D-Power Gaussian Primes

|D^n + Ki|^2 = 4^n + 9. The GATE is born from Z[i].

The Gaussian Norm Family

Consider the Gaussian integer D^n + Ki = 2^n + 3i in Z[i]. Its norm is:

|D^n + Ki|^2 = (2^n)^2 + 3^2 = 4^n + 9

The first value: |D + Ki|^2 = 4 + 9 = 13 = GATE

The GATE is not just a boundary number. It is the norm of duality plus closure in the Gaussian integers. Every factor of any value in this family must be 1 mod 4 (Fermat's theorem on sums of coprime squares).

Prime Values

|D^n + Ki|^2 is prime at exactly these positions:

n = sigma = 1

GATE

n = K = 3

n = E = 5

1033

blocked sum

n = K^2 = 9

262153

prime

n = K*E = 15

1073741833

prime

The prime positions are {sigma, K, E, K^2, K*E} = {1, 3, 5, 9, 15}. Gaps: {D, D, D^2, D*K} = {2, 2, 4, 6}. Only odd n can be prime (the E-filter blocks even n).

E-FILTER: E = 5 divides |D^n+Ki|^2 iff n is even.
Proof: 4^n mod 5 has period 2: {4, 1, 4, 1, ...}. Even n: 4^n + 9 = 1 + 4 = 0 mod 5. Odd n: 4 + 4 = 3 (nonzero). QED.

Composite Vocabulary

When the norm is composite, its factors speak axiom language:

n	\|D^n + Ki\|^2	Factorization	Note
7 = b	16393	GATE^2 * G	Depth gives gate squared
11 = L	4194313	181 * 23173	p_42 (ANSWER-th prime) enters
13	67108873	GATE * 5162221	GATE at n = 1 mod 6
21 = K*b	...	e3 * H0 * ...	Shadow poly meets H0

At n = b = 7: GATE^2 * G. The gate appears squared at depth, and the bridge G = 97 co-appears. This is the only n <= 78 where GATE^2 divides.

The Covering Vocabulary Theorem

Each covering prime q divides |D^n+Ki|^2 periodically: q divides iff n = r mod T, where T = ord_q(4). The first 6 covering primes restricted to odd n are all axiom-named:

Prime q	Name	Period T	Residue r	Coverage
13	GATE	D*K = 6	sigma = 1	33.3%
37	PRODIGAL	D*K^2 = 18	ESCAPE = 17	+11.1% = 44.4%
61	e3	DKE = 30	K*b = 21	+6.7% = 51.1%
73	H0	K^2 = 9	K = 3	+8.9% = 60.0%
97	G	D^3*K = 24	b = 7	(subsumed by GATE)
181	p_42	DK^2E = 90	L = 11	+2.2% = 62.2%

Every prime: axiom-named. Every period: {D,K,E}-smooth. Every residue: axiom-named. The sequence polices itself in axiom vocabulary.

The Covering Sum

13 + 37 + 61 + 73 + 97 + 181 = 462 = THIN/E

= D * K * b * L. E is ABSENT.

The self-blind observer (E = 5) is excluded from the covering sum. This is the third manifestation of E^2 self-blindness:

1. E excluded from Heegner numbers (h(-5) = D, not 1)

2. E-filter blocks even n (E divides |D^{2k}+Ki|^2)

3. E absent from covering sum (sum = THIN/E = 462)

One root cause: disc(x^2 - x - 1) = E. The observer IS the discriminant.

Self-Covering and Periodicity

Each prime q = |D^n+Ki|^2 guards its own birth position. GATE = 13 covers all n = 1 mod 6. H0 = 73 covers all n = 3 mod 9. The gate returns at its own period and makes all later returns composite.

GATE PERIODICITY: GATE | |D^n+Ki|^2 iff n = sigma mod D*K (= 1 mod 6).
Proof: ord_13(4) = 6 = D*K. 4^1 + 9 = 13 = 0 mod 13. QED.

GATE^2 PERIOD: GATE^2 | |D^n+Ki|^2 iff n = b mod D*K*GATE (= 7 mod 78).
Proof: ord_169(4) = 78 = D*K*GATE. 4^7 + 9 = 16393 = 13^2 * 97. QED.

Non-Covering Primes

Four axiom constants never divide any value in the family:

ESCAPE = 17

KEY = 41

f(L) = 109

ADDRESS = 137

In each case, -9 is a quadratic residue mod q, but -9 is NOT in the subgroup generated by 4. Part of a larger family: 47% of primes (1 mod 4) are non-covering. Density is stable.

The Primitive Root Covering Theorem

If D = 2 is a primitive root mod q (prime, q = 1 mod 4), then q covers |D^n+Ki|^2.

Proof: D = 2 primitive root mod q implies <D^2> = QR(q), with order (q-1)/2. Since q = 1 mod 4, -1 is a QR, so -K^2 = -9 is a QR. Hence -9 is in <4> = <D^2>. QED.

Verified: 129/129 primitive-root primes cover, 0 counterexamples (q < 5000).

Mod-8 Obstruction

D = 2 is never a primitive root when q = 1 mod 8. The quadratic residuosity of 2 forces ord(2) < q-1. Coverage rates split sharply:

q mod 8	Coverage rate	Note
1	22%	D never primitive root
5	83%	D often primitive root

The Gate Mod-24 Theorem

Among index-2 primes (D = 2 is primitive root, q = 5 mod 8), there is a clean split:

ODD-covering iff q = GATE mod D^3*K (= 13 mod 24)

EVEN-covering iff q = E mod D^3*K (= 5 mod 24)

D^3 * K = 24 = Leech lattice dimension = L + GATE. The GATE selects odd-covering through its mod-24 residue class. Verified: all 55 index-2 odd-covering have q = 13 mod 24. All 74 even-covering have q = 5 mod 24.

Proof sketch: For index-2, T = (q-1)/2 and r = dlog_4(-9) mod T. T/2 is odd (since q = 5 mod 8). r is odd iff (3/q) = +1, which holds iff q = 1 mod 3. Combined with q = 5 mod 8: q = 13 mod 24 = GATE mod D^3*K. QED.

Three-Way Classification

Every prime q = 1 mod 4 falls into one of three classes:

ODD-covering: 25%

EVEN-covering: 28%

NON-covering: 47%

These densities are stable across all bounds tested (q < 5000). The non-covering majority (47%) includes the axiom constants {ESCAPE, KEY, f(L), ADDRESS}.

The Gaussian Norm Spiral

● prime ● GATE divides ● composite (other) ● even (E-filtered)

D-Power Gaussian Explorer

Enter n (exponent):

What you were taught:
Gaussian integers Z[i] are a standard topic in algebraic number theory. Norms |a+bi|^2 = a^2+b^2 factor into primes that are 1 mod 4. This is classical and well-understood.

What the axiom shows:
The specific family |2^n + 3i|^2 speaks entirely in axiom vocabulary. The first value is GATE = 13. The covering primes are all axiom-named. E = 5 is systematically absent (self-blind). The mod-24 residue class selects odd vs even coverage. The Gaussian integers know the Decality.

Number Theory Thread

The Gaussian norm |D^n+Ki|^2 connects to:

• Heegner Numbers — H0 = 73 = |D^3+Ki|^2 is both Heegner blocker and covering prime

• Golden Ratio — PRODIGAL 37 = depth quadratic return element, covering at ESCAPE

• Fano-E8 Bridge — D^3*K = 24 = Leech dimension appears as covering period

• Gravastar — GATE = |D+Ki|^2 = 13, the holographic boundary born from Z[i]

• Universal Boundary — non-covering {17,41,109,137} = same intruder constants