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: "SSM — Spectrum Shape Method for gA Extraction"
+date: 2026-05-14
+tags:
+  - gA/extraction/algorithm
+  - gA/ssm
+  - beta-spectrum/forbidden
+status: active
+aliases: [Spectrum Shape Method, gA fitting, shape factor fit]
+related_notes:
+  - gA-extraction/02-ssnme
+  - gA-extraction/03-tc99-case-study
+  - gA-extraction/04-implementation-todo
+  - nuclear-structure/01-shape-factors
+  - corrections/05-radiative
+---
+
+# Spectrum Shape Method (SSM) — Step-by-Step Algorithm for gA Extraction
+
+## 1. Physical Basis
+
+For **forbidden nonunique β decays**, the shape factor $C(w_e)$ decomposes into three energy-dependent components with distinct coupling-constant dependence:
+
+$$C(w_e) = g_V^2\, C_V(w_e) + g_A^2\, C_A(w_e) + g_V g_A\, C_{VA}(w_e) \tag{1}$$
+
+where $w_e$ is the total electron energy in units of $m_e c^2$. The components $C_V$, $C_A$, $C_{VA}$ are computed from nuclear many-body theory and are **independent of the coupling constants**. The spectrum shape is sensitive to $g_A$ through the interference between vector, axial-vector, and mixed terms — a purely nuclear-structure effect unique to nonunique forbidden decays.
+
+> [!note] Why nonunique decays?
+> Unique forbidden decays have a single dominant NME → universal spectrum shape independent of nuclear structure. Nonunique decays involve multiple NMEs → the relative weight of vector vs. axial contributions depends on $g_A$, creating observable spectral shape differences.
+
+## 2. The Full Spectrum Model
+
+The differential spectrum (Paulsen et al., 2024, Eq. 5):
+
+$$N(W) = \mathcal{N} \cdot F(Z,W) \cdot p W (W_0 - W)^2 \cdot X(W) \cdot S(Z,W) \cdot R(W) \cdot C(W; g_A, g_V) \cdot \rho(Z,W)$$
+
+where:
+- $\mathcal{N}$ — overall normalization (absorbs $G_F^2 \cos^2\theta_C / 2\pi^3$)
+- $F(Z,W)$ — Fermi function (Coulomb correction) — [[corrections/01-fermi-function]]
+- $p W (W_0 - W)^2$ — phase space — [[02-phase-space]]
+- $X(W)$ — atomic exchange correction — [[corrections/04-exchange]]
+- $S(Z,W)$ — atomic screening — [[corrections/03-screening]]
+- $R(W)$ — outer radiative correction — [[corrections/05-radiative]]
+- $C(W; g_A, g_V)$ — **shape factor** — [[nuclear-structure/01-shape-factors]]
+- $\rho(Z,W)$ — atomic overlap correction — [[corrections/08-atomic-overlap]]
+
+## 3. The Algorithm
+
+### Step 1: Compute shape factor components from nuclear model
+
+**Input:** Nuclear many-body model (NSM, MQPM, etc.) with appropriate valence space and Hamiltonian for the transition of interest.
+
+**Output:** Three energy-dependent lookup tables: $C_V(W)$, $C_A(W)$, $C_{VA}(W)$ for $W \in [1, W_0]$.
+
+**Method:**
+1. Run shell model (or MQPM) to obtain one-body transition densities (OBTDs)
+2. For each OBTD, compute the contribution to each shape factor component using the Behrens-Bühring formalism (truncated lepton current approximation)
+3. Sum over all contributing multipole orders $K$ and lepton quantum numbers $k_e, k_\nu$
+
+**CVC constraint for vector current:**
+The relativistic vector current includes the small relativistic NME (sNME), which is related to the large vector NME (l-NME) via the CVC hypothesis:
+
+$$\text{sNME} = \frac{1}{2} \left[ W_0 - (m_n - m_p) + E_C \right] \cdot \text{l-NME}$$
+
+where $E_C$ is the Coulomb displacement energy. The sNME is critical for small-$Q$ decays like $^{99}\text{Tc}$ ($[W_0 - (m_n - m_p)] = -0.487$ MeV).
+
+**Practical note:** For $^{99}\text{Tc}$, the dominant transition is $2d_{5/2} \to 1g_{9/2}$ with OBTD $\approx 0.478$. The sub-dominant $1g_{7/2} \to 1g_{9/2}$ transition contributes OBTD $\approx 0.010$. See Kostensalo & Suhonen (2017), Table VII.
+
+### Step 2: Build theoretical spectrum for trial $g_A$
+
+For each trial value of $g_A$ (with $g_V = 1.0$ fixed by CVC):
+
+1. **Compute shape factor:**
+   $$C(W; g_A) = C_V(W) + g_A^2\, C_A(W) + g_A\, C_{VA}(W)$$
+   (using $g_V = 1.0$)
+
+2. **Compute full spectrum:**
+   $$N_{\text{theo}}(W; g_A) = \mathcal{N} \cdot F(Z,W) \cdot p W (W_0 - W)^2 \cdot X(W) \cdot S(Z,W) \cdot R(W) \cdot C(W; g_A) \cdot \rho(Z,W)$$
+
+3. **Normalize:** Choose $\mathcal{N}$ so that $\int_1^{W_0} N_{\text{theo}}(W; g_A)\, dW = 1$ (or match the total counts in the experimental spectrum)
+
+### Step 3: Compare with experimental spectrum
+
+**Input:** Experimental spectrum data $(W_i, N_i, \sigma_i)$ for $i = 1, \dots, N_{\text{bins}}$.
+
+**χ² function:**
+$$\chi^2(g_A) = \sum_{i \in \text{fit range}} \frac{\left[N_i - N_{\text{theo}}(W_i; g_A)\right]^2}{\sigma_i^2}$$
+
+**Fit range selection (Paulsen et al., 2024):**
+- **Lower bound:** ~20 keV — below this, detector threshold effects and atomic exchange modeling dominate
+- **Upper bound:** ~275 keV for $^{99}\text{Tc}$ — above this, endpoint region with poor statistics and resolution distortion
+- Rationale: the $g_A$-driven shape differences are most pronounced in the mid-energy range
+
+### Step 4: Minimize χ² and extract $g_A$
+
+**Grid search:** Evaluate $\chi^2(g_A)$ on a grid (e.g., $g_A \in [0.4, 1.8]$ with step 0.01).
+
+**Refinement:** Use `scipy.optimize.minimize_scalar` (Brent's method) around the grid minimum.
+
+**Uncertainty:** Find $g_A$ values where $\chi^2 = \chi^2_{\text{min}} + 1$. These define the 68% confidence interval.
+
+**Output:** $(g_A^{\text{best}}, g_A^{\text{low}}, g_A^{\text{high}})$
+
+### Step 5 (optional): Simultaneous $g_A$ and $g_V$ fit
+
+If $g_V$ is not fixed to 1.0 (e.g., testing CVC violation):
+
+$$\chi^2(g_A, g_V) = \sum_i \frac{\left[N_i - N_{\text{theo}}(W_i; g_A, g_V)\right]^2}{\sigma_i^2}$$
+
+Fit both parameters simultaneously. The correlation between $g_A$ and $g_V$ is typically strong.
+
+### Step 6 (optional): Half-life normalization — quenched $g_A$
+
+The $g_A$ extracted from **shape alone** can be enhanced ($>1.27$) because the normalization constant $\mathcal{N}$ absorbs the overall rate. To obtain the **quenched** values consistent with half-life:
+
+1. Compute the theoretical partial half-life for the best-fit $(g_A, g_V)$:
+   $$t_{1/2}^{\text{theo}} = \frac{\kappa}{\tilde{C}}, \quad \tilde{C} = \int_1^{W_0} C(W; g_A, g_V) \cdot p W (W_0 - W)^2 \cdot F(Z,W) \cdot X(W) \cdot S(Z,W) \cdot R(W) \, dW$$
+
+2. Compare with experimental partial half-life:
+   $$t_{1/2}^{\text{exp}} = \frac{t_{1/2}^{\text{total}}}{\text{branching ratio}}$$
+
+3. Extract quenched values:
+   $$g_i^{\text{quenched}} = g_i^{\text{shape}} \times \sqrt{\frac{t_{1/2}^{\text{theo}}}{t_{1/2}^{\text{exp}}}}$$
+
+For $^{99}\text{Tc}$: $t_{1/2}^{\text{theo}} \approx 30 \times 10^3$ a vs. $t_{1/2}^{\text{exp}} \approx 212 \times 10^3$ a → quenching factor $\approx 0.38$.
+
+## 4. The sNME Problem and ESSM
+
+### The Problem
+
+The sNME (small relativistic vector NME) is extremely difficult to compute in nuclear shell model because it involves contributions outside the valence shell. The CVC prediction gives a "canonical" value, but this assumes a perfect many-body theory — which finite valence spaces are not.
+
+### The Solution: Enhanced SSM (ESSM)
+
+Kumar et al. (2021) proposed treating the sNME as a **fitting parameter** alongside $g_A$. This is the **Enhanced Spectrum Shape Method (ESSM)**:
+
+1. Fix $g_A$ at a trial value
+2. Vary the sNME to simultaneously fit the **experimental half-life** (not just the shape)
+3. Because the half-life depends quadratically on sNME, there are typically **two solutions** for each $g_A$ value:
+   - **sNME(c):** closer to the CVC value
+   - **sNME(f):** further from the CVC value
+4. Compare both solutions' spectral shapes with the data
+
+The "optimal" choice (closest to CVC) is theoretically motivated, but experimental validation is needed. See [[gA-extraction/02-ssnme]] for details.
+
+## 5. Uncertainty Budget
+
+| Component | Uncertainty on $g_A$ | Relative | Dominant for |
+|---|---|---|---|
+| Fit method (statistics) | 0.0745 | 4.9% | All SSM analyses |
+| Coulomb displacement energy | 0.0425 | 2.8% | Small-$Q$ decays |
+| Nuclear model | 0.0294 | 1.9% | Model comparison |
+| Lepton current treatment | 0.0127 | 0.8% | High-$Z$ nuclei |
+| Fit range | 0.0061 | 0.4% | All |
+| Endpoint energy | 0.0038 | 0.2% | Precision $Q$-value needed |
+| Radiative corrections | 0.0033 | 0.2% | All |
+| Atomic exchange | 0.0029 | 0.2% | Low-energy region |
+| **Combined** | **0.0919** | **6.0%** | — |
+
+Source: Paulsen et al. (2024), Table VIII.
+
+## 6. Key References
+
+| Ref | Citation | Use |
+|---|---|---|
+| [SSM-orig] | Kostensalo, Haaranen, Suhonen, *Phys. Rev. C* **95**, 044313 (2017) | SSM introduction; 26 decays |
+| [SSM-Tc94-98] | Kostensalo, Suhonen, *Phys. Rev. C* **96**, 024317 (2017) | gA-driven evolution; $^{94,98}$Tc, $^{99}$Tc |
+| [Paulsen2024] | Paulsen et al., *Phys. Rev. C* **110**, 055503 (2024) | $^{99}$Tc MMC measurement; $g_A$ extraction (enhanced value) |
+| [Song2025] | Song et al., arXiv:2506.17544 (2025) | $^{99}$Tc MMC measurement; quenched $g_A$ via ESSM |
+| [ESSM] | Kumar, Srivastava, Suhonen, *Eur. Phys. J. A* **57**, 225 (2021) | Enhanced SSM formalism |
+| [Ramalho2024] | Ramalho, Suhonen, *Front. Phys.* **12**, 1455778 (2024) | sNME analysis; five second-forbidden decays |
+| [Hayen2017] | Hayen et al., *Rev. Mod. Phys.* **90**, 015008 (2017) | BB formalism; shape factor theory |
@@ -0,0 +1,106 @@
+---
+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 |
@@ -0,0 +1,105 @@
+---
+title: "$^{99}$Tc Case Study — gA Extraction Results"
+date: 2026-05-14
+tags:
+  - gA/extraction/tc99
+  - beta-spectrum/forbidden/second
+status: active
+aliases: [Tc99 gA, 99Tc beta spectrum, SSM Tc99]
+related_notes:
+  - gA-extraction/01-ssm-algorithm
+  - gA-extraction/02-ssnme
+  - refs/2024_Paulsen_T99_beta_spectrum/summary
+  - refs/2025_Song_tc99_measurement_for_g_v/summary
+---
+
+# $^{99}$Tc — The Premier Laboratory for gA Extraction
+
+## Decay Properties
+
+| Property | Value |
+|---|---|
+| Transition | $^{99}\text{Tc}(9/2_1^+) \to {}^{99}\text{Ru}(5/2_1^+)$ |
+| Forbiddenness | Second-forbidden nonunique |
+| Branching ratio | 99.99855(30)% |
+| Q-value | 295.82(16) keV (Paulsen et al.) |
+| Half-life | 211.5(1.1) $\times 10^3$ a (ENSDF) |
+| Dominant transition | $2d_{5/2} \to 1g_{9/2}$ (OBTD $\approx 0.478$) |
+| Sub-dominant | $1g_{7/2} \to 1g_{9/2}$ (OBTD $\approx 0.010$) |
+
+## Two Contradictory Results
+
+### Result 1: Paulsen et al. (2024) — Enhanced $g_A$
+
+**Measurement:** MMC at PTB Berlin + LNHB Paris. Two independent setups, consistent results.
+
+**Method:** Fixed sNME to CVC value (CloseCVC).
+
+**Result:**
+$$g_A^{\text{eff}} = 1.526(92)$$
+
+This is an **enhanced** value — significantly above the free-nucleon value of 1.27.
+
+**Follow-on values:**
+- $g_V^{\text{eff}} = 0.376(5)$ — quenched (via half-life normalization)
+- $g_A^{\text{eff, quenched}} = 0.574(36)$ — quenched
+- $\bar{E}_\beta = 98.51(23)$ keV
+- $\log ft = 12.3478(23)$
+
+**Key claim:** This resolves an inconsistency — first-forbidden nonunique transitions show $g_A^{\text{eff}} \approx 0.9$, while fourth-forbidden show $g_A^{\text{eff}} \approx 0.9$. The $^{99}\text{Tc}$ result suggests $g_A$ is **enhanced** in second-forbidden decays.
+
+### Result 2: Song et al. (2025) — Quenched $g_A$
+
+**Measurement:** MMC at LLNL (KRISS sensor). Independent of Paulsen et al.
+
+**Method:** ESSM — sNME fitted as free parameter (FarCVC).
+
+**Result:**
+$$g_A^{\text{eff}} \approx 1.0\text{--}1.2$$
+
+This is a **quenched** value, consistent with first-forbidden and fourth-forbidden nonunique decays.
+
+**Best fit:** glekpn Hamiltonian, $g_A^{\text{eff}} = 1.2$, sNME = −0.0698 fm³, $\chi^2_\nu = 1.066$, p-value = 0.22
+
+**Key claim:** The Paulsen result is an artifact of fixing sNME to CVC. When sNME is fitted, the data prefer quenched $g_A$, consistent with the broader pattern.
+
+## The Crucial Difference: sNME Treatment
+
+| Aspect | Paulsen et al. | Song et al. |
+|---|---|---|
+| sNME treatment | Fixed to CVC value | Fitted as free parameter (ESSM) |
+| $g_A^{\text{eff}}$ | 1.526(92) (enhanced) | 1.0–1.2 (quenched) |
+| Nuclear models | NSM (GLEKPN) only | NSM (glekpn, jj45pnb) + MQPM |
+| Half-life constraint | Separate normalization | Simultaneous shape + half-life fit |
+| Consistency with 113Cd, 115In | Inconsistent | Consistent |
+
+## What This Means for Your Experiment
+
+The $^{99}\text{Tc}$ case demonstrates that **the treatment of sNME is the dominant theoretical systematic** in gA extraction. The two experiments (Paulsen and Song) measured essentially the same spectrum (Paulsen's data was used as input for Song's analysis) but obtained **contradictory results** because of different theoretical assumptions.
+
+**For your data analysis:**
+1. **Do not fix sNME to CVC** — treat it as a fitting parameter
+2. **Use ESSM** — simultaneous fit to spectral shape + half-life
+3. **Try multiple nuclear models** — glekpn, jj45pnb, and MQPM (if available)
+4. **Report both sNME solutions** (c and f) for each model
+5. **The "optimal" choice** (closest to CVC) is theoretically motivated but requires experimental validation
+
+## Your Polynomial Fit: Connecting to SSM
+
+You've fitted $C(W) = A \cdot W + B \cdot W^{-1} + C \cdot W^2$. This empirical parametrization can be **cross-checked** against the SSM prediction:
+
+1. Compute $C_{\text{SSM}}(W; g_A)$ for a range of $g_A$ values
+2. Fit each SSM curve to the polynomial form $aW + bW^{-1} + cW^2$ over your fit range
+3. Extract the effective $(A, B, C)$ as a function of $g_A$
+4. Compare your fitted $(A, B, C)$ to the $g_A$-dependent curve to identify which $g_A$ range is consistent
+
+This provides a **model-independent bridge** between your empirical fit and the physical $g_A$ extraction.
+
+## References
+
+| Ref | Citation |
+|---|---|
+| [Paulsen2024] | Paulsen et al., *Phys. Rev. C* **110**, 055503 (2024) |
+| [Song2025] | Song et al., arXiv:2506.17544 (2025) |
+| [Kostensalo2017] | Kostensalo & Suhonen, *Phys. Rev. C* **96**, 024317 (2017) |
+| [ESSM] | Kumar et al., *Eur. Phys. J. A* **57**, 225 (2021) |
@@ -0,0 +1,175 @@
+---
+title: "Implementation TODO — gA Extraction Pipeline"
+date: 2026-05-14
+tags:
+  - gA/extraction/todo
+  - implementation-plan
+status: active
+aliases: [gA pipeline, SSM implementation, shape factor module]
+related_notes:
+  - gA-extraction/01-ssm-algorithm
+  - gA-extraction/02-ssnme
+  - gA-extraction/03-tc99-case-study
+---
+
+# Implementation TODO — gA Extraction Pipeline
+
+> [!info] Status
+> **Research phase.** Experimental data to be added by user. Implementation to follow after documentation is complete.
+
+## Phase 1: Nuclear Model Input (P0 — prerequisite)
+
+- [ ] **1.1** Obtain precomputed shape factor components $C_V(W)$, $C_A(W)$, $C_{VA}(W)$ for $^{99}\text{Tc}$
+  - [ ] Option A: Use Kostensalo & Suhonen (2017) MQPM results (published in PRC 96, 024317)
+  - [ ] Option B: Use Ramalho & Suhonen (2024) NSM results (Front. Phys. 12, 1455778)
+  - [ ] Option C: Run shell model with KShell or another available code
+  - [ ] **Decision:** Which source to use? (See [[gA-extraction/03-tc99-case-study]] for comparison)
+- [ ] **1.2** Store components as lookup tables (CSV or NumPy):
+  - `data/Tc95_CV.csv` — $C_V(W)$ vs. $W$
+  - `data/Tc95_CA.csv` — $C_A(W)$ vs. $W$
+  - `data/Tc95_CVA.csv` — $C_{VA}(W)$ vs. $W$
+  - Columns: `W (unitless)`, `C_V`, `C_A`, `C_VA`
+  - $W$ range: $[1.001, 1.580]$ (corresponds to $T \in [0.5, 295]$ keV)
+- [ ] **1.3** Decide on sNME handling:
+  - [ ] Fix to CVC value (Paulsen et al. approach)
+  - [ ] Fit as free parameter (ESSM — Song et al. approach) **RECOMMENDED**
+  - If ESSM: also need l-NME from nuclear model
+
+## Phase 2: Shape Factor Module (P1 — depends on 1.1)
+
+Create `beta_spectrum/components/shape_factor_gA.py`:
+
+- [ ] **2.1** `ShapeFactorGA` class:
+  ```python
+  class ShapeFactorGA:
+      def __init__(self, C_V_table, C_A_table, C_VA_table, sNME=None):
+          # Load tables, interpolate to arbitrary W
+          # sNME: if None, use CVC value; if float, use fitted value
+      
+      def evaluate(self, W, gA, gV=1.0):
+          """Compute C(W; gA, gV) = gV²·CV(W) + gA²·CA(W) + gV·gA·CVA(W)"""
+      
+      def evaluate_components(self, W):
+          """Return (C_V(W), C_A(W), C_VA(W))"""
+  ```
+- [ ] **2.2** Interpolation: Use `scipy.interpolate.interp1d` with `kind='cubic'` for smooth evaluation
+- [ ] **2.3** Validation: Check that $C(W)$ is monotonically reasonable for $g_A \in [0.4, 1.8]$
+- [ ] **2.4** Unit test: `test_shape_factor_gA_consistency` — verify decomposition matches Eq. (1)
+
+## Phase 3: Fitting Pipeline (P2 — depends on 1.2 + 2.1)
+
+Create `beta_spectrum/gA_fit.py`:
+
+- [ ] **3.1** `fit_gA_spectrum(W_exp, N_exp, sigma_exp, shape_factor, W_min, W_max)`:
+  ```python
+  def fit_gA_spectrum(W_exp, N_exp, sigma_exp, sf, W_min=0.020, W_max=0.275):
+      """
+      Fit gA to experimental spectrum using χ² minimization.
+      
+      Parameters
+      ----------
+      W_exp : array_like
+          Electron total energy (unitless, W = 1 + T/m_ec²)
+      N_exp : array_like
+          Measured counts (or normalized spectrum)
+      sigma_exp : array_like
+          Uncertainties on N_exp
+      sf : ShapeFactorGA
+          Precomputed shape factor
+      W_min, W_max : float
+          Fit range in unitless W
+      
+      Returns
+      -------
+      gA_best, gA_low, gA_high, chi2_min, chi2_curve
+      """
+  ```
+- [ ] **3.2** Grid search: $g_A \in [0.4, 1.8]$ with 0.01 step
+- [ ] **3.3** Brent's method refinement around grid minimum
+- [ ] **3.4** Confidence interval: find $g_A$ where $\chi^2 = \chi^2_{\text{min}} + 1$
+- [ ] **3.5** χ² curve output for visualization
+- [ ] **3.6** Unit test: `test_fit_gA_reproduces_input` — inject known $g_A$, verify recovery
+
+## Phase 4: ESSM Extension (P2 — optional but recommended)
+
+- [ ] **4.1** `fit_gA_sNME(W_exp, N_exp, sigma_exp, sf, l_NME, t_half_exp)`:
+  - Simultaneous fit of $g_A$ and sNME
+  - Match both spectral shape AND experimental half-life
+  - Return both (sNME_c, sNME_f) solutions
+- [ ] **4.2** Half-life computation:
+  ```python
+  def compute_half_life(sf, gA, gV, sNME, W0):
+      """Compute t_1/2 from integrated shape factor"""
+      C_integral = integrate(C(W; gA, gV, sNME) * phase_space * Fermi * ... dW)
+      return kappa / C_integral
+  ```
+- [ ] **4.3** Validation against Song et al. (2025) results for $^{99}\text{Tc}$
+
+## Phase 5: Validation (P3 — depends on 3.1 + 4.1)
+
+- [ ] **5.1** Reproduce Paulsen et al. (2024) result: $g_A^{\text{eff}} = 1.526(92)$ with CVC-fixed sNME
+- [ ] **5.2** Reproduce Song et al. (2025) result: $g_A^{\text{eff}} \approx 1.0\text{--}1.2$ with ESSM
+- [ ] **5.3** Cross-check: polynomial fit $A \cdot W + B \cdot W^{-1} + C \cdot W^2$ vs. SSM for same $g_A$
+- [ ] **5.4** Sensitivity analysis: vary $W_{\text{min}}$, $W_{\text{max}}$, nuclear model, sNME
+
+## Phase 6: Experimental Data Analysis (P4 — depends on user providing data)
+
+- [ ] **6.1** Import raw spectrum data (format TBD)
+- [ ] **6.2** Apply detector response correction (if needed) — [[corrections/07-detector-response]]
+- [ ] **6.3** Run `fit_gA_spectrum` with appropriate fit range
+- [ ] **6.4** Run `fit_gA_sNME` (ESSM) if nuclear model input available
+- [ ] **6.5** Generate publication-quality plots:
+  - Measured spectrum + best-fit theoretical curve
+  - Residuals plot
+  - χ² vs. $g_A$ curve
+  - Comparison with polynomial fit
+- [ ] **6.6** Uncertainty budget: propagate detector, model, and systematic uncertainties
+
+## Phase 7: Paper Draft Support
+
+- [ ] **7.1** Generate tables: fitted $g_A$, sNME, $\chi^2$, confidence intervals
+- [ ] **7.2** Generate figures for paper
+- [ ] **7.3** Cross-reference with literature values
+- [ ] **7.4** Draft methodology section using [[gA-extraction/01-ssm-algorithm]] as reference
+
+## File Structure (target)
+
+```
+beta_shape_pnpi/
+├── beta_spectrum/
+│   ├── components/
+│   │   └── shape_factor_gA.py        ← Phase 2
+│   └── gA_fit.py                      ← Phase 3 + 4
+├── data/
+│   ├── Tc95_CV.csv                    ← Phase 1
+│   ├── Tc95_CA.csv
+│   ├── Tc95_CVA.csv
+│   └── experimental/                  ← Phase 6 (user-provided)
+├── tests/
+│   └── test_shape_factor_gA.py        ← Phase 2 + 3
+│   └── test_gA_fit.py
+└── docs/
+    └── gA-extraction/
+        ├── 01-ssm-algorithm.md
+        ├── 02-ssnme.md
+        ├── 03-tc99-case-study.md
+        └── 04-implementation-todo.md
+```
+
+## Decision Points
+
+1. **Nuclear model source:** Which precomputed $C_i(W)$ to use? (MQPM vs. NSM)
+2. **sNME strategy:** CVC-fixed or ESSM? (ESSM recommended)
+3. **Fit range:** What $W_{\text{min}}$, $W_{\text{max}}$ for your detector?
+4. **Detector corrections:** Do you need response deconvolution before fitting?
+
+## References for Implementation
+
+| Item | Source |
+|---|---|
+| Shape factor formalism | Kostensalo & Suhonen, PRC 96, 024317 (2017) |
+| ESSM formalism | Kumar et al., EPJA 57, 225 (2021) |
+| sNME analysis | Ramalho & Suhonen, Front. Phys. 12, 1455778 (2024) |
+| $^{99}\text{Tc}$ MMC data | Paulsen et al., PRC 110, 055503 (2024) |
+| $^{99}\text{Tc}$ ESSM analysis | Song et al., arXiv:2506.17544 (2025) |
+| BB formalism | Behrens & Bühring (1982) |