The CKM Complex Phase and the Jarlskog Invariant from the McGucken Principle of a Fourth Expanding Dimension dx₄/dt = ic: Compton-Frequency Interference, the Kobayashi-Maskawa Three-Generation Requirement as a Geometric Theorem, and Numerical Verification at Version 1 Scope

Elliot McGucken, PhD elliotmcguckenphysics.com — April 2026

“More intellectual curiosity, versatility and yen for physics than Elliot McGucken’s I have never seen in any senior or graduate student. . . Originality, powerful motivation, and a can-do spirit make me think that McGucken is a top bet.”

— John Archibald Wheeler, Princeton’s Joseph Henry Professor of Physics, on Dr. Elliot McGucken

Abstract

CP violation in the quark sector is characterized by the Jarlskog invariant J = Im[V_us V_cb V_ub* V_cs*], a rephasing-invariant quantity measured experimentally to be |J| ≈ 3 × 10⁻⁵. The Standard Model attributes this CP violation to a single complex phase in the Cabibbo-Kobayashi-Maskawa mixing matrix V, whose existence requires at least three generations of quarks (Kobayashi-Maskawa, 1973). Both the specific value of J and the three-generation requirement are treated as features of nature in the Standard Model — measured rather than derived, with the KM argument showing that three generations are needed but not why the universe has this particular chiral/generational structure. Here we provide an explicit derivation of the Compton-frequency-interference mechanism — sketched in the broken-symmetries companion paper [4] and in the Dirac paper [1] §X.6 — from the McGucken Principle dx₄/dt = ic, the foundational principle of Light, Time, Dimension Theory (LTD) which states that the fourth dimension is expanding at the rate of light. The paper operates at what we call Version 1 scope: we derive the Jarlskog invariant formula J in terms of the six quark masses and three Cabibbo-like mixing angles, with the mixing angles remaining as inputs rather than derived quantities. The paper delivers three concrete results. First, we establish geometrically (§III) that the CKM matrix V arises in LTD as the overlap between the mass-eigenstate basis (diagonalizing the x₄-phase frequencies k_f = m_f c/ℏ via condition (M)) and the weak-eigenstate basis (diagonalizing the SU(2)_L gauge coupling established in [4]). Second, we derive the Kobayashi-Maskawa three-generation requirement as a geometric theorem (§V): with two generations, the mixing matrix has a phase that can be absorbed into field redefinitions; with three generations, one irreducible phase δ remains, because 2n − 1 = 5 of the n² = 9 phases of a unitary 3 × 3 matrix can be absorbed, leaving one for n = 3. Third, we verify numerically (§VI) that the LTD-derived formula for J reproduces the experimental value using the Particle Data Group’s measured quark masses and mixing angles: inserting the PDG 2024 values sin θ_12 = 0.22500, sin θ_13 = 0.00369, sin θ_23 = 0.04182, and δ = 1.144 rad gives |J|_LTD = 3.08 × 10⁻⁵, matching the direct measurement |J|_exp = (3.08 ± 0.14) × 10⁻⁵. We are explicit throughout that this is Version 1 scope: the mixing angles θ_12, θ_13, θ_23 and the phase δ are inputs, not outputs. Version 2 — deriving the mixing angles from quark masses alone, which would constitute genuine parameter reduction — would require a geometric argument for mass-ratio-to-mixing-angle relationships that is not provided here and is flagged as the natural next step (§VIII). What is established is the geometric origin of CP violation in LTD (Compton-frequency interference between mass-eigenstate quarks with different k values under SU(2)_L mixing) and the geometric origin of the three-generation requirement (rephasing counting of unitary matrices), not a prediction of the specific value of J from the McGucken Principle alone.

I. Introduction

The Cabibbo-Kobayashi-Maskawa (CKM) matrix V describes the mixing between the mass eigenstates and the weak eigenstates of the quarks in the Standard Model. The matrix is unitary (V V† = 1) and, for three generations of quarks, contains three real mixing angles (θ_12, θ_13, θ_23) and one complex phase δ. The Jarlskog invariant [5]:

J = Im[V_us V_cb V_ub V_cs]**

is a rephasing-invariant measure of CP violation — its value does not depend on field-redefinition choices for the quark phases. Experimentally, |J| has been measured from kaon and B-meson oscillation asymmetries to be |J| = (3.08 ± 0.14) × 10⁻⁵ [6].

In the Standard Model, the existence of a nonzero J is traced to the three-generation structure of the quark sector via the Kobayashi-Maskawa counting argument [7]: a unitary n × n matrix has n² real parameters, of which n(n−1)/2 are mixing angles, the remainder are phases, and 2n − 1 of the phases can be absorbed into field redefinitions. For n = 2, the number of physical phases is n² − n(n−1)/2 − (2n−1) = 0. For n = 3, it is exactly 1. CP violation requires at least three generations because only then is there an unabsorbable complex phase available.

This accounts for the structural fact of CP violation but leaves two questions open:

Why is the CKM matrix not diagonal? Why do the mass-eigenstate quarks not couple diagonally to the weak interaction? In the Standard Model this is a feature of the Yukawa coupling structure, but the Yukawa couplings themselves are free parameters.
What fixes the specific value of J? Why is |J| ≈ 3 × 10⁻⁵ and not zero or some other number?

The Dirac paper [1] and the broken-symmetries paper [4] argued that both questions have geometric answers in Light, Time, Dimension Theory (LTD), whose foundational principle is the McGucken Principle dx₄/dt = ic — the principle that the fourth dimension is expanding at the rate of light. This paper develops the quantitative version of that argument at what we call Version 1 scope — and we are explicit about the scope choice from the outset so that the reader does not come away with expectations the paper does not deliver.

I.1 Version 1 and Version 2: Scope Distinctions

The Dirac paper’s §X.6 distinguished two possible versions of the CP-violation derivation program in LTD:

Version 1 (modest). Derive J in terms of the six quark masses and the three Cabibbo-like mixing angles, with the mixing angles remaining as inputs. Explains:

Why J is nonzero rather than identically zero (Compton-frequency interference mechanism)
Why three generations are required (Kobayashi-Maskawa rephasing counting, derived geometrically)
Why the mass and weak bases differ (distinct diagonalizations of distinct operators in LTD)

Does not reduce parameter count below the Standard Model.

Version 2 (ambitious). Derive both J and the three mixing angles from quark masses alone. Would constitute genuine parameter reduction (from the 9 CKM parameters of the Standard Model down to 6 mass parameters). Requires a geometric argument for why the mass-basis-to-weak-basis rotation angles take their specific observed values — typically some variant of “mixing angles are determined by mass ratios.” No currently available mechanism in the LTD framework delivers this, and its specification is an open problem.

This paper delivers Version 1. Version 2 is discussed but not claimed. A reader looking for Version 2 should understand that it is not what this paper provides. A reader assessing the LTD program should weigh Version 1’s delivery (geometric origin of an observed phenomenon, with quantitative agreement on parametric formula) as evidence at the appropriate level — not as a proof of foundational correctness, and not as nothing. The accumulation of Version 1 results (Dirac equation, second quantization, QED, now the Jarlskog formula) is evidence that the framework has structural reach; the absence of Version 2 results (so far) is a limitation to be honestly noted.

I.2 Preview of the Three Concrete Deliverables

This paper delivers three specific results:

The geometric origin of the CKM matrix (§III). V arises as the overlap between the mass-eigenstate basis (diagonalizing the x₄-phase frequencies k_f = m_f c/ℏ from condition (M) in [1] §II.2) and the weak-eigenstate basis (diagonalizing the SU(2)_L gauge coupling from [4] §III).
The Kobayashi-Maskawa three-generation requirement as a geometric theorem (§V). The rephasing-counting argument is derived from LTD’s gauge-theory structure. With two mass eigenstates, all phases in the mixing matrix are absorbable; with three, exactly one is irreducible.
Numerical verification at Version 1 scope (§VI). Inserting experimental values for the mixing angles and phase into the LTD-derived formula for J produces the experimental |J| ≈ 3 × 10⁻⁵, establishing quantitative consistency.

The argument is developed in nine sections. Section II briefly recaps the prerequisites from [1]–[4] (with Appendix A providing sketches for independent readability). Section III establishes the geometric origin of the CKM matrix from mass-eigenstate vs. weak-eigenstate basis mismatch. Section IV develops the Compton-frequency-interference mechanism precisely, explicitly confronting the issue — raised honestly in [1] §X.6 — that the naive triple-exponential interference integrand vanishes identically. Section V derives the three-generation requirement. Section VI performs the numerical Version 1 verification. Section VII is explicit about what has and has not been delivered. Section VIII discusses open problems and the path toward Version 2. Section IX concludes.

II. Prerequisites from the Companion Papers

We briefly recapitulate the results from the companion papers [1]–[4] that this paper rests on. Full sketches are in Appendix A.

II.1 From [1]: Single-Particle Dirac and the Matter Orientation Condition (M)

Each fermion species f with rest mass m_f carries x₄-phase at the Compton frequency k_f = m_f c/ℏ via condition (M):

Ψ_f(x, x₄) = Ψ_f₀(x) · exp(+I · k_f x₄), k_f > 0

The positivity of k_f distinguishes matter (k > 0) from antimatter (k < 0). Different fermion species have different k_f values because they have different masses. For quarks:

Up-type (u, c, t): k_u = m_u c/ℏ, k_c = m_c c/ℏ, k_t = m_t c/ℏ
Down-type (d, s, b): k_d = m_d c/ℏ, k_s = m_s c/ℏ, k_b = m_b c/ℏ

The numerical values of these Compton frequencies, using PDG-recommended quark masses [6], are (in natural units where c = ℏ = 1):

Quark	Mass (MeV)	k_f (fm⁻¹)
u	2.16 ± 0.49	0.0110
c	1270 ± 20	6.43
t	172500 ± 760	874
d	4.67 ± 0.48	0.0237
s	93.4 ± 8.6	0.473
b	4180 ± 30	21.2

The wide range of k_f values across six orders of magnitude is the “quark mass hierarchy” whose origin is one of the Standard Model’s open problems. In the LTD framework, the mass hierarchy sets the interference pattern that produces J; the hierarchy itself is not derived at Version 1 scope.

II.2 From [2]: Second Quantization

The second-quantized Dirac field operator creates and annihilates localized x₄-standing-wave modes. For quarks in the mass eigenbasis, this gives operators a†_{f,p,s} that create a quark of species f with momentum p and spin s, each carrying the Compton-frequency phase exp(+I · k_f x₄) in the mass-basis field expansion.

II.3 From [3]: QED Gauge Structure

The QED U(1) gauge field couples to the electromagnetic current, but does not mix quark species — it is diagonal in mass basis (electric charge is a good quantum number). The paper [3] established that A_μ is the connection on the x₄-orientation U(1)-bundle, with the coupling determined by electric charge (which in LTD is the integer-quantized net-x₄-orientation charge).

II.4 From [4]: The Weak Interaction and SU(2)_L

The weak interaction is the gauged version of SU(2)_L — rotations within the spatial triple (x₁, x₂, x₃), transverse to the x₄-expansion axis. Under SU(2)_L, the left-chiral components of matter fields (which, by condition (M) and §[1] §IV.3, are the only components that transform in a way consistent with single-sided action) form doublets:

(u_L, d_L), (c_L, s_L), (t_L, b_L)

The W^± bosons mediate transitions within these doublets. Critically, the weak interaction is not diagonal in the mass basis — the SU(2)_L eigenstates are not the same as the mass eigenstates. The mismatch between these two bases is what produces the CKM matrix, and its irreducible complex phase is what produces CP violation.

These four results together — the Compton-frequency structure from [1], the second-quantized machinery from [2], the QED gauge structure from [3], and the SU(2)_L weak coupling from [4] — are the foundation on which the present paper builds.

III. The Geometric Origin of the CKM Matrix

III.1 The Mass Basis and the Weak Basis

Each quark species has two natural bases in which it can be described.

The mass eigenbasis diagonalizes the x₄-phase frequency operator. In this basis, a quark state has a definite mass m_f and a definite Compton frequency k_f. The time evolution under the free Hamiltonian (in the rest frame) is:

|q_f(t)⟩_mass = |q_f⟩ · exp(−i m_f c² t/ℏ) = |q_f⟩ · exp(−i k_f c² t)

where we use natural units with ℏ = 1 throughout. The mass-basis states propagate with definite, distinct frequencies.

The weak eigenbasis diagonalizes the SU(2)_L gauge coupling. In this basis, a quark couples to the W boson via a specific flavor index: an “up-type weak eigenstate” couples to the “down-type weak eigenstate” via W emission/absorption with unit amplitude (up to the overall weak coupling constant g_W). In LTD, as established in [4], this gauge coupling acts on the spatial triple x₁x₂x₃ transverse to x₄, and it treats the weak doublets as symmetric under SU(2)_L rotations.

III.2 Why the Two Bases Differ

The mass basis diagonalizes the x₄-phase operator (which is tied to the Compton-frequency structure, i.e. the x₄-direction). The weak basis diagonalizes the SU(2)_L gauge coupling (which acts on the spatial triple transverse to x₄). These two operators act on different geometric structures in the LTD framework — one on the x₄-phase, the other on the spatial-triple rotation group. There is no a priori reason these two diagonalizations would coincide, and in general they do not.

This is the LTD-geometric origin of the CKM matrix: it is the unitary transformation relating two bases that diagonalize operators acting on geometrically distinct structures.

III.3 Formal Definition of V in the LTD Framework

Let U_u be the unitary matrix that transforms the up-type mass eigenstates (u, c, t) to the up-type weak eigenstates. Let U_d be the corresponding matrix for down-type quarks. The CKM matrix is defined as:

V = U_u† · U_d

This is a 3 × 3 unitary matrix that tells us how a down-type quark in the weak basis (which couples diagonally to an up-type partner via W emission) is distributed over the down-type mass eigenstates.

The matrix V is unitary because U_u and U_d are both unitary (they are basis-transformation matrices). It has nine real parameters: three real mixing angles θ_12, θ_13, θ_23 and six phases. Five of the six phases can be absorbed into field redefinitions (2n − 1 = 5 for n = 3), leaving one physical phase δ.

This is all standard Standard Model content; what is distinctive about the LTD framing is that U_u and U_d are given a specific geometric interpretation: they are the basis-transformation matrices between the x₄-phase eigenbasis and the SU(2)_L eigenbasis. The values of U_u and U_d (and hence of V) are not predicted at Version 1 scope; they are measured.

III.4 The Standard CKM Parametrization

For explicit calculation, we use the standard CKM parametrization:

V = [[c_12 c_13, s_12 c_13, s_13 e^(−iδ)], [−s_12 c_23 − c_12 s_23 s_13 e^(iδ), c_12 c_23 − s_12 s_23 s_13 e^(iδ), s_23 c_13], [s_12 s_23 − c_12 c_23 s_13 e^(iδ), −c_12 s_23 − s_12 c_23 s_13 e^(iδ), c_23 c_13]]

where s_ij ≡ sin θ_ij and c_ij ≡ cos θ_ij for the three mixing angles, and δ is the CP-violating phase. This is the Wolfenstein parametrization [8] in its exact form; the approximate Wolfenstein parametrization expanded in the Cabibbo angle λ ≈ 0.22 gives a more intuitive form but the exact form is needed for quantitative work.

In the LTD framework, the three mixing angles θ_12, θ_13, θ_23 and the phase δ remain inputs — they are the angles that parametrize the mismatch between the mass and weak bases, and their specific values reflect whatever physics determines U_u and U_d individually. The LTD framework at Version 1 scope does not predict these specific values.

IV. The Compton-Frequency-Interference Mechanism

IV.1 Honest Confrontation with the Vanishing-Integrand Issue

The Dirac paper’s §X.6 noted a subtlety that deserves explicit confrontation here. If one naively attempts to compute a “Compton-frequency interference integral” by writing:

∫ dx₄ exp(i(k_u − k_c) x₄) · exp(i(k_c − k_t) x₄) · exp(i(k_t − k_u) x₄) · (mixing factors)

the exponentials’ arguments sum identically to zero: (k_u − k_c) + (k_c − k_t) + (k_t − k_u) = 0. So the integrand is 1 · (mixing factors), and the integral gives (volume) · (mixing factors) with no genuine interference. This would not produce a CP-violating phase from the Compton frequencies alone.

The resolution is that the correct framework for the calculation is not “x₄-phase integration” but meson-oscillation time evolution with complex amplitudes, which is how the Jarlskog invariant actually enters observable physics. Specifically: oscillating mesons like K^0 ↔ K̄^0 or B^0_d ↔ B̄^0_d are coherent superpositions of mass-eigenstate quark-antiquark bound states. The interference that produces CP-violation appears in the time evolution of these oscillations, where the mass differences Δm_K = m_K_L − m_K_S or Δm_B = m_B_H − m_B_L provide the phase-accumulation rate.

In the LTD framework, these mass differences ultimately come from the Compton-frequency differences of the underlying quarks (k_d, k_s, k_b for down-type; analogous for up-type) and from the mixing-matrix factors. The Jarlskog invariant emerges as the rephasing-invariant CP-violating contribution to the oscillation asymmetries, and its specific value depends on both the Compton frequencies (masses) and the mixing angles.

IV.2 The Meson Oscillation Framework

Consider a K^0–K̄^0 system. The mass eigenstates K_S and K_L are superpositions of the flavor eigenstates:

|K_S⟩ = p|K^0⟩ + q|K̄^0⟩ |K_L⟩ = p|K^0⟩ − q|K̄^0⟩

with |p|² + |q|² = 1. If CP were a good symmetry, |p| = |q| = 1/√2 exactly, and the mass eigenstates would be CP eigenstates. CP violation is the deviation: |p|² − |q|² ≠ 0, parametrized by a small complex number ε.

The mass eigenstates evolve with their own definite frequencies ω_S = m_S c² and ω_L = m_L c² (in natural units, ω_S = m_S, ω_L = m_L). The mass difference is:

Δm_K = m_L − m_S ≈ 3.48 × 10⁻¹⁵ GeV

(in natural units, which is a frequency scale). This Δm_K is the physical phase-accumulation rate for K^0–K̄^0 oscillations, and it is itself a calculable quantity — one that depends on the CKM matrix elements, the up-type quark masses running in the box diagram, and hadronic matrix elements.

IV.3 The Jarlskog Invariant as the Rephasing-Invariant CP Measure

The Jarlskog invariant J is defined to capture the CP-violating content of the CKM matrix in a rephasing-invariant way. The definition:

J ≡ Im[V_us V_cb V_ub V_cs]**

is not the only such quantity — any of the nine possible 2 × 2 subdeterminants of V has an imaginary part that captures the same information — but J is singled out as the “canonical” choice.

Theorem (Jarlskog): For any 3 × 3 unitary matrix V, the quantities |Im[V_ij V_kl V_il* V_kj*]| for i ≠ k and j ≠ l are all equal to the same value |J|, and this is the unique rephasing-invariant CP-violating quantity.

This is a purely mathematical fact about 3 × 3 unitary matrices. Its relevance here is that once we accept (from §III) that the CKM matrix is a 3 × 3 unitary matrix arising from the LTD mass-vs-weak basis mismatch, the Jarlskog invariant is automatically defined, and its value depends on the specific mixing angles and phase of V. The LTD framework does not predict V’s specific form, but it does predict that V will be a 3 × 3 unitary matrix (from the three-generation structure), and therefore that J is well-defined and generically nonzero.

IV.4 The Compton-Frequency-Interference Origin of Nonzero J

What the LTD framework does establish is the geometric origin of J being nonzero rather than identically zero. The argument is:

The mass-basis operator (diagonalizing Compton frequencies k_f) and the weak-basis operator (diagonalizing SU(2)_L couplings) act on geometrically distinct structures in LTD: the first on the x₄-direction, the second on the spatial-triple transverse to x₄. There is no geometric reason for these two diagonalizations to coincide.
With three generations, the mixing matrix V has one irreducible phase (§V), which is a real parameter that can take any value in [0, 2π). There is no geometric requirement that forces δ = 0.
Therefore, generically, J = Im[V_us V_cb V_ub* V_cs*] is nonzero.

The specific value of J is an empirical input, determined by whatever (unknown in Version 1) mechanism sets U_u, U_d individually. But the fact that J is nonzero is a consequence of the geometric structure of LTD.

IV.5 What This Establishes vs. What It Doesn’t

Established:

The CKM matrix V is a 3 × 3 unitary matrix arising from mass-basis vs weak-basis mismatch, with geometric content in LTD.
The three-generation structure produces an irreducible complex phase (Section V below).
J is generically nonzero.
The mechanism is Compton-frequency-driven (in the sense that the mass basis is defined by Compton frequencies), but operationally the CP violation is seen in meson oscillations.

Not established:

The specific numerical value of J from dx₄/dt = ic alone.
The specific values of θ_12, θ_13, θ_23, δ from first principles.
A derivation of the quark mass hierarchy.

These open items are the content of Version 2 and are the subject of §VIII.

V. The Kobayashi-Maskawa Three-Generation Requirement as a Theorem

V.1 The Counting Argument, Made Explicit

A unitary n × n matrix has n² real parameters (the complex n² entries minus the n² constraints from unitarity don’t actually reduce the count by n², but by n(n-1) — let us be careful).

More precisely: a general n × n complex matrix has 2n² real parameters. Unitarity V V† = 1 imposes n² real constraints (the matrix V V† is Hermitian with n² independent real entries, and each must equal the corresponding entry of the identity). So a unitary n × n matrix has 2n² − n² = n² real parameters.

These n² parameters split as:

Mixing angles: n(n − 1)/2 (the number of independent rotations in n dimensions)
Phases: n(n + 1)/2 (the remainder)

For n = 2: 3 phases + 1 angle = 4 parameters total. ✓ (n² = 4) For n = 3: 6 phases + 3 angles = 9 parameters total. ✓ (n² = 9) For n = 4: 10 phases + 6 angles = 16 parameters total. ✓ (n² = 16)

V.2 Rephasing Absorption

The field redefinitions that can absorb phases: each of the n “up-type” quark fields can be multiplied by an overall phase, and each of the n “down-type” quark fields can be multiplied by an overall phase. This is 2n phases. However, one overall common phase (multiplying all quarks by the same factor) is a physically unobservable global U(1), so it does not reduce the counting. The net number of absorbable phases is 2n − 1.

For n = 2: 2n − 1 = 3 phases absorbable. Physical phases = 3 − 3 = 0. No CP violation possible at two generations.

For n = 3: 2n − 1 = 5 phases absorbable. Physical phases = 6 − 5 = 1. Exactly one irreducible phase at three generations.

For n = 4: 2n − 1 = 7 phases absorbable. Physical phases = 10 − 7 = 3. Three phases at four generations.

V.3 The Geometric Content of the Counting in LTD

In the Standard Model, the counting argument is purely algebraic: it concerns the parameter space of unitary matrices. In LTD, the counting has a geometric interpretation that gives additional content.

The absorbable phases 2n − 1 correspond to the global U(1) phase rotations of each quark field — that is, to the x₄-phase rotations that the matter orientation condition (M) allows. Each quark has a well-defined overall x₄-phase that is physically unobservable (only differences in phases between quarks are observable), and redefining this overall phase absorbs one phase per quark species. This is 2n redefinitions total, minus 1 for the overall global U(1) that multiplies everything by the same factor, giving 2n − 1 absorbable phases.

The non-absorbable phases are those that cannot be removed by rephasing. For n = 3, exactly one such phase exists, and it is the CP-violating phase δ.

The three-generation requirement in LTD: For CP violation to exist, the number of physical (non-absorbable) phases in the CKM matrix must be at least 1. The counting formula gives:

Physical phases = (phases in unitary matrix) − (absorbable phases) = n(n+1)/2 − (2n − 1) = (n − 1)(n − 2)/2

For n = 1: 0 × (−1)/2 = 0 phases. No mixing matrix exists; CP violation trivially absent. For n = 2: 1 × 0 / 2 = 0 phases. CP violation impossible. For n = 3: 2 × 1 / 2 = 1 phase. CP violation possible with exactly one phase. For n = 4: 3 × 2 / 2 = 3 phases. Three CP-violating phases possible (if four generations existed).

V.4 Why Three Generations and Not Four

The LTD framework at Version 1 scope does not force exactly three generations; it forces at least three for CP violation to exist. Observed fact: three generations exist. The observation is consistent with LTD but not predicted by LTD at this scope.

Whether three is the maximum number of generations consistent with LTD is an open question. The Standard Model supplies an experimental constraint: the Z boson decays into exactly three light neutrinos (the LEP measurement of the number of light neutrino species: N_ν = 2.984 ± 0.008 [6]). This rules out any fourth generation with a neutrino lighter than m_Z/2 ≈ 45 GeV. A fourth generation with a much heavier neutrino is not ruled out by Z decay, but is constrained by other measurements. The LTD framework, to the extent developed here, does not predict the number of generations — it predicts the minimum number (three) needed for CP violation, and it predicts the structural consequences of however many generations exist.

VI. Numerical Verification at Version 1 Scope

VI.1 Experimental Inputs

We now perform the Version 1 numerical check: given the LTD geometric derivation that J is well-defined, nonzero, and computable from the CKM parametrization, we insert the experimental values of the CKM parameters and verify that the resulting J agrees with the directly measured value.

The CKM parameters, from the 2024 Particle Data Group [6]:

Parameter	Value
sin θ_12	0.22500 ± 0.00067
sin θ_13	0.00369 ± 0.00011
sin θ_23	0.04182 ± 0.00085
δ	1.144 ± 0.027 rad (≈ 65.5°)

Computing the corresponding CKM matrix elements:

c_12 = cos θ_12 = √(1 − 0.225²) = 0.97435
c_13 = cos θ_13 = √(1 − 0.00369²) ≈ 0.99999 ≈ 1
c_23 = cos θ_23 = √(1 − 0.04182²) = 0.99912

VI.2 The Four CKM Elements Entering J

Using the standard parametrization from §III.4:

V_us = s_12 c_13 = 0.22500 × 0.99999 = 0.22500

V_cb = s_23 c_13 = 0.04182 × 0.99999 = 0.04182

V_ub = s_13 e^(−iδ) = 0.00369 × (cos 1.144 − i sin 1.144) = 0.00369 × (0.4140 − 0.9103 i) = 0.001528 − 0.003359 i

Check: |V_ub| = √(0.001528² + 0.003359²) = √(2.33 × 10⁻⁶ + 1.128 × 10⁻⁵) = √(1.36 × 10⁻⁵) = 0.003690 ✓

V_cs = c_12 c_23 − s_12 s_23 s_13 e^(iδ)

The second term is small: s_12 s_23 s_13 = 0.22500 × 0.04182 × 0.00369 = 3.473 × 10⁻⁵. With e^(iδ) = cos 1.144 + i sin 1.144 = 0.4140 + 0.9103 i, this gives s_12 s_23 s_13 e^(iδ) = (1.438 × 10⁻⁵) + (3.161 × 10⁻⁵) i.

And c_12 c_23 = 0.97436 × 0.99912 = 0.97350.

So V_cs = 0.97350 − 1.438 × 10⁻⁵ − 3.161 × 10⁻⁵ i ≈ 0.97349 − 3.161 × 10⁻⁵ i.

VI.3 Computing J

J = Im[V_us V_cb V_ub V_cs]**

Product step by step:

V_us · V_cb = 0.22500 × 0.04182 = 0.009410 (real)

V_ub* = 0.001528 + 0.003359 i (complex conjugate of V_ub)

V_cs* = 0.97349 + 3.161 × 10⁻⁵ i (the imaginary part is negligible at the precision of the Jarlskog calculation)

V_ub* · V_cs* ≈ 0.97349 × (0.001528 + 0.003359 i) = 0.001487 + 0.003270 i

V_us · V_cb · V_ub* · V_cs* = 0.009410 × (0.001487 + 0.003270 i) = 1.399 × 10⁻⁵ + 3.077 × 10⁻⁵ i

|J| = Im[V_us V_cb V_ub V_cs] = 3.077 × 10⁻⁵**

Cross-check via the compact formula. The Jarlskog invariant has a well-known compact expression in the standard parametrization:

J = s₁₂ c₁₂ s₁₃ c₁₃² s₂₃ c₂₃ sin δ

Plugging in the PDG values:

J = 0.22500 × 0.97436 × 0.00369 × (0.99999)² × 0.04182 × 0.99912 × sin(1.144) = 0.22500 × 0.97436 × 0.00369 × 0.99998 × 0.04182 × 0.99912 × 0.9103 = 3.077 × 10⁻⁵

The compact formula agrees exactly with the direct computation.

VI.4 Comparison with Experiment

The directly measured value [6]:

|J|_exp = (3.08 ± 0.14) × 10⁻⁵

Our computation:

|J|_LTD Version 1 = 3.08 × 10⁻⁵

The agreement is excellent: to three significant figures, the LTD Version 1 calculation reproduces the directly measured Jarlskog invariant.

This is, of course, not a surprising result: we have taken the experimental CKM parameters and inserted them into the standard CKM parametrization. What the calculation establishes is not a new prediction, but the consistency of the LTD Version 1 framework with experiment. The LTD framework predicts that V is a 3 × 3 unitary matrix with one irreducible phase; that J, defined by the rephasing-invariant formula, is nonzero; and that its numerical value, computed from the parametrization, is whatever experiment measures. All three predictions are confirmed.

VI.5 What the Numerical Verification Shows

The verification shows three things:

The LTD framework is consistent with observed CP violation. If the LTD framework had predicted (incorrectly) that J = 0 exactly, or that V is not a 3 × 3 unitary matrix, the verification would have failed. Both the nonzero value and the 3 × 3 structure are correctly reproduced.
The Kobayashi-Maskawa three-generation requirement is correctly derived. The counting formula (n − 1)(n − 2)/2 = 1 for n = 3 gives exactly one physical phase, and this phase is what generates |J| ≈ 3 × 10⁻⁵ via the standard parametrization.
The parametric structure of the Jarlskog invariant is correctly reproduced. The formula J = Im[V_us V_cb V_ub* V_cs*] follows from the LTD-derived fact that V is 3 × 3 unitary, and the numerical value matches experiment.

What the verification does not show:

It does not predict |J| from first principles in LTD.
It does not derive the specific mixing angles or the phase δ.
It does not resolve the mass hierarchy problem.

These are Version 2 items. The present paper’s Version 1 scope delivers what it claims: consistency, parametric correctness, and geometric origin of the mechanism; not a first-principles value for |J|.

VII. What Has Been Established and What Has Not

To forestall misunderstanding about the scope of this paper’s claims, we summarize explicitly.

VII.1 What Is Established by This Paper

The CKM matrix has a geometric origin in LTD (§III). V = U_u† U_d is the overlap between the mass-eigenstate basis (diagonalizing x₄-phase frequencies via condition (M) from [1]) and the weak-eigenstate basis (diagonalizing SU(2)_L gauge coupling from [4]). Because these two operators act on geometrically distinct structures in LTD (x₄-direction vs. spatial-triple), the two bases generically differ.
CP violation has a geometric origin in LTD (§IV). The mismatch between mass and weak bases produces mixing; with three generations, the mixing matrix has one irreducible complex phase; this phase is the source of CP violation. The mechanism is Compton-frequency-driven in the sense that the mass basis is defined by Compton frequencies k_f = m_f c/ℏ.
The three-generation requirement is derived as a theorem (§V). The number of physical phases in an n × n unitary mixing matrix is (n−1)(n−2)/2. This vanishes for n ≤ 2 and equals 1 for n = 3. CP violation requires n ≥ 3, with the minimum case n = 3 giving exactly one irreducible phase.
Numerical consistency with experiment is verified (§VI). Inserting the experimental CKM parameters (sin θ_12, sin θ_13, sin θ_23, δ) into the standard parametrization gives |J|_LTD = 3.08 × 10⁻⁵, matching the direct measurement |J|_exp = (3.08 ± 0.14) × 10⁻⁵ to within three significant figures.

VII.2 What Is Not Established by This Paper

The specific values of the CKM mixing angles θ_12, θ_13, θ_23. These remain experimental inputs at Version 1 scope. Deriving them from the quark masses alone would be Version 2.
The specific value of the CP-violating phase δ. This also remains an experimental input. Deriving δ from the quark masses would be Version 2.
A first-principles prediction of |J| from the principle dx₄/dt = ic alone. The paper establishes that |J| can be computed from the CKM parameters and matches experiment; it does not predict |J| in absolute terms.
The quark mass hierarchy. The six quark masses are experimental inputs; deriving their specific ratios from the LTD framework is an open problem.
The number of generations being exactly three. LTD forces at least three generations for CP violation to exist; it does not predict that exactly three exist.

VII.3 Framing the Deliverables

A reader might reasonably ask: “Does this paper deliver a real prediction, or does it just show that the LTD framework is consistent with known CP violation?” The honest answer is: the latter, strictly speaking. At Version 1 scope, the paper does not deliver a genuinely new quantitative prediction.

But the paper does deliver the following structural predictions, which would be falsifiable by observations that contradicted them:

CP violation must come from a single irreducible phase in an n × n unitary mixing matrix (rather than from some entirely different mechanism like parity doubling or CPT violation). Falsifiable status: consistent with all current data.
Three generations must exist if CP violation is observed (where “three” means “at least three”; four would also be compatible). Falsifiable status: consistent with observed three generations.
The mass eigenbasis and weak eigenbasis must not coincide (generically, given that they diagonalize geometrically distinct operators). Falsifiable status: consistent with observed CKM matrix being non-diagonal.
Strong CP violation must be absent (from the symmetric action of x₄-expansion across x₁x₂x₃, per [4] §VII). Falsifiable status: consistent with θ_QCD < 10⁻¹⁰.

None of these are genuinely new predictions in the sense of predicting novel phenomena; they are predictions about the structure of known phenomena. The LTD framework explains why things are the way they are (Version 1), without yet predicting specific numerical values that differ from the Standard Model’s measured values (Version 2).

The accumulation of successful Version 1 derivations — the Dirac equation, second quantization, QED, CKM phase origin, and the three-generation requirement — is the evidence for the LTD framework’s structural reach. Version 2 remains open work.

VIII. Open Problems and the Path Toward Version 2

Version 2 of the Jarlskog derivation — predicting the mixing angles θ_12, θ_13, θ_23 and the phase δ from the quark masses alone — would constitute a genuine parameter reduction from the Standard Model’s nine CKM parameters to the six quark masses. We outline what would be needed to deliver it, without claiming to deliver it here.

VIII.1 The Geometric Structure That Version 2 Would Need

Version 2 requires a geometric mechanism that determines U_u and U_d individually — not just V = U_u† U_d. The mechanism must tie these rotation matrices to the quark masses in a specific, testable way.

Candidate approaches:

Candidate 1: Mass-ratio hierarchies fixing rotation angles. The quark masses exhibit a strong hierarchy: m_u ≪ m_c ≪ m_t, and m_d ≪ m_s ≪ m_b. If the mass-basis-to-weak-basis rotation angles are determined by ratios like √(m_d/m_s) ≈ 0.22 (close to the Cabibbo angle!) or m_s/m_b ≈ 0.02, a quantitative match might emerge. The traditional approach to deriving CKM mixing from mass hierarchies goes back to Gatto, Fritzsch, and others [9, 10]. In the LTD framework, one might ask whether the mass hierarchy itself has a geometric origin — e.g., from some discrete-ladder structure in the Compton frequencies — that simultaneously fixes the mixing angles.

Status: speculative. The Cabibbo angle sin θ_12 ≈ 0.22 is numerically close to √(m_d/m_s) ≈ 0.224 at the level of coincidence, but deriving this relation rigorously from dx₄/dt = ic has not been done.

Candidate 2: Grand unified structure. If the quarks and leptons are unified at a high scale via some LTD-compatible GUT structure, the mixing angles might be fixed by the representation-theoretic structure of the GUT group. This is analogous to how standard GUTs fix Yukawa coupling ratios, but would require LTD-compatible GUT machinery that has not been developed.

Status: more speculative than Candidate 1.

Candidate 3: Compton-frequency ladder structure. In the LTD framework, a natural question is whether the Compton frequencies k_f of the quarks and leptons form a specific mathematical ladder — e.g., a geometric or arithmetic progression with a specific ratio — that fixes both the mass values and the mixing angles. Observationally, the quark masses do not obviously form such a ladder (the hierarchy is too steep), but some nontrivial functional relationship may exist.

Status: speculative; no current evidence.

VIII.2 What a Successful Version 2 Would Predict

If any of these approaches succeeded, it would predict |J| from the six quark masses as a definite number (rather than requiring the experimental mixing angles as input). The prediction could then be compared to the measured |J| = (3.08 ± 0.14) × 10⁻⁵. Agreement to within experimental error would be strong evidence for the framework; significant disagreement would falsify the specific mechanism used.

This is the kind of predictive test that would move the LTD framework from “consistent alternative foundation” to “distinguished from the Standard Model by quantitative predictions.” The present paper does not provide it.

VIII.3 Other Open Work

Beyond Version 2, several other open items in the LTD program connect naturally to this paper’s content:

The quark mass hierarchy. Why are the masses m_u, m_c, m_t spread over four orders of magnitude, and similarly for the down-type quarks? No LTD-derived explanation currently exists.
The lepton CKM (PMNS) matrix. Neutrinos mix with a separate mixing matrix U_PMNS, with its own irreducible phases. An analogous LTD derivation should apply, but the PMNS matrix has different phenomenology (e.g., possible Majorana phases if neutrinos are Majorana particles), and the full LTD treatment of neutrinos is not developed.
Electroweak symmetry breaking and the Higgs mechanism. The Higgs mechanism that gives mass to the W and Z bosons and to the fermions operates via a specific mechanism in the Standard Model (spontaneous breaking of SU(2)_L × U(1)_Y). The LTD version, sketched in [4], is not fully worked out.
Hadronic matrix elements. The experimental determination of CKM matrix elements involves calculating hadronic matrix elements in QCD — itself an unsolved problem in most cases. An LTD-derived QCD would be needed for a genuinely complete alternative foundation.

These are all open. The present paper does not address them.

IX. Conclusion

The CKM matrix and the Jarlskog invariant have been given a geometric origin in Light, Time, Dimension Theory (LTD) — whose foundational principle is the McGucken Principle dx₄/dt = ic, the principle that the fourth dimension is expanding at the rate of light — at Version 1 scope. Specifically:

The CKM matrix arises as the overlap V = U_u† U_d between the mass-eigenstate basis (diagonalizing x₄-phase frequencies via condition (M) from the Dirac paper [1]) and the weak-eigenstate basis (diagonalizing SU(2)_L gauge coupling from the broken-symmetries paper [4]). The two bases differ generically because they diagonalize operators acting on geometrically distinct structures — the x₄-direction versus the spatial triple transverse to x₄.

CP violation is the irreducible complex phase in the 3 × 3 unitary CKM matrix. Its origin is the Compton-frequency-driven structure of the mass basis combined with the SU(2)_L-driven structure of the weak basis.

The three-generation requirement is derived as a geometric theorem: the number of physical phases in an n × n unitary mixing matrix is (n − 1)(n − 2)/2, which vanishes for n ≤ 2 and equals 1 for n = 3. CP violation requires n ≥ 3.

Numerical verification shows that the LTD Version 1 framework reproduces the experimental |J| = (3.08 ± 0.14) × 10⁻⁵ to ~0.3%, using the experimentally measured mixing angles and phase as inputs. The framework is consistent with observation, with the structural predictions (unitary 3 × 3 mixing matrix, single irreducible phase, geometric origin of mass-vs-weak-basis mismatch) all confirmed.

What remains open is Version 2: deriving the specific mixing angles and phase from the quark masses alone, which would constitute genuine parameter reduction from the Standard Model. Three candidate approaches are outlined (§VIII) — mass-ratio hierarchies, GUT-compatible unified structure, Compton-frequency ladders — but none is currently worked out to the level of a concrete quantitative prediction. This is the natural next step in the LTD program’s development of the quark sector.

The present paper adds one more rung to the LTD derivation ladder. The single-particle Dirac equation [1] established spin-½ and the matter-antimatter structure; the second-quantization paper [2] established fermion statistics and the Pauli exclusion principle; the QED paper [3] established the U(1) gauge structure and Maxwell’s equations. The present paper now adds the geometric origin of the CKM matrix and the Jarlskog-invariant structure of CP violation. Each rung is a Version 1 result at this stage — deriving the structural origin of phenomena without yet reducing the parameter count below the Standard Model. The cumulative structural reach of the framework — the Dirac equation, second quantization, QED, and now CP violation all from the McGucken Principle dx₄/dt = ic alone — is the evidence on which the LTD program’s claim to be a foundational theory rests.

Version 2 of each of these derivations remains open work. The accumulation of Version 1 successes, combined with the concrete path forward to Version 2 in each sector, is the standard by which the program should be judged.

Historical Note: The Origin of the McGucken Principle

“More intellectual curiosity, versatility and yen for physics than Elliot McGucken’s I have never seen in any senior or graduate student. Originality, powerful motivation, and a can-do spirit make me think that McGucken is a top bet for graduate school in physics. He could and did, and wrote it all up in a beautifully clear account.”

— Dr. John Archibald Wheeler, Joseph Henry Professor of Physics, Princeton University [11]

The McGucken Principle traces to Dr. Elliot McGucken’s undergraduate research with John Archibald Wheeler at Princeton University in the late 1980s and early 1990s. Two Wheeler-supervised projects — an independent derivation of the time factor in the Schwarzschild metric, and a study of the Einstein-Podolsky-Rosen paradox and delayed-choice experiments — planted the seeds of the theory. The first written formulation appeared in an appendix to McGucken’s 1998 NSF-funded UNC Chapel Hill dissertation on an artificial retina chipset [12]. The principle appeared on internet physics forums (2003–2006) as Moving Dimensions Theory, received formal treatment in five FQXi papers (2008–2013) [13–17], was consolidated in seven books (2016–2017) [18–23], and has been extensively developed at elliotmcguckenphysics.com (2024–2026).

The present paper extends the program to the quantitative treatment of CP violation and the CKM matrix structure, establishing that the three-generation requirement for CP violation is a geometric theorem from the LTD framework and that the Jarlskog invariant structure is reproduced consistently, while being explicit about the Version 1 scope limitation.

Appendix A: Prerequisites from the Companion Papers

For independent readability, we sketch the prerequisites from the four companion papers [1]–[4] that this paper rests on.

A.1 From [1]: The Matter Orientation Condition (M) and Compton Frequencies

Each fermion species f with rest mass m_f carries x₄-phase at the Compton frequency k_f = m_f c/ℏ. Mass eigenstates are by definition states with definite k_f. Different fermion species have different k_f values; the spread is six orders of magnitude for the quark sector. Full derivation in [1] §II.2 and the matter orientation condition (M) in §IV.2.

A.2 From [2]: The Second-Quantized Dirac Field

The second-quantized Dirac field operator creates localized x₄-standing-wave modes. In the mass basis, the creation operators a†_{f,p,s} produce quarks of species f; in the weak basis, a different linear combination applies. The second-quantization framework provides the machinery to compute mixing amplitudes and oscillation probabilities. Full construction in [2] §VII.

A.3 From [3]: QED and the U(1) Gauge Structure

Electromagnetic coupling is diagonal in mass basis (electric charge is a good quantum number independent of mixing). The QED gauge structure is therefore not the source of quark mixing; it is flavor-diagonal. Quark mixing comes from the weak interaction. Full treatment in [3].

A.4 From [4]: The Weak Interaction and SU(2)_L

The weak interaction couples through SU(2)_L — the rotation group transverse to x₄-expansion. Under SU(2)_L, left-chiral matter fields form doublets (u_L, d_L), (c_L, s_L), (t_L, b_L) in the weak basis. The mismatch between this weak basis and the mass basis (diagonalizing x₄-phase frequencies) is the origin of the CKM matrix. Full treatment in [4] §III–V.

A.5 What These Prerequisites Provide for the Present Paper

A.1 provides the Compton frequencies k_f that define the mass basis (§III.1).
A.2 provides the field-operator machinery that allows computation of oscillation amplitudes (§IV.2).
A.3 confirms that electromagnetism is not the source of mixing (§II.3).
A.4 provides the SU(2)_L gauge coupling that defines the weak basis (§III.1).

A reader convinced of these four prerequisites can verify the derivations of the present paper without consulting [1]–[4]. Readers seeking full derivations should consult the companion papers.

References

[1] McGucken, E. The Geometric Origin of the Dirac Equation: Spin-½, the SU(2) Double Cover, and the Matter-Antimatter Structure from the McGucken Principle of a Fourth Expanding Dimension dx₄/dt = ic. elliotmcguckenphysics.com (April 2026).

[2] McGucken, E. Second Quantization of the Dirac Field from the McGucken Principle of a Fourth Expanding Dimension dx₄/dt = ic: Creation and Annihilation Operators as x₄-Orientation Operators, Fermion Statistics as a Theorem, and Pair Processes as x₄-Orientation Flips. elliotmcguckenphysics.com (April 2026).

[3] McGucken, E. Quantum Electrodynamics from the McGucken Principle of a Fourth Expanding Dimension dx₄/dt = ic: Local x₄-Phase Invariance, the U(1) Gauge Structure, Maxwell’s Equations, and the QED Lagrangian. elliotmcguckenphysics.com (April 2026).

[4] McGucken, E. How the McGucken Principle of the Fourth Expanding Dimension (dx₄/dt = ic) Accounts for the Standard Model’s Broken Symmetries, Time’s Arrows and Asymmetries, and Much More. elliotmcguckenphysics.com (April 2026).

[5] Jarlskog, C. Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Nonconservation. Phys. Rev. Lett. 55, 1039 (1985).

[6] Workman, R. L. et al. (Particle Data Group). Review of Particle Physics. Prog. Theor. Exp. Phys. 2024, 083C01 (2024). CKM matrix values and uncertainties from the 2024 PDG edition.

[7] Kobayashi, M. & Maskawa, T. CP Violation in the Renormalizable Theory of Weak Interaction. Prog. Theor. Phys. 49, 652 (1973).

[8] Wolfenstein, L. Parametrization of the Kobayashi-Maskawa Matrix. Phys. Rev. Lett. 51, 1945 (1983).

[9] Fritzsch, H. Calculating the Cabibbo Angle. Phys. Lett. B 70, 436 (1977).

[10] Gatto, R., Sartori, G. & Tonin, M. Weak Self-Masses, Cabibbo Angle, and Broken SU(2) × SU(2). Phys. Lett. B 28, 128 (1968).

[11] Wheeler, J. A. Letter of Recommendation for Elliot McGucken. Princeton University, Department of Physics (late 1980s / early 1990s).

[12] McGucken, E. Multiple Unit Artificial Retina Chipset to Aid the Visually Impaired and Enhanced Holed-Emitter CMOS Phototransistors. Ph.D. Dissertation, University of North Carolina at Chapel Hill (1998).

[13] McGucken, E. Time as an Emergent Phenomenon: Traveling Back to the Heroic Age of Physics. FQXi Essay Contest, August 2008. https://forums.fqxi.org/d/238

[14] McGucken, E. What is Ultimately Possible in Physics? Physics! A Hero’s Journey… towards Moving Dimensions Theory. FQXi Essay Contest, September 2009. https://forums.fqxi.org/d/511

[15] McGucken, E. On the Emergence of QM, Relativity, Entropy, Time, iℏ, and ic from the Foundational, Physical Reality of a Fourth Dimension. FQXi Essay Contest, February 2011. https://forums.fqxi.org/d/873

[16] McGucken, E. MDT’s dx₄/dt = ic Triumphs Over the Wrong Physical Assumption that Time is a Dimension. FQXi Essay Contest, August 2012. https://forums.fqxi.org/d/1429

[17] McGucken, E. It from Bit or Bit From It? What is It? Honor!. FQXi Essay Contest, July 2013. https://forums.fqxi.org/d/1879

[18] McGucken, E. Light Time Dimension Theory: The Foundational Physics Unifying Einstein’s Relativity and Quantum Mechanics. 2016. ASIN: B01KP8XGQ6.

[19] McGucken, E. Relativity and Quantum Mechanics Unified in Pictures. 2017. ASIN: B01N2BCAWO.

[20] McGucken, E. The Physics of Time: Time & Its Arrows in Quantum Mechanics, Relativity, The Second Law of Thermodynamics, Entropy, The Twin Paradox, & Cosmology Explained via LTD Theory’s Expanding Fourth Dimension. 2017. ASIN: B07695MLYQ.

[21] McGucken, E. Quantum Entanglement and Einstein’s Spooky Action at a Distance Explained: The Nonlocality of the Fourth Expanding Dimension. 2017. ASIN: B076BTF6P3.

[22] McGucken, E. Einstein’s Relativity Derived from LTD Theory’s Principle. 2017. ASIN: B06WRRJ7YG.

[23] McGucken, E. The Triumph of LTD Theory and Physics over String Theory, the Multiverse, Inflation, Supersymmetry, M-Theory, LQG, and Failed Pseudoscience. 2017. ASIN: B01N19KO3A.

Submitted to elliotmcguckenphysics.com, April 2026.

Author: Elliot McGucken, PhD — Theoretical Physics Undergraduate research with John Archibald Wheeler, Princeton University (late 1980s) Ph.D., University of North Carolina at Chapel Hill (1998) Website: elliotmcguckenphysics.com