Common scales across particle sectors

The maintained mass law is

$$m_{n,d}=\mu_d S(n,d),\qquad S(n,d)=\binom{n+d-1}{d}.$$

The dimensional unit belongs to a kind of pattern. The historical notation $\mu_d$ labels the maintained depth assignments, but does not assert that depth alone determines the unit. In particular, the electron, muon and tau share one proposed charged-lepton fundamental despite the tau's different depth. In angular-frequency units,

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

The conversion between mass and frequency is ordinary physics. The unfinished IDWT problem is the same-field mechanism that produces and sustains these frequencies and selects the occupied patterns. The known target frequencies are not unknown observations awaiting discovery.

Three statements have different evidential status: the binomial is an exact mathematical count; its use as a particle-frequency multiplier is the maintained law; and the numerical prescription for the dimensional units remains conditional. A candidate action that retains supplied units does not thereby derive them. The sector evidence gives the complete numerical comparison and the constraints supplied by the shared assignment.

Reviewed Connections as of 5 November 2026

The newer support and property calculations do not replace this scale prescription. The centered-only and centered-window laws supply their rest clock Omega. The six-depth density comparison supplies m0=g=1 and omega/m0=0.99; these are uncalibrated comparison units, not new predictions of sector scales. The cross-depth radial result and selected-weight spatial law improve compatibility while leaving the physical source of the fundamental unfinished. Electron, muon and tau retain their common kind-dependent unit, and the neutrino absolute calibration remains separate.

The maintained non-neutrino prescription

The stated dimensionless sector coefficients are

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

With fixed reference counts

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

the proposed scale chain is

$$\mu_d=\frac{m_e}{C_d^\star} \sqrt{\frac{g_{dd}}{g_{66}}},\qquad \mu_{10}=\mu_6.$$

The electron is the sole measured dimensional reference when evaluating this prescription. The $g_{dd}$ are sector coefficients here. Their identification with physical interaction strengths would require a further derivation.

Using the archived reference $m_e=0.51099895000$ MeV/$c^2$, the scale prescription gives:

The current equality $\mu_{10}=\mu_6$ is the common-kind fundamental premise. An earlier overlap interpretation required equality of normalized overlaps; the numerical coefficients alone did not derive that equality. The resulting tau mass is approximately $1775.78576$ MeV/$c^2$, about $0.0644\%$ below the archived $1776.93$ comparison. That is close on a percent scale but not agreement at the quoted precision of the reference. Top differs by about $2.20\%$ under this prescription. These residuals belong in the empirical comparison; the current operator-construction target is the formula law itself, without fitting its residuals. Bottom's count is the arithmetic mean of its two neighbouring rungs, 892.5, for the exact geometric reason given in volume 5; at this unit it gives 4,196 MeV, and at the measured down-type unit below it gives 4,168 MeV, against $4{,}180\pm30$ MeV.

Quark-mass parameters depend on the stated mass definition, scheme and scale; the archived comparison does not turn them into directly observed free-quark rest masses. A common action must ultimately connect its frequency predictions with the relevant observables. The scale-free family comparisons also remain distinct from the prescription: fitting a fresh common scale at each trial depth is a diagnostic of the ratios.

Separate empirical calibration of one reference per sector is a legitimate alternative when the cross-sector equations are not justified. It supplies additional inputs and removes the corresponding absolute cross-sector predictions. It must not be presented as the same one-reference prescription.

Neutrinos retain their own absolute-scale question

The maintained mass-state counts are

$$(C_1,C_2,C_3)=(2002,11628,65780),\qquad m_i=\mu_5 C_i.$$

They predict the scale-free relation

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

An absolute unit can be inferred conditionally from one measured splitting:

$$\mu_5^{(21)} =\sqrt{\frac{\Delta m_{21}^2}{11628^2-2002^2}}.$$

The neutrino module retains its dated solar input $7.49\times10^{-5}$ eV$^2$, giving $\mu_5^{(21)}=7.5556190385\times10^{-7}$ eV/$c^2$. The sector evidence separately retains the archived $7.53\times10^{-5}$ solar benchmark. The resulting scale-free residuals are approximately $2.68\%$ and $1.98\%$, respectively, because the atmospheric benchmarks differ as well. They are not two error estimates from identical inputs.

This calibration gives conditional absolute masses of order $1.5,8.8,50$ meV/$c^2$. It is not three independently measured absolute masses or a derived neutral-sector unit. The earlier expression

$$\mu_5=\frac34\frac{\mu_6^3}{\mu_4^2}$$

is a phenomenological scale ansatz. Its powers and factor must come from a physical equation before it becomes a derived scale relation. Neutrino mass states must also remain distinct from flavor states and from the response that gives nonzero weak interactions despite zero ordinary electric charge. A small mass unit alone explains none of those distinctions.

The measured family units

Dividing each measured mass by its count gives the mass of one portion in each family. With 2026 Particle Data Group values:

The down-type unit is the down quark's own mass, because its count is one. The up-type unit is the up quark's mass divided by fifteen.

Two of these units connect to something outside their family.

The lepton unit is the hartree to 1.16%. The hartree, the energy scale of the atom, is 27.211 eV. It equals $\alpha^2m_e$, so the same statement reads

$$C_e\,\alpha^2=\frac{18564}{137.036^2}=0.98856 .$$

The electron's count is the inverse square of the fine-structure constant, 18,779, to 1.14%.

The boson unit is a million lepton units to 0.26%. Its ratio to the lepton unit is $0.99739\times10^6$, and its wavelength of 45.16 fm is the lepton unit's 45.03 nm scaled down by a million.

If each family's unit were the binding energy of a hydrogen-like state of one member, the coupling would be $\sqrt{\text{unit}/\text{mass}}$. For the electron this is 0.0073, which is $\alpha$. For the up quark it is 0.258, for the down quark 1.000 and for the W 0.0185; those are not their families' couplings. The relation belongs to the leptons alone.

Planck's constant as the field norm

Every property worked out for a particle so far is proportional to one number, the norm of its stationary solution:

$$N=\int(F^2+G^2)\,d^dx .$$

Spin is $N/2$, charge is a fixed multiple of $N$, and energy is $\omega N$ less a small binding share. Physics states the same three facts with $\hbar$ in place of $N$. If every particle carries the same norm, Planck's constant is that norm, and the count law becomes a mass law: the norm stays fixed while the frequency rises with the count.

Near threshold the norm of the density-law branch obeys

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

with, at $m_0=g=1$,

A universal norm therefore fixes one gap at each depth,

$$s_d=\left(\frac{\hbar}{c_d}\right)^{1/(9-d/2)},$$

and ties the coupling to it. Indices and counts enter through the family unit. Depth eighteen is the neutral case of the scaling: below it, which includes every particle depth from 2 to 10, the norm vanishes as the gap closes.

The field law does not yet select the value of the universal norm. That selection is the same question as what selects the levels $n$: whether the allowed levels are the frequencies at which the norm takes the universal value is open, as are the 1.14% between the electron's count and $1/\alpha^2$ and the 0.26% between the boson unit and a million lepton units.

The current common-fundamental target

For a kind $a$, write its fundamental as $\Omega_a$. The target is $\omega_m=C\Omega_a$, with no independently adjustable frequency for each portion. Same-kind patterns at different counts share $\Omega_a$; different kinds need not share it. Arbitrary phase offsets retain the additive long-time count, provided the registration counts each portion's event with multiplicity. This does not establish a pure Fourier line at $C\Omega_a$ for uneven phases.

The proposed common size within a kind could help explain its common clock, but an actual spatial operator must relate size, shape and $\Omega_a$. The newer canonical construction supplies $B=\Omega_a[P+\hbar(C-1)]+U$. On $P=\hbar,U=0$, a supplied relativistic kinetic law gives $m=C\hbar\Omega_a/c^2$. This leaves the fundamental, source leaf, connectivity law and selected count to be explained. It also does not require the old supporting field energy to equal the mass-frequency before changing the action. See common clocks and mechanical mass.

For statistics, the current joint mass-and-ratio assessment scores W, down and up as targets, not additional anchors. It retains the one-electron normalization and shared prescription. Its conditional coverage and dependent ratio information must not be turned into a global probability by multiplying the same mass evidence twice.

Earlier energy-per-mode inversions

In the earlier integrated-density construction, the count enumerates available states rather than establishing their occupation. For $N=n-1$,

$$D_{d,N}=\binom{N+d}{d},\qquad E_{d,N}=\varepsilon_dD_{d,N},\qquad \varepsilon_d=\mu_dc^2$$

is the proposed complete-energy interpretation. $\varepsilon_d$ is energy per counted mode. The dynamics must explain the occupied cumulative band, appropriate flat allocation, common participating clock and complete energy including its support. These requirements are developed in IDOS energy and supporting dynamics.

A useful reduced canonical example uses occupied actions $I_k$ and a positive angular-frequency coefficient $\varpi$, distinct from the mass unit $\mu_d$,

$$n=\sum_k I_k,\qquad R_I=\sum_k I_k^2,\qquad H_{\rm part}=\varpi\frac{n^3}{R_I}.$$

On a uniformly occupied band of size $D$, $I_k=n/D$, this gives

$$H_{\rm part}=\varpi nD,\qquad \frac{\partial H_{\rm part}}{\partial I_k}=\varpi D \quad(k\text{ occupied}).$$

Thus a specified participation Hamiltonian can convert a supplied uniform count into one clock. It does not by itself select $D$, derive $\varpi$, identify the action normalization $n$, or construct physical spatial modes. Classical quadrature variables here describe field dynamics, not electron-position probabilities.

Likewise a spatially covariant candidate can retain actual count clocks and a common supplied action unit while supporting exact local preparations. Its complete global profile, support stress and normalization must still be checked under that same action. A homogeneous core or a finite mode sum is not automatically an isolated particle. Retaining the prescribed units in a candidate action does not establish their derivation.

The overall target is one law applied at depths $2,3,4,5,6,10$, preserving distinct isolated particle frequencies in interactions rather than imposing synchronization between unlike particles. A per-particle amplitude fit or a coefficient retuned for each row is not that common law. Amplitude may enter nonlinear consistency and frequency selection when the equations require it; its chosen value is not itself the particle mass.

Spatial lengths cannot be read from a unit without an operator

Size and stiffness are useful hypotheses for the sector unit, while cumulative counting constrains the maintained harmonic multiplier. Their conversion must be derived from the actual spatial operator. If a common wave-speed model has $u_d=\kappa_dc/L_d$, then a frequency-unit ratio determines a length ratio only with the geometric factors $\kappa_d$ retained. A common size across depths is a further hypothesis, not a consequence of one same-field law.

The dimensional accounting of the curvature carrier makes this explicit. With tangential length $R_d$, normal-profile length $L_d$, dimensionless coefficient $g_d$, and dimensionless eigenvalue $\lambda_{n,d}$,

$$\gamma_d=\frac{g_dR_d}{L_d},\qquad \omega_{n,d}=\frac c{R_d}\lambda_{n,d}(\gamma_d).$$

In that carrier's large-hopping limit,

$$\omega_{0,d}= \frac{c\,d!}{g_d^{d-1}\prod_lh_l} \frac{L_d^{d-1}}{R_d^d},\qquad \omega_{n,d}=\omega_{0,d}S(n,d) [1+O(\gamma_d^{-2})].$$

The $h_l$ are dimensionless computed adjacent-profile integrals in that construction. A measured common unit therefore constrains a combination of spatial lengths, coefficients and integrals. It does not separately determine them. Those lengths are ordinary spatial lengths in the shared manifold, not compactification radii.

For example, retaining common normal width and coefficient at depths six and ten but allowing different tangential radii gives the conditional anchor equation

$$\frac{\lambda_{23,10}(r\gamma_6)} {r\lambda_{13,6}(\gamma_6)} =\frac{64512240}{18564},\qquad r=R_{10}/R_6.$$

The archived finite-matrix calculation gives $r=0.04125312$ at $\gamma_6=10000$. It matches that chosen cross-depth anchor under the stated additional assumptions; the finite within-depth muon/electron correction remains approximately $-1.10\times10^{-5}$. These are constraints on a possible spatial realization, not measured particle sizes or a replacement scale law.

A useful constraint on a common supporting response

An inverse-current comparison supplies a more specific scale-response condition. Its reference functions are

$$Z_d(h)=\sum_{j=0}^{d-1}\frac{h^{2j}}{(j!)^2},\qquad \widetilde\mu_d(h)=\frac1{(d-1)!\sqrt{Z_d(h)}},$$

$$h(R)=\frac{g(R)R}{c},\qquad u_d(R)=\frac cR d!\widetilde\mu_d(h(R)).$$

Here $u_d$ is the angular-frequency unit; $\widetilde\mu_d$ is a dimensionless comparison function, distinct from the physical mass unit $\mu_d$. Define

$$\eta(R)=-\frac{d\log g}{d\log R},\qquad q_d=-\frac{d\log\widetilde\mu_d}{d\log h}.$$

For $h>0$ at the maintained depths $d\ge2$, the positive finite-sum weights give $0<q_d<d-1$. Write $h_d=h(R_d)>0$ for evaluation at a candidate stationary radius $R_d$; a stationary unit requires

$$\eta(R_d)=1+\frac1{q_d(h_d)}.$$

Within the inverse-current response calculation, the occupied branch bounds and an additional nondecreasing $\eta$ assumption imply

$$\frac{u_{10}}{u_6}<3.788522\times10^{-6},$$

whereas the benchmark tau/electron unit ratio is approximately $1.000644$. This is an obstruction to that common monotone-elasticity realization, not to the physical number of spatial directions or to every common supporting law.

The constructive comparison is also useful. The positive rational response

$$\frac{g(R)}{g_0}= \frac{1+(R/bR_0)^4} {[1+(R/R_0)^4][1+(R/CR_0)^4]}$$

uses independent positive scales $R_0$ (a spatial length) and $g_0$ (a frequency-valued reference edge), with dimensionless shape parameters $b,C$. Its dimensionless overall scale is $g_0R_0/c$. It can have the required nonmonotone logarithmic elasticity while $g$ itself decreases. The source's fitted common example $b=1.34$, $C=18.00753138$, with depth-two gap $h_2=500$, retains reference minima at all six maintained depths and gives $R_{10}/R_6=0.07325$. It demonstrates conditional compatibility after changing the supporting-response assumption. It does not derive that response from $\Psi_\infty$, select between its multiple radial branches, or include their complete supporting energy. The parameters were fitted to known targets, and the comparison must retain that status.

The remaining physical bridge

A successful scale construction must connect the declared count clocks, class-dependent units, actual spatial gradients, reciprocal coupling and complete energy normalization in one action. It must retain the neutrino calibration freedom until it actually resolves it, and it must account for bottom's two-rung count and the common lepton unit across depths six and ten. A fixed normalization or a common scale inserted into an action is an input to that construction; a common positive complete inertia, if proved for an actual isolated solution, is a necessary physical consistency result rather than a derivation of the units.

The useful results above constrain what that construction must contain. They preserve both the proposed numerical scale chain and alternative calibrated/inverse branches with their changed assumptions explicit. They do not merge different candidate actions or turn a radius fit, mode norm or supporting-response coefficient into a completed mass operator.