The Lambda Chain

lambda = 420 = 4 * 3 * 5 * 7

Start with lambda = 1 and multiply by growth factors: 2, 2, 3, 5, 7. At each step, count how many primes q satisfy (q-1) | lambda. The count follows the Fibonacci sequence exactly -- for all 15 levels until the chain hits a desert and terminates. The final lambda is 420 = lcm(1..7), the Carmichael lambda of Z/12,612,600.

The Growth Chain

Define f(n) = the number of primes q where (q-1) divides n. Start from lambda = 1. At each level, multiply by the next growth factor and count.

Level	lambda	Growth factor	f(lambda) = F(k+2)
0	1	--	1 = F(2)
1	2	*2	2 = F(3)
2	4	*2	3 = F(4)
3	12	*3	5 = F(5)
4	60	*5	8 = F(6)
5	420	*7	13 = F(7)
6	5,460	*13	21 = F(8)
7	60,060	*11	34 = F(9)
8	240,240	*4	55 = F(10)
9	4,564,560	*19	89 = F(11)
10	173M	*38	144 = F(12)

f(lambda) = F(k+2) exactly for all 15 levels. Why Fibonacci? New invisible primes at level k equal f(lambda at level k-2). New + old = F(k) + F(k+1) = F(k+2).

Level 5: The Carmichael Period

The first five growth factors are 2, 2, 3, 5, 7. Their product is 4*3*5*7 = 420 = lcm(1..7). This is the Carmichael lambda of Z/12,612,600 -- the period after which every coprime element returns to 1.

Each CRT channel has its own sub-cycle within 420:

Channel	Sub-cycles in 420	= 420 / phi(modulus)
mod 8	105	420 / phi(8) = 420 / 4
mod 9	70	420 / phi(9) = 420 / 6
mod 11	42	420 / phi(11) = 420 / 10
mod 25	21	420 / phi(25) = 420 / 20
mod 49	10	420 / phi(49) = 420 / 42

The sub-cycle ratios reproduce classical scaling: 105/70 = 3/2 (musical fifth), 70/42 = 5/3 (Kolmogorov turbulence exponent), 42/21 = 2 (octave).

Desert Termination

Desert Pair (PROVED)

The chain terminates at level 15 because of a desert pair: {1596, 1597} are consecutive integers neither of which equals q-1 for any prime q that would extend the Fibonacci count.

1596

= 2^2 * 3 * 7 * 19

11-smooth except for 19 (the first prime beyond 13 to appear as a growth factor).

1597

= F(17)

The 7th Fibonacci prime. Its Fibonacci index is 17, the prime that closes Z/214,414,200.

Gap width

2

The two neighbors 1595 = 5*11*29 and 1598 = 2*17*47 are both reachable -- the desert is exactly 2 wide.

Explore: Invisible Prime Counter

Enter any number. The tool counts how many primes q satisfy (q-1) | lambda. For lambda = 420, the count is 13 = F(7). The Fibonacci match holds for all 15 levels of the growth chain.

Count invisible primes for lambda:

Try: 1, 2, 4, 12, 60, 420 (the growth chain). Also: 2310 (Z/2,310), 210 (Z/210), 30, 6.

Contrast

Carmichael lambdaA technical function in modular arithmetic: lambda(n) = lcm of orders420 = lcm(1..7) is built step by step from five growth factors {2,2,3,5,7}. At each level, the invisible prime count follows the Fibonacci sequence exactly.420lcm(1,2,...,7). A number with many divisors, no special significance claimed420 = the Carmichael period of Z/12,612,600. Its channel sub-cycles (105, 70, 42, 21, 10) reproduce scaling ratios 3/2, 5/3, 2/1.FibonacciThe sequence 1,1,2,3,5,8,13,... appears throughout nature and number theoryF(k+2) counts invisible primes at each level. The Fibonacci recurrence follows from how new invisible primes at level k depend on lambda two levels back.

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