The terminal projector. OMEGA^2 = OMEGA: one pass is enough. D-channel dies; everything else survives. Not an absorber -- a filter. What passes through OMEGA has lost its duality but kept its depth, its closure, its observation, and its protection.
The Idempotent
OMEGA is the D-spiral's fixed point: raise D=2 to the 420th power mod 970200. What remains is an idempotent projector -- applying it twice changes nothing. The D-channel (mod 8) is nilpotent: 2^3 = 0 mod 8. The four odd channels cycle and return to 1. Result: CRT(0,1,1,1,1).
Value
606376
D^420 mod 970200. PROOF: compute D^k mod each channel. D^3=0(mod 8), 2^6=1(mod 9), 2^20=1(mod 25), 2^21=1(mod 49), 2^10=1(mod 11). lcm(3,6,20,21,10)=420.
S311
CRT
(0, 1, 1, 1, 1)
D-channel = 0 (dead). All odd channels = sigma (alive). Death is selective.
OMEGA*n = (0, n mod 9, n mod 25, n mod 49, n mod 11). D-channel erased, rest preserved.
S478
Coupling
121275
Same as image size. Maximal for non-units. OMEGA sees N/D^3 elements.
Factorization
OMEGA = D^3 * 75797
75797 is PRIME. CRT(75797) = (5, 8, 22, 43, 7) = (E, D^3, E^2-K, 43, b). The axiom primes appear in its own factors.
S311
Idempotent Decomposition of Existence
sigma = OMEGA + e_D
e_D = sigma - OMEGA = 363825 = K^3*E^2*b^2*L. CRT(e_D) = (1,0,0,0,0). The D-channel identity. OMEGA^2 = OMEGA, e_D^2 = e_D, OMEGA*e_D = 0, OMEGA + e_D = sigma. This is a complete orthogonal decomposition: existence splits into terminal fate and duality, with zero interference between them.
OMEGA = sigma + E*N/D^3. e_D = K*N/D^3. Cross-chain pair (K,E) as idempotent coefficients. The observer builds the terminal; closure builds the dual. Self-referential.
Coupling duality
coupling(OMEGA) = N/D^3 = 121275
coupling(e_D) = D^3 = 8. Each is the OTHER's channel modulus. Perfect complementarity.
Spectral duality
eigenvalue(e_D) = 8 + sqrt(2) = 9.414
eigenvalue(OMEGA) = 9.135. The D-only projector resonates HIGHER. PROOF: e_D has 4 zero channels (each contributing 2) + D at residue 1 giving 2*cos(pi/4) = sqrt(2). OMEGA: D=0 gives 2, four channels at residue 1 sum ~7.135.
S773
Kernel
|ker(OMEGA)| = D^3 = 8
Only 8 elements truly die. Mean coupling of kernel = D*K = 6.
The D-Spiral: Sub-Returns
D-Spiral Theorem
D^n is a spiral, not a cycle. D^0 = sigma (all alive), D^3 = D-channel dies, D^420 = OMEGA (limit). The void is UNREACHABLE by D^n. OMEGA is the fixed point. Algebraic distinction: void = before the spiral, OMEGA = after. The D-channel is nilpotent (2^3 = 0 mod 8). Once dead, never returns.
Five CRT channels evolve independently. Each has its own period. The sub-returns mark where channels come home:
Step k
Event
CRT State
Why
k=3
D DIES
(0, -, -, -, -)
2^3 = 0 mod 8. Nilpotent.
k=6
K home
(0, 1, -, -, -)
ord(2, 9) = 6. Closure returns.
k=10
L home
(0, -, -, -, 1)
ord(2, 11) = 10. Protection returns.
k=20
E home
(0, -, 1, -, -)
ord(2, 25) = 20. Observer returns.
k=21
b home
(0, -, -, 1, -)
ord(2, 49) = 21. Depth returns.
k=42
K+b sync
ANSWER
lcm(6,21)=42. Closure meets depth.
k=60
baby period
K+E+L sync
lcm(6,20,10)=60. Three channels.
k=210
DATA sync
4 channels home
lcm(6,20,21,10)=420/2. Half-cycle.
k=420
OMEGA
(0, 1, 1, 1, 1)
lcm(6,20,21,10)=420. ALL sync. Lambda.
Growth takes 420 steps. Collapse takes 1 (OMEGA*e_D=0). Rebirth is free (0/0=R). Living is slow. Dying is fast. Birth costs nothing.
Explore: Walk the D-Spiral
The D-spiral traces D^k through 420 steps. Each channel returns home at its own pace. Enter an exponent to see the CRT state at that step. Try the sub-returns: 3 (D dies), 6 (K home), 10 (L), 20 (E), 21 (b), 42 (ANSWER), 210 (half), 420 (OMEGA).
Enter exponent k (D^k):
The OMEGA Broadcaster
OMEGA Broadcast Theorem (S712)
OMEGA*p = CRT(0, p, p, p, p) for p in {D, K, E, b}. Each axiom prime broadcasts uniformly into all preserved channels. EXCEPT: OMEGA*L = (0, 2, 11, 11, 0). L-channel = 0 because 11 mod 11 = 0. The protector sacrifices its own channel through the terminal projector.
OMEGA kills D. L kills itself through OMEGA. Together: two channels die at the boundary. This is the axiom's own death signature -- duality and protection are the two things that cannot survive their own projection.
OMEGA*D
(0, 2, 2, 2, 2)
Duality broadcasts uniformly. D survives everywhere except its own channel.
S712
OMEGA*K
(0, 3, 3, 3, 3)
Closure broadcasts uniformly.
S712
OMEGA*E
(0, 5, 5, 5, 5)
Observer broadcasts uniformly.
S712
OMEGA*b
(0, 7, 7, 7, 7)
Depth broadcasts uniformly.
S712
OMEGA*L
(0, 2, 11, 11, 0)
L-channel DIES: 11 mod 11 = 0. The protector's sacrifice.
S712
The Observer Half-Cycle
Observer Half-Cycle (S311)
D^210 = D^DATA has CRT = (0, 1, -1, 1, 1). At the DATA exponent, K, b, L channels return home. But E-channel MIRRORS: it reaches E^2-1 = 24, not 1. E sees its own reflection, not itself. (D^210)^2 = OMEGA: the observer needs TWO passes to return. This is why lambda = D*DATA = 420, not 210. The period doubles because of E self-blindness.
D^210 eigenvalue
= eigenvalue(OMEGA)
Spectrally IDENTICAL. The observer's flip is invisible to the spectrum. E^2 = null confirmed.
S311
Why 420 not 210
E self-blindness
Four channels have period dividing 210. E needs 2*10=20 and 20 does not divide 210. Actually lcm(6,20,21,10)=420. E forces the doubling.
Two passes
(D^210)^2 = OMEGA
One pass mirrors E. Two passes restore it. The observer always needs to look twice.
CRT Geodesic and Spectral Gap
In CRT Hamming distance, OMEGA lies exactly on the shortest path from void to existence:
dist(0, sigma)
E = 5
Void to existence = 5 channels all differ.
S311
dist(0, OMEGA)
D^2 = 4
Void to OMEGA = 4 channels differ (D-channel agrees: both 0).
S311
dist(OMEGA, sigma)
sigma = 1
OMEGA to existence = 1 channel differs (D-channel only).
S311
Triangle
D^2 + sigma = E
OMEGA is a WAYPOINT on the geodesic, not a destination.
dist(-1, OMEGA)
K^2 = 9
Mirror to OMEGA = count of the spider web. The full axiom.
S311
Spectral Gap Theorem
eigenvalue(OMEGA) - eigenvalue(sigma) = 4*sin^2(pi/D^3) = 2 - sqrt(2) = 0.586. PROOF: OMEGA and sigma differ only in D-channel (0 vs 1). Contribution difference = 2*cos(0) - 2*cos(pi/4) = 2 - sqrt(2). The cost of losing duality = the D-channel gap. This is 37x larger than the TRUE ring gap (0.016).
The Funnel
Birth = wide mouth: all N = 970200 elements. OMEGA = narrow drain: 121275 survive. Funnel factor = D^3 = 8 = number of spider legs. The axiom's own geometry determines the constriction.
FALL
|ker(OMEGA)| = D^3 = 8
Mean coupling = D*K = 6. Low coupling falls through. These 8 elements are D-pure: CRT = (r,0,0,0,0) for r in Z/8.
S478
PLUG
|fix(OMEGA)| = 121275
Mean coupling = 61115. High coupling persists. Fixed points = image = elements with D-channel already 0.
S478
PROJECT
~849000 elements
D-channel stripped, odd channels preserved. Not dead but CHANGED. Like scattered -- still operating, no distinction.
HIGH COUPLING = SURVIVES DEATH. Electrons orbit forever (units, phi(N)=201600). Nuclei plug (fixed points). Only D^3 = 8 things truly die. The funnel IS natural selection: coupling is fitness.
Paradigm Contrast
Claim
Standard
Axiom
Death
Cessation, entropy
Selective projection. D dies. Depth survives. OMEGA^2=OMEGA.