Complete Sector Assignments Under the Fixed Mass Prescription

Result

Only the maintained complete assignment matches or improves every tested particle's present mass accuracy under the stated prescription. The comparison lets entire particle blocks change depths, requires distinct occupied depths, retains the original sector scales and measured electron reference, and covers every positive integer mode index. It examines all 120 permutations before applying the reference and scale equations.

The surviving assignment is

$$(d_{WZH},d_{dsb},d_{uct},d_{e\mu},d_\tau)=(2,3,4,6,10).$$

The closest admissible different assignment has approximately 8.88588 times the RMS logarithmic mass error. The result gives the rigidity argument in the sector-support paper a complete-assignment test: improving an isolated family match is insufficient if the other families can no longer occupy distinct sectors under the same equations.

Two explicit uniqueness theorems below give a direct elimination proof for the depths and an exact proof for all twelve integer indices. The uniqueness is conditional on this numerical prescription and benchmark. The calculation leaves the origin of the scale coefficients, physical mode selection and bottom's mass unresolved. It does not assign zero probability to coincidence. The exact proof and the separate finite statistical controls are given below, with complete executable results.

1. The Mass Law and Fixed Inputs

The positive-index law is

$$S(n,d)=\binom{n+d-1}{d},\qquad P(n,d)=m_e a_d S(n,d),\qquad n\ge1.$$

Here $P$ is a predicted mass and $a_d=\mu_d/m_e$ is a dimensionless sector factor. The unchanged factors are

$$a_2=\sqrt{2890},\quad a_3=\sqrt{32\sqrt7},\quad a_4=\frac{\sqrt{48/\sqrt7}}{15},\quad a_6=a_{10}=\frac1{18564}.$$

These follow from the proposed coefficient and reference-count prescription displayed in the support paper's scale appendix. They are held fixed throughout the primary comparison. Their physical derivation remains open; they were developed in work seeking to explain previously fitted scales. Evaluating this recipe with one measured mass does not establish that its construction was independent of the other target masses.

The dated benchmark is retained unchanged. All masses in this article are in MeV/$c^2$.

The electron mass is the same measured input in every trial. It supplies the common dimensional reference and is excluded from the eleven-mass aggregate error. The other masses are comparison targets, not separately adjustable normalization inputs. The calculation treats their recorded decimals exactly for arithmetic purposes; this does not remove measurement uncertainty or quark mass-scheme dependence. It is not an uncertainty-weighted phenomenological fit or a new set of world averages.

Tau occupies a separate block because the maintained sector assignment places it at a different depth. This bookkeeping does not make it a different empirical charged-lepton family.

2. What a Competing Assignment Must Preserve

The primary comparison imposes the following simultaneous requirements:

1. Each listed block moves intact to one of depths $\{2,3,4,6,10\}$.
2. Different blocks occupy different depths.
3. All sector factors above and all observed benchmark masses remain fixed.
4. The electron must satisfy its reference equation $a_{d_e}S(n_e,d_e)=1$.
5. Tau and the electron–muon block retain their common sector mass unit, $a_{d_\tau}=a_{d_{e\mu}}$.
6. Every particle may select any positive integer mode index.

There are $5!=120$ depth permutations. The lepton placements are included in this enumeration before equations 4 and 5 are checked. Different particles' nearest indices are chosen independently; allowing repeated indices broadens the alternative class, so it cannot produce a false exclusion of a class with distinct indices required. The maintained assignment and the closest admissible alternative use distinct indices within their blocks.

Depth five has no independently specified absolute scale in this prescription. Its scale-free neutrino splitting comparison remains in the support paper; this test invents neither an absolute neutrino mass nor a depth-five scale. The massless photon is outside the positive-mass logarithmic comparison. Other depths and changed coefficient laws define different problems.

Exact Uniqueness Theorems

The complete assignment result can be stated as two theorems. The first establishes uniqueness of the depths under considerably wider tolerances than the maintained fit. The second establishes uniqueness of both depths and indices under the original individual accuracy requirement.

Let

$$\mathcal B=(WZH,dsb,uct,e\mu,\tau),\qquad \mathcal D=\{2,3,4,6,10\}.$$

An admissible placement consists of a bijection $\sigma:\mathcal B\to\mathcal D$ and a positive integer $n_i$ for each of the twelve particles. With $b(i)$ its block, set

$$P_i=m_ea_{\sigma(b(i))}S(n_i,\sigma(b(i))).$$

The $a_d$ and all benchmark masses are exactly those in section 1. In addition to the bijection, impose the two retained reference equations

$$P_e=m_e,\qquad a_{\sigma(e\mu)}=a_{\sigma(\tau)}.$$

These are hypotheses of the theorems. There is no upper limit on any $n_i$ and no independently readjusted sector scale.

Theorem 1 — Sector Uniqueness Under Wider Tolerances

For every admissible placement, if

$$\left|\frac{P_W}{M_W}-1\right|\le\frac1{200}, \qquad \left|\frac{P_{\rm up}}{M_{\rm up}}-1\right|\le1,$$

then necessarily

$$\boxed{(\sigma(WZH),\sigma(dsb),\sigma(uct), \sigma(e\mu),\sigma(\tau))=(2,3,4,6,10).}$$

At least one admissible placement satisfies these inequalities. This theorem fixes the block depths; by itself it does not fix every particle's index.

Proof. The electron equation is $a_dS(n,d)=1$. At depths two and three, $a_d>1$, so even the first rung is too large. At depth four,

$$a_4^4=\frac{256}{39375}<1 <\frac{160000}{39375}=(5a_4)^4,$$

and the first two counts are one and five. At depth ten,

$$S(7,10)=8008<18564<19448=S(8,10).$$

Strict monotonicity of $S(n,d)$ excludes every other index at those depths. At depth six, $S(13,6)=18564$ supplies the unique exact electron solution. Thus $\sigma(e\mu)=6$ and $n_e=13$. Only depth ten among the other four depths has $a_d=a_6$; shared scale and distinct occupancy therefore force $\sigma(\tau)=10$.

The remaining blocks must occupy depths two, three and four. To locate the W mass relative to their rungs, write the exact rational number

$$t=\frac{m_e}{M_W}=\frac{10219979}{1607384000000}.$$

For depth $d$ and index $k$, its fourth-power W ratio is $a_d^4S(k,d)^4t^4$. Direct integer multiplication gives the following rational certificates:

The simple rational bounds satisfy

$$\frac45<\frac{81}{100}<\left(\frac{199}{200}\right)^4, \qquad \left(\frac{201}{200}\right)^4<\frac{41}{40}<\frac{26}{25}.$$

Consequently, at both depths three and four, one adjacent W prediction is below $199M_W/200$ and the next is above $201M_W/200$. Strict monotonicity excludes the entire positive integer ladder from the allowed W interval. Hence $\sigma(WZH)=2$.

The up-type block must now occupy depth three or four. At depth three, its first-rung prediction already exceeds twice its target, because

$$\left(\frac{m_ea_3}{M_{\rm up}}\right)^4 =7168\left(\frac{10219979}{43200000}\right)^4>16.$$

Every higher rung is larger. Thus depth three violates even the stated 100% relative-error allowance, forcing $\sigma(uct)=4$ and, by exclusivity, $\sigma(dsb)=3$. The maintained electron, W and up indices $13,76,3$ satisfy the reference and widened inequalities, while the remaining positive indices can take their maintained values. Existence follows. This proves the theorem. $\square$

Theorem 2 — Uniqueness of the Complete Assignment and Indices

Let $\sigma_*$ be the arrangement in Theorem 1 and let $n_i^*$ be the indices in the table below. Define

$$P_i^*=m_ea_{\sigma_*(b(i))}S(n_i^*,\sigma_*(b(i))), \qquad \varepsilon_i=\left|\log(P_i^*/M_i)\right|.$$

Then an admissible placement satisfies

$$\left|\log(P_i/M_i)\right|\le\varepsilon_i \quad\text{for every particle }i$$

if and only if $\sigma=\sigma_*$ and $n_i=n_i^*$ for every $i$.

Bottom's $n^*=17$ defines its ordinary-rung comparison tolerance here, approximately +8.9194%. It is not asserted to be a physically explained bottom assignment.

Proof. The maintained W distortion obeys

$$\exp(4\varepsilon_W)<(201/200)^4, \qquad \exp(4\varepsilon_{\rm up})<16.$$

These are rational inequalities using $\exp(4\varepsilon_i)=\max((P_i^*/M_i)^4,(M_i/P_i^*)^4)$. The W log-error bound puts its ratio strictly between $200/201$ and $201/200$, hence inside the wider relative interval in Theorem 1. The up bound puts its ratio strictly between $1/2$ and two. Theorem 1 therefore forces $\sigma=\sigma_*$.

It remains to prove uniqueness of each index. For any strictly increasing positive ladder $P(n)$, logarithmic distance to $M$ is minimized uniquely at $k>1$ if

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

For $k=1$ only the upper inequality is required. These inequalities place $\log M$ strictly between the adjacent midpoints in the increasing sequence $\log P(n)$. Every other index is farther away, including indices outside the adjacent pair.

For the present law, put $A_d=a_d^4$ and $u_i=M_i/m_e$. The midpoint inequalities are exactly the rational comparisons

$$A_d^2S(k-1,d)^4S(k,d)^4 <u_i^8 <A_d^2S(k,d)^4S(k+1,d)^4.$$

The lower comparison is omitted at $k=1$. Substitution of the fixed benchmark rationals and the following integer counts verifies both strict inequalities for every applicable row:

Thus $n_i^*$ is the unique global minimizer of that particle's absolute log error at its now-fixed depth. No other index can be within a tolerance equal to that minimum. This proves necessity of every $n_i=n_i^*$. The maintained assignment itself attains every threshold, proving sufficiency. $\square$

What Has Been Proved

Theorem 1 establishes a unique depth arrangement without imposing tight accuracy on the other nine non-reference masses. Theorem 2 establishes the exact complete-assignment statement at the original individual accuracies, with unrestricted positive indices. Neither result requires a probability interpretation, a numerical upper cutoff or assuming that a finite search happened to find every alternative.

The hypotheses include the five scale-defined depths, block grouping, distinct occupancy, numerical sector factors and shared lepton unit. The electron factor already contains its reference count. These theorems prove uniqueness within those hypotheses; deriving the hypotheses physically, resolving bottom and selecting particle states dynamically remain separate problems. The aggregate RMS comparison later in this article is an additional reported calculation, not a consequence of componentwise uniqueness alone.

All finite arithmetic in the proofs is independently reproducible with the theorem certificate checker and exact rational certificates, using the same public sector-inputs.json. Each strict comparison records its exact positive rational difference; no rounded displayed percentage is used to decide the theorem.

3. Exact Coverage of Every Positive Integer Index

For fixed positive $d$,

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

Consequently, for a target $M>0$, either the first predicted rung already exceeds $M$, or two adjacent rungs bracket it. Every index below the bracket predicts an even smaller mass; every index above it predicts an even larger one. Only the two bracket endpoints can minimize $|\log(P/M)|$.

Doubling an upper trial index until its mass exceeds the target, then using integer binary search, finds this bracket in finite time. There is no chosen maximum index. This matters in practice: the closest top-quark trial at depth two is $n=112$, which a search ending at 100 would miss.

Rational Comparisons Eliminate Rounding Decisions

Although some factors contain radicals, all their fourth powers are rational:

$$a_2^4=2890^2,\qquad a_3^4=7168,\qquad a_4^4=\frac{256}{39375},\qquad a_6^4=a_{10}^4=\frac1{18564^4}.$$

The benchmark decimals are finite rational numbers as well. Comparing

$$P(n,d)^4=m_e^4a_d^4S(n,d)^4$$

with $M^4$ therefore determines the bracket exactly. For adjacent predictions $L\le M\le U$, the lower is at least as close in logarithmic error precisely when

$$\log(M/L)\le\log(U/M) \quad\Longleftrightarrow\quad M^2\le LU \quad\Longleftrightarrow\quad M^8\le L^4U^4.$$

The last comparison uses rational numbers only. An equality would be a two-rung tie; none occurs among the 60 particle/depth comparisons here.

For a selected prediction define

$$R=\left(\frac PM\right)^4,\qquad D=\max(R,R^{-1}),\qquad \left|\log\frac PM\right|=\frac14\log D.$$

Since the logarithm is increasing, comparing two errors is exactly equivalent to comparing their rational $D$ values. No numerical tolerance enters the accept/reject decision. Decimal roots and logarithms are used to display predicted masses and aggregate scores.

4. The Reference and Shared Scale Restrict the Lepton Placements

The electron equation requires $a_dS(n,d)=1$.

Thus 24 permutations satisfy the electron equation. Among the remaining distinct depths, only ten shares the depth-six mass unit. Six permutations satisfy both equations: the three remaining blocks exchange depths two, three and four.

The factor $1/18564$ already contains the electron reference count. Recovering $(d_e,n_e)=(6,13)$ here is consistency with that prescribed input, not an independent prediction of the electron's depth or mass. The substantive comparison is whether the remaining masses can match simultaneously under the complete retained prescription.

5. The Simultaneous Accuracy Test

For each non-anchor particle, the threshold is its absolute logarithmic error in the maintained assignment. A competitor must match or improve every threshold:

$$D_i^{\mathrm{trial}}\le D_i^{\mathrm{maintained}} \quad\hbox{for every tested particle }i.$$

These are the existing individual accuracies, not an assertion that all current predictions are exact. In particular, bottom's nearest ordinary rung is $n=17$, predicting approximately $4556.10038$ against $4183$, a $+8.91944\%$ error. The test grants alternatives this same bottom tolerance. It does not introduce the historical geometric-mean prescription or claim a solution for bottom.

The aggregate score, reported alongside the exact individual comparison, is

$$\epsilon=\sqrt{\frac1{11}\sum_{i\ne e} \left[\log\frac{P_i}{M_i}\right]^2}.$$

This dimensionless RMS log error includes bottom. It is not a significance measured in experimental standard deviations.

Every Admissible Complete Assignment

The electron–muon and tau depths in this table are six and ten after the reference equations above have been applied.

Every entry already gives each target its closest possible positive-integer rung at that depth. If any one target fails, changing its index cannot repair the complete assignment.

The closest different assignment exchanges the two quark sectors. Some individual predictions improve, but the following comparisons fail:

The RMS ratio is

$$\frac{0.238903041943638176}{0.0268856900548956481} \approx8.88588098.$$

The other alternatives already fail at W: its globally closest depth-three prediction has a $+1.18719\%$ error and its depth-four prediction a $+0.71607\%$ error, compared with $+0.0123439\%$ at depth two. The complete output includes every mass, nearest index, rational distortion and excluded permutation.

This exhausts the stated comparison: one complete arrangement meets every existing accuracy, and the same arrangement uniquely minimizes the aggregate score. Moving a family into a depth already occupied by another family cannot evade the other family's displacement or the shared scale equations.

6. Broader Normalization Controls

A separate finite comparison retains the same mass targets and relative sector factors while allowing one common normalization to be determined afresh by the electron's trial depth and index. Write

$$P_i=\lambda_c a_{d_i}S(n_i,d_i),\qquad \lambda_c=\frac{m_e}{a_{d_e}S(n_e,d_e)}.$$

The original prescription is recovered only when $\lambda_c=m_e$. Otherwise the absolute scale equation has changed, even though the measured electron mass has not. This control considers all 120 permutations and electron indices $1\le n_e\le100$; the other particle indices also range from 1 to 100.

The maintained assignment and electron index thirteen uniquely win both the raw RMS error and the calibrated score below in all four classified subsets. Broader freedom nevertheless produces closer competing fits than the unchanged prescription permits:

• With both original constraints relaxed, the closest different raw fit has block depths $(3,6,4,10,2)$, electron index nine and normalization multiplier $\lambda_c/m_e=14/33$. Its RMS error is approximately 0.0308852, against 0.0268857 for the maintained arrangement. It loses both the original absolute normalization and the common charged-lepton unit.
• With the common lepton unit retained but absolute normalization relaxed, the best different arrangement puts electron/muon at ten and tau at six. Electron index eight gives $\lambda_c/m_e=21/22$; the muon error is about $-18.71\%$ and the aggregate error approximately 0.0686336.

These controls show exactly which constraints matter. Neither is a counterexample satisfying the original complete prescription. The finite cap also distinguishes them from the uncapped proof in sections 3–5.

7. A Separate Finite Reference Null

The uniqueness result requires no probability interpretation. For completeness, the finite calculation also evaluates a specified reference-null model. Condition on the same electron mass and draw the eleven labelled logarithmic mass ratios independently and uniformly from $[0,L]$, with $L=\log(10^6)$ for the main comparison. The particle blocks, sector factors and candidate classes remain fixed.

For a fixed candidate $c$, the possible logarithmic predictions for particle $i$ are

$$\mathcal C_{ci}=\left\{ \log\frac{a_{d_i}S(n,d_i)}{a_{d_e}S(n_e,d_e)}: 1\le n\le100\right\}.$$

Define its distance and exact clipped interval-union cumulative distribution:

$$T_{ci}(u)=\min_{z\in\mathcal C_{ci}}|u-z|,\qquad F_{ci}(t)=\frac{\operatorname{length}\left( [0,L]\cap\bigcup_{z\in\mathcal C_{ci}}[z-t,z+t] \right)}{L}.$$

For continuous uniform $U_i$, the distance has a continuous distribution. The probability integral transform gives $p_{ci}=F_{ci}(T_{ci}(U_i))$ uniformly distributed on $[0,1]$. Independence of the eleven targets then gives

$$X_c=-\sum_{i=1}^{11}\log p_{ci}\sim\operatorname{Gamma}(11,1), \qquad q(x)=e^{-x}\sum_{j=0}^{10}\frac{x^j}{j!}.$$

The nearest-index optimization is already included in $F_{ci}$; it is not followed by another factor of $100^{11}$. For a fixed class of $K$ candidates, the union bound gives

$$q(x)\le\Pr\{\max_cX_c\ge x\}\le\min(1,Kq(x)).$$

Different assignments need not be independent for this bound. At the common observed maximum $X=44.5021538277$, $q=5.05999500197\times10^{-10}$.

These classes are overlapping subsets: the first six candidates belong to every row, and the final row contains all candidates. The first upper bound is approximately one in 329 million under this finite reference null with the numerical recipe fixed. It is not the probability that the theory is correct or the probability of its historical discovery. Selection of the formula, coefficient values, grouping and reference-null window is not calibrated here. A seven-decade window is also recorded as a sensitivity check.

These bounds concern $n\le100$. Extending the available rungs changes the null coverage, so the finite bounds cannot be transferred to the uncapped uniqueness proof. Nor does finding only one arrangement mean a zero coincidence probability. The primary conclusion remains exact conditional uniqueness of the observed benchmark comparison.

8. Meaning for the Sector Proposal

The result adds a simultaneous restriction to the familiar adjacent-rung rigidity. The current arrangement preserves close matches across the defined blocks under one numerical scale chain. Every rearrangement either fails the reference and shared-scale equations or sacrifices at least one of those existing mass accuracies, even when arbitrarily large mode indices are permitted.

That is useful support for investigating this particular sector prescription. Its restrictive behavior survives a complete rearrangement test; it is more than a collection of isolated nearby integer values. It also makes the research target sharper: explain why one spatial law should produce these scales and occupied counts while retaining the observed common properties of the charged leptons.

The calculation does not derive the numerical coefficients, dynamically forbid the unoccupied rungs, remove the measured residuals, explain bottom or identify the physical mass and coupling operator. The additional spin, orbit, helicity and charge constructions in the support paper were not used as extra numerical scores here. Their shared geometric assumptions should be assessed together, without multiplying their consequences as independent probabilities.

9. Reproduction

The downloadable evidence contains the complete exact permutation classifications, all 60 particle/depth nearest-rung certificates and every surviving assignment's mass errors. The finite record retains both null windows and all candidate scores, including the broader normalization controls. The code uses Python's standard library and public input files only.

Download the six original reproduction files below and keep them in one directory. The theorem checker and its certificates are linked immediately afterward.

With Python 3.11 or later, run:

python3 exact-sector-check.py
python3 finite-sector-check.py

Each script reads the input beside it and replaces its corresponding result file. They run independently; the order above puts the primary exact result first. Leave assertions enabled. The outputs contain SHA-256 fingerprints of their public script and input file. Their scientific records reproduce the reviewed calculations; executable checks verify the stated arithmetic and enumeration, not the physical truth of the scale prescription.

For the formal theorems, also download the theorem checker beside sector-inputs.json and run python3 uniqueness-theorem-check.py. The certificate record contains all 40 strict rational comparisons, their positive differences and the twelve unique-nearest-index records.