Neutrino mass splittings and the simplex spectrum

Current 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. Particle patterns have different dimensional involvement in that space. The ten-dimensional matter domain is presently modeled by flat $\mathbb R^{10}$. Nested layers describe proposed spatial support, not additional manifolds. Depth counts a pattern's dimensional involvement, not a separate containing space. The familiar three dimensions describe the observer's view, not the ambient dimension. All spatial directions are of the same physical kind. Confinement and persistence of the layers remain dynamical questions.

Mathematical scope: Projective, round-sphere and prescribed global-nesting calculations below retain their stated additional premises. Their curvature, topology and finite-volume normalizations are conditional comparisons; they do not establish the current flat spatial geometry. Field components and mode spaces are distinct from physical position. IDWT has no compactified dimensions or mediator particles; interactions must come from $\Psi_\infty$. See spatial meaning and mathematical scope.

Current Reading and Scope — 1 November 2026

The assigned neutrino mass states share one unknown fundamental with counts 2002, 11628 and 65780. The scale-free splitting ratio tests those counts, while an absolute splitting calibration supplies additional empirical input. Neutrality, handedness and rare interactions remain separate constraints. A proposed neutral depth-five geometric branch does not by itself derive all of them.

The new particle-geometry and sector-depths case provides the current derivations, shared assumptions, glossary and reproduction instructions. The twenty-volume research map connects the complete account. Calculations below retain their own stated mathematical premises and do not inherit results from a different action.

1. Result

In the V2 neutrino sector, the three occupied modes are

$$(n_{\nu_1},n_{\nu_2},n_{\nu_3})=(10,15,22),$$

at depth

$$d_\nu=5.$$

The bare mass law is

$$m_i=\mu_5 S(n_i,5), \qquad S(n,d)=\binom{n+d-1}{d}.$$

For the three neutrino modes,

$$S(10,5)=2002,$$

$$S(15,5)=11628,$$

$$S(22,5)=65780.$$

Therefore

$$(m_1,m_2,m_3) =\mu_5(2002,11628,65780).$$

The common normalization $\mu_5$ cancels exactly from the ratio of the two independent mass-squared splittings. The V2 neutrino assignments therefore make a parameter-free prediction:

$$\boxed{ \frac{\Delta m_{21}^2}{\Delta m_{31}^2} = \frac{11628^2-2002^2}{65780^2-2002^2} =0.030349842234897634. }$$

The depth-scan companion uses a different PDG-2024 benchmark and reports $1.98\%$ for that comparison. The $2.68\%$ below belongs specifically to the two inputs displayed here; these benchmarks must not be interchanged when quoting accuracy.

Using the benchmark oscillation values

$$\Delta m_{21}^2=7.49\times10^{-5}\ {\rm eV}^2,$$

$$\Delta m_{31}^2=2.534\times10^{-3}\ {\rm eV}^2,$$

the observed ratio is

$$\frac{\Delta m_{21}^2}{\Delta m_{31}^2} =0.029558011049723762.$$

The V2 simplex prediction is therefore high by approximately

$$\boxed{2.68\%}.$$

This comparison contains no fitted neutrino mass scale.

2. Independent reconstruction of the neutrino scale

The same mass law permits $\mu_5$ to be inferred independently from either measured splitting.

From the solar splitting,

$$\Delta m_{21}^2 =\mu_5^2\left[S(15,5)^2-S(10,5)^2\right],$$

so

$$\mu_5^{(21)} = \sqrt{ \frac{\Delta m_{21}^2} {11628^2-2002^2} } =7.5556190385\times10^{-7}\ {\rm eV}.$$

From the atmospheric splitting,

$$\Delta m_{31}^2 =\mu_5^2\left[S(22,5)^2-S(10,5)^2\right],$$

so

$$\mu_5^{(31)} = \sqrt{ \frac{\Delta m_{31}^2} {65780^2-2002^2} } =7.6561541256\times10^{-7}\ {\rm eV}.$$

Their ratio is

$$\frac{\mu_5^{(21)}}{\mu_5^{(31)}} =0.9868687221.$$

Thus the two experimentally independent splittings infer the same sector normalization to approximately

$$\boxed{1.31\%}.$$

This is a direct internal consistency test of the one-scale neutrino-sector hypothesis.

3. Absolute neutrino masses

Once either splitting fixes $\mu_5$, the simplex law determines the absolute neutrino masses.

Using $\Delta m_{21}^2$ gives

$$m_1=0.00151263\ {\rm eV},$$

$$m_2=0.00878567\ {\rm eV},$$

$$m_3=0.04970086\ {\rm eV},$$

and hence

$$\boxed{ \sum_i m_i=0.05999917\ {\rm eV}. }$$

Using $\Delta m_{31}^2$ instead gives

$$m_1=0.00153276\ {\rm eV},$$

$$m_2=0.00890258\ {\rm eV},$$

$$m_3=0.05036218\ {\rm eV},$$

with

$$\boxed{ \sum_i m_i=0.06079752\ {\rm eV}. }$$

The two independent absolute-spectrum reconstructions therefore lie in the narrow interval

$$\sum_i m_i\simeq0.0600\text{--}0.0608\ {\rm eV}.$$

The absolute spectrum is not an additional fit once one measured splitting fixes the common normalization. The second splitting then tests the mode pattern.

4. Joint one-scale reconstruction

A single common scale may also be chosen to reproduce both measured splittings simultaneously. If

$$a=S(15,5)^2-S(10,5)^2,$$

$$b=S(22,5)^2-S(10,5)^2,$$

and $x=\mu_5^2$, the unweighted least-squares value minimizing

$$(xa-\Delta m_{21}^2)^2 +(xb-\Delta m_{31}^2)^2$$

is

$$x= \frac{a\Delta m_{21}^2+b\Delta m_{31}^2} {a^2+b^2}.$$

This gives

$$\mu_5=7.6560622136\times10^{-7}\ {\rm eV},$$

and

$$(m_1,m_2,m_3) = (0.00153274,0.00890247,0.05036158)\ {\rm eV}.$$

The corresponding mass sum is

$$\boxed{ \sum_i m_i=0.06079679\ {\rm eV}. }$$

The reconstructed splittings are

$$\Delta m_{21}^2=7.69047\times10^{-5}\ {\rm eV}^2,$$

$$\Delta m_{31}^2=2.53394\times10^{-3}\ {\rm eV}^2.$$

Because the atmospheric splitting is numerically much larger, an unweighted fit in absolute squared residuals is dominated by it. The scale-free ratio in Section 1 is the cleaner test of the simplex pattern itself.

5. Why the result is nontrivial

For three masses obeying

$$m_i=\mu_5 S(n_i,5),$$

the two measured splittings are

$$\Delta m_{21}^2 =\mu_5^2 \left[S(n_2,5)^2-S(n_1,5)^2\right],$$

$$\Delta m_{31}^2 =\mu_5^2 \left[S(n_3,5)^2-S(n_1,5)^2\right].$$

Their ratio removes the only continuous parameter:

$$R_\nu = \frac{S(n_2,5)^2-S(n_1,5)^2} {S(n_3,5)^2-S(n_1,5)^2}.$$

Once the discrete mode triple $(10,15,22)$ and depth $d=5$ are specified, $R_\nu$ is fixed. The comparison with oscillation data is therefore not a three-mass fit and not a one-parameter fit. It is a test of the discrete spectrum shape.

The second form of the same test is to infer $\mu_5$ separately from each measured splitting. If the simplex mode assignments were badly wrong, the two inferred scales would generally disagree. Instead they differ by only about $1.31\%$.

6. Relation to depth selection

The same scale-free ratio can be evaluated while keeping the mode indices fixed and varying the candidate depth:

$$R_\nu(d)= \frac{S(15,d)^2-S(10,d)^2} {S(22,d)^2-S(10,d)^2}.$$

For the neighboring depths,

$$R_\nu(4)\approx0.05550,$$

$$R_\nu(5)\approx0.03035,$$

$$R_\nu(6)\approx0.01686.$$

Thus the observed oscillation hierarchy selects $d=5$ sharply relative to the adjacent integer depths. This result is also summarized in the mass-spectrum depth test.

7. What is established and what remains conditional

Established mathematics

Given the V2 mass law and the stated neutrino mode assignments:

1. the simplex counts are exactly $(2002,11628,65780)$;
2. the predicted splitting ratio is exactly $0.030349842234897634$;
3. the ratio is independent of $\mu_5$;
4. the two measured splittings independently infer nearly the same $\mu_5$;
5. fixing $\mu_5$ from either splitting determines the complete absolute spectrum;
6. the resulting mass sum is approximately $0.060$--$0.061\ {\rm eV}$.

Conditional inputs

The physical interpretation remains conditional on

• the mode assignments $(10,15,22)$ being selected independently of the measured neutrino splittings;
• the neutrino sector obeying the same one-scale simplex mass law as the other V2 sectors;
• normal mass ordering;
• the benchmark oscillation measurements used in the numerical comparison.

The result does not by itself derive the mode-selection rule or the projected normalization $\mu_5$ from the master geometry.

8. Discrete null tests

The immediate null test is discrete rather than continuous. For all ordered integer triples

$$1\leq n_1<n_2<n_3\leq N,$$

at fixed depth $d=5$, define

$$R(n_1,n_2,n_3) = \frac{S(n_2,5)^2-S(n_1,5)^2} {S(n_3,5)^2-S(n_1,5)^2}.$$

8.1 Unrestricted integer triples

The unrestricted search is deliberately permissive: every increasing integer triple is allowed, without imposing the V2 mode-selection construction.

Using the benchmark ratio

$$R_{\rm obs}=0.02955801105,$$

the committed V2 triple

$$(10,15,22)$$

has relative error

$$2.6789\%.$$

It is good but not unique in the unrestricted search. Its rank remains near the best $0.6\%$ of triples as the cutoff grows:

Many arbitrary triples fit the oscillation ratio more closely. For example,

$$(5,10,15)$$

gives a relative error of about $0.099\%$, and by $N=200$ some accidental triples match at the $10^{-3}\%$ level.

Therefore the oscillation ratio by itself is not strong evidence for the particular integers $(10,15,22)$. The unrestricted null does exactly what it should: it shows that one real-valued ratio can be approximated well by many freely chosen discrete triples.

8.2 Null constrained by the V2 mode-generation rule

The physically relevant comparison is narrower because the V2 indices are not arbitrary integers. The mode-selection construction generates the first two neutrino indices from a common seed $k$ through adjacent simplex firings,

$$n_1(k)=S(k,3),$$

$$n_2(k)=S(k,4),$$

and generates the third through the inclusion--exclusion pushout

$$n_3(k)=n_1(k)+n_2(k)-k.$$

For the committed seed $k=3$ this gives

$$S(3,3)=10, \qquad S(3,4)=15, \qquad 10+15-3=22.$$

The resulting index triple is then passed through the independent depth-five mass law,

$$m_i(k)=\mu_5 S(n_i(k),5).$$

Scanning the seed family $1\leq k\leq100$ gives the following leading results:

Within this constrained family,

$$\boxed{k=3\ \text{ranks first among all tested seeds }1\leq k\leq100.}$$

This distinction is essential. The unrestricted null shows that arbitrary integer triples can fit the measured ratio easily. The constrained null shows that the specific triple generated by the V2 contact-resonance construction is the best member of the tested seed-generated family.

The evidential content therefore lies not in the ratio alone but in the composition of two independent structures:

$$\text{mode selection} \longrightarrow (10,15,22) \longrightarrow \text{depth-five simplex masses} \longrightarrow R_\nu.$$

The remaining question is whether the seed rule and pushout are themselves uniquely forced by the projected master interaction rather than chosen after the spectrum is known.

9. A caution on extrapolated higher neutrino modes

The simplex law exists for every positive integer $n$, but this does not imply that every integer mode corresponds to a physical neutrino state. Physical states require the independent V2 mode-selection mechanism.

For example,

$$S(36,5)=658008,$$

so if $n=36$ were independently shown to be an occupied neutrino mode, its bare mass would satisfy

$$\frac{m(36,5)}{m(22,5)} = \frac{658008}{65780} =10.0031620553.$$

The large-$n$ approximation

$$\left(\frac{36}{22}\right)^5 \approx11.7327$$

overestimates the exact simplex ratio by about $17.3\%$.

No heavy-neutrino prediction should therefore be attached to $n=36$, or to any other unoccupied integer, until the selection mechanism derives that mode independently.

10. Falsifiable next calculation

The neutrino sector now presents two separate mathematical problems:

1. Discrete uniqueness: determine whether $(10,15,22)$ is exceptional among competing integer triples under fair null ensembles.
2. Geometric normalization: calculate $\mu_5$ from the projected master kernel rather than from either oscillation splitting.

The second problem is especially decisive. If the master projection independently produces

$$\mu_5\approx7.6\times10^{-7}\ {\rm eV},$$

then the three absolute neutrino masses and both oscillation splittings become outputs of the same geometry. If the projected normalization is incompatible with that scale, the neutrino realization of the V2 simplex mass law is false.