# V4 Coulomb Floor — \(S_{2n}\) Residual after Free-\(n\) L1

### Freeze engages hard Fl n-drip · table/light FAIL · preferred OPEN · promote NOT APPROVED

**Date:** 2026-08-11  
**Status:** **RESULTS FREEZE** · `PASS_COULOMB_FLOOR_CENSUS_OPEN`  
**Script:** `scripts/investigations/s3_geometry/v4_coulomb_floor_s2n_census.mjs`  
**Data:** `scripts/investigations/s3_geometry/data/v4_coulomb_floor_s2n_census.json`  
**Priors:** free-\(n\) L1 tight capacity (§0bn166) · SHE dual-edge · analytic dilution floor  

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

---

## 0. One-line

> **After L1 free-\(n\) saturation, residual \(S_{2n}\sim 3\,\mathrm{MeV}\) is pure Coulomb dilution \(\partial E_C/\partial N\); freezing dilution past free capacity (\(A_{\mathrm{cap}}=A-\mathrm{excess}\)) drives Fl \(S_{2n}\to 0\) at \(N\simeq 174\) — hard n-drip mechanism works — but light over-cap (O‑24) and measured mid-heavy table (Pb/U) FAIL gates; no promote-safe book; residual OPEN.**

---

## 1. Analytic floor (why L1 alone never hard-drips)

Stock dilution (engine form):

\[
\Delta E_C = A_C\,Z(Z-1)\Bigl[A_{NZ}^{-1/3}-A^{-1/3}\Bigr],\quad A_C=0.7,\ A_{NZ}=4n_\alpha.
\]

Two-neutron separation from dilution alone at Fl‑298:

\[
S_{2n}^{\mathrm{coul}} = A_C\,Z(Z-1)\bigl[(A-2)^{-1/3}-A^{-1/3}\bigr] \approx \mathbf{3.034\,\mathrm{MeV}}.
\]

Matches L1 census floor exactly (§0bn166). As \(A\to\infty\), \(S_{2n}^{\mathrm{coul}}\to 0^+\) but **never** \(\le 0\). Hard drip therefore requires **either**:

1. **Freeze / cut** further dilution once free-\(n\) is over capacity, or  
2. An **extra negative** channel (e.g. isospin asymmetry) that grows with \((N-Z)^2/A\).

---

## 2. Books (diagnostic BE rewrite: free + coulomb ± asym)

| ID | Free | Coulomb | Asym past cap |
|----|------|---------|---------------|
| F0 | stock | stock | — |
| F1 | **L1** \(k_{\mathrm{cap}}=n_\alpha\) | stock | — |
| **F2** | L1 | **freeze** at \(A_{\mathrm{cap}}\) | — |
| F3 | L1 | freeze (alias of F2) | — |
| F4 / F4b / F4c | L1 | freeze | \(a_{\mathrm{sym}}=B_e,\;B_e/2,\;B_e/4\) |
| F5 | L1 | stock | \(a_{\mathrm{sym}}=B_e/2\) |

**Capacity mass (bugfix):**

\[
A_{\mathrm{cap}} = A - \max(0,\,n_{\mathrm{free}}-k_{\mathrm{cap}}).
\]

For pure free-\(n\) Fl: \(A_{\mathrm{cap}}=4n_\alpha+k_{\mathrm{cap}}=285\).  
Earlier draft used \(A_{\mathrm{cap}}=2n_\alpha+k_{\mathrm{cap}}=171 < A_{NZ}=228\) → freeze was a no-op; fixed before freeze.

Freeze replacement (over-cap only):

\[
E_C^{\mathrm{book}} = E_C^{\mathrm{engine}} - c_{\mathrm{stock}}(A) + c_{\mathrm{stock}}(A_{\mathrm{cap}}).
\]

---

## 3. Fl‑298 / \(N=184\) scoreboard

| Book | \(S_{2n}(184)\) | Hard \(S_{2n}\le 0\) | \(N_{\mathrm{cross}}\) | O‑24 \(S_{2n}\) | Table K2 | mean \(\Delta\%\) |
|------|----------------:|:-------------------:|---------------------:|---------------:|:--------:|-----------------:|
| F0 stock | 10.88 | no | — | 8.72 | PASS | 0 |
| F1 L1 only | **3.03** | no | — | 0.40 | STRESS | 0.49 |
| **F2 freeze** | **0** | **yes** | **174** | **0** | **FAIL** | 0.61 |
| F4c freeze+¼ asym | −0.51 | yes | 173 | −0.64 | FAIL | 0.63 |
| F4b freeze+½ | −1.02 | yes | 173 | −1.27 | FAIL | 0.65 |
| F4 freeze+bond | −2.03 | yes | 173 | −2.55 | FAIL | 0.69 |
| F5 asym only | 2.02 | yes | **326** (far) | −0.87 | FAIL | 0.53 |

**Finding C1 [E]:** F2 freeze **does** zero the coulomb floor at Fl once over L1 cap (\(A_{\mathrm{cap}}=285\), engine \(E_C: 126\to 106\,\mathrm{MeV}\)). First hard cross \(N=174\) (just past \(N_{\mathrm{cap}}=171\)).

**Finding C2 [E honesty]:** Every hard-drip book **fails** either O‑24 (\(S_{2n}\le 0.5\)) or table K2 (mean \(\Delta\% > 0.5\)), usually both. **No preferred promote-safe book.**

**Finding C3 [O]:** Table FAIL on F2 is **not** uniform — light/mid measured spots ~0 dpp; damage is **Pb‑208** (\(+1.45\,\mathrm{pp}\)) and **U‑238** (\(+2.81\,\mathrm{pp}\)) from L1 free underbind on over-cap skins, not from freeze alone.

---

## 4. Light control (Z=8, \(N=16\) ≡ \(^{24}\)O class)

| Book | \(S_{2n}\) |
|------|----------:|
| F0 | 8.72 |
| F1 L1 | 0.40 |
| F2 freeze | **0** |
| F4* + asym | **negative** |

L1 already soft-underbinds \(^{24}\)O; freeze sets the residual coulomb floor to 0 → exact margin loss. Asymmetry books unbind light oxygen.  
Gate `O16_ok` (\(S_{2n}>0.5\)) fails all non-stock books except pure F0.

---

## 5. Preferred / park

| Class | Choice | Status |
|-------|--------|--------|
| **Hard + safe** | — | **OPEN** (none) |
| **Hard mechanism (diagnostic)** | **F2 L1 + freeze coulomb** | Candidate only · table/light FAIL |
| **Soft n-edge (prior)** | L1 free + soft thr \(S_{2n}>5\,\mathrm{MeV}\) | Still valid dual diagnostic (§0bn166) |
| **Promote** | — | **NOT APPROVED** |

**Do not** silently promote freeze into `predictBE` / separation engines.

---

## 6. Kills

| Kill | Claim | Verdict |
|------|-------|---------|
| K1 | Some book achieves Fl \(S_{2n}\le 0\) | **PASS_some** (F2–F5) |
| K2 | Table mean \(\Delta\%\le 0.5\) for hard books | **FAIL** (all hard) |
| K3 | O‑24 \(S_{2n}>0.5\) under hard books | **FAIL** |
| K4 | Promote freeze/asym | **FIRE_not_yet** |

---

## 7. Integrity

- Sp path **unchanged** (PRC‑A separate).  
- BE totals are **diagnostic rewrites** only.  
- \(a_{\mathrm{sym}}\) scales = \(\{B_e,\,B_e/2,\,B_e/4\}\) geometry-led, not fit.  
- Half-lives / island theorem **FORBIDDEN**.  
- Hard n-drip **mechanism** now known; **safe packaging** still OPEN.

---

## 8. Natural next (not done here)

1. **SHE-local freeze:** apply coulomb freeze only for high \(Z\) or high \(n_{\mathrm{free}}/k_{\mathrm{cap}}\) so Pb/U measured skins stay stock-ish.  
2. **Graded freeze** \(f\in(0,1]\): partial removal of \(\Delta E_C\) — alone cannot hard-cross if free is saturated (\(S_{2n}\propto(1-f)\cdot 3\,\mathrm{MeV}\)); needs \(f=1\) or asym.  
3. **Keep dual soft thr** for island membership without hard drip claim.  
4. Optional: re-express freeze as geometry of **fixed cage radius** once free packing channels are full (same physics as F2, cleaner ontology note).

---

## 9. One-line (banked)

> **Coulomb freeze past L1 capacity zeros Fl \(S_{2n}\) at \(N\simeq 174\) (hard mechanism PASS); table Pb/U + O‑24 gates FAIL; promote NOT APPROVED; hard n-drip packaging remains OPEN.**