Sector Structure Technical Evidence Map

The new case for particle geometry and sector depths explains the current argument, with a glossary and complete reproduction formulas. The earlier sector case remains a separate record. This map preserves access to every substantive section of its earlier extended version, including useful results whose full equations belong in focused technical notes. The headings identify the results; the links give their maintained derivations, assumptions and reproducible evidence. Existing research is reused rather than replaced by a new chronological report.

Three reading rules apply throughout. First, a projective or round-sphere calculation retains its actual geometry; it is not automatically a flat-space theorem. Second, each action has its own dynamics, energy, domains and persistence scope. Third, a shared capability at six and ten can support the lepton comparison without selecting either depth. Particles occupy nested dimensional layers in the shared higher-dimensional physical space; candidate localization dynamics are available, while their physical identification and the observed depth assignments remain to be derived.

Current Developments

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. Older projective and separately resonating occupation constructions remain scoped comparisons; they do not change the current single-field, common-fundamental portion premise.

Earlier Additions

The revised case also links the count-ratio mixing and cumulative-band test, literal flat precession, higher-dimensional static response, and local ambient action with complete inertia. The latter incorporates its full changed action, constrained stress, reference quadratic forms, weighted exterior domain and matching traces. Earlier nonlinear persistence theorems are not transferred to it.

The complete record also retains the descriptive Pascal and Hockey-Stick identities among the counts 10, 15, 20 and 35, the old project's explicit embedding/readout construction, and the mass-comparison code and dated benchmark receipts. Those are mathematical patterns and reproducibility tools, not physical rules for adding particle indices or independent evidence for occupation.

Appendix B. Exact Counting Patterns Among Some Assigned Values

Some maintained index values are themselves interior binomial counts. Their verified Hockey-Stick expansions are

$$\begin{aligned} \binom52&=\sum_{j=1}^{4}\binom j1=10,& \binom53&=\sum_{j=2}^{4}\binom j2=10,\\ \binom62&=\sum_{j=1}^{5}\binom j1=15,& \binom64&=\sum_{j=3}^{5}\binom j3=15,\\ \binom63&=\sum_{j=2}^{5}\binom j2=20,\\ \binom73&=\sum_{j=2}^{6}\binom j2=35,& \binom74&=\sum_{j=3}^{6}\binom j3=35. \end{aligned}$$

The associated Pascal partitions are

$$\binom63=\binom52+\binom53,\qquad \binom74=\binom63+\binom64.$$

These operands count subsets; they are not rules for adding particle indices. Complementary subsets explain repeated columns: seven expansions describe four count values. Among the assigned values through 95, only 10, 15, 20 and 35 occur in the interior family $2\le k\le r-2$. For $r\ge15$, its smallest value is at least $\binom{15}{2}=105$.

This is descriptive recurrence of the same combinatorial identities used in the integrated count. It is a lead for physical interpretation, with no established occupation mechanism or independent probability attached.

Count Recurrence and Mass Diagnostics

The following retained definitions and controls reproduce the numerical diagnostics. A freely optimized common scale at fixed indices is a different test from the constrained fixed-scale complete-assignment theorem. Neither test derives the sector units dynamically.

What Is Being Counted?

The count has a standard, explicit meaning:

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

It counts oscillator configurations through total excitation $n-1$, equivalently monomials of degree at most $n-1$ in $d$ variables. The extra coordinate is a slack variable, not an additional physical dimension. The sum is the Hockey-Stick Identity, and adjacent counts obey Pascal recursion:

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

Thus the mass proposal uses an established count of available configurations. In IDWT's law this count is exactly the proposed frequency multiplier: $\omega_{n,d}=\omega_{0,d}S(n,d)$, with $\omega_{0,d}=\mu_dc^2/\hbar$. The open spectral problem is which spatial operator and same-field dynamics produce this frequency spectrum and select the occupied rungs. The combinatorial count identifies the target spectrum; the operator must explain it.

The indices and grouping are maintained hypotheses developed using known particle data. They have not been selected by a derived field equation. Once specified, however, they impose the simultaneous numerical constraints shown here. A fair evaluation should retain both facts: the construction has a selection history, and its shared scales leave genuine restrictions to explain.

Fixed-Index Optimized-Scale Diagnostic

The first comparison holds the indices fixed and allows every trial depth from 1 to 15 its own best common scale. For benchmark masses $M_i$,

$$\log\widehat\mu_d=\frac1N\sum_i\log\frac{M_i}{S(n_i,d)}, \qquad \epsilon_d=\sqrt{\frac1N\sum_i \left[\log\frac{\widehat\mu_d S(n_i,d)}{M_i}\right]^2}.$$

Within this test, a scale adjustment does not erase the depth preference. The proposed depths give much closer simultaneous family ratios. This makes dimensionality a meaningful part of the numerical hypothesis.

This table tests the specified groupings and indices with independently optimized diagnostic scales. It does not discover the groupings or exhaust simultaneous changes of indices and depths. The complete assignment study instead tests complete rearrangements and variable indices under the original absolute scale prescription.

Depth Ten: A Demanding Charged-Lepton Comparison

The additional proposal $\mu_{10}=\mu_6$ gives

$$\frac{m_\tau}{m_e}=\frac{S(23,10)}{S(13,6)}.$$

It yields 1775.78576288 MeV/$c^2$, approximately 0.0643940% below the archived tau benchmark. Keeping the same index and scale at depths nine and eleven gives about 554.93 and 5327.36 MeV/$c^2$. The shared-scale assumption is doing real work: a freely adjustable scale could fit a single tau mass at any depth.

This raises a further physical test. A ten-dimensional tau construction must retain the charge and spin shared by charged leptons despite the change of depth. The revised case and its linked shared-property constructions develop concrete parts of that question.

Absolute Masses, Scale Assumptions and All Adjacent-Rung Steps

The scale-free family ratio tests do not require the proposed non-neutrino scale prescription. The complete assignment test in the linked assignment study explicitly retains it. The prescription is

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

These are sector coefficients; their identification with physical coupling strengths is open. With specified reference counts $C_d^\star$,

$$M_d^\star=\mu_d C_d^\star,\qquad \frac{(M_d^\star)^2}{g_{dd}}=\frac{m_e^2}{g_{66}},\qquad \mu_d=\frac{m_e}{C_d^\star}\sqrt{\frac{g_{dd}}{g_{66}}}.$$

Only the archived $m_e=0.51099895000$ MeV/$c^2$ is a measured dimensional input to this evaluation; W, down and up are outputs. The expressions were developed to explain previously fitted scales. Their one-input evaluation is therefore distinct from an independently derived scale law. The self-contact mass-operator module gives the conditional construction. The tau relation $\mu_{10}=\mu_6$ additionally requires equal normalized overlaps; equality of the listed sector coefficients alone is insufficient. The absolute neutrino scale remains separate.

The chosen coefficients imply $g_{33}g_{44}=96$, $g_{44}/g_{33}=3/14$ and the reference up/down ratio $\sqrt{3/14}$. Those are consequences of this prescription, not additional empirical confirmations. No arithmetic on particle indices is used to derive them here.

The electron is the calibration input and is not a predicted row. The table preserves the program's dated benchmarks so the calculation remains reproducible. It is not a table of newly updated world averages or a common-scheme uncertainty-weighted fit. Quark masses need their mass definitions and renormalization conventions; top is visibly the least close non-neutrino comparison. The down row's formal step to zero is not another positive-index massive branch. Empirical context is supplied by the Particle Data Group tables and NIST constants reference.

The Z/W ratio residual is approximately +0.0337405%, distinct from the absolute Z residual above. Using W as a diagnostic ratio reference does not make its measured mass an input to this prescription. The optimized family scales in the fixed-index diagnostic above are likewise diagnostics, not the procedure used here.