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: A singlet plus nine gapped contact excitations generate a positive orientation-kernel slope and a calculable tensor susceptibility, with a self-consistent rotating solution in the full finite source model.

A specified source instead of an unspecified susceptibility

Introduce one singlet contact state |0> and nine degenerate excited states |i,alpha> transforming as (3,3) under the nuclear-relative SO(3) times SO(3). Their gap Delta is positive. Ten is the number of response states here, not a particle depth, not a claim of ten nuclear spatial dimensions, and not an experimentally identified nuclear multiplet.

The presence of this cross-depth excitation channel is a new explicit effective-model premise. Neither the source carrier nor Delta nor its coupling g has been derived from the V2 master operator. This forward construction tests what such a source would actually generate; it does not claim to have discovered that source in the existing scalar vertex.

Define T_i_alpha=|0><i,alpha|+|i,alpha><0|. For the electron mixed block Q,

hN(J)=ΔΠexc−g∑i,αQiαTiα. h_N(J)=\Delta\Pi_{\rm exc}-g\sum_{i,\alpha}Q_{i\alpha}T_{i\alpha}.

The scalar contraction is invariant under simultaneous frame rotations. The unperturbed singlet has no oriented mean tensor, but it has a nonzero tensor susceptibility. Thus a vanishing unperturbed source is not the same as a vanishing induced response.

Exact static source and kernel

Only the excited combination along Q couples to the singlet. Eight orthogonal excited modes remain at Delta, while the two bright energies are

E±=Δ±Δ2+4g2∥Q∥22. E_\pm=\frac{\Delta\pm\sqrt{\Delta^2+4g^2\|Q\|^2}}2.

For the lower branch, using ||Q||²=3−2s,

Veff(s)=Δ−Δ2+4g2(3−2s)2,Veff′(s)=2g2Δ2+4g2(3−2s)>0. V_{\rm eff}(s)=\frac{\Delta-\sqrt{\Delta^2+4g^2(3-2s)}}2, \qquad V'_{\rm eff}(s)=\frac{2g^2}{\sqrt{\Delta^2+4g^2(3-2s)}}>0.

The induced expectation is

Xiα=⟨Tiα⟩=2gQiαΔ2+4g2∥Q∥2. X_{i\alpha}=\langle T_{i\alpha}\rangle =\frac{2gQ_{i\alpha}}{\sqrt{\Delta^2+4g^2\|Q\|^2}}.

The tensor norm is bounded below one. In the weak-coupling limit, chi=2/Delta and kappa=2g²/Delta, reproducing the quadratic inverse kernel up to its constant offset. The exact kernel is a saturating, nonlinear completion, not a permanently harmonic response.

For a general source vector f=g vec(Q), the differential susceptibility is 2I/D−8ff^T/D³ with D=sqrt(Delta²+4||f||²). Its transverse eigenvalue is 2/D and its longitudinal eigenvalue is 2 Delta²/D³, both positive.

The response timescale follows from the gap

Linearizing about the unperturbed singlet gives, with retarded boundary conditions,

χ(ω)=2ΔΔ2−(ℏω+i0)2. \chi(\omega)=\frac{2\Delta}{\Delta^2-(\hbar\omega+i0)^2}.

Equivalently the leading oscillator response has

χ(0)=2/Δ,MX=ℏ2/(2Δ),ΩX=Δ/ℏ. \chi(0)=2/\Delta,\qquad M_X=\hbar^2/(2\Delta),\qquad \Omega_X=\Delta/\hbar.

This relates the response inertia and susceptibility instead of choosing them independently once the gap and operator normalization are specified. It introduces no phenomenological damping. A finite-width resonance would require additional loss channels. The geometric electron action gives I_e=hbar/2 in the convention of the operator-map note.

Full reciprocal source dynamics

Let n be the normalized ten-component contact state and Z the normalized geometric four-component electron lift. With X=_n, use the one energy

E(Z,n)=Δ∥nexc∥2−gtr⁡(XTQ(Z)). E(Z,n)=\Delta\|n_{\rm exc}\|^2-g\operatorname{tr}(X^TQ(Z)).

Variation of the two projective action terms gives

iℏn˙=hN(J)n,iℏZ˙=g∑i,αXiασi⊗σα Z. i\hbar\dot n=h_N(J)n,\qquad i\hbar\dot Z=g\sum_{i,\alpha}X_{i\alpha} \sigma_i\otimes\sigma_\alpha\,Z.

The interaction is counted once in E although it appears in both variation equations. Both norms and the joint energy are conserved. There is no prescribed orbit-driving matrix: the response expectation drives Z and the electron Q drives n. This remains a finite effective contact model, not the full spatial electron–nucleus field equations.

A solved collective rotation without the harmonic approximation

Take a=.6, q=.8, hbar=2, Delta=4, and g²=25/32. These are dimensionless diagnostic choices. Let the O plane rotate at rate omega, and denote the source's rotating-frame eigenvalue by epsilon. Put d=Delta−epsilon and den=d²−(hbar omega)². The singlet occupation p0 and two consistency equations are

p0=[1+g2(2q2(d2+(ℏω)2)den2+1d2)]−1, p_0=\left[1+g^2\left( \frac{2q^2(d^2+(\hbar\omega)^2)}{\mathrm{den}^2} +\frac1{d^2}\right)\right]^{-1},

ϵ=−g2(2q2dden+1d),ω=4g2ap0dIeden. \epsilon=-g^2\left(\frac{2q^2d}{\mathrm{den}}+\frac1d\right),\qquad \omega=\frac{4g^2a p_0d}{\mathcal I_e\mathrm{den}}.

These equations come respectively from the contact-state eigenproblem and the electron precession equation. Solving them gives approximately epsilon=−.41150923 and omega=.40069857. Both rotating-frame eigenvector conditions and the full coupled time evolution are checked independently. The rate is an output for these chosen parameters and preparation, not an observed atomic frequency or a fitted integer lock.

For the same kappa and response frequency, the harmonic-response model has the low branch omega=.5. The difference is a concrete finite-response correction from the explicit bounded source; replacing the exact source with its linear susceptibility is not harmless in this example. The linear-stability result for the harmonic model does not automatically transfer to this full contact-state solution.

Full source stability diagnostic

An additional check linearizes the coupled projective equations on all six real CP3 tangent directions and eighteen real CP9 source directions. The two common phases are excluded. The analytic 24x24 Jacobian agrees with independent centered finite differences to approximately 3.5e-10. Its largest computed real eigenvalue is about 1.3e-8, associated with the near-zero sector; the computed nullities of the matrix and its square are three and four at threshold 1e-9. This is consistent with neutral drift and no resolved exponential instability in this example, not an exact stability proof or a claim of bounded nonlinear perturbations.

Finite differences alone initially gave an apparent positive rate near 5.6e-5 at step 1e-4. It shrank in proportion to the finite-difference step and was traced to the near-zero sector using the analytic Jacobian. That apparent rate was not promoted to a physical instability.

Verification and open physics

gapped_contact_source_checks.py checks exact source spectra, induced tensors, positive susceptibility, the orientation slope, the linear step response, the coupled rotation root, both rotating eigenvectors, direct integration of both states, and the full tangent linearization. Normal and optimized runs each complete 375 evaluations across 21 runtime conditions. The response carrier, gap and coupling are still hypotheses. Their derivation from an allowed cross-depth current vertex is the next physical test, followed by stability and the physical observer/current map.