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])