D-Power Gaussian Primes

|D^n + Ki|^2 = 4^n + 9

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

13 = GATE

The gate. First value of the family.

n = K = 3

73 = H0

Heegner blocker. 4^3+9=73.

n = E = 5

1033

Prime. 4^5+9=1033.

n = K^2 = 9

262153

Prime at the stop number.

n = K*E = 15

1073741833

Prime at closure*observer.

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

The E-Filter

E-Filter (PROVED)

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.

The self-blind observer E = 5 systematically blocks even indices. Only odd n survive the filter.

The Covering Vocabulary Theorem

Each covering prime q divides |D^n+Ki|^2 periodically. The first 6 covering primes restricted to odd n are all axiom-named:

Prime q	Name	Period T	Residue r
13	GATE	D*K = 6	sigma = 1
37	PRODIGAL	D*K^2 = 18	ESCAPE = 17
61	e3	DKE = 30	K*b = 21
73	H0	K^2 = 9	K = 3
97	G	D^3*K = 24	b = 7
181	p_42	DK^2E = 90	L = 11

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

The Covering Sum

Covering Sum (PROVED)

13 + 37 + 61 + 73 + 97 + 181 = 462 = THIN/E = D*K*b*L. E is ABSENT from the covering sum. Third manifestation of E^2 self-blindness: (1) E excluded from Heegner numbers, (2) E-filter blocks even n, (3) E absent from covering sum.

Depth Gives GATE Squared

|D^b + Ki|^2

= 16393 = GATE^2 * 97

Depth gives gate squared. 97 = G is the bridge prime.

GATE periodicity

GATE | iff n = 1 mod 6

ord_13(4) = 6 = D*K. Gate returns at its own period.

GATE^2 period

GATE^2 | iff n = b mod 78

ord_169(4) = 78 = D*K*GATE. Depth is the key.

Non-covering

{17, 41, 109, 137}

ESCAPE, KEY, f(L), ADDRESS. Never divide any value.

The Gate Mod-24 Theorem

Mod-24 Classification (PROVED)

Among index-2 primes (D=2 is primitive root, q = 5 mod 8): 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. Verified: 55/55 odd-covering, 74/74 even-covering.

Three-way classification: ODD-covering 25%, EVEN-covering 28%, NON-covering 47%. Densities stable across all bounds tested.

Contrast

Aspect	Standard view	Through the axiom
Gaussian norms	Z[i] norms factor into primes 1 mod 4. Classical	Family \|2^n+3i\|^2 speaks entirely in axiom vocabulary
First value	4 + 9 = 13, the next prime	\|D+Ki\|^2 = GATE. The gate IS the norm of duality+closure
Covering sum	462 is just a number	= THIN/E = DKb*L. E systematically absent (self-blind)
Mod-24	Quadratic residue classification	GATE mod D^3*K selects odd-covering. Leech dimension.

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