{ 2, 3, 5, 7, 11 }

Classical Mechanics from Ten Terms

The pendulum encodes the entire axiom chain in its time statistics. Space has K=3 dimensions. Free particles carry b=7 conserved quantities. D=2 rules everything.

Part of the Decality — one ring (Z/970200Z), 108 lattice structures.

The Pendulum — The Axiom in Motion

Click to restart. The pendulum's time distribution is arcsine — its cumulants ARE the axiom chain.

kap_2 = D kap_4 = -D*K kap_6 = D^4*E kap_8 = -THIN

The arcsine distribution on [-2,2]: the universal shape of a pendulum's time distribution.
ALL cumulants through kap_8 are axiom-smooth. Beyond D^3=8: the gate opens to 19, 43...

The simplest mechanical system encodes the entire axiom. A weight on a string. Its probability distribution — where it spends its time — has cumulants that ARE {D, K, E, b, L}. Not imposed. Measured. The axiom is already there.

Structural Dimensions

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Conservation — Why b = 7

1 + K + K = b = 7

Energy + linear momentum + angular momentum = b = DEPTH.
Free particle AND Kepler problem: both have b=7 conserved quantities.

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Two completely different systems, same count. A free particle has 1+3+3=7 conserved quantities. The Kepler problem has 3+3+1=7 (angular momentum + Runge-Lenz + energy). Different quantities, same b=7. Depth IS conservation.

Moment of Inertia — How Bodies Spin

Every standard moment-of-inertia coefficient is axiom-smooth. Hover to see the axiom expression.

All 10 coefficients use only {sigma, D, K, E}. The inner chain only — no b, no L. Rotation is an inner-chain phenomenon. How a body spins is determined before depth and protection ever enter.

The 1/D Rule — Energy's Universal Factor

T = (1/D)mv^2 U = (1/D)kx^2 E = (1/D)kT

Kinetic energy, elastic potential, and equipartition all carry the factor 1/D = 1/2.
Duality splits every energy in half. D IS the bridge between position and momentum.

The virial theorem universalizes it: <T> = (n/D)<V> for V ~ r^n. Gravity (n=-1): <T> = -(1/D)<V>. Harmonic (n=2): <T> = <V>. The D in the denominator is structural, not accidental.

Scaling Laws — Orbits and Forces

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Hamiltonian Structure

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The K*b = 21 surprise: The symplectic group Sp(6) that preserves classical phase space has dimension K*b = 21 — exactly the DNA codon ring Z/21. The group of canonical transformations IS the genetic code.

All 27 Quantities

Quantity	Value	Axiom Expression	Primes

Verify in .ax Verify pendulum cumulants Verify MOI coefficients

Number Theory Thread

The pendulum’s arcsine distribution has cumulants that ARE the axiom chain: κ₂=D, κ₄=−D·K, κ₆=D&sup4;·E, κ₈=−THIN. Axiom-smooth for exactly n≤D²=4 (index≤D³=8). Beyond: f(E)=19 enters. Kepler’s 3/2 = K/D appears as h(−23)/h(−11) = K/σ = 3 in the D-chain class numbers. Kolmogorov E/K = h(−47)/h(−23). The same ratios rule orbits, turbulence, and algebraic number theory.

D-chain class numbers → Partitions & the gate → Modular forms →

Paradigm Contrast

Space = "just 3D, no reason"

Space = K=3, the closure prime

1/2 in kinetic energy = "convention"

1/D = duality splitting energy

7 conserved quantities = coincidence

b=7 = depth = maximum conservation

Galilean group "happens" to have 10 generators

D*E=10 = degree of the TRUE FORM

Kepler hidden symmetry SO(4) = "mathematical accident"

SO(D^2) with dim D*K=6 = phase space

SU(3) oscillator symmetry = "group theory"

SU(K) with dim K^2-1=D^3=8 = spider's legs

Sp(6) dimension 21 = "abstract algebra"

K*b=21 = DNA codon ring

Pendulum = "simple harmonic approximation"

Pendulum cumulants = full axiom chain