Four Forces Paper VI: Why GUT-Predicted Proton Decay Remains Absent — Spin(10) as Classification Algebra and the GUT Prediction Blind Spot
Writing Declaration: This paper was independently authored by Han Qin. All intellectual decisions, framework design, and editorial judgments were made by the author.
Han Qin (ORCID: 0009-0009-9583-0018)
§1 Introduction: Half a Century of Silence
In 1974, Georgi and Glashow proposed the minimal SU(5) grand unified model, predicting a proton lifetime of approximately 10³⁰–10³¹ years. In the same year, Pati and Salam introduced the semi-simple unification scheme SU(4)×SU(2)_L×SU(2)_R. Over the following half century, Grand Unified Theories (GUTs) became a central paradigm in high-energy physics. Their core reasoning is: since all Standard Model fermions can be embedded in irreducible representations of a simple Lie group (such as SU(5) or SO(10)), that group should be promoted to a gauge symmetry.
This reasoning generates a series of distinctive predictions: proton decay, magnetic monopoles, gauge coupling unification at a single energy scale, and numerous gauge bosons beyond the Standard Model. However, to date, experimental results have not provided positive support:
- Super-Kamiokande, monitoring 50,000 tons of pure water for over two decades, has pushed the partial lifetime lower bound to τ(p → e⁺π⁰) > 2.4 × 10³⁴ years; the minimal SU(5) model has long been excluded.
- Since 1982, large-scale magnetic monopole searches have yielded no signal.
- Under the bare Standard Model (without SUSY), the three running coupling constants do not meet at a single point.
- The LHC at 13 TeV center-of-mass energy has found no gauge bosons beyond the Standard Model.
To improve compatibility between GUTs and experimental data, supersymmetry (SUSY) has been introduced as an extended framework: SUSY can achieve coupling unification, stabilize the hierarchy, and extend proton lifetime predictions. However, SUSY's own predictions — superpartners, additional Higgs particles — have likewise gone entirely unobserved.
Could it be that the proton simply does not undergo GUT-mediated decay? Could the X/Y bosons responsible for proton decay be altogether absent from the fundamental structure of the universe? This paper attempts a structural answer within the SAE framework.
Skepticism toward GUTs is not new. The Anti-GUT program of Nielsen and Laperashvili replaces a single large group with multiple copies of the Standard Model group, but remains within the framework of gauge unification. The present paper takes a step further back, exploring a more fundamental possibility: that the mechanism of force generation may not be determined by the adjoint representation of a unification group at all.
§2 From DD to Spin(10)
2.1 Prior Results
This section builds on the core theorems of Four Forces Paper IV. The 12 4DD blocks divide into L and R sides, 6 each. Each side's 6 blocks form 3 pair axes. From 6 pair-oriented blocks:
- Λ²V₆ ≅ so(6) ≅ su(4), dimension 15 = C(6,2)
- Complex structure J yields V₆ ≅ C³, giving su(3) ⊕ u(1)
- Spinor 4 = 3 ⊕ 1, where 1 = Λ³U = color-singlet volume line
- Within this framework, the lepton is not a fourth color but the complete antisymmetrization of three color directions
2.2 Four Z₂ Factors from DD
Each 4DD pair axis possesses a non-degenerate frequency ratio asymmetry: T₁ ≠ T₂ (established as a theorem in the Four Forces Prequel). The choice T₁ > T₂ or T₁ < T₂ endows each pair axis with a canonical orientation. Three pair axes independently make this choice, providing three Z₂ factors ε₁, ε₂, ε₃.
The fourth Z₂ factor ε₄ comes from L/R chirality — the fundamental binary split of the 3DD structure.
The four Z₂ factors together yield Z₂⁴ with 16 states.
2.3 Two Weyl Spinors of Spin(6)
Upon identifying the three pair axes as the Cartan basis of Spin(6) — a natural choice within the DD structure, as the pair axes provide three independent, canonical binary directions — the 8 sign states of ε₁ε₂ε₃ split by parity into two groups:
Even number of minus signs (4 states) → 4:
- (+,+,+): all axes positively oriented = positive volume form = Λ³U = lepton
- (+,−,−), (−,+,−), (−,−,+): two axes flipped each = three color quarks
Odd number of minus signs (4 states) → 4̄:
- (−,−,−): all axes negatively oriented = anti-lepton
- (−,+,+), (+,−,+), (+,+,−): one axis flipped each = three anti-color quarks
These are precisely the two Weyl spinors of Spin(6) ≅ SU(4), consistent with Paper IV's spinor decomposition 4 = 3 ⊕ 1. In the even sector, (+,+,+) corresponds to Λ³U (full alignment = positive orientation of the volume form), while the remaining three states correspond to three colors — a mapping that holds upon fixing the holomorphic orientation of U.
2.4 Weyl Condition and One-Generation Fermions
ε₄ = L/R chirality corresponds to Spin(4) ≅ SU(2)_L × SU(2)_R. In standard Spin(10) representation theory, the branching rule under Spin(6) × Spin(4) is:
16 = (4, 2, 1) ⊕ (4̄, 1, 2)
The Weyl condition is equivalent to the factorization of the 10-dimensional chirality operator: Γ₁₀ = Γ₆ · Γ₄. In DD language, this locks the parity of ε₁ε₂ε₃ to ε₄:
- Even(ε₁ε₂ε₃) × L → (4, 2, 1): left-handed color-lepton quartet
- Odd(ε₁ε₂ε₃) × R → (4̄, 1, 2): right-handed anti-color-lepton quartet
Principal Bridge Assumption: Canonical pairing L-a ↔ R-a. Each L-side block maps uniquely to its R-side counterpart, with chirality preserving structure. This assumption is falsifiable — if L/R pairing is not unique, the entire (4,2,1) ⊕ (4̄,1,2) DD dictionary fails.
Note: In SAE's mathematical foundation (ZFCρ recursion), Ω = 3 lies in the disordered phase (order parameter h > 0), where spontaneous symmetry breaking is not permitted. If the correspondence between DD levels and ZFCρ's Ω parameter holds, strict symmetry between the two 3DD sides is a structural consequence of the disordered phase, and this assumption can be upgraded to a corollary. The rigorous establishment of this correspondence is an independent open problem.
Under this assumption, the 16 states exactly cover all Weyl degrees of freedom of one fermion generation. In particular, the 16th state — the right-handed neutrino ν_R — is not an additional input but a natural output of the Z₂⁴ structure. This cross-validates the independent prediction of ν_R (y = 0) from the hypercharge table in Paper II.
2.5 Note on Particle/Antiparticle Labels
In the DD construction, the distinction between 4 and 4̄ depends on the observer's time-arrow choice: from block a, (+,+,+) = lepton and (−,−,−) = anti-lepton; from block b (with reversed time arrow), the labels swap. This provides an observer-dependent reinterpretation of particle/antiparticle labels, but constitutes an interpretive layer and cannot replace the CPT derivation given by the Jost theorem in standard quantum field theory. A fuller discussion is deferred to future work.
§3 Classification Algebra vs. Gauge Algebra
3.1 Central Thesis
Spin(10) emerges naturally from the DD structure as a classification algebra for 16 one-generation Weyl states. But classification does not imply force mediation.
In SAE, the mechanism of force generation is determined by the four-step axiomatic chain:
- Step 1 (select) → 1DD → U(1)_Y, 1 generator
- Step 2 (orient) → 2DD → SU(2)_L, 3 generators
- Step 3 (unfold) → 3DD → SU(3)_c, 8 generators
- Step 4 (close) → 4DD → gravity (AND readout)
Steps 1–3 correspond to the OR readout of the generating function Z(t) (low-order union, linear); Step 4 corresponds to the AND readout (full coincidence, exponential). Within the current DD analysis, these two readout modes cover the full range of known force-generation mechanisms.
Under SAE's current OR/AND readout framework, the total number of gauge directions is 1 + 3 + 8 = 12 = N_blocks. This correspondence — each block associated with exactly one gauge degree of freedom — is a notable structural feature of the DD framework, though its rigorous derivation depends on the OR/AND non-mixing working principle (§3.4).
3.2 12 vs. 45
When GUTs promote Spin(10) to a gauge symmetry, the adjoint representation dim so(10) = 45 yields 45 gauge bosons. Under the Pati-Salam branching:
45 → (15, 1, 1) ⊕ (1, 3, 1) ⊕ (1, 1, 3) ⊕ (6, 2, 2)
Dimension check: 15 + 3 + 3 + 24 = 45.
Here (15,1,1) is the adjoint of SU(4), (1,3,1) and (1,1,3) are the adjoints of SU(2)_L and SU(2)_R respectively, and (6,2,2) contains the cross-sector mixing generators between the color-lepton space and spacetime.
From the DD perspective, axiomatic steps 1–3 correspond to U(1), SU(2), and SU(3), requiring 12 gauge directions. Comparing with the Pati-Salam branching, only 12 of the 45 adjoint generators belong to SAE's currently allowed force-carrier set. The remaining 33 directions — including 7 in (15,1,1) beyond SU(3), 2 in (1,1,3) beyond U(1), and all 24 cross-sector mixing generators in (6,2,2) — have no corresponding force carriers in the DD structure.
In particular, the 24 mixing generators in (6,2,2) include the X/Y bosons required for GUT-mediated proton decay. In DD's axiomatic chain, these mixing directions correspond to cross-OR/AND mixing — which, under SAE's working principle (§3.4), does not support physical force carriers.
GUTs predict 45 gauge bosons. SAE's current framework accommodates only 12 gauge directions. The experimentally observed number of gauge bosons is exactly 12.
3.3 The Categorical Distinction Between Classification and Mediation
An analogy may be helpful. The periodic table classifies 118 elements by (period, group), with the classification structure arising from the quantum-mechanical shell model. But the shell model does not generate new forces — forces remain electromagnetic, strong, and weak. Classification tells us "what exists"; forces tell us "how things interact."
From SAE's perspective, GUTs may conflate two mathematically distinct types of objects: classification algebras and gauge algebras. Because fermions are classified by Spin(10), GUTs further assume that forces also arise from Spin(10)'s adjoint representation — but this inferential step is not logically necessary. Historically, the eightfold way (SU(3) flavor classification) was indeed promoted to QCD (SU(3) color gauge symmetry), constituting the strongest precedent for GUT reasoning. SAE's reading is that this promotion succeeded possibly because color SU(3) corresponds to Step 3 of the DD axiomatic chain — within the DD structure, it naturally assumes the role of a gauge symmetry. Embedding SU(3)_color into a larger unified group and gauging all generators may go beyond the force-generation mechanisms described by the DD structure.
3.4 OR/AND Non-Mixing
The absence of cross-OR/AND readout mixing is currently a working principle of SAE, not a strict theorem. Paper V's formal analytic generator Z(t) provides the framework for OR and AND as two inequivalent readouts, but deriving a strict selection rule from DD axioms remains an open problem. This paper marks this principle as a working principle and notes that if it were overturned — i.e., if cross-mode mixing force carriers were discovered — the core thesis of this paper would require revision.
§4 The Identity of the Lepton: Fourth Color or Volume Form?
GUT's answer to the identity of the lepton is unambiguous: the lepton is a fourth color. In the Pati-Salam framework, SU(4) ⊃ SU(3)_color × U(1)B−L; quarks carry red, green, and blue, while the lepton carries the fourth color. X/Y boson-mediated proton decay is essentially a "color-change" process — switching a quark from one of three colors to the fourth (lepton).
The picture under the SAE framework is entirely different. Paper IV has established that lepton = Λ³U = complete antisymmetrization of three color directions = color-singlet volume line. In ε-language, (+,+,+) is not a fourth option alongside (+,−,−); it is the unique result when all three pair axes are fully aligned. The lepton is a derived object from three colors, not a fourth color.
This identity difference bears directly on proton stability. The argument within the SAE framework requires two steps in conjunction:
Step 1: The lepton's singlet-volume identity. In SAE's exterior algebra dictionary, quark states live in Λ¹U or Λ²U, while lepton states live in Λ³U. Under the Standard Model's internal gauge symmetries (SU(3)_c × SU(2)_L × U(1)_Y), no gauge direction maps component states to volume-form states. GUT's X/Y bosons are precisely designed to bridge this gap — they are not SU(3) generators, but additional generators in a larger unified group, specifically responsible for cross-exterior-algebra-level transitions.
Step 2: Spin(10) is not fully gauged. Per the argument of §3, SAE does not gauge all 45 adjoint generators of Spin(10). Therefore, even though Spin(10) as a classification skeleton contains mathematical directions for cross-level transitions, these directions do not correspond to physically existing force carriers. Without X/Y bosons, there is no mediator for the quark → lepton cross-level jump.
Neither step alone suffices: Step 1 alone (exterior algebra level difference) cannot exclude additional generators from a larger group mediating the jump — this is precisely GUT's logic. Step 2 alone (not gauging Spin(10)) lacks the deeper reason for why leptons and quarks are fundamentally different objects. Together, the two steps eliminate unified gauge-boson-mediated proton decay channels within SAE.
GUT's "lepton as fourth color" hypothesis requires additional assumptions or mechanisms to account for three phenomena:
- Color confinement binds only the first three colors; the fourth (lepton) flies freely — why?
- Within the same quartet (e.g., third generation), the top mass is ~173 GeV while the tau mass is ~1.8 GeV, differing by two orders of magnitude — if they are components of the same representation, why can the mass difference be so large?
- Why does SU(4) break specifically to 3+1 rather than 2+2 or another pattern?
- 1DD freezes at q² → 0 (photon is massless)
- 2DD freezes at v ~ 246 GeV (W/Z have mass)
- 3DD freezes at Λ_QCD ~ 200 MeV (gluon confinement)
- 4DD freezes at M_Planck (gravity)
These phenomena are typically handled within the GUT framework through the details of symmetry-breaking chains.
Under the SAE framework, these three phenomena admit relatively natural readings: confinement acts on components (quarks) but not on the volume form (lepton); masses differ because they occupy different exterior algebra levels; the 3+1 pattern is not a result of breaking but the unique algebraic structure of Λ³U as the three-color volume form.
§5 Anti-Predictions and Experimental Status
5.1 Hard Anti-Predictions
If any of the following three anti-predictions is overturned, the core thesis of this paper is falsified.
Anti-prediction 1: No unified gauge-boson-mediated proton decay.
SAE does not predict absolute proton stability — other non-X/Y mechanisms for extremely rare processes may exist (e.g., gravitational effects or Planck-scale topological processes). SAE's specific prediction is: no GUT unified gauge boson (X/Y) mediated p → e⁺π⁰ or p → K⁺ν̄ type decay.
Experimental status: Super-Kamiokande gives partial lifetime lower bounds of τ(p → e⁺π⁰) > 2.4 × 10³⁴ years, τ(p → μ⁺π⁰) > 1.6 × 10³⁴ years (90% CL).
Falsification condition: If Hyper-Kamiokande at ~10³⁵ year sensitivity observes a p → e⁺π⁰ signal, this version of SAE is falsified.
Anti-prediction 2: No GUT magnetic monopoles.
No GUT phase transition occurs, hence no 't Hooft-Polyakov type topological defects are produced.
Experimental status: Since the 1982 Cabrera event, large-scale monopole searches have yielded no signal.
Falsification condition: Discovery of a magnetic monopole with GUT magnetic charge.
Anti-prediction 3: No extra gauge bosons beyond the Standard Model.
SAE predicts a total of 12 gauge bosons = N_blocks, no more, no less. No W_R, Z', gauged B−L, or any gauge boson belonging to a gauged SU(4)/SO(10) should exist.
Experimental status: The LHC at 13 TeV center-of-mass energy has found no extra gauge bosons.
Falsification condition: Discovery of W_R, Z', or any gauge boson clearly belonging to an extended gauge group.
5.2 Soft Expectations
The following two do not constitute strict falsification conditions but mark explanatory-power differences between SAE and GUTs.
Expectation 1: Gauge couplings do not unify at a single scale.
In SAE's freezing-scale picture, the three gauge couplings freeze at different scales and need not meet at a single point. Under the bare Standard Model (without SUSY), the three running curves indeed do not intersect. However, SAE has not independently derived the non-existence of SUSY; if SUSY were discovered and achieved coupling unification, this expectation would require revision but would not directly falsify the core thesis of this paper.
Expectation 2: No neutron-antineutron oscillation.
ΔB = 2 processes can be mediated not only by gauge bosons but also by scalars or higher-dimensional operators. Without a no-go theorem covering all possible mechanisms, this expectation does not constitute a hard anti-prediction.
§6 Dissolution of the Hierarchy Problem
A fundamental difficulty of GUTs is the hierarchy problem: why is the electroweak breaking scale v ~ 246 GeV lower than the GUT unification scale ~10¹⁶ GeV by 14 orders of magnitude? GUTs have not yet provided a natural explanation for this enormous gap — the unification scale and each step in the breaking chain are typically introduced as model parameters, rather than derived from more fundamental principles. Why are the W/Z bosons at ~80–91 GeV rather than near 10¹⁶ GeV? GUT's answer is "because there is a chain of successive symmetry breakings," but the scale of each step requires additional assumptions.
SUSY as an extended framework can stabilize the hierarchy (superpartners cancel quadratic divergences), but SUSY's own predictions have gone entirely unobserved.
Within the SAE framework, this problem does not arise. In the freezing-scale picture:
Each DD level has its own freezing scale, with inter-level ratios determined by DD combinatorial numbers and α_em. There is no "unification scale" and no 14-order-of-magnitude gap requiring explanation. The dissolution of the hierarchy problem in SAE is not an additional derivation but rather a natural consequence of its structure — in a framework where forces do not originate from a unified group, the gap between a unification scale and the electroweak scale simply does not arise as a problem requiring explanation.
§7 Concluding Remarks
Spin(10) emerges naturally from the DD structure as a classification skeleton for the 16 Weyl states of one fermion generation. This fact holds for both GUTs and SAE — both give the same classification for the same set of fermions. The divergence appears at the next step: GUTs choose to promote the classification algebra to a gauge symmetry, while SAE holds that classification itself is the complete role of Spin(10) within the DD structure.
The distinction between these two positions is entirely experimentally testable: GUT's distinctive predictions (proton decay, magnetic monopoles, extra gauge bosons) are all observable. Fifty years of searches have yet to yield a positive signal. Does this persistent silence have a structural origin?
This paper does not claim that any model — including the present one — is correct or incorrect, but rather seeks, through fewer prior assumptions, to explain more posterior observations, and thereby to deepen our understanding of the universe's fundamental nature.
One possible answer offered by this paper is: classification does not imply mediation, the volume form is not a fourth color, and 12 blocks give 12 gauge bosons — no more, no less.
Dependency and Assumption List
| Item | Status | Source |
|---|---|---|
| so(6) ≅ su(4) | Theorem | Four Forces Paper IV |
| Spinor 4 = 3 ⊕ 1 | Theorem | Four Forces Paper IV |
| T₁ ≠ T₂ | Theorem | Four Forces Prequel |
| ε₁ε₂ε₃ parity = Spin(6) Weyl spinors | Standard representation theory (B− level) | This paper §2.3 |
| L-R canonical pairing | Principal Bridge Assumption (falsifiable) | This paper §2.4 |
| Z(t) OR/AND readout | Formal analytic generator (B− level) | Four Forces Paper V |
| OR/AND non-mixing | Working principle (not theorem) | This paper §3.4 |
| 12 = N_blocks = gauge boson count | A priori derivation + posterior match | This paper §3.2 |
| Lepton = Λ³U | Theorem | Four Forces Paper IV |
Appendix A: Distinction from the Standard Pati-Salam Model
The standard Pati-Salam model also uses SU(4) × SU(2)_L × SU(2)_R, but fully gauges SU(4). It should be noted that the minimal Pati-Salam gauge bosons do not automatically mediate proton decay at the renormalizable level — the canonical proton decay channels require full SO(10)-level X/Y bosons.
The precise distinction between SAE and Pati-Salam is: SAE does not gauge SU(4). SU(4) ≅ SO(6) serves in SAE as the classification structure of the color-lepton space, not as a mediator of forces. The experimental discriminant is: the discovery of gauge bosons belonging to a gauged SU(4) (such as leptoquark gauge bosons) would falsify SAE.
Appendix B: Spin(10) DD Dictionary — Complete Mapping of 16 States to One-Generation Fermions
| ε₁ε₂ε₃ | ε₄ | Weyl | Spin(6) | Spin(4) | Fermion | Note |
|---|---|---|---|---|---|---|
| (+,+,+) | L | ✅ | 4 (even) | (2,1) | ν_L | Lepton = Λ³U |
| (+,−,−) | L | ✅ | 4 (even) | (2,1) | u_L^r | Color quark |
| (−,+,−) | L | ✅ | 4 (even) | (2,1) | u_L^g | Color quark |
| (−,−,+) | L | ✅ | 4 (even) | (2,1) | u_L^b | Color quark |
| (+,+,+) | R | ❌ | — | — | — | Weyl excluded |
| (+,−,−) | R | ❌ | — | — | — | Weyl excluded |
| (−,+,−) | R | ❌ | — | — | — | Weyl excluded |
| (−,−,+) | R | ❌ | — | — | — | Weyl excluded |
| (−,−,−) | L | ❌ | — | — | — | Weyl excluded |
| (−,+,+) | L | ❌ | — | — | — | Weyl excluded |
| (+,−,+) | L | ❌ | — | — | — | Weyl excluded |
| (+,+,−) | L | ❌ | — | — | — | Weyl excluded |
| (−,−,−) | R | ✅ | 4̄ (odd) | (1,2) | e_R | Anti-lepton |
| (−,+,+) | R | ✅ | 4̄ (odd) | (1,2) | d_R^r | Anti-color quark |
| (+,−,+) | R | ✅ | 4̄ (odd) | (1,2) | d_R^g | Anti-color quark |
| (+,+,−) | R | ✅ | 4̄ (odd) | (1,2) | d_R^b | Anti-color quark |
Upper block (even × L) = (4, 2, 1): left-handed color-lepton quartet. Lower block (odd × R) = (4̄, 1, 2): right-handed anti-color-lepton quartet. Middle 8 states are excluded by the Weyl condition. The 16 states exactly cover all Weyl degrees of freedom of one fermion generation, including ν_R.
Note: The precise assignment of specific fermion labels (u_L vs d_L, etc.) depends on mass eigenstate mixing after electroweak symmetry breaking; the convention adopted here assigns the upper component of SU(2)_L doublets.