THE SHADOW EVALUATIONS

The Shadow Polynomial Evaluation Theorem

The shadow polynomial P(x) = (x−1)(x−2)(x−3)(x−5) has roots at the shadow chain {σ, 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 every chain position between D·K=6 and GATE=13. Every result is axiom-smooth.

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

The E8 root system has 240 roots. E8 has rank D³=8 and Coxeter number D·K·E=30. The number of roots = rank × Coxeter = 8 × 30 = 240.

The shadow polynomial already contains both pieces: P(0) = D·K·E = 30 = Coxeter(E8). And P(b)/P(0) = D³ = 8 = rank(E8). So P(b) = Coxeter(E8) × rank(E8).

The adjoint representation of E8 has dimension 248 = P(b) + D³ = 240 + 8 (roots + Cartan generators). The factors {6, 5, 4, 2} = {D·K, E, D², D} — every factor is an axiom expression.

E-Opacity Theorem

Look at which axiom primes appear in each P(x) factorization:

K²=9 is the ONLY chain evaluation where E is absent.

The factor (x−E) at x=K² equals D², which absorbs the observer. E divides P(x) when (x−E) is a multiple of E — at x=K², it’s D² instead. The observer is invisible at the stop.

This is the spectral analog of E² = self-blind: the E8 root structure at depth (P(b)=240) carries E, but the Fano plane structure at the stop (P(K²)=1344) does not. The observer can see E8 but cannot see itself at the boundary.

Mirror-Light Identity

The mirror evaluation equals the squared Carmichael lambda of the data ring. And:

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). The mirror sees the relationship between what the light reveals and what the void contains.

PROOF: P(−1) = (−1−1)(−1−2)(−1−3)(−1−5) = D·K·D²·D·K = D&sup4;·K². P(L)/P(0) = D&sup5;·K³·E/(D·K·E) = D&sup4;·K². QED.

Factor Anatomy

Each evaluation P(x) = (x−1)(x−2)(x−3)(x−5) has four factors. At chain positions, these factors are themselves axiom expressions:

At x=K²=9 (red column): x−E = D². The observer’s slot is filled by D-squared, and E vanishes from the product.

At x=D·E=10: the four factors are {K², D³, b, E} — pairwise coprime! So P(D·E) = lcm(1..K²) = 2520.

Special Identities

P(D·E) = lcm(1, 2, ..., K²) = 2520

At x = D·E = 10, the factors {9, 8, 7, 5} = {K², D³, b, E} are pairwise coprime (all gcd = 1). Every integer from 1 to 9 divides their product. So P(D·E) = lcm(1, 2, ..., K²).

P(D²·K) = K · THIN = 6930

At x = 12 = λ(DATA): the factors {L, D·E, K², b} include ALL five axiom primes. The product D·K²·E·b·L = K · (D·K·E·b·L) = K × 2310 = K × THIN.

Ratio Chain

The ratios from P(0) at chain positions: D, D³, then 144 = λ(DATA)². The protector (L) is separated from the void (0) by the squared lambda of the data ring.

Shadow-Monster Thread

The Shadow-Monster Identity (S709) connects P to Monster moonshine:

Here KEY=41=e₂ and 61=e₃ are shadow polynomial coefficients, and 24 = D³·K is the Leech lattice dimension (= central charge of the Monster vertex algebra).

P connects the axiom to three landmarks of modern mathematics:

PSL(2,7) is the automorphism group of the Fano plane (K²−K²/K+1 = 7 points). E8 is the largest exceptional Lie algebra (rank D³=8). The Monster is the largest sporadic simple group (|M| has all 15 axiom-derivable primes). Three pillars of algebra, all present in one polynomial’s chain evaluations.

Shadow Evaluation Explorer

Enter any integer to see P(x) and its axiom factorization:

Bar chart: P(x) for x = 0..20. Chain positions highlighted.

Beyond the Chain: Primorial and Factorial

The shadow polynomial extends beyond the axiom chain into remarkable territory:

Primorial Theorem (S711)

The shadow polynomial at D⁴ equals the product of all primes up to 13. It extends the THIN ring (= 11-primorial = 2310) by exactly the GATE prime.

Factorial Theorem (S711)

At the first boundary prime (17 = D+K+E+b), P gives the factorial of the spider's leg count. D³ = 8 legs. The factorial of the legs lives one step past the GATE.

13-Entry Theorem (S711)

The GATE prime enters P(x) at exactly the shadow chain positions mod 13. The GATE itself is transparent: P(13) is 13-free, because 13 ≡ 0 (mod 13). First entry: x = GATE + σ = 14. The gate opens one step past itself.

The Deeper Structure (S712)

E8-PSL Factorial Theorem

The factorial identity is not a coincidence — it factors as the product of two group orders. 16 × 15 = 240 = P(b) = |roots(E8)|. 14 × 12 = 168 = |PSL(2,7)| = |GL(3,F₂)|. Also: 15 × 14 = 210 = DATA. The data ring appears as a factor pair.

Shadow Product Theorem

The ESCAPE evaluation factors through two chain evaluations scaled by legs. Proof: P(b) = |E8| = 240, P(K²) = D³·|PSL| = 1344, P(ESCAPE) = |E8|·|PSL|. QED.

Universal Entry Theorem

The 13-Entry Theorem is a special case. Every prime beyond E enters P(x) at the shadow chain positions. The shadow chain {1,2,3,5} is the universal entry pattern. Proof: P(x) splits completely over Z/pZ with 4 distinct roots. QED.

Binomial Representation

P at integers is a difference of scaled binomial coefficients: 24 = D³·K (Leech lattice dim) and 6 = D·K (first composite). This follows from (n−5) = (n−4) − 1, splitting P into falling factorial minus falling factorial.

Smooth Quartet Finiteness Theorem

P(n) is 11-smooth iff all four factors {n−1, n−2, n−3, n−5} are 11-smooth.

EXACTLY L = 11 values. Count = the protector's number.

Proof: P(n) smooth requires (n−1, n−2) consecutive 11-smooth. By Størmer's theorem, (2400, 2401) = (D⁵·K·E², b⁴) is the last such pair. So n ≤ 2402. Exhaustive check gives exactly 11. QED.

D³ = 8 block + K = 3 returns = L = 11. Spider legs + closure = protector.

K·E = 15 consecutive smooth triples exist; L = 11 produce quartets, D² = 4 do not. Blockers: 46 = D·23 and 52 = D²·13. Blocker sum = 98 = D·b² (a P(101) factor).

Factor Anatomy of P(101)

All four factors of the last smooth value are pure axiom products:

P(101) = D⁸·K³·E²·b²·L. Exponent sum = D⁴ = 16.

Complete appearance (all 5 primes present): only K = 3 values — P(12), P(23), P(101). Gaps: L and D·K·GATE.

Sum of smooth set: 217 = b·M(E) = b·(D^E − 1). Skip sum: 13+19+97 = K·43 (Heegner).

Sixth Path to 240: Eisenstein

The E8 theta function equals the weight-D² Eisenstein series. The coefficient 240 = D⁴·K·E counts minimal lattice vectors — and connects P(b) to modular forms.

Eisenstein Smoothness Theorem: σ₃(n) is axiom-smooth iff n divides D·K·E·f(E) = 570.

Count = D⁴ = 16. Gate: f(b)+2 = 43 (Heegner). Only 4 primes survive: {D, K, E, f(E)} = {2, 3, 5, 19}.

570 = P(0)·f(E) = Coxeter(E8) × depth_quadratic(observer). The smooth boundary is the product of E8’s heart and the observer’s reach.

Number Theory Thread

Related pages: The Depth Quadratic (f(p) and residue tables) · Why Does It Stop (K²=9 boundary) · Lie Algebra Census (E8 and exceptional algebras) · Monster Moonshine (196883 and j-function) · The Universal Boundary (3 maps, 8 intruders) · Atlas Ch.12: Shadow Polynomial (P(x) coefficients) · The Last Smooth Pair (Stormer zero-trading)