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 intopredictBE / 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.