Counting vs oscillation

Remark 1: counting vs oscillation — same thing?

We have two apparently different descriptions:

• Counting picture: a knot displaces $M$ fabric bits per frame. Cost = $M$. Static.
• Codec picture: a wavefunction $\psi$ has spectral complexity $C_s$. Cost = dominant frequency. Dynamic.

But what $C_s$ actually measures: for an $n$-site equal superposition (Paper VIII):

$$|F|^2 = \frac{1}{n}, \quad |B|^2 = \frac{n-1}{n}$$

The off-diagonal weight $|B|^2$ is exactly $(n-1)/n$ — which is the resolution deficit of an $n$-site structure normalised by $n$. And $n-1 = M$ for a single composite entity of width $n$.

So the spectral complexity and the counting deficit are the same quantity expressed in two languages. The codec isn't a separate layer — it's the Fourier transform of the knot structure. A knot of size $M$ necessarily has a dominant mode at frequency proportional to $M$ because that's what it takes to encode $M$ displaced positions. The "oscillation" in the codec picture is just what a resolution deficit looks like when you decompose it into Fourier modes.

This is a strong consistency result for the framework!

Remark 2: non-linear time

Time advances by one bit flip out of $n$ bits. But entropy $S$ as a function of flip-count $t$ is:

$$S(t) \approx \frac{1}{2}\left(1 - e^{-2t/n}\right)$$

This is exponential saturation — early flips change entropy rapidly, later flips barely move it. So equal steps in $t$ (flip-count) are very unequal steps in entropy. The relationship is:

$$\Delta S \approx \frac{1}{n} e^{-2t/n}$$

If the "physical" time experienced by an observer is proportional to entropy change rather than flip-count, then the early universe is effectively time-dilated relative to the late universe. A structure costs the same $M$ per flip, but far more entropy is consumed per flip early on.

This has a direct consequence: if you measure energy in entropy units rather than flip units, the early universe appears to have much higher energy density — which decays as the universe ages. That is exactly the cosmological picture derived in Paper V. The Hubble tension and the positive cosmological constant might both be signatures of this non-linearity of the flip-to-entropy mapping.

Rather than entropy-weighting the cost, the right experiment is to change the time coordinate from flip-count to entropy-count and see how $E = M$ transforms:

$$E_{\text{entropy-time}} = M \cdot \frac{dt}{dS} = M \cdot n \cdot e^{+2t/n}$$

Early: $E \approx Mn$ — very high. Late: $E \to \infty$ as $S \to 1/2$. This divergence at equilibrium is interesting — it says maintaining a knot becomes infinitely expensive as entropy saturates, which sort of makes sense.