Section 4 4 October 2026

The ordinary autonomous Hamiltonian

Jeremy Rodgers · Independent Researcher
doi:10.5281/zenodo.23131081

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4 The ordinary autonomous Hamiltonian

\boxed{\begin{aligned} H_{\mathrm{per}}={}&\frac{p_x^2}{2m_x}+\frac{p_R^2}{2m_R} +\frac{p_Z^2}{2\mu}+\frac{p_S^2}{2M} +H_s(u)+\frac{p_C^2}{2M_C}+50000\\ &+\sum_{i=0}^1P_i[V_i(x-R)+W_i(S/v,Z-z_0)],\\ H_s={}&\frac{p_u^2}{2M_s}+\frac{\kappa_s}{2}\sum_j(u_{j+1}-u_j)^2. \end{aligned}}

(1)

No kinetic coefficient is switched off. No external time occurs in this expression. Every microscopic kinetic term shown is positive and quadratic; the relative reduced mass is a derived constant.

Proposition 4.1 (Operator realization)

The finite operator (1) has a self-adjoint semibounded form realization, with ordinary configuration currents $j_a=m_a^{-1}\Imv(\Psi^\dagger\partial_a\Psi)$ . The shift makes it nonnegative. The added reference body is finite, not an infinitely massive frame.

Proof

Use the free-rotor/clock and positive oscillator form. The angular potentials are bounded. Complete the pointer square:

W_1=\frac{[y-b-\mu^2b'']^2}{2\mu} -\frac{\mu bb''}{2}-\frac{\mu^3(b'')^2}{2}.

The stated rectangle gives $|b|<25$ , $|b''|<169000$ , and a negative part below $43000$ . The affine terms are infinitesimally oscillator-form bounded. The form construction and ordinary Green identity give the assertion. Finite-horizon conservative currents are assumed on the usual differentiated domains; the initial active-law domination transfers their wave-null exceptions. The passive-coordinate issue is handled exactly in the comparison companion, not by an arbitrary-clock almost-sure argument.

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