Appendix G: Technical Details of the Quantum Completion

This appendix collects technical constructions and clarifications underlying Section 29. Its purpose is to make explicit the mathematical assumptions and approximations used in defining the quantum completion of the informational framework, without overloading the main text.

G.1 Functional Configuration Space and Measure

The configuration space of informational fields consists of coarse-grained functions $I(x)$ satisfying positivity and integrability constraints. The formal functional measure $\mathcal{D}I$ is understood as the continuum limit of a lattice discretization:

where $I_i$ denotes the informational density on lattice site $i$.

All functional integrals appearing in the text should be interpreted in this regulated sense.

G.2 Self-Adjointness and Unitarity of $\hatH _I$

The generator

is formally self-adjoint provided:

• $D(I,x,t) \ge 0$,
• $D$ and $V$ are sufficiently smooth functionals,
• boundary terms vanish under functional integration.

Under these conditions, the time evolution operator $\exp(-i\hat{H}_I t/\hbar)$ is unitary on $\mathcal{H}_I$.

G.3 Semiclassical Expansion and Ehrenfest-Type Arguments

To recover the classical emergent dynamics, consider a wavefunctional of the form:

where $S[I]$ varies rapidly compared to $A[I]$.

Inserting this ansatz into the functional Schr"odinger equation yields, to leading order in $\hbar$, a functional Hamilton–Jacobi equation for $S[I]$. The associated classical trajectories correspond to extrema of $S[I]$, leading to deterministic evolution of expectation values.

Higher-order terms generate fluctuations that are suppressed under decoherence and coarse-graining.

G.4 Emergence of the Reaction–Diffusion Equation

Under the semiclassical and decohered approximation, the expectation value $\langle I(x,t) \rangle$ obeys:

With

this reduces exactly to the classical equation used throughout the paper.

G.5 Decoherence Timescales

Decoherence arises from coupling between informational degrees of freedom and untracked environmental modes. The decoherence timescale $\tau_{\rm dec}$ depends on the functional distance between configurations:

where $\Gamma(x)$ encodes environmental sensitivity.

Macroscopically distinct informational configurations decohere extremely rapidly, justifying the classical treatment at large scales.

G.6 Relation to Standard Quantum Field Theory

In regimes where informational configurations admit an approximate embedding into a background spacetime, the present formalism reduces to an effective quantum field theory description. However, spacetime is not fundamental in this construction; it emerges only after informational stabilization.

Thus, the present approach is not a modification of quantum field theory, but a deeper framework from which QFT arises as a limit.

G.7 Limitations of the Present Construction

The formalism presented here does not yet address:

• renormalization at arbitrarily high informational resolution,
• interacting multi-field informational sectors,
• fully relativistic covariance at the quantum level.

These limitations define concrete directions for future mathematical development rather than conceptual gaps.

G.8 Consistency with the Selection Functional $\Xi$

The quantum completion respects the existential selection principle. Only informational dynamics that admit stable semiclassical limits and preserve structural coherence are compatible with $\Xi$.

Quantum theories that generate generic instabilities or information loss are excluded by the same selection logic that excludes unviable classical worlds.

Conclusion. Appendix G demonstrates that the quantum completion outlined in Section 29 rests on standard and defensible mathematical constructions. No additional postulates are introduced, and all approximations are controlled.