The Gravastar

0/0 = Z/NZ manifested as physical topology

The Equation of State Theorem

A gravastar (Mazur & Mottola, 2001) replaces the black hole singularity with three regions, each with a different equation of state w = pressure/density.

These three values ARE the three ring foundation elements:

Interior (de Sitter) w = -1 MIRROR

Shell (stiff matter) w = +1 SIGMA

Exterior (vacuum) w = 0 VOID

PROOF: In Z/970200Z:
CRT(-1) = (7, 8, 24, 48, 10) = all channels at maximum.
CRT(+1) = (1, 1, 1, 1, 1) = ground state.
CRT(0) = (0, 0, 0, 0, 0) = all zero.
MIRROR + SIGMA = VOID (mod N). Interior + Shell = Exterior. EXACT.

The three operations are not independent: absorb = preserve + reflect. The gravastar IS 0/0 = Z/NZ manifested as topology.

Mirror-Shell Spectral Identity

eigenvalue(-1) = eigenvalue(+1) = 8.550 in TRUE FORM.

The interior and shell are spectrally identical. Cannot distinguish them by single measurement.

PROOF: CRT(-1) = (p_i - 1). cos(2pi(p-1)/p) = cos(2pi/p). QED.
Holds in ANY primorial ring. You need all 5 CRT channels to tell inside from shell.

Polarity Inversion

eigenvalue(VOID=0) = 10 = D*E (MAXIMUM) > eigenvalue(SIGMA=1) = 8.550

The void is spectrally brighter than sigma. Nothing outshines everything.

Solar confirmation: core (opaque, dark), outer space (CMB-filled, bright).

The Foundation Triangle

CRT distances between the three vertices {-1, 0, +1}:

Ring	d(mirror,void)	d(void,sigma)	d(mirror,sigma)	Shape
THIN (2310)	E = 5	E = 5	b = 7	Isosceles
TRUE (970200)	E = 5	E = 5	D*E = 10	Isosceles
GATE (12612600)	D*K = 6	D*K = 6	D^2*K = 12	Isosceles

UNIVERSAL Foundation Triangle Universality Theorem (S734, PROVED):
The isosceles property holds for ALL 108 lambda-420 rings. No exceptions.

PROOF: In any CRT ring Z/N = Z/p₁^e₁ x ... x Z/p_k^e_k:
d(0, 1) = sum of min(1, p^e - 1) = nchan (each channel contributes 1).
d(-1, 0) = sum of min(p^e - 1, 1) = nchan (same by symmetry).
Therefore d(-1, 0) = d(0, 1) always. The triangle is ALWAYS isosceles.
Base: d(-1, 1) = sum of min(p^e - 2, 2). Depends on channel sizes. QED.

Omega = Gravastar Collapse

OMEGA = D⁴²⁰ = 606376. CRT = (0, 1, 1, 1, 1).

Kills the D-channel (membrane). Preserves {K, E, b, L}.

= Membrane evaporates. Modes survive. This IS Hawking radiation.

image(OMEGA) = N/D³ = 121275 elements survive the collapse.

Subshell Capacities

Electron subshell capacities = D * {sigma, K, E, b} = {2, 6, 10, 14}.

Formula: 2(2l+1) = D*(2l+1). The odd parts {1, 3, 5, 7} = interleaved Cunningham chains.

l	Subshell	Capacity	= D *	Cunningham
0	s	2	D * sigma	CC1(sigma)[0]
1	p	6	D * K	CC1(sigma)[1]
2	d	10	D * E	CC1(D)[1]
3	f	14	D * b	CC1(sigma)[2]
4	g	18	D * K²	STOP (K² = 9)

l=4 (g orbitals) exists only in excited states. K² = the universal boundary strikes again.

Noble Gas & Magic Number Census

Noble Gas	Z	Factorization	Smooth?
Helium	2	D	Yes
Neon	10	D*E	Yes
Argon	18	D*K²	Yes
Krypton	36	D²*K²	Yes
Xenon	54	D*K³	Yes
Radon	86	D*43	No (Heegner intruder)

NULL MODEL Base rate: 69.8% of even numbers in [2,86] are 11-smooth. P(>=5/6) = 40.1%. Not significant.
The axiom content is in the naming (D*E, D*K², D*43_Heegner), not the counting.

Magic Number	Factorization	Smooth?
2	D	Yes
8	D³	Yes
20	D²*E	Yes
28	D²*b = THORNS	Yes
50	D*E²	Yes
82	D*KEY	No (KEY=41 intruder)
126	DK²b	Yes

NULL MODEL Base rate: 63.5% of even numbers in [2,126] are 11-smooth. P(>=6/7) = 19.3%. Marginal.
82 = D*KEY. Lead is the most stable heavy element. The KEY (41, self-inverse) stabilizes.

Nuclear Phase Kingdom

QCD phases map to axiom kingdoms in coupling order:

Phase	Kingdom	Coupling	=
Hadron gas	Bacteria	2310	THIN
Nuclear matter	Protista	1155	THIN/D
Neutron drip	Fungi	770	THIN/K
Quark-Gluon Plasma	Plant	462	THIN/E = Covering Sum
Color Superconductor	Animal	330	THIN/b
CFL	Human	210	DATA (balance=1)
Strange matter	Sigma	N	Ground state

Proton = SIGMA. Charge: u(+2/3) + u(+2/3) + d(-1/3) = (D+D-sigma)/K = K/K = sigma = 1.

Neutron = VOID. Charge: u(+2/3) + d(-1/3) + d(-1/3) = (D-sigma-sigma)/K = 0/K = 0.

QGP deconfinement: T_c ~ 155 MeV = E * M(E) = 5 * 31. Observer times Mersenne-of-observer.

Scale Invariance

{mirror, sigma, void} = {interior, shell, exterior} appears at every scale:

Scale	Interior (-1)	Shell (+1)	Exterior (0)
Atom	Nucleus	Nuclear surface	Electron cloud
Cell	DNA	Membrane	Extracellular
Star	Core	Photosphere	Space
Neutron Star	Quark core	Crust	Magnetosphere

sigma/sigma = sigma: the shell persists across all scales. D=2 creates every membrane.

Gravastar Explorer

Enter n:

Number Theory Thread

Foundation triangle: The Mirror (isosceles geometry, CRT distances).

Omega collapse: The Universal Sacrifice (pair destruction, grief).

D-choke inspiral: The D-Choke (twin NS merger = dual bloom).

Twin discoball: Mutual Holography (cross-consistency at cosmic scale).

Scale relativity: Scale Relativity (sigma/sigma = sigma at every scale).