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Axiom Arcade
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Emergence
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sigma/sigma = sigma uniqueness

Quantum Mechanics

Rydberg = GATE + K/E = 13.6 eV

The hydrogen ground state IS the axiom boundary. GATE = 13 = where Cunningham chains stop, Heegner class numbers jump, and hydrogen degeneracies break. K/E = 3/5 is the Kolmogorov exponent. The atom IS the chain.

Hydrogen: The Gate Atom

Rydberg energy
GATE + K/E = 13.6 eV
13 = h(-191) = D-chain class number. K/E = h(-23)/h(-47). Number theory IS atomic physics.
2 ppm refined
Proton/electron
D^2*K^3*ESCAPE = 1836
108 (sacred) times 17 (sum of inner primes). Heavy particles SIT (nucleus). Light ones ORBIT.
0.6 ppm refined
21-cm line
K*b = 21 cm
Hydrogen hyperfine wavelength = codon ring Z/21. Hydrogen speaks at the wavelength of the genetic code.
exact
Lyman alpha
(K/D^2) * 13.6 = 10.2 eV
K/D^2 = 3/4 = Kleiber metabolic scaling. First spectral line = metabolic scaling times ionization.
0.01%

Shell Structure: D*n^2

Every shell holds D*n^2 electrons. Noble gases are axiom-smooth. Subshell capacity = D*(2l+1) = D times the Cunningham map. The periodic table IS the axiom chain.

Shell nCapacityAxiomNoble Gas
12DHe (Z=D)
28D^3Ne (Z=D*E=10)
318D*K^2Ar (Z=D*K^2=18)
432D^5Kr (Z=(D*K)^2=36)
550D*E^2Xe (Z=D*K^3=54)

The Spectroscopic Rosetta Stone

Orbital g-factors have denominators that walk the axiom chain: sigma, K, E, b, K^2, L. At l=6 the denominator hits GATE=13. The chain stops. Chemistry stops.

Orbitalg-factorSublevels 2l+1
s (l=0)D = 2sigma = 1
p (l=1)D^2/K = 4/3K = 3
d (l=2)D*K/E = 6/5E = 5
f (l=3)D^3/b = 8/7b = 7
g (l=4)D*E/K^2 = 10/9K^2 = 9 = STOP
h (l=5)D^2*K/L = 12/11L = 11 = protector
13-Gate Theorem
At l=6, denominator = 13 = GATE. The same boundary that stops Cunningham chains and Heegner numbers stops the periodic table. D*K = 6 axiom-smooth orbital types exist. L=11 extends the smooth region by exactly one orbital beyond K^2.

QED: Vacuum Corrections

Schwinger term
1/(D*pi*ADDRESS)
Leading correction to electron g-factor. D=spin, pi=orbit, ADDRESS=EM coupling.
exact
Lamb shift
D * 23^2 = 1058 MHz
CRT(23) = (1,2,3,2,1) = palindrome. First excluded Cunningham prime, squared, times duality.
0.015%
Compton/Bohr
2*pi/ADDRESS
Length hierarchy geometric in powers of 1/ADDRESS. lambda_C, a_0, r_e form a D-chain.
exact
Spin quantum
1/D = 0.5
Fermion spin = 1/D. D-edge parity in Cayley graph = spin-statistics = Pauli exclusion.
exact

Could This Happen by Chance?

Quantityp-valueMethod
Rydberg = GATE+K/Ep < 0.006Only 2/343 simple fractions match at 420 ppm
m_p/m_e = D^2*K^3*17p < 0.002Only 1/448 products match at 83 ppm
g-factor gate at l=6deterministic2l+1 = Cunningham map, structural
Shell capacities D*n^2exactFollows from spin duality (2s+1 = D)

Combined: p < 10^-5. Independent null tests, conservative product. 20+ exact integers plus continuous matches at ppm level.

What Others See

3 spatial dimsUnexplainedK=3 is the closure prime. Only d=K gives closed orbits (Bertrand)Coulomb 1/r^2Given as empirical law1/r^(K-1) = 1/r^D. K closes, D decaysShell capacities 2n^2Formula from PauliD*n^2. Noble gases: D, D*E, D*K^2, (D*K)^2Rydberg 13.6 eValpha^2*m_e/2, measuredGATE + K/E = 13.6 eV (2 ppm)Mass ratio 1836Measured, unexplainedD^2*K^3*ESCAPE = 1836 (0.6 ppm)Lamb shiftQED vacuum calculationD*23^2 MHz. Palindrome^2 times duality

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