The Constants of Nature as Vacuum Closure Conditions: ℏ, H0, and the Modular Architecture of the MEG Vacuum

The MEG uncertainty principle paper raises a question: can the entropic temperature $\Theta$ be derived from vacuum geometry alone, without specifying either $\hslash$ or $\Theta$ as input? This paper answers that question, interprets its consequences, and identifies what it may be pointing toward.

The dimensional impossibility of deriving $\Theta$ without $\hslash$ is established: any energy requires $\hslash$ as input, and MEG is no exception. $\hslash$ is the irreducible quantum input of every quantum theory, and reaching this boundary is not a failure but a confirmation that the framework is correctly structured.

The positive result is more striking. The coherence scale ℓ0=(c/H0)e−4π\ell_0 = (c/H_0)e^{-4\pi} ℓ0=(c/H0)e−4π is derived from the de Sitter modular back-reaction using only the Hubble constant H0H_0 H0 — no $\hslash$ appears. Substituting into Θ=ℏc/2ℓ0\Theta = \hbar c/2\ell_0 Θ=ℏc/2ℓ0 gives Θ=ℏH0e4π/2\Theta = \hbar H_0 e^{4\pi}/2 Θ=ℏH0e4π/2. Given the de Sitter geometry, $\hslash$ and $\Theta$ are not independent — they are a single degree of freedom of the MEG vacuum. The structural relation ℏH0=2Θe−4π\hbar H_0 = 2\Theta e^{-4\pi} ℏH0=2Θe−4π is a new result connecting the quantum of action to the cosmological expansion rate through the modular back-reaction factor e−4πe^{-4\pi} e−4π. The MEG entropic temperature is related to the Gibbons-Hawking temperature of the de Sitter vacuum by Θ=πe4πkBTGH\Theta = \pi e^{4\pi} k_B T_{\rm GH} Θ=πe4πkBTGH, where the factor e4πe^{4\pi} e4π is the same modular back-reaction factor that determines the coherence scale.

This result is interpreted as a parameter compression: what appear to be independent quantum and thermal scales collapse to a single degree of freedom once the vacuum geometry is specified. The relation ℏH0=2Θe−4π\hbar H_0 = 2\Theta e^{-4\pi} ℏH0=2Θe−4π is placed in the context of the other closure conditions the MEG programme has established — the scale relation 3A=ℓ0−13\sqrt{A} = \ell_0^{-1} 3A=ℓ0−1, the electroweak hierarchy, the dominant-mode consistency conditions — as a closure condition at the level of fundamental constants. The triangle connecting quantum scale ($\hslash$), cosmological scale (H0H_0 H0), and coherence scale (ℓ0\ell_0 ℓ0) is made explicit: knowing any two determines the third, and the modular back-reaction factor e−4πe^{-4\pi} e−4π is what places ℓ0\ell_0 ℓ0 at galactic rather than cosmological scales.

The paper then develops a speculative but precisely stated conjecture: the apparent independence of the fundamental constants $\hslash$, $c$, H0H_0 H0, and GNG_N GN may not be real. Each may be a structural requirement of the self-consistent MEG vacuum — a closure condition imposed by the vacuum distinguishability kernel. In this picture, the smallness of H0H_0 H0 in natural units, usually treated as fine-tuning, emerges as the modular back-reaction factor e−4πe^{-4\pi} e−4π, a calculable geometric quantity. Three specific calculations are identified that would develop or refute the conjecture: the derivation of GNG_N GN from the kernel, the Planck-scale modular back-reaction, and the connection between the microscopic origin of the compact phase variable U=eiSU = e^{iS} U=eiS and the value of $\hslash$. The last of these connects the two deepest remaining open questions of the programme into a single target.

The paper maintains a clear distinction between established results (Sections 2–3), interpretation (Section 3), and speculation (Section 4, explicitly labelled).