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)

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` |

g2_e7_quark_basis_dictionary_results.md

G₂ e₁…e₇ Quark Basis Dictionary (Gap Fill)

Gap statement (answered)

D0 — Representation guards [E]

D1–D2 — Preferred construction [E]

D3 — Frozen light dictionary [E structure]

Not a basis-vector equality

Light quarks (3) — preferred Cartan eigenvalues

Antiquarks (3̄)

Working freeze (use this)

D4 — Six-quark 1–1 map [CLOSED F]

T1 — Is a single e_k a flavour eigenstate?

T2–T3 — Generations and cardinality

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

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

D7 — Integrity board

Verdict

Do not redo

Related bank

Changelog