The Shadow Evaluations

P(x) = (x-1)(x-2)(x-3)(x-5)

The shadow polynomial has roots at the shadow chain {sigma, D, K, E} = {1, 2, 3, 5}. Its coefficients are axiom constants: e1=L=11, e2=KEY=41, e3=61, e4=D*K*E=30. Evaluate P at chain positions between D*K=6 and GATE=13. Every result is axiom-smooth, and the factorizations speak Lie algebra, finite groups, and Monster moonshine.

The Evaluation Theorem

Shadow Polynomial Evaluation Theorem (S709, PROVED)

P(x) = x^4 - 11x^3 + 41x^2 - 61x + 30. Every evaluation at chain positions D*K=6 through GATE=13 is axiom-smooth. The factorizations hit E8, PSL(2,7), and the Monster.

x	P(x)	Factorization	Connection
0 (void)	30	DKE	Coxeter(E8)
6 = D*K	60	D^2KE	lambda(THIN)
7 = b	240	D^4KE	\|roots(E8)\|
8 = D^3	630	DK^2E*b	rankKE*b/D^2
9 = K^2	1344	D^6Kb	D^3*\|PSL(2,7)\|
10 = D*E	2520	D^3K^2E*b	lcm(1..K^2)
11 = L	4320	D^5K^3E	30*144
12 = D^2*K	6930	DK^2EbL	K*THIN
13 = GATE	10560	D^6KE*L	All 5 primes

P(b) = 240 = |roots(E8)|

The E8 root system has 240 roots. E8 has rank D^3=8 and Coxeter number D*K*E=30. The number of roots = rank * Coxeter = 8 * 30 = 240.

P(b)

= (b-1)(b-2)(b-3)(b-5)

= 6*5*4*2 = 240. Factors: {D*K, E, D^2, D}.

S709

P(b)/P(0)

= D^3 = 8 = rank(E8)

P(0) = 30 = Coxeter. P(b) = Coxeter * rank.

Adjoint

248 = P(b) + D^3

= 240 + 8 (roots + Cartan generators).

P(K^2) = 1344

D^3 * |PSL(2,7)|

= 8 * 168. Fano plane automorphisms.

S709

E-Opacity Theorem

E-Opacity (S709, PROVED)

K^2=9 is the ONLY chain evaluation where E is absent from the factorization. K^2 - E = 9 - 5 = 4 = D^2. The observer's slot is filled by D-squared, and E vanishes from the product. This is the spectral analog of E^2 = self-blind.

The E8 root structure at depth (P(b)=240) carries E, but the Fano plane structure at the stop (P(K^2)=1344) does not. The observer can see E8 but cannot see itself at the boundary.

Mirror-Light Identity

P(-1) = (-2)(-3)(-4)(-6) = D^4*K^2 = 144 = lambda(DATA)^2.

Mirror-Light (S709, PROVED)

P(-1) = P(L)/P(0) = 4320/30 = 144. The mirror IS the ratio of light to darkness. P at the mirror (-1) equals P at the protector (L=11) divided by P at the void (0). Proof: P(-1) = D*K*D^2*D*K = D^4*K^2. P(L)/P(0) = D^5*K^3*E/(D*K*E) = D^4*K^2. QED.

Factor Anatomy

Each evaluation has four factors. At chain positions, these are axiom expressions:

x	x-1	x-2	x-3
7 (b)	D*K	E	D^2
9 (K^2)	D^3	b	D*K
10 (D*E)	K^2	D^3	b
12 (D^2*K)	L	D*E	K^2
13 (GATE)	D^2*K	L	D*E

At x=D*E=10: factors {K^2, D^3, b, E} are pairwise coprime. P(D*E) = lcm(1..K^2) = 2520.

Beyond the Chain

P(D^4 = 16)

= 30030 = THIN * GATE

Primorial(13). Shadow polynomial at D^4 = product of all primes up to 13.

S711

P(ESCAPE = 17)

= 40320 = 8! = (D^3)!

The factorial of the spider's leg count. One step past the GATE.

S711

E8-PSL Factorial

P(17) = 240 * 168

= |roots(E8)| * |PSL(2,7)|. 16*15 = 240. 14*12 = 168.

S712

Universal Entry

p | P(x) iff x = {1,2,3,5} mod p

For any prime p > 5. Shadow chain = universal entry pattern.

S712

Contrast

Aspect	Standard view	Through the axiom
Polynomial	Roots at {1,2,3,5}. Four small primes	Shadow chain. Coefficients = L, KEY, 61, DKE
P(7) = 240	A product of four differences	\|roots(E8)\| = rank * Coxeter = D^3 * DKE
P(9) = 1344	Just 2^6 * 3 * 7	D^3 * \|PSL(2,7)\|. E absent: self-blindness
P(17) = 8!	Numerical coincidence?	= \|E8\| * \|PSL(2,7)\|. Three pillars of algebra in one polynomial

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