = \{2 \times 3, \; 2 \times 5, \; 3 \times 5\}$ and the hardest gaps to fill by semigroup combinations.

The $E_8$ exponents have a dual role: they are both the eigenvalue phases of the Coxeter element in the root system and a distinguished subset of the forbidden KK harmonics on the Poincaré homology sphere. The McKay correspondence⁷ — the bijection between finite subgroups of $\mathrm{SU}(2)$ and simply-laced Dynkin diagrams — provides the bridge. Under this correspondence, the representation graph of $\mathrm{2I}$ has the connectivity of the extended $\hat{E}_8$ Dynkin diagram. The KK spectrum inherits $E_8$ structure because the branching problem and the McKay graph encode the same algebraic data.

6. The Coxeter Threshold

The Coxeter number $h = 30$ marks a sharp transition in spectral structure. Below $l = 30$, the density of surviving modes increases stepwise:

The transition at $l = 30$ is also the first appearance of $n_l = 2$: the second independent $\mathrm{2I}$-invariant harmonic appears at the same threshold where the tower achieves completeness. Below the Coxeter number, every surviving mode has a single invariant ($n_l = 1$).

7. Spectral Dimension and Weyl Scaling

The Weyl counting function $N(\lambda) = \#\{\text{eigenvalues} \leq \lambda\}$, counted with multiplicity, satisfies

$$N(\lambda) \sim \frac{\mathrm{Vol}(\Sigma)}{6\pi^2} \, \lambda^{3/2}$$

asymptotically on any Riemannian three-manifold $\Sigma$⁹. For $\Sigma = S^3/\mathrm{2I}$ with unit radius, $\mathrm{Vol}(\Sigma) = 2\pi^2/120$, giving a Weyl coefficient of $1/360$. At spin $l = 500$ ($\lambda = 1{,}002{,}000$), the Weyl prediction is $N_{\mathrm{Weyl}} \approx 2{,}786{,}115$, while the exact count gives $N_{\mathrm{exact}} = 2{,}794{,}378$ — a ratio of 1.003, confirming the multiplicity formula and three-dimensional Weyl asymptotics to within 0.3%.

Restricting the fit to the sparse regime below the Coxeter number yields a different exponent:

The deficit $\Delta d_{\mathrm{eff}} \approx 0.45$ between Region I and the asymptotic regime arises from the semigroup sparsity: 50% of spin values are forbidden in $[0, 30)$, each surviving level contributes multiplicity $(2l+1) \times 1$, and the resulting growth of $N(\lambda)$ is sub-Weyl.

The running Weyl exponent — computed cumulatively from $l = 6$ up to successive cutoffs — shows the crossover:

The crossover from $\sim$2.55 to $\sim$3.0 occurs in a window around the Coxeter number. The Poincaré homology sphere is topologically three-dimensional everywhere, but its low-energy spectral content is thinner than a generic three-manifold. The semigroup gap structure creates a spectral bottleneck: modes are absent, and the cumulative eigenvalue count grows as if the manifold had effective dimension $\sim$2.55. Above $h$, all levels are present, degeneracies grow normally, and standard Weyl behavior recovers. The crossover at $l = h$ is a spectral signature of the $E_8$ algebraic structure imprinted on the geometry.

8. Branch Comparison

The semigroup structure generalizes across the ADE classification. For each binary polyhedral group $\Gamma \subset \mathrm{SU}(2)$, the Klein invariant ring has specific generator degrees, and the KK gap set on $S^3/\Gamma$ is controlled by the corresponding semigroup.

The Penrose branch is extremal in protection — 56× versus 8× for the dodecagonal branch — but intermediate in Weyl deficit. The dodecagonal branch has a larger deficit (0.53 vs 0.45) despite having far fewer forbidden levels. The explanation is geometric: $\mathrm{BD}_6$ has only 24 elements, so $S^3/\mathrm{BD}_6$ has 5× the volume of $S^3/\mathrm{2I}$, and the 3 missing levels create a proportionally larger distortion in a shorter Coxeter window ($h = 14$ vs 30).

The cases $E_6$ and $D_8$ have the same group order, genus ($g = 3$), and Frobenius number ($F = 5$), yet their forbidden sets differ: $\{1, 2, 5\}$ for $E_6$ versus $\{1, 3, 5\}$ for $D_8$. These produce distinct protection factors (16× vs 8×) and distinct Region I effective dimensions (2.67 vs 2.47). The placement of the gap set within the spectral tower, not merely its cardinality or Frobenius number, governs the low-energy effective dimension.

For the binary dihedral family (McKay type $D_n$, $n \geq 4$), the semigroup generators reduce to $\langle 2, p \rangle$ with $p$ the unique odd generator. The gap set is $\{1, 3, 5, \ldots, p-2\}$ — the first $g = (p-1)/2$ odd integers — so the first excitation occurs uniformly at $l = 2$ ($\lambda = 24$, protection factor 8×) for the entire family. As $n \to \infty$, $F/h \to 1/2$ and $g/h \to 1/4$, so $g = h/2$ is never achieved in the $D$-family.

For $E_8$, the genus equals $h/2$ exactly. The three generators $\{6, 10, 15\}$ are pairwise coprime ($\gcd(6,10) = 2$, $\gcd(6,15) = 3$, $\gcd(10,15) = 5$, while $\gcd(6,10,15) = 1$), producing a maximally sparse semigroup for the given generator sizes. The normalized Frobenius number $F/h = 29/30 = 0.967$ — algebraic protection extends through nearly the entire Coxeter range. For $D_8$, $F/h = 5/14 = 0.357$. The Penrose branch is the unique ADE branch whose spectral protection saturates nearly the full window below $h$.

9. The LCD Question

The negative selection argument in the constraint geometry forces $C_{10}$ as the discrete sector symmetry. The 6D lattice is the minimum embedding dimension for a periodic lattice whose projection to $\mathbb{R}^3$ carries icosahedral symmetry (Kramer–Neri, 1984¹⁰). The $E_8$ algebraic structure enters through a separate channel: $C_{10}$ on $S^3$ gives $\mathrm{2I}$, and $\mathrm{2I}$ acting on $\mathbb{C}^2$ gives Klein invariants of degrees 12, 20, 30 satisfying $T^2 = H^3 - 1728f^5$ — the $E_8$ singularity. The McKay correspondence confirms the identification. Every link is a theorem.

The two dimensions answer different questions. 6D is the minimum embedding dimension for the periodic parent lattice that projects to a 3D icosahedral quasicrystal — standard quasicrystallography. $E_8$ (whose root lattice lives in 8D) is the algebraic structure governing the compactification geometry. It enters through the McKay correspondence, which identifies finite subgroups of $\mathrm{SU}(2)$ with ADE Dynkin diagrams, and determines the spectral content of $S^3/\mathrm{2I}$: which KK modes survive, how the gap set is organized, and what effective dimension the low-energy spectrum carries.

The chain is forced. Negative selection gives $C_{10}$. $C_{10}$ on $S^3$ gives $\mathrm{2I}$. $\mathrm{2I}$ acting on $\mathbb{C}^2$ gives the Klein invariants. The Klein invariants satisfy the $E_8$ surface equation. The McKay correspondence confirms the identification. The spectral content — 15 forbidden levels, protection factor 56×, genus $= h/2$, Frobenius number 29 — follows from the invariant ring structure alone.

10. Attack Surface

The computation rests on the character orthogonality formula applied to the known conjugacy classes of $\mathrm{2I}$¹¹. The conjugacy class data — sizes, orders, and half-angle parameters — are standard results in the representation theory of finite groups. The semigroup identification is verified by direct enumeration up to $l = 500$; discrepancy at any single level would falsify the correspondence.

The equivalence between the branching multiplicities $n_l$ and the Molien series is a theorem — it follows from the identification of $\mathrm{2I}$-invariant spherical harmonics with $\mathrm{2I}$-invariant homogeneous polynomials via the restriction map $\mathrm{Sym}^{2l}(\mathbb{C}^2) \to L^2(S^3)$. The semigroup structure of the gap set then follows from the structure of the invariant ring $\mathbb{C}[x,y]^{\mathrm{2I}}$ as a graded algebra with three generators and one relation.

The multiplicity formula $m_l = (2l+1) \cdot n_l$ follows from the Peter–Weyl decomposition: the eigenspace at KK level $k = 2l$ decomposes as $D_l^L \otimes D_l^R$ under $\mathrm{SU}(2)_L \times \mathrm{SU}(2)_R$, with $\mathrm{2I}$ acting on one factor. The invariant subspace retains the full $(2l+1)$-dimensional representation on the opposite factor. This is confirmed by the Weyl law check at $l = 500$: the predicted count $N \approx 2{,}786{,}115$ matches the computed $N = 2{,}794{,}378$ to within 0.3%.

The Weyl exponent fits are standard log-log regressions with 14 data points in Region I and 301 in the deep asymptotic regime. The deficit $d_{\mathrm{eff}} \approx 2.55$ is robust to variations in the fitting window and quantifies spectral thinning caused by the semigroup gap structure.

The computation does not address whether $S^3/\mathrm{2I}$ appears as an internal space in any consistent compactification of string theory or supergravity. The Poincaré homology sphere is not Kähler, not Calabi–Yau, and not a group manifold. The spectral results reported here follow from representation theory alone and hold for any scalar field equation on $S^3/\mathrm{2I}$ with the round metric. The extension to $p$-form spectra (vectors, spinors, symmetric tensors) requires analogous branching rules and has not been carried out.

The dynamical selection question — whether the $E_8$ branch is favored over competing ADE branches — is addressed in ADE Domain Walls and Branch Selection. The gap sets nest: $\mathrm{gap}(E_6) \subset \mathrm{gap}(E_7) \subset \mathrm{gap}(E_8)$ and $\mathrm{gap}(D_8) \subset \mathrm{gap}(E_8)$, so $E_8$ is the unique maximal element under gap-set inclusion. Domain wall tensions between branches are finite (bounded by $\max(h_1, h_2)$), with $E_8$ walls the most expensive by a factor of $\sim$456×. Thermal stability analysis shows $E_8$ walls dissolve last during cosmological cooling, and zero-temperature bounce actions satisfy $B \gg 1$ for all downhill transitions — tunneling between branches is frozen out. Selection occurs during the compactification epoch and is irreversible.

References

Footnotes

1. Poincaré, H. (1904). Cinquième complément à l’Analysis Situs. Rendiconti del Circolo Matematico di Palermo, 18, 45–110. ↩

2. Ikeda, A. (1997). On the spectrum of homogeneous spherical space forms. Kodai Mathematical Journal, 20(3), 259–274. ↩

3. Vilenkin, N. Ja. & Klimyk, A. U. (1991). Representation of Lie Groups and Special Functions. Volume 1. Springer. ↩

4. Lachièze-Rey, M. & Caillerie, S. (2005). Laplacian eigenmodes for spherical spaces. Classical and Quantum Gravity, 22(3), 695–708. ↩

5. Klein, F. (1884). Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade. Teubner, Leipzig. English translation: Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, Dover (1956). ↩ ↩²

6. Du Val, P. (1934). On isolated singularities of surfaces which do not affect the conditions of adjunction. Proceedings of the Cambridge Philosophical Society, 30(4), 453–459. ↩

7. McKay, J. (1980). Graphs, singularities, and finite groups. Proceedings of Symposia in Pure Mathematics, 37, 183–186. ↩ ↩²

8. Humphreys, J. E. (1990). Reflection Groups and Coxeter Groups. Cambridge University Press. ↩ ↩²

9. Weyl, H. (1911). Über die asymptotische Verteilung der Eigenwerte. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 110–117. ↩

10. Kramer, P. & Neri, R. (1984). On periodic and non-periodic space fillings of Eⁿ obtained by projection. Acta Crystallographica Section A, 40(5), 580–587. ↩

11. Coxeter, H. S. M. & Moser, W. O. J. (1972). Generators and Relations for Discrete Groups, 3rd ed. Springer. ↩