# G₂ e₁…e₇ Quark Basis Dictionary (Gap Fill)

**Date:** 2026-08-08
**Script:** `scripts/investigations/s3_geometry/g2_e7_quark_basis_dictionary.py`
**JSON:** `scripts/investigations/s3_geometry/data/g2_e7_quark_basis_dictionary.json`
**Priors:** `octonion_flavour_knowledge_arc.md` · `paper_octonion_flavour.md` · `paper_np_mass_splitting_g2.md` · `flavour_residuals_ledger.md` · preferred-basis scripts

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

---

## Gap statement (answered)

| | |
|--|--|
| **Arena** | 8D Im(𝕆), Aut = G₂ ⊃ SU(3) |
| **Known** | n–p Δm structural G₂ √3 rhyme **[M]** (`paper_np_mass_splitting_g2.md`); light SU(3) slot **[E]** (octonion arc) |
| **Gap wording** | Explicit e₁…e₇ → (u,d,s,c,b,t) dictionary under G₂→SU(3) |
| **Honest reformulation** | Explicit preferred-basis map of Im(𝕆) under Stab(e₄) to the **light** 3 ⊕ 3̄; score six-quark 1–1 as representation theory |
| **Status** | **Light freeze [E]**; **six-quark 1–1 [CLOSED F]**; **heavy packaging [M/O]** |

---

## D0 — Representation guards **[E]**

| Object | Dimension / content |
|--------|---------------------|
| Im(𝕆) = span{e₁,…,e₇} | 7 real |
| G₂ = Aut(𝕆) | 14 |
| SU(3) = Stab(one imaginary) | 8, rank 2 |
| Preferred invariant | Cayley–Dickson doubling unit **e₄** |
| Remaining 6 units | **3 ⊕ 3̄** of SU(3) (Günaydin–Gürsey 1973) |

Why a bare map e₁…e₇ → (u,d,s,c,b,t) is ill-posed:

1. **Count:** 7 real basis vectors ≠ 6 quark flavours.
2. **Rep:** SU(3) content is **3 quarks + 3 antiquarks**, not 6 quarks.
3. **Invariant:** one unit (e₄ preferred) is **not** a flavour state.
4. **Generations:** (c,b,t) are **not** states of light-flavour SU(3)_f **3**.

---

## D1–D2 — Preferred construction **[E]**

```text
H = (1, e1, e2, e3)     quaternion physical space
O = H ⊕ H·e4           Cayley–Dickson; e4 = doubling unit
SU(3) = Stab(e4)
I3 = Lz : e1→e2, e2→−e1, fix e3 and e4   (physical rotation)
Y  = unique Cartan partner commuting with I3 in su(3)
J  = L_{e4}           complex structure on O
3  = +i eigenspace of J in P = span(e1,e2,e3,e5,e6,e7)
3̄ = −i eigenspace of J in P
```

Equilateral weight-triangle spread (Killing): **6.66e-16** (0 → genuine **3**).
G₂ generators found: 14; SU(3) generators: 8; Lz⊂Stab(e₄) residual 1.83e-16.

---

## D3 — Frozen light dictionary **[E structure]**

### Not a basis-vector equality

| Object | Role |
|--------|------|
| **e₄** | SU(3)-invariant doubling unit — **not** a quark |
| **P = span{e₁,e₂,e₃,e₅,e₆,e₇}** | Carrier of **3 ⊕ 3̄** (complex linear combinations) |
| **I₃** | Rotation **within** physical (e₁,e₂,e₃) |
| **Y** | Generator rotating physical ↔ extra (e₅,e₆,e₇) |

### Light quarks (3) — preferred Cartan eigenvalues

| Flavour | I₃ | Y | Q = I₃+Y/2 | phys(e₁e₂e₃) | extra(e₅e₆e₇) | Top |c|² basis |
|---------|-----|---|------------|--------------|---------------|-------------------|
| **u** | +0.5000 | +0.3333 | +0.6667 | 0.500 | 0.500 | e2(0.25), e6(0.25), e1(0.25) |
| **s** | -0.0000 | -0.6667 | -0.3333 | 0.500 | 0.500 | e3(0.50), e7(0.50), e1(0.00) |
| **d** | -0.5000 | +0.3333 | -0.3333 | 0.500 | 0.500 | e2(0.25), e6(0.25), e1(0.25) |

### Antiquarks (3̄)

| Flavour | I₃ | Y | Q = I₃+Y/2 | phys | extra | Top |c|² |
|---------|-----|---|------------|------|-------|---------|
| **dbar** | +0.5000 | -0.3333 | +0.3333 | 0.500 | 0.500 | e2(0.25), e6(0.25), e1(0.25) |
| **sbar** | +0.0000 | +0.6667 | +0.3333 | 0.500 | 0.500 | e3(0.50), e7(0.50), e1(0.00) |
| **ubar** | -0.5000 | -0.3333 | -0.6667 | 0.500 | 0.500 | e2(0.25), e6(0.25), e1(0.25) |

**Reading:** each flavour eigenstate is a **50/50-class** mixture of physical and extra units (classic GG / preferred-basis result). There is **no** pure `e_k = u` assignment.

### Working freeze (use this)

```text
G2 dictionary (preferred, light only)
─────────────────────────────────────
e4                 :  SU(3) singlet / doubling unit
3 ⊂ P(e1,e2,e3,e5,e6,e7)  :  (u, d, s)   via I3 = Lz, Y partner
3̄ ⊂ P                  :  (ū, d̄, s̄)
I3 ladder             :  (+1/2, −1/2, 0)  =  (u, d, s)
Y structure           :  (+1/3, +1/3, −2/3)
Absolute I3 axis      :  knot J locks global SO(3) convention [E]
(c,b,t)               :  NOT in this 3  — see generation options
```

---

## D4 — Six-quark 1–1 map **[CLOSED F]**

### T1 — Is a single e_k a flavour eigenstate?

| Unit | weight in 3 | weight in 3̄ | I₃ purity in 3 | Pure eigenstate? |
|------|-------------|--------------|----------------|------------------|
| e1 | 0.5000 | 0.5000 | 0.5000 | **no** |
| e2 | 0.5000 | 0.5000 | 0.5000 | **no** |
| e3 | 0.5000 | 0.5000 | 1.0000 | **no** |
| e4 | 0.0000 | 0.0000 | 0.0000 | **no** |
| e5 | 0.5000 | 0.5000 | 0.5000 | **no** |
| e6 | 0.5000 | 0.5000 | 0.5000 | **no** |
| e7 | 0.5000 | 0.5000 | 1.0000 | **no** |

**Finding D4a [F as eᵢ=quark]:** pure single-eᵢ eigenstates in the **3**: **0/7**. Quarks are complex mixtures.

### T2–T3 — Generations and cardinality

| Test | Result | Tag |
|------|--------|-----|
| Room for (c,b,t) in light **3** | dim_ℂ(3)=3 already filled by (u,d,s) structure | **[F]** |
| Bijection 7 real → 6 flavours | must drop one unit + ignore 3̄ / complex structure | **[F]** |
| Naive e₅,e₆,e₇ = c,b,t | confuses Y-direction with generation; fails purity | **[F as derivation]** |

---

## D5 — Generation packaging for (c,b,t) **[M/O]** · **reframe 2026-08-09**

**Ontology update:** quarks are **not** CPT primitives — they are SM-equivalent labels for **lobe / multipole interaction channels**. Primary residual is **FL-HEAVY** (multiplet membership + scale), not “store \(c,b,t\) in \(G_2\)”. See `flavour_heavy_lobe_reframe.md`.

| ID | Option | Tag | Status |
|----|--------|-----|--------|
| **G-A** | Light-only GG freeze: Only (u,d,s)⊕antiquarks from G2→SU(3); heavy flavours elsewhere | [E scope of GG] / [O for heavy] | preferred default for this gap fill |
| **G-B** | Three copies of SU(3)_f: Generations as three isomorphic flavour SU(3)s (not second Im(O)) | [M packaging] / no second O (Hurwitz wall) | motivated, not constructed from G2 alone |
| **G-C** | S7 / Koide generation track: Heavy ladder from S7 generation programme (separate from light GG slot) | [M join] S7 catch-up banked | **L3 scale** track; not e_i map |
| **G-D** | Lattice / multiplet membership for heavy: Charm/bottom as multiplet labels after lattice membership (FL-O1 wall) | [M/O] FLAV-RES; lattice→multiplet still wall | **L2 primary residual** under reframe |
| **G-E** | Naive e5,e6,e7 = c,b,t: Identify extra units with heavy flavours by fiat | [F as derivation] | fails D4a purity; confuses Y-direction with generation |

**Preferred default for G₂ dictionary freeze:** **G-A** (light-only GG). Heavy work proceeds as **lobe/seat multiplet + scale** (G-D then G-C), with SM \((c,b,t)\) only as **export labels [M]** — **not** a second Im(𝕆) and **not** a forced eᵢ map.

---

## D6 — Hypercharge / GMN (light) **[E structure]**

With preferred Cartan scales, light GMN charges match SM (u,d,s):

| q | I₃ | Y | Q_GMN | Q_SM |
|---|-----|---|-------|------|
| u | +0.5000 | +0.3333 | +0.6667 | +0.6667 |
| s | -0.0000 | -0.6667 | -0.3333 | -0.3333 |
| d | -0.5000 | +0.3333 | -0.3333 | -0.3333 |

CKM matrix angles are **not** outputs of this light dictionary alone **[O / not claimed]**.

---

## D7 — Integrity board

| Claim | Tag |
|-------|-----|
| G2 ⊃ SU(3); six units = 3 ⊕ 3̄ (Günaydin–Gürsey) | [E] |
| Preferred invariant = Cayley–Dickson e4 | [E] |
| I3 = physical spatial rotation fixing e4 | [E] |
| Y = physical→extra Cartan partner | [E] |
| Explicit 3 eigenstates constructed in e1..e7 basis | [E process] |
| Light dictionary: complex combos → (u,d,s) ⊕ antiquarks | [E structure] |
| Single e_i = single quark flavour | [F] |
| Bijection e1..e7 → (u,d,s,c,b,t) | [F representation] |
| (c,b,t) inside light SU(3) 3 | [F] |
| LS (I3,Y) fit as derivation of labels | [F tautology] |
| paper_np G2 √3 as n–p structural rhyme (not quark labels) | [M] |
| Heavy generation packaging G-B/C/D as *quark warehouse* | [DEMOTED] → FL-HEAVY lobe/multiplet reframe |
| FL-HEAVY: heavy multiplet membership + scale | [O] · `flavour_heavy_lobe_reframe.md` |
| CKM angles from this e_i map alone | [O] / not claimed |
| Gap 'explicit G2 basis dictionary' status | LIGHT FREEZE [E]; six-quark 1–1 CLOSED F; heavy = multiplet residual not G₂ store |

---

## Verdict

1. **Explicit dictionary formulated and frozen for light flavours [E]:**
   e₄ fixed; P → 3 ⊕ 3̄ → (u,d,s) ⊕ (ū,d̄,s̄) with I₃ = physical Lz, Y = physical→extra partner.
2. **Gap inventory  e₁…e₇ → (u,d,s,c,b,t)  as basis-vector bijection is [CLOSED F]:**
   wrong count, wrong rep (3⊕3̄ not 6), invariant unit not a quark, generations not in one **3**.
3. **Heavy (c,b,t):** packaging options G-B/C/D **[M/O]**; naive e₅=c etc. **[F as derivation]**.
4. **Join to n–p paper:** G₂ √3 rhyme for Δm remains **[M structural]** and is **orthogonal** to quark-label assignment (different use of G₂).

> **Gap fill one-liner:** The G₂ basis dictionary is the preferred-basis **3 ⊕ 3̄** map to light quarks and antiquarks; it is **not** a seven-to-six list of eᵢ = quark flavours.

---

## Do not redo

- Least-squares (I₃,Y) as “derivation” of u,d,s labels (tautology)
- Claim e_k = single quark for each k
- Force (c,b,t) into the same light **3**
- Claim CKM from this dictionary alone without a generation mechanism
- Conflate paper_np G₂ √3 mass rhyme with flavour eigenstate labels

## Related bank

| Doc | Role |
|-----|------|
| `octonion_flavour_knowledge_arc.md` | Structure arc; Q1–Q12 |
| `paper_octonion_flavour.md` | Paper write-up |
| `paper_np_mass_splitting_g2.md` | n–p Δm; G₂ √3 rhyme **[M]** |
| `flavour_residuals_ledger.md` | FLAV-RES |
| `fl_o1_flavour_assignment_results.md` | Lattice→multiplet wall |
| `octonion_preferred_basis.py` | Preferred I₃ construction |

---

## Changelog

| Date | Change |
|------|--------|
| 2026-08-08 | Gap fill: explicit preferred G₂→SU(3) dictionary; close six-quark 1–1 |
| 2026-08-09 | D5 reframe: heavy = lobe/multiplet residual; link `flavour_heavy_lobe_reframe.md` |