Discernible Discreteness vs. Apparent Continuity
Each quantum “rung” differs from its neighbor by just 1 / 2n of the total state-space. Whether we see that step depends on our instruments’ ability to resolve a fractional change of that size. The percent gap between adjacent rungs is
Δ% = 100 / 2n.
The Numbers Behind What We Notice
| Scale | n (bits) | Δ% = 100 / 2n | Typical Instrument Resolution* | Discernible? |
|---|---|---|---|---|
| Planck surface | 5 | 3.1 % | > 0.01 % | Yes (huge step) |
| Hydrogen atom | 22 | 0.000 024 % | 0.0001 % | Yes (spectral lines) |
| Molecule (~1 nm) | 32 | 0.000 000 023 % | 0.000 001 % | Borderline—needs cryogenic, high-R gear |
| Human-scale object | 200 | ≈10-58 % | 10-6 % (best metrology) | No—far below noise floor |
*Resolution shown as the smallest fractional change modern lab equipment can reliably detect at that scale.
Why Atoms Click but Apples Glide
- Atomic rungs (n ≈ 22). Electron transitions differ by ≈ 10 eV—
two orders of magnitude above thermal noise (kT ≈ 0.026 eV). Even a modest spectrometer (R ≈ 100 000) hears the “click.” - Molecular rungs (n ≈ 32). Rotational levels are spaced by millielectron-volts. You must cool the sample and push resolution past a million to separate the steps.
- Macroscopic rungs (n ≫ 100). Adjacent states differ by energies smaller than 10-30 eV—
utterly swamped by ambient vibration and thermal jitter. The staircase masquerades as a smooth ramp.
Metaphysical Angle: Visible Bands vs. Hidden Bits
Our vision misses infrared and ultraviolet not because the spectrum ends, but because our receptors cannot register those bands. Likewise, at the coarse-grain human scale the ladder from 232 up to 2408 packs rungs so tightly that the Cosmic Dance feels continuous. Yet at the pure von Neumann fine-grain level every octave is an integer log2 count of bits—crisp, discrete, and ultimately discernible whenever an instrument’s resolution beats the 1 / 2n hurdle. The smoothness we perceive is not a property of reality, but a limit of our detectors.