Full title: Search for proton decay via $p\to e^{+}{\pi }^0$ and $p\to {\mu }^{+}{\pi }^0$ with an enlarged fiducial volume in Super-Kamiokande I-IV. Primary experimental analysis describes the water-detector signatures, event selection and atmospheric-neutrino background.

Particle Data Group review. Section 18 develops parton distributions, their number and momentum sum rules, scale evolution and factorized scattering expressions.

Institutional explanation of beta-minus and beta-plus transformations and their emitted leptons.

Institutional history of Chicago Pile-1 and its controlled self-sustaining chain-reaction demonstration on 2 December 1942. Used for the experiment’s purpose and qualitative apparatus, not construction specifications.

Reproduction of the authors’ 1939 Nature paper. Interprets the barium findings as division of a heavy nucleus and relates released energy to mass differences. Original publication: Nature 143, 239–240, doi.org/10.1038/143239a0.

Physical Review Letters 34, 1472–1474. The inspected primary abstract reports a gravitationally induced neutron-interferometer phase shift. The elementary rectangular-path phase calculation in the concept article is explicitly presented as a model, rather than as a reconstruction of the original apparatus.

Institutional apparatus description. Overview, Fig. 2 and NIOF subsection document perfect-crystal silicon blades with a common base and environmental isolation for phase stability. Accessed 2026-10-31.

Primary research paper. Introduction, printed pp. 2–3, equations (1.1)–(1.2), defines codimension q+1 symmetry generators and q-dimensional charged operators. A one-form symmetry therefore has codimension-two generators.

Primary research paper. The abstract and introduction discuss D8-branes as domain walls of Romans massive IIA supergravity and matching between the two bulk regions.

Authored Cambridge lecture notes. Chapter 3 defines Dp-branes, their transverse positions and the type-II dimensionality distinction; chapter 8 develops compactification and T-duality.

Collaboration detector documentation: superconducting solenoid, tracking, electromagnetic and hadronic calorimeters, and outer muon detectors.

Institutional detector description: general-purpose collision detector with tracking, calorimetry and muon measurement.

Institutional accelerator description: 27-kilometre ring, superconducting magnets and accelerating structures.

Nature 353, 412–414. The inspected abstract connects rotational cooling with molecular orientation and restricted angular motion in strong electric fields.

Nature 116, p. 278. The author's response describes simultaneous recoil and photoelectric events, directional correlations, and the consequences for BKS theory.

Detector institution description: water tank, photomultiplier coverage, underground location and physics purposes.

Science Advances 3, e1602258. Figure 1 and Results and Discussion explain skimmer interference and its experimental suppression by cooling. Preprint: arxiv.org/abs/1607.04844

Review of Scientific Instruments 73, 4201–4205. The inspected author-institution abstract describes the heated ribbon and position-sensitive detector, with measured resolution and efficiency.

CERN Accelerator School lecture, CERN-2013-007, pp. 463–502. Sections 2 and 3 define molecular flow, pumping speed, gas load and pressure-balance models.

The Stejskal-Tanner equation relates diffusion-sensitive echo attenuation to a rectangular gradient-pair experiment:It assumes free Gaussian diffusion with a constant scalar diffusion coefficient $D$, ideal refocusing and two equal nonoverlapping rectangular gradient pulses of magnitude $g$ and duration $\delta$, whose starting times are separated by $\Delta$. The gyromagnetic ratio $\gamma$ is angular. The baseline $S_0$ is an otherwise matched echo without the applied diffusion-encoding gradients. These definitions are part of the formula, rather than optional choices of notation.^[ref]

The finite-duration term follows directly from the encoding history. Write $q_0=\gamma g\delta$. The effective encoding rises linearly from zero to $q_0$, stays constant for $\Delta -\delta$, then falls linearly to zero. Each ramp contributes $q_0^2\delta /3$ to the diffusion weighting b-value, soThe two physical gradients can have the same polarity: the intervening $180^{\circ }$ pulse reverses the effective phase-encoding sign.

For the illustrated $\delta =2\mathrm{ms}$ and $\Delta =30\mathrm{ms}$, the normalized integral $b/q_0^2$ is $29\frac{1}{3}\mathrm{ms}$, rather than $30\mathrm{ms}$. This isolates the pulse-duration correction without assigning an unmeasured gradient amplitude.

The 1965 work combined theory with apparatus tests, including slow self-diffusion in dry glycerol.^[ref] Restricted diffusion, flow, background gradients and measurement noise require their own interpretation; fitting a single attenuation to this equation does not establish all of its assumptions.