V4 Topology-Local Freeze Gate + Dual Membership Convention
\(n_\alpha\ge 50\) replaces raw \(Z\) · hard-safe · dual: soft-L1 floor not freeze-\(S_{2n}>0\)
Date: 2026-08-11 Status: RESULTS FREEZE ·PASS_TOPO_LOCAL_FREEZE_HARD_SAFE
Script: scripts/investigations/s3_geometry/v4_topo_local_freeze_dual_census.mjs
Data: scripts/investigations/s3_geometry/data/v4_topo_local_freeze_dual_census.json
Priors: SHE-local S3z100 (§0bn168) · Coulomb floor F2 (§0bn167) · L1 free (§0bn166)
Tags: [E process] · [E honesty] · [M packaging] · [O residual] · [F overclaim]
0. One-line
Topology gate \(n_\alpha\ge 50\) (equiv. cage \(A_{NZ}\ge 200\)) matches S3z100 hard-safe Fl drip without a raw \(Z\) cut; dual island membership under freeze must not use \(S_{2n}>0\) (plateau zeros the strip) — freeze pre-cross empties classic \(N=184\); preferred dual = PRC‑A \(S_p>0\) and global‑L1 \(S_{2n}>S_{\mathrm{soft}}\) with \(S_{\mathrm{soft}}\simeq 2.5\)–\(3\,\mathrm{MeV}\) (floor-aware); promote NOT APPROVED.
1. Topology probe (gate design inputs)
| Nuclide | \(Z\) | \(n_\alpha\) | \(n_{\mathrm{free}}\) | L1 over? | |---------|------:|------------:|---------------------:|:--------:| | \(^{24}\)O | 8 | 4 | 8 | yes | | \(^{208}\)Pb | 82 | 41 | 44 | yes | | \(^{238}\)U | 92 | 46 | 54 | yes | | \(Z=100\) | 100 | 50 | 50 | edge | | Fl‑298 | 114 | 57 | 70 | yes | | \(Z=126\), \(N=184\) | 126 | 63 | 58 | no |
\(A_{NZ}=4n_\alpha\): U → 184, \(Z100\) → 200, Fl → 228.
2. Gate scoreboard (L1 + freeze when engaged)
| Book | Gate | Hard Fl \(N_{\mathrm{cross}}\) | O‑24 | U eng | Table | hard-safe | |------|------|-------------------------------:|-----:|:-----:|:-----:|:---------:| | B_Z100 | \(Z\ge 100\) (baseline) | 174 | stock | no | PASS | yes | | B_nA50 | \(n_\alpha\ge 50\) | 174 | stock | no | PASS | yes | | B_nA54 | \(n_\alpha\ge 54\) | 174 | stock | no | PASS | yes | | B_nA57 | \(n_\alpha\ge 57\) | 174 | stock | no | PASS | yes | | B_nA50_over | \(n_\alpha\ge 50\) ∧ over | 174 | stock | no | PASS | yes | | B_nA50_e01 | \(n_\alpha\ge 50\) ∧ \(r\ge 0.1\) | 177 | stock | no | PASS | yes | | B_cage200 | \(A_{NZ}\ge 200\) | 174 | stock | no | PASS | yes | | B_nA46 | \(n_\alpha\ge 46\) | 174 | stock | yes | STRESS | no | | B_cage184 | \(A_{NZ}\ge 184\) | 174 | stock | yes | STRESS | no | | B_global | always | 174 | 0 | yes | FAIL | no |
Finding T1 [E]: B_nA50 is hard-safe and topology-native; numerically coextensive with S3z100 on the tested strip (Fm+ cage size). Finding T2 [E]: \(n_\alpha\ge 46\) / cage 184 engages U → table STRESS — same failure mode as S3z90. Finding T3 [M]: \(n_\alpha\ge 50\) is still a threshold dial, but it is stated in α-count / cage size (engine topology), not bare proton number. Cleaner packaging than \(Z_{\mathrm{thr}}\). Preferred topo package: B_nA50 (primary); B_nA50_over if engagement should require free over-capacity explicitly; B_cage200 equivalent (\(4\times 50\)).3. Dual membership conventions (under B_nA50 @ \(N=184\))
| \(Z\) | \(n_\alpha\) | over | \(S_{2n}^{\mathrm{freeze}}\) | \(S_{2n}^{L1}\) | \(N_{\mathrm{cross}}\) | |------:|------------:|:----:|-----------------------------:|---------------:|---------------------:| | 108 | 54 | yes | 0 | 2.80 | 164 | | 110 | 55 | yes | 0 | 2.88 | 167 | | 112 | 56 | yes | \(\approx 0\) | 2.95 | 170 | | 114 | 57 | yes | 0 | 3.03 | 174 | | 118 | 59 | yes | 0 | 3.19 | 179 | | 120 | 60 | yes | \(\approx 0\) | 3.28 | 182 | | 126 | 63 | no | 10.76 | 10.76 | 191 |
Convention census (grid \(Z=100\)–130 even, \(N=184\); \(p\)-bound = 16)
| Convention | Definition | Strip hits | Grid count | Island use? | |------------|------------|------------|------------|-------------| | dual_hard_gt0 | \(S_p>0 \wedge S_{2n}^{\mathrm{fr}}>0\) | 126 only | 6 | NO — empties over-cap classic strip | | dual_ge0 | \(S_p>0 \wedge S_{2n}^{\mathrm{fr}}\ge 0\) | most strip | 13 | OK for freeze plateau membership | | dual_precross | \(S_p>0 \wedge N
Preferred dual convention (frozen)
\[ \boxed{ S_p^{\mathrm{PRC\text{-}A}}(Z,N)>0 \;\wedge\; S_{2n}^{L1}(Z,N) > S_{\mathrm{soft}} } \]
with \(S_{\mathrm{soft}} = 3\,\mathrm{MeV}\) (primary, above analytic coulomb floor contact) or \(2.5\,\mathrm{MeV}\) (inclusive floor band).
Hard neutron edge (separate map):
\[ N_{\mathrm{drip}}(Z) = N_{\mathrm{cross}}\bigl(Z;\; B_{n\alpha50}\bigr) \quad (S_{2n}^{\mathrm{freeze}}\le 0). \]
4. Ontology note (fixed cage radius)
Once free packing channels are full (\(n_{\mathrm{free}}>k_{\mathrm{cap}}=n_\alpha\)) and the cage is large enough (\(n_\alpha\ge 50\)):
- freeBE saturates (L1), - coulomb dilution radius freezes at \(A_{\mathrm{cap}}=A-\mathrm{excess}\),
i.e. fixed cage radius — no further neutron-induced Coulomb relief. Domain restriction \(n_\alpha\ge 50\) is the SHE-local packaging of that geometry, not a new force law.
5. Preferred package (diagnostic)
| Layer | Law | |-------|-----| | Engage freeze | \(n_\alpha \ge 50\) (or cage \(A_{NZ}\ge 200\)) | | Free rewrite | L1 \(k_{\mathrm{cap}}=n_\alpha\) when engaged | | Coulomb | freeze at \(A_{\mathrm{cap}}\) when over | | Proton edge | PRC‑A Sp-path (unchanged) | | Dual island | PRC‑A \(S_p>0\) ∧ L1 \(S_{2n}>3\,\mathrm{MeV}\) | | Hard n-drip | \(N_{\mathrm{cross}}\) under B_nA50 | | Promote | NOT APPROVED |
6. Kills
| Kill | Verdict | |------|---------| | K1 Topology hard-safe without raw \(Z\) | PASS (B_nA50 family) | | K2 \(n_\alpha\ge 46\) spares U | FAIL (engages U) | | K3 dual \(S_{2n}^{\mathrm{fr}}>0\) populates island | FAIL_empty on classic over-cap | | K4 dual_precross as island law | KILL for island (excludes Fl‑298) | | K5 Promote | FIRE_not_yet |
7. Integrity
- Sp PRC‑A untouched. - Diagnostic BE only. - Half-lives FORBIDDEN. - \(n_\alpha\ge 50\) still a dial — promote only after broader integrity / ontology lock. - \(Z=126\) @ \(N=184\) under L1 cap (\(n_{\mathrm{free}} 1. Broader dual grid (vary \(N\), not only 184) under soft-L1 thr. 2. Optional: derive \(n_\alpha^\star=50\) from packing/shell boundary (else keep dial explicit). 3. Closeout note for V4 SHE dual-edge package (p PRC‑A + n L1 soft + hard freeze edge). 4. Still no half-life claims.
8. Natural next
9. One-line (banked)
Topo \(n_\alpha\ge 50\) L1+freeze hard-safe @ Fl \(N\simeq 174\); dual = PRC‑A \(S_p\) + L1 \(S_{2n}>3\,\mathrm{MeV}\); freeze \(S_{2n}>0\) / pre-cross are not island laws; promote NOT APPROVED.