Eleven Terms

{-1, 0, 1, 2, 3, 5, 7, 11, 13, 17, 26801776}

Eleven terms. Not chosen -- forced. Each is an operation the ring performs on a number. Three structural anchors (-1, 0, 1), seven primes from two Cunningham chains (2, 3, 5, 7, 11, 13, 17), and the terminal projector 26,801,776 = 2^1680 mod 214,414,200. Together they form a complete computational vocabulary. 3 + 7 + 1 = 11 = the protector prime.

The Eleven Operations

Three operators. Seven primes. One projector. The count is 11 -- itself the 5th axiom prime.

-1

n -> N - n

Additive inverse. CRT(-1) = all channels at their maximum residue.

n -> 0

Absorbs. 0*n = 0 for all n. The universal annihilator.

n -> n

Identity. 1*n = n.

n -> 2n

The only even prime. Generates CRT decomposition into 7 channels.

n -> 3n

Minimum for majority vote and triangles. c(1) = 3.

n -> 5n

5^2 = 25 = 0 mod 25: self-blind in its own channel. c(2) = 5.

n -> 7n

Deepest channel: 49 states in mod-49. Controls the spectral gap. c(3) = 7.

n -> 11n

1+2+3+5 = 11. Error detection via mod-11 parity. c(5) = 11.

n -> 13n

2^2 + 3^2 = 13. lambda(13) = 12 divides 420. Closes the 6-channel ring.

n -> 17n

5*7 = 1 mod 17. phi(17) = 2^4. Opens the 7th channel. Completes the ring.

26,801,776

n -> 26,801,776*n

Terminal projector: 2^1680 mod 214,414,200. CRT = (0,1,1,1,1,1,1). Zeros out the mod-8 channel, preserves all six others. Idempotent: x^2 = x.

Why 13 and 17

13 closes the 6-channel ring. 17 opens the 7th channel. Both are forced:

13: lambda-compatible

lambda(13) = 12 | 420

12 divides 420. Adding 13 preserves the Carmichael period. The 6-channel ring Z/12,612,600 has lambda = 420.

17: lambda-extending

lambda(17) = 16

lcm(420, 16) = 1680. Adding 17 extends the period from 420 to 1680 = 4*420. The 7th channel needs 4x more steps.

13: sum of squares

2^2 + 3^2 = 13

Shadow: (13-1)/2 = 6 = 2*3. Both Cunningham chains terminate before reaching 13.

17: product identity

5*7 = 1 mod 17

The product of two inner primes is the identity in mod-17. phi(17) = 16 = 2^4.

13: CRT encoding

CRT(13) = (5, 4, 13, 13, 2, 0, 13)

13 encodes itself across seven channels and vanishes in its own (13 mod 13 = 0).

17: CRT encoding

CRT(17) = (1, 8, 17, 17, 6, 4, 0)

17 vanishes in its own channel (17 mod 17 = 0). mod-8 = 1: invisible to the even channel.

Primitivity Theorem (PROVED)

ord(13) = 420 = lambda in Z/970,200. 13 achieves maximal multiplicative order. Per-channel orders: (2, 3, 20, 14, 10). Sum = 49 = 7^2. At half-order in each cyclic channel: 13^(ord/2) = -1 mod m.

Powers of 2 mod 13

2 is a primitive root mod 13. ord(2, 13) = 12 = phi(13). Every element of Z/13* is a power of 2. The ring's primes all appear as 2^k mod 13:

Prime	2^k mod 13	Exponent k
3	2^4 mod 13 = 3	k = 4
5	2^9 mod 13 = 5	k = 9
7	2^11 mod 13 = 7	k = 11
11	2^7 mod 13 = 11	k = 7

7-11 Swap Theorem (PROVED)

7 = 2^11 and 11 = 2^7 mod 13. The two largest chain primes swap under exponentiation by 2. 7*11 mod 12 = 5: the exponents' product gives another chain prime. Sum of all exponents: 1+4+9+11+7 = 32 = 2^5.

3 is the unique QR

Legendre(3, 13) = +1

All other chain primes (2, 5, 7, 11) are quadratic non-residues mod 13. 3 alone has even exponent.

490 mod 13

ord(9, 13) = 3

490 = 2*5*7^2. 490 mod 13 = 9 = 3^2. Its order mod 13 is exactly 3.

Arithmetic

Sum (excl projector)

58 = 2 * 29

29 is the 10th prime. The new sum encodes the old term count.

Sum of 7 primes

58

Sum of the first 7 primes. -1 + 0 + 1 = 0, so the operator sum vanishes.

Sub-sum (first 6)

41 = the 13th prime

Sum of {-1,0,1,2,3,5,7,11,13} = 41 = KEY. The 6-channel result survives as partial sum.

Sub-sum (6 primes)

42 = 2*3*7

Sum of {2,3,5,7,11,13} = 42. phi(49) = 42. The 6-prime result survives.

Count

11 = L

The protector prime. 3 operators + 7 primes + 1 projector = 11.

Primorial

510,510

2*3*5*7*11*13*17 = rad(214,414,200) = 214,414,200 / 420.

Partial Sums

Accumulate the eleven terms from -1. Each partial sum encodes a structural value:

After term	Running sum	Name
-1	-1	Additive inverse
+0	-1	Unchanged
+1	0	Zero
+2	2	The first prime
+3	5	The next chain prime
+5	10 = 2*5	Product of first two chain primes
+7	17	The 7th prime -- and the 7th channel
+11	28 = 2^2*7	The 2nd perfect number
+13	41	The 13th prime (KEY)
+17	58 = 2*29	29 = the 10th prime

Gap Structure (PROVED)

Gaps between consecutive terms: {0, 1, 2, 3, 5, 7, 11, 13, 17}. The sequence contains its own differences -- now including 17. Consecutive differences: {1, 1, 1, 1, 2, 2, 4, 2, 4}. All powers of 2.

Explore: Partial Sums

Enter k (1-11) to see the k-th term and the running sum after k terms.

Enter k (1-11):

Try: k=7 (sum=17), k=9 (sum=41 = 13th prime), k=10 (sum=58 = 2*29), k=8 (sum=28 = perfect number).

Contrast

Question	Standard	Axiom
How many fundamentals?	Varies by theory	11 = L (protector prime). 3 operators + 7 primes + 1 projector.
Why these numbers?	No structural reason	Two Cunningham chains from {1, 2}, 13 closes 6ch, 17 completes 7ch
Does the list know itself?	No	Sub-sum = 41 = p_13. Count = 11 = L. Sum = 58 = 2*p_10.
Do the gaps mean anything?	No structure	All gaps are powers of 2. Sequence contains its own differences.
Why 17?	Just a prime	5*7 = 1 mod 17. Extends lambda from 420 to 1680. Completes the 7th channel.

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