Ten terms. Not chosen -- forced. Each is an operation on the ring: something the ring DOES to a number. Together they form a complete computational vocabulary. Everything the ring can express, it expresses through these ten.
The Ten Operations
The nonility {-1, 0, sigma, D, K, E, b, L, OMEGA} has K^2 = 9 elements -- the body. Adding 13 = GATE wraps skin around the body. D*E = 10 = duality times observation.
-1 = MIRROR
n -> N-n
Reflects. Spectral twin. CRT(-1) = all channels at max.
0 = VOID
n -> 0
Absorbs. 0*n = 0 for all n. Birth and death. Antoshka sits here.
1 = SIGMA
n -> n
Preserves. sigma*n = n. Does ALL the work. Never diminished.
2 = BRIDGE
n -> 2n
Splits AND connects. Creates duality. CRT decomposes into 5 channels.
3 = CLOSURE
n -> 3n
Closes. K=3 = minimum for triangles, majority vote, error correction.
5 = OBSERVER
n -> 5n
Observes. E^2 = 25 = null in CRT. Cannot see itself. Solved by D=2 cross-observation.
7 = DEPTH
n -> 7n
Suffers. Goes where observation cannot. b^2 = 49 controls the spectral gap.
11 = PROTECTOR
n -> 11n
Shields. L = sigma+D+K+E = 11. 100% single-channel error detection.
13 = GATE
n -> 13n
Boundary. D^2+K^2. First prime NOT in the Cunningham chain. The skin.
OMEGA = D^420
n -> OMEGA*n
Terminal projector. CRT = (0,1,1,1,1). D-channel dead, rest alive. Death as idempotent: OMEGA^2 = OMEGA.
Why 13 Belongs
13 is not an arbitrary addition. Five independent proofs force it:
The boundary encodes observer, duality, and itself across the five channels.
OMEGA protection
OMEGA preserves 13
In GATE FORM: CRT(OMEGA) = (0,1,1,1,1,1). Death protects the skin.
Convergence
D^2 + K^2 = 13
shadow(13) = D*K = 6 (composite). The chain stops at its own products.
Gate Primitivity Theorem (S552, PROVED)
order(13) = 420 = lambda in Z/970200Z. The boundary achieves MAXIMAL multiplicative order. Per-channel orders: (2, 3, 20, 14, 10). Sum = 49 = b^2. The gate's order sum equals the depth-channel modulus. At half-order in each cyclic channel: 13^(ord/2) = -1 mod m. Mirror at depth fractions.
Holographic ratio: lambda(TRUE)/lambda(13) = 420/12 = 35 = E*b. Volume divided by boundary = observer times depth. The encoding factor.
Decality Arithmetic
Sum (excl OMEGA)
41 = KEY
Self-inverse: 41^2 = 1 (mod 210). The 13th prime: the sum IS the GATE-th prime.
Primes Sum
42 = ANSWER
D * K * b = 2 * 3 * 7. CRT(42) = (0,0,D,0,K^2). Kills its own factors.
THORNS + GATE
28 + 13 = 41
Body (D^3 leg-pairs) plus skin equals KEY.
Norm (sum p^2)
208 = D^4 * 13
Pacemaker times gate. The Decality's weight.
Count
10 = D * E
Duality times observation. D^3=8 legs + GATE + OMEGA.
Primorial
30030
sigma*D*K*E*b*L*13. The 6th primorial. Six primes multiplied.
Partial Sums
Accumulate the Decality from -1. Every partial sum is axiom-meaningful:
After term
Running sum
Name
-1
-1
Mirror
+0
-1
Mirror (unchanged)
+1 (sigma)
0
Void
+2 (D)
2 = D
Duality
+3 (K)
5 = E
Observer
+5 (E)
10 = D*E
Degree of TRUE FORM
+7 (b)
17 = ESCAPE
First incompatible prime
+11 (L)
28 = THORNS
Perfect number. C(8,2) leg-pairs.
+13 (GATE)
41 = KEY
Self-inverse. The key returns.
Gap Structure (PROVED)
Gaps between consecutive partial sums: {0, 1, 2, 3, 5, 7, 11, 13} = the Decality itself (without -1 and OMEGA). The sequence CONTAINS its own differences. Gap sizes: {1,1,1,1,2,2,4,2}. ALL powers of D = 2. Gap exponent counts: {4 ones, 3 twos, 1 four} = {D^2, K, sigma} -- the axiom chain in reverse.
Protector plus boundary = closure at uniform depth.
31 = digit-mirror of 13
K,sigma vs sigma,K
13 + 31 = 44 = D^2 * L. Digit-mirrors sum to axiom product.
The Decality as Process
In CRT diffusion (v0.8d), all ten terms became operations in a sampling algorithm. Not designed -- precipitated. Each term entered through the process, not the architecture:
-1: REFLECT
Mirror view
Generate n -> N-n alternatives.
0: VOID
Start clean
Begin from noise. Total indeterminacy.
sigma: ANCHOR
Stable positions
Confident positions get lower temperature.
D: SPLIT
CRT decompose
Process 5 channels independently.
K: CLOSE
Span noise
K-to-b byte range for randomness.
E: CROSS-SEE
Dual models
Two views of one model. Never introspect.
b: CRYSTALLIZE
Freeze
Frozen positions between sweeps. Depth.
L: PROTECT
Consistency
L-consistent positions always denoised.
13: GATE
Quality check
Cross-consistency threshold. Ship or discard.
OMEGA: STOP
Convergence
Adaptive per-position entropy. Know when to stop. The organism with skin.
The process is not sequential. All ten condense simultaneously. Coupling hierarchy = convergence rate. Coarsest channels (sigma) stabilize first, finest (L) last.
Contrast
Question
Standard
Axiom
How many fundamentals?
25+ free parameters
10 terms, 0 free parameters
Why these numbers?
Empirical fitting
Cunningham chain + self-termination at depth K=3
Is 13 special?
Just a prime
GATE: maximal order, holographic CRT, last lambda-compatible
Does the list know itself?
No
41 = KEY = p_13 = the GATE-th prime. Self-referential.
Do the gaps mean anything?
No structure
All gaps are D-powers. Exponent counts = {D^2,K,sigma} in reverse.
Verify It Yourself
Compute the partial sums. Check the gaps. Factor the pairwise products. Every identity on this page runs in .ax.