Microphysical Extensions: Topology, Spectrum, Confinement, and Scale Continuity (Complete Derivations and Step-by-Step Analysis)

1. Topological Classification and Particle Generation

Unified Einstein Equation (QIR)

For reference, we record the canonical QIR Einstein equation:

$$G_{\mu \nu }={\displaystyle \frac{8\pi G}{c{\left(x\right)}^4}}Z\left(\widehat{p}\right)\left(T_{\mu \nu }+I_{\mu \nu }\right)$$

(1.1)

This labeled instance is used throughout the manuscript via (1.1).

1.1. Motivation

In QIR, matter does not preexist as an independent substance; particles arise as stationary or topological configurations of the informational field $\widehat{p}\left(x\right)$. Gauge bosons correspond to collective phase modes of this field, and internal symmetries emerge dynamically from its local informational structure. The fundamental scalar action reads

$$\mathcal{S}[p;g]={\int }^{4}x\sqrt{-g}\left[\frac{1}{2}Z\left(p\right)g^{\mu \nu }{\partial }_{\mu }p{\partial }_{\nu }p-V\left(p\right)\right],$$

(1.2)

coupled to gravity through the unified Einstein equation $G_{\mu \nu }={\displaystyle \frac{8\pi G}{c{\left(x\right)}^4}}Z\left(\widehat{p}\right)\left(T_{\mu \nu }+I_{\mu \nu }\right)$.

1.2. Phase Decomposition and Target Map

Writing $p\left(x\right)=\rho \left(x\right)e^{i\theta \left(x\right)}$, and compactifying spatial infinity into $S^3$, we define a normalized map

$$\Psi \left(x\right)\equiv {\displaystyle \frac{\widehat{p}\left(x\right)}{\sqrt{\langle {\widehat{p}}^{†}\widehat{p}\rangle }}}:S^3\to {\mathcal{M}}_{\mathrm{target}},$$

(1.3)

where ${\mathcal{M}}_{\mathrm{target}}$ is the manifold of internal informational states (e.g. $S^2$ for an SU(2)-like sector or $SU\left(3\right)/U{\left(1\right)}^2$ for color degrees of freedom). This compactification allows the definition of homotopy classes of configurations.

1.3. Phase Current and Winding Number

The local $U\left(1\right)$ phase current is defined as

$$J_i\left(x\right)=i{\Psi }^{†}{\partial }_i\Psi ={\partial }_i\theta \left(x\right),F_{ij}={\partial }_i{J}_j-{\partial }_j{J}_i.$$

(1.4)

The associated topological invariant (Hopf index) is

$$Q_{\mathrm{top}}={\displaystyle \frac{1}{16{\pi }^2}}{\int }_{{\mathbb{R}}^3}^{3}x{ϵ}^{ijk}{J}_i{F}_{jk},$$

(1.5)

which classifies field configurations into discrete topological sectors.

1.4. Informational Spin and Vorticity

For a multi-component decomposition $p_i={\rho }_i{e}^{i{\theta }_i}$, one defines an informational angular momentum density

$$\mathit{S}\left(x\right)=\sum _{i=1}^n{\rho }_i^2\left(x\right)\nabla {\theta }_i\left(x\right),$$

(1.6)

whose integral gives the effective spin. Quantized vorticity of ${\theta }_i$ naturally produces half-integer or integer spin states depending on the winding number.

1.5. Charge as Local Phase Generator

For a local phase transformation $\widehat{p}\mapsto e^{iq\alpha \left(x\right)}\widehat{p}$, the covariant derivative

$${\mathcal{D}}_{\mu }\widehat{p}={\partial }_{\mu }\widehat{p}-iq{A}_{\mu }\widehat{p},A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\alpha ,$$

(1.7)

ensures invariance. The charge q acts as the generator of local phase rotation. In the non-Abelian generalization (SU(2), SU(3)), $q{T}^a\to g{T}^a$, and $A_{\mu }\to A_{\mu }^a{T}^a$ with curvature $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }-ig[A_{\mu },A_{\nu }]$.

1.6. Topological Classes and Particle Families

We propose the following correspondence:

$$Q_{\mathrm{top}}=1\leftrightarrow \mathrm{Leptons},Q_{\mathrm{top}}=3\leftrightarrow \mathrm{Quarks}(\mathrm{color}\mathrm{triplet}),Q_{\mathrm{top}}=0\leftrightarrow \mathrm{Bosons}(\mathrm{collective}\mathrm{modes}).$$

(1.8)

Thus, fermions and bosons emerge as different topological sectors of $\widehat{p}\left(x\right)$, while internal SU(2) or SU(3) structures refine the classification through the geometry of ${\mathcal{M}}_{\mathrm{target}}$.

1.7. Energy Functional and Soliton Existence

The static energy reads

$$E\left[p\right]={\int }^{3}x\left[\frac{1}{2}Z\left(p\right){(\nabla p)}^2+V\left(p\right)\right],$$

(1.9)

subject to a conserved charge $Q={\int }^{3}x{\left|p\right|}^2$. The saturation property of $Z\left(p\right)$ at high density prevents collapse and allows localized, stable stationary solutions $p(x,t)=\varphi \left(r\right)e^{-i\omega t}$.

Proposition 1  

(Existence of informational solitons). If $Z\left(p\right)>0$ decreases to zero for large $\left|p\right|$ and $V\left(p\right)$ is bounded below with a nontrivial minimum, then the action (1.2) admits stationary finite-energy solutions of the form $p(x,t)=\varphi \left(r\right)e^{-i\omega t}$ minimizing (1.9) under fixed Q.

Sketch. 

Scaling $r\to \lambda r$ and studying $E\left[{\varphi }_{\lambda }\right]$ shows that the saturating factor $Z\left(p\right)$ balances gradient energy at large amplitudes. A fixed Q constraint sets $\omega$, preventing collapse. Compactness arguments ensure a minimizer in $H^1\left({\mathbb{R}}^3\right)$.    □

1.8. Spin–Vorticity and Charge–Curvature Links

With internal structure $\widehat{p}\left(x\right)=\rho \left(x\right)U\left(x\right)\xi$ and $U\left(x\right)\in SU\left(N\right)$, one defines the pure-gauge connection $A_{\mu }=i{U}^{-1}{\partial }_{\mu }U$. Then

$$\begin{array}{cc}\hfill S^k\left(x\right)& ={\rho }^2{\xi }^{†}{U}^{-1}{\Sigma }^{k}U\xi ,\hfill \end{array}$$

(1.10)

$$\begin{array}{cc}\hfill j_{\left(U\right(1\left)\right)}^{\mu }& =\Im \left({\widehat{p}}^{†}{\mathcal{D}}^{\mu }\widehat{p}\right)={\rho }^2({\partial }^{\mu }\theta -q{A}^{\mu }),\hfill \end{array}$$

(1.11)

showing that spin and charge arise as geometric observables of the underlying field.

1.9. Homotopy Classes and Quantum Numbers

Let the normalized field map $\Psi :S^3\to {\mathcal{M}}_{\mathrm{target}}$, cf. Equation (3). Two representative target manifolds are:

• SU(2)-like sector: ${\mathcal{M}}_{\mathrm{target}}\simeq S^2$. Then ${\pi }_3\left(S^2\right)=\mathbb{Z}$ (Hopf fibration), yielding an integer topological index $Q_{\mathrm{top}}$ via Equation (1.5). Minimal $|Q_{\mathrm{top}}|=1$ configurations correspond to leptonic sectors (colorless).
• SU(3)-like color sector: ${\mathcal{M}}_{\mathrm{target}}\simeq SU\left(3\right)/U{\left(1\right)}^2$ (flag manifold). It admits nontrivial ${\pi }_3$; the embedding of three fundamental directions induces a triply–linked structure, naturally assigning $Q_{\mathrm{top}}=3$ to color–triplet states (quarks). The detailed construction proceeds by restricting $\Psi$ to the Cartan torus and counting linked preimages of regular values.

Proposition 2  

(Topological assignment, sketch). If Ψ is smooth with compact support, the Hopf invariant computed from the $U\left(1\right)$ phase connection $J_i$ (Equation (1.5)) equals the linking number of preimages of regular values of Ψ. Embedding three independent phases associated with the $U{\left(1\right)}^2$ Cartan subgroup of SU(3) yields a minimal triply–linked configuration, $Q_{\mathrm{top}}=3$, for color-triplet excitations.

This formalizes the dictionary $\{Q_{\mathrm{top}}=0,1,3\}\leftrightarrow \{$bosons, leptons, quarks}.

1.10. Summary

Elementary particles correspond to topological or stationary classes of the informational field $\widehat{p}\left(x\right)$. Spin arises from quantized vorticity; charge is the generator of local phase symmetry; and gauge groups such as SU(2) and SU(3) emerge from internal transformations of $\widehat{p}$, anchoring microphysics in informational geometry.

2. Perturbative Quantization and Mass Spectrum

2.1. Background and Motivation

Having established the topological and solitonic nature of localized configurations, we now turn to their quantum fluctuations. Perturbative quantization around a stable classical solution provides the effective mass spectrum of the elementary excitations of the informational field.

Let the classical stationary solution be

$$p_0(x,t)={\varphi }_0\left(r\right)e^{-i{\omega }_{0}t},{\varphi }_0\left(r\right)\in {\mathbb{R}}^{+},$$

(2.1)

which satisfies the Euler–Lagrange equation derived from the action (1.2). We will expand $p(x,t)$ as a background plus a small fluctuation:

$$p(x,t)=p_0(x,t)+\delta p(x,t).$$

(2.2)

2.2. Linearization of the Field Equation

The field equation obtained by varying $\mathcal{S}[p;g]$ is

$${\nabla }_{\mu }\left(Z\left(p\right){\nabla }^{\mu }p\right)-{\displaystyle \frac{1}{2}}Z{\left(p\right)}^{\prime }\left({\nabla }_{\mu }p\right)\left({\nabla }^{\mu }p\right)+V{\left(p\right)}^{\prime }=0,$$

(2.3)

where the prime denotes $\partial /\partial p$. We expand this equation to first order in $\delta p$.

The first-order expansion gives

$$\begin{array}{cc}\hfill Z\left(p\right)\square p& =Z\left(p_0\right)\square \delta p+Z{\left(p_0\right)}^{\prime }\delta p\square p_0+2Z{\left(p_0\right)}^{\prime }\left({\nabla }_{\mu }{p}_0\right){\nabla }^{\mu }\delta p+\mathcal{O}\left(\delta p^2\right),\hfill \end{array}$$

(2.4)

$$\begin{array}{cc}\hfill V{\left(p\right)}^{\prime }& =V{\left(p_0\right)}^{\prime }+V{\left(p_0\right)}^{\prime \prime }\delta p+\mathcal{O}\left(\delta p^2\right).\hfill \end{array}$$

(2.5)

Substituting into (2.3) and using the background equation for $p_0$, we obtain the linearized fluctuation equation:

$$\mathcal{O}\delta p=0,\mathcal{O}\equiv Z\left(p_0\right)\square +\left[Z{\left(p_0\right)}^{\prime }\square p_0+2Z{\left(p_0\right)}^{\prime }\left({\nabla }_{\mu }{p}_0\right){\nabla }^{\mu }-{\displaystyle \frac{1}{2}}Z{\left(p_0\right)}^{\prime \prime }{\left({\nabla }_{\mu }{p}_0\right)}^2+V{\left(p_0\right)}^{\prime \prime }\right].$$

(2.6)

This defines the self-adjoint fluctuation operator $\mathcal{O}$ acting on $\delta p$.

2.3. Mode Decomposition and Effective Mass

In a locally flat region (neglecting curvature of g for simplicity), we write the fluctuation as a mode expansion:

$$\delta p(x,t)=\sum _n{\psi }_n\left(\mathit{x}\right)e^{-i{\Omega }_{n}t}.$$

(2.7)

Substituting into (2.6), the temporal dependence yields the eigenvalue equation for spatial modes:

$$-Z\left(p_0\right){\nabla }^2{\psi }_n+U_{\mathrm{eff}}\left(r\right){\psi }_n={\Omega }_n^2{\psi }_n,$$

(2.8)

where the effective potential is

$$U_{\mathrm{eff}}\left(r\right)=V{\left(p_0\right)}^{\prime \prime }-{\displaystyle \frac{1}{2}}Z{\left(p_0\right)}^{\prime \prime }{(\nabla p_0)}^2-Z{\left(p_0\right)}^{\prime }{\nabla }^2{p}_0.$$

(2.9)

Bound states of (2.8) correspond to discrete frequencies ${\Omega }_n$, interpreted as energy levels of the quantized excitations.

The lowest (nonzero) mode determines the effective mass:

$$m_{\mathrm{eff}}={\displaystyle \frac{\hslash {\Omega }_0}{c_0^2}}\simeq {\displaystyle \frac{\hslash }{c_0^2}}\sqrt{V{\left(p_0\right)}^{\prime \prime }+{\partial }_{p_0}Z\left(p_0\right){\omega }_0^2},$$

(2.10)

where we used the stationary condition ${\partial }_t{p}_0=-i{\omega }_0{p}_0$.

2.4. Canonical Quantization

The conjugate momentum derived from (1.2) is

$$\pi (x,t)\equiv {\displaystyle \frac{\partial \mathcal{L}}{\partial \left({\partial }_{t}p\right)}}=Z\left(p\right){\partial }_{t}p.$$

(2.11)

For small fluctuations around the background, we have $\pi \simeq Z\left(p_0\right){\partial }_t\delta p$. The canonical commutation relations read

$$[\delta \widehat{p}\left(\mathit{x}\right),\widehat{\pi }\left(\mathit{y}\right)]=i\hslash {\delta }^{\left(3\right)}(\mathit{x}-\mathit{y}).$$

(2.12)

Inserting the mode expansion (2.7), we quantize each mode as a harmonic oscillator:

$$\begin{array}{cc}\hfill \delta \widehat{p}(\mathit{x},t)& =\sum _n{\displaystyle \frac{1}{\sqrt{2{\Omega }_n}}}\left(a_n{\psi }_n\left(\mathit{x}\right)e^{-i{\Omega }_{n}t}+a_n^{†}{\psi }_n^{*}\left(\mathit{x}\right)e^{i{\Omega }_{n}t}\right),\hfill \end{array}$$

(2.13)

$$\begin{array}{cc}\hfill {\widehat{a}}_n,{\widehat{a}}_m^{†}]& ={\delta }_{nm}.\hfill \end{array}$$

(2.14)

The quadratic part of the Hamiltonian becomes

$$H_{\mathrm{eff}}=\sum _n\hslash {\Omega }_n\left(a_n^{†}{a}_n+{\displaystyle \frac{1}{2}}\right),$$

(2.15)

yielding discrete excitation energies $\hslash {\Omega }_n$.

2.5. Renormalization and Field Rescaling

Because $Z\left(p\right)$ acts as a field-dependent kinetic factor, we can perform a local field redefinition to normalize the kinetic term:

$$\chi \left(p\right)={\int }^p\sqrt{Z\left(p^{\prime }\right)}{p}^{\prime }.$$

(2.16)

The Lagrangian in terms of $\chi$ becomes canonical:

$$\mathcal{L}=\frac{1}{2}{\left({\partial }_{\mu }\chi \right)}^2-V_{\mathrm{eff}}\left(\chi \right),V_{\mathrm{eff}}\left(\chi \right)=V\left(p\left(\chi \right)\right).$$

(2.17)

The effective mass is then directly

$$m_{\mathrm{eff}}^2={\displaystyle \frac{{\mathrm{d}}^2{V}_{\mathrm{eff}}}{\mathrm{d}{\chi }^2}}{|}_{{\chi }_0}={\displaystyle \frac{1}{Z\left(p_0\right)}}\left[V{\left(p_0\right)}^{\prime \prime }-{\displaystyle \frac{1}{2}}{\displaystyle \frac{Z{\left(p_0\right)}^{\prime }}{Z\left(p_0\right)}}V{\left(p_0\right)}^{\prime }\right],$$

(2.18)

which explicitly shows how the informational modulation $Z\left(p\right)$ renormalizes the mass.

2.6. Spectrum Structure and Physical Interpretation

The discrete eigenvalues ${\Omega }_n$ from (2.8) form a ladder of informational excitations. In analogy with Q-balls or Skyrmions, we identify:

• $n=0$ ground state → stable particle (mass $m_0$),
• $n=1,2,\dots$ higher excitations → excited leptons/quarks,
• continuum states → unbound modes (decay or radiation).

Since $Z\left(p\right)$ decreases with density, the self-coupling naturally limits the energy spacing, producing a finite discrete spectrum rather than a continuum.

2.7. Energy Spectrum Example

Assume the potential $V\left(p\right)=\frac{1}{2}{\mu }^2{p}^2+\frac{1}{4}\lambda p^4$ and modulation $Z\left(p\right)={(1+\beta p^2)}^{-1}$. Then at equilibrium $p_0^2={\mu }^2/\lambda$, and from (2.10) we find

$$m_{\mathrm{eff}}^2\simeq {\displaystyle \frac{{\mu }^2(1+\frac{1}{2}\beta p_0^2)}{1+\beta p_0^2}}\approx {\mu }^2\left(1-\frac{1}{2}\beta p_0^2\right)={\mu }^2\left(1-\frac{1}{2}{\displaystyle \frac{\beta {\mu }^2}{\lambda }}\right).$$

(2.19)

The modulation thus induces a small correction $\Delta m/m\simeq -\frac{1}{4}\beta {\mu }^2/\lambda$, providing a testable prediction once parameters are fitted.

2.8. Worked Example: Discrete Spectrum in a Quartic Potential

Consider $V\left(p\right)=\frac{1}{2}{\mu }^2{p}^2+\frac{1}{4}\lambda p^4$, $Z\left(p\right)={(1+\beta p^2)}^{-1}$, and a spherically–symmetric background $p_0={\varphi }_0\left(r\right)e^{-i{\omega }_{0}t}$ approximately constant inside radius R and rapidly decaying outside (bag–like solution of Sec. Section 4). Linear modes satisfy Equation (2.8). Approximating ${\varphi }_0\left(r\right)\approx p_{\mathrm{in}}$ for $r<R$ and 0 for $r>R$, the radial equation inside reads

$$-{\displaystyle \frac{Z\left(p_{\mathrm{in}}\right)}{r^2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}r}}\left(r^2{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}\right)+U_{\mathrm{in}}u={\Omega }^{2}u,U_{\mathrm{in}}\equiv V^{\prime \prime }\left(p_{\mathrm{in}}\right)-\frac{1}{2}{Z}^{\prime \prime }\left(p_{\mathrm{in}}\right){(\nabla p_0)}^2\approx {\mu }^2+3\lambda p_{\mathrm{in}}^2.$$

With $u\left(r\right)\propto j_{\ell }\left(kr\right)$, $k^2={\displaystyle \frac{{\Omega }^2-U_{\mathrm{in}}}{Z\left(p_{\mathrm{in}}\right)}}$, and Dirichlet boundary $u\left(R\right)=0$, the lowest $\ell =0$ modes obey $k_{n}R\approx n\pi$. Thus

$${\Omega }_n^2\approx U_{\mathrm{in}}+Z\left(p_{\mathrm{in}}\right){\displaystyle \frac{n^2{\pi }^2}{R^2}},m_n=\hslash {\Omega }_n/c_0^2,n=0,1,2,\dots$$

(2.20)

which yields a concrete mass ladder. A sample (illustrative) set:

Table 1. Illustrative masses with $\mu =0.5\mathrm{GeV}$, $\lambda ={10}^{-2}$, $R=1.0\mathrm{fm}$, $p_{\mathrm{in}}^2={\mu }^2/\lambda$, $\beta p_{\mathrm{in}}^2=0.05$.

Mode	${\mathit{\Omega }}_{\mathit{n}}$ [GeV]	${\mathit{m}}_{\mathit{n}}{\mathit{c}}_0^2$ [GeV]	Comment
$n=0$	$\approx 0.50$	$\approx 0.50$	ground state (stable)
$n=1$	$\approx 0.50+0.20$	$\approx 0.70$	first excitation
$n=2$	$\approx 0.50+0.80$	$\approx 1.30$	second excitation

Table 1. Illustrative masses with $\mu =0.5\mathrm{GeV}$, $\lambda ={10}^{-2}$, $R=1.0\mathrm{fm}$, $p_{\mathrm{in}}^2={\mu }^2/\lambda$, $\beta p_{\mathrm{in}}^2=0.05$.

Mode	${\mathit{\Omega }}_{\mathit{n}}$ [GeV]	${\mathit{m}}_{\mathit{n}}{\mathit{c}}_0^2$ [GeV]	Comment
$n=0$	$\approx 0.50$	$\approx 0.50$	ground state (stable)
$n=1$	$\approx 0.50+0.20$	$\approx 0.70$	first excitation
$n=2$	$\approx 0.50+0.80$	$\approx 1.30$	second excitation

The Z–factor compresses level spacings relative to a pure Laplacian.

2.9. Summary of Section 2

Perturbative quantization of the informational field around a stationary solution leads to a discrete spectrum of excitations. The operator $\mathcal{O}$ in (2.6) governs the mass eigenmodes, and the informational modulation $Z\left(p\right)$ acts as a natural regulator and mass renormalization mechanism. In QIR, particle masses thus emerge from the local curvature of the informational potential $V\left(p\right)$ weighted by $Z\left(p\right)$, establishing a purely informational origin for the inertial mass spectrum.

3. Parametric Correspondence and Observables

3.1. Overview

The informational Lagrangian of QIR depends on two fundamental scalar functions: the modulation $Z\left(p\right)$ and the potential $V\left(p\right)$. Both are characterized by a small set of parameters that determine the particle mass spectrum and, at large scales, cosmological observables.

We denote:

$$\begin{array}{cc}\hfill Z\left(p\right)& ={\displaystyle \frac{1}{1+\beta p^2}},\hfill \end{array}$$

(3.1)

$$\begin{array}{cc}\hfill V\left(p\right)& ={\displaystyle \frac{1}{2}}{\mu }^2{p}^2+{\displaystyle \frac{1}{4}}\lambda p^4,\hfill \end{array}$$

(3.2)

where $\beta$ (dimension of ${\mathrm{length}}^2$) encodes the informational saturation scale, $\lambda$ is the self-interaction coupling, and $\mu$ sets the curvature of the potential near the minimum.

In addition, a large-scale parameter $\alpha$ will appear when we relate the local informational density to the cosmological curvature.

3.2. Dimensional Analysis and Natural Units

Using $\left[p\right]={\mathrm{L}}^{-1}$ (inverse length) so that ${\int \left|p\right|}^2{\mathrm{d}}^{3}x$ is dimensionless, we have:

$$\begin{array}{cccccc}\hfill \left[\beta \right]& ={\mathrm{L}}^2,\hfill & \hfill \left[\lambda \right]& ={\mathrm{L}}^2{\mathrm{T}}^{-2},\hfill & \hfill \left[\mu \right]& ={\mathrm{T}}^{-1}.\hfill \end{array}$$

(3.3)

A convenient set of natural units for QIR is obtained by introducing the informational scale

$${\ell }_{\mathrm{info}}\equiv \sqrt{\beta },E_{\mathrm{info}}\equiv {\displaystyle \frac{\hslash c_0}{{\ell }_{\mathrm{info}}}},$$

(3.4)

which plays a role analogous to the Planck scale but determined by information saturation rather than gravity.

3.3. Units and Dimensional Consistency

We use $\left[x\right]=\mathrm{L}$, $\left[t\right]=\mathrm{T}$, $\left[c_0\right]=\mathrm{L}{\mathrm{T}}^{-1}$, and choose $\left[p\right]={\mathrm{L}}^{-1}$ so that ${\int \left|p\right|}^2{\mathrm{d}}^{3}x$ is dimensionless. Then:

Table 2. Dimensions of primary QIR quantities in SI base units (L,T).

Quantity	Symbol	Dimension
Field	p	${\mathrm{L}}^{-1}$
Modulation	$Z\left(p\right)$	1 (dimensionless)
Potential parameters	$\mu$, $\lambda$	$\mu :{\mathrm{T}}^{-1}$;   $\lambda :{\mathrm{L}}^2{\mathrm{T}}^{-2}$
Saturation scale	$\beta$	${\mathrm{L}}^2$
Cosmic modulation	$\alpha$	${\mathrm{T}}^{-2}$
Effective mass	$m_{\mathrm{eff}}$	via $E=\hslash \Omega$, $m=E/c_0^2$
Covariant derivative	${\mathcal{D}}_{\mu }$	${\partial }_{\mu }-ig{A}_{\mu }$ (standard)

Table 2. Dimensions of primary QIR quantities in SI base units (L,T).

Quantity	Symbol	Dimension
Field	p	${\mathrm{L}}^{-1}$
Modulation	$Z\left(p\right)$	1 (dimensionless)
Potential parameters	$\mu$, $\lambda$	$\mu :{\mathrm{T}}^{-1}$;   $\lambda :{\mathrm{L}}^2{\mathrm{T}}^{-2}$
Saturation scale	$\beta$	${\mathrm{L}}^2$
Cosmic modulation	$\alpha$	${\mathrm{T}}^{-2}$
Effective mass	$m_{\mathrm{eff}}$	via $E=\hslash \Omega$, $m=E/c_0^2$
Covariant derivative	${\mathcal{D}}_{\mu }$	${\partial }_{\mu }-ig{A}_{\mu }$ (standard)

Consistency checks: Equations (3.10), (3.13) and (3.16) are dimensionally correct with these assignments.

3.4. Effective Mass and Coupling Relations

From the previous section, the effective mass of a stationary excitation is

$$m_{\mathrm{eff}}^2\simeq {\displaystyle \frac{1}{Z\left(p_0\right)}}\left[V{\left(p_0\right)}^{\prime \prime }-\frac{1}{2}{\displaystyle \frac{Z{\left(p_0\right)}^{\prime }}{Z\left(p_0\right)}}V{\left(p_0\right)}^{\prime }\right],$$

(3.5)

with $p_0^2={\mu }^2/\lambda$ from equilibrium.

Using (3.1)–(3.2), we obtain explicitly

$$\begin{array}{cc}\hfill V{\left(p_0\right)}^{\prime }& ={\mu }^2{p}_0+\lambda p_0^3=p_0({\mu }^2+\lambda p_0^2),\hfill \end{array}$$

(3.6)

$$\begin{array}{cc}\hfill V{\left(p_0\right)}^{\prime \prime }& ={\mu }^2+3\lambda p_0^2,\hfill \end{array}$$

(3.7)

$$\begin{array}{cc}\hfill Z{\left(p_0\right)}^{\prime }& =-{\displaystyle \frac{2\beta p_0}{{(1+\beta p_0^2)}^2}}.\hfill \end{array}$$

(3.8)

Substituting yields

$$m_{\mathrm{eff}}^2=({\mu }^2+3\lambda p_0^2)(1+\beta p_0^2)-\beta p_0^2({\mu }^2+\lambda p_0^2)+\mathcal{O}\left({\beta }^2\right).$$

(3.9)

At leading order in $\beta$,

$$m_{\mathrm{eff}}^2\approx {\mu }^2+2\lambda p_0^2-\beta p_0^2({\mu }^2+\lambda p_0^2),$$

(3.10)

recovering the correction term already observed in (2.19).

3.5. Numerical Estimates

Assuming $\mu$ corresponds to an intrinsic energy scale $E_{\mu }=\hslash \mu$ of the order of $1\mathrm{GeV}$ and $\lambda \sim {10}^{-2}$, $\beta$ must satisfy

$$\beta {\mu }^2\lesssim {10}^{-2}\Rightarrow {\ell }_{\mathrm{info}}\lesssim {10}^{-17}\mathrm{m},$$

(3.11)

to keep the correction small. This value lies between the electroweak and Planck lengths, suggesting that the informational cutoff acts as an intermediate scale bridging quantum and gravitational domains.

3.6. Relation to Gauge Couplings

Expanding the kinetic term of $\widehat{p}$ minimally coupled to a gauge field with ${\mathcal{D}}_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$ gives

$${\mathcal{L}}_{\mathrm{int}}=gZ\left(p\right)A_{\mu }{j}^{\mu }+{\displaystyle \frac{1}{2}}{g}^{2}Z\left(p\right)A_{\mu }{A}^{\mu }{\left|p\right|}^2.$$

(3.12)

The effective gauge coupling thus depends on the local informational density:

$$g_{\mathrm{eff}}=g\sqrt{Z\left(p_0\right)}={\displaystyle \frac{g}{\sqrt{1+\beta p_0^2}}}.$$

(3.13)

Hence, interaction strengths decrease in regions of high informational density. This provides a natural explanation for asymptotic freedom: strong interactions weaken as informational density saturates.

3.7. Large-Scale Modulation and Cosmological Parameters

At cosmological scales, $p\left(x\right)$ is nearly homogeneous:

$$p(x,t)=\overline{p}\left(t\right)+\delta p(x,t),$$

(3.14)

with $\overline{p}\left(t\right)$ driving the effective cosmological expansion via the averaged Einstein equation

$${\overline{G}}_{\mu \nu }={\displaystyle \frac{8\pi G}{c_0^4}}Z\left(\overline{p}\right)({\overline{T}}_{\mu \nu }+{\overline{I}}_{\mu \nu }).$$

(3.15)

Identifying the 00 component gives a modified Friedmann equation:

$$H^2={\displaystyle \frac{8\pi G}{3c_0^2}}Z\left(\overline{p}\right){\rho }_{\mathrm{tot}},$$

(3.16)

where ${\rho }_{\mathrm{tot}}$ includes both matter and informational energy.

If $Z\left(\overline{p}\right)={(1+\alpha t^2)}^{-1}$ with small $\alpha >0$, integration of (3.16) yields an expansion rate

$$H_0^{\left(\mathrm{QIR}\right)}=H_0^{\left(\mathrm{GR}\right)}(1+\frac{1}{2}\alpha t_0^2),$$

(3.17)

producing an increase of $H_0$ relative to standard $\Lambda$CDM and potentially resolving the observational tension.

Likewise, the growth rate of structure $S_8$ is reduced by an effective drag from $Z\left(\overline{p}\right)$, yielding $\Delta S_8/S_8\simeq -3\alpha t_0^2/4$, consistent with the correlations derived in the Minimal QIR framework.

3.8. Summary Table of Parameters

Table 3. Fundamental QIR parameters and physical correspondence.

Symbol	Physical meaning	Typical scale	Observational effect
$\lambda$	Self-interaction strength	${10}^{-2}$–${10}^{-3}$	Determines particle mass hierarchy
$\beta$	Informational saturation scale	${\left({10}^{-17}\text{–}{10}^{-19}\mathrm{m}\right)}^2$	Regulates coupling; induces confinement
$\alpha$	Cosmological modulation	${10}^{-36}\text{–}{10}^{-34}{\mathrm{s}}^{-2}$	Shifts $H_0$ and $S_8$ tensions

Table 3. Fundamental QIR parameters and physical correspondence.

Symbol	Physical meaning	Typical scale	Observational effect
$\lambda$	Self-interaction strength	${10}^{-2}$–${10}^{-3}$	Determines particle mass hierarchy
$\beta$	Informational saturation scale	${\left({10}^{-17}\text{–}{10}^{-19}\mathrm{m}\right)}^2$	Regulates coupling; induces confinement
$\alpha$	Cosmological modulation	${10}^{-36}\text{–}{10}^{-34}{\mathrm{s}}^{-2}$	Shifts $H_0$ and $S_8$ tensions

3.9. Interpretation and Testability

The QIR parameters $(\lambda ,\beta ,\alpha )$ form a unified set governing both microphysical and cosmological behavior:

• $\lambda$ controls the curvature of $V\left(p\right)$ and hence particle masses.
• $\beta$ controls the informational stiffness and sets the effective cutoff.
• $\alpha$ connects the slow time variation of $Z\left(p\right)$ to large-scale expansion.

All three appear in measurable quantities:

$$m_{\mathrm{eff}}(\lambda ,\beta ),g_{\mathrm{eff}}\left(\beta \right),H_0\left(\alpha \right),S_8\left(\alpha \right),$$

making the theory falsifiable. Future high-precision cosmological surveys (Euclid, LSST) and particle experiments sensitive to slight deviations in coupling constants could constrain $(\lambda ,\beta ,\alpha )$ simultaneously.

3.10. Covariant Origin of the Cosmological Modulation

We adopt a covariant FLRW background with proper time $\tau$ and metric $s^2={\tau }^2-a{\left(\tau \right)}^2{\mathit{x}}^2$. The homogeneous expectation value $\overline{p}\left(\tau \right)=\langle \widehat{p}\left(\tau \right)\rangle$ obeys the background EOM obtained by varying the action in Equation (1.2):

$$\ddot{\overline{p}}+3H\dot{\overline{p}}+{\displaystyle \frac{1}{Z\left(\overline{p}\right)}}{V}^{\prime }\left(\overline{p}\right)-{\displaystyle \frac{1}{2}}{\displaystyle \frac{Z^{\prime }\left(\overline{p}\right)}{Z\left(\overline{p}\right)}}{\dot{\overline{p}}}^2=0,$$

(3.18)

with $H=\dot{a}/a$ and overdots denoting derivatives with respect to $\tau$. The cosmological modulation is then a function of the scalar $\overline{p}\left(\tau \right)$:

$$Z\left(\overline{p}\right)=\mathcal{Z}\left(\overline{p}\left(\tau \right)\right),$$

independent of coordinate choices. The phenomenological ansatz $Z\left(\overline{p}\right)\simeq {(1+\alpha {\tau }^2)}^{-1}$ used in Sec. Section 3 is the small–variation limit of $\mathcal{Z}\left(\overline{p}\left(\tau \right)\right)$ when $\overline{p}$ evolves slowly under Equation (3.18). This makes the Friedmann form (3.16) manifestly covariant in the FLRW frame.

3.11. Summary of Section 3

The parameters $\lambda$, $\beta$, and $\alpha$ establish the quantitative bridge between QIR microphysics and cosmology. Masses, couplings, and cosmological rates are all determined by the same informational functions $Z\left(p\right)$ and $V\left(p\right)$, making the theory both self-consistent and observationally predictive.

4. Informational Confinement and Strong Interactions

4.1. Quantum Field Setting

At the fully quantized level, the informational field is an operator $\widehat{p}\left(x\right)$ acting on the Hilbert space of informational states. Its dynamics follow from the operator-valued action

$$\mathcal{S}[\widehat{p},g]={\int }^{4}x\sqrt{-g}\left[\frac{1}{2}Z\left(\widehat{p}\right)g^{\mu \nu }{\left({\mathcal{D}}_{\mu }\widehat{p}\right)}^{†}\left({\mathcal{D}}_{\nu }\widehat{p}\right)-V\left(\widehat{p}\right)\right],$$

(4.1)

with ${\mathcal{D}}_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$ the non-Abelian covariant derivative. Expectation values are taken with respect to the physical vacuum 0 such that $p_0\left(x\right)=0\widehat{p}\left(x\right)0$.

4.2. Informational Saturation Regime $Z\to 0$

The function $Z\left(\widehat{p}\right)$ saturates to zero when the local informational density $0{\widehat{p}}^{†}\widehat{p}0$ becomes large. In this limit, the kinetic term in (4.1) vanishes, freezing propagation:

$$\underset{Z\to 0}{lim}Z\left(\widehat{p}\right){\left({\mathcal{D}}_{\mu }\widehat{p}\right)}^{†}\left({\mathcal{D}}^{\mu }\widehat{p}\right)⟶0.$$

(4.2)

This halts the dispersion of informational energy and creates self-bound domains the analogues of hadrons in which $Z\left(\widehat{p}\right)$ acts as an effective confining wall.

4.3. Hamiltonian Density and Energy Localization

The quantum Hamiltonian density reads

$$\widehat{\mathcal{H}}={\displaystyle \frac{1}{2Z\left(\widehat{p}\right)}}{\widehat{\pi }}^{†}\widehat{\pi }+{\displaystyle \frac{1}{2}}Z\left(\widehat{p}\right){(\nabla \widehat{p})}^{†}·(\nabla \widehat{p})+V\left(\widehat{p}\right),$$

(4.3)

with the conjugate momentum defined by

$$\widehat{\pi }={\displaystyle \frac{\partial \mathcal{L}}{\partial \left({\partial }_t\widehat{p}\right)}}=Z\left(\widehat{p}\right){\partial }_t\widehat{p}.$$

(4.4)

In regions where $Z\left(\widehat{p}\right)\ll 1$, both gradient and temporal kinetic contributions are suppressed: substituting $\widehat{\pi }=Z\left(\widehat{p}\right){\partial }_t\widehat{p}$ into the first term gives ${\left(2Z\left(\widehat{p}\right)\right)}^{-1}{\widehat{\pi }}^{†}\widehat{\pi }=\frac{1}{2}Z\left(\widehat{p}\right){\left({\partial }_t\widehat{p}\right)}^{†}\left({\partial }_t\widehat{p}\right)$, so the energy becomes dominated by $V\left(\widehat{p}\right)$.

Operator ordering and hermiticity.

Since $Z\left(\widehat{p}\right)$ is an operator–valued function, one may adopt a pointwise hermitian ordering for the kinetic terms to make the construction manifestly self–adjoint:

$$\widehat{\mathcal{H}}={\displaystyle \frac{1}{2}}{\left(Z{\left(\widehat{p}\right)}^{-1/2}\widehat{\pi }\right)}^{†}\left(Z{\left(\widehat{p}\right)}^{-1/2}\widehat{\pi }\right)+{\displaystyle \frac{1}{2}}{\left(Z{\left(\widehat{p}\right)}^{1/2}\nabla \widehat{p}\right)}^{†}·\left(Z{\left(\widehat{p}\right)}^{1/2}\nabla \widehat{p}\right)+V\left(\widehat{p}\right).$$

(4.5)

With the canonical definition $\widehat{\pi }=Z\left(\widehat{p}\right){\partial }_t\widehat{p}$, the symmetrized form (4.5) reduces to the expression in (4.3) in the classical (c–number) limit where $Z\left(\widehat{p}\right)\to Z\left(p\right)$ and commutators vanish. At the operator level, (4.5) ensures hermiticity even if $[Z\left(\widehat{p}\right),\widehat{\pi }]\ne 0$ or $[Z\left(\widehat{p}\right),\widehat{p}]\ne 0$. In the $Z\left(\widehat{p}\right)\ll 1$ regime, both time– and space–like kinetic contributions remain suppressed, and the energy is still dominated by $V\left(\widehat{p}\right)$.

4.4. Confinement Potential and Bag Analogy

Let $\widehat{p}\left(\mathit{x}\right)=\varphi \left(r\right)\widehat{U}$ with $\widehat{U}\in SU\left(3\right)$. The static expectation value of (4.3) is

$$E\left[\varphi \right]=4\pi \int r^{2}r\left[\frac{1}{2}Z\left(\varphi \right){\left(\varphi r\right)}^2+V\left(\varphi \right)\right].$$

(4.6)

When $Z\left(\varphi \right)\to 0$ at large $\varphi$, the Euler–Lagrange equation admits “bag-like” solutions:

$$\varphi \left(r\right)=\left\{\begin{array}{cc}{\varphi }_{\mathrm{in}},\hfill & r<R,\hfill \\ {\varphi }_{\mathrm{out}}\left(r\right)\to 0,\hfill & r>R,\hfill \end{array}\right.$$

with R determined by the balance between gradient and potential energies. The surface tension

$$\sigma ={\int }_0^{{\varphi }_{\mathrm{in}}}\varphi \sqrt{2Z\left(\varphi \right)V\left(\varphi \right)}$$

(4.7)

acts as the confining pressure maintaining the stable radius R.

4.5. Field Correlator and Confinement Criterion

The equal-time two-point correlator

$$C\left(r\right)=0{\widehat{p}}^{†}\left(\mathit{x}\right)\widehat{p}(\mathit{x}+\mathit{r})0$$

(4.8)

obeys, in the Gaussian approximation,

$$C\left(r\right)\sim exp[-r/\xi ],{\xi }^{-2}=m_{\mathrm{eff}}^2/Z\left(p_0\right).$$

(4.9)

When $Z\to 0$, $\xi \to 0$ and correlations vanish beyond the domain size, indicating perfect confinement. Thus the informational field intrinsically confines itself through the saturation of its own modulation function.

4.6. Color Degrees of Freedom

Including the non-Abelian gauge structure $\widehat{p}=({\widehat{p}}_r,{\widehat{p}}_g,{\widehat{p}}_b)$, the gauge field $A_{\mu }^a$ mediates informational exchange between color components. The effective coupling derived earlier, $g_{\mathrm{eff}}=g\sqrt{Z\left(p_0\right)}$, vanishes at high density,

$$\underset{Z\left(p_0\right)\to 0}{lim}{g}_{\mathrm{eff}}=0,$$

(4.10)

ensuring that individual color states cannot propagate outside the saturated domain—the informational analogue of color confinement.

4.7. Deconfinement Transition

At high temperature T or low density, the expectation value $\langle Z\left(\widehat{p}\right)\rangle$ increases. Expanding $Z\left(\widehat{p}\right)={(1+\beta \langle {\widehat{p}}^{†}\widehat{p}\rangle )}^{-1}$ and using the thermal average ${\langle {\widehat{p}}^{†}\widehat{p}\rangle }_T\sim T^2$, the transition occurs when

$$\beta T_c^2\sim 1\Rightarrow k_B{T}_c\sim {\displaystyle \frac{\hslash c_0}{\sqrt{\beta }}}.$$

(4.11)

Hence the confinement scale corresponds exactly to the informational scale ${\ell }_{\mathrm{info}}=\sqrt{\beta }$ introduced in (3.4). For ${\ell }_{\mathrm{info}}\sim {10}^{-17}$m, this yields $T_c\sim 1$ TeV, close to the electroweak transition scale.

4.8. Vacuum Energy and Bag Pressure

Inside a saturated domain ($Z\to 0$), the vacuum energy density approaches

$${\rho }_{\mathrm{vac}}^{\left(\mathrm{in}\right)}\approx V\left({\varphi }_{\mathrm{in}}\right),{\rho }_{\mathrm{vac}}^{\left(\mathrm{out}\right)}\approx 0,$$

(4.12)

so the confining pressure is

$$P_{\mathrm{conf}}={\rho }_{\mathrm{vac}}^{\left(\mathrm{in}\right)}-{\rho }_{\mathrm{vac}}^{\left(\mathrm{out}\right)}.$$

(4.13)

This parallels the MIT bag model, but the pressure here arises from the informational potential itself, not from an external constant.

4.9. Conservation and Gauge Consistency

Even with $Z\to 0$, gauge invariance and energy conservation hold:

$${\nabla }_{\mu }{\widehat{I}}^{\mu \nu }=0,[{\mathcal{D}}_{\mu },{\mathcal{D}}_{\nu }]\widehat{p}=-ig{F}_{\mu \nu }\widehat{p},$$

ensuring the confined configurations remain dynamically consistent with the unified Einstein Equation (1.1).

Ward–Bianchi consistency (derivation).

Varying the action with respect to $g_{\mu \nu }$ and using diffeomorphism invariance gives the on–shell identity

$${\nabla }_{\mu }\left(Z\left(\widehat{p}\right)[{\widehat{T}}^{\mu \nu }+{\widehat{I}}^{\mu \nu }]\right)=0.$$

(4.14)

Sketch: (i) Gauge invariance implies ${\mathcal{D}}_{\mu }{\widehat{J}}^{\mu }=0$ and $[{\mathcal{D}}_{\mu },{\mathcal{D}}_{\nu }]\widehat{p}=-ig{F}_{\mu \nu }\widehat{p}$. (ii) The metric variation yields ${\nabla }_{\mu }{\widehat{I}}^{\mu \nu }=-({\partial }^{\nu }lnZ)({\widehat{\mathcal{L}}}_{\mathrm{mat}}+{\widehat{\mathcal{L}}}_{\mathrm{info}})$, which cancels against ${\nabla }_{\mu }\left[Z{\widehat{T}}^{\mu \nu }\right]$ when the Euler–Lagrange equation for $\widehat{p}$ holds. (iii) Contracted Bianchi identity ${\nabla }_{\mu }{G}^{\mu \nu }=0$ enforces (4.14) for the source of the Einstein equation (1.1).

4.10. Deconfinement Scale: Numerical Prefactor

In Sec. Section 4 we used ${\langle {\widehat{p}}^{†}\widehat{p}\rangle }_T\sim T^2$ as a scaling estimate. Including a numerical prefactor $\kappa =\mathcal{O}\left(1\right)$,

$${\langle {\widehat{p}}^{†}\widehat{p}\rangle }_T\simeq \kappa {\displaystyle \frac{{\left(k_{B}T\right)}^2}{{(\hslash c_0)}^2}}.$$

The deconfinement condition $\beta {\langle {\widehat{p}}^{†}\widehat{p}\rangle }_T\sim 1$ then gives

$$k_B{T}_c\simeq {\displaystyle \frac{\hslash c_0}{\sqrt{\kappa \beta }}},\kappa \in [0.1,10]\Rightarrow T_c\in [0.3,3]\times {\displaystyle \frac{\hslash c_0}{\sqrt{\beta }}}.$$

(4.15)

This band captures model–dependent details of the microscopic spectrum without affecting the qualitative QIR prediction that $T_c$ is set by the informational length ${\ell }_{\mathrm{info}}=\sqrt{\beta }$.

4.11. Summary of Section 4

In QIR, confinement is not imposed by a potential well but emerges naturally from the vanishing of the modulation function $Z\left(\widehat{p}\right)$ at high informational density. The same mechanism that stabilizes solitons also traps informational energy, producing colorless, finite-size bound states. The confinement scale and deconfinement temperature are set by $\beta$ through ${\ell }_{\mathrm{info}}=\sqrt{\beta }$, linking microphysics and cosmology in a unified informational picture.

5. Macro–Micro Continuity

5.1. Overview

At the macroscopic scale, the universe is described by the smooth metric $g_{\mu \nu }\left(x\right)$ obeying the averaged Einstein equation. At the microscopic scale, informational energy is carried by quantum excitations of the operator field $\widehat{p}\left(x\right)$. The goal of this section is to derive the macroscopic gravitational equations as expectation values of the quantum informational dynamics, and to identify the emergent components corresponding to ordinary matter and to dark energy.

5.2. Decomposition of the Quantum Field

We separate the operator field into its mean value and fluctuations:

$$\widehat{p}\left(x\right)=\overline{p}\left(x\right)+\delta \widehat{p}\left(x\right),\langle \delta \widehat{p}\left(x\right)\rangle =0,$$

(5.1)

where the expectation value $\overline{p}\left(x\right)=\langle \widehat{p}\left(x\right)\rangle$ defines the classical background. All observable quantities (energy, curvature, density) are expectation values of operator expressions.

5.3. Informational Energy–Momentum Tensor

The full operator tensor is

$${\widehat{I}}_{\mu \nu }=Z\left(\widehat{p}\right){\left({\mathcal{D}}_{\mu }\widehat{p}\right)}^{†}\left({\mathcal{D}}_{\nu }\widehat{p}\right)-g_{\mu \nu }\left[\frac{1}{2}Z\left(\widehat{p}\right)g^{\alpha \beta }{\left({\mathcal{D}}_{\alpha }\widehat{p}\right)}^{†}\left({\mathcal{D}}_{\beta }\widehat{p}\right)-V\left(\widehat{p}\right)\right].$$

(5.2)

Taking the quantum expectation value gives the macroscopic informational tensor

$${\overline{I}}_{\mu \nu }=\langle {\widehat{I}}_{\mu \nu }\rangle =I_{\mu \nu }^{\left(\mathrm{cl}\right)}+I_{\mu \nu }^{\left(\mathrm{q}\right)},$$

(5.3)

where the first term arises from $\overline{p}$ (classical mean field) and the second from the fluctuations $\delta \widehat{p}$.

5.4. Averaged Einstein Equation

Averaging the unified Einstein equation $G_{\mu \nu }={\displaystyle \frac{8\pi G}{c{\left(x\right)}^4}}Z\left(\widehat{p}\right)\left(T_{\mu \nu }+I_{\mu \nu }\right)$ gives

$${\overline{G}}_{\mu \nu }={\displaystyle \frac{8\pi G}{c_0^4}}〈Z\left(\widehat{p}\right)({\widehat{T}}_{\mu \nu }+{\widehat{I}}_{\mu \nu })〉.$$

(5.4)

Expanding $Z\left(\widehat{p}\right)$ around $\overline{p}$ and keeping second order terms:

$$\langle Z\left(\widehat{p}\right)\rangle \simeq Z\left(\overline{p}\right)+\frac{1}{2}Z{\left(\overline{p}\right)}^{\prime \prime }\langle {\left(\delta \widehat{p}\right)}^2\rangle .$$

(5.5)

We then obtain

$${\overline{G}}_{\mu \nu }={\displaystyle \frac{8\pi G}{c_0^4}}Z\left(\overline{p}\right)\left({\overline{T}}_{\mu \nu }+I_{\mu \nu }^{\left(\mathrm{cl}\right)}+{\tau }_{\mu \nu }^{\left(\mathrm{q}\right)}\right),$$

(5.6)

where ${\tau }_{\mu \nu }^{\left(\mathrm{q}\right)}$ represents the informational backreaction of quantum fluctuations:

$${\tau }_{\mu \nu }^{\left(\mathrm{q}\right)}={\displaystyle \frac{1}{2}}{\displaystyle \frac{Z{\left(\overline{p}\right)}^{\prime \prime }}{Z\left(\overline{p}\right)}}\langle {\left(\delta \widehat{p}\right)}^2\rangle ({\overline{T}}_{\mu \nu }+I_{\mu \nu }^{\left(\mathrm{cl}\right)})+\langle \delta {\widehat{I}}_{\mu \nu }\rangle .$$

(5.7)

5.5. Effective Energy Components

We identify three macroscopic contributions:

$$\begin{array}{cc}\hfill {\rho }_{\mathrm{m}}& \equiv {\displaystyle \frac{1}{c_0^2}}({\overline{T}}_{00}+I_{00}^{\left(\mathrm{cl}\right)})(\mathrm{ordinary}+\mathrm{informational}\mathrm{matter}),\hfill \end{array}$$

(5.8)

$$\begin{array}{cc}\hfill {\rho }_{\mathrm{q}}& \equiv {\displaystyle \frac{1}{c_0^2}}{\tau }_{00}^{\left(\mathrm{q}\right)}(\mathrm{quantum}\mathrm{correction}),\hfill \end{array}$$

(5.9)

$$\begin{array}{cc}\hfill {\rho }_{\Lambda }& \equiv {\displaystyle \frac{1}{c_0^2}}V\left(\overline{p}\right)(\mathrm{vacuum}\mathrm{contribution}).\hfill \end{array}$$

(5.10)

In the homogeneous and isotropic limit, inserting these into the Friedmann equation (3.16) yields

$$H^2={\displaystyle \frac{8\pi G}{3c_0^2}}Z\left(\overline{p}\right)({\rho }_{\mathrm{m}}+{\rho }_{\mathrm{q}}+{\rho }_{\Lambda }).$$

(5.11)

The quantum correction ${\rho }_{\mathrm{q}}$ acts as a small, time-dependent component behaving like a dynamic dark energy term.

5.6. Perturbative Backreaction

Let $\delta \widehat{p}$ satisfy the linearized equation $\mathcal{O}\delta \widehat{p}=0$ (see Equation (2.6)). Using the Green’s function $G(x,x^{\prime })$ of $\mathcal{O}$,

$$\langle \delta \widehat{p}\left(x\right)\delta \widehat{p}\left(x^{\prime }\right)\rangle =i\hslash G(x,x^{\prime }),$$

(5.12)

the expectation value $\langle {\left(\delta \widehat{p}\right)}^2\rangle$ contributes to the stress tensor as

$${\tau }_{\mu \nu }^{\left(\mathrm{q}\right)}\approx {\displaystyle \frac{\hslash }{2}}Z{\left(\overline{p}\right)}^{\prime }\int {\displaystyle \frac{{\mathrm{d}}^{4}k}{{\left(2\pi \right)}^4}}{\displaystyle \frac{k_{\mu }{k}_{\nu }}{k^2-m_{\mathrm{eff}}^2+iϵ}},$$

(5.13)

which is finite thanks to the natural cutoff provided by $Z\left(\overline{p}\right)$. This eliminates the usual UV divergences of quantum fields in curved space.

5.7. Emergent Newtonian Limit

In weak fields and low velocities, expanding $Z\left(\overline{p}\right)=1-\beta {\overline{p}}^2$, the 00 component of (5.6) gives

$${\nabla }^2\Phi =4\pi GZ\left(\overline{p}\right){\rho }_{\mathrm{m}},$$

(5.14)

so that the effective gravitational constant becomes

$$G_{\mathrm{eff}}=GZ\left(\overline{p}\right).$$

(5.15)

A spatial variation of $\overline{p}$ thus induces a small variation of G, suggesting possible observable signatures in precision measurements of the local gravitational constant.

5.8. Continuity Equation

From ${\nabla }_{\mu }{\overline{I}}^{\mu \nu }=0$ we obtain

$${\dot{\rho }}_{\mathrm{m}}+3H({\rho }_{\mathrm{m}}+P_{\mathrm{m}})=-\dot{\overline{p}}{\displaystyle \frac{\partial lnZ\left(\overline{p}\right)}{\partial \overline{p}}}({\rho }_{\mathrm{m}}+P_{\mathrm{m}}),$$

(5.16)

which describes a small exchange of energy between matter and the informational background. This term acts as a source or sink depending on the sign of $\dot{\overline{p}}$, potentially explaining late-time cosmic acceleration.

5.9. Macro–Micro Energy Equivalence

Combining Equations (5.11) and (5.16), one finds that the integrated informational energy stored in confined microstates equals the dark energy density:

$$\int {\rho }_{\mathrm{conf}}{\mathrm{d}}^{3}x\simeq {\rho }_{\Lambda }{V}_{\mathrm{H}},$$

(5.17)

where $V_{\mathrm{H}}$ is the Hubble volume. This provides a natural quantitative equivalence between the microscopic informational energy trapped in particles and the macroscopic vacuum energy driving expansion.

5.10. Summary of Section 5

The decomposition $\widehat{p}=\overline{p}+\delta \widehat{p}$ unifies quantum microphysics and macroscopic gravitation. Averaging over quantum fluctuations yields the standard Einstein equation with an effective source containing ordinary matter, informational energy, and vacuum terms. The modulation $Z\left(\overline{p}\right)$ links local physics (particle masses, couplings) to global cosmological parameters ($H_0$, $S_8$), achieving complete macro–micro continuity within the informational framework.

6. Conclusions and Perspectives

6.1. Synthesis of Results

Throughout this volume, we have extended the formalism of Quantum Informational Relativity (QIR) to the microphysical regime and established a consistent bridge between the informational field $\widehat{p}\left(x\right)$, its geometric modulation $Z\left(\widehat{p}\right)$, and the emergent gravitational and cosmological structure.

Main achievements:

• Topological origin of particles: Elementary particles appear as stationary or topological excitations of the informational field $\widehat{p}$, characterized by quantized winding numbers and internal symmetries derived from the geometry of the target space ${\mathcal{M}}_{\mathrm{target}}$. Spin and charge emerge from vorticity and phase rotation.
• Perturbative quantization and spectrum: Linearization around a stable background leads to a discrete mass spectrum governed by the curvature of $V\left(p\right)$ and modulated by $Z\left(p\right)$, providing an informational origin for inertial mass and natural mass renormalization.
• Parametric correspondence: The three parameters $(\lambda ,\beta ,\alpha )$ determine both microscopic and cosmological observables: $\lambda$ controls the self–interaction and particle masses, $\beta$ fixes the confinement and renormalization scale, and $\alpha$ links slow cosmological variations of $Z\left(p\right)$ to large–scale acceleration.
• Informational confinement: The saturation $Z\left(\widehat{p}\right)\to 0$ generates automatic confinement of informational energy, producing colorless bound states with deconfinement temperature $T_c\sim \hslash c_0/\sqrt{\beta }$.
• Macro-micro continuity: Averaging over quantum fluctuations yields the Einstein equation with effective sources reproducing matter, dark energy, and small variations of G and $H_0$, demonstrating full unification between microscopic dynamics and cosmic expansion.

6.2. Physical Implications

The informational framework replaces the notion of “material substance” with quantized informational energy. All physical quantities mass, charge, curvature, and coupling arise from informational relations rather than external postulates.

Key implications:

• Natural UV regularization: The saturation of $Z\left(\widehat{p}\right)$ at high informational density provides a built–in cutoff, eliminating ultraviolet divergences without ad hoc renormalization.
• Unification of interactions: Gauge couplings become local functions of $Z\left(\widehat{p}\right)$; their common informational dependence offers a unified description of SU(3)×SU(2)×U(1) dynamics under a single scalar modulation.
• Cosmological coherence: The same field governing microstructure drives large–scale acceleration, providing a natural resolution to $H_0$ and $S_8$ tensions without introducing a new constant $\Lambda$.
• Testable predictions: Observable signatures include slight variations of the effective gravitational constant $G_{\mathrm{eff}}$, deviations in coupling constants, and a characteristic deconfinement energy $\sim 1$–10 TeV.

6.3. Theoretical Extensions

Several natural extensions follow from the present work:

1. Supersymmetric Informational Field:

A superfield $\widehat{P}(x,\theta )$ including spinorial infons could incorporate fermionic degrees of freedom directly and unify matter and geometry at the informational level.

2. Quantum Coherence and Decoherence:

The evolution of informational entanglement $\langle \widehat{p}\left(x\right){\widehat{p}}^{†}\left(y\right)\rangle$ in curved backgrounds could explain the emergence of classicality and gravitational decoherence without external assumptions.

3. Black–Hole Information and Entropy:

Applying the QIR energy tensor to horizons may yield an intrinsic statistical origin of black–hole entropy, $S=k_B\int Z{\left(\widehat{p}\right)}^{3}x$, linking horizon area to informational density.

4. Holographic and Renormalization Structure:

The informational flux across scales might realize a concrete form of the holographic principle, where $Z\left(\widehat{p}\right)$ plays the role of a running coupling in a renormalization–group flow of spacetime itself.

6.4. Experimental and Observational Prospects

• Laboratory tests: Precision measurements of coupling constants at high energy (LHC, ILC) could detect the predicted small decrease $g_{\mathrm{eff}}/g={(1+\beta p_0^2)}^{-1/2}$.
• Astrophysical observations: Time–dependent $G_{\mathrm{eff}}$ or modified luminosity distances in supernova data would provide macroscopic evidence of $Z\left(p\right)$ variation.
• Cosmological surveys: Future missions (Euclid, LSST, CMB–S4) could constrain $\alpha$ through joint fits of $H_0$, $S_8$, and growth rate, testing QIR at cosmological scales.

6.5. Final Statement

The Quantum Informational Relativity framework offers a single, coherent mathematical structure from which space–time, matter, and gravitation emerge from the dynamics of information itself. By quantizing the informational field $\widehat{p}$ and tracing its effects from microscopic to cosmological scales, we achieve a conceptually minimal yet experimentally testable unification of quantum mechanics and general relativity.

Information is not carried by matter; itismatter.

Geometry does not contain information; itemergesfrom it.