Heisenberg - Planck - Einstein: The Logic Chain of Collapse

Overview. This document outlines the causal sequence in the Gravity as Collapse model, demonstrating how the probabilistic Quantum Future (Heisenberg) transitions through the Planck Interface (Inflection) to produce the Geometric Present (Einstein). This three-stage process unifies quantum mechanics and general relativity, showing gravity and time as emergent records of wavefunction collapse.

1. The Logic Chain: From Probability to Geometry

Quantum Future: Heisenberg (Uncertainty & Superposition)

Before realization, systems are described by a wavefunction $ \Psi $ on Hilbert space. Position and momentum are not simultaneously sharp; the non-commutation of observables reflects a pre-geometric description where "where" and "when" are not yet fixed coordinates.

\[ \Delta x\,\Delta p \ge \tfrac{\hbar}{2}, \quad [\hat x,\hat p]=i\hbar \]

Superposition: $ \Psi = \sum_i c_i\,|i\rangle $ encodes multiple coherent futures.
Entanglement: Non-factorizable states carry relational structure without spatial locality.
Density matrix: $ \rho=|\Psi\rangle\langle\Psi| $ has off-diagonals that track coherence.

Inflection: Planck Interface (Collapse & Dimensionalization)

At Planck scales, interaction and decoherence suppress off-diagonals ($ \rho_{ij}\!\to\!0 $ for $ i\ne j $), effectively selecting a definite outcome. This collapse maps probabilistic amplitudes to dimensional observables—events with concrete coordinates and energy–momentum.

\[ \ell_P=\sqrt{\tfrac{\hbar G}{c^3}}, \quad t_P=\sqrt{\tfrac{\hbar G}{c^5}}, \quad E_P=\sqrt{\tfrac{\hbar c^5}{G}} \]

Decoherence: $ \rho \to \text{diag}(p_i) $ in a pointer basis; phases disperse into the environment.
Projection: $ C:\ \mathcal H \to \mathcal M $ is a mapping from Hilbert space to spacetime events.
Clocking: Time increments with realized collapse, $d\tau \propto dC$ (at the Planck-time scale).

Geometric Present: Einstein (Curvature as Record)

The deterministic manifold $ (\mathcal M, g_{\mu\nu}) $ records realized outcomes as curvature. Persistent collapse density corresponds to persistent stress–energy, which sources gravity. The ensemble of realized events defines the classical spacetime we inhabit.

\[ G_{\mu\nu}=8\pi G\,T_{\mu\nu}, \quad \nabla^\mu G_{\mu\nu}=0 \Rightarrow \nabla^\mu T_{\mu\nu}=0 \]

Gravity: Curvature generated by realized energy–momentum (Einstein field equations).
Mass: Stable patterns of realized energy, where $ E=mc^2 $ is the conversion rate.
Vacuum: Background state of realized fields; only realized contributions gravitate.
Time: Procedural ordering of collapses (proper time accrued along worldlines).

Summary: Together, these steps connect the *probabilistic future* to the *geometric present*: collapse at the Planck interface instantiates the quantities that classical geometry measures.

2. Emergent Features: What Collapse Geometry Produces

Light, Gravity, Mass, Vacuum, Time

Light ($c$): Null propagation sets the causal horizon of realization; photons trace $ ds^2=0 $ and ferry collapse records.
Gravity (curvature): Persistent realized energy–momentum bends geodesics; curvature accumulates as the macroscopic trace of many collapses.
Mass (inertia): Localized, stable realization of energy patterns; $ E=mc^2 $ gives the conversion rate from energy to inertial mass.
Vacuum (ground state): Background of realized fields; only realized (not purely potential) contributions enter $ T_{\mu\nu} $ and gravitate.
Time (proper): Sequence of realized events gives procedural ordering; proper time accumulates along worldlines as collapse proceeds.

3. Foundational Pillars: Classical Physics in the Chain

The following classic theories are integrated into the three-stage logic chain, providing context for the quantum-to-classical transition:

Pillar (Role)	Core Concept
Schrödinger Dynamics	Unitary evolution of futures in Hilbert space; the continuous transformation that precedes collapse: $ i\hbar\,\partial_t \Psi = \hat H \Psi $
Born Rule (Probabilities)	Connects amplitudes to outcomes; weights of futures realized upon collapse at the interface: $ P(i)=\|\langle i\|\Psi\rangle\|^2 $
Bohr Complementarity	Mutually exclusive classical descriptions (wave/particle); context fixes the realized description at collapse.
Copenhagen Projection	Operational rule for selecting a definite result; interpreted here as the $C:\mathcal H\to\mathcal M$ map at the Planck inflection.
de Broglie Matter Waves	Wave–particle duality before collapse; pre-geometric phases interfere to set realized patterns: $ \lambda = \tfrac{h}{p} $
Maxwell Electromagnetism	Classical field structure for light; null geodesics at $c$ define the causal edge of realization.
Noether Symmetries	Conservation laws in the realized domain from spacetime symmetries recorded by geometry.
Decoherence (Zurek)	Explains pointer states and suppression of interference; the mechanism that readies states for collapse at the interface.

Conclusion: The Schrödinger + Born + Complementarity concepts shape the Heisenberg future; Decoherence & Projection operate at the Planck interface; and Maxwell, Noether, and Dirac structure the realized fields and geometry in the Einstein present.