Appendix E: Dimensional Analysis

This appendix clarifies the dimensional structure of the theory and demonstrates that all equations are internally consistent without invoking conventional energy–mass units as fundamental.

E.1 Fundamental Quantities

The primary dynamical variable of the theory is the informational field:

which represents coherent informational density.

$I$ is taken to be dimensionless. All physical dimensions emerge from relational and dynamical scales rather than being imposed a priori.

The coordinates $x$ and $t$ acquire effective dimensions only after locality and temporal ordering emerge.

E.2 Effective Length and Time Scales

Diffusion introduces an effective length scale through the diffusion coefficient:

which carries dimensions:

Time $t$ is defined operationally through the ordering of dynamical states. No absolute time unit is assumed prior to evolution.

E.3 Emergent Acceleration Scale

Acceleration appears through the informational potential:

Since $\Phi=-\log I$ is dimensionless (required for the logarithm to be well-defined), $\nabla\Phi$ has dimensions of $L^{-1}$, not $LT^{-2}$; the equation of motion $\ddot x=-\nabla\Phi$ (Section 11) is therefore dimensionally inconsistent as written. Repair: introduce an explicit dimensional constant $\Lambda_a$ (dimensions $L^2T^{-2}$) and define the physical potential entering the dynamics as $\Phi_{\rm phys}:=\Lambda_a\,(-\log I)$, distinguishing it from the dimensionless informational potential used elsewhere as an interpretive quantity. The equation of motion becomes

and the acceleration scale

inherits its $LT^{-2}$ dimensions from $\Lambda_a$, not from $\nabla\Phi$ alone.

No mass or energy scale is fundamental in the theory.

E.4 Absence of Fundamental Energy Units

Unlike conventional physics, the theory does not assign fundamental dimensions to energy or mass. What appears as mass is a stable localized concentration of information. What appears as energy is a rate of informational reconfiguration.

Dimensional quantities are therefore emergent bookkeeping devices, not ontological primitives.

E.5 Consistency Across Regimes

All regimes analyzed—galactic dynamics, cosmological expansion, and black-hole formation—use the same dimensional structure. No scale-dependent redefinitions are required.

This confirms that the framework is dimensionally self-consistent and free of hidden unit assumptions.