A conditional frame model of electron–nucleus binding

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 — 1 November 2026

The electron remains a six-dimensional particle moving on a literal external trajectory around the nucleus; the maintained precessing curve uses four spatial directions. Its internal field, external position and detector average are different quantities. Orienting one bound direction constrains the complementary free set but does not fix the complete orientation flag. The attractive interaction and any resulting frequency lock remain to be derived.

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.

2. One field, two sector projections

The fundamental field is the Dirac field on the full manifold,

$\Psi_\infty\in\Gamma(\Sigma M_\infty).$

The electron and nucleus are modes of this one field, bound on nested spatial submanifolds. The proposed sector maps refer to those physical submanifolds:

$\Pi_6:M_\infty\to Q_6, \qquad \Pi_N:M_\infty\to Q_N.$

The electron field is a spatial spinor on its sector,

$\psi_e=P_{6,e}\Psi_\infty \in\Gamma(\Sigma Q_6\otimes L_{\rm em}),$

while the electron's sector coordinate is independently

$q=[Z]\in Q_6.$

Both arise from the master theory. Here $Z$ is a homogeneous coordinate for the electron's position in $Q_6$ , while $\psi_{e,+}(q)$ is the value of its chiral Dirac section.

3. The chosen observer-relative coordinates

Here $O$ groups directions shared with the chosen observer, and $H$ groups the remaining directions. All six are ordinary spatial dimensions of the same kind. The model vector space $V_6$ represents these directions; its use in a homogeneous-space formula does not create another physical space or establish a global tangent-frame identification.

Let $P$ be the orthogonal projector onto the nuclear-relative three-plane $O\subset V_6$ , and let $1-P$ project onto $H=O^\perp$ . These projectors implement the chosen coordinate grouping; they do not establish two physically distinct kinds of electron dimension.

Selecting $O$ reduces the model rotation symmetry as

$SO(6)\longrightarrow SO(3)_O\times SO(3)_H.$

This gives two coordinate three-planes within the model. An additionally assumed frame lock reduces their product symmetry to a diagonal subgroup.

5. Exact block algebra

The equation $J^2=-I_6$ gives

$A^2-QQ^T=-I_3,$

$C^2-Q^TQ=-I_3,$

and

$AQ+QC=0.$

Define the invariant

$\boxed{ s(q;P)=\frac12\lVert A\rVert_F^2 =\frac12\lVert PJP\rVert_F^2 }.$

For an antisymmetric $3\times3$ matrix, $A=[a]_\times$ for a vector $a\in\mathbb R^3$ , and

$\frac12\lVert A\rVert_F^2=|a|^2.$

The complex-structure identities imply

$0\le s\le1$

and fix the singular values of $Q$ :

$\boxed{ \sigma(Q)= \left\{1,\sqrt{1-s},\sqrt{1-s}\right\}. }$

They also give

$\frac12\lVert C\rVert_F^2=s.$

Thus one scalar labels the $SO(3)_O\times SO(3)_H$ orbit relevant to locking.

6. The locked orbit

$s=0,$

the antisymmetric blocks vanish:

$A=C=0.$

The block identities then force

$QQ^T=Q^TQ=I_3.$

After choosing orientation, $Q\in SO(3)$ . A selected value $Q_0$ is preserved by pairs satisfying

$R_OQ_0R_H^T=Q_0,$

which form a diagonal $SO(3)$ . Hence

$\boxed{ s=0 \Longrightarrow SO(3)_O\times SO(3)_H \longrightarrow SO(3)_{\rm diag}. }$

Under the spin double cover, the homogeneous-coordinate representation obeys

$\mathbf4=(\mathbf2,\mathbf2) \longrightarrow \mathbf1\oplus\mathbf3.$

This is the exact $s\oplus p$ branching used in the chemistry construction.

7. Two-factor coordinate decomposition

Under

$Spin(3)_O\times Spin(3)_H,$

the normalized homogeneous coordinate $Z\in\mathbb C^4$ has a tensor-product representation in

$\mathbb C^2_O\otimes\mathbb C^2_H.$

Here $Z$ labels a physical point in the six-dimensional sector. The two factors are coordinate representations, not two probabilistic electron subsystems. By independent frame rotations it has the singular-value (Schmidt) form

$Z_\lambda = \sqrt\lambda\,|00\rangle +\sqrt{1-\lambda}\,|11\rangle, \qquad \frac12\le\lambda\le1.$

The locking invariant is

$s=(2\lambda-1)^2,$

while the coordinate invariant conventionally called concurrence satisfies

$\mathcal C=2\sqrt{\lambda(1-\lambda)}, \qquad \boxed{\mathcal C^2=1-s}.$

The assumed frame lock therefore has equal squared singular values in this two-frame representation. The same algebra describes maximal entanglement when used for quantum states, but no such physical identification or Born probability rule is assumed here. The representation does not divide the physical electron into two parts or replace its deterministic trajectory.

8. A canonical one-parameter family

Let

$K= \begin{pmatrix} 0&-1&0\\ 1&0&0\\ 0&0&0 \end{pmatrix}, \qquad D_s=\operatorname{diag} \left(\sqrt{1-s},\sqrt{1-s},1\right).$

Then

$J(s)= \begin{pmatrix} \sqrt s\,K&D_s\\ -D_s&-\sqrt s\,K \end{pmatrix}$

obeys

$J(s)^T=-J(s), \qquad J(s)^2=-I_6.$

This family crosses every value of the locking invariant and provides the shortest variational test of a proposed electron--nucleus interaction.

9. The projected binding energy

Project the master interaction onto the actual sector manifolds:

$K_{N6}:Q_N\times Q_6\longrightarrow\mathbb R.$

For a fixed nuclear bound state with density $\rho_N(u)$ , the electron sees

$V_N(q) = \int_{Q_N} \rho_N(u)K_{N6}(u,q)\,d\mu_N(u).$

If the nuclear configuration is invariant under $SO(3)_O\times SO(3)_H$ , then $V_N$ may depend on the electron coordinate only through $s$ at lowest orbit level:

$V_N(q)=V_0+\kappa s(q;P)+O(s^2).$

The decisive coefficient is

$\boxed{ \kappa = \left.\frac{dV_N(J(s))}{ds}\right|_{s=0}. }$

Its sign has an immediate meaning:

$\begin{array}{c|c} \kappa>0& s=0\text{ is locally selected},\\ \kappa=0& \text{the interaction does not select the lock at this order},\\ \kappa<0& s=0\text{ is locally disfavoured}. \end{array}$

Because $q=[Z]$ is the electron's sector coordinate, a scalar interaction on $Q_6$ may depend on $s(q;P)$ . The projected kernel determines that dependence and hence the sign of $\kappa$ .

Here “selected” means locally preferred in energy, not automatic dynamical capture. A fixed-nucleus reduction with the invariant Hamiltonian structure on $SO(6)/U(3)$ gives the explicit conditional flow

$\dot J=\frac{2V'(s)}{\mathcal I}[PJP,J],\qquad \dot A=\dot C=0,\qquad Q(t)=\exp\!\left(\frac{2V'(s_0)}{\mathcal I}A_0t\right)Q_0.$

The positive action scale $\mathcal I$ fixes the symplectic normalization; it is not a derived nuclear inertia. This reduced contribution generates precession for $0<s_0<1$ and nonzero $V'(s_0)$ , but conserves $s$ rather than attracting the electron to $s=0$ . At $s=0$ the contribution vanishes. The derivation, endpoint qualifications, and independent numerical checks are recorded in Flat nucleus precession. This is a trajectory on the proposed electron-sector manifold, not a stationary-orbital calculation. The nuclear kernel, the connection to the relative orbital clock, and the physical three-coordinate readout remain to be supplied before it yields the existing petal construction.

An explicit inverse candidate for the orientation channel is now available:

$K_{N6}^{\rm orient}(u,J) =\frac{\gamma(u)}2\|P(u)JP(u)\|_F^2.$

For a common nuclear plane and normalized density this gives $V_N=\bar\gamma s$ , with $\bar\gamma=\int\rho_N\gamma\,d\mu_N$ . This reconstructs the reduced precession energy but does not derive $\gamma$ , radial binding, or a unique microscopic interaction. The inverse-kernel calculation also gives the reciprocal projector variation and distinguishes this quadratic channel from the oriented linear source needed to produce the visualization's particular fixed rotation generator.

A forward effective-source construction now produces this inverse channel: a positive-cost mixed-plane response with energy $\|X\|^2/(2\chi)-g\operatorname{tr}(X^TQ)$ gives $\kappa=\chi g^2$ after minimizing over X. Its reciprocal dynamics admit checked collective precession branches; see Induced cross-plane response. A gapped contact source also gives a saturating kernel and a self-consistent rotation without the harmonic-response approximation. The existence of that contact channel, its gap and coupling remain hypotheses, not derived nuclear properties. These constructions generate an orientation interaction; they do not replace the outstanding spatial binding and physical current calculations.