Simplex spectrum and sector assignments

The maintained mass-frequency rule uses the cumulative simplex count, interpreted in the current physical proposal as the number of portions sharing one fundamental. The unit is shared within a kind of pattern; the electron, muon and tau share it across depths six and ten. Its numerical restrictions are substantial, but they answer a different question from the field dynamics that must sustain the patterns. Known masses are legitimate inputs to the constrained inference of depths and indices; those inferred assignments then become inputs to tests of candidate actions.

Depth describes a pattern's involvement in ordinary physical spatial directions. The physical setting has infinitely many such directions plus time; matter occupies up to ten, with every remaining direction unbound. Different dimensional layers can belong to this one nested manifold. A coefficient count is not itself proof of confinement to a layer. No compactification, additional configuration-space dimensions or mediator particles enter the count law.

Reviewed Connections as of 5 November 2026

There is now a genuine conditional energetic selector, beyond a supplied list of admissible counts. At zero phase-link coupling J=0, on a supplied cone with a normalized shape energy, its global minimum has completed support

$$C=\binom{k+d}{d},\qquad n=k+1,\qquad \binom{k+d}{d+1}<\frac{2\lambda}{\kappa} \le\binom{k+d+1}{d+1}.$$

The positive ratio fixes one completed rung at the given depth. The selector and its assumptions retain a common supplied fundamental independently of total norm. Other wrong-count stationary states can coexist; observed multiple rungs at one depth and physical formation are not derived by this minimum criterion.

A matrix-valued cell realization retains prepared spin/color/charge carriers. The later centered-window law controls the approach to first-simplex weights with a positive gradient term. Small nonzero exterior weights there are not absent portions; exact observed high counts have not followed.

The count and the frequency law

For a positive integer index $n$, put $N=n-1$ and define

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

The configurations being counted are explicit:

$$K_N^{(d)}=\{k\in\mathbb N^d:|k|\le N\},\qquad D_{d,N}=\sum_{j=0}^{N}\binom{j+d-1}{d-1}.$$

Here $\mathbb N$ includes zero. The degree-$j$ shell has $\binom{j+d-1}{d-1}$ labels; summing all shells gives the Hockey-Stick identity. Equivalently this counts monomials of degree at most $N$ in $d$ variables. A slack coordinate converts $|k|\le N$ to an equality in $d+1$ integer entries, without introducing another physical spatial direction. The recursion is Pascal's identity,

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

The proposed mass and angular-frequency laws are

$$m_{n,d}=\mu_dS(n,d),\qquad \omega_{n,d}=\omega_{0,d}S(n,d),\qquad \omega_{0,d}=\frac{\mu_dc^2}{\hbar}.$$

Mass and frequency here are equivalent units through $\omega_m=mc^2/\hbar$. The open mechanism is the dynamics producing and selecting these frequencies, not that conversion. A count of available modes does not establish their occupation, equal participating energy or a temporal frequency under an arbitrary kinetic law.

These are fifteen massive-state comparisons. The separate photon convention $n=0,d=2$ gives zero rest count, not a zero physical photon oscillation frequency. Bottom needs a count near 895 and lies between the rungs 816 and 969; its count is their arithmetic mean, 892.5, as the section on bottom below sets out. The count law does not imply an observed particle at every positive index.

The sector case places this numerical organization alongside the separate spin, charge, orientation and spatial-construction evidence. Those related tests retain their own premises.

Shared units and the absolute mass comparison

The maintained non-neutrino numerical prescription uses dimensionless sector coefficients

$$g_{22}=\frac{1445}{2},\qquad g_{33}=8\sqrt7,\qquad g_{44}=\frac{12}{\sqrt7},\qquad g_{66}=g_{10,10}=\frac14,$$

with fixed reference counts

$$C_2^\star=C_3^\star=1,\qquad C_4^\star=15,\qquad C_6^\star=18564,$$

and

$$\mu_d=\frac{m_e}{C_d^\star}\sqrt{\frac{g_{dd}}{g_{66}}} \quad(d=2,3,4,6),\qquad \mu_{10}=\mu_6.$$

The last equality additionally assumes the proposed equal normalized overlaps; equal coefficients alone do not prove equal units. In MeV/$c^2$, the evaluated units are approximately

$$(\mu_2,\mu_3,\mu_4,\mu_6) =(27.47064959,\ 4.701857978,\ 0.1451024874,\ 0.00002752633861).$$

The electron is the sole measured dimensional reference when evaluating this recipe. Their possible interpretation as physical coupling constants remains open. One empirical reference per depth is a separate calibration option; it relaxes the cross-depth constraints and must not inherit their evidential strength. The neutrino absolute unit is a separate problem.

Nine of these ten non-reference, non-neutrino outputs lie within 1% of their dated benchmarks; top is about 2.20% high. These are reproduction values, not current world averages or a common-scheme uncertainty-weighted fit. Quark mass conventions matter. A small percentage residual need not be agreement within experimental uncertainty: the muon discrepancy greatly exceeds its measurement uncertainty.

The electron is not the depth-six fundamental. It and the muon multiply the same unit by 18564 and 3838380 respectively. The shared-unit comparison gives

$$\frac{m_\mu}{m_e}=\frac{3838380}{18564}=\frac{3515}{17} \simeq206.7647059.$$

Its residual is about −17.3 ppm. Exact adjacent-rung spacings are

$$\frac{S(n+1,d)-S(n,d)}{S(n,d)}=\frac dn,\qquad \frac{S(n,d)-S(n-1,d)}{S(n,d)}=\frac d{n+d-1}.$$

At the muon rung they are 17.14% upward and 15% downward. Even the smaller step is roughly 8,700 times the residual. Changing the index cannot supply a small correction, and changing the unit moves the electron too. This is numerical rigidity; it is not a chance probability.

What the depth tests establish

Two comparisons must be kept distinct. A diagnostic holds the maintained indices fixed and optimizes one common scale at each trial depth from 1 through 15. It favors depths 2, 3, 4 and 6 for W/Z/H, down/strange, up/charm/top and electron/muon. Their RMS log errors are approximately $0.0001771,0.002667,0.006303,0.000008650$; the next-best values are $0.09185,0.2825,0.9795,0.3846$. These separately optimized diagnostic scales are not inputs to the electron-based prescription above.

The stronger complete assignment theorems move the intact blocks $(WZH,dsb,uct,e\mu,\tau)$ bijectively among $\{2,3,4,6,10\}$, retain the prescribed scales and all benchmark masses, and permit every positive integer index, without an upper cutoff. They impose the exact electron reference equation and equality of the electron/muon and tau units.

Within these hypotheses, even the widened requirements of at most 0.5% W error and 100% up-quark error force the maintained depth arrangement. The reference equation uniquely places the electron at $(d,n)=(6,13)$; shared lepton unit and distinct depths then place tau at ten. Exact adjacent-rung inequalities exclude W at three or four, leaving W/Z/H at two. The first up-quark rung at three already exceeds twice its benchmark, forcing that family to four and the down-type family to three.

A second theorem requires every individual absolute log error to be no worse than its maintained value. It uniquely fixes both that arrangement and every comparison index. Monotonicity covers all positive indices:

$$\frac{S(n+1,d)}{S(n,d)}=\frac{n+d}{n}>1.$$

For any increasing ladder $P(n)$, a unique nearest log-rung $k>1$ is certified by

$$P(k-1)P(k)<M^2<P(k)P(k+1).$$

Only the upper inequality is needed for $k=1$. The maintained study checks these inequalities with exact rational certificates after eliminating its algebraic scale factors. In these theorems bottom enters at its nearest single rung, index 17, count 969, with about +8.9194% error. That is the tolerance of a single-rung comparison; bottom's own count is the two-rung mean given below.

Of 120 permutations, the electron reference admits 24 and the shared lepton unit leaves six. Only the maintained arrangement meets every individual accuracy. The closest different admissible arrangement swaps the up- and down-type depths, with approximately +8.06% down, +10.96% strange and +117.68% up errors; its reported aggregate RMS log error is about 8.89 times larger. That aggregate comparison is additional numerical evidence.

These are conditional uniqueness results for the stated depths, blocks, reference equation and scale law. The reference count already contains 18564, so recovering the electron is input consistency. The proofs neither derive those hypotheses nor exclude every possible alternative scale law.

The separate neutrino test

For normal ordering and the maintained depth-five triple,

$$R_\nu=\frac{\Delta m_{21}^2}{\Delta m_{31}^2} =\frac{11628^2-2002^2}{65780^2-2002^2} =0.0303498422349.$$

The scale cancels. The sector-case benchmark $\Delta m_{21}^2=7.53\times10^{-5}$ and $\Delta m_{32}^2=2.455\times10^{-3}$ eV$^2$ gives $\Delta m_{31}^2=\Delta m_{32}^2+\Delta m_{21}^2$ and an observed ratio $0.0297593170770$, making the count result 1.98% high. The neutrino module retains a different dated pair, $7.49\times10^{-5}$ and $2.534\times10^{-3}$ eV$^2$ for the 21 and 31 splittings, giving a 2.6789% residual. These benchmark sets must not be interchanged. The first JUNO measurement of the solar splitting, $7.50\pm0.12$ in the same units, is compared in neutrino controls.

At the same indices the neighboring-depth ratios are approximately 0.05550 at four and 0.01686 at six. That is a fixed-index depth discriminator. It is not unrestricted uniqueness of the triple: many freely chosen integer triples approximate one measured ratio. The neutrino indices also enter the generation law below, which ties each of them to a charged lepton and a quark.

One splitting can calibrate $\mu_5$, after which the other tests the pattern. For the module's solar benchmark,

$$\mu_5=\sqrt{\frac{\Delta m_{21}^2}{11628^2-2002^2}} \simeq7.55562\times10^{-7}\ {\rm eV}/c^2,$$

giving approximately $(1.513,8.786,49.701)$ meV/$c^2$, with sum $0.0600$ eV/$c^2$. These are conditional inferred masses, not three measured absolute masses or a derivation of the neutrino unit. See neutrino controls for the wider comparison.

Real spatial realizations of the count

The count does not require complex projective physical space. On a real amplitude sphere $S^d$, the parity band

$$V_N=\mathcal H_N\oplus\mathcal H_{N-2}\oplus\cdots, \qquad \dim\mathcal H_j=\binom{j+d}{d}-\binom{j+d-2}{d}$$

has telescoping dimension $D_{d,N}$. This is not the full spherical harmonic band through $N$. Under uniform normalized sphere measure its evaluation kernel has diagonal $K_N(w,w)=D_{d,N}$. The real spatial contact construction maps these functions to finite-norm profiles in ordinary flat space. The amplitude coordinates describe shapes; they are not physical position. Under its spatial map, the projector diagonal is $D\rho_L(x)$ and integrates to $D$. A constant normalized shape diagonal therefore does not imply uniform contact per physical volume.

For a normalized covariant rank-one contact resolving the identity, its bright eigenvalue is $D$. A real canonical pair with $H=\mu(q^T\Lambda q+p^T\Lambda p)/2$ then has bright frequency $\mu D$. A second-order oscillator with the same stiffness and fixed inertia instead gives a frequency proportional to $\sqrt D$. The kinetic law is essential. The shape-symmetry, norm and contact premises do not follow from ordinary spatial rotation symmetry alone.

A different, actual spatial carrier uses weighted polynomials on the ordinary ball $B^d$. With

$$w=\sqrt{1-|x|^2},\qquad L=-w^{-1}\nabla\cdot[w(I-xx^T)\nabla],$$

the degree-$N$ orthogonal shell has eigenvalue $\lambda_N=N(N+d+1)$ and cumulative dimension $D_{d,N}$. There is an explicit positive polynomial $P_d$ with $P_d(\lambda_N)=D_{d,N}^2$. For example,

$$P_2(L)=\frac{(L+2)^2}{4},\qquad P_3(L)=\frac{(L+4)(L+3)^2}{36},\qquad P_6(L)=\frac{(L+6)^2(L+10)^2(L+12)^2}{518400}.$$

Writing $\mu_d$ as the frequency unit in this action,

$$\mathcal A[f]=\frac12\int dt\int_{B^d}w \big[(\partial_t f)^2-\mu_d^2fP_d(L)f\big]d^dx$$

gives $\partial_t^2f+\mu_d^2P_d(L)f=0$ and exactly $\omega_N=\mu_dD_{d,N}$. Thus a real spatial operator and an action can realize the complete count-frequency ladder. Within the family $w_a=(1-r^2)^a$, demanding the exact infinite squared-count law as a polynomial in the corresponding $L_a$ fixes $a=1/2$.

The flat-ball construction is inverse and constitutive: it supplies the material law, boundary domain and high-order stiffness. Interior high-frequency dispersion grows as $|k|^d$ and does not establish finite propagation speed. The physical radius, sustaining stress, transverse envelope and responding support require their own dynamics. Its scalar spectrum does not derive spin, charge or the massless photon. Compatibility with every assigned depth is useful; it is not selection of those depths or their occupied cutoffs.

Occupation, flat clocks and the remaining physical bridge

In the current interpretation, $C=S(n,d)$ counts physical portions, each with proper fundamental $\Omega$. Counting each portion's cycle crossing gives average rate $C\Omega/(2\pi)$, including arbitrary phase offsets if simultaneous crossings retain their multiplicity. This does not require the summed waveform to have Fourier fundamental $C\Omega$. It does require a physical registration mechanism.

One subdivided $(d-1)$-simplex sheet contains $\binom{k+d-1}{d-1}$ sites. Summing simultaneous sheets from degree zero through $k=n-1$ gives $\binom{k+d}{d}=C$. Moving one sheet through those positions is not the same simultaneous count. Geometry and the field's support must establish which construction is actually present.

The following occupation equations belong to earlier spectral comparisons. They remain useful mathematical constructions, but their count-dependent component phase rates must not be presented as the current common fundamental of the physical portions.

An integrated density of states counts what is available. To obtain total energy $E=\varepsilon_dD$, a filled band needs equal participating energy per counted mode, not equal total energy for every band. A coherent classical field also has interference terms; an incoherent trace cannot replace its actual spatial energy density.

The occupation and coherence constructions separate these requirements. A common nonnegative occupation potential can have exactly the uniformly occupied cumulative prefixes as its zero set. An inverse participation clock then uses classical component weights $p_k$, $\sum p_k=1$, and

$$r=\sum_kp_k^2,\qquad m=\frac{\mu}{r}.$$

Uniform occupation of $D$ labels gives $r=1/D$. More explicitly, for canonical actions $I_k$, $n=\sum I_k$ and $R=\sum I_k^2$,

$$H_{\rm count}=\mu\frac{n^3}{R},\qquad \partial_{I_k}H_{\rm count} =\mu\left(\frac{3n^2}{R}-\frac{2n^3I_k}{R^2}\right).$$

On the occupied uniform prefix $I_k=n/D$, every occupied phase rate is $\Omega=\mu D$ and $H=n\Omega$. This gives a flat occupied spectrum under an explicit interacting Hamiltonian; empty intensity derivatives are instead $3\mu D$. Rescaling total action changes energy without changing the frequency. A common integrated action normalization, including the proposed value $\hbar$, is an additional physical requirement. Occupation weights are classical field data, not position probabilities.

Inverting the fitted count-ratio mixing rule adds a separate requirement: lower-rung bands must nest inside higher-rung bands with the stipulated occupation and projection. Alternating sphere parity bands can have the correct dimension but fail that nesting. Uniform component occupation, nested coefficients and uniform complete energy per physical volume are three different statements; none follows merely from the count. See spatial carriers and the mixing analysis.

For the proposed local energy clock $|\Omega(x)|=h(x)/\sigma_0$, one coherent rate requires uniform participating energy density. An equipartitioned band's trace density is proportional to $K_N(x,x)w(x)$, so its integrated trace $D$ is insufficient. If the participating volume is $V$, the uniform branch gives $E/|\Omega|=\sigma_0V$; identifying this with the common action unit requires $\sigma_0V=\hbar$. This is a condition on that local-clock branch, not a general prohibition on nonuniform field patterns. A collective clock responding to integrated energy is a distinct branch and must retain its own action and normalization. Sphere comparisons and flat spatial carriers cannot exchange these conclusions without the bridge.

Later candidate actions establish particular reference clocks, complete translation responses, local persistence estimates and constrained occupation behavior. Each result belongs to its written action and domain. They do not jointly constitute one proof of the physical mass operator. The current action comparison and shared-scale analysis retain those distinctions. The six units, observed cutoffs, common normalization, physical confinement and reciprocal coupling still need a common same-field explanation. Bottom's physical level description and the neutrino calibration remain open.

Finally, an electron's internal classical field and its spectral coefficients remain distinct from its physical position. The baseline electron follows a literal deterministic trajectory in four ordinary spatial directions. Neither a count projector nor its spatial energy density replaces that trajectory with an electron cloud. The count law constrains the particle frequency; the orbit-plus-precession and reciprocal same-field dynamics must explain how the corresponding spatial particle moves and interacts.

Counting on Complete Straight-Fiber Sections

An extended density can have ridges rather than isolated maxima in all coordinates. A precise conditional count is now available when its free directions are constant straight directions. Write the density as $\rho(b,z)=q(b)$, with bound coordinates $b$ and free coordinates $z$. A complete section chooses one point on every free fiber, $z=h(b)$. Pulling back the scalar density gives exactly $q(b)$, so its strict maxima and nondegenerate Hessian signs agree on every such section. An arbitrary observer plane need not be a complete section, and an area-weighted density is a different observable.

For the supplied positive product $\prod_j[1+a\cos(2\pi h b_j/L_j)]$, with $0<a<1$, the strict interior maximum count is $(h-1)^d$. This independently assessed example defines a usable ridge-count diagnostic. It does not identify $h$ with the particle rung, turn that product into $S(n,d)$, or select observed counts. Its extruded-domain evolution statement retains finite free-volume factors and a regular-solution uniqueness premise. Curved or contact-ended extensions need their own construction. The finite regulator does not compactify physical space or make a nonzero uniform infinite extension have finite ordinary norm.

The case's count-selection discussion keeps this diagnostic separate from an actual portion-selection law.

The geometry of the count

The count $S(n,d)=\binom{n+d-1}{d}$ is the number of lattice points of a $d$-dimensional simplex with $n$ levels. Reading it as a shape gives exact relations among the assigned patterns, and the later flavor and decay results use them directly.

Faces. A $j$-dimensional face of the $n$-level simplex holds $S(n,j)$ points.

Corners. A corner of side $m$ is itself a simplex with $m+1$ levels. Cutting three levels off strange leaves down at its corner: $1$ point of $20$. Cutting 17 levels off charm leaves up, $15$ of $8855$. Cutting 69 levels off top leaves up, $15$ of $1215450$, and cutting 52 leaves charm.

Sides. Pascal's identity $S(n,d)-S(n,d-1)=S(n-1,d)$ removes one side.

Layers. Summing the slices gives $S(n,d)=\sum_{k\le n}S(k,d-1)$. A simplex at depth $d$ is a stack of the simplices at depth $d-1$. The layers of a depth-four pattern are depth-three patterns: down is the first layer of up and strange is the fourth layer of charm.

Interiors. Removing every side leaves an interior with $n-d-1$ levels. Tetrahedra therefore step down by four at each peel.

Combination. Vandermonde's identity is the coefficient of $z^d$ in $(1-z)^{-a}(1-z)^{-b}$:

$$S(a+b,d)=\sum_{k=0}^{d}S(a,k)\,S(b,d-k).$$

Two level sets joined into one have a count assembled term by term from the counts of their parts at complementary depths.

Shape statistics

The electron's 18,564 sites are the weak compositions of 12 into 7 parts. For side $k$ and $m=d+1$, the regular embedding has the isotropic second moment $\lambda=k(k+m)/(2m(m+1))$. Isotropy is a property of the regular embedding: the corner embedding has an anisotropy of $d+1$ to 1, and the fourth moment is anisotropic in both. The electron simplex has rms radius 3.495, circumradius 7.856 and inradius 1.309 in lattice units. The simplex is the reading of the label count; these numbers describe that reading and are not yet the field's spatial profile.

Bottom

Bottom's measured mass needs a count near 895 at the down-type unit of 4.67 MeV. The neighbouring rungs are $S(16,3)=816$ and $S(17,3)=969$. Their arithmetic mean is 892.5, which gives

$$m_b=892.5\times4.67\ \text{MeV}=4{,}168\ \text{MeV},$$

against the measured $4{,}180\pm30$ MeV. The mean is exact in the geometry of the three-dimensional simplices:

$$S(2n+2,3)=4\,[S(n,3)+S(n+1,3)],\qquad\text{so}\qquad \frac{S(34,3)}{8}=\frac{7140}{8}=892.5 .$$

Bottom is the 34-level tetrahedron taken three octaves down. A ring of 7,140 portions with a phase ripple of about two even steps puts its strongest component at harmonic 895.

The two rungs are also tied to the other quarks by exact peeling. A side of charm is a 20-level tetrahedron; peeling it gives 16, 12, 8, 4, so its interior is exactly the bottom-16 tetrahedron of 816 points. The bottom-16 core is strange (12, 8, 4) and the bottom-17 core is down (13, 9, 5, 1). A side of top passes through 20, 16, 12, 8, 4. Bottom therefore sits inside charm and inside top, with strange or down at its own core. What physically holds a pattern between two rungs remains open.

The generation law

Each charged lepton is a quark plus a neutrino:

$$13=3+10,\qquad 35=20+15,\qquad 23=1+22 .$$

The electron's index is the up quark's plus the first neutrino's; the muon's is the charm quark's plus the second neutrino's; the tau's is the down quark's plus the third neutrino's. Each neutrino is used once, in order. Vandermonde's identity applies to the whole generation: the lepton's count is assembled from its members' counts at their depths. In the tau expansion, for example, the term $k=5$ is $S(22,5)S(1,5)=65780$, which is the third neutrino's count, while the term carrying the actual depth-three factor is $S(22,7)S(1,3)$. The generation law, the decay product groups of volume 19 and the neutrino mass ratios of volume 16 are three separate results.

The depth tie

With the electron fixed at $(d,n)=(6,13)$, the muon's mass is matched to 0.0017% only at depth six among the sparse depths, and its index there is the generation law's 35. The tau's mass is matched to 0.064% only at depth ten, and its index there is the generation law's 23. A third generation at depth six would need $n=94$ and give 30.8 GeV. The neutrino indices give the splitting ratio only at depth five. The mass fit and the generation law select the same depths.

The tau is a complete simplex only at depth ten

The tau count 64,512,240 is a complete simplex count at exactly three places for $d=1,\dots,40$: the trivial $d=1$, then $d=10$ with $n=23$, and $d=22$ with $n=11$. For $2\le d\le 21$ depth ten is the only one, and no six-direction simplex has that count.

The exact branch at every depth

For $d$ bound directions, the stationary branch of

$$i\Phi_t=-i\gamma\cdot\nabla\Phi+M\beta\Phi-g\rho^{1/9}\Phi$$

at the continuum edge frequency $-M$ has the closed form

$$f=A\left(1+\frac{r^2}{a^2}\right)^{-5},\qquad a=\frac{10-d}{2M},\qquad g(s_0A^2)^{1/9}=\frac{20M}{10-d},$$

with energy per norm

$$\frac{E}{N}=-\frac{8M}{9}\,\frac{9-d}{10-d}.$$

Its tail is algebraic, $f\sim r^{-10}$. The energy per norm vanishes at $d=9$ and is positive for $9<d<10$. At $d=10$ the width $a$ is zero, so depth ten exists on this branch only as a limit. A direct Clifford check at depths 4, 6 and 8 leaves residuals of $10^{-11}$ or smaller.

The norm of the branch family below the edge follows

$$N=\int(F^2+G^2)\,d^dx=c_d\,(m_0-\omega)^{9-d/2},\qquad N\propto g^{-9},$$

with fitted exponents 8.0000, 7.4998, 6.9995, 6.4991, 5.9984 and 3.9922 at depths 2, 3, 4, 5, 6 and 10, which agree with $9-d/2$. The coefficients and their use in fixing one frequency per depth are given in volume 9.