The Resonant Theory: A Hydrodynamic Framework for the Quantum Plenum

Author: Justin Time Miller
Version: 7.0
Date: October 2026

Abstract

Contemporary theoretical physics relies on non-baryonic dark-sector parameterizations—dark matter and dark energy—to reconcile macroscopic cosmological observations with the Standard Model. This paper formulates a physical framework that replaces the empty inviscid vacuum with a continuous, compressible, two-fluid hydrodynamic substrate termed the Quantum Plenum, modeled on the Tisza–Landau two-fluid formalism of superfluid $^{3}\text{He}$ and $^{4}\text{He}$ [1, 2, 5]. From the electromagnetic fine-structure constant $\alpha$ and natural logarithmic attenuation $e$, we define two strictly dimensionless hydrodynamic coupling invariants: the Universal Acoustic Susceptibility $A_{\text{RT}} = \alpha^{-1}/e = 50.412727$ and its reciprocal damping ratio, the Acoustic Venting Coefficient $\beta_{\text{TRT}} = e / A_{\text{RT}} = 0.05392055$, which defines the equilibrium normal-fluid mass fraction $\chi_n \equiv \rho_{n,0}/\rho_0 = \beta_{\text{TRT}}$. Conditioning on explicitly declared constitutive postulates and discrete structural indices (Table 1) and anchoring to the local present-epoch acoustic wave speed $c_0 = 2.99792458 \times 10^8\text{ m s}^{-1}$ and carrier frequency $f_0 = 7.64496 \times 10^7\text{ Hz}$ ($L_0 = c_0 / f_0 = 3.92144\text{ m}$), we establish a dimensionally homogeneous system of characteristic Plenum scales (Table 2) for kinematic viscosity ($\nu_0 = c_0 L_0 / A_{\text{RT}} = 2.3320 \times 10^7\text{ m}^2\text{s}^{-1}$), angular relaxation time ($\tau_0 = (2\pi f_0)^{-1} = 2.0818 \times 10^{-9}\text{ s}$), and minimum cavitation energy ($E_{\text{cav}} = A_{\text{RT}} h f_0 = 2.5537 \times 10^{-24}\text{ J}$). To ensure complete claim-calibration transparency, all quantitative results are categorized up front into four distinct epistemological tiers (Table 4): the framework offers one prospective, out-of-sample falsification prediction—the 2028–2030 Europa Clipper outer-cruise Doppler residual ($\bar{a}_{\text{plenum}} = (8.47 \pm 0.34) \times 10^{-10}\text{ m s}^{-2}$, falsifiable if $|\mathbf{a}_{\text{res}}| < 2.0 \times 10^{-10}\text{ m s}^{-2}$)—while the subatomic mass ratio, apparent Hubble shift, galactic acceleration floor, and lunar recession are retrodictive consistency benchmarks conditioned on explicitly declared constitutive postulates and discrete structural integers (Table 1). Within this calibrated framework: (1) under a 6-sheeted branched covering torus $\widetilde{\mathbb{T}}^5 \to \mathbb{T}^5$ of covering degree $d = pq = 6$ corresponding to the $(2,3)$ Trefoil torus knot, where the metric Jacobian $\mathcal{J}_5 = \det(\partial\boldsymbol{\vartheta}/\partial\boldsymbol{\xi}) = \pi^5$ maps the unknotted unit cycle cell $\text{Vol}_5(\Omega_{\text{core}}) \equiv \int_{[0,1]^5} d^5\xi = 1$ to physical half-wave phase radians $[0,\pi]^5$ and the normal-fluid Stokes boundary-layer dipole projection adds $\frac{2}{\pi}\beta_{\text{TRT}}$, the constitutive proton-to-electron rest-energy ratio $\mu_{\text{TRT}} \equiv \mathcal{H}_{\text{sol}}(2,3)/\mathcal{H}_{\text{sol}}(1,1) = 6\pi^5 + \frac{2}{\pi}\beta_{\text{TRT}} = 1836.152436$ approximates the CODATA 2022 value $1836.15267343(11)$ to $0.13\text{ ppm}$ ($|\Delta\mu| = 2.37 \times 10^{-4}$); (2) a dimensionally homogeneous Bernoulli cavitation threshold defines the invariant topological core mass floor $\mu_0 = E_{\text{cav}} / c_0^2 = 2.8414 \times 10^{-41}\text{ kg}$ for super-threshold excitations; (3) introducing the solenoidal Leray–Helmholtz projected cubic venting term $-\beta_{\text{TRT}}\tau_0 \mathbb{P}(|\boldsymbol{\omega}|^2\boldsymbol{\omega})$ into the 3D vorticity equation preserves $\nabla \cdot \boldsymbol{\omega} = 0$ identically and admits a complete a priori global $L^2$ enstrophy bound for smooth solutions with zero spatial mean on $\mathbb{R}^3$ and $\mathbb{T}^3$ (Theorem 1); (4) solving the 1D line-of-sight quarter-wave Helmholtz resonator $\delta P''(r) + k^2\delta P(r) = 0$ with Neumann antinode $\delta P'(0)=0$ and Dirichlet horizon node $\delta P(R_H)=0$ yields $\delta P(r) = \delta P_0\cos(k_1 r)$ with exact fundamental eigenvalue $k_1 R_H = \pi/2$ and a constitutive achromatic cavity action-transfer coupling $\gamma_{\text{cav}} = k_1 \beta_{\text{TRT}} = \frac{\pi}{2 R_H}\beta_{\text{TRT}}$ retrodicting the $+8.47\%$ apparent local Hubble shift ($H_{\text{late, TRT}} = H_{\text{early}}(1 + \frac{\pi}{2}\beta_{\text{TRT}}) = 73.11\text{ km s}^{-1}\text{Mpc}^{-1}$ from $H_{\text{early}} = 67.40\text{ km s}^{-1}\text{Mpc}^{-1}$) and geometric-mean BAO scale $\lambda_{\text{BAO}} = 152.53\text{ Mpc}$; (5) a linearized Poisson hydrostatic pressure-depletion law $\nabla^2 P = 4\pi G\rho_0\rho_M$ reproduces Newton's inverse-square law $\mathbf{g} = -\frac{1}{\rho_0}\nabla P = -\frac{GM}{r^2}\hat{\mathbf{r}}$ with exact outward pressure gradient $\frac{dP}{dr} = +\frac{\rho_0 GM}{r^2} > 0$; (6) an emergent cosmological acoustic acceleration floor $a_0 \equiv c_0 H_{\text{early}}\beta_{\text{TRT}} = 3.531 \times 10^{-11}\text{ m s}^{-2}$ and circular disc boundary threshold $a_{\text{gal}} \equiv \pi a_0 = 1.109 \times 10^{-10}\text{ m s}^{-2} \approx 1.2 \times 10^{-10}\text{ m s}^{-2}$ in Section 7 recovering the flat galactic rotation curve plateau and baryonic Tully–Fisher scaling under a constitutive deep-field boundary transition (with multi-galaxy SPARC dataset fitting designated for Paper 2); and (7) a Master Classification Ledger (Table 4) separates retrodictive benchmarks (including lunar recession $\dot{r}_{\text{net}} = 3.815 \pm 0.001\text{ cm yr}^{-1}$, spanning the $[3.814, 3.816]\text{ cm yr}^{-1}$ truncation range between exact and linearized sheath roots) from a prospective out-of-sample 2028–2030 Europa Clipper Doppler falsification test ($\bar{a}_{\text{plenum}} = (8.47 \pm 0.34) \times 10^{-10}\text{ m s}^{-2}$ across the radial transition band $[8.124, 8.812] \times 10^{-10}\text{ m s}^{-2}$).

1. Introduction & Literature Positioning

1.1 Physical Picture: Space as an Acoustic Two-Fluid Medium

To grasp the core physical intuition of The Resonant Theory (TRT) before entering its formal differential equations, consider the difference between an empty geometric stage and a submerged acoustic chamber. In standard cosmology and quantum theory, empty space is treated as a frictionless void; when galaxies rotate faster than visible matter allows, or when distant supernovae appear dimmer than early-universe expansion predicts, invisible "dark matter" particles and repulsive "dark energy" fields must be added to balance the equations. TRT investigates a mechanical alternative: what we call "empty space" is modeled as a continuous, compressible two-fluid medium—the Quantum Plenum—analogous to laboratory superfluid helium ($^{3}\text{He}$ and $^{4}\text{He}$). At equilibrium, $94.61\%$ of this medium acts as an inviscid superfluid condensate ($\rho_{s,0} / \rho_0 = 1 - \beta_{\text{TRT}}$) that transmits electromagnetic and gravitational waves without shear resistance, while the remaining $5.39\%$ ($\chi_n = \rho_{n,0}/\rho_0 = \beta_{\text{TRT}} = 0.05392055$) acts as a viscous "normal fluid" boundary component. Just as a vortex ring in liquid helium is a self-sustaining topological structure of the fluid itself, elementary particles in TRT are modeled as quantized toroidal vortex solitons in the Plenum, while macroscopic orbital and cosmological residuals emerge from the $5.39\%$ normal-fluid attenuation and standing-wave boundary geometry of the medium.

1.2 The Cosmological Crisis and Theoretical Lineage

The persistent tension between quantum field theory and general relativity has required auxiliary dark-sector parameterizations across cosmological scales. Late-time cosmic acceleration and the $5\sigma$ Hubble tension between early-universe Cosmic Microwave Background (CMB) inferences ($H_{\text{early}} = 67.40 \pm 0.50\text{ km s}^{-1}\text{Mpc}^{-1}$ [8]) and local Type Ia supernova distance-ladder measurements ($H_{\text{late}} = 73.04 \pm 1.04\text{ km s}^{-1}\text{Mpc}^{-1}$ [9]) are conventionally attributed to dark energy ($\Lambda$), while flat galactic rotation curves [7] and outer heliospheric angular-momentum residuals are attributed to cold dark matter halos.

Rather than introducing new particle species, The Resonant Theory (TRT) investigates whether these anomalies can be organized within a single constitutive hypothesis: that the physical vacuum acts as a compressible, two-fluid electro-hydrodynamic substrate. This inquiry synthesizes three established physics lineages while introducing an explicit normal-fluid dissipation closure:

1. Superfluid Vacuum & Analogue Gravity Theories: Following Tisza (1938) [1] and Landau (1941) [2], Volovik (2003) [5] showed that quasiparticles in superfluid $^{3}\text{He-A}$ reproduce emergent Lorentz invariance, gauge fields, and effective acoustic metrics [4]. However, standard superfluid-vacuum models treat the vacuum ground state as strictly inviscid ($\nu = 0$) below the Landau critical velocity, providing no quantitative mechanism for secular orbital drag or cosmological redshift attenuation.
2. Topological Soliton Models of Baryons: Following Kelvin's vortex-atom hypothesis and Skyrme's (1961) nonlinear meson theory [3], Faddeev and Niemi (1997) [6] demonstrated that knotted toroidal vortex solitons—specifically the $(2,3)$ Trefoil knot $3_1$—emerge as stable energy minima in three-dimensional field theories, though standard Skyrmion models require tunable coupling parameters to match empirical baryon-to-lepton mass ratios.
3. Galactic Phenomenology Scope & Separation of Program Papers: While this foundational paper focuses on subatomic rest energies, cosmological horizon attenuation, and heliospheric drag, Section 7 derives the background acoustic acceleration floor $a_0 \equiv c_0 H_{\text{early}}\beta_{\text{TRT}} = 3.531 \times 10^{-11}\text{ m s}^{-2}$ and effective galactic disc boundary threshold $a_{\text{gal}} \equiv \pi a_0 = 1.109 \times 10^{-10}\text{ m s}^{-2} \approx 1.2 \times 10^{-10}\text{ m s}^{-2}$ underlying flat rotation curves; full parameter-by-parameter fitting across the 175-galaxy SPARC database [7] is formally designated for the companion paper (Paper 2) in this research series.
4. Methodological Transparency & Separation of Postulates from Predictions: TRT models the vacuum as a viscous-normal plus inviscid-superfluid two-component substrate governed by two dimensionless electro-hydrodynamic invariants, $A_{\text{RT}} = \alpha^{-1}/e = 50.412727$ and $\beta_{\text{TRT}} = e / A_{\text{RT}} = 0.05392055$. To maintain strict distinction between constitutive axioms, discrete topological/harmonic integers, retrodictive benchmarks, and prospective out-of-sample predictions, Section 2 declares all six foundational postulates (P1–P6) in Table 1, and Section 8 classifies every quantitative claim in Table 4.

2. Foundational Postulates, Two-Fluid Constitutive Equations & Closure Relations for $\beta_{\text{TRT}}$

Physical Picture: Why does a single dimensionless number, $\beta_{\text{TRT}} = 0.05392055$, appear across the boundary-layer skin of a subatomic proton, the damping of a spinning fluid vortex, and the attenuation of light across cosmological distances? Within the Tisza–Landau two-fluid framework, all three effects represent standard constitutive closures of the equilibrium normal-fluid mass fraction ($\chi_n = 5.392\%$). At the medium's characteristic relaxation resonance ($\omega\tau_0 = 1$), the $5.392\%$ normal-fluid fraction sets the per-radian wave attenuation rate; in a spinning vortex tube, centrifugal phase separation expels that same $5.392\%$ normal-fluid fraction into an annular boundary sheath of leading-order thickness $\delta^*/R \approx \frac{1}{2}\beta_{\text{TRT}} = 2.696\%$ (exact nonlinear root $1 - \sqrt{1-\beta_{\text{TRT}}} = 2.733\%$); and in high-vorticity cores, Hall–Vinen mutual friction proportional to $\chi_n$ vents excess spin into acoustic waves. Proposition 1 formalizes these three constitutive closure relations.

2.1 State Variables and Tisza–Landau Two-Fluid Decomposition

Postulate P1 (Two-Fluid Quantum Plenum): Physical space is occupied by a continuous, compressible two-fluid substrate described at every spacetime point $(\mathbf{x}, t)$ by the Tisza–Landau mass and momentum decomposition [1, 2]: \begin{equation} \rho(\mathbf{x}, t) = \rho_s(\mathbf{x}, t) + \rho_n(\mathbf{x}, t), \qquad \mathbf{j}(\mathbf{x}, t) = \rho_s \mathbf{u}_s + \rho_n \mathbf{u}_n \tag{1} \end{equation} where:

• $\rho_s$ $[\text{kg m}^{-3}]$ and $\mathbf{u}_s$ $[\text{m s}^{-1}]$ denote the density and velocity field of the superfluid condensate component, which is irrotational outside quantized vortex cores ($\nabla \times \mathbf{u}_s = 0$) and has zero intrinsic shear viscosity ($\eta_s = 0$);
• $\rho_n$ $[\text{kg m}^{-3}]$ and $\mathbf{u}_n$ $[\text{m s}^{-1}]$ denote the density and velocity field of the normal viscous component, which carries thermal/acoustic excitations and dynamic shear viscosity $\eta_n = \rho_n \nu_0$.

At the present cosmological epoch ($t = t_0$, redshift $z = 0$), the undisturbed background Plenum has uniform total density $\rho_0 = \rho_{s,0} + \rho_{n,0}$, equilibrium normal-fluid mass fraction $\chi_n \equiv \rho_{n,0} / \rho_0 = \beta_{\text{TRT}} = 0.05392055$, and barotropic equation of state $P(\rho) = c_0^2 \rho$. Linearizing the mass continuity equation $\partial_t \rho + \nabla \cdot (\rho \mathbf{u}) = 0$ and Euler momentum equation $\rho \partial_t \mathbf{u} = -\nabla P$ around $\rho_0$ yields the wave equation $\partial_t^2 \delta\rho - c_0^2 \nabla^2 \delta\rho = 0$, where the present-epoch local longitudinal acoustic wave speed is the invariant vacuum speed of light: \begin{equation} c_0 = \sqrt{\left.\frac{\partial P}{\partial \rho}\right|_{\rho_0}} = 2.99792458 \times 10^8\text{ m s}^{-1} \quad [\text{L T}^{-1}] \tag{2} \end{equation}

Operational Distinction Between Present-Epoch Local $c_0$ and Cosmological Lookback Attenuation: Throughout this framework, $c_0$ is a fixed, invariant local constant at the present epoch ($z = 0$) used in all atomic, laboratory, and solar-system equations (preserving the constancy of $\alpha$ in Section 3.1). Over cosmological lookback paths ($z > 0$), normal-fluid Stokes–Kirchhoff damping along the line of sight attenuates propagating photon frequency $\omega(r)$ without altering local present-day $c_0$, as formulated in Section 5.1.

2.2 Proposition 1: Constitutive Closure Relations Unifying the Operational Roles of $\beta_{\text{TRT}}$

In TRT, the dimensionless constant $\beta_{\text{TRT}} = 0.05392055$ is defined fundamentally as a single physical state ratio: the equilibrium normal-fluid mass fraction of the Plenum, \begin{equation} \chi_n \equiv \frac{\rho_{n,0}}{\rho_0} \equiv \beta_{\text{TRT}} = 0.05392055 \quad [\text{dimensionless: } 1] \tag{3} \end{equation} Proposition 1 (Two-Fluid Constitutive Closures): Under the standard two-fluid closure hypotheses of (i) unit relaxation quality factor $Q_n(\omega_0\tau_0 = 1) = 1$ at the angular relaxation resonance $\tau_0 \equiv \omega_0^{-1}$, (ii) centrifugal phase expulsion of the normal fluid into a peripheral annular vortex sheath, and (iii) Hall–Vinen cubic mutual friction at high strain rates $\tau_0|\boldsymbol{\omega}| \gtrsim 1$, the equilibrium normal-fluid fraction $\chi_n = \beta_{\text{TRT}}$ in Eq. (3) determines three secondary transport coefficients:

1. Per-Radian Acoustic Energy & Single-Quantum Frequency Attenuation Rate ($\Gamma_{\text{rad}} = \beta_{\text{TRT}}$): Consider a longitudinal wave mode $\mathbf{u}(\phi) = \mathbf{u}_0 e^{i\phi}$ ($\phi = k r - \omega t$) coupled to the Plenum's angular relaxation resonance $\omega_0\tau_0 = 1$. In the two-fluid mixture (1), the superfluid fraction $\rho_{s,0}/\rho_0 = 1 - \chi_n$ oscillates without viscous loss, whereas the normal-fluid fraction $\chi_n = \rho_{n,0}/\rho_0$ undergoes Debye/Stokes–Kirchhoff relaxation damping with unit loss tangent $Q_n^{-1}(\omega_0\tau_0=1) = 1$. Thus the fraction of classical wave energy $E = \frac{1}{2}\rho_0 |\mathbf{u}|^2$ dissipated per radian of phase advance ($d\phi = 1\text{ rad}$) is: $\frac{1}{E}\frac{dE}{d\phi} = -\frac{\rho_{n,0}}{\rho_0} Q_n^{-1}\big|_{\omega_0\tau_0=1} = -\chi_n = -\beta_{\text{TRT}}$ For a single coherent electromagnetic/acoustic wave packet (photon/phonon) of energy $E_\gamma = \hbar\omega$ propagating along a forward number-conserving ray ($N_\gamma = E / (\hbar\omega) = 1$, $\hbar = \text{const}$), adiabatic action transfer to the normal-fluid background satisfies $\frac{dE_\gamma}{E_\gamma} = \frac{d(\hbar\omega)}{\hbar\omega} = \frac{d\omega}{\omega}$, yielding the exact per-radian frequency attenuation law used in Section 5.1: $\frac{1}{\omega}\frac{d\omega}{d\phi} = \frac{1}{E_\gamma}\frac{dE_\gamma}{d\phi} = -\beta_{\text{TRT}} \quad \left[\text{rad}^{-1}\right]$
2. Annular Vortex Boundary-Layer Thickness Ratio ($\delta^*/R$ — Exact vs. Leading-Order Linearization): In a cylindrical or toroidal vortex tube of outer cross-sectional radius $R$, centrifugal pressure expels the normal-fluid fraction $\chi_n = \beta_{\text{TRT}}$ into a peripheral annular shear sheath of radial thickness $\delta^*$ surrounding the inviscid superfluid core of radius $R - \delta^*$. Equating the annular cross-sectional area fraction $A_{\text{sheath}}/A_{\text{total}}$ to $\chi_n = \beta_{\text{TRT}}$ yields the quadratic equation: $\frac{A_{\text{sheath}}}{A_{\text{total}}} = \frac{\pi R^2 - \pi (R - \delta^*)^2}{\pi R^2} = 2\left(\frac{\delta^*}{R}\right) - \left(\frac{\delta^*}{R}\right)^2 = \chi_n = \beta_{\text{TRT}}$ Solving this quadratic exactly and comparing with its leading-order thin-sheath linearization ($\delta^*/R \ll 1$) gives: $\left(\frac{\delta^*}{R}\right)_{\text{exact}} = 1 - \sqrt{1 - \beta_{\text{TRT}}} = 0.0273339, \qquad \left(\frac{\delta^*}{R}\right)_{\text{lin}} = \frac{1}{2}\beta_{\text{TRT}} = 0.0269603$ The quadratic truncation residual is $\left(\frac{\delta^*}{R}\right)_{\text{exact}} - \frac{1}{2}\beta_{\text{TRT}} = 3.74 \times 10^{-4}$ ($\mathcal{O}(\beta_{\text{TRT}}^2)$), confirming that the linear thin-sheath closure $\delta^*/R \approx \frac{1}{2}\beta_{\text{TRT}}$ is accurate to $1.37\%$ relative ($0.037\%$ absolute).
3. Cubic Vorticity Mutual-Friction Coefficient ($\beta_{\text{TRT}}\tau_0$): Under the Hall–Vinen mutual-friction closure for a two-fluid vortex core of vorticity $\boldsymbol{\omega}$ $[\text{s}^{-1}]$, coupling between the normal-fluid fraction $\chi_n = \beta_{\text{TRT}}$ and quantized vortex lines transfers rotational energy into longitudinal Kelvin/acoustic modes at a nonlinear rate proportional to $\chi_n \tau_0^{-1} (\tau_0|\boldsymbol{\omega}|)^2 \boldsymbol{\omega} = \beta_{\text{TRT}}\tau_0 |\boldsymbol{\omega}|^2\boldsymbol{\omega}$ $[\text{s}^{-2}]$. Applying the solenoidal Leray–Helmholtz projector $\mathbb{P}$ yields the curl-compatible damping term $-\beta_{\text{TRT}}\tau_0 \mathbb{P}(|\boldsymbol{\omega}|^2\boldsymbol{\omega})$ analyzed in Section 6.1. $\blacksquare$

2.3 Summary of Declared Constitutive Postulates & Discrete Structural Indices

To ensure complete methodological transparency and avoid conflating continuous parameter fitting with discrete topological/harmonic indexing, Table 1 lists all six foundational postulates and discrete structural indices used in this framework.

Table 1: Declared Constitutive Postulates and Discrete Structural Indices of the TRT Framework

3. Dimensionless Coupling Constants & Characteristic Plenum Scales

Physical Picture: In engineering and fluid mechanics, dimensionless numbers (like the Reynolds or Mach number) describe how a fluid behaves regardless of scale, whereas dimensional scales (meters, seconds, kilograms) set the physical size of the system. In TRT, the fine-structure constant $\alpha^{-1} \approx 137.036$ measures how many electromagnetic wave cycles the Plenum sustains per natural logarithmic attenuation factor $e \approx 2.718$, giving the dimensionless Acoustic Susceptibility $A_{\text{RT}} = \alpha^{-1}/e = 50.412727$. Once we anchor a single macroscopic carrier frequency $f_0 = 76.4496\text{ MHz}$ (corresponding to a fundamental standing wavelength of $L_0 = 3.92144\text{ meters}$), every dimensional transport property of the vacuum—its cycle period $T_0$, its angular relaxation time $\tau_0 = \omega_0^{-1}$, its kinematic viscosity $\nu_0$, and the minimum energy $E_{\text{cav}}$ required to nucleate a cavitated vortex core—is fixed with exact SI units.

We maintain a strict separation between dimensionless coupling ratios (Section 3.1) and dimensional physical scales carrying SI units of length $[\text{L}]$, time $[\text{T}]$, mass $[\text{M}]$, and kinematic viscosity $[\text{L}^2\text{T}^{-1}]$ (Section 3.2).

3.1 The Two Strictly Dimensionless Coupling Invariants ($A_{\text{RT}}$ and $\beta_{\text{TRT}}$)

Using the CODATA 2022 inverse fine-structure constant $\alpha^{-1} = 4\pi \varepsilon_0 \hbar c_0 / e_c^2 = 137.035999177$ and Euler's number $e = 2.71828182846$, Postulate P2 defines the Universal Acoustic Susceptibility ($A_{\text{RT}}$) as the dimensionless ratio of electromagnetic coupling capacity to single-neper logarithmic attenuation: \begin{equation} A_{\text{RT}} \equiv \frac{\alpha^{-1}}{e} = \frac{137.035999177}{2.71828182846} = 50.412727 \quad [\text{dimensionless: } 1] \tag{4} \end{equation} Its reciprocal damped ratio defines the Acoustic Venting Coefficient ($\beta_{\text{TRT}}$): \begin{equation} \beta_{\text{TRT}} \equiv \frac{e}{A_{\text{RT}}} = \frac{e^2}{\alpha^{-1}} = 0.05392055 \quad [\text{dimensionless: } 1] \tag{5} \end{equation} Both $A_{\text{RT}}$ and $\beta_{\text{TRT}}$ are strictly dimensionless pure numbers everywhere in this work (never carrying degree, radian, or dimensional units).

3.2 Characteristic Dimensional Scales of the Plenum

Heuristic binary-octave scaling down from the cyclic Planck frequency $f_P \equiv \frac{\omega_P}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{c_0^5}{\hbar G}} = 2.95209 \times 10^{42}\text{ Hz}$ across $N_0 = 114.85$ octaves plus the hexagonal close-packed ($\theta_{\text{HCP}} = \pi/6 = 30^\circ$) normal-fluid mass-loading shift $\Delta N_{\text{visc}} = \beta_{\text{TRT}} \cos(\pi/6) = 0.046697\text{ octaves}$ gives: \begin{equation} f_{\text{oct}} \equiv \frac{f_P}{2^{114.85 + \beta_{\text{TRT}}\cos(\pi/6)}} = \frac{2.95209 \times 10^{42}\text{ Hz}}{2^{114.8967}} = 7.64239 \times 10^7\text{ Hz}, \qquad f_0 = 7.64496 \times 10^7\text{ Hz} \quad [\text{T}^{-1}] \tag{6} \end{equation} (where $N_0 = 114.85 = 117.5 - \log_2(2\pi)$ formulates the octave chain directly in cyclic frequency units without any implicit $2\pi$ conversion factor). Because $\omega_P \propto G^{-1/2}$, the $2.2 \times 10^{-5}$ experimental uncertainty in $G$ propagates to $\delta\omega_P / \omega_P = \frac{1}{2}(\delta G / G) = 1.1 \times 10^{-5}$, whereas the $3.36 \times 10^{-4}$ ($0.034\%$) offset between $f_{\text{oct}}$ and $f_0$ corresponds to second-order normal-fluid dispersion ($\frac{1}{8}\beta_{\text{TRT}}^2 = 3.63 \times 10^{-4}$); accordingly, $f_0 = 7.64496 \times 10^7\text{ Hz}$ is treated explicitly as a declared empirical dimensional anchor (Postulate P4). From $f_0 = 7.64496 \times 10^7\text{ Hz}$, $c_0 = 2.99792458 \times 10^8\text{ m s}^{-1}$, and Planck's constant $h = 6.62607015 \times 10^{-34}\text{ J s}$, the characteristic dimensional scales of the Quantum Plenum follow directly from two-fluid kinetic transport theory:

1. Fundamental Carrier Wavelength ($L_0$): The wavelength of a longitudinal carrier wave propagating at speed $c_0$ with frequency $f_0$ is: \begin{equation} L_0 \equiv \frac{c_0}{f_0} = \frac{2.99792458 \times 10^8\text{ m s}^{-1}}{7.64496 \times 10^7\text{ s}^{-1}} = 3.92144\text{ m} \quad [\text{L}] \tag{7} \end{equation}
2. Fundamental Carrier Period ($T_0$) & Characteristic Angular Relaxation Time ($\tau_0$): The single-cycle period of the carrier wave is: \begin{equation} T_0 \equiv \frac{1}{f_0} = \frac{L_0}{c_0} = 1.30805 \times 10^{-8}\text{ s} \quad [\text{T}] \tag{8} \end{equation} The characteristic angular relaxation timescale associated with carrier angular frequency $\omega_0 \equiv 2\pi f_0 = 4.80347 \times 10^8\text{ rad s}^{-1}$ is: \begin{equation} \tau_0 \equiv \frac{1}{\omega_0} = \frac{1}{2\pi f_0} = \frac{T_0}{2\pi} = 2.08183 \times 10^{-9}\text{ s} \quad [\text{T}] \tag{9} \end{equation} guaranteeing that at the carrier angular frequency, the Debye relaxation resonance condition $\omega_0 \tau_0 \equiv 1$ is satisfied identically.
3. Plenum Kinematic Viscosity ($\nu_0$): By Chapman–Enskog and Landau–Khalatnikov kinetic transport theory, a compressible fluid with sound speed $c_0$ and fundamental wave-packet mean free path $L_0$ possesses bare acoustic diffusivity $D_0 = c_0 L_0$. Dividing $D_0$ by the dimensionless acoustic susceptibility quality factor $A_{\text{RT}}$ defines the equilibrium kinematic viscosity of the Plenum: \begin{equation} \nu_0 \equiv \frac{D_0}{A_{\text{RT}}} = \frac{c_0 , L_0}{A_{\text{RT}}} = \frac{(2.99792458 \times 10^8\text{ m s}^{-1})(3.92144\text{ m})}{50.412727} = 2.33199 \times 10^7\text{ m}^2\text{ s}^{-1} \quad [\text{L}^2\text{T}^{-1}] \tag{10} \end{equation}
4. Hydrodynamic Cavitation Energy Threshold ($E_{\text{cav}}$) and Mass Floor ($\mu_0$): A single carrier phonon carries quantum energy $E_1 = h f_0$. Nucleating a persistent cavitated vortex core against the Plenum's acoustic susceptibility barrier $A_{\text{RT}}$ requires the coherent superposition of $A_{\text{RT}}$ carrier quanta within a single wavelength volume, defining the minimum cavitation energy $E_{\text{cav}}$ and its Einstein–de Broglie mass floor $\mu_0 = E_{\text{cav}}/c_0^2$: \begin{equation} E_{\text{cav}} \equiv A_{\text{RT}} h f_0 = 2.55373 \times 10^{-24}\text{ J}, \qquad \mu_0 \equiv \frac{E_{\text{cav}}}{c_0^2} = \frac{A_{\text{RT}} h f_0}{c_0^2} = 2.84140 \times 10^{-41}\text{ kg} \tag{11} \end{equation}

Table 2: Characteristic Scale System of the Quantum Plenum (Exact Arithmetic Verification)

4. Topological Soliton Particles & Constitutive Rest-Energy Ratio Formula

Physical Picture: Why is a proton $1836.15267$ times heavier than an electron, and why do atomic nuclei destabilize when too many nucleons crowd into a shell? Picture an electron as a single, unknotted toroidal vortex ring ($0_1$) in the Plenum. A proton, by contrast, is modeled as the simplest self-locking 3D torus knot: the $(2,3)$ Trefoil knot ($3_1$), which winds twice through its poloidal core ($p=2$) and three times around its toroidal perimeter ($q=3$). Those $p \times q = 2 \times 3 = 6$ standing half-wave cells sweep across five cyclic degrees of freedom ($\Omega_{\text{cell}} = [0,\pi]^5 \subset \mathbb{T}^5$), giving a bare phase-space rest-energy ratio of $6\pi^5 = 1836.118109$ relative to the unknotted unit electron core. Because the knotted vortex core also couples to the $5.392\%$ normal-fluid boundary sheath across its meridian dipole profile ($\frac{2}{\pi}\beta_{\text{TRT}} = 0.034327$), its total rest-energy ratio rises to $1836.152436$—matching the experimental proton-to-electron mass ratio to $0.13$ parts per million. Furthermore, when nucleons pack together like spheres around a central core (as in Carbon-12, where 12 spheres surround 1), their normal-fluid boundary skins reduce the empty interstitial space from $25.95\%$ to $19.80\% \approx 20\%$. Twelve spheres fit around a central core with $>60^\circ$ ($>1\text{ radian}$) between centers, whereas forcing a $13\text{th}$ sphere onto the inner shell compresses neighbors closer than 1 radian ($57.2958^\circ$), causing adjacent acoustic wakes to interfere constructively and trigger nuclear fission.

4.1 Topological Duality & Relative Soliton Winding Functional on $\mathbb{T}^5$

Under Postulate P3 (Table 1), elementary fermions are modeled as phase-locked toroidal vortex solitons whose internal configuration space is parameterized by five normalized cyclic coordinates $\boldsymbol{\xi} = (\xi_1, \xi_2, \xi_3, \xi_4, \xi_5) \in [0, 1]^5$ representing normalized cycle fractions of: (1) toroidal azimuth, (2) poloidal twist, (3) radial core acoustic standing-wave phase, (4) superfluid condensate phase, and (5) normal-fluid shear phase:

• The Electron (Unknotted Unit Reference State $0_1$, $(p_e, q_e) = (1, 1)$): The electron is the fundamental unknotted point-like vortex state whose core configuration cell is the unit hypercube $\Omega_{\text{core}} \equiv [0, 1]^5$ equipped with canonical Lebesgue cycle measure $\text{Vol}_5(\Omega_{\text{core}}) \equiv \int_{[0,1]^5} d^5\xi = 1$, quantized superfluid circulation $\Gamma_e = h / m_e$, canonical reference rest energy $E_e = m_e c_0^2$, and zero knotted boundary-layer crossing ($\Delta M_{\text{visc}} = 0$). Under Postulate P3, its rest energy is identically: \begin{equation} \mathcal{H}_{\text{sol}}(1, 1) \equiv E_e \left[ \frac{\text{Vol}5(\Omega{1,1})}{\text{Vol}5(\Omega{\text{core}})} + 0 \right] = E_e [1 + 0] = E_e = m_e c_0^2 \tag{12} \end{equation}
• The Proton (Knotted Trefoil Torus Soliton $3_1$, $(p, q) = (2, 3)$): Modeled as the minimal non-trivial torus knot—the $(p, q) = (2, 3)$ Trefoil knot ($3_1$) [3, 6], which winds $p = 2$ times poloidally and $q = 3$ times toroidally (forming 3 azimuthal scattering lobes).
• Branched Covering Degree and Phase Metric Jacobian ($\mathcal{J}_5 = \pi^5$): In algebraic knot topology [3, 6], a closed $(p, q)$ torus knot on the 2-torus cannot be deformed into a single unknotted loop without self-intersection; its configuration space defines a $d$-sheeted branched covering torus $\widetilde{\mathbb{T}}^5 \to \mathbb{T}^5$ of covering degree $d = p \cdot q$: \begin{equation} d_{\text{cover}}(2, 3) = N_{\text{sheets}} = p \cdot q = 2 \times 3 = 6 \tag{13} \end{equation} Across each sheet, the physical standing waves are parameterized in half-wave phase radians $\vartheta_k = \pi \xi_k \in [0, \pi]$ ($k = 1, \dots, 5$). The coordinate transformation from normalized cycle coordinates $\boldsymbol{\xi} \in [0, 1]^5$ to physical radian phase coordinates $\boldsymbol{\vartheta} \in [0, \pi]^5$ introduces the exact metric volume element Jacobian: \begin{equation} \mathcal{J}5 \equiv \det\left(\frac{\partial\boldsymbol{\vartheta}}{\partial\boldsymbol{\xi}}\right) = \prod{k=1}^5 \left(\frac{d\vartheta_k}{d\xi_k}\right) = \pi^5 = 306.019685 \tag{13a} \end{equation} Thus, each fundamental physical sheet of the proton spans metric phase volume $\text{Vol}_5(\Omega_{\text{sheet}}) = \mathcal{J}_5 \int_{[0,1]^5} d^5\xi = \pi^5$ relative to the unit cycle measure of the electron reference state.

4.2 Commensurable Rest-Energy Ratios for the Proton-to-Electron Mass Ratio ($\mu_{\text{TRT}} = m_p / m_e$)

By Einstein–de Broglie relativistic hydrodynamics ($m = E_{\text{rest}} / c_0^2$), the dimensionless proton-to-electron mass ratio $\mu_{\text{TRT}} \equiv m_p / m_e$ is defined strictly as the ratio of soliton rest energies evaluated by the constitutive winding functional $\mathcal{H}_{\text{sol}}$ relative to the electron reference state $\mathcal{H}_{\text{sol}}(1, 1) = E_e$: \begin{equation} \mu_{\text{TRT}} \equiv \frac{\mathcal{H}{\text{sol}}(2, 3)}{\mathcal{H}{\text{sol}}(1, 1)} = \frac{E_{\text{geom}}(2, 3)}{E_e} + \frac{E_{\text{visc}}(\beta_{\text{TRT}})}{E_e} = M_{\text{geom}}(2, 3) + \Delta M_{\text{visc}}(\beta_{\text{TRT}}) \tag{14} \end{equation}

1. Bare 5-Torus Phase-Space Rest-Energy Ratio ($M_{\text{geom}} = 6\pi^5$): Under Postulate P3, the total phase volume of the $(2, 3)$ Trefoil soliton is the canonical measure of its 6-sheeted covering manifold $\widetilde{\Omega}_{2,3} = \bigsqcup_{j=1}^6 \Omega_{\text{sheet}, j}$, where each sheet sweeps the physical half-wave hypercube $\Omega_{\text{sheet}} = [0, \pi]^5$ with metric volume $\mathcal{J}_5 = \pi^5$. Evaluating the relative volume against the unknotted electron reference cell $\Omega_{\text{core}} = [0, 1]^5$ (with unit measure $\text{Vol}_5(\Omega_{\text{core}}) \equiv \int_{[0,1]^5} d^5\xi = 1$) yields the exact dimensionless geometric ratio: \begin{equation} M_{\text{geom}}(2, 3) \equiv \frac{\text{Vol}5(\widetilde{\Omega}{2,3})}{\text{Vol}5(\Omega{\text{core}})} = \frac{\sum_{j=1}^{d_{\text{cover}}} \mathcal{J}5 \int{[0,1]^5} d^5\xi}{\int_{[0,1]^5} d^5\xi} = d_{\text{cover}} \cdot \mathcal{J}_5 = 6\pi^5 = 1836.118109 \tag{15} \end{equation} (Consistency with Electron Reference State: For the unknotted electron $(p_e, q_e) = (1, 1)$, the covering degree is $d_{\text{cover}}(1, 1) = 1$, and its point-vortex core is defined directly in the base unit cycle space without internal spatial half-wave extension ($\mathcal{J} = 1$), identically satisfying $\mathcal{H}_{\text{sol}}(1, 1) = E_e [1/1 + 0] = E_e$.)

2. Normal-Fluid Meridian Boundary-Layer Rest-Energy Ratio ($\Delta M_{\text{visc}} = \frac{2}{\pi}\beta_{\text{TRT}}$): In an inviscid fluid ($\chi_n = \beta_{\text{TRT}} = 0$), Eq. (15) gives $6\pi^5 = 1836.118109$, leaving a residual of $0.034564$ below experiment. To evaluate the normal-fluid boundary-layer rest energy $E_{\text{visc}}(\beta_{\text{TRT}})$ in units directly commensurable with $E_e$, consider the meridian polar angle $\theta \in [0, \pi]$ of the toroidal vortex core. Relative to the unattenuated core meridian energy density $e_0 \equiv E_e / \pi$ $[\text{J rad}^{-1}]$ (normalized such that $\int_0^\pi e_0\,d\theta = E_e$), the peripheral normal-fluid mass fraction $\chi_n = \rho_{n,0}/\rho_0 = \beta_{\text{TRT}}$ couples under Postulate P3 to the fundamental poloidal half-wave dipole mode $\sin\theta$ across $\theta \in [0, \pi]$, storing differential boundary-layer rest energy $dE_{\text{visc}}(\theta) = \chi_n e_0 \sin\theta\,d\theta$. Integrating $dE_{\text{visc}}(\theta)$ over $\theta \in [0, \pi]$ and dividing by the reference rest energy $E_e = \int_0^\pi e_0\,d\theta$ yields the exact dimensionless rest-energy increment: \begin{equation} \Delta M_{\text{visc}}(\beta_{\text{TRT}}) \equiv \frac{E_{\text{visc}}(\beta_{\text{TRT}})}{E_e} = \frac{\int_0^\pi \chi_n e_0 \sin\theta,d\theta}{\int_0^\pi e_0 , d\theta} = \frac{\beta_{\text{TRT}}}{\pi}\int_0^\pi \sin\theta,d\theta = \frac{2}{\pi}\beta_{\text{TRT}} = \frac{2}{\pi}(0.05392055) = 0.0343269 \tag{16} \end{equation}

Summing the two dimensionless rest-energy ratios (15) and (16) yields the constitutive TRT mass-ratio formula: \begin{equation} \mu_{\text{TRT}} \equiv \frac{\mathcal{H}{\text{sol}}(2, 3)}{\mathcal{H}{\text{sol}}(1, 1)} = 6\pi^5 + \frac{2}{\pi}\beta_{\text{TRT}} = 1836.118109 + 0.034327 = 1836.152436 \tag{17} \end{equation} Precision Calibration vs. CODATA 2022: Comparing $\mu_{\text{TRT}} = 1836.152436$ to the CODATA 2022 experimental value $\mu_{\text{exp}} = 1836.15267343(11)$ gives an absolute residual of $|\mu_{\text{TRT}} - \mu_{\text{exp}}| = 2.37 \times 10^{-4}$, or a relative approximation accuracy of $0.13\text{ ppm}$ ($99.999987\%$). Because this $0.13\text{ ppm}$ residual exceeds the $6 \times 10^{-11}$ CODATA experimental uncertainty, Eq. (17) is classified in Table 4 as a leading-order constitutive soliton benchmark (prior to $\mathcal{O}(\beta_{\text{TRT}}^3)$ toroidal aspect-ratio curvature corrections) rather than an exact identity.

4.3 Secondary Bjerknes Nuclear Binding, Void Fraction ($\epsilon_{\text{eff}}$) & The 1-Radian ($57.2958^\circ$) Limit

In a compressible fluid of density $\rho_0$, two spherical solitons pulsating in phase at angular frequency $\omega_0$ with volume amplitudes $\dot{V}_1, \dot{V}_2$ experience an attractive secondary Bjerknes acoustic radiation force: \begin{equation} \mathbf{F}_{\text{Bj}}(r) = -\frac{\rho_0 \langle \dot{V}1 \dot{V}2 \rangle}{4\pi r^2} \exp!\left(-\frac{r}{\lambda{\text{screen}}}\right)\hat{\mathbf{r}}, \qquad \lambda{\text{screen}} \equiv \frac{c_0}{\omega_0} = \frac{L_0}{2\pi} = 0.62412\text{ m} \tag{18} \end{equation} At contact, 12 equal spheres packed around a central sphere of radius $R$ form the Carbon-12 12-around-1 close-packed cluster:

1. Effective Interstitial Void Fraction ($\epsilon_{\text{eff}}$ — Both Linearized and Exact Sheath Roots): By Kepler–Hales close-packing, bare rigid spheres of radius $R$ have maximum FCC/HCP packing density $\phi_{\text{FCC}} = \frac{\pi}{3\sqrt{2}} = 0.740480$ (bare void fraction $1 - \phi_{\text{FCC}} = 0.259520$). Including the normal-fluid boundary-layer sheath derived in Proposition 1 expands the effective hydrodynamic exclusion radius to $R_{\text{eff}} = R(1 + \delta^*/R)$, yielding the Effective Interstitial Void Fraction for both the leading-order linear closure ($\delta^*/R = \frac{1}{2}\beta_{\text{TRT}} = 0.026960$) and the exact quadratic root ($\delta^*/R = 1 - \sqrt{1-\beta_{\text{TRT}}} = 0.027334$): \begin{equation} \epsilon_{\text{eff}}^{\text{(lin)}} = 1 - \phi_{\text{FCC}}\left(1 + \frac{1}{2}\beta_{\text{TRT}}\right)^3 = 0.19800 \approx 0.200, \qquad \epsilon_{\text{eff}}^{\text{(exact)}} = 1 - \phi_{\text{FCC}}\left(2 - \sqrt{1 - \beta_{\text{TRT}}}\right)^3 = 0.19712 \approx 0.200 \tag{19} \end{equation} Both roots agree to within $8.8 \times 10^{-4}$ ($0.4\%$ relative), establishing an effective interstitial void fraction of $\approx 20\%$.
2. Spatial Hard-Core Euclidean Chord Distance ($d_{\text{chord}}$) vs. Geodesic Surface Wave Path ($s(\theta)$): On a spherical coordination shell of radius $R_{\text{coord}} = 2R$, two distinct geometric separation measures govern physical stability:
   • Spatial Hard-Core Euclidean Chord Distance ($d_{\text{chord}}$): The straight-line Euclidean distance through 3D space between two sphere centers separated by angle $\theta$ is $d_{\text{chord}}(\theta) = 2 R_{\text{coord}}\sin(\theta/2) = 4R\sin(\theta/2)$. Hard-sphere non-penetration requires $d_{\text{chord}}(\theta) \ge 2R \iff \sin(\theta/2) \ge 1/2 \iff \theta \ge \pi/3 = 60.0000^\circ$, which is satisfied exactly by standard close-packed FCC kissing configurations.
   • Geodesic Surface Acoustic Boundary-Wave Path ($s(\theta)$ & The 1-Radian Limit): Along the coordination shell surface, the acoustic boundary-layer shear wave propagates along the geodesic arc $s(\theta) = R_{\text{coord}}\,\theta = 2R\,\theta$. By the Helmholtz–Rayleigh acoustic interference criterion, independent secondary Bjerknes pressure wells require the geodesic boundary wave path to exceed the acoustic exclusion diameter: $s(\theta) \ge d_{\text{ex}} \iff 2R\,\theta \ge 2R \iff \theta \ge 1\text{ radian} = \frac{180^\circ}{\pi} = 57.2958^\circ$ Because the scale factor $2R$ cancels identically, $\theta \ge 1\text{ rad}$ is an exact geometric boundary condition for acoustic surface sheath independence. For $N = 12$ outer spheres, the Tammes icosahedral maximum $\theta_{\text{Tammes}}(12) = 63.4349^\circ$ and FCC angle $60.0000^\circ$ both safely satisfy both the chord constraint ($60^\circ$) and the surface wave criterion ($57.30^\circ$). Forcing a $13\text{th}$ sphere compresses the maximum angular separation to $\theta_{\text{Tammes}}(13) = 57.1367^\circ < 1\text{ rad}$ ($57.2958^\circ$), simultaneously violating both criteria and triggering acoustic shear cavitation fission.

4.4 Bernoulli Derivation of the Hydrodynamic Cavitation Mass Floor

To derive the cavitation mass-floor step function from the Euler–Bernoulli pressure law, consider a localized rotational wave packet of kinetic energy $E_k$ $[\text{J}]$ confined within a fundamental wavelength core volume $V_{\text{core}}$ in the Plenum of ambient hydrostatic pressure $P_0 \equiv E_{\text{cav}} / V_{\text{core}}$. By the steady compressible Bernoulli theorem, the rotational kinetic energy density induces a core pressure drop $\Delta P_{\text{core}} = P_0 - P_{\text{core}} = E_k / V_{\text{core}}$. A hollow phase-locked vortex core nucleates (cavitates) if and only if the core absolute pressure drops to the vacuum vapor floor ($P_{\text{core}} \le 0 \iff \Delta P_{\text{core}} \ge P_0$), which yields the exact Cavitation Nucleation Condition $E_k \ge P_0 V_{\text{core}} = E_{\text{cav}} = A_{\text{RT}} h f_0$:

• For $E_k < E_{\text{cav}}$ ($P_{\text{core}} > 0$), the perturbation cannot nucleate a cavitated boundary wall and disperses as massless longitudinal acoustic radiation ($m = 0$).
• For $E_k \ge E_{\text{cav}}$ ($P_{\text{core}} \le 0$), cavitation nucleates a stable topological vortex boundary wall whose minimum nucleation threshold rest energy $E_{\text{min}} = E_{\text{cav}}$ defines the invariant core mass floor $\mu_0 = E_{\text{cav}}/c_0^2 = 2.84140 \times 10^{-41}\text{ kg}$:

\begin{equation} \mu_0 \equiv \frac{E_{\text{cav}}}{c_0^2} = \frac{A_{\text{RT}} h f_0}{c_0^2} = 2.84140 \times 10^{-41}\text{ kg} \tag{20} \end{equation} For an arbitrary super-threshold wave packet of kinetic energy $E_k \ge E_{\text{cav}}$, the total mass is $m(E_k) = \frac{E_k}{c_0^2}\,\Theta(E_k/E_{\text{cav}} - 1) \ge \mu_0$, while sub-threshold excitations ($E_k < E_{\text{cav}}$) fail to cavitate and disperse as massless acoustic radiation ($m = 0$). Every factor is dimensionally homogeneous ($[E_k / (A_{\text{RT}} h f_0)] = \text{J}/\text{J} = 1$ and $[\mu_0] = \text{kg}$). (Scope Qualification: Eq. (20) formulates a classical hydrodynamic cavitation mass-floor mechanism in the two-fluid Plenum rather than a constructive quantum field theory proof for pure $SU(N)$ Yang–Mills gauge fields on $\mathbb{R}^4$.)