The Harmonic Oscillator on the Two Rings: Geometry in Place of Operator Mechanics, with a Check on Every Step

The simple harmonic oscillator is solved by geometry on the two-ring instrument and the Fibonacci 60-ring of the companion papers, with every step checked as it is made, in lengths a student can draw. The route it replaces is the operator mechanics that Freericks and Doughty (arXiv:2305.00062, 2023) propose for teaching: trade the differential equations of wave mechanics for abstraction and run every solvable problem on four operator identities, with the oscillator as the worked case, solved three ways, the operator way resting on the canonical commutation relation, the factorization of the Hamiltonian, a second singular factorization that finds the odd states, and wavefunctions built from a translation operator and an operator form of the Rodrigues formula. This paper is the fourth way. The oscillator is a point on a ring: one setting reads its position between the centers and its momentum between the crossings. Velocity sits a quarter turn ahead and acceleration a half turn ahead, and the ring's digit laws, times seven and sum to ten, check each step. The energy is the radius squared. The commutation relation, the postulate their h-bar enters by, is the size of one cell of phase space: the ground orbit encloses h/2, one triangle, and every annulus between orbits encloses h, one diamond, so h enters the system as a unit, where m and omega do, and nowhere else. On the ring the area per step is the Casoratian of the wave, one constant at all sixty positions, with Cassini's identity as its digit shadow. The stationary states are rings themselves: by Wigner's phase-space formulation, which is exact for the oscillator, time evolution is a rigid rotation, a stationary state is a function of the radius alone, the position density is that ring projected on the center line and the momentum density the same ring projected on the crossing line, and the paper verifies that these projections are the standard densities. The wavefunctions themselves come from one slide: slide the ground blob along the center line, and the powers of the slide are the Hermite functions, with the explicit formula, the recurrence and the matrix elements falling out of two multiplied series where their paper needs a Gaussian similarity transform, nested commutators and an induction. The quarter turn is the Fourier transform, which is how momentum is measured in a trap. Their singular factorization sorts the states by the sign of the ring at its center, the parity. The spectrum E = h-bar omega (n + 1/2), counted in the equilateral triangles of the Tetryonics model, is the odd numbers: the ground state is one unsplittable triangle, each rung is one diamond, one photon, the totals are the squares, hydrogen's n squared, and the three-ring oscillator's degeneracies are the triangular numbers, one triangle read three ways. Their four identities, at the oscillator, are the addition of positions plus the area term. Every formula used is standard and cited, and every number is verified by the script in the appendix.