Sleep: A CRT Model

A hypothesis: if sleep stages correspond to progressive CRT channel locking, the eigenvalue class counts form the sequence 288, 144, 72, 24, 6, 1.

This is a MODEL, not a proved correspondence. The mathematics below is real algebra. The mapping to neuroscience is interpretive.

The Model

Z/2,310 has five CRT channels (mod 2, mod 3, mod 5, mod 7, mod 11). If each sleep stage corresponds to locking one channel -- setting its residue to zero -- the number of eigenvalue classes drops at each step. The locking order follows coupling cost: cheapest channel first.

This mapping is a hypothesis. The class-count arithmetic is proved. The biological correspondence is not.

Class Count Sequence

All open
288 classes
All channels active. Mean eigenvalue = 0. Full spectrum.
Lock mod 2
144 classes
288/2. The mod-2 channel has 2 eigenvalue classes.
Lock mod 3
72 classes
144/2. The mod-3 channel has 2 eigenvalue classes.
Lock mod 5
24 classes
72/3. The mod-5 channel has 3 eigenvalue classes.
Lock mod 7
6 classes
24/4. The mod-7 channel has 4 eigenvalue classes.
Lock mod 11
1 class
6/6. The mod-11 channel has 6 eigenvalue classes. All locked.

Eigenvalue Mathematics

CRT Class Counting (PROVED -- ring algebra)
In Z/2,310: 288 eigenvalue classes = 2 * 2 * 3 * 4 * 6 (one factor per channel). Locking channels: mod 2: 288/2=144. mod 3: 144/2=72. mod 5: 72/3=24. mod 7: 24/4=6. mod 11: 6/6=1. Mean eigenvalue at each stage: {0, 1, 3, 5, 7, 9}. Each locked channel contributes its maximum (cos(0)=1).
Positivity Threshold (PROVED -- ring algebra)
All eigenvalues become positive exactly when the mod-5 channel locks. With mod-2 and mod-3 locked but mod-5 open: minimum eigenvalue = -2.339 < 0. With mod-5 also locked: minimum = 5 - 2cos(pi/7) - 2cos(pi/11) = 1.279 > 0. The threshold is exact.
Information Content (PROVED -- ring algebra)
Per-channel information: mod 2 = 1 bit, mod 3 = 1 bit, mod 5 = log2(3) = 1.585 bits, mod 7 = log2(4) = 2 bits, mod 11 = log2(6) = 2.585 bits. Total = log2(288) = 8.17 bits. Locking sheds the cheapest channel first.

Explore: Channel Locking

Enter a locking stage (0-5). See which channels are locked at each stage and the eigenvalue class count. Stage 0 = all 5 channels open (288 classes). Stage 5 = all locked (1 class).

Stage (0-5):

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