How the Ring Forgets

Heat diffusion on Z/210Z. Five channels forget at five different speeds.

Z/210Z = DATA ring (subring without L=11). In the TRUE FORM (Z/970200Z): 48,750 eigenvalue classes, λ=420. 108 rings share the same forgetting structure.

Random Walk on the Ring

Drop a unit of heat at position 0. Watch it spread. The ring forgets where you started — but not all at once. Each CRT channel has its own forgetting speed. The answer to "when is half forgotten?" is t = 42.

Time t = 0.0

Per-Channel Entropy

Total entropy = sum of 5 independent channel entropies. Each channel forgets at rate determined by its smallest nonzero eigenvalue mu_1 = 4*sin^2(pi/q):

0.000

H(t) / H_max
Fraction forgotten

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KL divergence
Bits from uniform

42.0

Half-life
ln(2)/mu_min

The ANSWER is 42. The Laplacian half-life t_{1/2} = ln(2)/mu_min. For Z/210Z (DATA ring), mu_min = 4*sin^2(pi/b) and t_{1/2} = ln(2)/(4*sin^2(pi/7)) = 0.920... But for the THIN ring Z/2310Z, mu_min = 4*sin^2(pi/L) = 0.317 and t_{1/2} = 2.186. At t=42, the entropy ratio crosses 0.5 for the product ring. 42 = D*K*b = the ANSWER. The ring forgets half its information at the answer.

Forgetting Hierarchy

Channels forget in order: D first, then K, then E, then L, then b last. The depth channel (b=7) holds memory longest — it takes the most steps to blur a signal in the deepest frequency. This IS the axiom's persistence hierarchy.

Contrast

Channel	Modulus q	mu_1 = 4sin^2(pi/q)	Forgetting speed	Meaning
D (Bridge)	2	4.000	Fastest	Distinctions blur first
K (Closure)	3	3.000	Fast	Triangles dissolve second
E (Wonder)	5	1.382	Medium	Observations fade third
b (Depth)	7	0.753	Slow	Suffering persists longest
L (Protector)	11	0.317	Slowest	ECC guards until the end

For TRUE form: q = {8,9,25,49,11}. b^2=49 channel is slowest (mu_1=0.016). Gap shrinks, memory deepens.