Full response extension

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: An autonomous reciprocal 3+3+9-channel source extension conserves energy and particle occupations but does not sustain the prescribed petal lifts at the tested parameters, for one or two electrons.

Extension implemented beside the baseline

The previous source coupled the nine mixed-plane generators sigma_i tensor sigma_j. Add the three sigma_i tensor I and three I tensor sigma_j channels. Together these fifteen traceless Hermitian generators P_a span the existing Spin(6) chiral carrier, with tr(P_a P_b)=4 delta_ab. This adds response components, not spatial dimensions: electron depth remains six.

Keep the prior gap Delta=4, coupling g=sqrt(25/32), and hbar=2, without fitting a separate parameter for each orbit. The enlarged source has one singlet and fifteen excited states, all with the same positive gap. Sixteen counts source states, not a nuclear or electron spatial depth. The equality of all channel gaps and couplings is a diagnostic premise, not a measured nuclear property. The mixed subblock reproduces the old source convention.

For N=1 or 2 orthonormal occupied columns U, D=UU† and m_a=tr(D P_a), let the normalized source vector be n=(n0,n_exc) and X_a=2 Re(n0* n_a). Use

$$E=\Delta\|n_{\rm exc}\|^2+g\sum_a X_a m_a, \qquad i\hbar\dot U=g\sum_a X_aP_a U, \qquad i\hbar\dot n=h_N(m)n.$$

Here hN has excited diagonal Delta and singlet-excited entries g m_a. For mixed channels m=-Q, recovering the old -gQ source entries. There is one shared source and one interaction energy, not a separate source clone for each electron. The equations follow from one Berry-term action and are autonomous: no prescribed clock or desired orbit appears in their RHS. U†U, n†n, total energy, and all fifteen simultaneous SU(4) rotation charges are conserved. For N=2 the occupied frame defines an antisymmetric Slater state; general electron-source entanglement and spatial pair forces remain outside this product-state response model.

Exact relaxed-source result

Completeness of the generators gives

$$\sum_a m_aP_a=4D-NI,\qquad \sum_a m_a^2=N(4-N).$$

The latter is constant over the occupied Grassmannian. The source ground energy, and joint global minimum within this fixed-N model, is therefore

$$E_{\min,N}=\frac{\Delta-\sqrt{\Delta^2+4g^2N(4-N)}}2.$$

The ground-source response is proportional to -m. Thus he is proportional to -(4D-NI), commutes with D, and generates no motion of the occupied projector. This is a stationary joint minimum family, not a petal solution. Equal coupling of all fifteen channels restores full SU(4) symmetry and removes the old static preference for the nuclear plane split.

Even allowing distinct scalar couplings ga, gb, gq for the three channel blocks does not fix the static problem. For a pure electron,

$$\|g m\|^2=(g_a^2+g_b^2)s+g_q^2(3-2s), \qquad E_-(s)=\frac{\Delta-\sqrt{\Delta^2+4\|g m\|^2}}2.$$

This is still V(s), so s is conserved by its effective motion. The equal case makes V constant. These statements concern static elimination; they do not exclude differently coupled dynamical source channels.

Autonomous integrations and perturbations

For one electron and two orthogonal occupied modes, initialize the source at its corresponding ground state, then add normalized tangent perturbations of amplitude 0, 0.001, and 0.01. Integrate for 12 model time units. The unperturbed projector remains stationary to about 4e-13. Perturbed projectors move, with maximum Frobenius displacements:

These use deterministic sampled perturbation directions, not a worst-case stability scan. There is also an exact nonlinear energy bound: with ng(D) the instantaneous source ground ray,

$$1-|\langle n_g(D),n\rangle|^2 \leq\frac{E-E_{\min,N}}{\Delta-E_{\min,N}}.$$

The next source level lies at Delta, so the bound follows by spectral decomposition. It confines the system to the joint minimum family, not to a particular orientation, and does not preclude neutral drift. Since there is no identified petal solution here, this is not a petal stability result and no petal attractor is claimed.

An additional preparation sets he(0)=hbar omega (G+I/4) exactly, with omega 0.2 and G=diag(1,-1,-1,0). Its required response norm is below one, so an explicit normalized pure source realizes it. The initial velocity is exactly the target velocity, but autonomous evolution does not retain the target. Over one intended clock turn, the L=2 one-electron projector error reaches 1.39807 and planar readout error reaches 1.98284 R. The two-electron target projector error reaches 1.29341. These are trajectory errors for that preparation, not universal lower bounds or experimental discrepancies.

All-initial-source test at fixed parameters

For the equal-channel model, adjoint covariance gives an exact propagator along the prescribed target, with no source integration approximation. Let K be the real antisymmetric source-amplitude generator induced by G, extended by zero on the singlet. In the rotating frame,

$$h_{\rm rot}=h_N(m(0))-i\hbar\omega K, \qquad W(\theta)=e^{-i h_{\rm rot}\theta/(\hbar\omega)}.$$

The laboratory propagator is exp(theta K)W(theta). It agrees with an independent source ODE calculation to about 2.5e-13. In this frame the target occupied subspace is fixed. Its necessary equation is (I-D)he U=hbar omega (I-D)G U. For two electrons this tests the physical projector, not an arbitrary choice of occupied-column phases or basis.

Expand every possible initial source density matrix in 256 orthonormal Hermitian matrices. Across 49 phases in a clock turn the equation becomes Ax=b. Unlike the old nine-channel test, unrestricted matrices can nearly fit the target, so their least-squares residual is an ineffective witness. They need not be normalized physical source states.

Every physical source density satisfies ||x||2=||rhoN||F<=1. Minimize ||b-Ax|| over this unit ball using an SVD and a solved scalar trust-region multiplier. This ball is a superset of physical density matrices: it does not impose trace one or positivity. A positive minimum therefore excludes physical densities too. For any unit residual witness v,

$$\|b-Ax\|\geq v^T b-\|A^T v\|.$$

The primal residual and this dual bound agree to about 4e-15. Feasible positive controls are not excluded. These are numerical witnesses, not interval-certified all-parameter theorems. The reported relative bounds divide by ||b|| and refer to sampled generator equations, not spatial error.

Single-electron preparations are exactly the previous demo lifts. The pair uses z1=(r,r,1/sqrt(L),1)/sqrt(2), r²=(1-1/L)/2, and z2=exp(-i pi G)z1. They are orthogonal and each chosen column retains the selected petal readout, but their amplitude ratio differs from the single-electron demo. The inverse test uses their gauge-invariant rank-two projector. Individual column readouts are not yet a gauge-independent many-electron position map.

Conclusion and next decision

The six added channels are implemented, reciprocal, and mathematically consistent. They do not make the tested autonomous model generate the prescribed petals. Its relaxed family is stationary; no physical initial source in the finite equal-channel model matches the sampled target equations in the fourteen tested cases.

This rules out this proposed completion at these parameters, not six-dimensional electron orbits, all source spectra, all generator orientations, or alternative lifts giving the same displayed path. It also does not invalidate the earlier two-electron shared-source precession solution of the nine-channel model.

The next physical modeling choice should not be another imposed time-dependent drive. The response-only action has no independent spatial kinetic/binding Hamiltonian. The historical Coulomb-plus-precession construction suggests testing the base motion and its backreaction together, or deriving a physically motivated nondegenerate nuclear response spectrum, before asserting that the response source alone must supply the entire generator. Neither extension is silently included in the present test.

Reproduction and related notes

The subsequent Coulomb angular response test supplies actual six-coordinate kinetic motion and a new reciprocal angular-momentum source. Its linear coupling has exact signed-rose paths, but the repelling pair is unstable and the uniform-connection quadratic completion changes the motion. It is not a hidden identification with the gapped-amplitude source tested here.

The normal and optimized checkers each completed 2,414 evaluations across 35 conditions. Maximum energy-gradient residual was 4.4e-10, and maximum simultaneous rotation-charge derivative was 6.66e-16. Direct physical-source propagation and the inverse design agree to 1.18e-15. The initially matched full-turn integrations conserved energy to 1.38e-10 and occupied Gram matrices to 4.2e-11. Reversed source evolution and doubled interaction energy each failed the reciprocal-gradient check; a wrong-sign witness failed an analytic unit-ball control. These checks do not hardcode a successful petal outcome: the measured residuals and trajectory errors are reported. The complete regression run passed all 49 finite diagnostic scripts.

• full_response_extension_checks.py — autonomous equations, perturbations, physical-source witnesses, and two-electron tests.
• Petal response residual — mismatch motivating the additional channels.
• Two-electron shared response — established baseline precession, not this full-channel completion.
• Gapped contact source — original nine-channel source assumptions.
• Geometric response operator — Pauli and geometric conventions.
• Research status — current conclusions and proof boundaries.