Four Forces Paper VIII: The Hierarchical Dissolution of Strong CP

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

Self-as-an-End Four Forces Series: Paper VIII

Han Qin

ORCID: 0009-0009-9583-0018

§1 The Correct Target of Strong CP

§1.1 Standard Formulation

The QCD Lagrangian admits a topological term:

$$\mathcal{L}_\theta = \frac{\theta}{32\pi^2} g^2 F^a_\mu\nu \tilde{F}^a,\mu\nu$$

This term violates P and CP. A nonzero θ would induce a neutron electric dipole moment d_n ∝ θ. Experiment constrains d_n < 1.8 × 10⁻²⁶ e·cm (90% CL), implying θ̄ < 10⁻¹⁰.

The true target of the strong CP problem is not bare θ but the rephasing-invariant physical quantity:

$$\bar{\theta} = \theta + \arg\det(M_u M_d)$$

where M_u, M_d are the quark mass matrices. Chiral rotations can transfer phase between θ and arg det(M), but θ̄ is invariant. Experiment constrains θ̄, not θ alone.

§1.2 Responses Within the Standard Model

Axion / Peccei-Quinn mechanism. A global U(1)_PQ symmetry is introduced along with its associated Goldstone boson (the axion), which dynamically drives θ to the CP-conserving point. This introduces a new particle (axion) and new symmetry (U(1)_PQ). No axion has been found experimentally, though searches continue (ADMX, IAXO, etc.).

Massless up quark. If m_u = 0, θ can be rotated away by a chiral rotation. Lattice QCD calculations have excluded m_u = 0.

Nelson-Barr mechanism. CP is an exact high-energy symmetry that is spontaneously broken through specific Yukawa/mediator structures, yielding tree-level θ̄ = 0 while preserving CKM phases. This faces the technical difficulty of radiative regeneration: loop corrections tend to reignite θ̄, requiring strong constraints on model parameters.

Parity symmetry solutions. Bare θ is set to zero by parity invariance, while arg det(M) is controlled through Hermitian Yukawa structures or similar mechanisms. These also face radiative stability difficulties.

§1.3 Position of This Paper

All the above solutions operate within the Standard Model framework: they accept the Yukawa coupling as a fundamental object, accept M = Yv, and then arrange parameters or introduce new particles within this framework to make θ̄ small or zero.

This paper offers a different type of response. SAE does not solve the strong CP problem within the Standard Model framework; rather, it asserts that SAE's ontology does not generate this problem. The Standard Model's Yukawa coupling simultaneously carries mass generation and weak mixing, so complex phases inevitably enter the mass matrix. This is the root of the strong CP problem. SAE assigns mass construction and weak mixing to different DD levels, severing the pathway for complex phases to enter the mass matrix.

§2 CP Stratification in the DD Hierarchy

§2.1 Symmetry Map of the Chisel-Construct Sequence

The DD hierarchy has the following structure:

The 2DD chiral splitting is the first explicit symmetry breaking in the chisel-construct sequence: it distinguishes L from R. The weak force couples only to left-handed fields; this is a defining property of the 2DD level.

3DD forms above 2DD. The L-side and R-side each independently produce their 3DD color structures.

§2.2 CP Transformation and DD Levels

The CP transformation consists of spatial inversion P (exchanging L and R) plus charge conjugation C. For the strong CP problem, the key effect is that of P:

The θ-term changes sign under P: θ → −θ.

Therefore, if a theory possesses exact P symmetry (L-R mirror) at the 3DD level, then θ = −θ, hence θ = 0.

This is a standard QCD result, not specific to SAE. SAE's contribution is to provide a structural reason for L-R 3DD mirror symmetry.

§2.3 L-R 3DD Mirror Symmetry in SAE

Proposition S1 (L-R 3DD mirror). The DD chisel-construct sequence produces 3DD structures on the two sides that are exact mirror images.

Argument.

(a) There is only one kind of chisel (negation), and it does not distinguish direction. Each step of the chisel-construct sequence applies the same operation to the L-side and R-side.

(b) The 2DD chiral splitting is a 1-bit scalar operation (L or R), granting no structural privilege to either side. It transmits exactly 1 bit of information: which side is which. It does not transmit orientation, phase, or other internal structure.

(c) The two sides of 3DD each independently execute the three-axis spatial splitting. By (a) and (b), the 3DD structures on the two sides are exact mirror images as wholes.

(d) The remainder conservation principle of ZFCρ provides independent mathematical support.

Key distinction. This proposition does not require 4DD-level axis pairing. We do not claim that axis 1 on the L-side equals axis 1 on the R-side (the Generation Paper's G1 has argued that no such canonical identification exists). We claim only that L-3DD as a whole and R-3DD as a whole are mirror-symmetric. θ is a global topological quantity of 3DD (the weight of the instanton number); it does not depend on axis labeling.

Level. Near-theorem level (ZFCρ + DD axiom chain, dual-line support).

§2.4 Parallel with G5: Bidirectional Information Isolation Between DD Levels

The Generation Paper's G5 (Isospin Universality) established a perfectly parallel argument:

G5 states: color (3DD) collapses to a color singlet under cross-block projection, but isospin (2DD) remains intact. The reason: isospin comes from the deeper 2DD shared layer; 3DD processes cannot destroy it.

The present paper's argument is the converse of G5: 3DD structure does not inherit 2DD's symmetry breaking. The 2DD chiral splitting introduced L-R asymmetry (the source of CP violation), but this asymmetry is encapsulated within 2DD and does not propagate to 3DD.

The two directions together constitute a general principle: information isolation between DD levels is bidirectional. Lower-level structure is not destroyed by higher-level processes (G5); higher-level structure does not inherit lower-level symmetry breaking (this paper).

§3 Core Argument

§3.1 Conditional Theorem (bare θ = 0)

Statement. Under L-R 3DD mirror symmetry (Proposition S1), the bare QCD vacuum angle θ = 0.

Proof.

1. L-R 3DD mirror symmetry holds (Proposition S1).
2. The θ-term changes sign under P: θ → −θ (standard QCD result).
3. Since 3DD possesses exact L-R mirror symmetry, any term appearing in the 3DD effective description must be invariant under P.
4. By 2 and 3: θ = −θ, therefore θ = 0. □

Level. Conditional theorem (conditional on Proposition S1).

§3.2 Positive-Real Determinant Proposition (det(M_u M_d) > 0)

Statement. Under SAE's mass construction mechanism, the quark mass operators M_u, M_d take values in GL⁺(3,ℝ) (real matrices with positive determinant), so that arg det(M_u M_d) = 0.

Argument. In SAE, the quark mass matrix is constructed in two steps (Generation Paper G3, G6):

Step one: topological barriers yield the natural mass. Different generations receive different masses from the topological distance between the observer block and source blocks. The natural mass matrix D_nat is a real positive diagonal matrix. det(D_nat) > 0.

Step two: the 3DD color-field redistribution operator R acts on D_nat. In SAE's 3DD dictionary, the pure color response operator is taken to be real. This is treated as a working interpretation under 3DD mirror symmetry: a complex phase would require a preferred orientation to define its sign, and 3DD mirror symmetry forbids any preferred orientation. The present paper treats "R is real" as a direct consequence of 3DD mirror; its full formalization is deferred to future work.

Furthermore, R evolves continuously from the identity operator I. The chisel-construct sequence is gradual: color-field redistribution is not a discontinuous jump, but a continuous evolution from "no redistribution" (R = I) to the actual redistribution strength. det(I) = 1 > 0. During continuous evolution, det(R) changes continuously. det(R) = 0 would mean some mass eigenvalue is driven to zero (matrix degeneracy). From the DD-level perspective, det(R) = 0 means the oriented volume of 3DD color space collapses to zero, reducing 3DD to 2DD. This directly violates the premise that 3DD has already completed its chisel-construct formation. det(R) < 0 would mean 3DD undergoes an inversion (reversal of spatial orientation), but 3DD is constructed within a subspace whose chiral direction has already been anchored by 2DD; it has no degree of freedom to execute a global odd-parity reversal. Therefore det(R) > 0. This conclusion also has a posteriori support: no zero-mass quarks exist experimentally (lattice QCD excludes m_u = 0), ruling out the degenerate case det(R) = 0.

In total: M = R · D_nat, where R is real with det > 0 and D_nat is real positive diagonal with det > 0. Therefore M is a real matrix with det(M) > 0. This holds for both M_u and M_d.

$$\det(M_u M_d) = \det(M_u) \cdot \det(M_d) > 0$$

$$\arg\det(M_u M_d) = 0$$

Level. Conditional proposition (conditional on Proposition S1 + Generation Paper G3/G6 mass construction + no zero-mass quarks).

§3.3 Yukawa Emergence Principle

Statement. Under SAE's constitutive ordering, the Yukawa coupling is not a fundamental object but an emergent low-energy projection superposition of 3DD mass structure and 2DD weak mixing structure. The CKM complex phases originate from the 2DD weak vertex's readout structure, not from the mass matrix's complex entries. Therefore observed CP violation does not conflict with arg det(M_u M_d) = 0.

Argument.

In the Standard Model, the Yukawa matrix Y simultaneously serves two functions: mass generation (M = Yv) and weak mixing (the CKM matrix arises from the diagonalization mismatch of Y_u and Y_d). Y is a complex matrix, so complex phases inevitably enter both M and CKM simultaneously. This is precisely the root of the strong CP problem: one cannot maintain CKM CP violation while ensuring arg det(M) = 0, unless additional mechanisms (axion, parity, Nelson-Barr) are introduced.

SAE's constitutive ordering (Generation Paper G7) separates these two functions:

1. Mass is constructed independently at the 3DD level. Topological barrier heights determine the natural mass; color-field redistribution determines the actual mass. The definition of mass does not depend on the existence of 2DD weak force. Even if the weak force were turned off, quarks would still have mass.

1. Weak mixing is independently provided at the 2DD level. The 2DD weak force reads out the physical mass states already constructed by 3DD, producing inter-generational transition amplitudes. The CKM matrix is the output of this readout process, not a byproduct of mass-matrix diagonalization.

1. The 2DD chiral splitting distinguishes L from R, permitting complex structure (CP violation) to reside in the weak vertex. This is the source of the experimentally observed CKM CP violation.

1. The low-energy experimentalist simultaneously observes mass and weak force (the W boson simultaneously changes flavor and transfers momentum) and uses a unified Yukawa matrix Y to describe both. This Y is a complex matrix because it superimposes the real mass information from 3DD with the complex mixing information from 2DD. But Y is not a fundamental object; it is a projection superposition of two different DD levels.

Therefore in SAE: M comes from 3DD (real matrix, det > 0), V comes from 2DD (unitary matrix, may carry complex phases). V does not participate in the construction of M and does not appear in the expression for det(M_u M_d). arg det(M_u M_d) = 0 and CKM CP violation coexist without contradiction.

Level. Structural argument, dependent on the G7 constitutive ordering.

§3.4 Main Conclusion: The 3DD Contribution to θ̄ Vanishes

Statement. Under Proposition S1 (L-R 3DD mirror) and the Yukawa emergence principle, all 3DD-level contributions to θ̄ vanish:

$$\bar{\theta}_\text{3DD} = \theta + \arg\det(M_u M_d) = 0 + 0 = 0$$

Argument chain.

§3.5 Cross-Level Leakage

The conclusion of §3.4 is that within the 3DD level, θ̄ = 0. However, SAE's level isolation is not an absolute prohibition. The remainder develops as it must is a general principle of the SAE framework. Cross-level leakage exists between DD levels, but is strongly suppressed by the "distance" between levels.

In the specific case of this paper: the CKM complex phase carried by the 2DD weak vertex can in principle leave a faint imprint on 3DD observables through cross-level effects. The neutron electric dipole moment d_n is the observable carrier of this imprint. Standard Model calculations (the higher-order CKM contribution to d_n) yield ~10⁻³² e·cm, six orders of magnitude below current experimental sensitivity and four orders below next-generation targets (~10⁻²⁸ e·cm).

In SAE, this extremely small CKM contribution is interpreted as 2DD → 3DD cross-level leakage. The leakage is profoundly weak: CP violation within 2DD is an O(1)-scale effect (CKM phase ~1 radian, producing significant CP asymmetries in K and B meson systems), yet when it crosses the level boundary and projects onto a 3DD entity (the neutron EDM), it is suppressed to ~10⁻³² e·cm. This suppression by tens of orders of magnitude is the quantitative manifestation of the extremely high strength of DD level isolation.

SAE's complete statement on θ̄ is therefore:

$$\bar{\theta} = 0 + \epsilon_\text{leak}, \qquad |\epsilon_\text{leak}| \ll 10^-10$$

where ε_leak is the 2DD → 3DD cross-level leakage term, far below any foreseeable experimental sensitivity. The experimental consequences are the same as for "θ̄ = 0 exactly," but the ontological statement differs: SAE does not claim absolute prohibition, only that level isolation renders the leakage negligibly small. This is fully consistent with SAE's general principle that the remainder develops as it must: levels can break through to each other, but doing so is extremely difficult.

§4 Comparison with Mainstream Solutions and Falsifiable Predictions

§4.1 Three-Way Comparison

§4.2 On Radiative Stability

Parity and Nelson-Barr solutions face the difficulty of radiative stability: tree-level θ̄ = 0 can be reignited by loop corrections. This is because they operate within the Standard Model EFT framework, where mass and weak mixing originate from the same Yukawa object, and loops can transport phase between them.

SAE's situation is different. The separation of M and V is not a tree-level approximation but a defining property of DD-level structure. DD levels are not a perturbative expansion of an effective theory. 3DD is not "tree-level 3DD plus loop corrections," just as 1DD is not "tree-level charge plus corrections to 2DD." Levels are ontological structure; there is no concept of "quantum corrections breaking level assignment."

Accordingly, SAE does not owe a radiative stability proof of the kind required within the Standard Model framework. SAE does, however, still owe a systematic projection rule from DD object stratification to low-energy observables. Section 3 of this paper provides the first step of this projection, from DD-level structure to the vanishing of θ̄'s 3DD contribution, rather than a complete DD-to-observable projection theory. Establishing the full projection rule is an open problem for the entire SAE physics series (§5.2), not specific to this paper.

§4.3 Falsifiable Predictions

Prediction 1: The 3DD contribution to θ̄ is zero. Within the 3DD level, the strong CP source term is exactly zero. Cross-level leakage (the projection of the 2DD CKM phase onto 3DD) is in principle nonzero but lies far below any foreseeable experimental sensitivity (the literature estimates the CKM contribution to d_n at ~10⁻³² e·cm). The experimental consequences are therefore the same as "θ̄ = 0."

This is a distinguishable prediction between SAE and parity/NB schemes. Parity/NB deviations in θ̄ arise from loop regeneration within the same framework and may range from 10⁻¹⁵ to 10⁻¹⁰ (depending on model details), far above SAE's cross-level leakage. If future neutron EDM experiments detect a nonzero signal at 10⁻²⁸ e·cm precision, both SAE and parity/NB would require explanation; but if the signal level is above 10⁻¹⁵, it would point toward parity/NB-type mechanisms rather than SAE's cross-level leakage.

Prediction 2: No strong-CP-motivated QCD axion is required. The basic motivation of the Peccei-Quinn mechanism, that θ needs a dynamical zeroing mechanism, does not exist in SAE. If axion search experiments detect a signal within the QCD axion's theoretically favored parameter space, this paper is falsified.

Narrowed statement: this paper predicts the nonexistence of a QCD axion required as a strong CP solution. Axion-like particles may have motivations independent of strong CP (e.g., dark matter); this paper makes no prediction about those.

Prediction 3: All experimentally observed CP violation is attributable to the weak sector. This is consistent with existing experimental data: all confirmed CP violations (K, B, D meson systems) have been observed in processes mediated by the weak force.

§4.4 Anti-Prediction Summary Table

Together with the three hard anti-predictions from Paper VI, the SAE Four Forces series has now issued five hard anti-predictions:

Five hard anti-predictions plus one structural expectation, all experimentally adjudicable.

§5 Discussion

§5.1 Dependency Chain Summary

This paper's argument depends on the following published or near-theorem-level results:

Does not depend on: L-R 4DD axis pairing (G1), OR/AND non-mixing assumption, Mass-Channel Proportionality, or any continuous parameters or boundary conditions.

§5.2 Weaknesses and Open Problems

1. Theorem-level proof of L-R 3DD mirror symmetry. Currently near-theorem level rather than theorem. If this symmetry has a tiny breaking, θ̄ may not be exactly zero but extremely small. Promoting L-R 3DD mirror from near-theorem to theorem is future work.

2. Formalization of the Yukawa emergence principle. Section 3.3 argued that the Yukawa coupling is an emergent effective description rather than a fundamental object in SAE. This principle is aligned with G7's constitutive ordering but has not yet been written as an independent theorem. More precise formalization, for instance proving that under the G7 constitutive ordering there exists no mechanism for CKM phases to flow back into det(M), is deferred to future work.

3. Electroweak baryogenesis. Can the CKM phase provide sufficient CP violation to drive baryon asymmetry? This is an independent question that does not affect this paper's argument about θ̄. However, if CKM is the only CP source, the sufficiency of baryogenesis requires separate argument.

4. Quantitative theory of cross-level leakage. Section 3.5 noted that 2DD → 3DD cross-level leakage is in principle nonzero but extremely small. The CKM contribution to d_n (~10⁻³² e·cm) provides a posteriori calibration of the leakage magnitude. However, SAE currently lacks an a priori theory for the leakage magnitude. Whether the leakage rate can be calculated from the "distance" between DD levels, analogous to how topological distance determines mass hierarchy in the Generation Paper, is a meaningful open problem.

5. Complete projection rule from DD object stratification to low-energy observables. This paper provides the DD-level to θ̄ and CKM correspondence. Establishing a more systematic DD-to-observable projection rule has significance for the entire SAE physics series, not just this paper.

§5.3 Broader Significance

The core message of this paper is not a new technical solution but an ontological reclassification:

The strong CP "problem" is a product of the Standard Model's Yukawa ontology. The Standard Model treats the Yukawa coupling as fundamental, so complex phases inevitably enter the mass matrix, arg det(M) can in principle be nonzero, and one must explain why it happens to be zero.

In SAE, mass comes from 3DD topological barriers (real matrices), and weak mixing comes from 2DD readout (which may carry complex phases). The two belong to different DD levels; complex phases do not enter the mass matrix. arg det(M) = 0 is not the result of fine-tuning, not the result of symmetry protection, not the result of dynamical relaxation. It is an automatic consequence of mass and weak mixing belonging to ontologically distinct levels.

If this picture is correct: the QCD axion's basic motivation as a strong CP solution collapses (axion-like particles may have independent motivations). The strong CP "problem" is reclassified: it is not a problem, but an artifact of the Standard Model's Yukawa ontology.

Proposition Status Table

References

Strong CP Problem and θ̄

[1] S. Weinberg, The Quantum Theory of Fields, Vol. 2: Modern Applications, Cambridge University Press (1996), Ch. 23.6.

[2] R.D. Peccei, "The Strong CP Problem and Axions," Lect. Notes Phys. 741, 3-17 (2008), arXiv:hep-ph/0607268.

[3] M. Dine, "The Strong CP Problem," SCIPP preprint, https://scipp.ucsc.edu/~dine/solutions_of_strong_cp.pdf

Axion / Peccei-Quinn

[4] R.D. Peccei, H.R. Quinn, "CP Conservation in the Presence of Instantons," Phys. Rev. Lett. 38, 1440 (1977).

[5] S. Weinberg, "A New Light Boson?", Phys. Rev. Lett. 40, 223 (1978).

[6] F. Wilczek, "Problem of Strong P and T Invariance in the Presence of Instantons," Phys. Rev. Lett. 40, 279 (1978).

Nelson-Barr / Parity Solutions

[7] A.E. Nelson, "Naturally Weak CP Violation," Phys. Lett. B 136, 387 (1984).

[8] S.M. Barr, "Solving the Strong CP Problem without the Peccei-Quinn Symmetry," Phys. Rev. Lett. 53, 329 (1984).

[9] J. de Vries, P. Draper, H.H. Patel, "Do Minimal Parity Solutions to the Strong CP Problem Work?", arXiv:2109.01630.

[10] A. Valenti, L. Vecchi, "The CKM Phase and θ̄ in Nelson-Barr Models," JHEP 07, 203 (2022), arXiv:2112.09122.

Neutron Electric Dipole Moment Experiments

[11] C. Abel et al. (nEDM Collaboration), "Measurement of the Permanent Electric Dipole Moment of the Neutron," Phys. Rev. Lett. 124, 081803 (2020). [Current best bound: |d_n| < 1.8 × 10⁻²⁶ e·cm]

[12] n2EDM Collaboration at PSI, https://www.psi.ch/en/nedm [Next-generation experiment; baseline sensitivity ~10⁻²⁷ e·cm, upgrade target ~10⁻²⁸ e·cm]

Standard Model CKM Contribution to d_n

[13] I.B. Khriplovich, A.R. Zhitnitsky, "What Is the Value of the Neutron Electric Dipole Moment in the Kobayashi-Maskawa Model?", Phys. Lett. B 109, 490 (1982). [d_n^CKM ~ 10⁻³² e·cm]

[14] N. Yamanaka et al., "Standard model contribution to the electric dipole moment of the neutron, deuteron, and helion," arXiv:1512.03013. [Modern recalculation confirming d_n^CKM ~ 10⁻³¹ to 10⁻³² e·cm]

SAE Series Published Papers

[15] H. Qin, "Self-as-an-End: The Chisel-Construct-Remainder Cycle" (SAE P1-P3), DOI: 10.5281/zenodo.18528813, .18666645, .18727327.

[16] H. Qin, "Four Forces Prequel: DD Splitting and α_G = α_em^65/4", DOI: 10.5281/zenodo.19341042.

[17] H. Qin, "Four Forces Paper I: nDD → SU(n) Correspondence", DOI: 10.5281/zenodo.19342106.

[18] H. Qin, "Four Forces Paper III: sin²θ_W = 3/13", DOI: 10.5281/zenodo.19379412.

[19] H. Qin, "Generation Paper: Topological Origin and Mass Structure of Three Fermion Generations", DOI: 10.5281/zenodo.19394500.

[20] H. Qin, "Four Forces Paper VI: Spin(10) Classification and Three Hard Anti-Predictions", DOI: 10.5281/zenodo.19426067.

[21] H. Qin, "Four Forces Paper VII: Three-Layer Correction Structure", DOI: 10.5281/zenodo.19433220.