The Septum

192 / 1001

Of every 1001 elements in Z/12,612,600, exactly 192 are units -- invertible, with all 6 CRT channels nonzero. The ratio 192/1001 = 19.18% comes from multiplying six Euler factors: (1-1/2)(1-1/3)(1-1/5)(1-1/7)(1-1/11)(1-1/13). One factor per prime. The other 80.82% are zero-divisors: at least one channel is zero, so they cannot be inverted.

The Derivation

An element n is a unit (invertible) if and only if gcd(n, N) = 1 -- it shares no prime factor with N. The count of such elements is Euler's totient:

Septum Theorem (PROVED)

phi(N)/N = (1-1/2)(1-1/3)(1-1/5)(1-1/7)(1-1/11)(1-1/13) = (1/2)(2/3)(4/5)(6/7)(10/11)(12/13) = 192/1001 = 19.18%. The ratio depends only on which primes divide N, not their exponents. phi(12,612,600) = 2,419,200 units. Z/970,200 (5 primes, no 13): 16/77 = 20.78%.

Numerator: 192 = 2^6 * 3. Denominator: 1001 = 7 * 11 * 13. The fraction of units is determined entirely by the six primes dividing the ring.

Units

2,419,200

phi(12,612,600). All 6 CRT channels nonzero.

Zero-divisors

10,193,400

N - phi(N). At least one channel is zero. 80.82%.

6-prime ratio

192/1001

= 2^6*3 / (7*11*13). Euler product of the six primes.

5-prime ratio

16/77

= 2^4 / (7*11). Without the mod-13 channel. 20.78%.

Projector Decomposition

The identity 1 splits into two orthogonal idempotent projectors:

1,576,576

CRT = (0, 1, 1, 1, 1, 1)

2^420 mod 12,612,600. Kills the mod-8 channel, preserves the other five. Idempotent: x^2 = x.

11,036,025

CRT = (1, 0, 0, 0, 0, 0)

The complementary projector. Preserves only the mod-8 channel.

Sum = 1

1,576,576 + 11,036,025 = 12,612,601 = 1 mod N

The two projectors add to the identity. Their product is 0. Orthogonal decomposition.

These are the two extremes of unit structure: the projector that kills one channel (mod-8) and the one that keeps only that channel. Together they partition the ring.

The Persistence Gradient

How long does each element survive before self-annihilating? The chain IS a persistence gradient:

Element	Steps to 0	Bits/step	Meaning
1	infinite	-	Never annihilates. 1^k = 1 forever.
2	3 steps	1.0	Most persistent prime. 2^3 = 0 mod 8.
3	2 steps	1.58	3^2 = 0 mod 9.
5	2 steps	2.32	5^2 = 0 mod 25.
7	2 steps	2.81	7^2 = 0 mod 49. Deepest 2-step pool.
11	1 step	3.46	11 = 0 mod 11. Instant.
13	1 step	3.70	13 = 0 mod 13. Instant.
0	0 steps	-	Already zero.

Persistence Theorem (PROVED)

p^k -> 0 mod p^e takes exactly e steps. The Pareto exponents {3, 2, 2, 2, 1, 1} are the persistence depths. 2 is the most persistent prime (3 steps). 11 and 13 are the most volatile (1 step each).

Information drip rate: 2 leaks 1 bit/step (slowest), 7 leaks 2.81 bits/step (deepest 2-step pool), 11 leaks 3.46 bits in 1 step.

The Seventh Prime

Adding the seventh prime (17) extends the ring from Z/12,612,600 to Z/214,414,200. The unit fraction gains one more Euler factor: (16/17). Everything above remains true -- the six-prime foundation is unchanged. The seventh prime deepens it.

TRANS Septum (PROVED)

phi(214,414,200) / 214,414,200 = 3072/17017 = 18.05%. Seven Euler factors: (1/2)(2/3)(4/5)(6/7)(10/11)(12/13)(16/17). Numerator: 3072 = 2^10 * 3. Denominator: 17017 = 7 * 11 * 13 * 17. Exactly 38,707,200 units in a ring of 214,414,200 elements.

6 primes

192/1001 = 19.18%

2,419,200 units. 64 idempotents. 128 involutions.

7 primes

3072/17017 = 18.05%

38,707,200 units. 128 idempotents. 256 involutions.

Factor

16/17

192/1001 * 16/17 = 3072/17017. Each new prime reduces unit density. phi(17) = 16 enters the numerator; 17 enters the denominator.

Projector

CRT = (0, 1, 1, 1, 1, 1, 1)

26,801,776 = 2^1680 mod 214,414,200. The seven-channel projector kills mod-8 and preserves six channels. Complement: 187,612,425 with CRT = (1, 0, 0, 0, 0, 0, 0).

Persistence extends: 13 and 17 both have exponent 1 (one step to self-annihilation), joining 11. The full persistence depths are {3, 2, 2, 2, 1, 1, 1} -- the Pareto exponents. 2 remains the most persistent prime. The three extension primes (11, 13, 17) are all instantaneous.

Explore: Euler Totient Ratio

Enter any N to compute phi(N)/N -- the fraction of the ring that is alive (units).

Enter N:

Try: N=214414200 (7 primes, 3072/17017), N=12612600 (6 primes, 192/1001), N=970200 (5 primes, 16/77), N=2310 (5 thin, 8/21), N=210 (4 primes, 8/35), N=30 (3 primes, 4/15).

Contrast

Question	Standard	Axiom
Unit density	phi(N)/N, a fraction	192/1001. Six Euler factors, one per prime. Ratio independent of exponents.
Zero-divisors	Elements without inverses	80.82%. At least one CRT channel is zero. Cannot participate in full multiplication.
Persistence	No standard notion	Exponents {3,2,2,2,1,1} = steps to self-annihilation. 2 survives longest (3 steps). 11 and 13 fastest (1 step each).
Projector split	Abstract algebra	1 = (0,1,1,1,1,1) + (1,0,0,0,0,0). Two orthogonal idempotents summing to the identity. Product = 0.
7th prime	Just another factor	3072/17017 = 18.05%. Adding 17 multiplies by 16/17. Idempotents double (64 to 128). Same structure, one more channel.