# FL-HEAVY H4/H5: Selection Principle + Tower Test

**Date:** 2026-08-09  
**Script:** `scripts/investigations/s3_geometry/fl_heavy_selection_tower.py`  
**JSON:** `scripts/investigations/s3_geometry/data/fl_heavy_selection_tower.json`  
**Priors:** `fl_heavy_lattice_dual_seat_results.md` · light 102 census features

Tags: **[E]** · **[E honesty]** · **[M]** · **[O]** · **[F]**

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## 0. Why H4

H3 produced empty Hopf seeds with competitive residuals. At heavy mass the lattice is dense, so many points fit. **Selection** is the non-vacuous residual.

Light prior: Hopf **gcd=1** fraction = **50/86 = 0.581**.

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## 1. H4 — Cloud uniqueness + rules

| Ground | cloud ≤2% | n ≤0.5% | n in 2×best | err2/err1 | empty∩comp | S0 seed | S2 gcd1 | S6 |
|--------|----------:|--------:|------------:|----------:|-----------:|---------|---------|-----|
| D | 25 | 3 | 4 | 1.13 | 2 | (12,24) 0.26% | (15,22) 0.51% | — |
| Ds | 26 | 9 | 2 | 1.73 | 9 | (11,26) 0.07% | (11,26) 0.07% | (11,26) 0.07% |
| Lc | 36 | 7 | 2 | 1.00 | 7 | (7,32) 0.04% | (7,32) 0.04% | (7,32) 0.04% |
| Jpsi | 62 | 15 | 2 | 1.00 | 15 | (11,43) 0.00% | (11,43) 0.00% | (11,43) 0.00% |
| etac | 60 | 10 | 1 | 2.15 | 10 | (8,42) 0.03% | (12,41) 0.12% | (12,41) 0.12% |
| B | 185 | 42 | 3 | 1.00 | 42 | (10,75) 0.01% | (37,66) 0.01% | (37,66) 0.01% |
| Bs | 189 | 46 | 2 | 1.00 | 46 | (21,74) 0.00% | (21,74) 0.00% | (21,74) 0.00% |
| Lb | 209 | 45 | 3 | 1.00 | 45 | (34,73) 0.02% | (34,73) 0.02% | (34,73) 0.02% |
| Upsilon | 589 | 149 | 4 | 1.00 | 149 | (31,132) 0.00% | (31,132) 0.00% | (31,132) 0.00% |

**S2 differs from S0:** 3/9  
**S6 available (gcd1∩≤0.5%):** 8/9  
**Grounds with >5 competitive points:** 8/9

| Finding | Tag |
|---------|-----|
| Competitive clouds are large — best-err not unique | **[E honesty]** |
| gcd=1 (light-like) re-seeds many grounds vs S0 | **[E process]** |
| **S6** = empty ∩ gcd=1 ∩ err≤0.5% preferred provisional package | **[M]** |
| S6 = finished multiplet theorem | **[F if claimed]** |

### Preferred provisional seed package (S6)

| Ground | S6 (w1,w2) | N | err% | empty |
|--------|------------|--:|-----:|:-----:|
| D | S2 fallback (15,22) | 709 | 0.507 | False |
| Ds | (11,26) | 797 | 0.069 | True |
| Lc | (7,32) | 1073 | 0.044 | True |
| Jpsi | (11,43) | 1970 | 0.005 | True |
| etac | (12,41) | 1825 | 0.117 | True |
| B | (37,66) | 5725 | 0.009 | True |
| Bs | (21,74) | 5917 | 0.001 | True |
| Lb | (34,73) | 6485 | 0.017 | True |
| Upsilon | (31,132) | 18385 | 0.001 | True |

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## 2. H5 — Tower clustering

For each family, nearest Hopf point of each excitation vs ground S0/S6 seed and a wrong light seed (5,6).

| Family | n | mean d(S0) | mean d(wrong) | frac closer to S0 than wrong |
|--------|--:|-----------:|--------------:|-----------------------------:|
| D | 12 | 11.4 | 28.8 | 1.00 |
| B | 7 | 18.2 | 74.1 | 1.00 |
| psi | 11 | 16.2 | 50.5 | 1.00 |
| Upsilon | 8 | 24.2 | 142.3 | 1.00 |
| Lc | 8 | 8.7 | 32.3 | 1.00 |

**Finding H5a [E honesty]:** Towers **do not** form tight clusters about a single ground lattice seed (mean \(d\sim 9\)–\(24\) winding units). Not a K*/K-style multiplet glued to one \((w_1,w_2)\).

**Finding H5b [E / weak M]:** Excitation best-points are **systematically closer** to the heavy ground seed than to a light wrong seed (frac \(=1\) in tested families). That is largely a **mass-neighbourhood / annulus** effect (states near \(M_g\) sit near \(N\sim(M/M_{0C})^2\)), **not** proof of multiplet flavour inheritance on the lattice plane.

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## 3. Integrity board

| Claim | Tag |
|-------|-----|
| Competitive lattice clouds large at heavy mass | [E] |
| S0 best-err unique seed | [F as unique] for most grounds |
| gcd=1 filter (light-like) changes seed vs S0 | [E process] |
| S6 empty∩gcd1∩competitive package | [M selection package] |
| S6 = multiplet theorem | [F if claimed] |
| Excitation tower clusters on ground seed | [F / weak] |
| Density caveat high N | [E] |
| Free c,b,t on e_i | [CLOSED F] |
| FL-HEAVY after H4/H5 | [O] selection package provisional; towers weak |

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## 4. Verdict

1. **H4 [E/M]:** Selection is mandatory; **S6** (empty ∩ gcd=1 ∩ competitive) is the best *provisional* package aligned with light priors.
2. **H5 [E]:** Excitation towers do **not** validate seeds as multiplet centres in \((w_1,w_2)\).
3. **FL-HEAVY residual:** still open — seeds without tower glue are incomplete membership.
4. **Next:** H6 multi-seat/binding geometry, or park until a tower-compatible invariant appears.

> **One-liner:** Light-like gcd=1 selects cleaner empty lattice seeds for heavy hadrons, but PDG towers do not cluster on those seeds — membership package provisional, not closed.

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## Changelog

| Date | Change |
|------|--------|
| 2026-08-09 | FL-H4 selection rules + H5 tower clustering |