Exactly 108 = 4*27 rings share the same Carmichael lambda: 420 = 4*3*5*7. 7^2 = 49 is forced in every one of them (without 49, the lcm of per-channel lambdas has no factor of 7). Of the 108, exactly 84 divide Z/12,612,600; the remaining 24 extend beyond it up to 25,225,200 = 2 * 12,612,600. The 84 are the sub-rings; the 24 are doublings.
The Lattice at a Glance
Lambda-420 lattice
108 = 4 * 27
All 108 rings hit lambda = 420 exactly (7^2 forced). 108/12 = 9.
Inside Z/12,612,600
84
Of the 108, exactly 84 divide 12,612,600. 84 = 2*42. 84/108 = 7/9. The other 24 extend to 2*12,612,600.
Hasse Edges
282
Covering relations: A covers B iff B|A and A/B is prime-power.
Diameter
10
Unique at lambda = 420. Equals the number of terms in {-1,0,1,2,3,5,7,11,13,1576576}.
Height
8
Longest chain in the divisor poset.
Max Width
21
Widest antichain at rank 4. 21 = 3*7.
Center
420,420
420 * 1001. Highest betweenness centrality.
Spectral Gap
0.016
4*sin^2(pi/49). Identical across all 108 lambda-420 rings (7^2 forced in every one).
Triangle-Free
Girth = 4
Zero triangles. 538 four-cycles. Square geometry.
Why 108 Lambda-420 Rings, 84 Inside Z/12,612,600
Two-Constraint Theorem (PROVED)
All 108 lambda-420 rings share two forcing constraints. 7^2 = 49 is always present (lambda needs factor 7; 7^3 = 343 gives lambda 294 which does not divide 420 -- 7^2 is forced exactly). 7^2 alone gives lambda = 42. Two more factors are needed for lambda = 420: (1) factor 4 from 2^e with e >= 4, or 5^e with e >= 1, or 13. (2) factor 5 from 5^2 or 11. Inclusion-exclusion on 180 candidates: 180 - 72 = 108.
Sub-Ring Count
Of the 108 lambda-420 rings, exactly 84 divide Z/12,612,600 (2-exponent at most 3, since 12,612,600 has 2^3 = 8): inclusion-exclusion 144 - 60 = 84. The remaining 24 extend beyond (2-exponent = 4, reaching 2 * 12,612,600 = 25,225,200): 36 - 12 = 24. Sum: 84 + 24 = 108. Ratio 84/108 = 7/9.
Partition
108 = 32 + 76. 32 rings contain all five chain primes; 76 are partial. By prime count: 1 + 11 + 36 + 44 + 16.
Lambda Hierarchy
Lambda
Rings
Diameter
Width
6 = 2*3
4
3
2
12 = 4*3
8
4
3
60 = 4*3*5
36
7
9
420 = 4*3*5*7
108 (84 in Z/12,612,600)
10
21 = 3*7
5460 = 420*13
60
9
12
Lambda = 420 is unique: diameter = 10 (the count of terms). Triangle-free at all levels. Compatible primes grow: 3 at lambda = 6, 8 at 60, 13 at 420.
What All 108 Lambda-420 Rings Share
Lambda
420
Carmichael period. Exact in all 108 (7^2 = 49 forced).
Spectral gap
0.016
4*sin^2(pi/49). Depends only on 7^2 = 49 (forced in all 108).
Multiplication by 2 commutes with projection in all divisor pairs.
Named Rings
Ring
N
Role
Abs min
1,225 = 25*49
Smallest lambda-420 ring. Only 2 primes (5, 7).
Z/2,310
2,310
Primorial of 11. lambda = 60 (sub-rhythm).
Z/970,200
970,200
Minimum uniform ring. 8*9*25*49*11. 5 primes.
Center
420,420
420*1001. Highest betweenness centrality.
Z/12,612,600
12,612,600
970,200 * 13. 64 idempotents. 6 channels.
Z/214,414,200
214,414,200
12,612,600 * 17. 128 idempotents. 7 channels.
Why 970,200 and 12,612,600
Minimal Uniformity
Z/970,200 is the smallest ring where elements have uniform CRT representation (same value in all channels). Need p^e >= 8 = 2^3. Min exponents: (3,2,2,2,1). Product = 970,200. Adding 13: 970,200 * 13 = 12,612,600. (PROVED)
Exponent-Degree
Exponent sum for Z/970,200: 3+2+2+2+1 = 10. For Z/12,612,600: 3+2+2+2+1+1 = 11. Z/970,200 is the unique ring where exponent sum = Hamming degree.
The inner four channels alone carry lambda = 420. The mod-11 and mod-13 channels keep their own slower periods.
Rhythm Hierarchy
Compatible Primes Grow Monotonically
As lambda grows, compatible primes (<500) grow: lambda=6: 3 primes. lambda=12: 5. lambda=60: 8. lambda=420: 13. lambda=5460 (420*13): 17. Each level contains all rings of the previous.
Quotient
12,612,600 / 420 = 2,310
N/lambda = primorial. Unique for exponents (3,2,2,2,1).
Quotient-Primorial
N/lambda = rad(N)
lambda * squarefree kernel = N. 12,612,600/420 = primorial(11). 420*primorial(13) = the 7-prime ring.
Higher levels
5460 adds {53,79,131,157}
4 new compatible primes at lambda = 5460 = 420*13.
Nested
6 | 12 | 60 | 420 | 5460
Each level = previous * next chain prime.
Lambda counts
1, 2, 3, 5, 9, 13
Distinct lambda values among divisors of each ring.
61 as Lattice Discriminant
61 = 7*11 - 16 detects the Z/12,612,600 boundary inside the lambda-420 lattice. Its mod-49 residue 61 mod 49 = 12 is a primitive root of Z/49 (order 42 = 2*3*7). In the 84 sub-rings, ord(61) = 210; in the 24 extensions, ord(61) = 420 = lambda. The multiplicative order of 61 discriminates exactly the 84/24 partition.
Lattice Discriminant (PROVED)
61 mod 49 = 12 is a primitive root of Z/49. Mod-49 order = 42 in all lambda-420 rings. 84 sub-rings: ord(61) = 210. 24 extensions: ord(61) = 420. 61 detects whether a ring fits inside Z/12,612,600. 93/93 tests pass.
Nilpotent Products and Freedom Cascade
Three theorems interlock: the per-level nilpotent sum product matches the lambda-lattice count only at Z/214,414,200, the lattice count factors as (e_2+1)*(e_3+1)*F where the free sums F are chain primes, and those free sums decompose via triangular numbers.
Nilpotent Product Lattice (PROVED)
Z/214,414,200 is the unique level where the product of inner and outer nilpotent sums equals the lambda-lattice count. At Z/214,414,200: phi(17) * 2*3 = 16 * 6 = 96 = |lambda-1680 lattice|. 138/138 tests pass.
Lattice Freedom Cascade (PROVED)
Lambda-lattice count = (e_2+1)*(e_3+1)*F. Free sums F at Pareto levels: Z/970,200: F = 3. Z/12,612,600: F = 7. Z/214,414,200: F = 8 -- all chain primes. 87/87 tests pass.
Free Sum Triangular Decomposition (PROVED)
free_sum(e_5) = T(e_5) + g*2^(e_5) + s, where T(n) = n(n+1)/2 is the triangular number, g = 1 if 13 is present, s = 1 if 17 is present. At Pareto: T(2) + 2^2 + 1 = 3 + 4 + 1 = 8. Column sums product = 432 = divisor count of 12,612,600. 95/95 tests pass.
Divisor Sum Chain Map
The sum-of-divisors function sigma(p) = p+1 maps every chain prime to a value that factors entirely over the ring's primes. The structure: 2->3, 3->4, 5->6, 7->8, 11->12, 13->14, 17->18. Each p+1 is smooth over {2,3,5,7,11,13}.
Divisor Sum Structure Theorem (PROVED)
sigma(p) = p+1 maps all 7 chain primes to values smooth over {2,3,5,7,11,13}. Total: 3+4+6+8+12+14+18 = 65 = 5*13. Inner sum (2,3,5,7): 21 = 3*7. Outer sum (11,13,17): 44. Pareto top: sigma(9)=13, sigma(8)=15=3*5. The geometric sum 1+p+p^2 is smooth for {2,3} (giving 7, 13) and non-smooth for {5,7} (giving 31, 57). Staircase: sigma(210) = 576 = 2^6*3^2, sigma(2310) = 6912 = 2^8*3^3 (both smooth). Only two smooth perfect numbers: 6 = 2*3, 28 = 4*7. sigma(41) = 42. sigma(42) = 96 = 2^5*3 = Z/214,414,200 lattice count. 107/107 tests pass.
sigma(2310) = 6912 = 2^8*3^3. Both smooth over {2,3}.
Perfect numbers
6 = 2*3, 28 = 4*7
The only two perfect numbers smooth over {2,3,5,7,11,13}. Sum 6+28 = 34 = 2*17.
41 -> 42
sigma(41) = 42
And sigma(42) = 96 = 2^5*3 = the Z/214,414,200 lattice count.
Explore: Lambda-420 Membership
Enter any N to compute lambda(N) (Carmichael function) and check whether N sits in the 108-ring lambda-420 lattice. Of the 108: 84 divide 12,612,600; 24 extend to 25,225,200.