Physical premises and nested spatial sectors

IDWT investigates particles as persistent or finite-lived structures of one field, $\Psi_\infty$, in infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; the remaining directions are unbound for every particle. Flat $\mathbb R^{10}$ is the spatial baseline for the ten-dimensional matter domain. All spatial patterns, frames and trajectories belong to this one spacetime.

“Ordinary” describes the physical kind of each direction: coordinates specify real positions, separations, gradients and motion. It does not mean that the containing space is three-dimensional. The familiar three dimensions belong to the observer's accessible view. The remaining directions are neither compactified nor replaced by a configuration space, a list of field components or a space of mode coefficients. IDWT has no Kaluza–Klein tower and no mediator-particle force mechanism. Interactions must arise from the dynamics of the same field.

A particle depth $d$ names the ordinary spatial directions involved in its bound pattern. The layer or nested-submanifold picture remains an admissible description of this involvement in one shared manifold. It does not supply separate containing worlds, a confinement mechanism, a global metric or a fixed preferred orientation. Those physical structures must be constructed and tested. The spatial-geometry module distinguishes these premises from its conditional mathematical realizations.

Reviewed Connections as of 5 November 2026

A pure seed and a compatible bound/free split supply an explicit new test of these premises. If p=floor(d/2) and q=floor((10-d)/2), the seed-fixing rotations preserving that split form the trace-zero blocks of u(p)+u(q). The maximal simple free factor is su(3) at the two assigned quark depths, three and four. Compatibility of the split with the seed's rotation structure is an extra geometric premise, not a coordinate relabeling or a proof of confinement.

The plane-count construction takes place in the same ordinary ten-space. Its common quark reference does not yet establish lepton singlets under one shared color action. The centered-window candidate also makes its global instantaneous interaction and fixed-rank assumptions explicit; those are candidate-specific constitutive choices, not additional universal IDWT premises.

What is spatial, and what describes a field?

Four objects must remain distinct:

A spatially extended field does not make a particle's position indeterminate. A squared field norm, a normalized coefficient share or an integration measure does not establish a Born position-probability law. Configuration space records several physical positions; events and forces still occur in the shared physical manifold.

Real amplitude and shape descriptions are the current convention. Complex notation may package two real quadratures or a rotation, but does not turn a coefficient ray into physical position. Older complex projective constructions retain additional mathematical assumptions. Their curvature, topology and finite-volume normalization are conditional comparisons; they are not established physical geometry.

The Dirac spinor is a working underlying-field proposal rather than a completed derivation or an irrevocable premise. Other classical field or spacetime constructions remain candidates when their effective Dirac behavior, localization, spin, frequency and interaction response are checked explicitly. A field-value metric is not automatically the metric of physical space.

Particle depth and orientation

The current oscillator proposal has $C=S(n,d)=\binom{n+d-1}{d}$ portions at one common fundamental per kind of pattern. Arbitrary phase offsets are permitted. Spatial and Fourier expansions describe their combined field; their basis components are not separately resonating physical objects. Counting one crossing per portion per cycle gives the mean rate $C\Omega/(2\pi)$ when the interaction registers coincident crossings with multiplicity. This registration law remains a physical premise to derive from the same field's interaction.

The electron, muon and tau share a fundamental in this proposal. A geometric mass coefficient, conserved field norm and detector event count remain different mathematical quantities whose proposed relations need one consistent action. In particular, changing the total conserved norm to make mass scale with count also changes norm-bilinear source charges unless the charge construction compensates dynamically.

The maintained assignments organize six depths:

These are hypotheses constrained by the sector evidence. A successful calculation at several higher depths can establish constructive compatibility without uniquely selecting one. Conditional evidence favoring a maintained depth must retain the same rule and normalization in the alternatives. Neither a count match nor a representation dimension alone proves spatial confinement or identifies an observed particle.

All ten matter directions are the same kind of physical direction. What distinguishes them is which patterns bind them, as the block geometry below sets out. The electron's six directions divide as the base four and one hidden pair; it has no $3+3$ split. Particles are independent field structures: an electron is not a constituent located inside a tau, and its solution need not be a restriction of a tau solution.

The block geometry of the ten matter directions

The ten matter directions fall into five blocks: the three observed directions, a fourth direction, and three hidden pairs $(5,6)$, $(7,8)$, $(9,10)$. A pattern binds some of these blocks and is spread evenly through the rest. For the four fermion depths,

$$d=3+[\text{fourth}]+2[\text{pair}],\qquad \Gamma_4=(-1)^{[\text{fourth}]+[\text{pair}]+1},\qquad j=2-2[\text{pair}],$$

where a bracket is one when that block is bound and zero otherwise, $\Gamma_4$ is the handedness sign on the base four directions and $j$ is the degree in the three pairs. The charge is $q=(\tfrac32\Gamma_4+j-\tfrac32)/3$:

The two quark families are the two depths with no pair bound, which is what colour requires: the three-colour symmetry needs all three pairs free. The tau binds every block. W, Z and Higgs bind one pair. The electron's six directions are the up quark's four together with one pair.

The partners of each weak doublet differ by one bound base direction. This is what survives of the suggestion that the fourth direction is itself the weak force: rotations of the base four commute with $\Gamma_4$, while each single Clifford matrix of the base four flips $\Gamma_4$ and changes the charge by exactly one unit, and the observed three and the fourth do this identically. The value algebra singles out one bound base direction, not the fourth one in particular.

A nucleus is uniform across the pairs. Only four of an electron's six coordinates can therefore change in an atom: the four directions of the orbit, which are the up quark's two planes. The fourth direction is $w=Ja$, the partner of the down quark's axis.

Particles differ only by frequency and geometry, because there is one field. Each particle label is to be read as a geometric statement about which blocks are bound, how many levels the pattern holds and how it oscillates. Polarization is by definition a spatial direction.

A pattern may rotate independently. Rotating its polarization within a chosen plane, changing the orientation of its occupied spatial subspace and changing a containing coordinate frame are different operations. Depth does not restrict a particle to a finite list of headings. A common physical construction must allow the compared patterns to coexist and interact in the same space, rather than combining independently optimized but incompatible geometries.

Literal electron motion around a nucleus

The six-dimensional electron itself follows a literal deterministic trajectory around the nucleus. The orbit is external motion through surrounding physical space; it is not motion inside the electron and is not replaced by a stationary orbital wavefunction or probability cloud. The existing orbit-plus-precession construction is the baseline. Its position $q(t)$, the electron's six-dimensional internal field pattern and a detector's accumulated image are separate descriptions.

The orbit is calculated in its four directions. It is not a three-dimensional Bohr or Kepler orbit with a fourth coordinate added. The electron's internal motion does not affect the orbit, and a spinor rotation belongs to the electron's own six-dimensional pattern, never to its external path. The light of a spectral line has the frequency of the orbit's precession, that is, of its motion along the fourth direction; it does not have the in-plane orbital frequency. In a molecule the electron's orbit goes around all the nuclei involved, and multiple bonds are electrons on equal orbits kept apart by phase. Volume 18 gives the calculations.

The neutron's charge is negated completely, in all four directions. Its opposite quark charges spread around one circle cancel in the three observed directions; the up quark's pull along the fourth direction would remain unless the negation is complete, and the deuterium line shows that it is complete to better than a part per million.

The current prepared flat curve uses four active real position coordinates embedded in the surrounding six-dimensional space. The orbit and observer module records the flat action equation

$$\ddot q=2\beta J\dot q-(\delta^2-\beta^2)q, \qquad J^T=-J,$$

with the gyroscopic Lagrangian term $+m\beta q^T J\dot q$. Here $\beta$ and $\delta$ are prescribed rotation and rotating-frame rates, $m>0$ is the positive collective trajectory inertia, and $J^T=-J$, $J^2=-I$ on the active four-dimensional position subspace. This is a concrete conditional trajectory family. It does not establish that nature selects its prescribed rates, its integer closures or four active directions.

A plane projection of a trajectory may display petals, while a time average and detector response may display a broader shape. The lowest circular prepared path and tuned higher-energy ringed shapes must not be conflated. In a detector description, a curve $\Gamma(t)$ and response kernel $K_\sigma$ can give an averaged registration

$$\rho_{\rm reg}(x)=\frac1T\int_0^T K_\sigma\!\left(x-\Gamma(t)\right)dt.$$

This is a conditional detector model. Its sampling law, width and interaction must be justified; the equation does not replace the trajectory or introduce a position-probability postulate. The projective observer calculations likewise retain their readout assumptions and their separate bridge from coefficients to physical position.

Bound within, free without

The working phrase distinguishes bound spatial structure from directions in which that structure is extended. It does not by itself fix a unique global realization. In the uniform-extension branch, local product coordinates for depths $a<b$ give

$$\Xi_b\simeq\Xi_a\times F_{b/a},\qquad f_a^{\uparrow b}(x,y)=f_a(x),\qquad \partial_y f_a^{\uparrow b}=0.$$

The coordinates $y$ are additional ordinary spatial directions. The displayed independence is a specified uniform-extension premise; localization in $x$ alone does not imply it. Whether the full kinetic and interaction operators preserve that premise is a dynamical question. Translation along a uniformly extended direction changes nothing in that pattern; distinguishable center motion occurs in directions where it varies. Reorientation of its bound and free directions is a separate possible motion.

The bound-within/free-without module gives an exact conditional overlap identity. For an allowed integrable local same-field term,

$$I_{ab}=\int d\mu_a(x)\,d\mu_F(y)\, \rho_a(x)\rho_b(x,y) =\int d\mu_a(x)\,\rho_a(x)\bar\rho_{b\to a}(x),$$

where

$$\bar\rho_{b\to a}(x)=\int\rho_b(x,y)\,d\mu_F(y).$$

This is Fubini marginalization under the stated cylindrical extension. “Complete marginal” means integrating all extra coordinates. It does not mean unit physical interaction strength: orientation, polarization, relative phase, the actual interaction kernel and the complete normalization can still suppress a channel.

On an infinite noncompact extra-coordinate domain, a nonzero uniform positive energy density generally has infinite total energy. A cylindrical field or generalized state and a finite-energy isolated particle therefore have distinct normalization questions. In a box of extra-coordinate volume $V_F$, a unit-normalized uniform density introduces $1/V_F$, and the corresponding overlap with a fixed normalized partner scales as $I_{ab}/V_F$. A physical source, flux or action normalization must justify any compensation. Marginalization itself provides none.

Consequently, the conditional contact identity does not construct a finite-inertia particle, prove universal cylindrical support, or replace the complete spatial energy calculation. Infinite cylinders, finite profiles and other support realizations must retain their separate assumptions. The proposed uniform extension used in electron-orbit comparisons must be tested for preservation under reciprocal interaction.

Free extension ends at contact

A free extension is uniform over its span and ends where it makes contact. It is not infinite, so the divergent-energy question of the infinite cylinder does not arise for it. Nothing travels along it: the object is already spread across the distance, so a connection through a free direction is already in place. This premise supplies the span, and the tension $T$ of a span is calculated in volume 12.

A pattern that is uniform along a free direction and also vibrates with its bound profile is linearly unstable to necking at every depth below ten that has been tested (volume 11). That calculation copies the vibrating profile along the free direction. It says nothing yet about a contact-ended or non-vibrating free extension, whose own stability is open.

The fourth direction as the path of alpha decay

Alpha decay needs no tunnelling and no imaginary time. The fourth direction is the natural path for that motion and the equation works the same way: for a field steady in time, a real spatial direction enters exactly where imaginary time enters the scalar equation. The alpha's down quarks are already spread along the fourth direction, so only its up quarks have a position to change there. The prediction is faster alpha decay with no electrons present. Volume 18 gives the exponent comparison and the present experimental limit.

Photon geometry and phase transfer

The photon is the explicit depth-two transverse control. Its two bound oscillation directions are distinct from its free directions in the shared higher-dimensional space. Linear, circular and elliptical polarization are physical spatial patterns to compare; the existence of an abstract two-state space is not a derivation of those patterns. The photon's massless case need not impose its transversality law on every higher-depth massive pattern.

The current spanning picture treats a photon as one source-to-target connection, with its line-like, spreading structure already present along its span and phase advancing through it. Observed timing is to be explained through phase accumulation, rather than the transit of a point photon. This is a physical hypothesis requiring a same-field derivation. A phase difference at a single frequency does not determine a delay; the frequency dependence and the full energy, stress, recoil and detection balance must be calculated.

Where the connection ends is likewise fixed by the field's actual configuration, not by an added probability law. The field's real configuration, relative phases and spatial suitability must supply that endpoint and the subsequent reconfiguration. Geometric coverage or a nonzero tail alone does not establish the required transfer channel, instantaneous completion or observed causal timing. The photon module develops the polarization comparison under its explicit assumptions.

A kinetic test of a spatial claim

A useful mathematical diagnostic survives the change of geometric interpretation. If a candidate field operator contains independent derivatives in $d$ physical directions, its principal symbol must retain their independent covector components. For a conditional Dirac-type construction,

$$\sigma_1(\mathcal D_d)(q,p)=i\gamma_d^a p_a, \qquad a=1,\ldots,d,$$

and

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

Adding algebraic polarization matrices changes components and couplings; it does not add physical spatial momenta. Conversely, suppressing derivatives by a chosen projector does not derive physical confinement. A tangential compression

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

has the desired symbol only if the assumed tangent geometry is present and the projected subbundle is invariant under tangential Clifford multiplication. Derivatives of a smooth order-zero projector enter lower-order connection terms. These are conditional mathematical facts, not a forward selection of the physical depth or proof that a particular curved topology is realized.

The same spatial metric, measure, support and normalization must ultimately enter the kinetic operator, spectral count and reciprocal interaction. Establishing that common physical structure is the bridge between the spatial premises and a complete mass operator. Existing count, representation and local-action compatibility results are useful constraints on this bridge; they do not supply the unfinished global spatial state, observed-state selection or interaction normalization.