Two-electron shared response

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. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Summary: Two orthogonal occupied modes admit self-consistent collective precession through one dynamical contact source, with conserved energy and antisymmetry; this extends the response model but does not yet derive interacting petal trajectories.

What is established

The finite response model has a two-electron collective rotating family. Both occupied modes drive the same source, and that source acts back on both. The second electron cannot be omitted while retaining this solution. This is a positive existence result in a specified reciprocal model, not two noninteracting copies of the one-electron calculation.

Electron depth remains six. The four complex components below are the existing chiral carrier, not four spatial dimensions. Nor do the four available carrier modes set atomic shell capacities: spatial and other degrees of freedom have not been supplied in this truncation.

Shared variational system

Let U be a complex 4 by 2 occupied frame with U†U=I2. Its two columns define a normalized antisymmetric Slater state. Put D=UU†, so D²=D and tr D=2. Apply the existing linear moment map to D and take its mixed block Q. Thus Q is the sum of the two individual mixed blocks; D is a number-density matrix, not a trace-one single-electron state.

For the normalized ten-state contact-source vector n, use the existing response X=2 Re(n0* n_exc), arranged as a 3 by 3 matrix. The energy is

$$E(U,n)=\Delta\|n_{\rm exc}\|^2-g\sum_{i,j}X_{ij}Q_{ij}.$$

Varying this energy with the occupied-state and source Berry terms gives

$$i\hbar\dot U=h_e(X)U,\qquad i\hbar\dot n=h_N(Q)n, \qquad h_e(X)=g\sum_{i,j}X_{ij}\sigma_i\otimes\sigma_j.$$

The source Hamiltonian has excited diagonal Delta and singlet-to-excited matrix entries -g Qij. The interaction appears once in the joint energy, not once in each subsystem energy followed by double counting. Hermiticity preserves U†U and n†n. Reciprocity conserves E.

This is a variational Slater-times-source product-state model. It includes shared-source feedback but not unrestricted electron-source entanglement, general electron correlation, or direct spatial electron-electron repulsion.

Explicit collective family

For 0<a<1, define q=sqrt(1-a²), c=sqrt((1+a)/2), s=sqrt((1-a)/2),

$$u_1=(0,c,-s,0)^T,\qquad u_2=(c,0,0,s)^T.$$

These vectors are orthonormal and their total Q is diag(0,2q,0). Let K rotate the first two axes of the first real three-plane, KN act as K tensor I3 on source excitations and vanish on the singlet, and Ze=(sigma_z tensor I2)/2. With d=Delta-epsilon and b=d²-(hbar omega)², solve

$$p_0=\left[1+4g^2q^2\frac{d^2+(\hbar\omega)^2}{b^2}\right]^{-1}, \qquad \epsilon=-\frac{4g^2q^2d}{b},\qquad \omega=\frac{4g^2a p_0 d}{(\hbar/2)b}.$$

The source initial state is proportional to (1, g[(Delta-epsilon)I9-i hbar omega KN_exc]^{-1}vec Q). It is an eigenvector of hN-i hbar omega KN with eigenvalue epsilon. The initial occupied subspace is invariant under he-hbar omega Ze; call its 2 by 2 restriction Lambda. Then

$$U(t)=e^{-i\omega t Z_e}U(0)e^{-it\Lambda/\hbar},\qquad n(t)=e^{-i\epsilon t/\hbar}e^{\omega t K_N}n(0)$$

solve the coupled equations. These are relative equilibria, not imposed forcing of a frozen source. For Delta=4, g=sqrt(25/32), hbar=2:

Rates are in the model's units. The previous one-electron a=0.6 example has rate 0.4006985724 with the same Delta, g, and hbar. The pair changes both total preparation and source response; this is not a universal electron-count scaling law.

Physical reading and remaining test

U and U R for any unitary 2 by 2 matrix R describe the same occupied projector and the same Slater ray. Individual occupied-column paths are therefore not uniquely labeled electron trajectories. The gauge-invariant collective projector evolves nontrivially; a physical position readout must respect this frame freedom. This does not replace the intended actual orbits by stationary orbitals: it identifies what the present carrier calculation can establish about them.

The proven extension is shared-source collective precession with two electrons. The source has not yet generated the visualization's complete petal generator and position readout, for either one or two interacting electrons. No stability conclusion follows from existence or an unperturbed integration. Next is a gauge-consistent spatial many-electron readout and interaction, followed by perturbation tests and shell observables. Neither p/d/f capacities nor a helium spectrum has been derived here.

The subsequent petal response residual makes the generator mismatch explicit: the prescribed demo lift varies s at three times the clock frequency, whereas every orientation-only V(s) conserves s. Full finite-source inverse tests also find nonzero residual bounds at the tested parameters. The existing selected map already produces the radial modulation from the prescribed lift; its source dynamics, not the algebraic readout formula, is the newly tested missing connection.

Reproduction

two_electron_shared_response_checks.py solves all four roots and independently integrates each coupled solution for a full collective revolution. It checks energy, source normalization, occupied orthogonality, projector rank, Slater normalization and exchange antisymmetry. Random preparations test reciprocity and occupied-frame invariance. Omitting the second electron is required to fail the source equilibrium equation. Checks raise exceptions rather than using assert.

The initial run completed 3,716 evaluations across 21 conditions. Maximum full-revolution occupied-state and source errors were 7.75e-12 and 1.30e-11; energy drift was 2.51e-12 and occupied Gram error was 1.31e-12.

Mutation testing exposed that conservation on a relative equilibrium alone does not detect a doubled energy coupling. Twelve arbitrary-direction finite-difference energy-gradient checks were added to compare the energy function directly with both evolution equations. The expanded checker rejects reversed source evolution, doubled interaction energy, and omission of the second electron from the source tensor. This validation correction does not change the rotating solutions.

The expanded checker completed 3,728 evaluations across 22 conditions in both normal and Python -O runs. The energy-gradient finite-difference residual was at most 6.41e-10; all three deliberately incorrect variants were rejected by that check.

Related

• Gapped contact source — source definitions and the one-electron construction.
• Research status — current proof boundaries and correction routes.