The gallium anomaly as a comparison of reclosure weights

Three radioactive-source experiments, GALLEX, SAGE and BEST, exposed gallium to the neutrino lines of ⁵¹Cr and ³⁷Ar and counted the germanium-71 atoms the captures produced. The count sits about 20% below the predicted capture rate, a deficit that now exceeds five standard deviations. Averaged by source, with the excited-state contribution subtracted, the measured ground-state cross sections are 4.44 ± 0.33 for ⁵¹Cr and 5.21 ± 0.72 for ³⁷Ar, in units of 10⁻⁴⁵ cm².

The standard prediction takes the nuclear matrix element from the reverse direction. Germanium-71 returns to gallium by electron capture with a half-life of 11.465 ± 0.003 days, and detailed balance equates the two matrix elements once the lepton waves are evaluated at a single point. Cadeddu et al. keep the lepton waves inside the radial integral with the Gamow–Teller transition density, fit four phenomenological profiles jointly to the half-life and both cross sections, and find that every profile with a sign change reaches both targets while the single Gaussian does not. They state that the profiles are not unique and leave a microscopic calculation of the transition density to nuclear-structure work.

Capture raises the whole atom by the step Q, and the raised configuration resettles through its one open channel: the germanium half-life is that resettlement. These are two events. Each is a many-to-many reclosure of the whole atom, settled by its own global optimisation, and the resettlement is not the capture run backward (0.1a, I-⁠5b). Equating the two nuclear matrix elements by detailed balance reads one event as the mirror of the other. FDT declines that reading; the detailed-balance expression remains the constant-lepton limit of the full integral.

In the closure classes of I-⁠5, capture takes a free two-strand closure and a seated two-strand closure and recloses them as a seated three-strand closure and a free three-strand closure: the neutrino and neutron readings in, the proton and electron readings out. Electron capture takes a bound three-strand closure and a seated three-strand closure and recloses them as a seated two-strand closure and a free two-strand closure. Each list is one event’s book, never a one-to-one particle reaction, and every reclosure runs over both rooms at once.

The half-life, the capture cross section and forward charge exchange are three profile-weighted means of one signed transition profile, each of its own event and each with its probe weight named (I-⁠3): the bound 1s electron for the half-life, the outgoing continuum electron for capture, a plane wave for charge exchange. The profile is signed, so none of these is a positive average. The charge radius is a different reading again, a skin-weighted second moment of the ground state.

A profile with a node nearly cancels against itself. Its net integral is 2.8% of its absolute weight for the double Gaussian and 1.2% for the modified double Gaussian, so small differences between the two lepton weights decide the capture reading. A finite-charge-sphere reconstruction makes the mechanism quantitative: at the skin the bound and continuum weights differ by 1.5 × 10⁻³, and the capture readings relative to the single Gaussian come out at 0.84 ± 0.02 (DG) and 0.86 ± 0.02 (mDG), against 0.839 and 0.854 in Table I. The measured targets sit at 0.84 ± 0.06 (⁵¹Cr) and 0.83 ± 0.11 (³⁷Ar) of the single-Gaussian reading.

FDT reads the source’s box radius, √(5/3) times the charge radius, as the skin radius of the nucleus: R_room = 5.205 fm (I-⁠3). The compact profiles place their principal nodes inside it, at 3.42 fm (mDG) and at 2.71 and 4.82 fm (mTG), with 13% and 14% of their absolute weight outside. The extended double Gaussian places its node at 5.61 fm with 52% outside, and the source treats it as a limiting case. As an exploratory reading, the compact nodes mark surfaces between the centre-side and the skin-side readings of the reclosure, and the sign change marks relative phase.

The analysis makes no prediction beyond consensus for gallium. Each candidate divergence needs a map from the core to an observable that is to be derived: the decay kernel, spectrum and rates (O-⁠12), the confinement operator and its spectrum (O-⁠9), the map from a charge to an input of the force law (O-⁠6), and the fission computation against the SNO neutral-current flux (O-⁠7). The reading supplies measurements that decide between the fitted profiles instead, each with its comparator named.

1. DM-1
   Independent forward charge exchange. Measure the forward ground-state ⁷¹Ga(p,n) or ⁷¹Ga(³He,t) cross section against a unit cross section fixed independently of germanium electron capture. At equal capture reading the nodal profiles carry 2.8 ± 0.4 (DG) and 3.1 ± 0.4 (mDG) times the single-Gaussian strength, against 0.99 for the charge shape, and their plane-wave form factors pass through zero at 0.070 and 0.091 fm⁻¹. The existing excited-state normalisation is referenced to electron capture, so it cannot serve as the unit.
2. DM-2
   Energy dependence at fixed target. Compare capture from the two ⁵¹Cr lines, the ³⁷Ar line and the ⁶⁵Zn line after an independently normalised excited-state subtraction. The mDG reading relative to SG falls from 0.95 at 0.427 MeV to 0.71 at 1.343 MeV; a sign-definite profile stays close to 1 at every line.
3. DM-3
   Charge state and hyperfine preparation. Count surviving bare and hydrogen-like germanium ions in a storage ring, with the populated hyperfine state measured. Bare germanium has no open channel. Hydrogen-like germanium has its allowed capture channel closed in hyperfine state zero and open in state one. This checks the exact books every theory shares.
4. DM-4
   Recomputed ground-state targets. Normalise each excited-state strength to the independently calibrated ground-state strength of DM-1 and subtract again. The comparator is the existing subtraction referenced to electron capture.
5. DM-5
   A microscopic transition profile. Pass a microscopic profile for ⁷¹Ga ↔ ⁷¹Ge through both measured lepton responses with the electron-capture normalisation fixed once, and compare it with both target ratios and their correlated errors.
6. X-⁠1
   Nodes against the skin. The exploratory observable: a profile’s principal sign change relative to R_room, with its share of absolute weight outside the skin.

With identical lepton functions, transition profile, line fractions, shell probabilities, corrections and nuisance values, the analysis returns the source’s equations and its Table I term for term: the limiting map is the identity on those inputs and that dictionary of observables. Extracting constant lepton factors returns the detailed-balance approximation as a limit, and the finite-charge-sphere reconstruction measures how far each profile departs from it. The measured half-life, the inferred capture targets and the BEST zone ratios are accepted in full. For the same stress tensor and data, the field equation returns the curvature of general relativity (I-⁠3). In-sample recovery corroborates no invariant; the measurements above decide between the fitted profiles.