6-Block Pair Algebra, 𝔰𝔬(6) ≅ 𝔰𝔲(4), and Lepton as a Color-Singlet Volume Line

Writing Declaration: This paper was independently authored by Han Qin. All intellectual decisions, framework design, and editorial judgments were made by the author.

Abstract

Several structural facts of the Standard Model — why SU(3)_c has exactly 3 colors, why the lepton is a color singlet, and why quarks and leptons always appear in matched sets — lack a unified dynamical explanation. This paper starts from a minimal input: 6 pair-oriented fundamental objects ("blocks," identified as 4DD blocks in the SAE framework) and derives the following results through standard representation theory.

(i) The pair algebra of 6 blocks is 𝔰𝔬(6) ≅ 𝔰𝔲(4), dimension 15. (ii) The pair orientation induces a complex structure J, yielding V₆ ≅ ℂ³. (iii) J selects 𝔲(3) = 𝔰𝔲(3) ⊕ 𝔲(1) ⊂ 𝔰𝔬(6), producing the adjoint decomposition 15 = 8 + 1 + 3 + 3̄. (iv) The 4-dimensional half-spinor representation of 𝔰𝔲(4), under the selected SU(3)-structure, naturally decomposes as 4 = 3 ⊕ 1, where 1 is carried by the holomorphic volume line L = Λ³U, with generator Ω = u₁ ∧ u₂ ∧ u₃ — the complete antisymmetrization of the three color directions. The lepton is not a "fourth color" entity but a color-singlet volume line. (v) The 6-dimensional vector representation satisfies 6 = Λ²(4): the 6 blocks are pairs of the 4 spinor components, with explicit map u_a ↔ s_a ∧ s₄, ū_a ↔ ½ε_abc s_b ∧ s_c.

The entire derivation uses only two inputs: the existence of 6 blocks and their pair orientation. The result naturally produces an 𝔰𝔲(4) ⊃ 𝔰𝔲(3) ⊕ 𝔲(1) color-lepton unification skeleton. At the mathematical level, the existence of the singlet volume line is a representation-theoretic theorem. At the physical level, identifying this singlet line with the lepton is an SAE-internal physical identification, compatible with the remote readout picture in the SAE framework.

Keywords: Pati-Salam, 𝔰𝔬(6), 𝔰𝔲(4), color-singlet, lepton, spinor representation, complex structure, Self-as-an-End

§1. Motivation and Problem

§1.1 Unsolved structural questions in the Standard Model

The Standard Model gauge group SU(3)_c × SU(2)_L × U(1)_Y is experimentally established, but the following structural questions remain unanswered within the SM framework:

(a) Why does SU(3)_c have 3 colors? The number 3 is a manual input, not a dynamical output.

(b) Why does one generation of fermions have exactly 16 Weyl components? Including ν_R: Q_L(6) + u_R(3) + d_R(3) + L_L(2) + e_R(1) + ν_R(1) = 16. The binary structure 16 = 2⁴ has no explanation. (This paper provides one half of the skeleton via 𝔰𝔲(4); the complete 16-state interpretation requires incorporating the R-side structure; see §7.3.)

(c) Why is the lepton a color singlet? In the SM this is a manually assigned quantum number.

(d) Why do quarks and leptons always appear in matched sets? Anomaly cancellation requires quark-lepton pairing within each generation, but the SM does not explain the origin of this pairing.

GUT approaches (SU(5), SO(10), Pati-Salam) address (b)–(d), but at the cost of introducing a unification scale and predictions such as proton decay that have not been observed.

§1.2 Approach of this paper

This paper does not assume any GUT. The starting point consists of exactly two inputs:

Input 1. There exist 6 fundamental objects ("blocks"), organized into 3 pairs, each pair carrying an orientation ("+" and "−" distinction).

Input 2. These 6 blocks reside on the L-side of a single space. Another 6 reside on the R-side (not treated in this paper; the R-side mirror structure does not affect the L-side derivation).

Within the SAE (Self-as-an-End) framework, these inputs have definite physical origins: the 6 L-side 4DD blocks are determined by spacetime dimension d = 4 via S³ packing (d − 1 = 3 orthogonal axis pairs, 2 blocks per pair). The pair orientation arises from frequency asymmetry (a consequence of the "remainder must develop" axiom). However, the mathematical derivation in this paper does not depend on these physical details — it requires only "6 pair-oriented objects."

§1.3 Preview of results

From these two inputs, purely through standard representation theory, we obtain:

• 𝔰𝔬(6) ≅ 𝔰𝔲(4) (color-lepton unification skeleton)
• SU(3)_c and its 8-dimensional adjoint representation
• The lepton as a color-singlet volume line
• The algebraic necessity of quark-lepton pairing
• 15 = 8 + 1 + 3 + 3̄ (adjoint decomposition)

§1.4 Proposition status table

Proposition	Status	Depends on
Λ²V6 ≅ 𝔰𝔬(V6, g)	Theorem	V6 + inner product g
(V6, J) ≅ ℂ3	Theorem	Pair orientation + J² = −id
U(3) ⊂ SO(6)	Theorem	g + orthogonal J
15 = 8 ⊕ 1 ⊕ 3 ⊕ 3̄	Theorem	𝔰𝔲(3) ⊂ 𝔰𝔲(4)
S = 4 ≅ 3 ⊕ 1	Theorem	Chosen half-spinor + Ω
6 = Λ2(4)	Theorem	Standard isomorphism
1 = Λ3U = color-singlet volume line	Theorem	All of the above
Lepton = singlet volume line	SAE identification	Spinor dictionary + remote readout
Physical necessity of quark-lepton pairing	SAE consequence	Above identification + irreducibility of 4

§2. From 6 Blocks to ℂ³

§2.1 The vector space V₆

Let the 6 blocks form the basis of a 6-dimensional real vector space:

V₆ = span_ℝ{e₁⁺, e₁⁻, e₂⁺, e₂⁻, e₃⁺, e₃⁻}

where a = 1, 2, 3 labels the 3 axes and ± labels the orientation within each pair. Equip V₆ with a positive-definite inner product g making {e_a^±} an orthonormal basis.

§2.2 Complex structure

The pair orientation defines a linear map J: V₆ → V₆:

J(e_a⁺) = e_a⁻,    J(e_a⁻) = −e_a⁺

One verifies directly that J² = −id. This is a complex structure on V₆. (On a vector space, J² = −id already constitutes a complex structure; the integrability question for almost complex structures arises only on manifolds.)

Physical origin: The "+" and "−" distinction endows each pair with an orientation. J implements a π/2 rotation within each dual plane; two applications yield the π sign reversal (J² = −id).

§2.3 Complexification

J turns V₆ into a 3-dimensional complex vector space. Define the holomorphic basis:

u_a = (1/√2)(e_a⁺ − i e_a⁻),   a = 1, 2, 3

Then (V₆, J) ≅ U := span_ℂ{u₁, u₂, u₃} ≅ ℂ³. The complexification yields V₆^ℂ = U ⊕ Ū, decomposing under U(3) as 6_ℂ = 3 ⊕ 3̄.

§3. Pair Algebra = 𝔰𝔬(6) ≅ 𝔰𝔲(4)

§3.1 Bivector space

The second exterior power of V₆:

dim Λ²V₆ = C(6,2) = 15

Via the inner product g, Λ²V₆ is naturally isomorphic to 𝔰𝔬(V₆, g) (the Lie algebra of g-antisymmetric linear transformations on V₆). Explicitly, each bivector e_i ∧ e_j corresponds to the antisymmetric map v ↦ g(e_j, v)e_i − g(e_i, v)e_j.

§3.2 Lie algebra isomorphism

𝔰𝔬(6) ≅ 𝔰𝔲(4) is a standard result (dimension 15, coincidence of Dynkin diagrams D₃ ≅ A₃). The pair space of the 6 blocks is not merely a combinatorial object — it is a 15-dimensional Lie algebra.

§3.3 The 𝔲(3) subalgebra

The complex structure J preserves the inner product g (i.e., g(Jv, Jw) = g(v, w)), making J an orthogonal complex structure. Within SO(V₆, g), J selects the subgroup preserving J: U(3) ⊂ SO(6). At the Lie algebra level:

𝔲(3) = 𝔰𝔲(3) ⊕ 𝔲(1) ⊂ 𝔰𝔬(6)

• 𝔰𝔲(3) (8-dimensional): the traceless part — the Lie algebra of the color group SU(3)_c.
• 𝔲(1) (1-dimensional): the overall phase — distinguishes color-charged states from the color singlet.

§3.4 Adjoint decomposition

The adjoint representation of 𝔰𝔬(6) decomposes under 𝔰𝔲(3) as:

15 = 8 ⊕ 1 ⊕ 3 ⊕ 3̄

• 8: 𝔰𝔲(3) itself = 8 gluon directions
• 1: 𝔲(1) direction = the color-singlet discriminator
• 3 ⊕ 3̄: the coset 𝔰𝔲(4)/𝔲(3) = directions connecting color triplets to the color singlet

§4. Spinor Representation and the Lepton

§4.1 The spinor representation 4

A key consequence of 𝔰𝔬(6) ≅ 𝔰𝔲(4): besides the 6-dimensional vector representation 6, 𝔰𝔬(6) possesses a 4-dimensional half-spinor representation.

Define the 1-dimensional volume line L := Λ³U, and take as generator Ω = u₁ ∧ u₂ ∧ u₃ ∈ L, so that L = ℂΩ. The half-spinor representation is:

S := U ⊕ L

S is a 4-dimensional complex vector space. Choose the basis: s₁ = u₁, s₂ = u₂, s₃ = u₃ (in U), s₄ = Ω (in L).

The vector and spinor representations are related by:

6 = Λ²(4)

That is: V₆^ℂ ≅ Λ²S. The 6 blocks are pairs of the 4 spinor components.

§4.2 𝔰𝔲(3) decomposition of the spinor

The 𝔰𝔲(3) ⊂ 𝔰𝔲(4) selected by J decomposes S as:

S = U ⊕ L ≅ 3 ⊕ 1

The first three components (s₁, s₂, s₃) ∈ U transform as the fundamental representation 3 of 𝔰𝔲(3). The fourth component s₄ ∈ L is an 𝔰𝔲(3)-singlet.

§4.3 The lepton as a color-singlet volume line

s₄ = Ω ∈ L = Λ³U. This is the holomorphic volume form of ℂ³ — the complete antisymmetrization of the three color directions. Under 𝔰𝔲(3), Λ³ℂ³ is a 1-dimensional singlet.

Mathematical theorem: The half-spinor representation of 𝔰𝔲(4) decomposes under 𝔰𝔲(3) as 4 = 3 ⊕ 1, where 1 is carried by the holomorphic volume line L = Λ³U.

SAE identification: Within the SAE framework, this singlet volume line is naturally identified with the physical role of the lepton — the remote readout / post-color-collapse singlet. Λ³U is the mathematical operation of "completely rolling the three color directions into a singlet," compatible with the SAE picture of "color collapse."

§4.4 Explicit map

Λ²S = Λ²U ⊕ (U ⊗ L). Using contraction with Ω and the Hermitian structure induced by g and J, one has the standard isomorphisms Λ²U ≅ U* ≅ Ū, giving Λ²S ≅ Ū ⊕ U ≅ V₆^ℂ.

Explicitly, the 6 basis bivectors of Λ²S (the ∧ here is in Λ²S, not in Λ^•U):

u_a ↔ s_a ∧ s₄   (in U ⊗ L ≅ 3)

ū_a ↔ ½ε_abc s_b ∧ s_c   (in Λ²U ≅ 3̄)

Translating back to the real basis:

e_a⁺ = (1/√2)(u_a + ū_a) ↔ (1/√2)(s_a ∧ s₄ + ½ε_abc s_b ∧ s_c)

e_a⁻ = (1/i√2)(u_a − ū_a) ↔ (1/i√2)(s_a ∧ s₄ − ½ε_abc s_b ∧ s_c)

Each block is a linear combination of a color-singlet pairing (s_a ∧ s₄) and a pure color-color pairing (ε_abc s_b ∧ s_c). Blocks within the same dual pair share the same axis index a, differing only in the relative phase between holomorphic and antiholomorphic components.

§4.5 Decomposition of Λ²(4)

Λ²(3 ⊕ 1) = Λ²3 ⊕ (3 ⊗ 1) = 3̄ ⊕ 3

• 3: pairing of a color direction with the singlet line
• 3̄: pairing between two color directions

Together these reconstruct the 3 ⊕ 3̄ decomposition of V₆^ℂ.

§5. Unification of the Two "3"s

Two structurally distinct "3"s appear in this construction:

3a: the number of pairs. The 6 blocks form 3 pairs because they reside on S³ (the direction sphere of 4-dimensional space, dimension d − 1 = 3), which accommodates at most 3 mutually orthogonal anti-aligned pairs.

3b: the complex dimension of ℂ³, i.e., the dimension of the defining representation of SU(3).

These two 3s are unified through the complex structure J: ℝ^2(d−1) →^J ℂ^d−1. For d = 4: ℝ⁶ → ℂ³. 3a determines that V₆ has 3 pairs; J converts each pair into one complex direction, yielding the complex dimension 3b. The stabilizer of J within SO(6) is U(3), whose traceless part SU(3) is the color group.

Therefore, the "3" of color SU(3)_c ultimately originates from the spacetime dimension d = 4.

§6. Necessity of Quark-Lepton Pairing

§6.1 Why quarks and leptons come in matched sets

In this structure, 4 = 3 ⊕ 1 is an irreducible representation of 𝔰𝔲(4). The 3 (quark) and 1 (lepton) cannot be separated — they reside in the same irreducible representation.

This provides an algebraic explanation for quark-lepton pairing: it is not a dynamical coincidence but a representation-theoretic necessity of 𝔰𝔬(6) ≅ 𝔰𝔲(4).

§6.2 The 3 ⊕ 3̄ directions

The 3 ⊕ 3̄ in the adjoint decomposition represents directions in 𝔰𝔲(4) connecting quarks and leptons. In the Pati-Salam model (under a gauged SU(4) interpretation), these directions correspond to leptoquark gauge bosons. In SAE, these directions correspond to block-pair components that "cross the color-singlet boundary." Their existence is algebraically automatic, requiring no manual introduction.

§7. Discussion

§7.1 Relation to and distinction from Pati-Salam

The algebraic structure obtained here coincides with the color part of the Pati-Salam gauge group SU(4)_c × SU(2)_L × SU(2)_R. The key distinction:

• Pati-Salam works top-down: SU(4)_c is postulated as a gauge symmetry, then decomposed.
• This paper works bottom-up: starting from 6 oriented blocks, 𝔰𝔲(4) emerges automatically.

This paper does not assume that SU(4)_c is a gauge symmetry, nor does it assume any unification scale. The 𝔰𝔲(4) structure is a mathematical necessity of the 6-block pair algebra, not a dynamical postulate.

§7.2 Non-canonical choices

The map from 6 real blocks to 4 spinor components requires two choices:

(a) The complex structure J: derived from pair orientation. In SAE this has a physical origin (frequency asymmetry), and is not a free parameter.

(b) The holomorphic volume form Ω: choosing u₁ ∧ u₂ ∧ u₃ over −u₁ ∧ u₂ ∧ u₃. This is an orientation choice corresponding to the phase of s₄. The second choice is standard in spin geometry: the stabilizer of a nonzero spinor is SU(3), and the choice of spinor defines the SU(3) structure.

§7.3 Outlook toward Spin(10)

Incorporating the R-side structure, Spin(4) ≅ SU(2)_L × SU(2)_R may emerge from the 4DD spacetime structure. The complete Pati-Salam skeleton would be 𝔰𝔲(4) ⊕ 𝔰𝔲(2)_L ⊕ 𝔰𝔲(2)_R. Further, Spin(6) × Spin(4) ⊂ Spin(10). The standard branching is:

16 = (4, 2, 1) ⊕ (4̄, 1, 2)

where the three factors correspond to SU(4) × SU(2)_L × SU(2)_R. The first summand gives 8 left-handed states; the second gives 8 right-handed states; 8 + 8 = 16. This direction requires further independent work.

§7.4 Relation to Paper I: partial unification of 15

Paper I of the Four Forces series established the structural correspondence nDD ⟝ SU(n): 3DD yields SU(3). This paper arrives at a compatible result from a different angle: the pair algebra of 6 blocks yields 𝔰𝔲(4) ⊃ 𝔰𝔲(3). The two routes are complementary and converge on the same 𝔰𝔲(3).

With this, the number 15 achieves a partial unification across the SAE series:

• Generation Paper: 15 = C(6,2) (pair count of 6 single-side blocks)
• This paper: 15 = dim 𝔰𝔬(6) (Lie algebra dimension of the pair algebra)

This paper provides the canonical map from pair counting to Lie algebra (§3.1), thereby unifying combinatorics and representation theory.

§7.5 Relation to the Generation Paper

This paper provides a clearer representation-theoretic background for the "single-side 15" in the Generation Paper (DOI: 10.5281/zenodo.19394500) and the geometric identity of the color-singlet line: 15 = dim 𝔰𝔬(6) = C(6,2). However, this paper does not independently derive the C₃ quotient (15 → 5) or the doublet/lepton mass ratio 13 from the Generation Paper's G10. Those results require additional physical input (the color-collapse mechanism).

§7.6 What this paper does not explain

This paper does not explain: why d = 4; the values of gauge coupling constants; the fermion mass spectrum; or the specific quantization of U(1)_Y hypercharge.

§8. Summary

Starting from 6 pair-oriented fundamental objects, through standard representation theory:

1. Pair algebra = 𝔰𝔬(6) ≅ 𝔰𝔲(4), dimension 15.
2. Pair orientation induces complex structure J, yielding V₆ ≅ ℂ³.
3. J selects 𝔰𝔲(3) ⊕ 𝔲(1) ⊂ 𝔰𝔬(6). Adjoint decomposition: 15 = 8 + 1 + 3 + 3̄.
4. 𝔰𝔲(4) spinor: 4 = 3 ⊕ 1. The fourth component = holomorphic volume form = color-singlet volume line = lepton.
5. 6 = Λ²(4). The 6 blocks are pairs of the 4 spinor components. Explicit map provided.
6. The "3" of color SU(3)_c originates from d − 1 = 3 (the spatial part of spacetime dimension 4).
7. Quark-lepton pairing is a necessary consequence of the irreducibility of the 𝔰𝔲(4) representation.

The entire derivation requires only two inputs: 6 blocks and pair orientation. No GUT is assumed, no unification scale, no proton decay.

Acknowledgments

Four-AI collaboration: ChatGPT/Gongxihua (proposal of the 𝔰𝔬(6) route, explicit construction of the spinor-block map), Claude/Zilu (derivation chain, paper writing), Gemini/Zixia, Grok/Zigong.

Appendix A: Physical Origin of the Inputs

6 blocks: 4DD space has d − 1 = 3 spatial dimensions. S^d−1 = S³ accommodates at most 3 mutually orthogonal anti-aligned direction pairs. Each direction hosts one dual pair of 2 blocks, giving 2 × 3 = 6 blocks per side. Including L/R sides, the total is 12 blocks.

Pair orientation: The SAE axiom "remainder must develop" guarantees that the two blocks in each dual pair have unequal frequencies. The higher-frequency block is labeled "+", the lower "−". This orientation is axiom-enforced, not a parameter choice.