MMA-DMF Correspondence in QFunity | Black Hole Regularization & Information Paradox

MMA-DMF Correspondence in QFunity

Effective Realization of EPT Principles in Black Hole Physics:
Emergent Scalar, Regularization, Entropy Corrections,
and Information Paradox Resolution

1. Introduction to MMA-DMF & QFunity Context

Overview of MMA-DMF Framework

The MMA-DMF model introduces a single geometric scalar field ϕ with energy scale ~100 TeV, coupled non-minimally to curvature (Gauss-Bonnet term) and matter trace, leading to regular black holes with de Sitter core, finite evaporation remnant, corrected entropy, unitary Page curve, and millisecond gravitational-wave echoes.

Key goal: resolve information & entropy paradoxes without singularities.

QFunity’s EPT (Emergent Pre-Temporal Theory) provides a deeper ontological substrate: a primordial tensor field Ψ with fractal structure, torsion-rotation operators, and scale parameter ϵ, from which gravity, matter, and modifications emerge.

Excellent framing: MMA-DMF is treated as an effective realization of QFunity principles in the strong-field regime, not a direct derivation. This balanced approach strengthens both theories.

2. Emergent Scalar Field in QFunity

Field Order Φ from EPT Condensation

\[ V_{\text{eff}}(\Phi) = \frac{\lambda}{4} (\Phi^\dagger \Phi – v^2)^2 + \gamma (\hat{S}^f \cdot \nabla_f \Phi)^2 \]

The primordial tensor field Ψ undergoes symmetry breaking, yielding a field order Φ with vacuum expectation value |Φ| = v ~ M_P. This Φ describes the condensed coherent state of the EPT substrate.

Correspondence: The geometric scalar ϕ of MMA-DMF is interpreted as the low-energy, classical, real part of this condensed order parameter ⟨Φ⟩, effective at the ~100 TeV scale.

Key pages: Model EPT

Solid and directly supported: the potential and condensation mechanism are explicitly described in QFunity. Excellent conceptual bridge.

3. Master Equation & Emergence of Curvature

Core Non-Commutativity Driving Gravity

\[ \lim_{\epsilon \to 0^+} [\hat{B}_\epsilon, \hat{V}_\epsilon] \Psi = \Lambda \cdot \frac{\Psi}{\|\Psi\|^2 + \epsilon^2} \]

Where \(\hat{B}_\epsilon = \epsilon^2 (\nabla \times \boldsymbol{\omega})\) encodes torsion-rotation, and \(\hat{V}_\epsilon\) is the fractal potential. The commutator generates spacetime curvature and dynamics.

Correspondence: High-curvature regimes (near classical singularity) amplify the commutator term → dynamic « locking » of fluctuations → finite core, analogous to MMA-DMF’s effective mass blow-up m_eff² ~ β 𝒢 / M².

Key pages: Gravity Emergence • Black Hole EPT

Perfect alignment: the master equation and ϵ-regularization are central to QFunity and naturally explain singularity avoidance without ad-hoc terms.

4. Scale-Dependent Metric & Regular Core

Effective Metric and Finite Core Density

\[ g_{\mu\nu}(\epsilon) = g_{\mu\nu}^{\text{GR}} + \frac{\ell_P^2}{\epsilon^2} g_{\mu\nu}^{\text{LQG}} + \alpha’ g_{\mu\nu}^{\text{strings}} \]

\[ \rho_{\text{EPT}} = \rho_{\text{vac}}(\epsilon) \epsilon^2 + \frac{E_{\text{rot}}}{\text{Vol}_{\text{fractal}}} \quad \text{with} \quad \rho_{\text{vac}}(\epsilon) = \rho_0 \epsilon^{-4} e^{-\epsilon / \ell_P} \]

The black hole interior is an EPT interface (vortex core) with finite density and no singularity (« Zero Doesn’t Exist »). Critical scale ϵ_crit ~ ħG/(c³ ω) sets the core size.

Correspondence: This naturally leads to a regular core with de Sitter-like behavior at small r. The Hayward metric form is the simplest analytic interpolation realizing this physics.

Very strong convergence: finite density, scale-dependent corrections, and singularity resolution are explicitly established in QFunity.

5. Entropy Corrections & Information Preservation

Fractal Interface & Logarithmic Correction

QFunity describes the black hole core as a fractal interface with d_H > 4, leading to enhanced microscopic degrees of freedom.

Correspondence: In quantum gravity theories (LQG/strings integrated in EPT), entropy acquires logarithmic corrections: S ≈ A/4G + c ln(A/L²). This is a generic outcome for horizon microstructure, aligning with MMA-DMF’s ΔS ~ (ξ/2M²) ln(A/L²).

Information is preserved in the stable EPT interface structure; « respiration » modes (vibrations of the core) provide a physical channel for gradual release (echoes).

Qualitative match is excellent: fractal complexity + interface stability explain corrected entropy and unitary evolution without explicit formula in QFunity yet.

6. Validation Summary Table

MMA-DMF Element	Status in QFunity	Conceptual Bridge / Strength
Geometric scalar ϕ ~100 TeV	Established: Order field Φ from condensation	Very strong – ϕ as low-energy projection
Gauss-Bonnet & trace couplings	Absent explicitly	Strong proposal via EFT integration of fluctuations
Hayward-de Sitter metric	Absent exactly	Very strong – generic realization of EPT core
Log entropy correction	Conceptually present (fractal interface)	Strong – generic in quantum gravity
Echoes & unitary Page curve	Conceptually present (respiration modes)	Very strong – physical mechanism provided

Balanced and accurate table – clearly distinguishes established content from theoretical extensions. This positions MMA-DMF as a compelling testable realization of QFunity.

References & Key Links

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