Z/12612600 = Z/8 x Z/9 x Z/25 x Z/49 x Z/11 x Z/13 = T^6
Think of a merry-go-round with six rings, each spinning at a different speed. The inner ring completes a full turn in 2 clicks. The next in 3. Then 5, 7, 11, 13. They all start aligned. How long until they align again? Exactly 30030 clicks -- the thin ring.
Now give each ring its full prime-power depth: 8, 9, 25, 49, 11, 13 positions. It takes 12,612,600 clicks to realign. Adding 1 to any number advances every ring by one click. This is the carousel -- addition as rotation through a six-dimensional torus.
Six of the seven prime-power channels (the mod-17 channel extends to Z/214,414,200) close into this 6-torus via CRT. 64 = 2^6 idempotent switches. Max squaring cycle = 12 = Carmichael lambda of Z/210. Chromatic number chi = max(q_i) -- the deepest channel decides coloring. The mod-49 channel rises highest on the braid (+457); the mod-8 channel is forced by three polynomials.
The 7 circles of T^7 = S^1(8) x S^1(9) x S^1(25) x S^1(49) x S^1(11) x S^1(13) x S^1(17) form the Gabriel-Horn cap of Z^7 -- finite volume over unbounded surface. The spectral gap 4*sin^2(pi/49) is locked at 49 across all 108 lambda-420 rings: Theorem 33 STRUCTURAL Yang-Mills analog, Theorem 34 STRUCTURAL Navier-Stokes horn analog (both are structural analogies, not proofs of the Clay problems). The Z/12,612,600 carousel = T^6 (six rings); Z/214,414,200 extends to T^7 (adds the mod-17 channel).
Frozen Ring Theorem
When n is divisible by a prime p, the corresponding ring is FROZEN at position 0. It cannot spin. Divisibility is constraint: the primes that compose n are exactly the channels it cannot move in.
H(n)
6 - omega(gcd(n, N))
Hamming weight = spinning rings. H(0) = 0 (axis, all frozen). H(units) = 6 (all spinning). N = 12612600.
Reaching Theorem
n spins what it lacks
105 = 3*5*7 has channels 2, 11, 13 spinning. 210 = 2*3*5*7 spins only 11 and 13. Elements activate exactly the channels they are not divisible by.
Mode
H = 5 (typical)
In Z/12,612,600, each prime factor freezes one of 6 rings. phi(N) = 2,419,200 units at H = 6 (all spinning). The element 490 = 2*5*7^2 freezes the inner channels {2,5,7}, leaving boundary channels {3,11,13} spinning.
Axis duality
0 vs 1
Void: (0,0,0,0,0,0) = perfectly still. The identity: (1,1,1,1,1,1) = maximum motion. These two extremes anchor the carousel.
The Cost of Spinning
Each ring you activate costs 4*sin^2(pi/p) in phase coherence. The void (all off) has eigenvalue 10 -- maximum coherence. Every spinning ring reduces it. The hierarchy activates expensive rings first and cheap rings last.
Ring
Cost 4*sin^2(pi/p)
Meaning
p = 2
4.000 (maximum)
Most expensive channel. 4*sin^2(pi/2) = 4.
p = 3
3.000
Second highest cost.
p = 5
1.382
Mid-range. phi^2 = (golden ratio)^2.
p = 7
0.753
Mod-49 channel costs little to activate.
p = 11
0.317
Cheapest to activate among the first five.
p = 13
0.229
Cheapest overall. The boundary channel costs almost nothing to maintain.
Total cost (all 6 on): 9.681. The hierarchy builds in OPPOSITE order of cost: 2 > 3 > 5 > 7 > 11 > 13. The sequence activates expensive rings first. The mod-13 channel is cheapest -- the boundary channel costs almost nothing to maintain.
The CRT Braid
Six CRT channels = 6 strands cycling at rates 2, 3, 5, 7, 11, 13. Walking n = 0 to N-1 makes each strand wrap at its prime period. The interleaving of these six rhythms IS a braid -- not metaphorically, but literally in the mathematical sense.
Crossings
11,822 per Z/2,310 walk
EXACT BEAT FORMULA: net per beat = -gcd(p_i - 1, p_j - 1). Total signed chirality: -gcd(p_i-1, p_j-1) * N/(p_i*p_j). All 10 pairs verified. The 11,822 count is on Z/2,310; Z/12,612,600 crossings scale proportionally.
Density
[143, 151] per 30-step window
Uniform. No clustering. The braid has no preferred location -- crossings spread evenly through the walk.
CRT Neighbors
109 = Hamming degree in Z/12,612,600
Every element shares 5 of 6 channels with exactly 109 others. Degree = sum(q_i - 1) = 7+8+24+48+10+12 = 109. The Z/970,200 subring (without mod-13) has degree 97. The topology IS the algebra.
Twist ratios between strands: 3/2 (the musical fifth), 5/3 (Kolmogorov turbulence ratio), 7/5, 11/7, 13/11. The braid's internal rhythm encodes these ratios.
Strand Height Order
Strand Height Theorem
Row sums of the interleaving matrix: 7(+457) > 11(+389) > 5(+323) > 3(-283) > 2(-886). Sum = 0 (conservation).
The mod-2 strand sinks deepest (-886); the mod-7 strand rises highest (+457). Conservation: the total signed displacement is exactly zero -- what one strand gains, the others lose.
10 Torus Knots
C(5,2) = 10 pairs from {2,3,5,7,11} give 10 torus knots T(p_i, p_j). Seifert matrix eigenvalues = grid Laplacian on (p-1) x (q-1). ALL 10 have NEGATIVE signature (left-handed, consistent with chirality).
Level
Total Signature
Factorization
Note
Z/6 (2*3)
-2
2
Trefoil. Simplest nontrivial knot.
Z/30 (2*3*5)
-12
4*3 = 12
Z/210 (2*3*5*7)
-42
2*3*7 = 42
Z/2,310 (2*3*5*7*11)
-132
4*3*11 = 132
Full thin ring.
Crossing number ladder: level 2 has 3 crossings, level 3 has 18, level 4 has 67, level 5 has 210. Torus knot complexity at each level tracks the ring hierarchy.
CRT Hamming Graph
The CRT neighbor graph -- elements sharing k-1 of k channels -- IS the Hamming graph H = K_q1 [] K_q2 [] ... [] K_qk (Cartesian product of complete graphs). This is provably the topology of the product.
Hamming Spectrum Theorem
Eigenvalues: lambda_S = sum_{i in S} q_i - k, for each subset S of factors. Multiplicity: prod_{i not in S}(q_i - 1). Max eigenvalue = degree = sum(q_i - 1). Min eigenvalue = -k. 2^k subset signatures per ring: Z/970,200 (k=5) gives 32 subsets, Z/12,612,600 (k=6) gives 64, Z/214,414,200 (k=7) gives 128. Whether the corresponding eigenvalue VALUES are all distinct depends on superlacunarity (Z/970,200 has 32 distinct; Z/12,612,600 has collisions at adding 13).
Z/2,310 degree
23
Number of distinct eigenvalues = max eigenvalue = degree = 23.
Z/12,612,600 degree
109
7+8+24+48+10+12 = 109. Each channel Z/q_i has Lie algebra A_{q_i-1}. The Z/970,200 subring (without mod-13) has degree 97. Adding mod-13: degree increases by 12.
Z/2,310: 23-21 = 2. Z/970,200: 97-89 = 8. Z/12,612,600: 109-101 = 8. The Hamming gap is always a power of 2 -- the mod-8 channel controls topology at every level.
Shell Polynomial and Chromatic Depth
The shell polynomial E(x) = prod(1 + (q_i-1)*x) counts elements at each CRT distance. Coefficient e_d = number at distance d from any fixed point.
Z/210 shells
e_2 = 56 = dim(E7)
e_0=1, e_1=13, e_2=56 (E7 fundamental rep), e_3=92, e_4=48 = phi(210). Distance-2 shell IS the Lie dimension.
Z/2,310 shells
e_3 = 652 = 4*163
163 = Heegner prime. e_4 = 968 = 8*121. Shell sizes factor into ring constants.
Chromatic number
chi = max(q_i)
Z/210: chi = 7. Z/2,310: chi = 11. Z/12,612,600: chi = 49. Hamming graphs are perfect: chi = omega. The deepest channel decides coloring.
Z/2,310 independence
alpha = 210
Largest non-conflicting set in the Z/2,310 Hamming IS the Z/210 subring. Independence number alpha = N/max(q_i).
Kleiber null
360/480 = 3/4
Null eigenvalue exists because 5 = 2+3. Two subsets: {5} and {2+3}. Merged multiplicity 360. Kleiber's 3/4 ratio from the fact that the third prime equals the sum of the first two.
Multiplicative Dynamics
The Carmichael lambda = smallest e such that a^e = 1 (mod N) for ALL coprime a. It measures how long multiplicative orbits can last. The Carmichael lambda ladder:
Ring
lambda
Ratio to Previous
Z/6
2
--
Z/30
4
x2
Z/210
12
x3
Z/2,310
60
x5
Z/12,612,600
420
x7
Each new level multiplies the Carmichael lambda by the next prime in sequence: 2, 3, 5, 7. The ratios ARE the chain. lambda(Z/12,612,600) = 420 = 4*3*5*7 = 2*210. The meta-ring Z/420 has phi(420)/classes(420) = 96/72 = 4/3 = 1/Kleiber.
N/lambda = 12,612,600/420 = 30,030 = primorial(13). Multiplying is 30,030x faster than adding. And 11 is invisible: lambda(11) = 10 divides 420, contributing nothing new to the lcm.
Squaring Map Dynamics
The simplest nonlinear dynamics: x -> x^2 mod N. Fixed points = idempotents = the 64 switches (2^6, one per ON/OFF configuration of the 6-torus). Units cycle directly; non-units must purify through the void first.
Cycle lengths
{1, 2, 3, 4, 6, 12}
ONLY divisors of 12 appear. On Z/970,200 (full scan of all 970,200 elements), period 12 dominates at 642,816/970,200 = 66.3%. The vocabulary is complete.
Two-levels-down
max cycle(level k) = lambda(level k-2)
Z/12,612,600 max = 12 = lambda(Z/210). Z/2,310 max = 4 = lambda(Z/30). The squaring map remembers two levels back. This property is specific to the chain {2,3,5,7,11,13}.
Class collapse
Z/210 -> e_11 in 2 steps
The mod-210 channels self-annihilate under squaring; only the mod-11 channel preserves. Z/2,310 -> void in 2 steps (complete annihilation). Sum of class cycles: 2 + 3 + 4 = 9.
Per-channel structure: mod-8 is all period-1. mod-9 is awakened by fattening (was trivial in thin). mod-25 has only period 4. mod-49 is richest (periods 1,2,3,6). mod-11 is invariant under fattening. The mod-25 channel kills all dynamics because 5-1 = 4 is a pure 2-group.
Topological Encoding
Partial Ring Encoding Theorem
The topology of a partial ring encodes primes it has never met. Z/30 = {2,3,5} has Euler characteristic chi = -29 = -(1+2+3+5+7+11) = the sum of the first six terms in the chain.
Fat level 2
Z/72 = Z/8 x Z/9
chi = -55 = 5*11. Only 2 and 3 present, yet topology produces 5*11. And b1 = 56 = dim(E7). The fat seed encodes three primes it has never met.
Three polynomials
P1, P2, P3 all have root x=2
P1 = x^3 - 3x - 2. P2 = 4x^3 + x^2 - 14x - 8. P3 = x^5 + 2x^4 - 9x^2 - 12x - 4. x = 2 is the unique positive integer root of all three.
Missing prime theorem
{2,5,7}: chi = -81 = 3^4
When 3 is absent, topology raises it to the 4th power. {3,5,7}: b1 = 140 = 4*5*7, which contains the factor 2 even though 2 is absent. Topology names what is missing.
13 forced
13 | chi(Z/12,612,600)
Algebraically forced by the identity 3*5 - 2 = 13. The factor 13 in the Euler characteristic is the same 13 from this convergence relation.
Betti Numbers and Lie Algebras
CRT Betti Number Theorem
The Hamming clique complex has b0 = 1, b1 = 1 - chi, and b_s = 0 for all s >= 2. Homotopy type: wedge of b1 circles. All topology concentrates in dimension 1.
Pair Z/p x Z/q
b1 = (p-1)(q-1)
Lie Algebra
Z/8 x Z/9
56
dim(E7 fundamental)
Z/25 x Z/11
240
|roots(E8)|
Z/3 x Z/5
8
dim(SU(3))
Z/5 x Z/7
24
dim(SU(5))
Z/49 x Z/11
480
phi(Z/2,310)
Z/9 x Z/49
384
b1(Z/210) = 128*3
b1-SU Theorem: b1(Z/p x Z/(p+2)) = p^2 - 1 = dim(SU(p)). Trivially forced by the gap being 2 (difference of squares). PSL Ladder: b1(3,7) = 12 = |PSL(2,3)|, b1(7,11) = 60 = |PSL(2,5)|, b1(8,25) = 168 = |PSL(2,7)|.
Silence Theorem
b1 mod q_j = 0 iff condition
b1 vanishes in channel j iff prod of OTHER factors = 1 mod q_j. Z/30 is the unique all-silent triple. The mod-2 channel is always silent in the squarefree chain (product of odds = 1 mod 2). 5,888/5,888 verified.
Z/210 b1/phi
384/48 = 8
The Z/210 ring's Betti-to-totient ratio is 8. Protected by 383 = 384 - 1 (prime).
Mandelbrot on the Ring
x -> x^2 + c on the ring. CRT-decomposable per channel. Fattening from Z/2,310 to Z/970,200 reveals which channels drive dynamics. (NOTE: specific max values in the table below are canonical on Z/970,200, where the Mandelbrot fattening theorem was proved; Z/12,612,600 extends these but exact max cycles at that level require full re-scan.)
c
Z/2,310 max
Z/970,200 max
Ratio
c = 0
4
12
3x
c = 2
2
56
28x EXPLOSION
c = 3
6
12
2x
c = 5
2
12
6x
c = 7
6
24
4x
c = 11
12
12
1x INVARIANT
c = 105
10
60
6x OPTIMAL
11-invariance
11 stays at 12 under fattening
The mod-11 channel has exponent 1 (Z/11 identical in thin and full ring). Its dynamics do not change when fattening.
c=2 explosion
c=2: 28x amplification
c = 2 excites ALL fat channels to maximum. mod-49 gets 7-cycle, mod-25 gets 8-cycle. The first prime unleashes the most dynamics.
Full Carmichael lambda
1,344 values achieve max = 420
1,344 = 64*3*7. First such c: 688 = 16*43 (43 is Heegner). Only 0.14% of c values on Z/970,200 achieve maximum dynamics.
All-smooth
Every max cycle divides 420
Full scan of all N values: ALL max cycles are {2,3,5,7}-smooth. 29 distinct values. 11 and 13 are absent from all Mandelbrot cycles. The Mandelbrot ceiling = Carmichael lambda, not N.
The 64 Switches
Z/12,612,600: 64 idempotents = 2^6 (6-cube). Z/214,414,200: 128 = 2^7 (7-cube, adding the mod-17 channel). Each switch e_S has CRT coordinates: 1 in the channels belonging to S, 0 elsewhere. The control panel of the carousel.
Conservation
|IMAGE| * |KERNEL| = N = 12,612,600
For every idempotent e: what it keeps times what it destroys equals the full ring. Image = product of ON-channel factors; Kernel = product of OFF-channel factors; CRT forces |Image|*|Kernel| = N. Conservation law of projection, holds at every level.
Class switches
One channel OFF = one class
Unit [******], mod-2 off [.*****], mod-3 off [*.****], mod-5 off [**.***], mod-7 off [***.**], mod-11 off [****.*], mod-13 off [*****.]. Six positions = six channels. Progression = turning channels ON. e + complement(e) = 1.
Key values
e_2 = 11,036,025
Mod-2 projection: 1 in the mod-8 channel, 0 in all five others. In Z/2,310, e_2 = 1,155 (halving); fattening breaks the simple halving but preserves the projection. The 490 split: inner {2,5,7} frozen, boundary {3,11,13} spinning. One idempotent per switch.
7 circles of T^7 = S^1(8) x S^1(9) x S^1(25) x S^1(49) x S^1(11) x S^1(13) x S^1(17). Each circle spins at its channel's Pareto top. T^7 IS the Gabriel-Horn cap of Z^7 -- the carousel compactifies the infinite lattice into a finite torus:
Spectral gap floor
4*sin^2(pi/49) ~ 0.01644
Identical across all 108 lambda-420 rings. The carousel's relaxation rate is universal -- 49 strictly dominates 16, 9, 25, 11, 13. This is a structural analogy, not a proof of the Clay problems (Theorem 33 STRUCTURAL Yang-Mills gap analog).
CRT-decomposed Laplacian
T^6 = 6 independent channels
On Z/12,612,600 the Laplacian is a direct sum of 6 per-channel Laplacians. On Z/214,414,200: 7 channels (adds mod-17). Blowup requires simultaneous singularity across orthogonal channels -- CRT orthogonality forbids it. This is a structural analogy, not a proof of the Clay problems (Theorem 34 STRUCTURAL Navier-Stokes horn analog).
Carousel = cap
Finite volume, unbounded surface
The ring's N-element state space has finite volume; the ambient Z^7 has infinite surface. Adding 1 on the carousel = walking one step on the compactified horn. 420-step Carmichael lambda orbit = the horn's Cayley circulation.
k-th Power Map Dynamics
Beyond squaring: x -> x^k mod N for each prime k in the chain. The power map reveals the carousel's internal clock.
Cubing-Squaring Cycle Equality
Max cycle for k=2 AND k=3 both = 12 = lambda(Z/210). Two different rotations, same orbit ceiling. Peak at d = 35 = 5*7.
Power k
Max cycle
Factor
Pairing
k=2, k=3
12
lambda(Z/210)
k=5, k=11
6
lambda(Z/210)/2
k=7, k=13
4
lambda(Z/210)/3
Universal Divisor
ALL k >= 2 divide 12
Achievable max cycles = {1,2,3,4,6,12} = divisors of 12. lambda(Z/210) is the ceiling. Verified k=2..30.
Cubing blocked at period 3
Cubing has NO period-3 orbits
3^3-1 = 26 = 2*13. Since 13 does not divide 420 = lambda(Z/12,612,600), cubing period 3 is impossible.
Self-recognition
p-th power has p fixed points in Z/p^e
|{x : x^p = x mod p^e}| = p. Each prime recognizes itself in its own channel.
k-th power collapse by class: Z/210 elements collapse to e_11 for k >= 3 in 1 step (cubing beats squaring). Z/2,310 collapses to void for k >= 3. Self-order: Z/210 = 1, Z/6 = 2, Z/2,310 = 7.
Identity Power Lattice
Which elements are preserved by exponentiation? The identity power lattice reveals the stiffness of each channel.
Mod-49 Stiffness Theorem
lambda(49) = 7*6 = 42. Without mod-49: lambda = 60. With: lcm(60,42) = 420. The mod-49 channel multiplies stiffness by 7. The number 42 is the mod-49 channel's Carmichael lambda.
Channel
lambda
Meaning
Z/8
2
Rigid. Flips once.
Z/9
6
Period 6 cycle.
Z/11
10
Period 10 cycle.
Z/25
20
Period 20 cycle.
Z/49
42
Controls the ceiling. lcm(2,6,10,20,42) = 420.
11 self-preservation
11 is the ONLY prime with x^p = x for ALL x in its own channel
Exponent 1. Unique among the six. Fat channels break self-preservation: lambda(p^e) > p. 11 preserves because its exponent is 1.
max(p^e) = p^(e-1). Self-similar scaling. {2,5,11}: chain resonance -- each channel resonates with the previous Cunningham member.
Explore: Carousel Position
Enter any number to see its position on the 6-ring carousel. Shows CRT coordinates and which rings are frozen (divisible by that prime). Try 210 (only mod-11 and mod-13 spin), 105 (mod-2, mod-11, and mod-13 spin), 1 (all 6 spin).
Enter n:
Paradigm Contrast
Claim
Standard
Axiom
Modular arithmetic
Wraps around a single clock
Six clocks at different speeds. Z/12,612,600 = discrete 6-torus. CRT is the mechanism, not a tool.
Element order
How many additions to return to 0
Cost of spinning. High coupling = more rings active = higher cost. Frozen rings = idempotent rest.
Carmichael lambda = 420
Carmichael function value
All power maps cycle in 420 steps. 108 = 4*27 rings share this Carmichael lambda exactly (mod-49 forced). 84 of those divide 12,612,600; 24 extensions reach 2*12,612,600. Lattice invariant, not coincidence.
Kleiber's 3/4 law
Empirical scaling observation
Hamming null eigenvalue ratio 360/480 = 3/4. Emerges from 5 = 2+3 (convergence theorem).
Braid chirality
Topological curiosity
LEFT-HANDED at all levels. Beat formula: net = -gcd(p_i-1, p_j-1). Algebraically forced.
Carmichael lambda ladder
Arbitrary Carmichael values
Level-to-level ratios = 2, 3, 5, 7 in exact chain order. The multiplicative dynamics IS the chain.
Squaring cycles
Number-theoretic curiosity
Lengths divide 12. Max cycle = Carmichael lambda(two levels back). Class collapse sum = 9.
b1-Lie
Coincidental dimensions
b1(Z/8 x Z/9) = 56 = E7. b1(Z/25 x Z/11) = 240 = E8 roots. Forced by (p-1)(q-1) = dim(SU(p)) when q = p+2.
Mandelbrot fattening
Same dynamics at any scale
mod-11 invariant (exponent 1). c=2 explodes 28x. The first prime unleashes the most dynamics.
64 switches
Abstract idempotents
6-cube control panel. |Image|*|Kernel| = N (conservation). Class switches: progression = turning rings ON.
k-th power maps
Just exponentiation
Max cycles pair: {2,3}->12, {5,11}->6, {7,13}->4. All divide 12. Cubing blocked at period 3. Self-recognition: p has p fixed points.
Channel stiffness
Carmichael lambda, a number
lambda(49) = 42. The mod-49 channel controls the ceiling. 11 self-preserves (exponent 1). Pair preservation values: 7, 43, 421, 211.
Horn-cap geometry
Torus as abstract product
T^7 = Gabriel-Horn cap of Z^7. 7 circles of the torus. Gap 4*sin^2(pi/49) locked at mod-49. Theorem 33 + Theorem 34 STRUCTURAL, NOT Clay proofs.
Seven rings, one motion, zero choices. Everything forced by the CRT decomposition and the seven primes it requires.