# Free-Neutron Path + WKB Toy (1-D Reaction Coordinate)

**Date:** 2026-07-24
**Script:** `scripts/investigations/s3_geometry/free_neutron_path_wkb_toy.py`
**Priors:** §0bl plug-in, §0bm 3D dynamics; N7-1/3 path dual
**Handover:** §0bn

---

---

## Closeout status (25 July 2026)

**Free-β derivation campaign CLOSED** (C0+C1 complete; **C2-A park**).  
Canonical freeze: `documents/free_neutron_beta_closeout_results.md` (FB-K known · FB-C closed · FB-O parked).  
**Known:** Coulomb instability cause; S-map eversion pattern; minutes-scale lifetime **inference**.  
**Parked:** unique path-action proof — not a denial of the physical account.  
**Do not:** reopen `c`-shopping in `(c/alpha) int v`.

## Goal

Build an explicit **free** reaction coordinate for n -> p packaging,
compute Coulomb/Weber landscape, and test WKB action dictionaries —
**without** fitting tau_n or setting Gamma_Ev = 1/tau_n.

---

## P1 — Path A: reverse topology **[E]**

Reaction coordinate t in [0,1]:

```
p = 3 - t,  q = 2 + t     # T(3,2) -> T(2,3)
```

At a/R = eps_der = 0.28868, D1 map E = alpha hbar c * UC / (L^2 R):

| Dictionary | E_n | E_p | E_max | B from n |
|------------|-----|-----|-------|----------|
| D1 R_p | 2.817 | 2.859 | 4.246 @ t=0.50 | **1.429 MeV** |
| D1 R_C | 11.264 | 11.433 | 16.978 | **5.714 MeV** |
| D1 R_p Weber-soft | 2.763 | 2.760 | 4.243 | **1.480 MeV** |

**Finding:** free reverse path has a **mid-path activation barrier** (N7 dual).
Endpoint shape gap is small — Delta m is not the D1 endpoint gap.

---

## P2 — Path B: multipole dissolve **[E / F alone]**

Fixed equilateral lobes; charges flow (e/3,e/3,e/3) -> (e,0,0).

| U(s=0) | U(s=1) | Shape |
|--------|--------|-------|
| 1.3177 MeV | 0.0000 MeV | **monotonic downhill** |

Pure charge dissolve has **no Coulomb barrier**. Classical release would give
tau ~ L/c ~ 1e-23 s — **[F]** as the free-beta minutes clock.
Free beta needs topology and/or weak packaging structure.

---

## P5 — Free effective potential

| Model | Residual barrier |
|-------|------------------|
| Shape D1 only | 1.429 MeV |
| Linear Q-tilt V - Delta_m * t | 0.783 MeV |
| Weber-softened shape | 1.480 MeV |

Free space: **no** N7 environment assist — only Q_beta budget.

---

## P6 — WKB dictionaries

### Nucleon-mass WKB **[F for minutes]**

```
S/hbar = 2 integral sqrt(2 m0 (V-E)) dx / (hbar c)
```

Through a ~1-2 MeV barrier over ~1 fm: **S = O(0.1-few)**, tau ~ 1e-23 s class.
**Cannot** produce e^{59} / minutes. Wrong action mass/scale.

### EM 1/alpha-class toy **[M family]**

Actions can sit in O(10-100+) depending on width — **right family** for historical
E0 ≈ 59.3, coefficient **not derived**.

### Structural identity vs path integral **[E / F as equality]**

E0 = m0/(2 pq U0) ≈ 59.34 is an **algebraic ansatz** with fixed lobe U0.
It is **not** equal to integral m0/(pq U_D1(t)) dt over the mid-path hump.

---

## Integrity board

| Claim | Tag |
|-------|-----|
| Path A reverse topology exists; mid barrier [E] | [E] |
| Endpoint D1 gap << mid barrier (not Delta m source) | [E] |
| Path B multipole dissolve downhill | [E] |
| Pure multipole as free-beta clock | [F] |
| Free space: no N7 environment assist | [E ontology] |
| Q-tilt may leave residual barrier (D1 Rp) | [E/M dict] |
| Nucleon-mass WKB -> minutes tau | [F] |
| EM 1/alpha-class action family | [M] |
| E0 = path integral of D1 hump | [F as equality] |
| E0 = m0/(2 pq U0) algebraic ansatz | [E algebra / M physics] |
| Gamma_Ev = 1/tau_n | [F] |
| Full free-beta path + weak packaging | [O] |

---

## Verdict

1. **Path exists [E]:** free reverse topology coordinate with mid-path Coulomb barrier.
2. **Multipole alone fails [F]:** downhill, wrong lifetime class.
3. **Nucleon WKB fails for minutes [F]:** MeV·fm barrier too transparent.
4. **Minutes still want EM e^{1/alpha} action [M]** — historical E0 encodes that family,
   but is **not** derived from this path integral.
5. **Open [O]:** hybrid charge+topology coordinate; EM-WKB measure from Weber current;
   explicit weak packaging coordinate (e, nu phase space).

---

## Next forks

1. Hybrid path: simultaneous charge asymmetry + (p,q) reverse
2. EM-WKB measure from Weber self-energy density along Path A
3. Weak packaging / phase-space factor (SM-matching layer, honest [O])