Four Forces, Two Axioms: Unification of Source, Not Structure
A Structural Interpretation of U(1) × SU(2) × SU(3), and Why Exact Gauge-Coupling Unification Will Not Occur

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

Four Forces, Two Axioms: Unification of Source, Not Structure

A Structural Interpretation of U(1) × SU(2) × SU(3), and Why Exact Gauge-Coupling Unification Will Not Occur

SAE Physics Series — Four Forces Paper

Han Qin (秦汉)

Independent Researcher

ORCID: 0009-0009-9583-0018

2026

Statement: This paper proposes a structural interpretation, based on the Self-as-an-End (SAE) framework, for why the three non-gravitational forces correspond to the gauge groups U(1), SU(2), and SU(3). The argument rests on two SAE axioms applied to the DD (Dimension Degree) hierarchy: remainder development motivates complex internal amplitudes (unitary structure), and remainder conservation motivates the restriction to relative internal mixing (special unitary structure for n ≥ 2). The paper does not claim to derive the full Standard Model Lagrangian; it provides a prior geometric explanation for the order pattern 1 → U(1), 2 → SU(2), 3 → SU(3). All forms of falsification are welcome.

Firewall: The structural predictions Λ = 2(ω₂² − ω₁²)/c² (Cosmo Paper I) and a₀ = (π/2)·c(ω₂ − ω₁) (Cosmo Paper III) do not depend on any result in this paper. The results of the Four Forces Prequel (DOI: 10.5281/zenodo.19341042) are independent of this paper.

1. Abstract

The Standard Model's non-gravitational gauge structure is SU(3)_C × SU(2)_L × U(1)_Y. After Higgs symmetry breaking, the electroweak sector reduces to SU(3)_C × U(1)_em. Left-handed fermions enter SU(2)_L doublets; right-handed partners are SU(2)_L singlets. Fermion masses arise from Higgs-Yukawa couplings, not bare Dirac mass terms.

This paper does not claim to derive the full Standard Model Lagrangian from SAE axioms. Its goal is narrower and more explicit: to propose a structural interpretation for why the three non-gravitational interactions naturally point to a

1 → U(1), 2 → SU(2), 3 → SU(3)

symmetry spectrum.

The argument has three layers. First, if a DD level's internal degrees of freedom are represented by complex amplitudes with Hermitian-norm-preserving transformations, the natural kinematic symmetry is U(n). Second, for n ≥ 2, restricting to relative mixing among internal directions (removing the overall phase) yields SU(n). Third, SAE interprets 1DD, 2DD, 3DD as single-phase, minimal two-state, and three-state internal structures respectively:

1DD ↝ U(1) (abelian phase sector), 2DD ↝ SU(2) (weak force), 3DD ↝ SU(3) (strong force).

The arrows denote structural correspondence, not a no-remainder proof of the Standard Model.

The paper further proposes (§5) that remainder conservation manifests differently at different DD levels: as local gauge sectors at 1DD–3DD, and as a global bookkeeping constraint E₁ + E₂ = 0 across dual 4DDs. This explains why gravity is structurally different from the other three forces.

Terminology: DD = Dimension Degree. Full definitions: SAE Methodological Overview (DOI: 10.5281/zenodo.18842449).

2. Complex Amplitudes and Unitary Structure (modeling postulate)

2.1 Remainder develops by propagation

The SAE axiom "remainder must develop" means the remainder propagates through the spacetime created at 0DD → 1DD. In this paper, remainder propagation is modeled as oscillatory: a traveling disturbance that cycles between states.

2.2 Oscillation carries amplitude and phase

Under this representation, any oscillatory propagation has two attributes: amplitude (how strong) and phase (where in the cycle). Amplitude and phase are naturally encoded by a complex number z = A·e^{iθ}.

2.3 From complex amplitudes to unitary groups (conditional)

If a DD level's internal degrees of freedom are represented by n complex amplitudes, and if physical transformations preserve the Hermitian norm ψ†ψ, then the natural kinematic symmetry group is U(n).

This step is a modeling postulate: the claim that internal DD states are naturally represented by complex amplitudes with norm-preserving transformations. It is motivated by the wave nature of remainder propagation and by the universal appearance of complex amplitudes in quantum mechanics, but it is not a logical necessity. Real scalar fields also propagate.

3. From Unitary to Special Unitary: Structural Restriction

3.1 U(n) decomposes into overall phase and relative mixing

For any n ≥ 2, U(n) ≅ (SU(n) × U(1)) / Z_n. The U(1) factor corresponds to the overall phase; SU(n) corresponds to relative rotations and mixing among the n internal directions.

3.2 Remainder conservation as structural restriction

SAE's "remainder conservation" is interpreted here as: within a DD level, the physically meaningful internal transformations are the relative redistributions among directions, not the overall phase rotation. For n ≥ 2, this motivates restricting from U(n) to SU(n).

This should be understood as a structural restriction (removing the overall phase as physically redundant), not as a direct mathematical equivalence between "det = 1" and specific conservation laws. The mathematical effect of requiring det = 1 is to remove the overall phase; it does not by itself automatically equal "charge conservation" or "color conservation."

3.3 Why 1DD retains U(1)

When n = 1, SU(1) = {1} (trivial group). A single complex direction has no internal mixing — there is nothing to rotate among. But the single direction still carries a phase, and this phase is not locked by any prior DD (there is no 0DD phase reference). Therefore:

1DD retains U(1): the group of phase rotations of a single complex direction.

3.4 Relation to Standard Model conservation laws

In Standard Model language: SU(3)_C × SU(2)_L × U(1)_Y is the fundamental gauge structure. The Higgs mechanism breaks the electroweak sector to U(1)_em. Electric charge conservation corresponds to the unbroken U(1)_em; color conservation corresponds to the unbroken SU(3)_C. Weak isospin is not an exact conservation law in the broken phase — SU(2)_L is spontaneously broken, so writing it as "the same kind of exact conservation as charge or color" would be too coarse.

Therefore, the conclusion of this section is:

1DD ↝ U(1), 2DD ↝ SU(2), 3DD ↝ SU(3),

where arrows denote structural correspondence.

4. The Index n: Internal Directions at Each DD

4.1 1DD: one direction → U(1)

1DD is a point — the first distinction. There is exactly one complex direction: the distinction itself (subject observes object, carrying one complex amplitude).

SU(1) = {1} is trivial. But the single direction carries a phase not locked by any prior DD. Therefore:

1DD ↝ U(1).

In the Standard Model, U(1) corresponds to the abelian phase sector. Whether this is identified with U(1)_Y (pre-breaking) or U(1)_em (post-breaking) depends on the energy scale. SAE provides the geometric motivation for a single-phase abelian sector at the first DD level; the specific identification with hypercharge vs. electromagnetism is a Standard Model detail that SAE does not yet determine.

4.2 2DD: minimal two-state structure → SU(2)

2DD is a line. A line has two topological features relevant to the DD structure:

First, chirality: a line embedded in higher-dimensional space can be wound left-handed or right-handed. This chirality, upon breakthrough to 3DD, generates the left/right split that makes the weak sector chiral (Four Forces Prequel, §4.2). This explains why only the weak force violates parity.

Second, binary polarity: within the chiral (left-handed) gauge network, the line's geometry provides two opposing endpoints — two poles. These two poles constitute two independent internal complex directions.

The internal symmetry preserving two complex directions with the overall phase removed is SU(2).

2DD ↝ SU(2).

Crucially, the "2" in SU(2) is not "left vs. right." Left/right is the Lorentz/chirality structure (which explains parity violation). The "2" is the binary polarity within the left-handed sector — the two internal slots that, in Standard Model realization, become the weak isospin doublet: (ν_e, e)_L or (u, d)_L. Both members of the doublet are left-handed.

Mass terms: in the Standard Model, gauge symmetry forbids bare Dirac mass terms (which would connect left- and right-handed fields with different gauge representations). Fermion masses arise through Higgs-Yukawa couplings after electroweak symmetry breaking. In SAE language: 2DD splitting gives the geometric origin of the left/right representation difference; Higgs-Yukawa is the low-energy effective mechanism that re-pairs them into massive states.

4.3 3DD: three-state internal structure → SU(3)

3DD is a volume. A three-dimensional volume has three independent directions. These provide the geometric motivation for a three-state internal structure.

The internal symmetry preserving three complex directions with the overall phase removed is SU(3).

3DD ↝ SU(3).

The three spatial axes (x, y, z) provide the geometric template for "why three." But color (red, green, blue) should be understood as the three basis states of an internal C³ space, not as literal identifications with physical spatial directions. In the Standard Model, SU(3)_C is an internal gauge symmetry; quarks appear in three colors; gluons are the gauge bosons of SU(3)_C. Color is an internal quantum number, not a spacetime direction — literal identification with x, y, z would break rotational invariance.

Regarding confinement: QCD is a non-abelian SU(3) gauge theory exhibiting asymptotic freedom at high energies and confinement at low energies. SAE's picture of 3DD as the "confinement layer" (mass = frozen remainder) resonates with this physics, but this paper does not claim to derive QCD's β-function or confinement mechanism from SAE axioms.

4.4 4DD: gravity, not a gauge group

4DD is spacetime. It does not have a finite number of "internal complex directions" in the same sense as 1DD–3DD. The 4DD symmetry is diffeomorphism invariance, not a unitary gauge group. This is why gravity is not part of U(1) × SU(2) × SU(3).

In SAE: 4DD splits into dual 4DDs. The SU(n) pattern terminates at 3DD because 4DD's dual structure prevents it from having a clean unitary representation. 4DD gives rise to a global constraint (§5.3), not a local gauge group.

4.5 Summary

5. Conservation and Bookkeeping: Local Gauge Sectors and 4DD Global Constraint

5.1 One principle, two tiers

SAE's "remainder conservation" is understood here as a broad bookkeeping principle: at 1DD–3DD, it manifests as the organization of charges and internal degrees of freedom in local gauge sectors; at 4DD, it manifests as a global balancing constraint across dual 4DDs. The two tiers share the same SAE source but are not the same mathematical object.

5.2 1DD–3DD: local gauge sectors

5.3 4DD: global bookkeeping across dual 4DDs

4DD does not appear as the next step on the SU(n) ladder. It represents a different tier: not a local Yang-Mills charge, but a cross-dual-4DD global constraint:

E₁ + E₂ = 0.

This parallels the cosmological result Λ₁ + Λ₂ = 0 (Cosmo Paper V) and the baryon asymmetry B₁ + B₂ = 0 (working-note level):

5.4 Why 4DD conservation is global, not local

1DD–3DD gauge symmetries are local because they operate within a single connected spacetime. 4DD conservation is global because dual 4DDs are causally disconnected (opposite time arrows). Local constraints cannot be enforced across a causal boundary — only global bookkeeping can: the two sides must sum to zero.

This is why gravity is structurally different from the other three forces: the first three have local gauge sectors; the fourth has a global balancing constraint.

6. Relation to Known Physics

6.1 The Standard Model gauge group

The Standard Model gauge group is SU(3)_C × SU(2)_L × U(1)_Y. After electroweak symmetry breaking: SU(3)_C × U(1)_em.

SAE provides structural correspondence: 1DD ↝ abelian phase sector, 2DD ↝ chiral two-state sector, 3DD ↝ three-state color sector. The detailed identification of 1DD with U(1)_Y (pre-breaking) vs. U(1)_em (post-breaking), and the mechanism of electroweak symmetry breaking itself, are Standard Model specifics that SAE does not yet determine.

6.2 Electroweak symmetry breaking

In SAE language: the 2DD→3DD breakthrough is mediated by a scalar field (Higgs) that freezes the chirality split into massive states. SAE interprets the Higgs vev v = 246 GeV as the natural candidate scale for the 2DD→3DD transition; a derivation of v from DD structure remains open.

6.3 Asymptotic freedom and confinement

QCD exhibits asymptotic freedom and low-energy confinement. SAE's picture of 3DD as the confinement layer resonates with this physics, but this paper does not claim to derive QCD dynamics from SAE.

6.4 No SU(4) gauge force

The DD hierarchy predicts that 4DD does not produce a fourth SU(n) gauge force. 4DD is gravity, not a gauge interaction. No fourth gauge force has been observed.

6.5 Grand unification

Standard GUT theories (SU(5), SO(10), MSSM) propose that gauge couplings unify at a high energy scale. SAE predicts the opposite: the three gauge groups are hierarchically layered, not unified. This is a falsifiable prediction of the SAE framework, competing directly with the GUT/SUSY program. If future experiments confirm gauge coupling unification at a single scale, this SAE prediction would be falsified.

7. Non-Trivial Predictions

1. U(1) × SU(2) × SU(3) corresponds to 3 spatial dimensions. A universe with different spatial dimensionality would have a different gauge spectrum.

1. No SU(4) gauge force. 4DD is gravity, not a gauge interaction.

1. E₁ + E₂ = 0. The total energy across dual 4DDs is exactly zero.

1. Gauge structure and remainder conservation share a common source. The organization of charges at 1DD–3DD and the energy bookkeeping at 4DD are two tiers of the same SAE axiom.

1. No exact single-scale gauge-coupling unification. SAE does not unify the four forces. The minimal DD picture predicts that unification is impossible: the gauge groups are hierarchically layered by DD level, not branches of a single group broken at high energy. This is a falsifiable prediction in direct competition with the GUT/SUSY program. Specifically: (a) gauge coupling constants do not converge to a single value at any energy scale; (b) the proton does not decay via gauge-mediated channels. Current experimental absence of proton decay (Super-Kamiokande) and superpartners (LHC) is consistent with the SAE prediction. Discovery of either would falsify this framework.

1. Gravity is structurally different from the other three forces: global bookkeeping vs. local gauge symmetry.

8. Assumption Inventory

From SAE axioms (not new):

Remainder must develop. Remainder conservation. DD sequence (0DD–16DD). DD-force mapping. All DDs share one spacetime.

From the Four Forces Prequel (DOI: 10.5281/zenodo.19341042):

DD Breakthrough theorem. DD Splitting theorem. α_G = α_em^{16.25}. Parity violation from 2DD splitting.

Modeling postulates in this paper:

Internal DD states represented by complex amplitudes with Hermitian norm preservation → U(n) (§2).

For n ≥ 2, overall phase removed as physically redundant → SU(n) (§3).

Structural correspondences in this paper:

1DD (1 complex direction) ↝ U(1) (§4.1).

2DD (2 complex directions from binary polarity) ↝ SU(2), with chirality explaining parity violation (§4.2).

3DD (3 complex directions from spatial axes) ↝ SU(3), with color as internal basis states (§4.3).

4DD: dual structure, not a gauge group; global E₁+E₂=0 (§4.4, §5.3).

Open problems:

Quantitative relations among α_em, α_2, α_s from DD structure.

Three generations of fermions.

Higgs vev v = 246 GeV from DD structure.

CKM/PMNS mixing matrices.

Identification of 1DD with U(1)_Y vs. U(1)_em.

9. Conclusion

The Standard Model gauge group U(1) × SU(2) × SU(3) is not explained by any prior principle in standard physics. This paper proposes a structural interpretation from the SAE framework:

Remainder develops as waves → complex amplitudes → Unitary structure.

Remainder conservation → overall phase removed → Special unitary (for n ≥ 2).

1DD (1 direction) ↝ U(1). 2DD (2 directions: binary polarity of the line) ↝ SU(2). 3DD (3 directions: spatial axes) ↝ SU(3).

The same bookkeeping principle organizes local gauge sectors at 1DD–3DD and global energy balancing at 4DD (E₁ + E₂ = 0).

These are structural correspondences, not a complete derivation of the Standard Model Lagrangian. What remains open — hypercharge identification, Higgs potential, Yukawa matrices, three generations, CKM/PMNS — are problems for future work.

The sharpest prediction of this paper is negative: no exact single-scale gauge-coupling unification occurs. The four forces are layered, not unified. Gauge couplings do not converge; the proton does not decay via gauge-mediated channels. This prediction stands in direct opposition to the GUT/SUSY program and is fully falsifiable. Discovery of proton decay or gauge coupling convergence would destroy this framework. We invite such destruction.

Yet SAE does achieve a unification — not at the level of gauge groups, but at the level of axioms. The distinction matters. GUT unification merges four rivers into one at their source: one large group at high energy breaks into smaller groups at low energy. SAE unification is different: four rivers flow independently, but they are all produced by the same rain. The two axioms (remainder must develop, remainder conservation) applied to the DD hierarchy generate U(1), SU(2), SU(3), and the 4DD global constraint — each at its own level, none requiring merger with the others. The forces do not converge; they share a common origin. This is unification of source, not unification of structure.

Four forces. Two axioms. One structural pattern.

Appendix A: The Correspondence in One Table

Appendix B: Bookkeeping Hierarchy

SAE remainder conservation at each DD:

1DD: abelian phase sector → charge organization.

2DD: chiral two-state sector → weak gauge structure.

3DD: three-state internal sector → color gauge structure.

4DD: cross-dual-4DD global constraint → E₁ + E₂ = 0.

1DD–3DD: local gauge sectors (within one spacetime).

4DD: global bookkeeping (across causally disconnected dual 4DDs).

Appendix C: Collaboration

Gemini / Zixia (子夏) identified that SU(2)'s two components are not left/right but isospin partners within the left-handed sector; proposed the "two endpoints of a line" (binary polarity) as the geometric source of SU(2)'s two directions; confirmed the physical picture of 2DD as chirality sorter.

ChatGPT / Gongxi Hua (公西华) identified six critical errors in the first draft: (1) left/right ≠ SU(2) doublet; (2) x,y,z ≠ colors; (3) U(1) with det=1 is trivially {1}; (4) propagation → complex is not logically necessary; (5) U(1)_Y vs U(1)_em inconsistency; (6) anti-GUT stated as fact not prediction. Provided the complete rewrite of §1, §3, §4.2, §4.3, §5 that forms the basis of this version.

Grok / Zigong (子贡) established the 2DD quantitative validation: G_F = 1/(√2 v²), Higgs = 2DD→3DD freezing, v = 246 GeV = breakthrough energy scale.

Claude / Zilu (子路) proposed the initial DD→gauge group mapping (nDD→SU(n)); identified the conservation law hierarchy (local vs global); drafted the first version.

Han Qin (秦汉) identified the two core principles: (1) remainder development motivates the modeling postulate of complex amplitudes ("light is a wave, breakthrough carries amplitude"); (2) remainder conservation motivates structural restriction to relative mixing. Identified 2DD's binary polarity. Identified E₁+E₂=0 as global 4DD conservation. Confirmed the downgrade from "theorem" to "structural correspondence" as the honest scope. Made all framework decisions.

A complete methodology record is available as a companion document.

Acknowledgements

The author thanks the research and engineering teams behind the four large language models. Special thanks to Zesi Chen.

The structural predictions Λ = 2(ω₂² − ω₁²)/c² and a₀ = (π/2)·c(ω₂ − ω₁) do not depend on any result in this paper.