index

Penrose tilings¹ 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 $|\varphi - p/q| < 1/(\sqrt{5} \cdot q^2)$². That extremal property of a single irrational number connects a 6D lattice to cosmic expansion through a chain of results.

A 6D periodic lattice, projected to 3D, simultaneously produces a quasicrystal, a spectral gap, and a triadic frustration. The spectral gap and the frustration are dual descriptions of one obstruction—the orbifold quotient removes degrees of freedom that would otherwise allow simultaneous relaxation of all three curvature sectors. That duality is not approximate; it is mediated by an exact algebraic identity. The Chebyshev norm ratio $\|U_{2\bullet}(\varphi/2)\|^2_5 / \|U_{2\bullet}(1/(2\varphi))\|^2_5$ equals $\varphi^2$ exactly over one period, and this single object delivers both inputs to the $\beta$-function: the effective dimension $\mathcal{D}$ through the semigroup-controlled eigenvalue count, and the coupling constant $u^*$ through the ground-state curvature $I = 4\pi\varphi^2$, which factorizes cleanly as Gauss–Bonnet times the Chebyshev norm ratio. Both feed into $\beta(\xi, \mathcal{D}) = -\xi(1-\xi)[u^* + (\mathcal{D}-2)\ln\varphi/2]$, whose flow from $(\xi, \mathcal{D}) = (0, 3)$ to $(1, 2)$ organizes the regime structure of compact objects through a geometric slack mapping $l_{\max} = \lfloor h(r/r_{\mathrm{crit}} - 1)\rfloor = \lfloor h(1/\xi - 1)\rfloor$ that connects macroscopic maintenance fraction $\xi = r_{\mathrm{crit}}/r$ to internal spin cutoff, with force-specific bankruptcy scales $r_{\mathrm{crit}}$ (QCD confinement, classical electron radius, Chandrasekhar scale, Schwarzschild radius). Bound systems occupy distinct walking plateaus along that flow and complete it only at the two-dimensional IR fixed point, where the Bekenstein–Hawking area law appears as a derived consequence—the geometric resolution of a gravitational ultraviolet catastrophe structurally parallel to Planck’s 1900 problem. The universe as a whole, which is unbound, responds to the same threshold by switching eigenmodes, producing accelerated expansion at $z \approx 0.63$.

The constraint geometry provides the $\beta$-function. The KK spectrum computation provides the effective dimension. The ADE domain wall analysis establishes why $E_8$ is selected over competing branches.

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 $d = 6$³. The physical motivation came from the experimental discovery of icosahedral quasicrystals⁴. 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 correspondence⁵ identifies this orbifold with the $E_8$ singularity, with Coxeter number $h = 30$.

Three independent requirements force the symmetry group and 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 orbifold $S^3/\mathrm{2I}$ determines which fields survive the quotient and how the surviving spectrum differs from that of the parent $S^3$.

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 series⁶,

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

The exponents 12, 20, 30 are the degrees of the Klein invariants⁷ $(H, T, f)$ satisfying the $E_8$ surface equation $T^2 = H^3 - 1728f^5$. The surviving spins are the elements of the numerical semigroup⁸ $\langle 6, 10, 15 \rangle$, with gap set $\{1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 17, 19, 23, 29\}$. The generators are the pairwise least common multiples of the icosahedral rotation axis orders $\{2, 3, 5\}$: $\operatorname{lcm}(2,3) = 6$, $\operatorname{lcm}(2,5) = 10$, $\operatorname{lcm}(3,5) = 15$—the spins at which the period-4, period-6, and period-10 character contributions first constructively interfere. There are 15 gaps, and the genus $g = 15 = h/2$.

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 law⁹ 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 $d_{\mathrm{eff}} = \varphi^2 \approx 2.618$. The orbifold is topologically three-dimensional but spectrally behaves as dimension $\varphi^2$ at low energies. As the cutoff increases past the Coxeter number, $d_{\mathrm{eff}}$ asymptotes back toward 3—the Weyl regime.

Character theory of $\mathrm{2I}$ decomposes the multiplicity sum into four components, of which only the icosahedral component carries $\varphi$. That component evaluates Chebyshev polynomials at the two icosahedral half-angles $\cos(\pi/5) = \varphi/2$ and $\cos(2\pi/5) = 1/(2\varphi)$—the dominant and subdominant eigenvalues of the Penrose substitution matrix $M = \bigl(\begin{smallmatrix}1&1\\1&0\end{smallmatrix}\bigr)$—each producing a period-5 Chebyshev cycle in $l$. The squared norm ratio of the two cycles evaluates to

$$\frac{\|U_{2\bullet}(\varphi/2)\|^2_5}{\|U_{2\bullet}(1/(2\varphi))\|^2_5} = \frac{1^2 + \varphi^2 + 0^2 + \varphi^2 + 1^2}{1^2 + (1/\varphi)^2 + 0^2 + (1/\varphi)^2 + 1^2} = \varphi^2$$

exactly, with the integer terms cancelling in the ratio and the algebraic proof reducing to $(1+\varphi^2)/(1+1/\varphi^2) = \varphi^2$ by multiplying numerator and denominator by $\varphi^2$.

The spectral gap delivers the first of two inputs to the $\beta$-function: the effective dimension $\mathcal{D}$. The projection from 6D forces some ADE orbifold; narrowing to $E_8$ specifically requires comparing all available branches.

Branch Competition

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 $\sim1\%$ 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 second input to the $\beta$-function—a coupling constant—comes from the same $E_8$ orbifold, read differently.

The Frustration

The orbifold structure that deletes 15 KK modes also 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 $\pi$–$\varphi$ clash is geometric. The $\varphi$-minimizer—the Penrose tiling itself—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 $m = 10$. Optimizing self-similarity forces angular structure that the $\pi$-sector penalizes.

The $\varphi$–$N$ clash is algebraic. 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 $\delta_{\varphi N}/A^2 = 2\sin^2(\pi\ln\varphi/\ln 10) = 0.745$. The golden-ratio inflation period and the decade symmetry period cannot be simultaneously satisfied.

The $\pi$–$N$ clash is harmonic. $C_{10}$ 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 $\approx 311$). Discrete resonance demands precisely the angular structure that isotropy forbids.

The ground-state curvature is $I = 4\pi\varphi^2 \approx 32.9$, and it factorizes cleanly into a topological factor and a spectral factor,

$$I \;=\; \underbrace{2\pi\chi(S^2)}_{=\,4\pi,\;\text{topological}} \;\times\; \underbrace{\frac{\|U_{2\bullet}(\cos(\pi/5))\|^2_5}{\|U_{2\bullet}(\cos(2\pi/5))\|^2_5}}_{=\,\varphi^2,\;\text{spectral}}.$$

The first factor is the Gauss–Bonnet invariant of $S^2$—the angular manifold at the endpoint of dimensional flow, whose topology is forced by compactness, orientability, and positive curvature. The second factor is the Chebyshev norm ratio at the two icosahedral half-angles. The product structure reflects the factorization of the configuration space $\mathcal{M}_\pi \times \mathbb{R}_\varphi$ (angular times log-radial), with spectral content on product spaces factorizing via the heat-kernel product identity.

The frustration delivers the second input to the $\beta$-function—a coupling constant derived from $I$. The 15 missing KK modes and the nonzero ground-state curvature converge in a single equation.

Convergence Through Tension

The spectral gap supplies $\mathcal{D}$. The frustration supplies $u^*$. They feed into the $\beta$-function through different channels, but both read the same Chebyshev norm ratio—the algebraic core of the duality. Three incompatible curvature sectors held under one projection converge on a single invariant, and the $\beta$-function is what the convergence becomes when taken as a law of motion.

The spectral path delivers the effective dimension. The thinned Weyl count reads the ratio as an exponent, giving $d_{\mathrm{eff}} = \varphi^2 \to \mathcal{D}$.

The frustration path delivers the coupling constant. The ground-state curvature reads the ratio as a factor in the product $I = 4\pi\varphi^2$. $C_{10}$ decade symmetry partitions each RG period into 10 equivalent shells; the per-shell curvature budget is

$$u^* = \frac{I}{10} = \frac{4\pi\varphi^2}{10} \approx 3.290.$$

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 components: 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 vertex selection rule enforces one-loop exactness. 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. Higher-order corrections vanish identically, and the one-loop $\beta$-function is exact.

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

with coupled dimensional flow $d\mathcal{D}/d(\ln\mu) = -(\xi/u^*)\ln\varphi$. At the Coxeter threshold $\mathcal{D} = \varphi^2$, the dimensional correction becomes $(\varphi^2 - 2)\ln\varphi/2 = \ln\varphi/(2\varphi)$, since $\varphi^2 - 2 = 1/\varphi$—the subdominant eigenvalue of the Penrose substitution matrix whose dominant eigenvalue squared sets the spectral dimension itself.

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 $\nu = 1/u^* = 10/(4\pi\varphi^2) \approx 0.304$. This same ratio defines the critical maintenance fraction $\xi_c = 1/u^* \approx 0.304$—the threshold where the maintenance multiplier $(1-\xi)^{-u^*}$ enters nonlinear divergence. Systems crossing $\xi_c$ undergo qualitative structural change; the universe as a whole, being unbound, responds with eigenmode transition rather than collapse.

Regime Structure of Bound Systems

Physical systems sit at different points along the flow from $(0, 3)$ to $(1, 2)$, and the $\beta$-function value at each point determines the local dynamics. The UV and IR endpoints are the only exact fixed points. Intermediate populations occupy walking plateaus where the flow lingers, organized by whichever force sets $r_{\mathrm{crit}}$ for the system at each rung.

The Maintenance Fraction as Curl Cost

Feasibility projection through incompatible curvature sectors generically produces irreducible curl—correction fields that cannot be written as gradients. A system with irreducible curl cannot globally relax. It must sustain ongoing redistributive dynamics: energy continuously committed to maintaining the constrained configuration rather than settling to a minimum—a thermodynamic cost locked into structural coherence and unavailable for any other purpose.

The maintenance fraction $\xi$ quantifies this cost as the ratio of the bankruptcy radius to the system radius,

$$\xi = \frac{r_{\mathrm{crit}}}{r},$$

with $r_{\mathrm{crit}}$ the bankruptcy radius where maintenance cost equals total energy budget. Four force-specific scales populate $r_{\mathrm{crit}}$: $r_{\mathrm{QCD}} \approx 1$ fm for strong-force confinement, the classical electron radius $r_{\mathrm{EM}} \approx 2.8 \times 10^{-15}$ m for electromagnetic binding, the Chandrasekhar scale for the gravitational electron-degeneracy rung, and the Schwarzschild radius $R_S = 2GM/c^2$ for the gravitational horizon rung. Each is an instance of the same universal mapping; the bankruptcy radius is where $\xi = 1$ and the system runs out of integrable degrees of freedom.

Equivalently, $\xi$ expresses the fraction of rest-mass energy committed to maintenance,

$$\xi \equiv \frac{E_{\mathrm{bind}}}{Mc^2}.$$

For a gravitational system, this is the share of rest mass removed from availability by the act of gravitational organization—the physical realization of the circulation cost that irreducible curl demands. A diffuse gas cloud has committed no energy to holding itself together: $\xi = 0$, everything is available, and the system sits at the UV fixed point $(0, 3)$. A black hole has committed its entire rest-mass energy to gravitational configuration: $\xi = 1$, nothing remains, and the system sits at the IR fixed point $(1, 2)$. Between these extremes, $\xi$ tracks how far along the flow from uncommitted to fully committed a system has traveled.

The logistic structure $\xi(1-\xi)$ in the $\beta$-function follows directly from this physical picture. At $\xi = 0$, there is nothing to maintain—the maintenance rate vanishes. At $\xi = 1$, there is no remaining budget from which to draw further maintenance cost—the rate vanishes again. Between these endpoints, the rate is positive and maximal near $\xi = 1/2$. This is the unique first-order polynomial that respects both boundaries.

The Decade Ladder and Its Populations

The $C_{10}$ partition of the ground-state curvature $I = 4\pi\varphi^2$ fixes the decade spacing of walking plateaus along the flow. Each decade in $\xi$ is one complete $C_{10}$ shell cycle, and physical systems inhabit the rungs according to whichever force sets their $r_{\mathrm{crit}}$. The universal slack mapping

$$l_{\max} = \left\lfloor h\left(\frac{r}{r_{\mathrm{crit}}} - 1\right)\right\rfloor = \left\lfloor h\left(\frac{1}{\xi} - 1\right)\right\rfloor = \left\lfloor 30\left(\frac{1-\xi}{\xi}\right)\right\rfloor$$

connects the macroscopic maintenance fraction to the internal KK spin cutoff on $S^3/\mathrm{2I}$, with the Coxeter number $h = 30$ setting the normalization. The radial slack $\Delta r / r_{\mathrm{crit}} = 1/\xi - 1$ is the operational envelope for internal degrees of freedom: diffuse objects carry near-infinite slack and access the full spectrum; systems at the bankruptcy radius carry zero slack and zero accessible internal modes. At $\xi = 1/2$—equivalently $r = 2 r_{\mathrm{crit}}$—the mapping gives $l_{\max} = 30$, the Coxeter boundary where the semigroup releases its last gap and the spectral dimension crosses $\varphi^2$. This wall is universal: the same algebraic boundary at $r/r_{\mathrm{crit}} = 2$ sits at QCD confinement ($r = 2$ fm), at EM bankruptcy ($r = 2 r_{\mathrm{EM}}$), and at gravitational compactness ($R = 2 R_S$), struck by any system whose organizing force drives its $\xi$ to $1/2$.

At the bottom of the ladder, the formal UV fixed point $(\xi, \mathcal{D}) = (0, 3)$ is spectrally perfect and physically inaccessible. QED vacuum coupling—self-energy, vacuum polarization, and Fermi-golden-rule coupling to photon and phonon modes—lifts the flow off $(0, 3)$ at the lowest accessible decade, anchoring elementary particles at $\xi \sim 10^{-6}$ with $r_{\mathrm{crit}}$ set by radiative self-dressing. Climbing the ladder, atoms at $\xi \sim 10^{-3}$ carry $r_{\mathrm{crit}}$ from electromagnetic binding; molecules at $\xi \sim 10^{-2}$ from chemical organization; biological systems at $\xi \sim 10^{-1}$ from metabolic thermodynamics. For biology, $r_{\mathrm{crit}}/r$ takes its operational form as the metabolic maintenance fraction—the share of energy committed to homeostasis, ion pumps, and repair against entropy. The universal rule $\xi = r_{\mathrm{crit}}/r$ generalizes across organizing mechanisms: geometric for gravity and electromagnetism, thermodynamic for biology, chemical for molecular.

Gravitational systems span the full ladder. Dark matter halos at $\xi \sim 10^{-5}$ sit bound but far below $\xi_c$; compact stellar remnants climb from white dwarfs at $\xi \sim 10^{-3}$ through the NICER neutron-star window; black holes close the flow at $\xi = 1$. Gravity is the only organizing force that reaches horizon-rung saturation; other forces stall at finite $r/r_{\mathrm{crit}}$ below the Coxeter boundary.

Neutron stars span $\xi \sim 0.31$–$0.49$, measured from NICER X-ray timing via simultaneous mass–radius fits¹⁰ (individual values in the table below). Every measured neutron star exceeds $\xi_c = 0.304$, having passed all three stages of the activation cascade: Stage 1 (kinematic binding), Stage 2 (organizational loading past $\xi_c$), and Stage 3 (geometric transition where the curl floor engages). The $\beta$-function is steep ($\beta \sim -0.83$ for the canonical $1.4\,M_\odot$/11 km case), and the effective dimension has begun dropping below 3.

Neutron stars are the highest-$\xi$ objects that remain below tri-sector saturation—the final metastable plateau of the gravitational regime before total saturation. At this loading the continuous isotropic sector ($\pi$-sector) runs out of physical capacity to absorb further compounding maintenance, and the system offloads the excess into the discrete, quantized internal topology of the substrate itself (the $N$-sector). The internal spin cutoff is the operational record of how much has been offloaded.

The NICER targets populate the mapping at the gravitational-Schwarzschild rung:

PSR J0740+6620—the most massive neutron star reliably measured—sits at $l_{\max} = 30$, exactly on the Coxeter boundary. The heaviest material body in the catalog saturates the internal spectrum at the last accessible spin level.

The Critical Approach Regime

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 the approach interval.

Cosmological Matter Partition

The cosmological $\bar{\xi}(z)$ diagnostic sums over gravitationally-bound matter: non-gravitational systems carry mass-energy density far below the cosmic total and contribute negligibly despite nonzero individual $\xi$. Within the gravitational subset, cosmic matter partitions into four regimes, each occupying a different region of the $(\xi, \mathcal{D})$ flow plane with its own RG evolution.

1. Diffuse, non-collapsed matter: $\xi_0 = 0$, $\mathcal{D}_0 = 3$. This matter has never entered the flow. At $z = 0$, roughly 56% of cosmic matter remains in this regime.

2. Halo gravitational channel: Dark matter halos have entered gravitational collapse but remain at $\xi \sim 10^{-5}$—bound but far below $\xi_c$. About 44% of matter at $z = 0$.

3. Compact stellar remnants: White dwarfs and neutron stars. Individual objects carry significant $\xi$ ($10^{-4}$ to $10^{-1}$), but their mass-weighted cosmological contribution is small ($\sim 10^{-3}$ of total matter).

4. Black holes: $\xi = 1$, $\mathcal{D} = 2$. At the IR fixed point. Every black hole has $R_S/R = 1$ regardless of formation channel—the Bekenstein–Hawking area law $S_{\mathrm{BH}} = A/(4\ell_P^2)$ confirms $\mathcal{D} = 2$.

Mass fractions satisfy $f_0(z) + f_h(z) + f_r(z) + f_{\mathrm{BH}}(z) = 1$, with compact remnants and black holes subtracted from halo mass to avoid double counting.

Each of these regimes requires its own construction. The halo regime’s $\xi$ is an integral over the halo population. Halos are modeled with NFW profiles¹¹, and the maintenance fraction uses the central potential depth normalized to $R_S/R$ in the point-mass limit,

$$\xi_{\mathrm{halo,grav}}(M,z) = \frac{2GM}{R_{\mathrm{vir}}c^2} \frac{c(M,z)}{\ln(1+c) - c/(1+c)},$$

where $c(M,z)$ is the concentration parameter. The $2GM/c^2$ in the numerator is the Schwarzschild radius, so $\xi_{\mathrm{halo,grav}}$ reduces to $R_S/R_{\mathrm{vir}}$ (times the concentration correction) in the point-mass limit—matching the universal $\xi = r_{\mathrm{crit}}/r$ rule at the gravitational-horizon rung.

The global halo maintenance load, mass-weighted over the Sheth–Tormen halo mass function¹² and normalized to total matter density, is

$$\xi_h(0) = 3.02 \times 10^{-5}.$$

This number is small. Halos are gravitationally bound—they pass Stage 1 of the activation cascade—but their binding-energy fraction is five orders of magnitude below $\xi_c$. The $\beta$-function evaluates to $\beta \sim -4 \times 10^{-5}$: weak flow, far from any structural threshold.

Compact stellar remnants populate the middle of the flow. White dwarfs approach information bankruptcy through the Chandrasekhar rung—organizational $\xi$ climbs toward unity as mass approaches the limit. The Gaia DR3 catalog contains 7,515 white dwarfs with a cooling anomaly at $r/r_{\mathrm{crit}} \approx 10^3$, at $14.5\sigma$ significance¹³; at that threshold the maintenance multiplier $(1-\xi)^{-u^*}$ enters nonlinear divergence. Neutron stars sit in the same regime at higher $\xi \sim 0.31$–$0.49$.

Stellar black holes belong with the black hole regime at the IR fixed point, where $R_S/R = 1$.

Although individual neutron star $\xi$ values are large, the neutron star mass density is small relative to total matter ($\rho_{\mathrm{NS}} \sim 10^{-4}\,\rho_m$), so the global compact-remnant maintenance load remains small,

$$\xi_r(0) \sim 10^{-5}.$$

Black holes close the tier at the IR fixed point, $\xi = 1$ and $\mathcal{D} = 2$, regardless of formation channel. The event horizon defines $R = R_S$, Bekenstein–Hawking entropy¹⁴¹⁵ scales as area not volume, and the system is confined to boundary-supported dynamics. The LIGO/Virgo catalog (GWTC, 164 binary black hole mergers) yields spin populations consistent with the $u^*$ prediction at $0.1\sigma$ deviation¹⁶.

Cosmic black hole mass density aggregates all populations. The Soltan argument¹⁷—integrating AGN luminosity across cosmic time with radiative efficiency $\epsilon \approx 0.1$—gives $4.3 \times 10^5\,M_\odot\,\mathrm{Mpc}^{-3}$ for the supermassive population; stellar evolution inventories contribute the rest. Since $\xi_{\mathrm{BH}} = 1$ for every black hole, maintenance contribution is the total mass fraction,

$$\xi_{\mathrm{BH}}(0) \approx 1.1 \times 10^{-4}.$$

Because $\xi_{\mathrm{BH}} = 1$, all energy is consumed by maintenance and the dimensional flow drives $\mathcal{D} \to 2$—horizons are boundary-supported dynamics at the IR fixed point. The flow factor $\xi(1-\xi) = 0$ confirms they have completed the RG trajectory: no further evolution is possible, and their contribution to the cosmological regime budget is purely their combined mass fraction.

Global Diagnostic and Tierwise Integration

The global diagnostic is the direct sum of regime contributions, each normalized to total matter density,

$$\bar{\xi}(z) = \xi_h(z) + \xi_r(z) + \xi_{\mathrm{BH}}(z).$$

This value is diagnostic only—it indicates the overall scale of cosmological maintenance burden. The nonlinearity of the flow equations requires regime-by-regime RG integration; by Jensen’s inequality, averaging $\xi$ first overestimates the result because the maintenance rate $d\xi/d\mu = \xi(1-\xi)\mathcal{W}(\mathcal{D})$ is concave in $\xi$. The error is proportional to the variance of $\xi$ across regimes—and since $\xi$ spans from 0 (diffuse) to 1 (black holes), the variance is maximal.

The diffuse regime ($\xi = 0$) contributes zero to the flow because no energy has been committed to maintenance—structure has not begun. The black hole regime ($\xi = 1$) also contributes zero further flow, but for the opposite reason: all energy is consumed by maintenance, the dimensional flow has driven $\mathcal{D} \to 2$, and the system has completed its RG trajectory. Both give $\xi(1-\xi) = 0$, but the physics is opposite—one has not started, the other has completed the flow at the IR fixed point. Only the intermediate regimes (halos and compact remnants) drive nonzero evolution.

The regime-wise dimension evolution is

$$\mathcal{D}_i(z) = 3 - \frac{\ln\varphi}{u^*} \int_{\mu(z_{\mathrm{ref}})}^{\mu(z)} \xi_i(\mu')\,d\mu',$$

with RG clock $\mu(z) = \ln[k_{nl}(0)/k_{nl}(z)]$ defined by the nonlinear scale where $\Delta^2(k_{nl}, z) = 1$. The mass-weighted global effective dimension at $z = 0$ gives

$$3 - \langle \mathcal{D}(0) \rangle = 1.4 \times 10^{-5}.$$

The universe remains extremely close to $\mathcal{D} = 3$: $R(t)$ increases faster than cumulative binding energy can grow in the bulk.

Linear Growth Modifications

The standard $\Lambda$CDM linear growth equation for matter density perturbations is

$$\delta'' + \left[2 + \frac{d\ln H}{d\ln a}\right]\delta' - \frac{3}{2}\Omega_m(a)\,\delta = 0,$$

where primes denote derivatives with respect to $\ln a$. Maintenance costs enter as additional friction—binding energy committed to structural coherence is unavailable for driving perturbation growth,

$$\delta'' + \left[2 + \frac{d\ln H}{d\ln a} + \Gamma_{\mathrm{TCG}}(z,k)\right]\delta' - \frac{3}{2}\Omega_m(a)\,\delta = 0.$$

The friction kernel $\Gamma_{\mathrm{TCG}}(z,k)$ combines the logistic maintenance rate $\mathcal{W}(\xi)\,\xi(1-\xi)$ from the $\beta$-function, a normalization set at criticality, a microstructural amplification factor from spectral concentration on Penrose defect vertices, and a Lorentzian cutoff at the cosmological coherence scale. Every factor is assembled from previously defined geometric objects.

Because $\bar{\xi}(z) \ll 1$ and diffuse mass dominates, the friction kernel forces deviations into the sub-ppm regime:

Across $z \leq 1000$, the maximum $(f\sigma_8)$ deviation reaches $1.3 \times 10^{-5}$. Strict binding-energy maintenance forces TCG corrections into the sub-ppm regime for linear observables. Constraint geometry coexists with $\Lambda$CDM in the linear regime, with its structural content confined to the nonlinear and boundary-saturated regimes.

The Gravitational Ultraviolet Catastrophe

PSR J0740+6620 sits at $l_{\max} = 30$ because of the structural boundary built into the slack mapping. One more small step in $\xi$ drives the system into a regime the internal geometry cannot mathematically inhabit.

When $\xi$ exceeds $0.5$, the slack mapping demands $l_{\max} < 30$. That request is mathematically invalid. Below the Coxeter threshold, the accessible spin levels are no longer the continuous tail of the spectrum; they are scattered across the $\langle 6, 10, 15 \rangle$ semigroup, with 15 forbidden levels $\{1,2,3,4,5,7,8,9,11,13,14,17,19,23,29\}$ between the remaining generators. The system is requesting storage for its maintenance information on a support whose admissible spin set has just been thinned to the semigroup generators alone. The Chebyshev destructive interference—driven by $\varphi$‘s worst-approximability, the extremal property responsible for the gap set—removes the modes the system requires. The internal dimensions lose coherence, the substrate forfeits the degrees of freedom required to store its own organizational overhead, and the object undergoes an information-theoretic bankruptcy. The only resolution is collapse to the IR fixed point ($\xi = 1$, $\mathcal{D} = 2$), at which the radial slack vanishes, the slack mapping returns $l_{\max} = 0$, and the internal KK spectrum is extinguished. Storage and processing capacity are permanently bounded by the holographic Bekenstein–Hawking area law, with processing saturating at $N_{\max} = 2 N_{BH}$.

This is the structural dual of Planck’s 1900 ultraviolet catastrophe. Both systems confront an infinite information request from a finite substrate, and both resolve the same way in principle and differently in kind. The electromagnetic catastrophe was a divergence in spectral density: classical mode counting gave $N(\nu) \sim \nu^3$, each mode costing $k_B T \ln 2$ to specify. The resolution was spectral—Planck’s quantum $h\nu$ exponentially suppressed high-frequency modes, capping the information budget without altering the geometry. The gravitational catastrophe is a divergence in coordination overhead: three-dimensional information processing at Planck-scale density demands communication channels that scale as $N^{4/3}$, exceeding the holographic area budget. The resolution is geometric—the radial dimension freezes, and the effective dimension drops from 3 to 2. Storage and processing become finite and proportional.

In constraint geometry, the mechanism is transparent. The system’s maintenance load saturates the isotropic sector, overflows into the discrete sector, and demands internal KK modes the semigroup cannot supply. The Bekenstein–Hawking area law $S_{\mathrm{BH}} = A/(4\ell_P^2)$ forces the information budget finite when the continuous sector reaches capacity. Collapse is the topological phase transition the substrate executes when the request becomes algebraically impossible.

Every system discussed so far—halos, white dwarfs, neutron stars, black holes—is gravitationally bound. The same $\beta$-function applies at the global cosmological scale, but the universe as a whole is unbound, and this changes the outcome.

Accelerated Expansion

Bound systems can complete the flow to $(1, 2)$. The universe fails Stage 1 of the activation cascade—it is unbound—so it responds differently. When the global matter fraction $\Omega_m$ approaches $\xi_c \approx 0.304$—the same divergence threshold that drives white dwarf information bankruptcy—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 $\xi_c = 0.304$. The upper boundary of the transition is set by $u^*/10 = 0.329$—the per-shell curvature budget—where the discrete sector becomes fully binding and the $\pi$-eigenmode takes over. The acceleration onset occurs when $\Omega_\Lambda(z)$ crosses this threshold, which happens at $z = 0.632$, matching the observed $q = 0$ redshift.

The present-day $\Omega_m = 0.315 \pm 0.007$ (Planck 2018)¹⁸ sits inside the approach window $[\xi_c, \, u^*/10] = [0.304, \, 0.329]$. The window is a structural attractor. When $\Omega_m$ drops below $\xi_c$, the N-sector desaturates, the $\pi$-eigenmode weakens, structure formation reactivates, and the matter fraction stabilizes. When $\Omega_m$ rises above $u^*/10$, the N-sector saturates, the $\pi$-eigenmode dominates, expansion accelerates, and the matter fraction relaxes downward. The feedback holds $\Omega_m$ in the window between the two thresholds derived from the same coupling constant $u^* = 4\pi\varphi^2/10$.

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 each structural feature descended from the projection. The mean-field exponent $n = 1/2$ is the linear-order saturation response. The background exponent $m = \varphi^2$ is the Chebyshev norm ratio at the icosahedral half-angles—the same object that fixes $d_{\mathrm{eff}} = \varphi^2$. The cap $1/\xi_c - 1$ marks the N-sector saturation limit, with $\xi_c = 1/u^*$ and $u^* = I/10 = 4\pi\varphi^2/10$. The amplitude $A = \mathcal{W}(3) + 1 = u^* + 1 + \ln\varphi/2 \approx 4.531$ is the cross-sector coupling, and it decomposes through the same Chebyshev structure that fixes $d_{\mathrm{eff}}$ and the spectral factor in $I$.

The period-5 cycle at the icosahedral half-angles has two classes of values: identity values ($U_l = \pm 1$) and Penrose eigenvalue values ($U_l = \pm\varphi$ at the dominant angle, $\pm 1/\varphi$ at the subdominant angle). Both classes appear in the norms

$$\|U_{2\bullet}(\varphi/2)\|^2_5 = 2(1 + \varphi^2), \qquad \|U_{2\bullet}(1/(2\varphi))\|^2_5 = 2(1 + 1/\varphi^2),$$

with identity contributions collecting into the "$1$" term and eigenvalue contributions collecting into "$\varphi^2$" or "$1/\varphi^2$". The identity piece is a polynomial invariant: $U_0(x) \equiv 1$ holds at every argument, and $U_8(\cos(\alpha)) = -1$ at every icosahedral half-angle by the trigonometric period structure. The eigenvalue piece is the spectral content that depends on which Penrose substitution eigenvalue—dominant $\varphi$ or subdominant $1/\varphi$—drives the cycle.

The pump amplitude is the RG-dressed generalization of the same decomposition:

$$A \;=\; \underbrace{1}_{\text{Chebyshev identity}} \;+\; \underbrace{\mathcal{W}(3)}_{\text{Chebyshev eigenvalue, RG-dressed}} \;=\; u^* + \frac{\ln\varphi}{2} + 1 \approx 4.531,$$

where $\mathcal{W}(3) = u^* + \ln\varphi/2$ is the eigenvalue contribution integrated under the $\beta$-function, carrying the decade partition $u^* = I/10 = 4\pi\varphi^2/10$ and the dimensional correction $(\mathcal{D}-2)\ln\varphi/2$ at $\mathcal{D} = 3$. The identity piece is untouched by the flow—there is no coupling for a polynomial invariant to run along. The static norm ratio $(1 + \varphi^2)/(1 + 1/\varphi^2) = \varphi^2$ delivers the spectral dimension; the dynamic sum $1 + \mathcal{W}(3)$ delivers the pump amplitude.

The resulting equation of state gives $w_0 = -0.724$ vs DESI DR2’s $-0.75 \pm 0.07$ ($0.4\sigma$) and $w_a = -0.746$ vs DESI’s $-0.86 \pm 0.25$ ($0.5\sigma$)¹⁹. As a 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%.

The eigenmode transition also makes contact with Type Ia supernova energetics. At the $\pi$-sector transition, the white dwarf’s organizational information must be erased and rewritten—a process whose minimum thermodynamic cost is set by Landauer’s principle²⁰ applied to the number of structural bits crossing the threshold. The resulting energy $E = 4.3 \times 10^{44}$ J matches the observed bolometric output of normal Type Ia events²¹.

Conclusion

A 6D periodic lattice projects to 3D. The projection simultaneously produces the orbifold $S^3/\mathrm{2I}$—the Poincaré homology sphere, whose McKay correspondent is $E_8$.

$6\text{D lattice} \xrightarrow{\;\text{project}\;} S^3/\mathrm{2I}.$

The Molien series of $\mathrm{2I}$ reads off which Kaluza–Klein modes survive the quotient. The surviving spins are exactly the elements of the numerical semigroup $\langle 6, 10, 15 \rangle$, with 15 gaps.

$S^3/\mathrm{2I} \xrightarrow{\;\text{Molien}\;} \langle 6,10,15 \rangle \xrightarrow{\;n_l = 0\;} d_{\mathrm{eff}} = \varphi^2.$

Those 15 missing modes thin the low-energy eigenvalue count. By Weyl’s law, the orbifold is topologically three-dimensional but spectrally behaves as dimension $\varphi^2 \approx 2.618$—the effective dimension $\mathcal{D}$ that enters the $\beta$-function. The equality is exact: $\varphi^2$ is the Chebyshev norm ratio at the two icosahedral half-angles, the algebraic object underneath the finite-sample regression.

A second path runs in parallel. The same orbifold quotient prevents simultaneous relaxation of three curvature sectors, producing a ground-state curvature $I = 4\pi\varphi^2 \approx 32.9$ that factorizes as Gauss–Bonnet ($4\pi$) times the same Chebyshev norm ratio ($\varphi^2$). Decade symmetry partitions each RG period into 10 equivalent shells, giving the coupling constant $u^* = I/10 \approx 3.29$.

$\text{projection} \xrightarrow{\;\text{frustration}\;} I = 4\pi\varphi^2 \xrightarrow{\;C_{10}\;} u^* = I/10.$

The spectral path delivers $\mathcal{D}$. The frustration path delivers $u^*$. Both read the same Chebyshev norm ratio, and they converge in the bracket $[u^* + (\mathcal{D}-2)\ln\varphi/2]$.

The flow from $(\xi, \mathcal{D}) = (0, 3)$ to $(1, 2)$ closes on the geometric slack mapping $l_{\max} = \lfloor h(r/r_{\mathrm{crit}} - 1)\rfloor$, which ties the macroscopic maintenance fraction to the internal spin cutoff at every bankruptcy scale—QCD, EM, Chandrasekhar, Schwarzschild. The Coxeter boundary $l_{\max} = 30$ lands at $\xi = 0.5$ (equivalently $r = 2 r_{\mathrm{crit}}$), where PSR J0740+6620 is observed at the Schwarzschild-rung instance. Beyond it, the semigroup’s gap set makes the internal request algebraically impossible, and the substrate executes the dimensional cutoff that closes the flow—a gravitational ultraviolet catastrophe resolved by reducing from 3D to 2D, structurally parallel to how Planck’s quantum resolved the electromagnetic one.

The projection forces some ADE orbifold. Branch selection completes the chain: $E_8$ has the highest spectral drive (40.7 vs 5.8 for $D_8$), the most expensive domain walls (factor 456×), and is the unique maximal element under gap-set inclusion. All tunneling rates are frozen ($B \gg 1$, $R$-independent). Selection is primordial and irreversible.

Two inputs enter the framework without derivation.

• 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. Constraint geometry begins at the lattice and derives everything downstream.
• The compactification radius $R$ is a free parameter. Every other quantity—$u^*$, $\xi_c$, the spectral gap, the regime structure, 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.

Everything else is structural.

Footnotes

1. Penrose, R. (1974). The role of aesthetics in pure and applied mathematical research. Bulletin of the Institute of Mathematics and Its Applications, 10, 266–271. ↩

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

3. 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. ↩

4. Shechtman, D., Blech, I., Gratias, D., & Cahn, J. W. (1984). Metallic phase with long-range orientational order and no translational symmetry. Physical Review Letters, 53(20), 1951–1953. ↩

5. McKay, J. (1980). Graphs, singularities, and finite groups. In The Santa Cruz Conference on Finite Groups, Proceedings of Symposia in Pure Mathematics, Vol. 37, AMS, 183–186. See also Baez, J. C. (2018). From the icosahedron to E₈. London Mathematical Society Newsletter, 476, 18–23. arXiv

   .06436. ↩

6. Molien, T. (1897). Über die Invarianten der linearen Substitutionsgruppen. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, 1152–1156. ↩

7. 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. ↩

8. Rosales, J. C. & García-Sánchez, P. A. (2009). Numerical Semigroups. Developments in Mathematics, Vol. 20, Springer. ↩

9. Weyl, H. (1912). Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen. Mathematische Annalen, 71, 441–479. ↩

10. Miller, M. C., et al. (2019). PSR J0030+0451 mass and radius from NICER data and implications for the properties of neutron star matter. The Astrophysical Journal Letters, 887(1), L24. Miller, M. C., et al. (2021). The radius of PSR J0740+6620 from NICER and XMM-Newton data. The Astrophysical Journal Letters, 918(2), L28. Choudhury, D., et al. (2024). A NICER view of the nearest and brightest millisecond pulsar: PSR J0437−4715. The Astrophysical Journal Letters, 971(1), L20. ↩

11. Navarro, J. F., Frenk, C. S., & White, S. D. M. (1997). A universal density profile from hierarchical clustering. The Astrophysical Journal, 490(2), 493–508. ↩

12. Sheth, R. K. & Tormen, G. (1999). Large-scale bias and the peak background split. Monthly Notices of the Royal Astronomical Society, 308(1), 119–126. ↩

13. 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. ↩

14. Bekenstein, J. D. (1973). Black holes and entropy. Physical Review D, 7, 2333–2346. ↩

15. Hawking, S. W. (1975). Particle creation by black holes. Communications in Mathematical Physics, 43, 199–220. ↩

16. 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. ↩

17. Soltan, A. (1982). Masses of quasars. Monthly Notices of the Royal Astronomical Society, 200(1), 115–122. ↩

18. Planck Collaboration (2020). Planck 2018 results. VI. Cosmological parameters. Astronomy & Astrophysics, 641, A6. ↩

19. DESI Collaboration (2025). DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints. Physical Review D, 112(8), 083515. ↩

20. Landauer, R. (1961). Irreversibility and heat generation in the computing process. IBM Journal of Research and Development, 5(3), 183–191. ↩

21. 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. ↩