docs: restructure knowledge base into tree layout + gA extraction module

- Organize into tree structure: corrections/, nuclear-structure/, gA-extraction/, refs/ - Move all correction notes to corrections/ (01-09) - Move nuclear structure to nuclear-structure/ with index - Add gA-extraction/ with SSM algorithm, sNME/ESSM formalism, Tc99 case study, and implementation TODO - Reorganize refs by topic (electromagnetic/, forbidden-decay/) - Update all wikilinks to new paths - Remove radiative_spec.md (code algorithm, not reference) - Lint: all wikilinks verified valid - Comprehensive gA extraction research based on Paulsen et al. (2024) and Song et al. (2025)
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+---
+title: "sNME — Small Relativistic Vector NME"
+date: 2026-05-14
+tags:
+  - gA/extraction/ssnme
+  - beta-spectrum/forbidden
+  - nuclear-structure/NME
+status: active
+aliases: [small relativistic vector nuclear matrix element, sNME, CVC vector current]
+related_notes:
+  - gA-extraction/01-ssm-algorithm
+  - gA-extraction/03-tc99-case-study
+  - nuclear-structure/01-shape-factors
+---
+
+# The Small Relativistic Vector NME (sNME) — Why It Matters
+
+## What is the sNME?
+
+In forbidden β decay, the relativistic vector current has two components:
+- **Large NME (l-NME):** Computed from the large components of nucleon wave functions. Reliably calculable in NSM.
+- **Small NME (sNME):** Couples the large and small components. **Extremely difficult** to compute because it involves contributions outside the valence shell.
+
+In a perfect many-body theory with infinite valence space, the sNME is related to the l-NME by the **CVC hypothesis**:
+
+$$\text{sNME} = \frac{1}{2} \left[ W_0 - (m_n - m_p) + E_C \right] \cdot \text{l-NME} \tag{1}$$
+
+where:
+- $W_0$ — endpoint energy in MeV
+- $m_n - m_p \approx 1.293$ MeV — neutron-proton mass difference
+- $E_C$ — Coulomb displacement energy (depends on nuclear charge distribution)
+- l-NME — large vector NME (computable in NSM)
+
+### Why is sNME important?
+
+Despite being "small" (typically $< 0.1$ in absolute units), the sNME can influence β-spectral shapes and half-lives **quite strongly** because:
+
+1. **Quadratic dependence:** The half-life $t_{1/2} \propto 1/(\text{l-NME} + \text{sNME})^2$, so even a small sNME shifts the total matrix element significantly.
+2. **Small Q-value amplification:** For $^{99}\text{Tc}$, $[W_0 - (m_n - m_p)] = -0.487$ MeV — the negative mass difference nearly cancels the endpoint energy, making the sNME contribution relatively large.
+3. **Shape dependence:** Different sNME values produce measurably different spectral shapes, especially below 100 keV.
+
+## The CVC Value — An Idealization
+
+The CVC prediction (Eq. 1) is an **idealization** — it pertains to a perfect many-body theory. In practice:
+- NSM calculations with finite valence spaces give sNME $\approx 0$ (the valence space truncation kills it)
+- The CVC value serves as a **reference**, not a physical prediction
+
+## ESSM: sNME as a Fitting Parameter
+
+### The Enhanced Spectrum Shape Method (ESSM)
+
+Kumar et al. (2021) proposed using the sNME as an additional fitting parameter:
+
+1. **Fix** $g_A$ at a trial value
+2. **Vary** the sNME to match the **experimental partial half-life**
+3. Because $t_{1/2} \propto (\text{l-NME} + \text{sNME})^{-2}$, the equation $t_{1/2}^{\text{theo}}(g_A, \text{sNME}) = t_{1/2}^{\text{exp}}$ typically yields **two solutions** for the sNME:
+   - **sNME(c):** closer to the CVC value
+   - **sNME(f):** further from the CVC value
+
+### Two-Solution Structure
+
+For each $g_A$ value, there are two sNME solutions (or none, if the experimental half-life cannot be reproduced). This creates two distinct spectral shape predictions:
+
+- The **sNME(c)** solution is theoretically preferred (closer to CVC)
+- The **sNME(f)** solution may also be physical — only experiment can decide
+
+### Application to $^{99}\text{Tc}$
+
+Song et al. (2025) applied ESSM to $^{99}\text{Tc}$ with three nuclear models:
+
+| Model | $g_A^{\text{eff}}$ | sNME (c) [fm³] | sNME (f) [fm³] | $\chi^2_\nu$ | p-value |
+|---|---|---|---|---|---|
+| **CloseCVC (CVC fixed)** | 1.1 | 0.0674 | — | 1.106 | 0.12 |
+| | 1.0 | 0.0681 | — | 1.244 | 0.004 |
+| | 1.0 | 0.0651 | — | 1.644 | <0.0001 |
+| **FarCVC (sNME fitted)** | 1.2 | −0.0698 | — | 1.066 | 0.22 |
+| | 1.0 | −0.0669 | — | 1.097 | 0.14 |
+| | 1.0 | −0.0683 | — | 1.095 | 0.14 |
+
+Key findings:
+- **Paulsen et al. (CloseCVC):** $g_A^{\text{eff}} = 1.526(92)$ — enhanced value
+- **Song et al. (FarCVC/ESSM):** $g_A^{\text{eff}} \approx 1.0$–1.2 — quenched value
+- The discrepancy arises from the **sNME treatment**, not from the data
+
+## sNME in Other Transitions
+
+Ramalho & Suhonen (2024) studied five second-forbidden nonunique decays:
+
+| Transition | $g_A$ Sensitivity | sNME Sensitivity |
+|---|---|---|
+| $^{60}\text{Fe} \to {}^{60}\text{Co}$ | Strong | Weak |
+| $^{94}\text{Nb} \to {}^{94}\text{Mo}$ | Strong (jj45pnb) | Weak |
+| $^{98}\text{Tc} \to {}^{98}\text{Ru}$ | Strong | Moderate (below 100 keV) |
+| $^{126}\text{Sn} \to {}^{126}\text{Sb}$ | Weak | **Strong** |
+| $^{129}\text{I} \to {}^{129}\text{Xe}$ | Weak | **Strong** |
+
+For $^{99}\text{Tc}$, the $g_A$ sensitivity is **strong** and the sNME sensitivity is **moderate** — making it an excellent candidate for combined $g_A$ + sNME fitting.
+
+## References
+
+| Ref | Citation | Use |
+|---|---|---|
+| [ESSM] | Kumar, Srivastava, Suhonen, *Eur. Phys. J. A* **57**, 225 (2021) | ESSM formalism |
+| [Song2025] | Song et al., arXiv:2506.17544 (2025) | $^{99}$Tc with ESSM |
+| [Ramalho2024] | Ramalho, Suhonen, *Front. Phys.* **12**, 1455778 (2024) | sNME in five second-forbidden decays |
+| [Behrens] | Behrens & Bühring, *Electron Radial Wave Functions and Nuclear Beta Decay* (1982) | sNME formalism |