Bound within, free without

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 $\Psi_\infty$. See spatial meaning and mathematical scope.

Current Reading and Scope — 5 November 2026

Exact uniform extension means that translation in a pattern’s own free directions leaves it unchanged. Its center moves where its profile varies. An already extended pattern supplies spatial coverage along its free directions. Changes in mutual influence follow the causal same-field evolution, while finite contact endpoints supply their own boundary data. A compact profile in all ten directions is a distinct support ansatz, not automatically a lower-depth realization.

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. Claim and status

Within the layer description, a particle's depth counts the macroscopic spatial directions in which it has bound structure in the one shared space. The historical notation $M_\infty$ below denotes the single containing space; $\mathbb R^{10}$ below is the ten-dimensional matter domain within it. A depth-$a$ mode has localized structure in $\Xi_a$, while the same physical manifold continues beyond it:

$$\Xi_a\hookrightarrow\Xi_b\hookrightarrow M_\infty, \qquad a<b.$$

The founding kinematic statement is:

A particle is bound within the dimensions in which it has localized spatial structure and spans every additional dimension in which it has no localized motion.

Here spans means coordinate independence: the sector mode has equal support across the additional coordinate and is localized in none of its values. This is the content of bound within, free without.

That statement is a premise of the sector ontology. Together with coordinate nesting, it yields the contact theorem below by pullback, pushforward, and Fubini integration.

2. Bound coordinates and free coordinates

For nested depths $a<b$, write the higher coordinate space locally as

$$\Xi_b\simeq\Xi_a\times F_{b/a},$$

with shared coordinates $x\in\Xi_a$ and additional coordinates $y\in F_{b/a}$. Globally, the same argument uses a fibre bundle projection

$$\pi_{ba}:\Xi_b\longrightarrow\Xi_a.$$

A depth-$a$ mode has a normalizable bound factor $\chi_a(x)$ on its own sector. Its continuation to depth $b$ is the cylindrical pullback

$$(\pi_{ba}^{*}\chi_a)(x,y)=\chi_a(x).$$

Therefore

$$\partial_{y^j}(\pi_{ba}^{*}\chi_a)=0 \qquad (j=1,\ldots,b-a).$$

The lower-depth particle has no profile that must be aimed at a point of the extra fibre. It spans that fibre because nothing in its spatial mode distinguishes one fibre point from another.

The same state is compatible with, but not forced by, a free outer Hamiltonian. In a direction with no binding potential,

$$H_{\mathrm{free}} =-\frac{\hbar^2}{2\mu}\partial_y^2.$$

Its generalized momentum eigenfunctions $e^{iky}$ have uniform magnitude. The zero-momentum member is constant; a nonzero-momentum member has changing phase but still no localized body being carried along the coordinate. A finite wavepacket may have a localized envelope, but it is then a superposition of outer momenta. Thus the translation eigenmode gives the spatial “already-there” sector state, while outer localization represents additional dynamical preparation.

More generally, for outer coordinates $y=(y^1,\ldots,y^{b-a})$,

$$F_K(y)=e^{iK_jy^j}, \qquad |F_K(y)|=1.$$

If the particle has no phase motion in direction $y^j$, then $K_j=0$ and

$$\boxed{\partial_{y^j}F_K=0}.$$

It therefore spans every direction with zero phase motion. Nonzero phase motion preserves the same uniform plane-wave magnitude.

3. The cross-depth contact theorem

Let $\rho_a(x)$ be a density or current carried by the depth-$a$ mode, and let $\rho_b(x,y)$ be one carried by the depth-$b$ mode. From the higher-depth side, the lower-depth quantity is cylindrical:

$$\rho_a^{\uparrow b}(x,y)=\rho_a(x).$$

For an allowed interaction local on $\Xi_b$, the overlap is

$$I_{ab} = \int_{\Xi_a}d\mu_a(x) \int_{F_{b/a}}d\mu_F(y)\, \rho_a(x)\rho_b(x,y).$$

Define the pushforward, or marginal, of the higher-depth density:

$$\bar\rho_{b\to a}(x) = (\pi_{ba})_*\rho_b(x) = \int_{F_{b/a}}\rho_b(x,y)\,d\mu_F(y).$$

Fubini's theorem then gives

$$\boxed{ I_{ab} = \int_{\Xi_a} \rho_a(x)\bar\rho_{b\to a}(x)\,d\mu_a(x) }.$$

This is the cross-depth contact theorem. The higher-depth particle contributes through its complete marginal over the extra coordinates, and the cylindrical lower-depth extension covers the entire fibre. The overlap therefore carries the full marginal weight.

If $\rho_b$ is a normalized nonnegative density, its whole weight survives the pushforward:

$$\int_{\Xi_a}\bar\rho_{b\to a}\,d\mu_a = \int_{\Xi_b}\rho_b\,d\mu_b =1.$$

This is the precise sense in which contact is full: the interaction integrates the extra coordinates over the complete fibre.

For a nonlocal or filtered interaction, replace the pointwise product by a kernel on the shared coordinates:

$$I_{ab}[K] = \int_{\Xi_a\times\Xi_a} \rho_a(x)K_{ab}(x,x')\bar\rho_{b\to a}(x') \,d\mu_a(x)d\mu_a(x').$$

Unequal depth determines how the modes meet. The kernel, coupling constant, and representation content determine what they do when they meet.

4. Noncompact normalization

On an infinite fibre, the cylindrical pullback $\pi_{ba}^{*}\chi_a$ can be used as a translation-invariant generalized state or as a field/observable. These uses need distinct normalizations:

• a cylindrical field or observable pulled back from $\Xi_a$;
• a generalized momentum eigenstate with delta normalization; or
• a box-normalized approximation, whose density contains an explicit inverse fibre-volume factor.

The particle's normalization over its spatial support remains

$$\int_{\Xi_a}|\chi_a(x)|^2d\mu_a(x)=1.$$

The cross-depth interaction of a cylindrical field is finite when the higher-depth density supplies an integrable fibre factor. This does not make it equal to the overlap of two probability-normalized states. In a box of fibre volume $V_F$, a normalized uniform extension has density $\rho_a(x)/V_F$; its overlap with a fixed normalized $\rho_b$ is $I_{ab}/V_F$, tending to zero as the box grows. Cancelling that factor requires a specified flux, source-density, or coupling normalization in the action. It does not follow from Fubini's theorem. The contact formula above uses the cylindrical-field convention, normalized on the physical spatial sector.

5. Reciprocity

Let the interaction part of the action be

$$S_{ab}=g_{ab}\int dt\,I_{ab}.$$

For the local form, functional differentiation gives

$$\frac{\delta S_{ab}}{\delta\rho_a(x)} =g_{ab}\bar\rho_{b\to a}(x),$$

and

$$\frac{\delta S_{ab}}{\delta\rho_b(x,y)} =g_{ab}\rho_a(x).$$

The lower-depth mode feels the higher-depth mode through the latter's complete marginal. The higher-depth mode feels the lower-depth mode uniformly across the additional fibre. These are two variations of the same interaction term, so reciprocity follows when the action and kernel have the corresponding exchange symmetry. Both objects remain in their respective sectors.

In IDWT's one-field language, $\rho_a$ and $\rho_b$ are sector components of one field $\Psi_\infty$. A cross-sector term is that field coupling two of its own depth-resolved structures on their shared coordinates.

6. Coupling conditions

An interaction requires three ingredients:

1. Coordinate incidence. An embedding or shared-coordinate map must identify the active lower-depth structure inside the higher-depth sector.
2. Coupling filter. The modes must carry compatible charge, representation, chirality, or another interaction handle. The projected current may be zero despite full contact.
3. Dynamics. The action supplies a coupling constant and an interaction kernel or connection.

In a fixed nested gauge the inclusion $\Xi_a\hookrightarrow\Xi_b$ is explicit. Rotating spatial frames require the corresponding embedding or projector in the interaction. Coordinate independence supplies complete fibre coverage; the polarization or representation projector on the shared coordinates determines whether the contact channel is nonzero. Numerical coupling ratios come from that projector and the normalized action.

7. Causal evolution

Uniform support specifies the mode state. A disturbance evolves through the retarded Green function of the local field equation, whose causal cone governs the propagation of controllable changes. Marginalization specifies the coordinates sampled by the interaction at an event. Thus “already there” supplies coverage while the field equation supplies causal evolution.

8. Photon–electron example

For a photon sector of depth $2$ and an electron sector of depth $6$, use

$$x\in\Xi_2, \qquad y\in F_{6/2}.$$

The photon's spatial current is cylindrical in the electron's four additional coordinates. The electron enters the shared interaction through

$$\bar J_{e\to2}(x) = \int_{F_{6/2}}J_e(x,y)\,d\mu_F(y).$$

An electromagnetic term therefore has the schematic form

$$S_{\gamma e} = e\int dt\int_{\Xi_2} A_\alpha(x)\bar J_e^\alpha(x)\,d\mu_2(x),$$

or its covariant analogue on the full manifold. The photon's coordinate independence across the four additional electron directions supplies the cylindrical extension, and the electron supplies the complete fibre marginal at the shared $d=2$ interaction. Electric charge and the normalized $U(1)$ connection determine the interaction coefficient $e$.

The same geometry underlies photon transversality. The photon oscillates in its physical spatial two-plane, while outer coordinates carry free phase or zero variation. Higher-depth embedding therefore preserves the spatial polarization plane.

9. Result

The defensible chain is

$$\text{spatial localization plus assumed outer translation invariance} \Longrightarrow \text{cylindrical extension across additional coordinates},$$

followed by

$$\text{cylindrical lower-depth current} + \text{integrable higher-depth current} \Longrightarrow \text{full fibre marginal on the shared depth}.$$

This establishes the marginal overlap formula under the specified cylindrical extension and interaction. Spatial localization alone does not imply outer translation invariance or select that extension. Sector selection, representation filters, coupling constants, masses, and force laws are supplied by the corresponding projected dynamics.

10. Mathematical domain

Marginalization also does not by itself make a point particle appear smeared. For an instantaneous point trajectory,

$$\rho_b(x,y,t)=\delta(x-X(t))\delta(y-Y(t)),\qquad \int\rho_b(x,y,t)\,dy=\delta(x-X(t)).$$

An extended visible density instead requires visible extent in the joint state, an ensemble or time average of the trajectory, or detector response. Consequently, observing a spread alone does not distinguish the cylindrical ontology from an unobserved-localized source. Cylindrical extension remains the selected V2 premise here; the claim that smearing uniquely selects it is withdrawn.

Coordinate nesting, outer translation invariance, full-coverage coupling, higher-to-lower marginalization, reciprocal response, and coupling filters form one consistent geometric mechanism.

On a compact outer fibre, uniform support admits ordinary normalization. On an infinite fibre, it is a cylindrical or generalized state. Coupling magnitudes are computed from the action, kernel, connection, and representation geometry.

The companion module Coupling as higher-dimensional polarization supplies the representation-theoretic criterion for full geometric contact to survive the polarization projection as a nonzero interaction.

Uniform Extension and Kinetic Invariance

A uniform extension must be preserved by the actual ambient kinetic and interaction operators. For a linear generator H and a defined uniform-sector projector P, this requires (I-P)HP=0.

Contact-Limited Free Extent

A further construction lets a pattern extend along an unbound direction until actual contact with another pattern. The already present field need not cross that distance anew. Its endpoints, profile and reciprocal response must be part of the field evolution; changes in mutual influence still propagate causally. This is a contact-limited support hypothesis, not an already solved finite-span particle.

The exact uniform-pattern translation identity requires the complete field, its phase and components, the energy functional and the boundary conditions to remain translation invariant. A finite endpoint introduces spatial gradients. With noncompact tails, endpoint effects can survive even when centres are far from the endpoint. Derive their size, norm and clock dependence from the actual same-field solution rather than assuming an exact flat interior or selecting an arbitrary stability length. A finite-span calculation must use its own physical boundary conditions; periodic instability controls do not determine those conditions.

Intrinsic Vibration and Contact Energy

The current hypothesis places the mass-setting vibration in the bound structure, while the free extension need not copy that time-dependent excitation over its complete extent. In a conditional one-field collective description H=H_core(I)+E_span(L)+U_int(I,L,...), the core phase rate is partial_I H. It remains contact-independent when partial_L partial_I H=0. Span energy can then vary without changing the supplied fixed-count intrinsic mass-frequency. A static extension can still carry gradients and stress; the complete field must establish this reduction, its normalization and the causal work at moving contacts. The real fourth-direction account gives the exact scope.