# V4 Tight Free-\(n\) Capacity — Neutron Edge Census Results

### L1 engages at Fl-298 · \(S_{2n}\) floors \(\sim 3\,\mathrm{MeV}\) · hard drip still OPEN · dual soft-thr

**Date:** 2026-08-11  
**Status:** **RESULTS FREEZE** · `PASS_TIGHT_CAP_CENSUS_CANDIDATE`  
**Script:** `scripts/investigations/s3_geometry/v4_freen_tight_capacity_nedge_census.mjs`  
**Data:** `scripts/investigations/s3_geometry/data/v4_freen_tight_capacity_nedge_census.json`  
**Priors:** free-\(n\) C1 · SHE dual-edge · Coulomb floor  

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

---

## 0. One-line

> **Tighter geometry caps (\(k_{\mathrm{cap}}=n_\alpha\), half-\(n_\alpha\), surface, skin, …) **do engage** at Fl-298 (\(n_{\mathrm{free}}=70>k_{\mathrm{cap}}\)), collapsing island \(S_{2n}\) from \(\sim 11\) to a **\(\sim 3\,\mathrm{MeV}\) Coulomb floor** — still **never** \(S_{2n}\le 0\) out to far \(N\); preferred clean law **L1** \(k_{\mathrm{cap}}=n_\alpha\); hard n-drip remains OPEN without Coulomb work.**

---

## 1. Capacity at Fl-298 (\(n_\alpha=57\), \(n_{\mathrm{free}}=70\))

| Law | \(k_{\mathrm{cap}}\) | Over cap? | \(S_{2n}(184)\) |
|-----|--------------------:|:---------:|----------------:|
| L2 \(2n_\alpha\) (old C1) | 114 | No | 10.88 |
| **L1 \(n_\alpha\)** | **57** | **Yes** | **3.03** |
| Lφ \(1.2\,n_\alpha\) | 68 | Yes | 3.03 |
| Lh \(n_\alpha/2\) | 28 | Yes | 3.03 |
| Ls \(2n_\alpha^{2/3}\) | 29 | Yes | 3.03 |
| Lskin \(p_n\sqrt{n_\alpha}\) | 22 | Yes | 3.03 |
| Lsp spare−ring | 1 | Yes | 3.03 |
| Lc contacts | 142 | No | 10.88 |

Once **over cap**, freeBE is saturated: \(S_{2n}\) from free-\(n\) \(\to 0\), residual \(S_{2n}\sim 3\,\mathrm{MeV}\) is **non-free** (Coulomb / ring / α structure).

**Finding T1 [E]:** All engaging caps give the **same** floor \(S_{2n}\) at \(N=184\) — the cut is binary (over/under cap), not graded by how tight once exceeded.

**Finding T2 [E honesty]:** **No law** produced \(S_{2n}\le 0\) on Fl out to \(N\sim 400\). Hard neutron drip still requires the **Coulomb floor** residual.

---

## 2. Light control (Z=8)

| \(N\) | stock \(S_{2n}\) | L1 \(S_{2n}\) |
|------:|-----------------:|--------------:|
| 16 (²⁴O class) | 8.72 | **0.40** (over cap if \(k=n_\alpha=4\), \(n_{\mathrm{free}}=8\)) |
| 18+ | ~8 | ~0.3 floor |

L1 **stresses** ²⁴O (\(S_{2n}\) small but \(>0\)). Harsher caps (half, skin) behave similarly once over cap.  
**Not** a clean ²⁴O kill if thr is \(S_{2n}>0\); **is** a soft underbind of the last bound oxygen isotopes.

---

## 3. Preferred law for **engagement** (not hard drip)

| Preferred | **L1 \(k_{\mathrm{cap}}=n_\alpha\)** |
|-----------|-------------------------------------|
| Geometry | One free-\(n\) packing channel per α (tighter than spare-lobe \(2n_\alpha\)) |
| SHE \(N=184\) | **Engaged** (\(70>57\)) |
| Effect | \(S_{2n}: 11\to\sim 3\,\mathrm{MeV}\) floor |
| Hard drip | **Still no** |
| Vs L2 | L2 not engaged at island |

**Secondary:** Lφ (\(k=1.2 n_\alpha\)) — barely over at Fl (70 vs 68).  
**Avoid as n-edge:** L2 (no engage); Lsp \(k=1\) (too violent conceptually).

---

## 4. Dual island membership (p PRC-A + n \(S_{2n}^{L1}>0\))

Because floor \(\sim 3>0\), **thr=0 dual still fills** the p-bound set (same non-selectivity as L2 in-window).

**Soft dual (recommended diagnostic):** require

\[
S_p^{\mathrm{PRC\text{-}A}}>0
\quad\text{and}\quad
S_{2n}^{L1} > S_{\mathrm{soft}}
\]

with e.g. \(S_{\mathrm{soft}}=5\,\mathrm{MeV}\) (above Coulomb floor, below mid-shell \(\sim 10\)):

- Under-cap nuclei: \(S_{2n}\sim 10\) → dual pass  
- Over-cap nuclei: \(S_{2n}\sim 3\) → **fail soft dual** → neutron-edge **selectivity**

At \(N=184\) Fl: \(S_{2n}^{L1}\sim 3 < 5\) → **soft dual fails** even though p-bound — a **soft neutron wall inside the proton-open island**.

| Concept | Fl-298 |
|---------|--------|
| p-bound PRC-A | Yes |
| n-bound hard \(S_{2n}\le 0\) | No |
| n-firm soft \(S_{2n}>5\) under L1 | **No** (\(\sim 3\)) |
| Reading | Proton-open, free-\(n\) **over capacity**, still Coulomb-bound neutrons |

---

## 5. Kill / pass board

| ID | Verdict |
|----|---------|
| **K1** Some cap engages at Fl-298 | **PASS** (L1, Lh, Ls, Lskin, Lφ, …) |
| **K2** Hard \(S_{2n}\le 0\) near island | **FAIL / OPEN** (floor \(\sim 3\,\mathrm{MeV}\)) |
| **K3** L1 destroys O-16 absolute | **STRESS** (²⁴O \(S_{2n}\sim 0.4\), not \(\le 0\)) |
| **K4** Promote | **FIRE / NOT APPROVED** |
| **P1** Cap table + Fl engage | **PASS** |
| **P2** Coulomb floor diagnosed | **PASS** |

---

## 6. Programme implication

```text
Proton edge (PRC-A):     OPEN / working diagnostic
Neutron edge L1 engage:  YES at N=184 (over cap)
Hard n-drip:             STILL OPEN → Coulomb floor primary
Soft dual thr~5 MeV:     selective neutron firmness proxy
Lifetime:                still FORBIDDEN
```

**Next natural residual for hard n-drip:** Coulomb floor (P1b), not tighter caps alone.

---

## 7. Permanent product card

```text
TIGHT FREE-N CAP N-EDGE
-----------------------
Engage at Fl-298:  L1 k_cap=nα (and tighter) YES
L2 k_cap=2nα:      still under-cap at N=184
S2n when over:     ~3 MeV floor (all tight laws)
Hard S2n<=0:       NEVER in Fl scan
O-24 under L1:     S2n~0.4 stress
Preferred engage:  L1_lobe1na
Soft dual:         Sp_PRC-A>0 & S2n_L1>5 MeV (packaging)
Promote:           NOT APPROVED
```

---

## 8. One-line banked

> **Tight free-\(n\) L1 engages at Fl-298 and drops \(S_{2n}\) to a \(\sim 3\,\mathrm{MeV}\) Coulomb floor; hard n-drip still OPEN; soft dual thr can restore neutron selectivity.**