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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.
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