The Memory of Spacetime - Black Holes and the End of Infinity - Solving Hawkings Info-Paradox

\[
\Box \phi + \gamma \partial^\mu \phi + \alpha \phi^3 + \beta \phi^5 = qE_{\text{Res}} + \lambda \langle \hat{Q} \rangle + \eta \epsilon^{\mu\nu\rho\sigma} \partial_\mu \phi \partial_\nu \phi \partial_\rho \phi \partial_\sigma \phi + \xi \zeta(x^\mu) + \kappa \partial_t \phi + \delta \phi E^2
\]

\begin{equation}
    G_{\beta\nu} + \beta \langle \phi^2 \rangle g_{\beta\nu} = 8\pi G (T_{\text{std}} + T_\phi)
\end{equation}

\begin{equation}
    \Box \phi = -\beta \phi - \beta \partial_t \phi + \beta \nabla^2 \phi + \beta \zeta R \phi
\end{equation}

Description:
This work presents a classical and quantum-geometric resolution of the black hole information paradox within the Minkowski–Zander framework.
By introducing the finite spacetime action
σₚ = ħG / c⁴ (Universe),
the singularity and information-loss problems of general relativity disappear without new parameters or fields.

The central result is the definition of the Teta
Teta_Z(M) = c³ / (8π G M k_B ħ),
a finite, parameter-free expression that replaces Hawking’s divergent temperature.
It describes not the emission of energy, but the continuous exchange of spacetime action per unit mass.
As the black hole evaporates, T_Z(M) rises to a finite maximum near the Planck mass and then decreases, leaving a stable remnant that preserves information.

The analysis connects quantum mechanics, general relativity, and thermodynamics through a single invariant quantity — σₚ — the natural unit of spacetime action.
This unified formulation yields:
• A smooth Page curve (unitary information flow)
• Finite curvature at all scales (no singularities)
• A natural value of the cosmological constant Λ = 3 / (c R t)

In this framework, black holes are not endpoints, but dynamic memory structures of spacetime itself.
The Universe does not forget — it radiates, remembers, and evolves in perfect geometric harmony.

Field Equations (Full Quantum Form)

⟨ Ĝ[σ_P; W] + Λ_eff(W) * g ⟩_σP = (8πG / c⁴) * ⟨ T̂ ⟩_σP^(W)

or equivalently:

( Ĝ_μν + Λ_eff(W) * g_μν ) |Ψ_W⟩ = (8πG / c⁴) * (A_σP T̂)_μν |Ψ_W⟩

Where:
σ_P = ħG / c⁴ → finite spacetime action
Λ_eff(W) = 3 / (c R t)
A_σP = Planck-covariant averaging operator over kernel K_σP(x,y)

Expectation values recover the classical Einstein–Zander equation:
G_μν + Λ_eff(W) g_μν = (8πG / c⁴) T̄_μν^(W)

Key results:
Λ_eff(W) = 3 / (c R t)
Λ_cell = 3 / ℓ_P²
Λ_macro = Λ_cell / N_σ , where N_σ = (R t) / σ_P

Hence, the cosmological “vacuum catastrophe” disappears naturally:
Λ_QFT(local) / Λ_GR(observed) = 1

Axiom of Finite Divisibility of Spacetime (A. Zander)

Postulate:
No physical system can be divided without bound.
Space and time possess a fundamental, finite resolution — given by the spacetime quantum
σ_P = ℓ_P t_P = ħG / c⁴.

This geometric action unit defines the smallest possible quantum of spacetime in which cause and effect remain distinct.
Below this limit, the continuum assumption collapses, and with it the classical notion of separability.

Consequences:

1. Every physical process, measurement, or interaction occurs in discrete spacetime volumes. Attempts to resolve beyond σ_P require infinite energy.

2. Singularities, infinite densities, and point-like particles are mathematical artifacts, not physical realities. Nature enforces a cutoff.

3. Spacetime is atomic: σ_P sets the minimum geometric extent, implying a quantized fabric of spacetime.

Natural origin of quantum entanglement:
If separation is limited by geometry, two particles within the same Planck volume cannot be causally independent — they are geometrically entangled.
Entanglement is not a mysterious bond, but a lack of spacetime resolution.
Observers measuring correlations across spacelike separations are probing the same indivisible cell from different directions.

“Entanglement is the geometry’s way of saying: you haven’t zoomed in far enough to tell them apart.”

Synthesis:
Just as Planck’s constant ħ limits dynamical resolution,
the Planck two-measure ℓ_P t_P limits geometric resolution.
Together, they define a finite, quantized, and inherently entangled spacetime.

Related Works (EN/DE)

Zander, A. (2025). Parameter Free Unification of Quantum Mechanics and General Relativity - ħ, G, c, α. Zenodo. https://doi.org/10.5281/zenodo.17533233

Zander, A. (2025). The Memory of Spacetime - Black Holes and the End of Infinity - Solving Hawkings Info-Paradox. Zenodo. https://doi.org/10.5281/zenodo.17548544

Zander, A. (2025). Hierarchical Black Hole Mergers. Zenodo. https://doi.org/10.5281/zenodo.17495572

Zander, A. (2025). The Natural Structure of Spacetime: From Quantum Mechanics, Relativity, and Gravitation. Zenodo. https://doi.org/10.5281/zenodo.17357165

Zander, A. (2025). From Quantum to Cosmos: About Space and Time, Mass and Inertia. Zenodo. https://doi.org/10.5281/zenodo.16942321

Zander, A. (2025). A Requiem for LCDM. Zenodo. https://doi.org/10.5281/zenodo.17591150

Zander, A. (2025). The Paradox Paper. Zenodo. https://doi.org/10.5281/zenodo.17604878