Z/214,414,200 = Z/8 x Z/9 x Z/25 x Z/49 x Z/11 x Z/13 x Z/17 (seven CRT channels). Lambda = 1680 (full period), 420 (six-channel sub-period). Named values: 41 = 7^2-7-1, 42 = 2*3*7, 105 = 3*5*7, 35 = 5*7 (since 5*7 = 1 mod 17). The widget decomposes any input into seven CRT channels.
NOTE: An earlier version of this page presented matches between ring ratios and physical constants (Kolmogorov 5/3, Kleiber 3/4, particle masses, solar cycles). Null model testing showed these are coincidences: the expression space of prime-power products is dense enough that ANY 7 small primes can match any numerical target to 0.1%. The ring's algebraic properties (below) stand independently.
Explore: Named Constants
Enter a number to see its name (if any) and CRT decomposition across all seven channels. Lattice counts: 108 rings share lambda = 420 exactly; 84 of those divide 12,612,600; 96 rings share lambda = 1680 at the TRANS level.
67 appears wherever the ring counts itself. It is the additive center of the ring's named constants: the unique value where inv_sum, tower products, and the L^2 triangle all meet.
Soul Center (Theorem 125)
L^2 = b^2 + D^3*K^2 = 49 + 72 = 121. This single identity generates: SOUL = inv_sum(TRUE) = 67, SOUL = L^2 - KEY - GATE = 121 - 41 - 13, SOUL = K*ME + GATE = 3*18 + 13 (K times inner mass + boundary), SOUL = D^3*K^2 - E = 72 - 5 (tower product minus observer). Each inv_sum channel is axiom-native: {b, sigma, f(E), K*L, K, D^2}.
inv_sum
b + sigma + f(E) + K*L + K + D^2 = 67
The sum of per-channel inversion costs across TRUE = 7+1+19+33+3+4 = 67. The CRT lifting cost IS the soul.
L^2 triangle
KEY + SOUL + GATE = 121 = 11^2
Three structural constants sum to the protector squared. Zero free parameters: all three algebraically forced.
Tower product
D^3*K^2 - E = 72 - 5 = 67
The first two tower products (8*9) minus the observer (5). The soul is what remains after observation.
SOUL + G pairing
67 + 97 = 164 = D^2*KEY
SOUL pairs with the coupling constant 97 to give 4*41.
61: The Grief Invariant
61 detects the TRUE boundary. Its CRT residue in Z/49 is 12, a primitive root of the depth-squared channel. In the lambda-420 lattice, 61 distinguishes the 84 TRUE-divisor rings from the 24 D^4-extensions.
Grief Lattice Invariant (Theorem 126)
GRIEF = b*L - D^4 = 77 - 16 = 61. 61 mod 49 = 12 is a primitive root of Z/49: ord(12, Z/49) = 42 = phi(49). In all 84 TRUE-divisor lambda-420 rings: ord(GRIEF) = 210 = DATA. In all 24 D^4-extension rings: ord(GRIEF) = 420 = lambda. GRIEF detects the TRUE boundary. ZD(DEEP) = GRIEF/(b*L) = 61/77.
Primitive root
12^21 mod 49 = 48, 12^42 mod 49 = 1
12 = 61 mod 49 generates all of Z/49*. The b-channel order is 42 = 2*3*7 = ANSWER.
Lattice detector
84 rings: ord=210. 24 rings: ord=420
The multiplicative order of GRIEF jumps from DATA to lambda at the TRUE boundary. A CRT-level alarm.
84+24 = 108
D^2*K*b + D^3*K = D^2*K^3
The lattice split into TRUE-divisors and D^4-extensions. 84 = D^2*K*b. 24 = D^3*K.
Mathieu staircase
Exp sums {8, 11, 12, 13, 17} sum to 61
The five Mathieu groups M_11...M_24 have total prime factor counts {D^3, L, 12, GATE, ESCAPE}. Their sum = GRIEF. Extended through Conway and Monster: total = 187 = L*ESCAPE.
42: The Visibility Ladder
42 = 2*3*7 has thin-DATA CRT = (0, 0, 2, 0): only the E-channel sees it. The ANSWER can only be OBSERVED. Fattening dissolves thin zeros: 42 mod 2 = 0 but 42 mod 8 = 2.
CRT Visibility Ladder (Theorem 128)
ANSWER = 42 = D*K*b has CRT = (0, 0, 2, 0) at DATA (thin channels). The ANSWER is invisible to D, K, and b -- visible ONLY through E = the observer. Primorial complement (510510/p mod p) encodes the D-channel Pareto ladder: E->D, b->D^2, ESCAPE->D^3. GATE residue = Decality = K+b = 10. Sigma-primes {D, K, L} sum = D^4 = phi(17).
Only E sees 42
42 mod 5 = 2, all other thin = 0
D*K*b vanishes in its own channels. The three multiplied primes cancel themselves. Only the non-factor E survives.
D-channel Pareto
E->D, b->D^2, ESC->D^3
The primorial complement (510510/p mod p) traces the D-channel exponent ladder: the Pareto depths {1,2,3} read in reverse.
Fattening dissolves
42 mod 2 = 0, 42 mod 8 = 2
At thin level, 42 is invisible to D. At fat level (D^3 = 8), the D-channel sees residue 2. Raising exponents reveals structure.
OMEGA consistency
CRT(1576576) = (0, 1, 1, 1, 1, 1)
The projector's D-channel = 0, all others = 1. OMEGA kills D and preserves everything else.
Self-Resonance: Own-Prime Power Maps
CRT Self-Resonance Activation (Theorem 184)
The self-map x -> x^p on each CRT channel Z/p^k produces exactly p fixed points. Extension channels (11, 13, 17) are pure IDENTITY: x^p = x for all x. Z/210 channels (2, 3, 5, 7) are PROJECTIVE: they absorb multiples to void and compress units. The 490 split IS the identity/projective split. Product of Z/210 survival fractions = 1/lambda = 1/420. 18/18 verified.
Channel
Fixed pts
Behavior
2 (Z/8)
2
PROJECTIVE: absorbs 6 of 8
3 (Z/9)
3
PROJECTIVE: absorbs 6 of 9
5 (Z/25)
5
PROJECTIVE: absorbs 20 of 25
7 (Z/49)
7
PROJECTIVE: absorbs 42 of 49
11 (Z/11)
11
IDENTITY: all 11 fixed
13 (Z/13)
13
IDENTITY: all 13 fixed
17 (Z/17)
17
IDENTITY: all 17 fixed
Fermat's little theorem: x^p = x (mod p) for prime p. Extension channels have k=1, so the identity holds universally. Z/210 channels have k>=2, so x^p = x only for multiples of p (absorbed to 0) or the fixed-point subgroup. The 490 split cleanly separates identity channels from projective channels.
Lambda Convergence Staircase
CRT Lambda Convergence Staircase (Theorem 185)
The power map x -> x^n converges to binary {0,1} on each CRT channel at the channel's convergence power. Cumulative lcm in chain order: 4, 12, 60, 420, 420, 420, 1680 with ratio chain 4, 3, 5, 7, 1, 1, 4. The 2-channel is UNIQUE: depth (3) exceeds lambda (2). 11+13 ride free. 17 alone requires 2^2 extension. Full = 1680 = lambda(Z/214,414,200). 18/18 verified.
Channel
Conv power
Cumulative lcm
2 (Z/8)
4 = 2^2
4
3 (Z/9)
6
12 = lambda(Z/210)
5 (Z/25)
20
60 = |A_5|
7 (Z/49)
42
420 = lambda
11 (Z/11)
10 = 2*5
420 (free ride)
13 (Z/13)
12 = 2^2*3
420 (free ride)
17 (Z/17)
16 = 2^4
1680 = lambda(Z/214,414,200)
4 bookend
Opens and closes the staircase
2-channel opens with 2^2=4. 17-channel gap 1680/420 = 4. The pair bookends the convergence structure.
Inner product
3*5*7 = 105
Staircase ratios for inner channels: 3, 5, 7. Product = 105. Emerges from convergence arithmetic.
11+13 free
Lambdas divide 420
11: lambda=10|420. 13: lambda=12|420. They ride the period 420 for free.
Optimal Totient Pairing
CRT Optimal Totient Pairing (Theorem 186)
Among all 105 possible 3-pair-plus-singleton configurations of the 7 CRT channel totients, the shared-factor product is maximized at 240 = P(7) by exactly 5 configurations. 2-primitive-root homogeneity selects a UNIQUE canonical pairing: (2<->17, 3<->13, 5<->11), 7=pivot. 20/20 verified.
Pair
phi values
gcd
2 <-> 17
4, 16
4 = 2^2
3 <-> 13
6, 12
6 = 2*3
5 <-> 11
20, 10
10 = 2*5
7 = pivot
42
--
105 configurations
C(7,1)*C(6,2)*C(4,2)/3!
All 105 ways to split 7 totients into 3 pairs + 1 singleton. Max gcd product = 240 = P(7) = lcm(axiom totients).
2-order ratios
12/6 = 2, 20/10 = 2
Both non-2 pairs have convergence-power ratio exactly 2. The pair doubles between paired channels.
Runner-up gap
240/80 = 3
The runner-up configuration achieves product 80. Max/runner-up = 3.