Count Ratios, Quark Mixing and the Cumulative-Band Test

The maintained particle counts give two economical candidates for quark mixing:

$$|V_{us}|^2=\frac{D_d}{D_s}=\frac1{20},\qquad |V_{cb}|^2=\frac{D_u}{D_c}=\frac{15}{8855}=\frac3{1771}.$$

Here $D=S(n,d)=\binom{n+d-1}{d}$ is the cumulative count attached to a mass assignment. The symbols $V_{ij}$ refer to measured charged-current mixing magnitudes in the conventional CKM parametrization; introducing this comparison does not introduce exchanged particles into IDWT. A same-field interaction must still derive what experimental transition amplitude it predicts.

The two candidates are fits to already known measurements. They give approximately 0.223607 and 0.041158. The PDG 2024 benchmarks used in the calculation are $0.22431\pm0.00085$ and $0.0411\pm0.0012$. The signed relative discrepancies are about −0.31% and +0.14%, respectively. The first is roughly 0.8 quoted standard deviations away; the second is much smaller than its quoted uncertainty. These input values are dated comparison inputs, not a claim to have refreshed the experimental averages. See the PDG 2024 CKM review.

The familiar external square-root mass-ratio form is not new by itself. Within the maintained common-scale families, the proposal sharpens it to the displayed exact rational squared values. The new research question is whether the same cumulative geometry that enters the mass count also enters a correctly normalized transition interaction.

A precise projection identity

Let $\mathcal B_1\subset\mathcal B_2$ be real mode spaces with dimensions $D_1<D_2$, and $P_1,P_2$ their orthogonal projectors. A unit classical coefficient vector with isotropic second moment on the larger band has

$$\langle aa^T\rangle=\frac{P_2}{D_2},\qquad \langle\|P_1a\|^2\rangle =\operatorname{tr}\left(P_1\frac{P_2}{D_2}\right) =\frac{D_1}{D_2}.$$

The brackets specify a classical preparation average. They do not define a probability distribution of electron positions or a Born law. Likewise the trace identity is not yet a transition amplitude. Identifying its square root with an observed mixing magnitude is the physical bridge being proposed.

For an orthonormal spatial basis $\phi_j(x)$, write the cumulative diagonal kernel as $K_N(x,x)=\sum_{j\le D_N}|\phi_j(x)|^2$. Its spatial integral is $D_N$. The ratio of the integrated kernels therefore reproduces $D_1/D_2$; the pointwise ratio generally does not. On the weighted ball it varies with position. On a homogeneous band whose diagonal kernels are constant, the ratio is pointwise constant as well. A physical contact must specify whether it samples an integrated, local or otherwise weighted response. These are different mechanisms.

Cumulative bands of all polynomial degrees through $N$ supply the required nesting. A single-parity real amplitude-sphere realization has a different property: degree-$N$ homogeneous polynomial restrictions contain harmonic degrees $N,N-2,\ldots$. Opposite parities are orthogonal. The proposed mixing pairs have degrees $(0,3)$, $(2,19)$ and $(2,71)$; they are all opposite-parity pairs. Their direct inner-product contact is consequently zero in that particular realization. Under the proposed direct overlap mechanism, the fitted nonzero mixing favors a genuinely cumulative realization. It does not rule out every sphere construction: a different interaction could mix parity, but would require its own derivation.

What the same rule fails to explain

The analogous up/top candidate gives

$$|V_{ub}|=\sqrt{\frac{15}{1215450}} =\frac1{\sqrt{81030}}\simeq0.003513.$$

Against the used benchmark 0.00382 this is about 8% low. The down-side third-generation candidate fails much more severely, and bottom has no established ordinary integer-count assignment. No bottom count is invented to complete the table. A two-angle approximation gives $|V_{cd}|\simeq|V_{us}|$ and $|V_{ts}|\simeq|V_{cb}|$; exact three-generation unitarity does not make these equalities exact or determine the CP phase.

Applying the same squared ratio rule to neutrino mass-state counts $2002,11628,65780$ gives approximately $0.172,0.177,0.030$, compared with the used PMNS values $0.307,0.561,0.022$. These are failed fits. The successful quark comparisons do not supply a general mixing matrix, the CP-violating phase or a neutrino mixing mechanism.

The exploratory candidate set contained the plain and square-root values of every ratio below one among fourteen maintained counts: 182 candidates per mixing element. A crude log-range estimate gave about 0.3 accidental hits within a 1% window per target. This is not a calibrated significance test; the ratios are correlated and the choice of family relationships carries a selection history. Cross-family numerical hits were not adopted as mechanisms. The credible contribution is two compact targets with a testable geometric bridge and explicit failures, not a discovery probability.

Bottom and the adjacent-depth hypothesis

The same inversion makes the bottom difficulty sharper. Its mass at the supplied down-sector unit corresponds to roughly 890 counted contributions, whereas a mixing denominator inferred from the approximate two-angle relations is $D_sD_c/D_u=35420/3\simeq11807$. These are fitted targets for two different observables, not an identified band with a fractional number of modes. In the unchanged residual-free occupation action, a participation count $C_m$ and origin weight $p_0$ obey

$$\frac H\mu\ge C_m[1+2(1-C_mp_0)^2].$$

A clock equal to its participation count therefore needs $C_mp_0=1$. The proposed mass/mixing mismatch cannot simply be assigned to one undressed state of that action. The bound is in the carrier's actual unit $\mu$, not automatically the down unit. Changed projections, dressed states or changed actions are possible research routes; this note does not adopt an unreviewed repair.

Likewise the pairs 3/4 and 5/6 suggest an adjacent-depth charged-current hypothesis, but a depth-ten tau has no assigned partner at nine or eleven. The mass fit and a universal depth-change-by-one law cannot both be asserted from these data. Adjacency is a conjecture suggested by two pairs, not an established selection rule.

Reproduction and next test

The arithmetic follows directly from the mass assignments and scale law. The exact trace calculation is reproducible by choosing any nested orthonormal bases and summing the retained squared components; the ball version additionally requires its actual kernel and integration measure. The original calculation used exact rational checks for the band ratios and parity controls. The finite candidate search and its failed comparisons are retained in the source-pinned research record.

The useful next calculation is a reciprocal same-field transition vertex on a cumulative spatial carrier, with its kinetic metrics and contact weighting varied as part of the action. It must yield these magnitudes while accounting for the third generation, bottom, unitarity and CP phase. A mode-space projection alone does not do that work.

Fixed Bottom Labels 16 and 17 — 5 November 2026

Keeping all other labels, targets and the original one-percent relative-error criterion fixed, the requested ordinary bottom counts are $S(16,3)=816$ and $S(17,3)=969$. The down and strange counts remain 1 and 20.

The n16 control fails, so n17 was then tested; it also fails. The original $|V_{us}|=\sqrt{1/20}$ and $|V_{cb}|=\sqrt{15/8855}$ expressions omit bottom and remain unchanged. The up/top expression for $|V_{ub}|$ remains about 8.04 percent below its original target. Adding the 28 plain/square-root ratios between either proposed bottom count and the 14 original distinct counts gives no new one-percent match among the nine dated CKM targets. No parameter was refitted. The targets and their uncertainties remain the original comparison inputs, not a refreshed joint unitary fit or a global significance test.

The independently assessed conclusion concerns this specified ratio comparison. It does not exclude either label under an independently derived spatial vertex, and it does not replace the unresolved two-band bottom proposal. The existing fixed binomial matrix control was evaluated separately, retaining its own explicit polar-normalization premise.

Other Bottom Rungs in the Fixed Controls

The bottom-only extension keeps every other label, depth and dated target unchanged. For $C=\binom{n+2}{3}$, the tested pair is $(|V_{td}|,|V_{ts}|)=(C^{-1/2},\sqrt{20/C})$. No positive integer rung matches both within the original one-percent relative-error window. The integer minimax control is $n=41$, $C=12341$: its two errors are approximately +4.6710 percent and -2.9955 percent. Only $n=40$ matches the second entry alone; the first entry has no one-percent integer hit.

Independent analysis strengthens that finite scan. The predicted ratio is always $\sqrt{20}$, whereas the two target windows require a ratio strictly above $(415/86)(99/101)>\sqrt{20}$. Even an arbitrary positive real count cannot satisfy both windows. Its best possible maximum relative error is

$$\epsilon_*= \frac{0.0415-0.0086\sqrt{20}}{0.0415+0.0086\sqrt{20}} \simeq0.0380.$$

The real-count relaxation proves an obstruction for these fixed formulas; it does not permit fractional physical portions. The original scalar expressions omitting bottom stay unchanged. This is neither a simultaneous mass fit nor an exclusion of other interaction dynamics or two-band bottom constructions. Complete author output and code are preserved with independent assessment SHA-256 137db93368f3e2006a7979f7043b44a9e16dc817fba61622781c8629f27bbd5c. No positive sector evidence or calibrated statistical significance follows.