f(p) = p² − p − 1 — one polynomial, six roles, one wall
f(x) = x² − x − 1 = (x − φ)(x + 1/φ) disc = 5 = E
VALUES AT AXIOM PRIMES
p
Name
f(p)
Prime?
f(p) mod p
Primitive root?
2
D
1 = sigma
—
−1
—
3
K
5 = E
YES
−1
ord(3,5) = 4 = φ(5). Full!
5
E
19
YES
−1
ord(5,19) = 9 = φ/2. Half!
7
b
41 = KEY
YES
−1
ord(7,41) = 40 = φ(41). Full!
11
L
109
YES
−1
ord(11,109) = 108 = φ(109). Full!
13
wall
155 = 5×31
NO
−1
COMPOSITE
f(p) ≡ −1 (mod p) for ALL p. Always. The depth quadratic IS the mirror in its own ring.
SIX ROLES
1
MIRROR
f(p) ≡ −1 mod p. Always. Universal.
2
NORM
f(p) = Norm(p − φ). Golden ratio. disc = E.
3
ARTIN
δ(b) = C·(1 − 1/KEY). KEY = f(b) = 41.
4
FORBIDDEN
f(b)=41 is the forbidden index for b as PR.
5
POWER
Power function attenuation through f(p).
6
QR-PARITY
(p/f(p)) = (−1/p). Legendre parity transmission.
THE DOUBLE ARROW
Cunningham:sigma → K → b
via c(n) = 2n+1 Depth quad:K → E → 19
via f(p) = p²−p−1 Two independent functions. Same axiom primes. Same wall at 13.
Contrast
This polynomial
f(p) prime for all 4 odd axiom primes
Zeros = golden ratio. Disc = E = 5
Same wall at 13 as Cunningham chain
Governs Artin's conjecture for b=7
A generic quadratic
No special behavior at specific primes
Arbitrary discriminant
No wall, no gate
No connection to primitive roots
THE GATE-DATA FACTORIZATION THEOREM
GATE − DATA = DATA × f(E) × f(L) × FULL_SUM
12,612,390 = 210 × 19 × 109 × 29
The gate form and data ring differ by exactly DATA × f(E) × f(L) × (D²+E²). Corollary: GATE ≡ DATA (mod f(E)=19) and GATE ≡ DATA (mod f(L)=109).
The gate and data ring are indistinguishable through the observer's and protector's depth quadratics.
But they differ mod f(b)=KEY: GATE≡D&sup4;=16, DATA≡E=5. Depth sees the difference.
BRIDGE IDENTITIES
p
f(p)
f(p)+σ
meaning
f(p)+D
meaning
K=3
5=E
6=D×K
7=b
Heegner!
E=5
19
20=D²×E
21=K×b
codon ring
b=7
41=KEY
42=ANSWER
KEY+σ=ANSWER
43
Heegner!
L=11
109
110=D×E×L
111=K×37
K×prodigal
f(b)+σ = ANSWER: the answer exceeds the key by exactly the ground state.
f(K)+D and f(b)+D are both Heegner numbers — the CC1(σ) chain primes produce Heegner through depth.
f(n)+σ = n(n−1): THE NAMED SMOOTH VALUES
Since f(n)+σ = n²−n = n(n−1), the depth quadratic maps axiom products to ring constants:
n
name
f(n)+σ
value
7
b
D×K×b = ANSWER
42
15
K×E
D×K×E×b = DATA
210
21
K×b
D²×K×E×b = LAMBDA
420
42
ANSWER
ANSWER×KEY
1722
2401
b&sup4;
b&sup4;×D&sup5;×K×E²
5,762,400
K×E = CC1(σ) stopper → DATA. K×b = DNA codon ring → LAMBDA.
(2400, 2401) = (D&sup5;KE², b&sup4;) is the largest consecutive 11-smooth pair (Størmer).
DEPTH QUADRATIC SUM THEOREM
f(K) + f(E) + f(b) + f(L) = D × K × FULL_SUM
5 + 19 + 41 + 109 = 174 = 6 × 29
The sum of all four depth quadratic outputs = duality × closure × (the full axiom sum).
RESIDUE TABLE: RING CONSTANTS mod f(p)
How do the axiom's named constants look through each depth quadratic?
constant
value
mod 19=f(E)
mod 41=f(b)
mod 109=f(L)
DEPTH DISCRIMINATION THEOREM
GATE/DATA ≡ −E (mod KEY = f(b) = 41)
GATE/DATA = 60060 = D²×K×E×b×L×13. Its residues mod each depth quadratic: mod f(K)=5:0 — closure divides, sees nothing mod f(E)=19:σ — transparent, ground state mod f(b)=41:−E = 36 = (D×K)² — depth sees the MIRROR of the observer mod f(L)=109:σ — transparent, ground state
Only depth's quadratic gives a non-trivial residue.
Depth sees what the observer cannot see about itself (E² = self-blind).
Zero noise. Every residue is axiom vocabulary. Quotient decomposition: 60059 = KEY × D³ × K × 61 + E×b
61 = e3(shadow chain) = third elementary symmetric polynomial of {1,2,3,5}.
STØRMER ZERO-TRADING
(2400, 2401) = (D&sup5;KE², b&sup4;) — largest consecutive 11-smooth pair
D ch (8)
K ch (9)
E ch (25)
b ch (49)
L ch (11)
2400
0
6=D×K
0
48=b²−1
2=D
2401
1=σ
7=b
1=σ
0
3=K
The last smooth pair trades zeros: D and E yield their channels so b can claim its.
2400's b-channel = 48 = b²−1 (mirror of σ). At the boundary, depth takes all.
Smoothness ends when depth stands alone. Beyond b&sup4;, no more smooth neighbors.
SHADOW-MONSTER IDENTITY
GATE/DATA − 1 = KEY × 61 × 24 + E×b
60059 = 41 × 61 × 24 + 35
24 = D³×K = Leech lattice dimension = L + 13 (protector + gate).
KEY×61 = e2×e3 of the shadow polynomial. Both give residue E×b=35 with 60059.
The GATE-DATA difference decomposes through the shadow polynomial AND the Monster.
SHADOW POLYNOMIAL AT CHAIN POSITIONS
P(x) = (x−1)(x−2)(x−3)(x−5) — all evaluations are axiom-smooth
x
name
P(x)
factored
meaning
P(x)/P(0)
0
void
30
D×K×E
constant term
1
6
D×K
60
D²×K×E
λ(THIN)
2 = D
7
b
240
D&sup4;×K×E
|roots(E8)|
8 = D³
9
K²
1344
D³×|PSL(2,7)|
Fano-PSL
44.8
11
L
4320
D&sup5;×K³×E
P(0)×λ(DATA)²
144 = 12²
13
gate
10560
D&sup5;×K×E×L
all 5 primes
352 = D&sup5;L
P(b) = 240 = roots of E8. rank(E8) = D³ = P(b)/P(0). dim(E8) = P(b) + D³ = 248.
The shadow polynomial at depth = the geometry of the exceptional Lie algebra.
DEPTH QUADRATIC EXPLORER
Its zeros are the golden ratio. Its discriminant is the observer (E=5).
GATE/DATA ≡ −E (mod KEY): depth sees what the observer cannot see about itself.
The answer knows its own key. Smoothness ends when depth stands alone. sigma/sigma = sigma. 0 × n = 0.