# FL-HEAVY H6: Dual / Multi-Seat Binding Geometry

**Date:** 2026-08-09  
**Script:** `scripts/investigations/s3_geometry/fl_heavy_binding_geometry.py`  
**JSON:** `scripts/investigations/s3_geometry/data/fl_heavy_binding_geometry.json`  
**Priors:** `fl_heavy_selection_tower_results.md` · `fl_heavy_lattice_dual_seat_results.md` · `a_geom_gap2_gs_b2_multi_seat_binding_mass_results.md` · `flavour_heavy_lobe_reframe.md`

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

---

## 0. Why H6

H3B showed \(2M_{\mathrm{open}}\) overshoots onium by GeV-scale binding.
H4/H5 left S6 seeds provisional without tower glue.
GS-B2 already killed L1c multi-seat as a *hadron* mass mechanism (\(O(\alpha)\)).

**Question:** Is there a *geometric* dual/multi-seat package that explains
heavyonium relative to open heavy seats — without free windings or free \(c,b,t\)?

---

## 1. H6A — Binding ledger

| Pair | \(M_{\mathrm{on}}\) | \(M_{\mathrm{op}}\) | \(2M_{\mathrm{op}}\) | Δ (MeV) | Δ/\(M_{\mathrm{on}}\) | Δ/\(2M\) |
|------|--------------------:|--------------------:|----------------------:|--------:|-----------------------:|---------:|
| 2D−J/ψ | 3096.9 | 1867.2 | 3734.5 | 637.6 | 20.59% | 17.07% |
| 2Ds−J/ψ | 3096.9 | 1968.3 | 3936.7 | 839.8 | 27.12% | 21.33% |
| 2D−ηc | 2984.1 | 1867.2 | 3734.5 | 750.4 | 25.15% | 20.09% |
| 2Ds−ηc | 2984.1 | 1968.3 | 3936.7 | 952.6 | 31.92% | 24.20% |
| 2B−Υ | 9460.4 | 5279.6 | 10559.1 | 1098.7 | 11.61% | 10.41% |
| 2Bs−Υ | 9460.4 | 5366.9 | 10733.9 | 1273.5 | 13.46% | 11.86% |

| Light control | Δ (MeV) | Δ/\(M_{\mathrm{on}}\) |
|---------------|--------:|-----------------------:|
| 2π−ρ | -496.1 | -64.0% |
| 2K−φ | -32.1 | -3.1% |

| Ratio | Value |
|-------|------:|
| Δ(2B−Υ)/Δ(2D−J/ψ) | 1.723 |
| Δ(2Bs−Υ)/Δ(2Ds−J/ψ) | 1.516 |
| \(m_B/m_D\) | 2.827 |
| \(m_\Upsilon/m_{J/\psi}\) | 3.055 |
| Δ/\(2M\) heavy mean±std | 0.1749±0.0498 |

| Finding | Tag |
|---------|-----|
| Dual residual is GeV-class, not O(α) | **[E]** |
| Δ/\(2M\) not universal across heavy pairs | **[E honesty]** |
| Bottom/charm Δ ratio ~ O(1–2), tracks mass hierarchy weakly | **[M family]** |

---

## 2. H6B — Lattice dual packages

Zero free parameters. Seeds from H4/H5.

| Pkg | Definition |
|-----|------------|
| P1 | \(2 M_{\mathrm{open}}^{\mathrm{lab}}\) |
| P2 | \(\sqrt{2}\,M_{\mathrm{Hopf}}(w_{\mathrm{open}})\) |
| P3 | \(M_{0C}\sqrt{2 N_{\mathrm{open}}}\) |
| P4 | \(2 M_{\mathrm{Hopf}}(w_{\mathrm{open}})\) |
| P5 | \(M_{\mathrm{Hopf}}(w_{\mathrm{onium}})\) own seed (control) |
| P6 | Hopf of doubled open windings |
| P7 | Free of doubled open windings |

### S6 seeds (provisional package)

| Pair | P1% | P2% | P3% | P4% | P5% | P6% | P7% | \(N_{\mathrm{on}}/2N_o\) | ΔN |
|------|----:|----:|----:|----:|----:|----:|----:|----------:|---:|
| 2D−J/ψ | 20.6 | 15.2 | 15.2 | 20.0 | 0.005 | 20.0 | 66.7 | 1.389 | +552 |
| 2Ds−J/ψ | 27.1 | 10.1 | 10.1 | 27.2 | 0.005 | 27.2 | 66.7 | 1.236 | +376 |
| 2D−ηc | 25.1 | 12.0 | 12.0 | 24.5 | 0.117 | 24.5 | 73.0 | 1.287 | +407 |
| 2Ds−ηc | 31.9 | 6.7 | 6.7 | 32.0 | 0.117 | 32.0 | 73.0 | 1.145 | +231 |
| 2B−Υ | 11.6 | 21.1 | 21.1 | 11.6 | 0.001 | 11.6 | 51.9 | 1.606 | +6935 |
| 2Bs−Υ | 13.5 | 19.8 | 19.8 | 13.5 | 0.001 | 13.5 | 40.1 | 1.554 | +6551 |

**Non-P5 win counts (S0+S6):** `{'P1': 0, 'P2': 8, 'P3': 0, 'P4': 4, 'P6': 0, 'P7': 0}`

| Finding | Tag |
|---------|-----|
| Dual packages miss onium by tens of percent | **[F as zero-param dual mass]** |
| P5 competitive is own-seed tautology (H3/H4) | **[E]** |
| Dual lattice packaging ≠ membership theorem | **[F if claimed]** |

---

## 3. H6C — GS-B2 / Coulomb scale kill

| Seat | \(M\) | Δ_bind | \(\lambda_c\) (fm) | \(\alpha M\) | \(\alpha\hbar c/0.5\mathrm{fm}\) | \(\alpha_s{=}0.3/0.5\mathrm{fm}\) | Δ/\(\alpha M\) |
|------|------:|-------:|-------------------:|-------------:|----------------------------------:|-----------------------------------:|-------------:|
| Jpsi | 3097 | 638 | 0.0637 | 22.6 | 2.9 | 118 | 28.2 |
| Upsilon | 9460 | 1099 | 0.0209 | 69.0 | 2.9 | 118 | 15.9 |
| D | 1867 | 638 | 0.1057 | 13.6 | 2.9 | 118 | 46.8 |
| B | 5280 | 1099 | 0.0374 | 38.5 | 2.9 | 118 | 28.5 |

| Finding | Tag |
|---------|-----|
| EM Compton/fm Coulomb ≪ GeV dual Δ | **[E]** |
| L1c multi-seat cannot set heavy dual binding (GS-B2 join) | **[F L1c⇒heavy dual]** |
| Hadronic dual residual needs sector/Route A-class scale, not exterior Coulomb | **[E honesty]** |

---

## 4. H6D — Structural invariants

| Seed | Label | \(N_{\mathrm{on}}/2N_o\) | ΔN | \(d(\mathrm{round}\sqrt{2}\cdot\mathrm{open})\) | gcd o→on | mass-\(N\) ratio |
|------|-------|--------------------------:|---:|-------------------------------------------------:|----------:|----------------:|
| S0 | ψ←D | 1.368 | +530 | 10.8 | 12→1 | 1.375 |
| S0 | ηc←D | 1.269 | +388 | 12.0 | 12→2 | 1.277 |
| S0 | Υ←B | 1.606 | +6935 | 31.1 | 5→1 | 1.605 |
| S0 | ψ←Ds | 1.236 | +376 | 7.8 | 1→1 | 1.238 |
| S0 | Υ←Bs | 1.554 | +6551 | 27.0 | 1→1 | 1.554 |
| S6 | ψ←D | 1.389 | +552 | 15.6 | 1→1 | 1.375 |
| S6 | ηc←D | 1.287 | +407 | 13.5 | 1→1 | 1.277 |
| S6 | Υ←B | 1.606 | +6935 | 44.3 | 1→1 | 1.605 |
| S6 | ψ←Ds | 1.236 | +376 | 7.8 | 1→1 | 1.238 |
| S6 | Υ←Bs | 1.554 | +6551 | 27.0 | 1→1 | 1.554 |

Mass-implied \(N_{\mathrm{on}}/(2N_o)\): mean **1.410** ± 0.147

| Finding | Tag |
|---------|-----|
| \(N_{\mathrm{on}}\neq 2N_{\mathrm{open}}\) | **[E]** |
| Ratio ~1.3–1.6 is rephrasing of Δ_bind, not new glue | **[E honesty]** |
| √2 open winding ≉ onium seed in \((w_1,w_2)\) | **[F dual-norm ID]** |

---

## 5. H6E — Vector dual

Exact open+open winding sum/diff → onium seed: **0/16**  
Min lattice distance any sum/diff: **13.9** winding units.

| Finding | Tag |
|---------|-----|
| No vector dual identity on S0/S6 seeds | **[F]** |

---

## 6. Integrity board

| Claim | Tag |
|-------|-----|
| Δ_bind ledger 0.6–1.1 GeV class | [E] |
| Δ/(2M) universal across heavy | [F] — std large |
| Δ ratio Υ/ψ tracks M_B/M_D | [M weak family] |
| Zero-param lattice dual (P1–P4,P6,P7) = onium | [F] |
| Own-seed P5 residual competitive | [E tautology of H3/H4] |
| N_on = 2 N_open | [F] |
| N_on/(2No) universal integer | [F] |
| √2·open winding = onium seed | [F] |
| Vector sum open+open = onium seed | [F] |
| GS-B2 / L1c Coulomb = GeV dual Δ | [F] join GS-B2 |
| α_s toy fm = exact dual theorem | [F if claimed] |
| Free c,b,t on e_i | [CLOSED F] |
| S6 multiplet theorem via dual | [F] |
| FL-HEAVY membership closed | [O still] |
| FL-HEAVY after H6 | [O park recommended] |

---

## 7. Verdict

1. **H6A [E]:** Dual binding residual is real and GeV-class; fractions not universal.
2. **H6B [F]:** Zero-param lattice dual packages do not predict onium masses.
3. **H6C [F join GS-B2]:** Exterior Coulomb / Compton multi-seat cannot supply the residual.
4. **H6D–E [F]:** No integer / vector dual invariant glues open seeds to onium seeds.
5. **FL-HEAVY:** Membership seeds (S6) remain **provisional [M]**; dual binding does **not**
   close membership or replace tower glue.

### Park recommendation

> **PARK FL-HEAVY** after H1–H6. Banked: hadron seats exist in PDG; native banks empty;
> lattice seeds empty+competitive with density caveat; S6 selection provisional;
> towers not tight; dual/multi-seat binding not zero-param geometric.
>
> **Reopen only with a new native object** (not nearest-lattice + dual toy; not free \(c,b,t\);
> not CKM-first). Same wall class as light FL-O1 lattice→multiplet.

> **One-liner:** Heavy dual binding is a real GeV residual that no lattice dual package or GS-B2 Coulomb multi-seat explains — FL-HEAVY stays open; park until a new native seat principle.

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## Changelog

| Date | Change |
|------|--------|
| 2026-08-09 | FL-H6 binding ledger + dual packages + GS-B2 kill + park |