smearing.py / smearing.json. An extended charge of mean-square radius <r^2> in the three ordinary directions shifts an nS level by (2/3)<r^2>/n^3 hartree (leading order, Z = 1). Measured-equivalent smearing: Dirac's s-level (Darwin) term equals <r^2> = 0.75 lambda^2 (lambda = reduced Compton wavelength); the measured 2S-2P1/2 Lamb shift (1057.845 MHz) adds 0.036, total 0.786 lambda^2. One unit of lambda^2 is 29,198 MHz at 2S. The proton's size contributes 5e-6. ASSUMPTIONS: the pattern's M is the electron's mass frequency (a = 2 lambda for six normal directions); the charge follows the named density; the ordinary-space share of the six-direction mean square is 3/6; the s-level shift is read as charge smearing (standard reading of Dirac theory, an interpretation).
| density of the exact branch (d = 6, aM = 2) | <r^2> / lambda^2 | ratio to measured 0.786 | 2S shift relative to Dirac, in Lamb shifts | aM that would match 0.786 |
|---|---|---|---|---|
| rho = f^2 (1+x) (norm density) | 1.2 | 1.53 | +12.4 | 1.62 |
| upper components f^2 | 1.0 | 1.27 | +6.9 | 1.77 |
| lower components f^2 x | 1.6 | 2.04 | +23.5 | 1.40 |
| scalar f^2 (1-x) | 0.4 | 0.51 | -9.7 | 2.80 |
| Four normal directions (d = 4, aM = 3): 2.25, 1.93, 3.38, 1.35: all further away. | ||||
| Reading: the branch's size is the right scale with no tuning (candidates bracket the measured value, 0.4 to 1.6 against 0.786), but no single candidate matches; each misses the 2S level by 7 to 23 Lamb shifts, far outside measurement error. A match would need aM between 1.4 and 2.8 depending on which density carries the charge; the branch fixes aM = 2. Not established: which density couples to the nuclear field; whether the s-level shift should be read as smearing at all in IDWT. |