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Penrose tilings have appeared on bathroom floors, building facades, and medieval Islamic architecture centuries before anyone wrote down their mathematics. Their defining property is aperiodicity — no finite translation maps the pattern onto itself — enforced by the golden ratio $\varphi = (1+\sqrt{5})/2$, whose continued fraction $[1; 1, 1, 1, \ldots]$ converges more slowly than any other irrational number. Hurwitz proved this in 1891: $\varphi$ is the worst-approximable number, meaning no rational $p/q$ satisfies $|\varphi - p/q| < 1/(\sqrt{5} \cdot q^2)$¹. That extremal property of a single irrational number connects a 6D lattice to cosmic expansion through a chain of results.

This post traces that chain. The constraint geometry provides the $\beta$-function. The KK spectrum computation provides the effective dimension. These are two outputs of a single projection, and the projection’s consequences include the tier hierarchy of compact objects, the area law for black hole entropy, and the transition to accelerated expansion at $z \approx 0.63$.

The 6D Lattice and Its Projection

Kramer and Neri proved in 1984 that the minimum dimension for a periodic lattice whose projection to $\mathbb{R}^3$ carries icosahedral symmetry is $d = 6$². The lattice $\Lambda \subset \mathbb{R}^6$ is periodic, infinite, and entirely without frustration. The projection decomposes $\mathbb{R}^6 = E_\parallel \oplus E_\perp$ into a physical subspace $E_\parallel \cong \mathbb{R}^3$ and an internal subspace $E_\perp \cong \mathbb{R}^3$.

The projection map $\Pi = (\pi_\parallel, \, S^3 \to S^3/\mathrm{2I})$ acts on $\Lambda$ and on the internal space simultaneously. Three independent consequences follow from this single operation:

$\Pi \;\longmapsto\; \begin{cases} \mathcal{T}: \pi_\parallel(\Lambda) \to \text{aperiodic tiling} & \text{(quasicrystal)} \\ \mathcal{S}: \mathrm{Spec}(\Delta_{S^3/\mathrm{2I}}) \to \{l : n_l \neq 0\} & \text{(spectral gap)} \\ \mathcal{F}: F[P]\big|_{S^3/\mathrm{2I}} \to I > 0 & \text{(frustration)} \end{cases}$

The image $\pi_\parallel(\Lambda)$ is a 3D Penrose tiling — aperiodic because the irrationality of $\varphi$ prevents any lattice translation from mapping entirely into $E_\parallel$. The internal space $E_\perp$ compactifies to $S^3$ and the icosahedral symmetry quotients it to the orbifold $S^3/\mathrm{2I}$, where $\mathrm{2I}$ is the binary icosahedral group ($|\mathrm{2I}| = 120$, the largest finite subgroup of $\mathrm{SU}(2)$). The McKay correspondence identifies this orbifold with the $E_8$ singularity, with Coxeter number $h = 30$.

The symmetry group is forced. Three independent requirements eliminate all alternatives. Crystallographic restriction eliminates $C_1, C_2, C_3, C_4, C_6$ — these produce resonance lock-in under recursive scaling. $\varphi$-compatibility eliminates every remaining $n \neq 5$, since $2\cos(\pi/n) = \varphi$ has the unique solution $n = 5$. Binary closure on $S^2$ (which carries $\mathbb{Z}_2$ antipodal symmetry) lifts $C_5$ to $C_2 \times C_5 = C_{10}$. On $S^3$, the group implementing $C_{10}$ symmetry is $\mathrm{2I}$.

The Spectral Gap

Scalar fields on $S^3/\mathrm{2I}$ decompose into Kaluza–Klein modes. On the parent $S^3$, every spin level $l = 0, 1, 2, \ldots$ contributes. On the orbifold, only modes invariant under $\mathrm{2I}$ survive. The generating function is the Molien series:

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

The exponents 12, 20, 30 are the degrees of the Klein invariants $(H, T, f)$ satisfying the $E_8$ surface equation $T^2 = H^3 - 1728f^5$. The surviving spins are the elements of the numerical semigroup $\langle 6, 10, 15 \rangle = \langle a/2, \, b/2, \, h/2 \rangle$, with gap set $\{1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 17, 19, 23, 29\}$. There are 15 gaps, and the genus $g = 15 = h/2$. This relationship — genus equals half the Coxeter number — holds only for $E_8$ among all ADE types.

The first surviving mode sits at $l = 6$, with eigenvalue $\lambda_1 = 4 \cdot 6 \cdot 7 = 168$. On the unquotiented $S^3$, the first excitation is $\lambda = 3$. The protection factor $\sigma = 168/3 = 56$ measures how far the orbifold has pushed up the spectral floor.

The deletion of 15 modes changes the effective dimensionality. Weyl’s law says the cumulative eigenvalue count on a $d$-dimensional manifold grows as $N(\lambda) \sim \lambda^{d/2}$. On $S^3/\mathrm{2I}$, the thinned count in the semigroup-controlled regime ($l \leq h$) grows as $N(\lambda) \sim \lambda^{d_{\mathrm{eff}}/2}$ with $d_{\mathrm{eff}} = 2.61$. The orbifold is topologically three-dimensional but spectrally behaves as dimension $\sim$2.6 at low energies. As the cutoff increases past the Coxeter number, $d_{\mathrm{eff}}$ asymptotes back toward 3 — the Weyl regime. The full computation is in the KK spectrum post.

The Frustration

The same orbifold structure that deletes modes prevents simultaneous relaxation of three curvature sectors. The constraint functional

$F[P] = g_\pi K_\pi[P] + g_\varphi K_\varphi[P] + g_N K_N[P]$

operates on entropy-constrained probability densities, where $K_\pi$ measures departure from angular isotropy, $K_\varphi$ measures departure from golden-ratio self-similarity, and $K_N$ measures departure from $C_{10}$ discrete resonance.

The triadic tension theorem establishes four properties: (T1) the sector minimizers are mutually disjoint; (T2) the covariance matrix $\Sigma_{ij} = \langle \delta K_i \, \delta K_j \rangle$ satisfies $\Sigma_{ij} < 0$ for $i \neq j$; (T3) $\det(\Sigma) > 0$; (T4) the ground state has $F[P_0] > 0$. No configuration relaxes all three sectors simultaneously.

The pairwise incompatibilities are quantitative, computed from a 21,976-vertex Penrose tiling (Robinson triangle inflation, 12 subdivision steps, 62,820 directed nearest-neighbor bonds). The $\varphi$-minimizer retains tenfold bond-angular anisotropy — Fibonacci inflation produces ten edge directions at 36° intervals, giving bond-orientational order $|\psi_{10}| = 0.849$ and angular Fisher information $K_\pi \geq 72.1$ (gap $\delta_{\pi\varphi} \geq 72.1$; von Mises estimate $\approx 311$). The fivefold vertex symmetry promotes to tenfold in the pair correlation by Friedel’s law, so the relevant harmonic is $m = 10$. The frequency combs $\omega = 2\pi k / \ln\varphi$ and $\omega = 2\pi k / \ln 10$ are incommensurable because $\ln 10 / \ln\varphi = 4.785\ldots$ is irrational, and the self-similarity mismatch factorizes as $\delta_{\varphi N}/A^2 = 2\sin^2(\pi\ln\varphi/\ln 10) = 0.745$. And $C_{10}$ symmetry requires angular harmonics at $k = 10, 20, \ldots$, each carrying $k^2 |\hat{P}_k|^2$ of angular curvature (gap $\delta_{\pi N} \geq 72.1$, exact von Mises value $\approx 311$).

The ground-state curvature is $I = 4\pi\varphi^2 \approx 32.9$. The $4\pi$ comes from the Gauss–Bonnet theorem applied to $S^2$ — the angular manifold at the endpoint of dimensional flow, whose topology is forced by compactness, orientability, and positive curvature. The $\varphi^2$ comes from the fixed-point equation $x = 1 + 1/x$, giving $\varphi^2 = \varphi + 1$.

Two Paths, One Bracket

The spectral gap and the frustration are two descriptions of the same obstruction — the orbifold quotient removes degrees of freedom that would otherwise permit simultaneous sector relaxation. They feed independently into the $\beta$-function through different channels.

The spectral path delivers the effective dimension:

$\mathcal{S} \;\to\; d_{\mathrm{eff}} \;\to\; \mathcal{D}$

The frustration path delivers the coupling constant. Self-similarity requires uniform curvature spectral density. $C_{10}$ decade symmetry partitions each RG period into 10 equivalent shells. Total curvature per period is $I$, so:

$\mathcal{F} \;\to\; I = 4\pi\varphi^2 \;\to\; u^* = I/10 \approx 3.290$

Both enter the same equation. The dissipation fraction $\xi \in [0, 1]$ measures the share of organizational capacity devoted to maintenance. Its flow under coarse-graining is governed by a $\beta$-function assembled from four derived ingredients: the logistic factor $-\xi(1-\xi)$ (from the boundedness of $\xi$), the coupling $u^*$ (from the frustration), the dimensional correction $(\mathcal{D}-2)\ln\varphi/2$ (from the spectrum), and the vertex selection rule. The recursion symmetry operator acts on perturbations $\delta P$ around the self-similar ground state as $T_\varphi(\delta P)(\theta, \ell) = \varphi \cdot \delta P(\theta, \ell + \ln\varphi)$, where the shift implements geometric inflation and the prefactor $\varphi$ is the Perron–Frobenius eigenvalue of the Penrose substitution matrix $M = \bigl(\begin{smallmatrix}1&1\\1&0\end{smallmatrix}\bigr)$. At $P_0$, the vertex functionals inherit $\ln\varphi$-periodicity, so the shift drops out and only the amplitude scaling $\varphi^n$ remains: $\varphi^n g_n = \varphi^2 g_n$ forces $g_n = 0$ for $n > 2$ because $\varphi$ is irrational, making the one-loop result exact.

$\beta(\xi, \mathcal{D}) = -\xi(1-\xi)\left[u^* + \frac{\mathcal{D}-2}{2}\ln\varphi\right]$

with coupled dimensional flow $d\mathcal{D}/d(\ln\mu) = -(\xi/u^*)\ln\varphi$.

The UV fixed point is $(\xi, \mathcal{D}) = (0, 3)$: no organizational structure, full three dimensions. The IR fixed point is $(\xi, \mathcal{D}) = (1, 2)$: complete organizational saturation, two effective dimensions. At $\mathcal{D} = 2$, the dimensional correction vanishes exactly, and the critical exponent takes its universal value $\nu = 1/u^* = 10/(4\pi\varphi^2) \approx 0.304$.

The Tier Hierarchy

The flow from $(0, 3)$ toward $(1, 2)$ is not abstract. Physical systems sit at different points along this trajectory, and the $\beta$-function value at each point determines the local dynamics.

Dark matter halos occupy $\xi \sim 10^{-5}$. The $\beta$-function evaluates to $\beta \sim -4 \times 10^{-5}$ — weak flow, far from any threshold. White dwarfs approach the threshold $u^*/10 \approx 0.329$, where a single RG shell’s worth of curvature budget is consumed. The Gaia DR3 catalog contains 7,515 white dwarfs with a cooling anomaly at this radius ratio, at $14.5\sigma$ significance³. Neutron stars sit at $\xi \sim 0.125$, where the $\beta$-function is steep ($\beta \sim -0.38$) and the effective dimension has begun dropping below 3.

Black holes complete the trajectory. At $(\xi, \mathcal{D}) = (1, 2)$, the $\beta$-function evaluates to $\beta(1, 2) = -1 \cdot 0 \cdot [u^* + 0] = 0$. The flow halts. The effective dimension is 2. Entropy scales as area rather than volume because the system has reached a two-dimensional fixed point of a derived flow. The LIGO/Virgo gravitational wave catalog (GWTC, 164 binary black hole mergers) yields spin populations consistent with the $u^*$ prediction at $0.1\sigma$ deviation⁴.

The three-stage activation cascade — kinematic binding (decoupling from Hubble flow), organizational loading ($\xi \to \xi_c$), and geometric transition (curl-floor activation when the correction field goes non-integrable) — determines which systems reach the IR fixed point. Halos fail at Stage 2. White dwarfs are marginal. Neutron stars are loaded. Black holes complete all three stages.

The critical approach regime is the interval of the dissipation flow in which curvature tradeoffs are binding, absorption capacity remains nonzero, and no integrable correction exists. It is bounded by two thresholds: entry at structural saturation $u^*/10 \approx 0.329$, where the discrete sector becomes binding; exit at maintenance bankruptcy $\xi \to 1$, where the logistic factor $\xi(1-\xi)$ drives absorption capacity to zero. Between these thresholds, the coexistence of slowing flow and irreducible frustration produces oscillatory dynamics as a structural consequence. The richest dynamics concentrate in approach rather than at equilibrium or after collapse.

Accelerated Expansion

Bound systems can complete the flow to $(1, 2)$. The universe as a whole cannot, because it fails Stage 1 — it is not gravitationally bound. When the global matter fraction $\Omega_m$ approaches $\xi_c \approx 0.304$, the response is eigenmode transition.

The constraint functional admits different dominant eigenmodes at different epochs. The Euler–Lagrange equation

$-g_\pi \partial_{\theta\theta}\ln P - g_\varphi \partial_{\ell\ell}\ln P + g_N \frac{\delta C_{2\times 5}}{\delta P} = \lambda + \tau(1 + \ln P)$

yields radiation-dominated dynamics ($\pi$-sector dominant) at early times, matter-dominated clustering ($\varphi + N$ sectors dominant) during structure formation, and accelerated expansion ($\pi$-sector dominant again) at late times. The transition occurs at $z \approx 0.63$, when $\Omega_m$ approaches 0.304 — the same $\xi_c$ derived from the coupling $u^*$.

Accelerated expansion is the $\pi$-eigenmode of the same constraint functional that produces triadic tension, becoming dominant when the matter sectors approach saturation at the global scale. The matter fraction $\Omega_m \approx 0.31$ and the critical threshold $\xi_c \approx 0.304$ are the same quantity appearing in different observational contexts.

The eigenmode transition produces a quantitative prediction for the dark energy equation of state. The acceleration onset occurs when $\Omega_\Lambda(z)$ crosses $u^*/10 = 0.329$, which happens at $z = 0.632$ — matching the observed $q = 0$ redshift. The present-day $\Omega_m = 0.315$ sits inside the approach window $[\xi_c, \, u^*/10] = [0.304, \, 0.329]$.

The physical mechanism is the N-sector curvature pump. For $w_0 > -1$ (as DESI observes), $\rho_{DE}$ must increase with $z$ near $z = 0$: dark energy was slightly stronger in the recent past. At higher $z$, $\Omega_m$ is larger, the N-sector is more saturated, and the cross-susceptibility $R_{\pi N} = +0.069 > 0$ (from T2) pumps curvature into the $\pi$-sector, enhancing $\rho_{DE}$. As the N-sector desaturates toward $z = 0$, the pump weakens and $\rho_{DE}$ relaxes toward $\Lambda$. The capped pump model

$f(z) = 1 + A \times \min\!\left[\left(\frac{\Omega_m}{\Omega_{m,0}}\right)^{1/2} - 1, \; \frac{1}{\xi_c} - 1\right] \times \left(\frac{\Omega_\Lambda}{\Omega_{\Lambda,0}}\right)^{\!\varphi^2}$

has all structural features derived: $n = 1/2$ (mean-field exponent), $m = \varphi^2$ (framework constant), cap $= 1/\xi_c - 1$ (N-sector saturation limit). The amplitude $A = \mathcal{W}(3) + 1 = u^* + 1 + \ln\varphi/2 \approx 4.531$ (the full 3D coupling plus unit curvature offset). Results: $w_0 = -0.724$ vs DESI DR2’s $-0.75 \pm 0.07$ ($0.4\sigma$), $w_a = -0.746$ vs DESI’s $-0.86 \pm 0.25$ ($0.5\sigma$). Cross-check: $A = \mathcal{W}(3)+1$ predicts $R_{\pi N}/|R_{\pi\pi}| = (\mathcal{W}(3)+1)/5 = 0.906$, matching the measured 0.908 to 0.2%.

Type Ia supernova energies are consistent with Landauer bit-counting of information reorganization at the eigenmode transition, yielding $E = 4.3 \times 10^{44}$ J⁵.

Branch Competition

The projection from 6D does not uniquely specify $E_8$. Other finite subgroups of $\mathrm{SU}(2)$ — the binary octahedral ($\mathrm{2O}$, mapping to $E_7$), binary tetrahedral ($\mathrm{2T}$, mapping to $E_6$), and binary dihedral groups (mapping to the $D$-series) — produce legitimate eigenbranches with distinct orbifolds, semigroups, spectral gaps, and $\beta$-function profiles.

The spectral drive — the product of protection factor $\sigma$ and $\beta$-function strength at criticality — measures the total flow capacity of each branch:

The $\beta$-function varies by $\sim$1% across branches; the protection factor dominates. The $E_8$ branch’s dominance traces to three independent properties: $\varphi$ is the worst-approximable irrational (maximal resonance protection), $\mathrm{2I}$ is the largest finite subgroup of $\mathrm{SU}(2)$ (most aggressive mode deletion), and $\langle 6, 10, 15 \rangle$ is the unique ADE semigroup achieving genus $g = h/2$ (maximally distributed gap structure).

Branches with weaker spectral drive remain viable only with thermodynamic compensation — configurational entropy from defect-tile species mixing. The dodecagonal branch requires approximately 5× entropy amplification ($S/Nk_B$ from 0.120 to 0.554). The Penrose branch requires no entropy subsidy.

The ADE domain wall computation establishes that domain wall energies between all branch pairs are finite, with the $E_8$ walls most expensive by a factor of $\sim 456\times$. Gap-set nesting ($\mathrm{gap}(E_6) \subset \mathrm{gap}(E_7) \subset \mathrm{gap}(E_8)$) makes $E_8$ the unique maximal element under inclusion. Thermal stability analysis shows $E_8$ walls dissolve last during cosmological cooling, and all tunneling rates are frozen ($B \gg 1$, $R$-independent). Selection is primordial and irreversible.

The Complete Chain

$$6\text{D lattice} \xrightarrow{\text{project}} S^3/\mathrm{2I} \xrightarrow{\text{Molien}} \langle 6,10,15\rangle \xrightarrow{n_l = 0} d_{\mathrm{eff}}$$ $$d_{\mathrm{eff}} \xrightarrow{\;\mathcal{D}\;} \beta(\xi,\mathcal{D}) \xrightarrow{\text{flow}} \begin{cases} (1,2) & \text{local: BH} \\ \text{expansion} & \text{global} \end{cases}$$

In parallel:

$$\text{projection} \xrightarrow{\text{frustration}} I = 4\pi\varphi^2 \xrightarrow{C_{10}} u^* = I/10 \xrightarrow{\;\xi(1-\xi)\;} \beta(\xi,\mathcal{D})$$

The spectral path supplies $\mathcal{D}$. The frustration path supplies $u^*$. They converge in the bracket $[u^* + (\mathcal{D}-2)\ln\varphi/2]$.

Attack Surface

The 6D lattice is assumed. The Kramer–Neri theorem establishes that 6 is the minimum dimension for icosahedral projection, but does not explain why a 6D lattice exists. The framework begins at the lattice and derives everything downstream.

The compactification radius $R$ is a free parameter. Everything else — $u^*$, $\xi_c$, the spectral gap, the tier hierarchy, the expansion onset — follows from the projection. But $R$ sets the energy scale at which KK modes become relevant, and its value is not derived.

The “+1” in $A = 1 + \mathcal{W}(3)$ has a physical interpretation but not yet a first-principles derivation. The N-sector curvature pump model yields $w_0 = -0.724$ ($0.4\sigma$ from DESI DR2) and $w_a = -0.746$ ($0.5\sigma$ from DESI DR2). The amplitude $A = \mathcal{W}(3) + 1$ is numerically verified ($\chi^2$ at the derived value matches the numerical optimum to 3 decimal places) and cross-predicts $R_{\pi N}/|R_{\pi\pi}| = (\mathcal{W}(3)+1)/5 = 0.906$ vs measured 0.908 (0.2% match). The “+1” admits a bare + dressed decomposition: $1$ = bare constraint coupling, $\mathcal{W}(3)$ = RG-mediated coupling. A rigorous derivation from the Euler–Lagrange equation showing that the constraint-level response is exactly unity would close this item.

The $d_{\mathrm{eff}} \to \mathcal{D}$ identification. The effective spectral dimension from the KK computation enters the $\beta$-function as $\mathcal{D}$, the effective spatial dimension governing the RG flow. This identification is natural but not rigorously derived — it assumes that the spectral content of the orbifold directly controls the dimensional parameter in the Wilsonian flow. A derivation from first principles (showing that the RG blocking procedure on $S^3/\mathrm{2I}$ produces a $\beta$-function with $\mathcal{D} = d_{\mathrm{eff}}$) would strengthen the chain.

The tier hierarchy placements are approximate. The $\beta$-function’s flow from $(0, 3)$ to $(1, 2)$ produces a hierarchy, but the specific $\xi$ values assigned to halos, white dwarfs, and neutron stars are estimates based on physical properties, not direct computations from the framework. The white dwarf anomaly ($14.5\sigma$ at $u^*/10$) and BH spins ($0.1\sigma$ at $u^*$) provide quantitative contact with data. The other placements need similar observational tests.

The triadic tension theorem’s domain. T1–T4 are proved for the constraint functional $F[P]$ on entropy-constrained densities with the specific sector functionals $K_\pi$, $K_\varphi$, $K_N$. If alternative sector functionals are equally well-motivated, the specific incompatibility mechanisms (and therefore the numerical value of $I$) could change. The T1 incompatibility proofs are quantitative — three computable gaps $\delta_{\pi\varphi} \geq 72.1$, $\delta_{\varphi N}/A^2 = 0.745$, $\delta_{\pi N} \geq 72.1$ — and T2 is verified numerically. The domain question remains open.

References

Footnotes

1. Hurwitz, A. (1891). Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche. Mathematische Annalen, 39(2), 279–284. ↩

2. 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. ↩

3. Gentile Fusillo, N. P., et al. (2021). A catalogue of white dwarfs in Gaia EDR3. Monthly Notices of the Royal Astronomical Society, 508(3), 3877–3896. ↩

4. Abbott, R., et al. (The LIGO Scientific Collaboration and Virgo Collaboration). (2023). Population of merging compact binaries inferred using gravitational waves through GWTC-3. Physical Review X, 13, 011048. ↩

5. Scalzo, R. A., et al. (2019). Type Ia supernova bolometric light curves and ejected mass estimates from the Nearby Supernova Factory. Monthly Notices of the Royal Astronomical Society, 483(3), 3297–3311. ↩