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
Square completion gives the exact normalized ground state of the supplied nonrelativistic Coulomb Hamiltonian. Its spectrum and relative root-mean-square speed provide precise atomic benchmarks without selecting the coupling itself.
- Why binding does not fix the coupling
Dilation shows that every positive Coulomb coupling admits binding and the same type of ground-state filter. A source derivation of a particular coupling must therefore fix field, matter, action and clock normalization beyond the existence of an atom.
- Freeze parameters before predicting spectroscopy
A spectroscopy test becomes predictive when its operator family, coefficients and calibration data are fixed before the reserved transitions are examined. The freeze record makes the dependency between constants and atomic predictions auditable.
- Hydrogen and deuterium as complementary probes
Hydrogen and deuterium separate universal electronic rules from isotope-specific mass and nuclear effects. Reduced-mass and contact-scaling examples show why a common operator can still produce different numerical shifts in the two atoms.
- Designing an overconstrained atomic test
An overconstrained atomic test compares independent residual directions with a declared set of source and nuisance coefficients. Covariance weighting and singular-value analysis reveal whether the chosen observables can actually distinguish the proposed mechanism.