Why Do Falling and Distant Observers Describe Different Signals?

How can two valid descriptions differ without either observer seeing magic?

You defined proper time τ, kept both local clocks explicit, followed a synthetic pulse train as it spread and faded, and replaced the permanent frozen-image myth with a detector-aware description.

A probe falling toward a black hole sends blue pulses toward a distant radio dish; the pulses become redder, dimmer, and farther apart. — Vastward artistic reconstruction. Pulse colour, spacing, and brightness communicate a distant receiver's changing signal. The image is a theory-led teaching visualization, not a recording of a real probe crossing a horizon.

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