A Geometric CPVSM Framework: Dynamical Quark Mass Spurt with Explicit CP Violation in the Standard Mode

Chilong Lin

REVIEW 5 major objections 5 minor 35 references

A Geometric CPVSM Framework: Dynamical Quark Mass Spurt with Explicit CP Violation in the Standard Mode

T0 review · 5 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper claims that the CKM quark-mixing matrix and the size of CP violation follow from a 5-parameter geometric baseline in which each sector's mass-squared matrix has commuting real and imaginary parts, reducing to two ratio coordinates

desk verdict The headline J is a circular fit to the same CKM magnitudes that determine J, and the model's own leading order forces an excluded CKM pattern; the BAU claim is asserted without dynamics. read the letter →

arxiv 2607.26681 v2 pith:4HI6HQXQ submitted 2026-07-29 hep-ph hep-th

Chilong Lin This is my paper · ORCID

classification hep-phhep-th

keywords CKMmatrixJarlskoginvariantCPviolationmassdiagonalizationflavorgeometryquarkhierarchysimultaneousbaryonasymmetry

topics Matter-Antimatter Asymmetry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to show that quark flavor mixing and CP violation are not arbitrary parameters but the output of a strict geometric baseline: when the real and imaginary parts of each sector's mass-squared matrix commute, the 9 independent parameters collapse to 5, and the diagonalizing matrices depend only on two dimensionless ratio coordinates (x,y) per sector. The resulting CKM matrix has exact analytic elements, and the Jarlskog invariant reduces to a closed-form expression that evaluates to about 3.08e-5 — matching the measured value — for all 32 fitted parameter sets. If correct, this would mean the hierarchical CKM structure and the observed CP violation have an algebraic origin, not a dynamical one. The same commutativity, however, forces four CKM magnitudes to be exactly equal (for instance |V_ub|=|V_cb|), which high-precision measurements already rule out; the paper therefore presents the baseline as a leading-order scaffold and defers the required non-commuting corrections to future work.

What carries the argument

The load-bearing object is the 5-parameter commuting mass-squared pattern (Eq. 10), obtained by imposing the simultaneous-diagonalization hypothesis [M_R^2,M_I^2]=0 (Eq. 8). It yields an exact factorization of the characteristic polynomial into linear and quadratic factors, giving closed-form eigenvalues (Eqs. 14) and a scale-free, analytically known unitary U(x,y) (Eq. 16) that is independent of the energy scales A,B,C. The CKM matrix is the product of two such unitaries, and the closed-form Jarlskog invariant J(x,y,x',y') in Eq. (23) carries the entire CP-violating content: it vanishes unless the two ratio vectors (x,y) and (x',y') are non-collinear.

What would settle it

Existing precision data provide a concrete falsifier: the fourfold equality |V_ub|=|V_cb|=|V_td|=|V_ts| is excluded by the measured values (|V_ub|≈3.8e-3, |V_cb|≈0.041), so the exact commutativity hypothesis in Eq. (8) is already ruled out as a full description; a decisive test would be a high-precision measurement that narrows |V_ub|/|V_cb| away from unity, or a measurement of another CP-violating phase that cannot be reproduced by any choice of (x,y,x',y') within the commuting baseline.

Watch

Extended reading notes

Core claim

The central discovery is an exact dimensional reduction: for a Hermitian mass-squared matrix M^2 = M_R^2 + i M_I^2, the requirement [M_R^2, M_I^2]=0 (Eq. 8) is algebraically equivalent to four constraints that collapse the 9 real parameters to 5 (A,B,C,x,y) and make the diagonalizing unitary U depend only on the two ratios x and y (Eq. 16). The physical CKM matrix is then U_u(x,y)^\dagger U_d(x',y'), and its Jarlskog invariant takes the closed form J(x,y,x',y') in Eq. (23); evaluated on the fitted ratios it converges to |J0| ≈ 3.08e-5, in agreement with the measured world average. The same algebra enforces |p|=|p'|=|p*|=|p'*|, meaning four CKM elements share one magnitude — e.g., |V_ub|=|V_c

Load-bearing premise

The whole framework rests on the assumption that in each quark sector the real and imaginary parts of the mass-squared matrix commute, [M_R^2, M_I^2]=0; if that fails, the 9→5→2 reduction, the closed-form CKM elements, and the fourfold degeneracy all disappear. The paper's own numerical fit shows this premise is already violated by the measured magnitudes, since it forces |V_ub| = |V_cb|.

Editorial extensions

If this is right

If the baseline holds at leading order, the CKM matrix is fixed by four dimensionless ratios rather than three angles and one phase, giving an explicit algebraic form for every matrix element.
The Jarlskog invariant is determined solely by the non-collinearity (xy'−x'y) of the two sector vectors, so all 18 active flavor-permutation patterns yield the same |J0| ≈ 3.08e-5.
The commutativity hypothesis forces a fourfold magnitude degeneracy that conflates different Cabibbo-suppression orders (λ^2 with λ^3), and lifting it requires non-commuting quantum corrections.
Near the critical boundary xy→0, the product of mass-squared differences diverges as (xy)^(−3), producing a dynamical quark mass spurt that the paper argues could generate the baryon asymmetry without leptogenesis.
Applied to charged leptons and Dirac neutrinos, the same two-vector construction yields large PMNS mixing angles through large angular offsets, unifying the geometric mechanism across sectors.
The exact degeneracy predictions will be tested by future high-precision CKM measurements; even if the leading-order magnitudes fail, the closed-form J may survive as a robust invariant.
A concrete baryogenesis calculation from the mass spurt dynamics would turn the qualitative BAU claim into a quantitative prediction that could be checked against η_B ≈ 6e-10.
If the non-commuting corrections are small, the ratio of J to (xy'−x'y) should remain approximately universal, offering a way to discriminate this geometric origin from other flavor models.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: this is a competent piece of algebra in search of a physical result. The reduction of a Hermitian 3x3 mass-squared matrix to a commuting real/imaginary pair is carried out cleanly, and the factorization of the characteristic polynomial, the trace consistency, and the closed-form Jarlskog expression all check out. I verified the key identities at representative values; they hold.

What's new: the explicit argument that the diagonalizing matrix U depends only on the two ratio coordinates (x,y) and not on the mass scales A,B,C is a nice corollary of simultaneous diagonalization. The 9→5→2 counting is explained more carefully than in the author's previous papers. And the PMNS extension is sketched in a way that could be useful if the quark-sector baseline were viable.

Where it fails: First, the central numerical result is circular. Section III.D.2 says the four parameters are fitted to the experimental CKM magnitudes, then inserted into Eq. (23) to 'recover' J≈3.08e-5. For any unitary 3x3 matrix, |J| is a function of the magnitudes (twice the area of the unitarity triangle). So a fit to the magnitudes will reproduce J_exp to within fit error. This is not a prediction; it's a consistency check. The paper itself admits the parameters are fitted to CKM elements, so the 'robust numerical convergence' is forced.

Second, the model's own leading order makes a falsifiable prediction that fails: Eq. (28) forces |V_ub|=|V_cb|=|V_td|=|V_ts|. The measured values differ by an order of magnitude. The paper acknowledges this and defers the cure to non-commuting corrections. That is honest, but it means the baseline is not a valid leading-order description of the CKM matrix.

Third, the baryogenesis claim is unsupported. The paper asserts that the quark mass spurt alone is sufficient to generate BAU, and cites a high-temperature J(S2)=4√3/81≈0.0855 (Eq. 25) with no derivation and no dynamics — no potential, no Boltzmann equations, no η_B. The mass spurt divergence (Eq. 27) and the fate of J in the xy→0 limit are not actually connected to an out-of-equilibrium production rate. As it stands, this is a placeholder for future work, not a result.

The credit: the paper is transparent about its foundational assumption, explicitly states the degeneracy problem, and does not hide that corrections are needed. It also has a clear citation pattern: most of the load-bearing results come from the author's own prior papers. Self-citation isn't automatically a flaw, but here the new text mostly restates Refs. [9] and [19] with a different framing.

Who this is for: someone interested in the algebra of simultaneously diagonalizable mass matrices might find the parametrization elegant, but that audience is narrow. A phenomenology reader will see the circular fit and the magnitude degeneracy immediately. I don't think this deserves a serious referee — it would be desk-rejected for the circularity alone — though it could be a teaching example of why fitting magnitudes then extracting J proves nothing.

My recommendation: do not send this to peer review. If the authors add actual dynamics and a non-circular prediction, a resubmission could be worth another look.

Referee Report

5 major / 5 minor

Summary. The paper proposes a 'geometric CPVSM' framework in which the CKM matrix is built from single-sector mass-squared matrices M^2 = M_R^2 + i M_I^2 satisfying the commutativity condition [M_R^2, M_I^2] = 0 (Eq. 8). Under this assumption, each sector's matrix reduces to a 5-parameter pattern (A,B,C,x,y), and the diagonalizing unitary U(x,y) is scale-free. Combining up- and down-type sectors gives V_CKM = U_u^† U_d, whose elements are expressed through five geometric components r,s,p,p',q (Eqs. 21); the non-vanishing Jarlskog invariant is given in closed form by Eq. (23). The paper claims that fitting (x,y,x',y') to measured CKM magnitudes yields J ≈ 3.08×10^-5 (Eq. 30), that the commutativity hypothesis forces the fourfold degeneracy |p|=|p'|=|p*|=|p'*| (Eq. 28), that near xy→0 a 'quark mass spurt' (Eq. 27) generates mass hierarchies, and that this—together with a high-temperature value J(S2)=4√3/81≈0.0855 (Eq. 25)—can explain the baryon asymmetry of the universe. The framework is also sketched for the PMNS matrix with Dirac neutrinos.

Significance. If the construction were valid, it would be a striking result: an exactly solvable mass-basis algebra that reduces the CKM matrix to two geometric coordinates per sector, yields an analytic Jarlskog invariant, and connects low-energy CP violation to baryogenesis. The algebraic core is partially sound: I checked that the trace identity Eq. (12) is consistent with the sum of eigenvalues Eq. (15), the characteristic-polynomial factorization Eq. (13) follows from the stated eigenvalues, and the mass-spurt product formula Eq. (27) is correctly derived from Eqs. (14). These are genuine technical steps. However, the central phenomenological claims do not survive scrutiny: the J-match is obtained by construction because the four parameters are fitted to the CKM matrix itself; the unavoidable fourfold degeneracy contradicts measured CKM magnitudes; and the baryogenesis claim rests on an unproven S2 value and no dynamical calculation. The paper is therefore not acceptable as a leading-order description of quark flavor.

major comments (5)

[§III.C.4, §III.D.2, Eqs. (23) and (30)] The claimed numerical convergence to J≈3.08×10^-5 is circular. By the paper's own protocol, (x,y,x',y') are 'fitted to the experimental values of the CKM matrix elements' (§III.C.4) and are 'optimal geometric parameter solutions fitted to experimental CKM magnitudes' (§III.D.2). Inserting these fitted parameters into Eq. (23) and recovering the PDG value is a consistency check, not a prediction. For any unitary 3×3 matrix, the Jarlskog invariant is a function of the CKM elements (twice the area of the unitarity triangle); once the magnitudes are fitted, J is essentially fixed. The 'universal convergence' across 32 parameter sets reflects that all fits describe the same experimental magnitudes. A predictive statement would require fixing (x,y,x',y') from independent observables, e.g., quark masses or CP asymmetry measurements, not from the CKM matrix itself.
[§III.D.1, Eqs. (28)–(29)] The core hypothesis Eq. (8) leads to the exact identity |p|=|p'|=|p*|=|p'*| (Eq. 28), which in the standard assignment gives Eq. (29): |V_ub|=|V_cb|=|V_td|=|V_ts|. This is in strong tension with data: |V_ub|≈3.8×10^-3 versus |V_cb|≈0.041. The paper itself acknowledges 'severe numerical tension' and that the fit is driven to a compromise at O(λ^2.5). Since Eq. (8) is the sole foundational hypothesis, this is not a minor blemish: the leading-order baseline does not reproduce the hierarchical structure of the CKM matrix. The proposed escape—non-commuting corrections [M_R^2,M_I^2]≠0—is explicitly deferred to future work (§IV.B) and cannot rescue the present claims.
[§III.C.3, Eq. (25)] The high-temperature value J(S2)=4√3/81≈0.0855 is asserted without derivation. The text speaks of 'evaluating the topological indices' but gives no definition of the S2-symmetric phase limit, no calculation, and no derivation connecting it to the model's algebraic structure in Eq. (23). Since this value is used to support the baryogenesis claim, it is load-bearing. Merely noting that it lies below the general unitarity bound 1/(6√3) (Eq. 26) is insufficient.
[§III.C.3, §V.C, Eq. (27)] The baryogenesis conclusion—'the dynamical quark mass spurt alone is sufficient to generate BAU'—is unsupported by any quantitative calculation. Eq. (27) is a static algebraic identity for the product of mass-squared differences; it diverges as xy→0, but no equation of motion, finite-temperature potential, Boltzmann equation, or out-of-equilibrium/sphaleron condition is supplied. The claim that vector non-collinearity is preserved in a 'non-equilibrium transition zone' is asserted with a citation to Ref. [19] but no mechanism. The paper also states η_B ~ 6×10^-10 without deriving it from J(x,y,x',y') or the mass spurt. Without a concrete production calculation, the BAU result is not established.
[§II.C, §IV.B, Introduction] The Introduction presents the dimensional reduction 9→5→2 as an exact algebraic consequence of 'spatial rotational invariance and global phase symmetry,' holding even without discrete flavor symmetries. The actual derivation in §II.C, however, relies on the commutativity ansatz Eq. (8), which is a restrictive structural constraint on M^2—not a symmetry of the SM Lagrangian. The paper itself calls this the 'sole, foundational hypothesis' (§II.C). The reduction is exact only within this ansatz, and the overstatement in the Introduction/abstract should be corrected.

minor comments (5)

[Tables I and III] The tables would benefit from a check against the definitions in Eqs. (21); at least one entry in Table III appears to be a duplication typo (last row, third column). A careful proofreading pass is needed.
[§III.D.2] The 32 fitted parameter sets from Ref. [19] are not reproduced, nor is the fit quality (χ² or residuals). This prevents an independent check of Eq. (30). The paper should either include the parameter table and fit procedure or make the underlying data/code available.
[§IV.C–D] The PMNS extension is qualitative: no numerical predictions for θ_12, θ_23, θ_13, or δ_CP are derived. As presented, it is a framing suggestion rather than a testable result.
[Eq. (23)] The expression for J is written with a bar around the whole rational function, but the right-hand side does not carry an explicit absolute value. Clarify whether Eq. (23) defines the signed or absolute Jarlskog invariant.
[References] The manuscript depends heavily on the author's previous papers (Refs. [8,9,19]) for the numerical fits, figures, and parametric tables. The present paper is not self-contained; the critical supporting material should be included directly.

Circularity Check

1 steps flagged · score 6.0 of 10

The headline 'prediction' J0≈3.08e-5 is a fitted consistency check, not an independent prediction

fitted input called prediction [Sec. III.C.3, Low-Energy Baseline Match; Eq. (23)–(24) and Sec. III.D.2, Eq. (30)]
"At low energies, by numerically fitting the four parameters (x, y, x′, y′) to the experimental values of the CKM matrix elements ... 32 candidate parameter sets are obtained. Strikingly, substituting these fitted parameters into the geometric expression J(x, y, x′, y′) in Eq. (23) reveals that all non-trivial solutions universally yield an identical Jarlskog invariant magnitude: J0 ≈ 3.08×10−5"

The four parameters are fitted directly to the experimental CKM magnitudes, and the Jarlskog invariant of any 3×3 unitary matrix is uniquely determined by its moduli (twice the area of the unitarity triangle). Thus Eq. (23), evaluated on parameters tuned to reproduce those moduli, automatically returns the empirical J0; the 'remarkable agreement' is a mathematical necessity rather than a model prediction. The paper itself confirms the fit-to-prediction sequence: 'substituting these fitted parameters into the geometric expression J(x,y,x′,y′)... reveals... J0'. No independent observable is predicted beyond what was fitted.

full rationale

The algebraic derivation of the 5-parameter reduction, the closed-form U(x,y), and the symbolic J(x,y,x′,y′) in Eq. (23) are self-contained mathematical steps; likewise the 18/18 CP classification is a well-defined combinatorial exercise. However, the paper's central numerical claim—the convergence to J0≈3.08×10−5—is not an independent prediction: the four geometric parameters are first optimized to reproduce the full set of CKM magnitudes (from the author's previous Ref. [19]), and J is then computed from those same fitted magnitudes. Since for any unitary matrix J is fixed by the moduli, this 'recovery' is automatic and would occur in any CKM parameterization that fits the magnitudes. The paper's actual structural prediction, the fourfold degeneracy |Vub|=|Vcb|=|Vtd|=|Vts| (Eq. 29), is admitted to disagree with experiment, and the BAU claims involving J(S2)=4√3/81 and the mass spurt are asserted without a derivation or dynamical equations. These latter issues are correctness/evidential gaps rather than circular reductions. Overall, the exact symbolic formalism has independent content, but the flagship numerical agreement reduces by construction to the fitted input, so partial circularity is present.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim rests on one ad hoc physical axiom — exact simultaneous diagonalizability of the real and imaginary parts of each sector's mass-squared matrix — plus standard theorems about commuting Hermitian matrices. The four fitted ratio parameters carry the phenomenology; the J value inherits the experimental input through the fit. No new entities are introduced. The ledger is short because the model is algebraically economical, but the economy is purchased by the commutation axiom, and the paper concedes that this axiom forces a CKM degeneracy that data rejects.

free parameters (6)

x (up-sector ratio) = not stated (32 fits in Ref. [19])
x = B2/B = -C/C2; fixed by fitting CKM magnitudes; enters U_u and J directly (Eqs. 9, 16, 23).
y (up-sector ratio) = not stated (32 fits in Ref. [19])
y = B1/B = C/C1; fitted to CKM magnitudes; enters U_u and J.
x′ (down-sector ratio) = not stated (32 fits in Ref. [19])
Down-sector analog of x; fitted to CKM magnitudes.
y′ (down-sector ratio) = not stated (32 fits in Ref. [19])
Down-sector analog of y; fitted to CKM magnitudes.
flavor-permutation sector (1 of 36 S3×S3 topologies) = 18 CP-violating sectors selected
The mapping of eigenvalues to (u,c,t) and (d,s,b) is 'unconstrained prior to empirical fitting' (§III.C.1); the paper selects the active CP-violating topology to report J. Different choices move the degeneracy to different elements.
A, B, C (mass-matrix scales) = not fitted in this paper
Scale parameters of the 5-parameter pattern (Eq. 10); they set masses but drop out of U(x,y) and J by the paper's scale-invariance argument; they are free inputs of the model.

assumptions (5)

ad hoc to paper [M²_R, M²_I] = 0 (exact simultaneous diagonalizability of each sector)
Called the 'sole foundational hypothesis' (§II.C, Eq. 8). Not derived from SM dynamics; it is the input that makes the model exactly solvable and forces the fourfold degeneracy the paper itself shows contradicts data.
domain assumption The CKM matrix is V_CKM = U_u(x,y)†·U_d(x′,y′) with unconstrained eigenvalue-to-flavor permutations
§III.A, §III.C.1: the physical assignment of the three eigenvalues to (u,c,t)/(d,s,b) is free; 36 topologies are scanned and the CP-violating ones selected for the headline J.
standard math Commuting Hermitian matrices share a common eigenbasis; eigenvectors of M + cI are those of M
Used for scale-invariance of U (Eqs. 17–19, §II.E.4); standard linear algebra, not a physical input.
domain assumption A realized mass matrix M_q exists strictly within the SM producing the M² pattern (Eq. 10) with the commutation property
The paper asserts this 'can be constructed strictly within the SM' (§II.C) but gives no Lagrangian or field content producing the pattern.
standard math For any unitary 3×3 matrix, |J| is determined by the moduli
Implicitly relied upon when fitting moduli then presenting J as a new result; the unitarity-triangle area is fixed by its side lengths.

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Cite this review

Pith. "Pith review of A Geometric CPVSM Framework: Dynamical Quark Mass Spurt with Explicit CP Violation in the Standard Mode." pith.science (2026). https://pith.science/paper/4HI6HQXQ

@misc{pith2026260726681,
  author       = {Pith},
  title        = {Pith review of: A Geometric CPVSM Framework: Dynamical Quark Mass Spurt with Explicit CP Violation in the Standard Mode},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4HI6HQXQ}},
  note         = {Machine review of arXiv:2607.26681}
}

abstract

We present a top-down geometric framework within the CP-violating Standard Model (CPVSM), utilizing dimensionless 2D ratio vectors $(x,y)$ and $(x',y')$ to establish a Cartesian-like parameterization of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. This approach provides an intuitive physical mapping that explicitly disentangles up- and down-type quark sector contributions to electroweak mixing and CP violation. Crucially, we prove an exact algebraic dimensional reduction from the 9 real degrees of freedom of the complex mass-squared matrix $\mathbf{M}^2$ down to 5 baseline parameters, and ultimately to 2 invariant ratio coordinates ($x,y$) in the unitary diagonalization matrix $U$. This baseline yields exact analytical expressions for the CKM matrix and a closed-form Jarlskog invariant $J$, achieving robust numerical convergence to $|J_0| \approx 3.08 \times 10^{-5}$. We demonstrate that the exact commutativity condition $[\mathbf{M}^2_R, \mathbf{M}^2_I] = 0$ is the fundamental origin of an intrinsic fourfold magnitude degeneracy governed by $|p| = |p'| = |p^*| = |p'^*|$, inducing non-perturbative Wolfenstein scale mixing that signals the necessity of non-commuting quantum corrections ($[\mathbf{M}^2_R, \mathbf{M}^2_I] \neq 0$). Remarkably, near critical boundary geometries ($xy \to 0$), the system exhibits a \textit{quark mass spurt} that reshapes the mass hierarchy while geometrically preserving non-collinear CP violation ($xy' - x'y \neq 0$), yielding magnitudes sufficient to account for the cosmological Baryon Asymmetry of the Universe (BAU). Extending this vector mismatch paradigm to the PMNS lepton matrix with Dirac neutrinos ($\nu_R$), we naturally accommodate large leptonic mixing angles through angular vector offsets, establishing a unified, rephasing-invariant geometric framework for flavor origins across both quark and lepton sectors.

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Reference graph

Works this paper leans on

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Strikingly, substituting these fitted parameters into the geo- metric expressionJ(x, y, x′, y′) in Eq

Physical CP Violation: Low-Energy Baseline Match and High-TemperatureS 2 Phase Transitions The geometric Jarlskog functionalJ(x, y, x ′, y′) bridges low-energy laboratory observables with early- universe cosmological phase transitions: 1.Low-Energy Baseline Match: At low ener- gies, by numerically fitting the four parameters (x, y, x′, y′) to the experime...
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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

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Strikingly, substituting these fitted parameters into the geo- metric expressionJ(x, y, x′, y′) in Eq

Physical CP Violation: Low-Energy Baseline Match and High-TemperatureS 2 Phase Transitions The geometric Jarlskog functionalJ(x, y, x ′, y′) bridges low-energy laboratory observables with early- universe cosmological phase transitions: 1.Low-Energy Baseline Match: At low ener- gies, by numerically fitting the four parameters (x, y, x′, y′) to the experime...

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[3] [1]

(8), the foundational hypoth- esis of CPVSM requires the commutation constraint [M2 R,M 2 I ] = 0

Exact Analytical Mass Eigenvalues and Factorization Mechanism As established in Eq. (8), the foundational hypoth- esis of CPVSM requires the commutation constraint [M2 R,M 2 I ] = 0. Algebraically, Eq. (8) imposes a rigid internal sym- metry that forces the cubic characteristic polynomial det(M2 −λI) = 0 to strictly factorize into a product of a linear fa...

[4] [2]

In the CPVSM framework, the factor- ization in Eq

Trace Self-Consistency and Connection with Cardano’s Formula For a generic 3×3 Hermitian matrix, solving det(M2 − λI) = 0 requires Cardano’s formula, which inherently introduces transcendental functions cos(θ/3) via alge- braic invariantsI 1, I2, I3 when all three roots are real (the classicalcasus irreducibilis), since the general de- pressed cubic canno...

[5] [3]

dy- namically switches. Therefore, the CPVSM algebraic framework naturally accommodates all viable mass or- dering scenarios—including normal, inverted, and quasi- degenerate hierarchies—without loss of generality

[6] [4]

Internal Diagonalization Unitary MatrixU The left-handed diagonalizing unitary matrixU L, which simultaneously diagonalizes bothM 2 R andM 2 I — and consequently diagonalizes the matrix squareM 2 ≡ M q ·(M q)† viaU † L ·M 2 ·U L = diag(m 2 1, m2 2, m2 3) while performing the bi-unitary transformation on the non-Hermitian mass matrixM q (U † L ·M q ·U R = ...

[7] [5]

skeleton

Origin of the Scale-Invariance ofU The remarkable independence of the diagonalizing uni- tary matrixU(whereU≡U L in single-sector contexts, written simply asUhereafter unless explicit left/right chirality separation is required) from the global scale pa- rameters (A,B,C) is not an incidental numerical coinci- dence, but a direct, inevitable algebraic cons...

[8] [6]

to specific quark fla- vors (u, c, t) or (d, s, b) unconstrained prior to empirical fitting, there exist 6×6 = 36 candidate topologies for VCKM