CKM count-domain audit

Physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Summary: Corrected rank, degree-loss, charged-current normalization, and CP tests constrain CKM candidates.

NOT A PHYSICS MODULE. This note replaces the earlier count-domain negative-results document. It preserves three candidate constructions, corrects their mathematical treatment, and follows the corrected premises through to their consequences. Exact results below apply to the stated models; they are not a general impossibility theorem for V2 mixing.

1. Scope, source inputs, and conventions

The earlier document tested a separable power-ratio kernel, a binomial degree-loss table, and a one-step creation operator. It described their unitary factors as trivial and concluded that the cross-sector remainder must supply mixing. Those claims require replacement, not just a change of wording.

The parent CKM proposal, Sections 2 and 8, specifies up labels $(3,20,72)$ with homogeneous degrees $(2,19,71)$, down labels $(1,4)$ with degrees $(0,3)$, and an exceptional bottom state involving labels $(16,17)$, hence degrees $(15,16)$. Its physical bottom vector is not fixed by selecting one of those degrees.

For reproducibility, the binomial calculation below retains the earlier single-band approximation $N_b=16$, meaning the upper label $n_b=17$. It is not silently promoted to the actual two-band bottom state. The power-ratio rank proof needs only a positive bottom count and is independent of its numerical choice.

Throughout, matrix rows are $(d,s,b)$ and columns $(u,c,t)$. The down-row charged-current convention is the adjoint of the usual up-row CKM convention; in particular, the magnitude of its $su$ entry corresponds to the usual $|V_{us}|$. The benchmark is a unitary three-generation charged-current matrix with nontrivial mixing and a nonzero rephasing-invariant CP phase. The 2025 PDG review supplies this benchmark and the definition of the Jarlskog invariant [E1]. No central-value scan or statistical exclusion is claimed here.

These are transition-operator tests using specified mode counts. No independently measured field amplitude is fitted or reconstructed. Eigenfrequency data and mode counts do not by themselves determine the relative orientation or phases of the two flavor bases, as the parent source already explains in Section 5.

2. First establish what normalization is allowed

2.1 Polar factorization is not automatically canonical normalization

For a square matrix $K$, the right polar decomposition is $K=QH$, with $H=(K^\dagger K)^{1/2}$. If $K$ is invertible, $Q=KH^{-1}$ is a unique unitary matrix. If $K$ is singular, the canonical polar factor is a partial isometry; its full unitary extensions are not generally unique [E2]. Discarding $H$ is a new physical prescription unless its removal follows from the action.

To make that issue precise, suppose the independently derived left-handed kinetic metrics are positive definite matrices $Z_u,Z_d$. Removing them gives

$$K_c=Z_d^{-1/2}KZ_u^{-1/2}.$$

If $L_u,L_d$ subsequently put the canonically normalized mass operators into their physical bases, the charged-current matrix is

$$V_\downarrow=L_d^\dagger K_cL_u.$$

Consequently, after factoring out the common weak coupling, unitarity of this square matrix is equivalent to

$$\boxed{K^\dagger Z_d^{-1}K=Z_u.}$$

This is a test of the derived vertex and kinetic metrics, not permission to choose metrics that force the answer. Choosing $Z_d=I$ and $Z_u=K^\dagger K$ reproduces the ordinary polar factor for invertible $K$, but the theory would have to derive precisely those metrics. For singular $K$, that choice is not positive definite.

An exact two-dimensional example, embeddable with a spectator third generation, shows the distinction. Let

$$S_d=\operatorname{diag}(2,1),\qquad S_u=\begin{pmatrix}2&1\\1&2\end{pmatrix},\qquad Z_d=S_d^2,\quad Z_u=S_u^2,$$

and

$$K=S_dS_u=\begin{pmatrix}4&2\\1&2\end{pmatrix}.$$

The actual canonical matrix is $S_d^{-1}KS_u^{-1}=I$. Yet the Euclidean polar factor of $K$ is

$$Q=\frac{1}{\sqrt{37}}\begin{pmatrix}6&1\\-1&6\end{pmatrix}\ne I.$$

Thus extracting a rotation by polar factorization can invent a mixing angle that is absent after the specified physical normalization. Conversely, unknown kinetic or mass-basis data cannot be ignored when interpreting a raw kernel.

2.2 Rank survives every invertible normalization

Invertible left and right transformations preserve rank, so

$$\operatorname{rank}V_\downarrow=\operatorname{rank}K.$$

A rank-one matrix cannot become a complete rank-three unitary charged-current vertex merely through legitimate nonsingular normalization. This conclusion is stronger than a poor numerical fit, but it does not forbid a rank-one term as one contribution to a larger vertex.

3. Test 1: the separable power-ratio family

Write the positive counts as $A_i$ and $B_j$, and take any finite real exponent $q$:

$$K_{ij}=(A_i/B_j)^q=x_i y_j,\qquad x_i=A_i^q,\quad y_j=B_j^{-q}.$$

This is exactly an outer product. Every two-by-two minor vanishes. The singular values are

$$\boxed{\|x\|\|y\|,\quad0,\quad0.}$$

The earlier description in terms of extreme singular values obscured two exact zeros. No exponent scan is needed. Positive sector units, a different positive scalar bottom count, or separate row and column phases do not remove this separability.

3.1 What the kernel does and does not determine

With normalized vectors $\widehat x,\widehat y$, the canonical partial isometry is $Q_0=\widehat x\widehat y^\dagger$. Every full unitary extension has the form

$$Q=\widehat x\widehat y^\dagger+E_xWE_y^\dagger, \qquad W\in U(2),$$

where $E_x,E_y$ are orthonormal bases of the complementary two-dimensional spaces. All such $Q$ give the same $K$ when multiplied by its positive polar factor. The arbitrary $W$ is missing information, not mixing predicted by the counts.

A further consequence matters for the CP claim. At $q=0$, $K$ is the all-ones matrix, with positive polar factor $H=K$. Set

$$w=(1,1,1)^T/\sqrt3,\quad r=(1,-1,0)^T/\sqrt2,\quad s=(1,1,-2)^T/\sqrt6.$$

Both $Q=I$ and

$$Q_*=ww^T+i rr^T-ss^T$$

are unitary and satisfy $QH=K$. But for the quartet convention

$$J(Q)=\operatorname{Im}(Q_{11}Q_{22}Q_{12}^*Q_{21}^*),$$

one obtains

$$J(I)=0,\qquad \boxed{J(Q_*)=-2/27.}$$

This is not a proposed physical CP phase. It proves that a blanket claim of zero J for every arbitrary unitary completion of the real rank-one kernel is false. A real completion has zero J; a chosen complex completion can introduce nonzero J, but that phase was not predicted by the original kernel.

3.2 Logical conclusion and size of the missing structure

As a complete vertex in canonical three-generation units, this family fails the rank test for every real $q$. As an input to an unspecified completion, it underdetermines mixing rather than predicting a unique wrong matrix.

If $K+\Delta$ is to have rank three, then

$$3\le1+\operatorname{rank}\Delta, \qquad\boxed{\operatorname{rank}\Delta\ge2.}$$

If the completed matrix is unitary, a unit vector $v\in\ker K$ satisfies $\|\Delta v\|=1$, so $\|\Delta\|_2\ge1$. The missing action on two directions cannot be supplied by a norm-small correction in these fixed canonical units. This bound is not a statement about raw unnormalized coefficients.

In particular, adding the rank-one one-step kernel of Test 3 to this rank-one family still gives rank at most two. At least three rank-one terms would be necessary in a purely separable sum, and even that would not guarantee unitarity, the correct hierarchy, or CP violation.

4. Test 2: the binomial degree-loss prescription

Preserve the displayed numerical prescription as

$$B_{ij}(p)=\binom{N_{u_j}}{N_{d_i}} p^{N_{d_i}}(1-p)^{N_{u_j}-N_{d_i}}, \qquad N_{d_i}\le N_{u_j},$$

and zero otherwise. Here $p$ is the surviving-direction weight. The parent source uses $p$ for the lost-direction weight in Section 11, so comparison requires $p\leftrightarrow1-p$. This convention change does not alter the support arguments.

For $N_u=(2,19,71)$ and $N_d=(0,3,16)$, the interior support is

$$\boxed{\begin{pmatrix}*&*&*\\0&*&*\\0&*&*\end{pmatrix}}.$$

The $b\leftarrow c$ entry is nonzero. The matrix is not triangular.

4.1 Interior rank and endpoint exceptions

Factoring powers from the rows and columns gives

$$\boxed{\det B(p)=\binom{19}{3} \left[\binom{71}{16}-\binom{71}{3}\right] p^{19}(1-p)^{73}>0},\qquad0<p<1.$$

Thus the whole interior family has rank three. At $p=0$ its rank is one; at $p=1$ the selected matrix is zero. An interior theorem about a unique polar factor must not be extended to these singular endpoints.

Taking the other single bottom degree, $N_b=15$, gives the positive coefficient

$$\binom{19}{3}\binom{71}{15}-\binom{19}{15}\binom{71}{3}$$

multiplying $p^{18}(1-p)^{74}$. The same support and interior-rank conclusions hold for that alternative single-band test. Neither test determines the actual bottom singular vector.

4.2 Triangularity would not imply a trivial polar factor anyway

The exact factorization

$$\begin{pmatrix}1&1\\0&1\end{pmatrix} =\frac1{\sqrt5}\begin{pmatrix}2&1\\-1&2\end{pmatrix} \frac1{\sqrt5}\begin{pmatrix}2&1\\1&3\end{pmatrix}$$

has a nontrivial orthogonal factor and a positive factor. It refutes the old triangular-to-permutation inference independently of the binomial calculation.

For the actual $B(1/2)$, calculations at 70 and 110 decimal digits agree on

$$|Q|\approx\begin{pmatrix} 0.999999999971&7.550453669\,10^{-6}&2.894612645\,10^{-11}\\ 5.340954801\,10^{-6}&0.707366017336&0.706847449942\\ 5.336998446\,10^{-6}&0.706847449922&0.707366017357 \end{pmatrix}.$$

Reconstruction and orthogonality are checked independently. This is a reproducible counterexample to a trivial polar factor, not a CKM fit and not a proof about every possible magnitude fit in $p$.

It also demonstrates that polar replacement does not preserve entrywise selection rules: $B_{su}=0$, but $Q_{su}\ne0$. A nonzero entry manufactured by this matrix operation is not proof that the physical loss process produces the forbidden transition.

4.3 Three interpretations must remain separate

A raw coherent vertex: the displayed real $B$ is not unitary. In fact,

$$(B^TB)_{12}=(1-p)^{21}>0.$$

Its first two columns are not orthogonal. Diagonal nonzero state normalizations and row/column rephasings cannot cancel this single shared-row overlap or create the absent $u\to s$ matrix element. More general kinetic and mass-basis maps require independent derivation and the test in Section 2; they cannot be assumed to be the polar prescription.

A defined polar-factor toy model: if one explicitly postulates $V_\downarrow=\operatorname{polar}(B)$ with Euclidean metrics, its interior polar factor is uniquely real orthogonal. Therefore every CP quartet has zero imaginary part. No $p$ in the interior supplies a nonzero J. Diagonal row and column phases cancel from those quartets and do not repair this. Nontrivial real mixing is possible; physical three-generation CP violation is not supplied by this prescription. This does not prove that no $p$ approximates selected magnitudes, and that stronger unverified scan claim is not retained.

A traced-out loss channel: the parent construction is naturally a map on density operators, not a single coherent matrix. A number-state realization of the same binomial weights has Kraus operators

$$A_r(p)=\sum_{N\ge r}\sqrt{\binom Nr} p^{(N-r)/2}(1-p)^{r/2}|N-r\rangle\langle N|,$$

and

$$\mathcal E_p(\rho)=\sum_r A_r\rho A_r^\dagger, \qquad\sum_rA_r^\dagger A_r=I.$$

For input $N$ and output $k$, the binomial entry is $\sum_r|\langle k|A_r|N\rangle|^2$. Different loss labels are distinct traced channels. The weights do not specify one coherent transition matrix, and taking their square roots and assigning phases does not uniquely reconstruct one. This channel distinction uses the standard Kraus framework [E3]; the displayed degree-loss realization is checked directly here.

For $0<p<1$, a one-quantum pure input produces a mixture of zero and one surviving quanta, so this loss channel is not a unitary channel. Restricting it to the selected flavor bands can also discard probability into unselected bands. An arbitrary polar factor of its probability table is not the same physical operation.

4.4 The genuine support conclusion

Every bare-loss Kraus operator has $N_{\rm out}\le N_{\rm in}$. Therefore none creates the specified $N_u=2\to N_s=3$ transition. This remains true when the two bottom degrees are retained, because neither changes the $u,s$ degree labels.

The conclusion is conditional on identifying those named physical bands with the input and output projectors of the loss model. Unknown non-diagonal basis maps are extra structure, not consequences of bare loss. A derived coherent transition operator must provide the missing degree-raising element or justify a different physical identification. Polarizing a probability table does neither.

5. Test 3: isolated one-step creation

For the stipulated support $N_{\rm out}=N_{\rm in}+1$, the selected matrix has only one possible nonzero cell:

$$K=\alpha|s\rangle\langle u|.$$

For $\alpha\ne0$, it has rank one. Including both bottom degrees 15 and 16 adds no allowed cell, since the candidate outputs of degrees $(2,19,71)$ are $(3,20,72)$. At $\alpha=0$, there is no transition and no selected polar mapping at all.

5.1 A stronger conclusion than the old permutation claim

Write $\alpha=|\alpha|e^{i\phi}$. Every unitary polar extension must satisfy

$$Q|u\rangle=e^{i\phi}|s\rangle.$$

The other two columns are an arbitrary unitary map between the complementary two-dimensional spaces. They need not form a permutation. Nevertheless, every extension has $|Q_{su}|=1$, not a small Cabibbo transition. This failure is independent of the chosen completion.

There is a second completion-independent consequence: the fixed unit column decouples one generation, leaving only a two-by-two unitary block. All three-generation CP quartets then have zero imaginary part. The remaining two-by-two quartet is real by orthogonality, and every quartet involving the fixed row or column vanishes. Thus J is zero for every completion, including complex ones. This is stronger than the corresponding conclusion for Test 1, where a complex arbitrary completion can have nonzero J.

5.2 The coefficient was not determined by the stated operator alone

For a specified common bosonic mode space, normalized coherent states, and

$$T=\sum_m t_m b_m^\dagger,$$

the matrix element is

$$\boxed{\langle v;N+1|T|u;N\rangle =\sqrt{N+1}\,(v^\dagger t)(v^\dagger u)^N.}$$

A magnitude $\sqrt{p(N+1)}$ follows only after specifying the relevant overlaps; for example, aligned unit coherent vectors and $p=|u^\dagger t|^2$. For the unnormalized sum with every $t_m=1$, $\|t\|^2$ is the number of summed modes, so that $p$ is not automatically a probability bounded by one. The original definition omitted these premises.

The equation above is a comparison calculation in a defined common mode space. It does not supply the missing cross-depth intertwiner between different V2 carriers. The reliable negative result is the sparse support under the stipulated one-step rule, not an undocumented coefficient formula.

6. The bottom state cannot be supplied by its mass alone

The parent proposal explicitly distinguishes its bottom two-band singular-value construction from an ordinary band. Even in a simplified orthonormal two-band representation,

$$|b\rangle=c_{15}|15\rangle+c_{16}|16\rangle$$

would give a row

$$K_{bj}=c_{15}^*K_{15,j}+c_{16}^*K_{16,j}.$$

The coefficients and their phases would have to come from the actual bottom operator. Its proposed mass or a geometric mean of two counts does not determine this row. The simple representation is illustrative, not a derivation of the physical bottom spinor structure.

The structural rank proof for a separable scalar-count family and the absence of the $u\to s$ cell in bare loss survive their stated assumptions. Detailed singular values, mixing angles, and realness of a fully completed two-band model cannot be inferred from one arbitrarily chosen bottom level. A new complex bottom construction would be a new model and would need a fresh CP audit.

7. Corrected conclusions and their limits

The full CKM problem is not solved by any of these prescriptions as stated. The reasons are now explicit and different: missing rank, missing predicted directions, absent coherent phase information, a forbidden physical transition, or an exactly unit-sized transition in the wrong place. None of those facts proves that all count-informed cross-sector operators fail.

For CP, the relevant condition is a nonzero rephasing-invariant quartet of the complete canonically normalized mass-basis vertex. Complex phases that are only row or column rephasings do not help. A numerical complex completion in an undetermined null space is not a predicted CP mechanism. Conversely, realness of one raw term does not establish a theorem about a larger model with independently derived complex kinetic, mass-basis, or interaction structure.

The missing dynamics is not proved to reside only in $\mathcal R_{34}$. The parent source also leaves the off-diagonal coherent-contact operator unevaluated. The leading contact, the remainder, their combination, or a derived relation between physical flavor frames could contribute. What is required is a complete coherent construction, including the actual bottom state and kinetic metrics, not the naming of a particular remainder as the only possible source.

The parent source's leading identity result retains its stated assumption of no generation mixing; the scalar mass counts alone do not prove that assumption. Its bare-loss support result applies to bare loss. Its proposed identification of canonical normalization with an unspecified unitary part needs the metric and operator conditions in Section 2. This note records those qualifications without silently rewriting the parent module.

8. Reproduction and source provenance

Run from the v2 directory:

python3 experiments/ckm_count_audit_checks_2026_09_05.py

The check script verifies exact rank and determinant identities, explicit real and complex polar completions, the kinetic-normalization counterexample, the binomial polar example at two precisions, finite loss-channel Kraus identities, and the normalized coherent creation formula. Analytic arguments in this note establish the all-exponent rank result, the all-interior-p CP result for the real polar prescription, and the all-completion consequences of the single forced column. Finite examples are not substituted for those universal arguments.

No undocumented exponent scan, p scan, target dataset, empirical significance calculation, or master-action derivation is claimed. The new calculations are this review's derivations, not statements silently attributed to the original note or parent source.

The superseded source was physics/ckm-count-domain-negative-results.md, SHA-256 5f5eb3d08780e238869fae0246ae2d2c3afbabc3ad3e8ecd9e36b837f9e4e455. The parent source at this review has SHA-256 c2c7ea7595147239125d66a25dcac07203ddbc1929ef59e54b60ab617b763664 and is unchanged by this rewrite.

External references checked for this review

[E1] Particle Data Group, CKM Quark-Mixing Matrix, 2025 edition, Sections 12.1--12.2: charged-current conventions, three-generation unitarity, rephasing-invariant CP benchmark, and nonzero Cabibbo mixing. This is a specified reference edition, not a claim to reconstruct the original note's unspecified target dataset.

[E2] Nicholas J. Higham, What is the Polar Decomposition?: polar factors, full-rank uniqueness, and singular-factor nonuniqueness. The displayed counterexamples and consequences are derived explicitly above.

[E3] IBM Quantum Learning, Channel representations: the distinction between Kraus sums and unitary channels. The degree-loss operators used here are the explicit comparison model, not a new assertion about the V2 master action.

Related

• Research status — current proof boundaries and correction guidance.

9. Fixed Bottom Labels in the Original Polar-Matrix Control

The existing single-band toy can be evaluated at both requested bottom labels without changing its parameter. Keep $p=1/2$, up degrees $(2,19,71)$, and down degrees $(0,3,n_b-1)$. Its real matrix is

$$B_{ij}=\binom{N_{u,j}}{N_{d,i}}2^{-N_{u,j}} \quad(N_{d,i}\leq N_{u,j}),$$

with zero otherwise. Take its unique Euclidean polar factor $Q$. The original matrix has down rows and up columns, so the standard CKM comparison is $V=Q^T$. Physical n16 uses bottom degree15; n17 uses degree16 and reproduces the original displayed audit baseline.

The evaluated matrices agree at 70 and 110 decimal digits within $10^{-50}$, with reconstruction and orthogonality residuals below that threshold. Independent review checked the source formula, degrees, transpose, saved comparisons and exact real-matrix consequence $J=0$; it did not independently rerun the high-precision computation. The one-percent threshold is borrowed explicitly from the earlier scalar-ratio comparison; the original matrix audit itself specified the hierarchy and nonzero CP requirement.

This retains the existing distinction between a mathematical polar replacement and derived kinetic normalization. Neither fixed-p case fits, while no scan over p or every possible coherent field interaction is implied. The two-band bottom state remains a separate unresolved construction. The complete matrices, nine target comparisons, angles, successful output and all failed checker attempts are preserved with their exact research-database provenance.

Complete Full-Rank Bottom-Rung Control

At the unchanged $p=1/2$, up degrees $(2,19,71)$ and down degrees $(0,3,k)$, with $k=n_b-1$, an exact determinant proof identifies the unique orthogonal polar-factor domain. Removing positive column factors,

$$\det B=2^{-92}\left[ 969\binom{71}{k}-57155\binom{19}{k}+56186\binom{2}{k} \right],$$

where a binomial vanishes above its top argument. The determinant vanishes at $k=0,3$, is nonzero at $k=1,2$, and is positive for $4\leq k\leq71$; for $k>71$ the bottom row vanishes. Thus exactly 70 positive integer rungs have full rank: $n_b=2,\ldots,72$ except four. The low-rung determinants are negative, as allowed for the specified ordinary orthogonal polar factor; no special-orthogonal rule was imposed.

The author evaluated every full-rank case at 70 and 110 decimal digits. Executed assertions check positive singular values, reconstruction and orthogonality residuals below $10^{-50}$, and successive polar factors agreeing within that tolerance. The saved JSON contains final-precision values; its receipt and inspected code support the two-precision assertion, not an independent numerical replay. The reviewer separately proves domain completeness and the exact real-matrix consequence $J=0$.

No rung matches all nine dated magnitudes. None places the first eight entries within the one-percent window; only the individually quoted $|V_{tb}|$ target passes, for $n_b=20,\ldots,72$. This window was borrowed from the earlier scalar diagnostic, not a matrix likelihood or joint experimental fit. Singular cases leave an unspecified null-direction completion; they were not tested under an invented completion rule.

Changing only bottom therefore does not repair this fixed real polar toy. Physical kinetic normalization, complex response and a common mass/mixing solution remain separate questions. Assessment SHA-256: 137db93368f3e2006a7979f7043b44a9e16dc817fba61622781c8629f27bbd5c.