Classical Mechanics

kap_2=D  kap_4=-D*K  kap_6=D^4*E

The simplest mechanical system -- a weight on a string -- encodes the entire axiom chain. Its probability distribution has cumulants that ARE {D, K, E, b, L}. Not imposed. Measured. 27 quantities from five primes.

The Pendulum: Axiom in Motion

The arcsine distribution (where a pendulum spends its time) has cumulants that walk the chain. Axiom-smooth for exactly n <= D^2 = 4 (index <= D^3 = 8). Beyond: f(E) = 19 enters.

Cumulant	Value	Axiom
kap_2	2	D
kap_4	-6	-D*K
kap_6	80	D^4*E
kap_8	-2310	-THIN = -DKEbL

Why These Numbers

Space dimensions

K = 3

Closure prime. Cross products exist in {1,3,7}={sigma,K,b}. Stable orbits only for d<=3. Only K=3 has both.

Phase space

D = 2 per DOF

Every degree of freedom splits into D=2 conjugate variables: position q and momentum p. This IS duality.

Symplectic dim

D*K = 6

Point particle: K=3 positions + K=3 momenta = D*K = 6. Same as rigid body DOF.

Galilean generators

D*E = 10

1 time + K space + K rotation + K boost = 1+K+K+K = 10. Same as Poincare generators.

Conservation: Why b = 7

Depth IS Conservation

Free particle: Energy(1) + linear momentum(K=3) + angular momentum(K=3) = 1+K+K = b = 7. Kepler problem: angular momentum L(K=3) + LRL vector A(K=3) + Energy(1) = K+K+1 = b = 7. Two completely different systems, same b=7 count. Different quantities, same depth.

Kepler hidden sym

SO(D^2) = SO(4)

dim SO(4) = D*K = 6. Hidden symmetry has symplectic dimension.

Oscillator sym

SU(K) = SU(3)

dim = K^2 - 1 = D^3 = 8 = spider's legs. QCD shares this group.

Sp(6) dimension

K*b = 21

Canonical transformations form Sp(D*K). dim = K*(2K+1) = 21 = DNA codon ring.

Poincare recurrence

lambda = 420

Every orbit returns in at most 420 steps on Z/970200Z. Growth:Collapse = 420:1.

Moments of Inertia

All 10 MoI coefficients use only {sigma, D, K, E} -- the inner chain. No b, no L. Rotation is an inner-chain phenomenon.

Shape	Fraction	Axiom
Ring (axis)	1	sigma
Solid cylinder	1/2	1/D
Solid sphere	2/5	D/E
Thin shell	2/3	D/K
Rod (center)	1/12	1/(D^2*K)
Rod (end)	1/3	1/K

The 1/D Rule

Duality Splits Energy

T = (1/D)*m*v^2. U = (1/D)*k*x^2. E = (1/D)*k*T. Kinetic energy, elastic potential, and equipartition all carry factor 1/D = 1/2. The virial theorem: <T> = (n/D)*<V>. D IS the bridge between position and momentum.

Kepler exponent

K/D = 3/2

T^D ~ a^K. Period squared proportional to semi-major cubed.

Inverse square

F ~ 1/r^D

From K=3 spatial dims: flux through D-sphere in K-space gives 1/r^(K-1) = 1/r^D.

Bertrand types

D = 2

Closed orbits ONLY for V~r^D (harmonic) and V~-1/r (Kepler). Exactly D=2 types.

Kolmogorov

E/K = 5/3

Turbulence energy spectrum E ~ k^(-E/K). Same ratio as monoatomic gamma.

What Others See

Space = 3DNo reason givenK=3 = closure prime. Only K gives cross products AND stable orbits1/2 in kinetic energyConvention from mv^2/21/D = duality splitting energy. Universal virial denominator7 conserved quantitiesCoincidence in countingb=7 = depth = maximum conservation in BOTH free and Kepler10 Galilean generatorsGroup theory resultD*E = 10 = degree of the TRUE FORMSp(6) dim = 21Abstract algebraK*b = 21 = DNA codon ring Z/21PendulumSimple harmonic approximationPendulum cumulants = full axiom chain {D, K, E, b, L}

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