The mass spectrum selects particle depth

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 fixed-index optimized-scale test below is one diagnostic. The stronger complete-assignment and joint mass-and-ratio tests retain the electron as sole measured anchor and score W, down and up as targets. Their conditional numerical coverage is not a global discovery probability. The common fundamental belongs to the pattern kind: the electron, muon and tau share it across their different depths.

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

For a mode with positive integer index $n$ in a particle sector of real depth $d$, define the simplex count

$$S(n,d)=\binom{n+d-1}{d}$$

and the bare mass law

$$m(n,d)=\mu_d S(n,d),$$

where one scale $\mu_d$ is shared by every particle in the same sector.

The dimensional test fits one common scale to every occupied mode in a family. Within-sector ratios eliminate that scale exactly:

$$\frac{m_i}{m_j}=\frac{S(n_i,d)}{S(n_j,d)}.$$

Holding the stated mode indices fixed and scanning integer depths gives an unambiguous result:

$$\boxed{ d_{WZH}=2,\quad d_{ds}=3,\quad d_{uct}=4,\quad d_\nu=5,\quad d_{e\mu}=6,\quad d_\tau=10. }$$

The multi-entry family tests have unique minima even when every candidate depth receives its own optimized common scale. The neutrino test uses a splitting ratio. The tau comparison additionally assumes that its sector shares the electron sector's scale, $\mu_{10}=\mu_6$; it is a cross-depth test with that normalization held fixed.

The scale-free result is:

Given the stated mode indices, one shared scale within each family, and the additional charged-lepton scale relation, the benchmark mass ratios select the proposed particle depths, with large residual gaps to the neighbouring integer depths.

2. Why the count depends so strongly on depth

$S(n,d)$ is a theorem of combinatorics. It has several equivalent forms:

$$S(n,d) =\sum_{k=0}^{n-1}\binom{k+d-1}{d-1} =\#\left\{(a_0,\ldots,a_d)\in\mathbb N^{d+1}: \sum_{r=0}^{d}a_r=n-1\right\}.$$

Thus it is simultaneously

• the cumulative degeneracy through level $n-1$ of $d$ independent bosonic oscillator directions;
• the number of lattice points in the standard $d$-simplex dilated by $n-1$;
• the dimension of the degree-$(n-1)$ symmetric power of a $(d+1)$-component vector space.

The hockey-stick and Pascal identities are exact:

$$S(n,d)=\sum_{k=0}^{n-1}S(k+1,d-1),$$

$$S(n,d)=S(n,d-1)+S(n-1,d).$$

At large $n$,

$$S(n,d)=\frac{n^d}{d!}\left[1+O\!\left(\frac1n\right)\right].$$

Changing $d$ changes the shape and polynomial degree of a family spectrum. A common $\mu_d$ changes its vertical scale. Widely separated mode indices are especially powerful depth discriminators.

3. Scale-free family tests

The benchmark masses below are the particle-property values used for the scan. Quark-mass scheme dependence broadens their quantitative uncertainty; their depth minima remain unchanged.

3.1 Electroweak bosons select $d=2$

With

$$(n_W,n_Z,n_H)=(76,81,95),$$

the two independent ratios are

$$\frac{m_Z}{m_W}=\frac{S(81,d)}{S(76,d)}, \qquad \frac{m_H}{m_W}=\frac{S(95,d)}{S(76,d)}.$$

At $d=2$ their fractional errors are approximately $+0.03\%$ and $+0.04\%$. At $d=1$ they become $-6.07\%$ and $-19.76\%$; at $d=3$ they become $+6.45\%$ and $+24.41\%$. These ratios are scale invariant.

3.2 Down and strange select $d=3$; bottom remains open

The ordinary occupied counts in this comparison are

$$S(1,d),\qquad S(4,d).$$

Relative to the down-quark benchmark, $d=3$ predicts the strange ratio to about $+0.53\%$. At $d=2$ the error is $-49.73\%$; at $d=4$ it is $+75.94\%$. These two particles share one sector scale.

The bottom quark remains an unresolved exception to the ordinary integer sequence. The historical numerical proposal

$$B_d=\sqrt{S(16,d)S(17,d)}$$

is a nearby geometric mean, not an established beat mechanism, occupied particle branch or physical explanation of bottom's mass. It is excluded from the ordinary down/strange depth diagnostic below.

3.3 Up-type quarks select $d=4$

For

$$(n_u,n_c,n_t)=(3,20,72),$$

the scale-free charm/up and top/up ratios at $d=4$ differ from the benchmark ratios by about $+0.17\%$ and $+1.42\%$. The corresponding errors at $d=3$ are $-73.87\%$ and $-91.89\%$; at $d=5$ they are $+243.43\%$ and $+1001.16\%$.

The bare ratios isolate $d=4$ before interaction corrections are introduced.

3.4 Neutrino splittings select $d=5$

The mass-squared splitting ratio is independent of the absolute neutrino-mass scale. For

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

the unknown scale cancels from the normal-ordering splitting ratio:

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

The atmospheric splitting is conventionally quoted as $\Delta m^2_{32}$. For this ratio it must first be converted to $\Delta m^2_{31}=\Delta m^2_{32}+\Delta m^2_{21}$. The PDG 2024 central values therefore give $R_\nu^{\rm obs}\simeq0.02976$. The predictions are

$$R_\nu(4)=0.05550, \qquad R_\nu(5)=0.03035, \qquad R_\nu(6)=0.01686.$$

Depth five misses this benchmark ratio by about $1.98\%$. The neutrino module uses $\Delta m^2_{21}=7.49\times10^{-5}$ and $\Delta m^2_{31}=2.534\times10^{-3}$ eV$^2$, giving about $2.68\%$ instead. These are different input benchmarks, not two errors for identical inputs. Neither percentage is an uncertainty-weighted exclusion. The adjacent depths miss the benchmark used here by about $86.49\%$ and $43.33\%$, respectively. This is a genuinely scale-free selection of the odd-dimensional neutrino sector.

3.5 Electron and muon select $d=6$

With

$$(n_e,n_\mu)=(13,35),$$

the prediction is simply

$$\frac{m_\mu}{m_e}=\frac{S(35,d)}{S(13,d)}.$$

At $d=6$, $S(35,6)/S(13,6)=3838380/18564=3515/17=206.764705882\ldots$. Using $m_e=0.51099895$ MeV and $m_\mu=105.6583755$ MeV gives a signed relative error of approximately $-0.00173\%$ (predicted minus observed, divided by observed). At $d=5$ it is $-55.00\%$ and at $d=7$ it is $+115.79\%$. Because the electron fixes the one common scale, each candidate depth makes a fixed muon-mass prediction.

3.6 The tau comparison with a shared charged-lepton scale

Assume the tau sector shares the charged-lepton scale. Anchoring that scale with the electron gives the depth scan

$$m_\tau(d)=m_e\frac{S(23,d)}{S(13,6)}.$$

The neighbouring results are

$$m_\tau(9)=554.93\ {\rm MeV},$$

$$m_\tau(10)=1775.79\ {\rm MeV},$$

$$m_\tau(11)=5327.36\ {\rm MeV},$$

compared with $1776.93\ {\rm MeV}$. The bare $d=10$ result is low by $0.064\%$. The scale was fixed in the six-dimensional charged-lepton sector, so the tau mass tests the ten-dimensional count.

4. Scale-optimized depth scan

For a family with benchmark masses $M_i$ and counts $C_i(d)$, allow every candidate depth to choose the common scale that minimizes logarithmic error:

$$\log\widehat\mu_d =\frac1N\sum_{i=1}^{N}\log\frac{M_i}{C_i(d)},$$

$$\epsilon_d =\sqrt{\frac1N\sum_{i=1}^{N} \left[\log\frac{\widehat\mu_dC_i(d)}{M_i}\right]^2}.$$

Scanning $1\leq d\leq15$ with an independently optimized scale at every depth gives:

For these four tabulated families, the runner-up depths have residuals larger by factors of roughly $106$ to $44{,}500$. The neutrino splitting ratio selects $d=5$ at its stated indices. The tau-to-electron ratio selects $d=10$ under the additional shared-scale assumption in section 3.6.

5. Result, inputs, and operator completion

Established mathematics

1. $S(n,d)$ is an exact simplex and cumulative-oscillator count.
2. Its dependence on $d$ survives optimization of a shared sector scale.
3. At the stated indices and scale relations, the family ratios and tau-to-electron comparison recover the six proposed depth labels.
4. These are the same depth labels whose minimal projection spaces generate the candidate coupling stabilizers and spinor branchings. The mass data therefore test labels that also organize the coupling geometry.

The depth fitting each family ratio is also the depth whose projection geometry supplies its candidate coupling structure. This is a cross-consistency condition between two independently computed uses of $d$.

The dimension $d$ is the rank of $T^*Q_d$ in the cumulative mode bundle. Its simultaneous occurrence in the kinetic principal symbol, mass growth, and contact projection is analyzed in particle depth as spatial geometry.

Conditional inputs

The result is conditional on the mode indices. If both $n$ and $d$ were allowed to vary independently for every particle after inspecting its mass, the search would acquire enough discrete freedom to generate accidental matches. The depth test has evidential force to the extent that the mode indices are fixed by one common, mass-independent selection construction.

The relation $\mu_{10}=\mu_6$ is a further input to the tau comparison. For a single tau mass and a freely adjustable scale at each candidate depth, choosing $\mu_d=M_\tau/S(23,d)$ gives zero fitted residual at every depth. The tau therefore contributes a depth test only when its scale is constrained independently of that one mass, as in the stated electron anchor.

The covariant self-contact construction derives $m=\mu_dS$ once the projected master interaction is local on the occupied coherent mode ray. One shared $\mu_d$ further requires that its scalar contact strength be independent of the band label $N=n-1$. The family ratios provide evidence for these two operator properties; their final proof belongs to the master-kernel projection.

6. Falsifiable next calculation

The decisive next step is to calculate

$$P_{d,N}\mathcal K_\infty P_{d,N}$$

and prove that its scalar contact channel obeys

$$P_{d,N}\mathcal K_\infty P_{d,N} =\mu_dS(N+1,d)|z;N\rangle\langle z;N|.$$

The projected calculation must supply

• one shared $\mu_d$ for all occupied modes of sector $d$;
• a self-interaction local on the occupied coherent mode ray;
• a selection rule for the stated $n$ values;
• how the odd-depth sphere sectors and even-depth projective sectors realize the same simplex law;
• the class of interaction corrections that preserves the bare depth test.

The rank, coherent normalization, and mass eigenvalue then follow as theorems. If the projection produces an identity, kinetic, or ray-averaged quartic operator instead, the numerical recovery remains real but the proposed self-contact origin is false.

7. Data reference

The benchmark values are drawn from the Particle Data Group summary tables; the scan should be rerun whenever the reference inputs or quark-mass schemes are changed: Particle Data Group, Review of Particle Physics 2024.

The complete scan and its assertions are executable in the V2 verification program.