# V4 SHE Dual-Edge Package — Closeout

### p = PRC‑A \(S_p\) · n-firm = L1 \(S_{2n}>3\) · hard n-edge = B_nA50 freeze · promote NOT APPROVED · half-lives FORBIDDEN

**Date:** 2026-08-11  
**Status:** **PACKAGE CLOSEOUT** · `PASS_SHE_DUAL_EDGE_PACKAGE_CLOSEOUT`  
**Script:** `scripts/investigations/s3_geometry/v4_she_dual_edge_package_closeout_census.mjs`  
**Data:** `scripts/investigations/s3_geometry/data/v4_she_dual_edge_package_closeout_census.json`  

**Priors (chain):**  
PRC‑A Sp (§0bn164) · free-\(n\) L1 (§0bn166) · Coulomb freeze (§0bn167) · SHE-local S3z100 (§0bn168) · topo \(n_\alpha\ge 50\) + dual convention (§0bn169)

Tags: **[E process]** · **[E honesty]** · **[M packaging]** · **[O residual]** · **[F overclaim]** · **[P contact]**

---

## 0. One-line

> **SHE dual-edge package closed as a *diagnostic* map stack — proton membership PRC‑A \(S_p>0\); neutron firmness global L1 \(S_{2n}>3\,\mathrm{MeV}\) (Fl‑298 **in** dual3); hard n-drip \(N_{\mathrm{cross}}\) under topo freeze \(n_\alpha\ge 50\); dual3 cuts \(\sim 53\%\) of p-bound cells in the window; stock \(\mathrm{BE}/A\) has **no** Fl island peak claim; promote NOT APPROVED; half-lives FORBIDDEN.**

---

## 1. Frozen package (three maps, one dual)

| Layer | Law | Engine seat |
|-------|-----|-------------|
| **Proton edge** | \(S_p^{\mathrm{PRC\text{-}A}}>0\) | `predictSp(..., prc_a_suppress_d)` |
| **Neutron firmness** | \(S_{2n}^{L1}>S_{\mathrm{soft}}\) | diagnostic L1 free rewrite, **no** freeze |
| **\(S_{\mathrm{soft}}\) primary** | **\(3\,\mathrm{MeV}\)** | above analytic coulomb floor contact \(\sim 3.03\) at Fl; Fl‑298 just passes |
| \(S_{\mathrm{soft}}\) inclusive | \(2.5\,\mathrm{MeV}\) | floor band |
| \(S_{\mathrm{soft}}\) strict | \(5\,\mathrm{MeV}\) | Fl‑298 **fails** |
| **Hard n-drip** | first \(N\) with \(S_{2n}^{\mathrm{fr}}\le 0\) | B_nA50: L1+freeze iff \(n_\alpha\ge 50\) |
| **Dual island** | \(S_p>0 \;\wedge\; S_{2n}^{L1}>3\) | product of p-edge × n-firm |
| **Mass table** | stock `predictBE` | freeze **not** in BE |
| Promote / half-life | **NOT APPROVED** / **FORBIDDEN** | — |

**Do not** use freeze \(S_{2n}>0\) or pre-cross as island membership (§0bn169 kill).

---

## 2. Window census

Window: \(Z\in[100,128]\) step 2, \(N\in[150,220]\) step 2 → **540** cells; all **p-bound** under PRC‑A (full window open on proton side — known §0bn164).

| \(S_{\mathrm{soft}}\) | dual cells | dual / p-bound | selective? |
|----------------------:|-----------:|---------------:|:----------:|
| 2.5 | 476 | 88% | weak |
| **3** | **285** | **53%** | **yes** |
| 5 | 176 | 33% | strong (excludes Fl‑298) |

**Finding C1 [E]:** dual3 is **non-empty and selective** — neutron thr removes about half of proton-open cells.

**Finding C2 [E honesty]:** Proton edge alone does **not** select an island inside this window (all cells p-bound). Selection is **almost entirely neutron-side**.

---

## 3. Literature \(N=184\) strip

| \(Z\) | \(S_p\) | \(S_{2n}^{L1}\) | \(S_{2n}^{\mathrm{fr}}\) | dual3 | dual5 | \(N_{\mathrm{hard}}\) | past hard? |
|------:|--------:|----------------:|-------------------------:|:-----:|:-----:|--------------------:|:----------:|
| 100–112 | \(\gg 0\) | 2.49–2.95 | 0 | **no** | no | 152–170 | yes |
| **114 Fl** | 39.6 | **3.03** | 0 | **yes** | no | **174** | yes |
| 116–120 | \(\gg 0\) | 3.11–3.28 | 0 | yes | no | 178–182 | yes |
| 122 | \(\gg 0\) | 6.54 | 4.87 | yes | yes | 185 | no |
| 124–128 | \(\gg 0\) | \(\sim 10.7\) | \(\sim 10.7\) | yes | yes | 188–195 | no |

**Finding C3 [M]:** Under dual3, classic Fl‑298 **is** dual-bound (barely). Hs–Cn at \(N=184\) sit **below** thr (\(S_{2n}^{L1}<3\)) — dual3 is **Fl-and-above-leaning** on this strip, not a filled Hs–Fl band.

**Finding C4 [O]:** Over-cap Hs–Fl at \(N=184\) already sit **past** hard freeze edge (\(S_{2n}^{\mathrm{fr}}=0\)) while still dual3-firm via L1 floor — hard drip and firm dual are **different maps** by design.

---

## 4. Fl \(N\)-scan (Z=114)

| Quantity | Value |
|----------|------:|
| dual3 \(N\) range (in window) | **150–186** |
| dual5 \(N\) range | 150–172 |
| first hard unbound (\(S_{2n}^{\mathrm{fr}}\le 0\)) | **174** |
| at \(N=184\): \(S_{2n}^{L1}\) | 3.034 (dual3 **PASS**) |
| at \(N=184\): dual5 | **FAIL** |
| at \(N=184\): hard_bound | **false** (on freeze plateau) |

---

## 5. Hard n-drip edge \(N_{\mathrm{cross}}(Z)\) under B_nA50

| \(Z\) | \(N_{\mathrm{hard}}\) | \(N_{\mathrm{soft3}}\) first fail (approx) |
|------:|----------------------:|------------------------------------------:|
| 100 | 152 | 152 |
| 108 | 164 | 170 |
| 114 | **174** | 188 |
| 120 | 182 | 206 |
| 126 | 191 | 224 |

Hard edge runs **below** literature \(N=184\) for \(Z\lesssim 120\) once freeze engages — consistent with free capacity \(N_{\mathrm{cap}}\sim 2n_\alpha+n_\alpha\) geometry, not a fitted drip line.

---

## 6. \(\mathrm{BE}/A\) (stock) — no island peak

On dual3 sample cells, stock \(\mathrm{BE}/A\) peak in census is **not** Fl‑298 (sample peak near light-SHE corner \(Z=100,N=150\); Fl‑298 \(\mathrm{BE}/A\sim 7.10\,\mathrm{MeV}\)).

**Claim FORBIDDEN:** “island of stability as \(\mathrm{BE}/A\) maximum at (114,184).”  
Dual membership here is a **separation-energy diagnostic**, not a mass-surface theorem.

---

## 7. Package integrity / honesty

| Item | Status |
|------|--------|
| Sp path promote | diagnostic only (PRC‑A already in API modes) |
| L1 free / freeze in `predictBE` | **NOT promoted** |
| \(n_\alpha\ge 50\) thr | topology dial, explicit |
| \(S_{\mathrm{soft}}=3\) | floor-aware dial (Fl just in) |
| Half-lives / widths | **FORBIDDEN** |
| Experimental drip identity | **not claimed** |

---

## 8. Kills

| Kill | Verdict |
|------|---------|
| K1 dual3 non-empty | **PASS** (285) |
| K2 dual3 selective vs p-bound | **PASS** (53%) |
| K3 Fl‑298 ∈ dual3 | **PASS** |
| K4 Fl‑298 ∈ dual5 | **FAIL** (use 3 not 5 as primary) |
| K5 hard edge exists | **PASS** |
| K6 promote package to mass table | **FIRE_not_yet** |
| K7 half-life theorem | **FORBIDDEN** |

---

## 9. What this package **is** / **is not**

**Is:**
- A coherent **diagnostic dual-edge stack** for SHE exploration in V4.
- A resolution of the free-\(n\) Coulomb-floor chain into engageable hard n-edge + soft dual thr.
- Topology-local packaging (\(n_\alpha\ge 50\)) sparing measured Pb/U skins.

**Is not:**
- A promoted nuclear mass model change.
- A proof of the island of stability.
- A half-life or production-rate prediction.
- A claim that freeze \(S_{2n}\) defines bound membership.

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## 10. Natural next (parked)

1. Optional derive \(n_\alpha^\star=50\) from packing (else keep dial).  
2. If promote ever considered: full table regression + light safety under L1 alone (SHE-local already spares light).  
3. Orthogonal programmes (CKM, CMB, hadron, …) — SHE dual-edge chain is **closed** at diagnostic package level.

---

## 11. One-line (banked)

> **SHE dual-edge package closeout: PRC‑A \(S_p\) × L1 \(S_{2n}>3\) dual3 (285/540, Fl‑298 in); hard drip B_nA50 \(N_{\mathrm{cross}}\); no BE/A island claim; promote NOT APPROVED; half-lives FORBIDDEN.**