Particle depth as spatial geometry

Literal macroscopic spatial dimensions

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 $\Psi_\infty$.

For the conditional global-nesting calculations in this module, write the shared physical manifold as $M_\infty$ and the depth layers as $Q_d$. The historical notation now denotes the ten-dimensional containing space. A proposed inclusion is:

$$Q_a\hookrightarrow Q_b\hookrightarrow M_\infty,\qquad a<b.$$

This inclusion relates spatial sectors. Individual particles are field structures associated with the sectors; it does not place one particle inside another or make it a constituent of another. In particular, an electron state need not be the restriction of a tau state. A restriction of bundles compares the available field components across sectors; identifying particular particle solutions through that restriction is an additional dynamical assumption.

Depth $d$ counts ordinary, literal macroscopic spatial dimensions of $Q_d$. They are all the same kind of physical dimension. Their coordinates specify position, separation, and motion. A sector is not an internal state space, a compactified auxiliary space, or a configuration-space substitute for physical space. All remaining spatial directions are unbound for every particle.

For the electron and tau, this means $Q_6\hookrightarrow Q_{10}$: the electron's six spatial dimensions are six of the tau's ten. All ten are the same kind of dimension for the tau. The electron is “flat” relative to it, meaning lower-dimensional, not that the tau has an inherent six-plus-four division. The same relationship holds between three-dimensional nuclei and six-dimensional electrons. “Flat” here describes relative dimensionality, not a claim of zero metric curvature.

The photon-transversality construction illustrates this at every containing depth. The photon retains two directions of bound oscillation. There is one perpendicular direction within a three-dimensional sector, four within a six-dimensional sector, and eight within a ten-dimensional sector. These are descriptions relative to the photon; they do not create different kinds of dimensions in the containing sector.

IDWT attributes the three-dimensional experience of ordinary matter and observers to the three-dimensional nuclear sector. This general premise does not require a physical division of any higher-depth particle into different kinds of dimensions. The observer map must describe what the observer's interactions register; its dynamics remains to be derived.

For example, the electron has six ordinary macroscopic spatial dimensions. Three of those dimensions are shared with us. That fact makes none of the six physically different in kind, and does not give the electron an inherent $3+3$ split. Grouping coordinates by an observer's access is a choice of description, not a division in the electron. Frame-lock models that use such a grouping have additional assumptions; they do not establish the general origin of three-dimensional experience.

This states the physical premise; it does not declare the proposed metric, embedding, lock, or observer response mathematically derived or experimentally established. The following vocabulary applies throughout the research record:

Inclusions of nested submanifolds, maps that marginalize spatial coordinates, and projectors acting on field components are different mathematical objects. One cannot establish the first merely by constructing either of the others. See the geometry and nesting audit for the resulting restrictions on existing calculations.

Current Reading and Scope — 1 November 2026

The newer geometric account combines flat full-profile half-spin, the minimal-carrier clock criterion, independent color and the normal exterior charge rule. It also tests free angular-charge conservation and its interaction limitation. These constructions share several compatible choices, but do not yet give one local field action realizing all properties or an occupied-count selector.

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. Geometric claim

Status: spatial depth and a named geometry are separate hypotheses. The candidate sphere/projective assignments elsewhere in V2 are not proved by the dimension and principal-symbol arguments below. Those arguments also assume the sector embedding and projection stated in their hypotheses; they do not independently select a physical depth. See the geometry premise audit.

A particle of depth $d$ lives on a spatial submanifold $Q_d$ of $M_\infty$ with real dimension $d$. Each spatial coordinate supplies a derivative direction and a conjugate covector component in the principal symbol of the kinetic operator.

In IDWT the depth $d$ appears simultaneously in

$$TQ_d, \qquad T^*Q_d, \qquad \gamma^ap_a, \qquad S(n,d), \qquad \pi_{ba}:Q_b\to Q_a,$$

and in the integration measure used for cross-depth contact. These are linked uses of one physical spatial depth.

The particle's physical spatial submanifold is $Q_d$. Its spinor bundle carries the spatial Dirac field. The auxiliary coherent-ray fibre $\mathbb{CP}^{d}$ organizes the phase modes constructed from the $d$ spatial oscillator directions, and embeds by the degree-$N$ Veronese map into the full projective mode space $\mathbb P(\mathcal E_{N,d})\cong\mathbb{CP}^{S(N+1,d)-1}$ at a point (for $N>0$). These three geometric layers are

$$Q_d, \qquad \Sigma Q_d, \qquad \mathbb P(\mathcal E_{N,d}).$$

Only $Q_d$ names physical spatial dimensions. The other two objects describe the field and its modes on $Q_d$; their component counts add no dimensions to $M_\infty$.

2. Principal symbol on a base manifold

Let $X$ be a spatial manifold and $F\to X$ a vector bundle carrying polarization or charge data. A Dirac-type field has kinetic operator

$$\mathcal D_{\rm int} =i\gamma^\mu\nabla_\mu\otimes I_F+A(x),$$

where $A(x)$ acts algebraically on $F$. Its principal symbol is

$$\sigma_1(\mathcal D_{\rm int})(x,k) =\gamma^\mu k_\mu\otimes I_F.$$

Additional physical spatial coordinates $y^a$ extend the kinetic operator to

$$\mathcal D_{\rm spatial} =i\gamma^\mu\nabla_\mu+i\Gamma^a\nabla_a+A(x,y),$$

and its principal symbol becomes

$$\sigma_1(\mathcal D_{\rm spatial})(x,y;k,p) =\gamma^\mu k_\mu+\Gamma^ap_a.$$

The covector components $p_a$ are their conjugate momenta. They determine

• the characteristic set and propagation directions;
• the short-time heat kernel and spectral dimension;
• localization and uncertainty relations;
• the measure over which contact and overlap are calculated;
• the kinetic response to gradients in those directions.

Thus base dimension is read invariantly from the number of independent covector components in the principal symbol. Fibre matrices enter the lower-order algebraic structure and representation content.

3. Application to a particle sector

The projected field of a depth-$d$ particle is

$$\psi_d\in\Gamma(\Sigma Q_d\otimes F_d), \qquad \dim_{\mathbb R}Q_d=d.$$

Its kinetic principal symbol is

$$\sigma_1(\mathcal D_d)(q,p) =\gamma_d^ap_a\otimes I_{F_d}, \qquad p\in T_q^*Q_d,$$

with Clifford relation

$$\{\gamma_d^a,\gamma_d^b\}=2g_d^{ab}I.$$

There are exactly $d$ independent local momentum components. They are tied to the $d$ tangent directions of the manifold where the particle lives.

The bundle $F_d$ carries spinor polarization, charge, flux, or a finite representation on top of the base geometry:

$$\underbrace{Q_d}_{\text{physical spatial submanifold}} \quad+\quad \underbrace{\Sigma Q_d}_{\text{spinor fibre}} \quad+\quad \underbrace{F_d}_{\text{field representation}}.$$

The plus signs separate mathematical descriptions; they are not a sum of physical dimensions. Any physical role assigned to $F_d$ must come from the one master field and geometry, rather than a separate internal sector.

4. Why the mass count tests the base dimension

At each $q\in Q_d$, the oscillator symbol has $d$ complex amplitudes

$$a_a =\frac1{\sqrt2} \left( \frac{x_a}{\ell_d} +i\frac{\ell_dp_a}{\hbar} \right), \qquad a=1,\ldots,d.$$

These are the position and momentum quadratures belonging to the tangent and cotangent directions of $Q_d$. Their shell-$k$ symmetric mode space has rank

$$\binom{k+d-1}{d-1}.$$

The cumulative band through $N=n-1$ is

$$\mathcal E_{N,d} = \bigoplus_{k=0}^{N} \operatorname{Sym}^k(T^*Q_d\otimes\mathbb C),$$

with

$$\operatorname{rank}\mathcal E_{N,d} =\binom{N+d}{d} =S(n,d).$$

Thus $d$ enters the mass formula as the rank of the cotangent space whose independent momenta generate the oscillator modes.

The homogenized expression

$$\mathcal E_{N,d} \cong \operatorname{Sym}^{N} \left(\mathbf1\oplus T^*Q_d\otimes\mathbb C\right)$$

introduces one slack occupation coordinate. Its ray fibre $\mathbb{CP}^{d}$ compactly organizes the cumulative phase modes. Its real dimension $2d$ reflects one position and one momentum quadrature for every base direction.

5. Why the coupling construction also uses base geometry

Suppose a depth-$a$ sector and a depth-$b$ sector are related by

$$\pi_{ba}:Q_b\to Q_a, \qquad b>a.$$

The lower-depth field reaches the higher sector by pullback:

$$\psi_a^{\uparrow b}=\pi_{ba}^*\psi_a.$$

The higher-depth current reaches the lower sector by pushforward:

$$\bar J_{b\to a}=(\pi_{ba})_*J_b.$$

In local product coordinates $(x,y)\in Q_a\times F_{ba}$,

$$\bar J_{b\to a}(x) =\int_{F_{ba}}J_b(x,y)\,d\mu_{F_{ba}}(y).$$

The interaction is then an overlap on the common base:

$$I_{ab} =\int_{Q_a} J_a(x)\bar J_{b\to a}(x)\,d\mu_{Q_a}(x).$$

This geometric contact uses the additional directions to determine which points can coincide, how a density is marginalized, which polarizations survive the normal-direction contraction, and how the interaction varies with separation and orientation.

6. Normal geometry after lower-depth projection

When a higher-depth field is restricted to $Q_a\subset Q_b$, the tangent bundle splits:

$$TQ_b|_{Q_a}=TQ_a\oplus N_{ba}.$$

A lower-depth observer resolves $TQ_a$. Rotations of the normal frame $N_{ba}$ act on the projected fields as representation transformations, and the normal connection transforms as a gauge connection.

The projection map is therefore

$$\text{higher-dimensional geometry} \quad\xrightarrow{\text{restriction}}\quad \text{projected polarization and gauge connection}.$$

The generators descend from the parent Clifford term $\Gamma^ap_a$ and the normal connection. Gauge-like structures are therefore observer restrictions of particle-sector geometry.

7. Combined dimensional evidence

The same integer $d$ performs four linked jobs:

1. it is $\dim_{\mathbb R}Q_d$;
2. it counts the independent momenta in the Dirac principal symbol;
3. it fixes the degree of the simplex mass growth;
4. it fixes the tangent, normal, and fibre dimensions entering cross-depth contact and polarization.

The observed mass hierarchies recover

$$d=2,3,4,5,6,10$$

from scale-free ratios. Those same values organize the coupling projection, its stabilizers, the symmetric-power spectrum, spinor branching, contact marginalization, and mass growth. The projected principal symbol provides the invariant kinetic test joining these structures.

8. Projected-principal-symbol theorem

The decisive calculation can be made exactly once the meaning of sector projection is stated. Let

$$i_d:Q_d\hookrightarrow M_\infty$$

be a sector embedding with an orthogonal tangent--normal splitting along $Q_d$. Let $P_d$ be a smooth zeroth-order orthogonal projector from the restricted master spinor bundle onto a sector bundle $\Sigma_d$. For a covector $p\in T_q^*Q_d$, let $\iota_dp$ denote its tangential extension that annihilates the normal bundle.

Assume the master operator is Dirac type, so

$$\sigma_1(\mathcal D_\infty)(q,\Xi) =i\,c_\infty(\Xi),$$

and define the sector operator by tangential compression,

$$\mathcal D_d =P_d\mathcal D_\infty^{\parallel}P_d.$$

Because $P_d$ is order zero, symbol composition gives

$$\sigma_1(\mathcal D_d)(q,p) =iP_d c_\infty(\iota_dp)P_d.$$

Derivatives of $P_d$ contribute to the order-zero connection and algebraic mixing terms. The principal momenta remain those in the compressed Clifford symbol.

Now impose the essential compatibility condition: $\Sigma_d$ is invariant under tangential Clifford multiplication and its identification with the spin or $Spin^c$ bundle of $Q_d$ obeys

$$P_dc_\infty(\iota_dp)P_d=c_d(p).$$

Choose a local coframe $e^1,\ldots,e^d$ on $Q_d$ and write $p=p_ae^a$. Then

$$\boxed{ \sigma_1(P_d\mathcal D_\infty^{\parallel}P_d)(q,p) =i\gamma_d^ap_a }$$

with exactly $d$ independent momentum components. The inherited Clifford relation is

$$c_d(p)c_d(r)+c_d(r)c_d(p) =2g_d^{-1}(p,r)I,$$

and hence

$$\sigma_1(\mathcal D_d)(q,p)^2 =-g_d^{-1}(p,p)I$$

for the displayed $i\mathcal D$ convention. Thus the same $d$-dimensional cotangent metric controls the characteristic kinetic response.

For time-dependent dynamics, the same evolution parameter $t$ is used for every sector. The count $d$ here concerns spatial momenta on the nested submanifold; introducing a time parameter does not postulate another spatial manifold or another particle depth.

8.1 Lower-order normal geometry

Tangential compression sets normal covectors to zero. The normal connection and normal Clifford products survive in the lower-order part:

$$\mathcal D_d =i\gamma_d^a(\partial_a+\omega_a+A_a^{\perp})+\Phi_d.$$

Here $A_a^{\perp}$ and $\Phi_d$ act algebraically on the lower-depth fibre as the projected gauge polarization, while the principal symbol contains the momenta tangent to $Q_d$.

8.2 The theorem's hypotheses

The result is conditional on three geometric facts:

1. $P_d$ is a smooth zeroth-order bundle projector;
2. sector restriction means tangential compression with normal covectors set to zero;
3. the projected spinor subbundle is invariant under tangential Clifford multiplication.

Tangential Clifford invariance gives a full-rank elliptic symbol. Its direct verification for every sector is the diagnostic that the projected geometry retains all $d$ spatial directions.

9. Master-geometry closure

Under the hypotheses just stated, the formal calculation is complete:

$$\sigma_1 \left( P_d\mathcal D_\infty^{\parallel}P_d \right)(q,p) =i\gamma_d^ap_a, \qquad p\in T_q^*Q_d.$$

Physical completion constructs the master operator and sector projectors with:

• exactly $d$ independent momentum components;
• the Clifford relation for the metric on $Q_d$;
• propagation or standing-wave gradients in every one of those directions;
• the same metric and measure in the self-contact and cross-depth kernels;
• reduction to the observed lower-depth dispersion after inaccessible momenta are marginalized.

The mass and coupling constructions use the same $T^*Q_d$, metric, and measure. The completed projection therefore joins kinetic propagation, spectral growth, and cross-depth contact in one $d$-dimensional base geometry.