The supplied Coulomb Hamiltonian gives v_rel,rms=αc in its ground state and v_e,rms=(μ/m_e)αc in the center-of-mass frame. This tests coupling normalization but does not determine alpha independently.
Define the speed before using the identity
A stationary hydrogen ground state has zero mean velocity vector and no probability current, yet it has nonzero momentum variance. These statements are compatible because the mean of a vector and the mean of its squared magnitude are different quantities. The relevant statistic is vrel,rms=√⟨p²⟩/μ for the relative coordinate, with reduced mass μ.
This definition avoids picturing an electron completing a sharp circular orbit. It also makes the SMM/DTT connection testable: a proposed generative delay must reproduce this particular momentum statistic under the same kinetic operator, rather than simply produce a number called a speed.
Derive the Coulomb virial relation
Write H=T+V with T=p²/(2μ) and V=−C/r. Under a normalized spatial dilation, kinetic energy scales quadratically with the dilation parameter and potential energy linearly. Stationarity of the expectation in an exact bound eigenstate makes the first derivative vanish, giving 2⟨T⟩+⟨V⟩=0.
The relation applies to the specified nonrelativistic Coulomb bound states. Relativistic, radiative, and additional potential terms change the corresponding identities and must be handled within their own operators.
Insert the spectrum and obtain the RMS speed
Set C=għc. The exact Coulomb spectrum is En=−μc²g²/(2n²). The virial theorem then fixes ⟨p²⟩/(2μ)=μc²g²/(2n²). Taking the positive square root gives the relative RMS speed in any bound level with principal quantum number n.
After electromagnetic matching sets g=α, the ground-state relation becomes vrel,rms/c=α. The factor n belongs to the supplied spectral family; it is not an independently derived hierarchy of source delays. All statements use the same action, clock, and interaction convention.
Separate relative motion from electron motion
In the center-of-mass frame, electron and nucleus carry opposite momenta p and −p. The electron’s velocity operator is p/me, whereas the relative velocity is p/μ. Thus the electron RMS speed differs from the relative statistic by a definite reduced-mass factor.
Only in the infinitely heavy nucleus limit does the factor become one. For an equal-mass oppositely charged pair, μ/me=1/2, so each constituent’s center-of-mass RMS speed is half the relative RMS speed. That simple counterexample shows why the two meanings cannot be silently interchanged.
This separation gives a useful isotope check. At fixed coupling, the relative RMS speed remains cg/n when the nuclear mass changes, while the electron center-of-mass speed changes through μ/me. Meanwhile the binding energy scales with μ and the radius with μ−1. A temporal construction claiming one universal electronic coupling should reproduce all of these different mass dependences from the same two-body reduction.
Worked ideal-model example with 137
Consider an ideal coupling label g0=1/137. In the ground state the relative RMS ratio is then 1/137≈0.00729927. For an illustrative nucleus with M=2000me, the electron ratio is 2000/(2001×137)≈0.00729562. These are arithmetic consequences of the chosen model parameters, not measured values or a determination of the physical fine-structure constant.
The same input gives the binding fraction |E1|/(μc²)=1/(2×137²). Radius, energy, and momentum therefore provide linked checks of one Hamiltonian. A source-level assignment must satisfy these relations simultaneously rather than choose separate normalizations for each observable.
For n=2 the relative RMS ratio is half the ground value, whereas the binding-energy magnitude is one quarter. The differing powers are consequences of the kinetic relation and provide another internal consistency check.
Why the identity does not select the coupling
Every positive g in the Coulomb family gives a normalized bound state and the corresponding speed ratio g. Consequently, requiring a stable atom or requiring the identity vrel,rms/c=g does not pick 1/137. Under rescaling, the radius changes as g−1 and energies as g², while the dimensionless wavefunction remains the same.
A structural recurrence producing 137 is a different result from a physical interaction strength. The missing bridge is a theorem connecting that structure to the normalized mediator response relative to coherent matter propagation. The ideal α0 label should also remain distinct from empirical low-energy or high-energy couplings until the holonomy, mass, and matching corrections are calculated.
A constructive route for the temporal program
The strongest DTT route would start with one autonomous source, its retained memory, and six nested inner operations. It would derive both spatial motion and the mediator response, then fix the physical clock and action dictionary at outer completion. The resulting ratio g=C/(ħc) would enter the atomic Hamiltonian before any hydrogen radius or transition was used to choose it.
Hydrogen would then supply several independent consequences: the radius, binding energy, relative momentum variance, and transition ratios. The relation to alpha is valuable precisely because it links these outputs. It becomes a prediction of source dynamics only when the coupling entering the common operator has been fixed upstream.
Continue with the articles
- A normalizable hydrogen-like ground state
Introduce the lower-bound and bound-state results of the effective model. Explain how the supplied kinetic and potential rules produce a meaningful atomic test case.
- Why binding does not fix the coupling
Introduce the dilation argument showing that a family of positive couplings remains compatible with binding and completion. Identify the extra source-selection law needed to obtain a specific interaction strength.
- Freeze parameters before predicting spectroscopy
Explain the post-freeze strategy proposed for atomic observables. A spectroscopy comparison becomes informative when its coefficients are fixed without using the same transitions as targets.
- Hydrogen and deuterium as complementary probes
A proposed isotope-comparison article within the differential-test program. Identify which effects are shared and which require separate mass and nuclear inputs before interpreting a residual.
- Designing an overconstrained atomic test
Turn the skeleton draft’s research direction into a practical analysis plan. Require more independent observables than fitted parameters and predeclare the cross-state prediction rules.