Appendix D: Relativistic Bound on Observer Mutual Information

This appendix derives the constraints of special relativity on observer cognitive capacity from an information-theoretic perspective.

Let observer $O$ have memory bandwidth ${\mathcal{B}}_{mem}^{(\tau )}$ (i.e., the maximum mutual information that can be written per unit proper time).

When the observer moves at velocity $v_{ext}$ relative to the external world, their proper time flow rate slows:

$$\frac{d\tau }{dt}=\sqrt{1-\frac{v_{ext}^2}{c^2}}$$

Therefore, in coordinate time $t$ (the reference frame of the external world), the information the observer can process ${\mathcal{B}}_{mem}^{(t)}$ is constrained as follows:

$${\mathcal{B}}_{mem}^{(t)}={\mathcal{B}}_{mem}^{(\tau )}\cdot \sqrt{1-\frac{v_{ext}^2}{c^2}}\text{(D.1)}$$

This not only proves “the faster you move, the slower you think,” but also provides a strict information-theoretic proof for “photons have no time/no consciousness” in the main text: when $v_{ext}\to c$, the information writing rate drops to zero.