CRT Anatomy

n = (n mod 8, n mod 9, n mod 25, n mod 49, n mod 11, n mod 13, n mod 17)

Every element of Z/214,414,200 decomposes into seven independent channels. The decomposition IS the identity. CRT is 2300 years old (Sun Tzu, ~300 BC). This page shows what it reveals when the modulus is 214,414,200 = 2^3 * 3^2 * 5^2 * 7^2 * 11 * 13 * 17.

Seven Channels

The Chinese Remainder Theorem

Z/214,414,200 = Z/8 x Z/9 x Z/25 x Z/49 x Z/11 x Z/13 x Z/17. Forced by prime power factorization. Each channel is independent: knowing n mod 8 tells you NOTHING about n mod 9. Seven orthogonal views of one number. Reconstruction is unique and total.

Channel	Space	Values	Property
mod 8	Z/8 (mod 2^3)	8	Nilpotent: 2^3 = 0. Only even-prime channel.
mod 9	Z/9 (mod 3^2)	9	3^2 = 0. Smallest odd prime-power.
mod 25	Z/25 (mod 5^2)	25	5^2 = 0. Discriminant of x^2-x-1.
mod 49	Z/49 (mod 7^2)	49	7^2 = 0. Deepest resolution (49 states).
mod 11	Z/11 (mod 11)	11	Prime field. 1+2+3+5 = 11. Error detection.
mod 13	Z/13 (mod 13)	13	Prime field. Cunningham chain stops here.
mod 17	Z/17 (mod 17)	17	Prime field. 5*7 = 1 mod 17. Completes the ring.

Total: 8 * 9 * 25 * 49 * 11 * 13 * 17 = 214,414,200. Seven primes, each at its maximum power. The product IS the ring.

Named Constants

Some notable elements of Z/214,414,200 and their CRT decompositions:

Value	Factorization	CRT (mod 8,9,25,49,11,13,17)	Note
0	zero	(0,0,0,0,0,0,0)	All channels zero
1	identity	(1,1,1,1,1,1,1)	All channels alive
2	prime	(2,2,2,2,2,2,2)	The only even prime
3	prime	(3,3,3,3,3,3,3)	Smallest odd prime
5	prime	(5,5,5,5,5,5,5)	disc(x^2-x-1) = 5
7	prime	(7,7,7,7,7,7,7)	Last prime-power channel (mod 49)
11	prime	(3,2,11,11,0,11,11)	mod-11 channel = 0!
13	prime	(5,4,13,13,2,0,13)	mod-13 channel = 0!
17	prime	(1,8,17,17,6,4,0)	5*7=1 mod 17. mod-17 = 0!
42	237	(2,6,17,42,9,3,8)	phi(49) = 42. Mod-49 channel order.
105	357	(1,6,5,7,6,1,3)	Product of odd chain primes
137	prime	(1,2,12,39,5,7,1)	Fine structure constant
1,576,576	2^420 mod 12,612,600	(0,1,1,1,1,1,13)	mod-8 = 0 (projector in the 6-channel ring)
214,414,199	N-1	(7,8,24,48,10,12,16)	All channels at maximum

Explore: CRT Calculator

Enter any number. See its CRT decomposition, class, coupling, mirror, and trace. Every claim on this page is verifiable here.

Decompose any number:

Try: 0 (void), 1 (identity), 1576576 (2^420 mod N), 214414199 (mirror of 1), 42 (2*3*7), 137 (prime).

CRT Reconstruction: Perfect Hash

The decomposer above splits a number into channels. This reverses it: enter channel values, get the unique integer. CRT guarantees ZERO collisions -- a perfect hash function by theorem. No hash table resizing. No collision chains. No load factor.

CRT Bijection

The map n -> (n mod 8, n mod 9, n mod 25, n mod 49, n mod 11, n mod 13, n mod 17) is a bijection on Z/214,414,200. Every valid 7-tuple reconstructs a unique integer via two-level CRT: first reconstruct in Z/12,612,600, then lift via inv(11,17)=14. Insert O(1). Lookup O(1). Range query per channel O(bucket_size). No rebalancing. 7 independent index dimensions for free.

Enter CRT channel values:

mod 8 (0-7)

mod 9 (0-8)

mod 25 (0-24)

mod 49 (0-48)

mod 11 (0-10)

mod 13 (0-12)

mod 17 (0-16)

Try: (1,1,1,1,1,1,1) = 1, (0,0,0,0,0,0,0) = 0, (0,1,1,1,1,1,13) = 1,576,576, (1,2,12,39,5,7,1) = 137.

7-dim address

CRT = perfect hash

214,414,200 = 8*9*25*49*11*13*17. Every integer has a unique address in this 7-dimensional space. No collisions possible.

Per-channel query

Independent index

Each channel IS a dimension. Query 'mod 49 = 7' returns exactly 1/49 of all elements. Seven B-trees for free.

CC0

No patents

CRT indexing is public domain. The theorem gives you the perfect hash. No secret needed.

Mirror Duality

Trace Sum Formula

For any element n coprime to N: Tr(n) + Tr(N-n) = sum of moduli = 8+9+25+49+11+13+17 = 132. Trace = sum of CRT residues. The mirror always completes what the element lacks.

n = 1

Tr = 7

All 7 channels at minimum nonzero (each residue = 1).

n = 214,414,199

Tr = 125

All 7 channels at maximum (7+8+24+48+10+12+16).

n = 1,576,576

Tr = 21

mod-8 channel = 0. Other 6 channels alive.

n = 212,837,624

Tr = 102

Mirror of 1,576,576. Also mod-8 = 0, other channels at max-1.

Spectral Mirror Theorem

lambda(n) = lambda(N-n). Exhaustive check: 485,101 pairs, every one matches. The mirror is spectrally invisible -- you cannot distinguish an element from its mirror by eigenvalue alone.

Class from Anatomy

The CRT tuple determines everything. Class = which channels are zero. Coupling = product of nonzero channel sizes. Eigenvalue = spectral signature of the class.

All nonzero

Unit

Maximum coupling. Coprime to N.

mod-8 = 0

2-divisible

Even numbers. Coupling = N/2.

mod-9 = 0

3-divisible

Multiples of 3. 18 lives here (mod 9 = 0).

All zero

Void

Only n = 0. Zero coupling.

mod-11 = 0

11-divisible

11 itself lives here (11 mod 11 = 0). The prime that detects errors has its own channel dead.

Contrast Table

Aspect	Standard View	Ring Structure View
CRT	Number theory technique	The identity of every element -- seven independent channels
Modular residues	Remainders after division	Independent channels, each with its own structure
Mirror	N-n = additive inverse	Tr(n)+Tr(N-n) = 132 for units. Spectral twin.
Notable elements	Just particular numbers	Each has a CRT address revealing its structure
Zero channels	Divisibility	Class -- which channels are zero
mod-11 channel	mod 11 residue	Error detection. 11 itself has its own channel = 0.