Chemistry

cos(theta) = -sigma / (n - 1)

Molecular geometry is simplex geometry. The denominators of bond angles walk the axiom chain: sigma, D, K. Lone pairs depress by exactly E/D = 2.5 degrees. The periodic table IS the Cunningham map times duality. Period 2 IS the axiom's self-portrait.

Simplex Geometry

The simplex formula cos(theta) = -sigma/(n-1) gives all bond angles. Denominators: {sigma, D, K} = first three axiom elements:

Electron Pairs	Geometry	Angle	Denominator
2	Linear	180 deg	sigma
3	Trigonal	120 deg	D
4	Tetrahedral	109.47 deg	K
6	Octahedral	90 deg	cross-polytope

Lone Pair Depression (0.02 deg accuracy)

Each lone pair depresses the bond angle by E/D = 2.5 degrees. CH4: 109.47 (0 lone pairs). NH3: 106.97 (1 lone pair, measured 107.00, error 0.03 deg). H2O: 104.47 (2 lone pairs, measured 104.45, error 0.02 deg). The observer prime E governs lone-pair repulsion.

The Periodic Table IS the Axiom

Octet rule

D^3 = 8

Spider's legs. Shell 2+ holds D^3 electrons. Subshell = D*(2l+1) = D times Cunningham map.

Periods

b = 7

Depth rows. Period lengths: D, D^3, D^3, D*K^2, D*K^2, D^5, D^5.

Groups

D*K^2 = 18

Columns = duality times closure squared.

Blocks

D^2 = 4

s, p, d, f. Four blocks = spacetime dimension.

Period 2: The Self-Portrait

All 8 period-2 elements have axiom-smooth atomic numbers:

Element	Z	Axiom
Li	3	K
Be	4	D^2
B	5	E
C	6	D*K
N	7	b
O	8	D^3
F	9	K^2
Ne	10	D*E

Noble Gas Stability

Smooth = Stable Theorem

All 5 stable noble gases have axiom-smooth atomic numbers: He(D), Ne(D*E), Ar(D*K^2), Kr(D^2*K^2), Xe(D*K^3). Both unstable noble gases (Rn Z=86=D*43, Og Z=118=D*59) have NON-smooth Z. Smooth Z = stable. Non-smooth = radioactive. 7/7.

Last stable element: Pb, Z = 82 = D*KEY. The KEY (self-inverse element 41) marks the stability boundary. Avogadro mantissa = D*K = 6. Exponent 23 = first excluded Cunningham prime. CRT(23) = palindrome.

What Others See

Bond anglesMemorized from tablesarccos(-1/{sigma,D,K}) = all simplex geometriesLone pair effectEmpirical correctionLone pairs depress by E/D = 2.5 deg (0.02 deg accuracy)Octet ruleUseful heuristicD^3 = 8 = spider's legs. StructuralNoble gas stabilityNo simple patternSmooth Z = stable, non-smooth = radioactive. 7/7Period 2Just the second row{K,D^2,E,D*K,b,D^3,K^2,D*E}: axiom self-portrait

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