Five-dimensional CP3 selection and spectral discrimination

Projective rays here describe field coefficients. Particle positions remain ordinary spatial coordinates in the infinite-dimensional containing space.

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 five-dimensional spinor-bivector model also selects a CP3 minimum family, but its exact spectral sum rule excludes the six-dimensional singlet-triplet gap within the linear-bivector operator class.

Why this comparison matters

The six-dimensional selection uses one pure normalized irreducible spinor, a real skew source, positive quadratic source energy, and linear spinor-bivector coupling. Here the same construction is performed for a five-dimensional Euclidean carrier. No inherited V1 spatial geometry is assigned to that carrier.

An irreducible Spin(5) spinor has four complex components, as does either irreducible half-spin representation of Spin(6). The general dimensions are $2^n$ for Spin$(2n+1)$ and $2^{n-1}$ for each half-spin of Spin$(2n)$. See Woit's representation construction. The explicit five-dimensional representation and its quaternionic structure are also discussed in Niedzialomski's five-dimensional spinor analysis.

The following energy and spectrum calculations are explicit local derivations, not claims taken from those references.

Exact five-dimensional construction

Using Pauli matrices, take the Hermitian Clifford generators

$$\gamma=(\sigma_1\otimes I,\sigma_2\otimes I, \sigma_3\otimes\sigma_1,\sigma_3\otimes\sigma_2, \sigma_3\otimes\sigma_3).$$

They obey $\{\gamma_a,\gamma_b\}=2\delta_{ab}I_4$. For $z^\dagger z=1$, define

$$H_{ab}=-i\gamma_a\gamma_b,\quad b_{ab}=z^\dagger H_{ab}z,\quad n_a=z^\dagger\gamma_a z, \quad S(b)=\sum_{a<b}b_{ab}H_{ab}.$$

Let $J$ be the skew matrix with upper entries $b_{ab}$, with $\|J\|^2=-\operatorname{tr}(J^2)/2=\sum b_{ab}^2$. Direct Clifford calculation gives

$$|n|^2=1,\qquad \|J\|^2=2,\qquad J^2=-I_5+nn^T,\qquad Jn=0.$$

Thus the source has two unit rotation planes and one null axis. It is not an invertible complex structure on an odd-dimensional vector space. Furthermore

$$S(b)z=2z,\qquad zz^\dagger=\frac{S(b)^2}{8}+\frac{S(b)}4.$$

The last identity reconstructs the ray from the bivector, so no ray degree is lost. These identities are verified as polynomials in eight independent real spinor components, without assuming normalization numerically.

For an initially unrestricted skew $B$, take

$$E(B,z)=\mu\|B\|^2-g\langle B,J(z)\rangle, \qquad \mu,g>0.$$

With $\rho=g/(2\mu)$, completion gives

$$E(B,z)=\mu\|B-\rho J(z)\|^2-\frac{g^2}{2\mu}.$$

All normalized rays minimize at $B=\rho J(z)$, and the injective reconstruction makes their minimum family CP3. This does not require assuming CP3 as the carrier's spatial base. It is the collective polarization/source family of this specified five-dimensional model.

The Berry term $i\hbar z^\dagger\dot z$ is the same nonzero projective symplectic structure as in the six-dimensional construction. Adding $M\|\dot B\|^2/2$ gives reciprocal dynamics and the exact conserved bound

$$\mu\|B-\rho J(z)\|^2+\frac{\|\Pi\|^2}{2M} =E_{\rm total}-E_{\min}.$$

This controls departure from the minimum family, not spatial localization or an electron-nucleus orbit. The ten-component source-space Hessian after ground-ray elimination has six flat directions and four positive normal directions $2\mu$; these are matrix-space counts, not extra spatial axes. The Hessian statement also follows analytically: minimizing the completed square over rays gives $E_{\rm eff}(B)-E_{\min}=\mu\,operatorname{dist} (B,\rho J(\mathrm{CP}^3))^2$. In a tubular neighborhood of this smooth embedded orbit, the squared-distance Hessian is twice the normal projector. The numerical Hessian is an independent finite-difference check of that result.

What differs from six dimensions

At the selected source, the five-dimensional spinor Hamiltonian has

$$\operatorname{spec}h_5(\rho J) =g\rho\,(-2,0,0,2),$$

whereas the six-dimensional chiral model has

$$\operatorname{spec}h_6(\rho J) =g\rho\,(-3,1,1,1).$$

Their gaps above the selected ray are respectively $(2,2,4)g\rho$ and $(4,4,4)g\rho$. Thus “CP3 minima, a gap, a Berry term, and conservative confinement” do not distinguish them, while the full gap multiplicity does.

Their induced source kinetic metrics also differ. For Fubini–Study unit horizontal tangents at $z=(1,0,0,0)$, the five-dimensional squared bivector derivative has weights $(4,4,4,4,8,8)$; the six-dimensional construction has weight $8$ in all six directions. The five-dimensional CP3 family is not being claimed to have the same induced metric or every same response.

A genuine no-go theorem for the five-dimensional competitor

Every real skew $5\times5$ source is orthogonally reducible to two rotation blocks and a zero direction. The corresponding irreducible spinor Hamiltonian has eigenvalues

$$\{\alpha+\beta,\alpha-\beta,-\alpha+\beta,-\alpha-\beta\},$$

up to an overall scalar shift and coupling scale. Order the positive block strengths $\alpha\geq\beta\geq0$. If its ground state is nondegenerate, the three gaps are

$$\Delta_1=2\beta,\qquad\Delta_2=2\alpha, \qquad\Delta_3=2(\alpha+\beta)=\Delta_1+\Delta_2.$$

Consequently three equal positive gaps are impossible for any source in this operator class, not just for the selected minimum or a sampled parameter range. Equivalently, the ordered eigenvalues always obey $\lambda_1+\lambda_4=\lambda_2+\lambda_3$.

This is robust: a target with eigenvalues $(E_0,E_0+\Delta,E_0+\Delta, E_0+\Delta)$, $\Delta>0$, has operator-norm distance at least $\Delta/4$ from every scalar-plus-five-dimensional-bivector Hamiltonian, even allowing arbitrary spinor bases. If the norm error is $\epsilon$, each ordered eigenvalue differs by at most $\epsilon$. The two pair sums of the target differ by $\Delta$, so the competitor's equality forces $\Delta\leq4\epsilon$. The bound is attained: for target $\operatorname{diag}(-3,1,1,1)$, the aligned canonical bivector spectrum $(-2,0,0,2)$ is at distance one.

Conditional exact-six theorem

Assume all of the following are independently justified:

1. The relevant polarization is one fundamental complex spatial spin module, or one irreducible half-spin module in even dimensions, of a Euclidean $d$-dimensional carrier. It has exactly four complex components. This means the module supplied by the spatial Clifford construction, not an arbitrary higher-spin representation, multiplicity space, or four-level internal system.
2. Its source Hamiltonian is linear in spatial bivectors, apart from a scalar shift; no extra vector or higher-degree operator is present.
3. A selected ray has one nondegenerate energy and three exactly equal higher energies separated by a nonzero gap.

The spin-representation dimension formula restricts $d$ to five or six. The spectral sum rule excludes five. Within these premises, $d=6$. Unlike a theorem that assumes chirality from the start, this argument uses the spectrum to discriminate the two four-component irreducible spinors. It does not establish that an actual electron satisfies premises 1–3. In the existing archive the six-dimensional operator produces this spectrum, but deriving a physical requirement for it independently of that operator remains necessary to avoid circular identification.

Frozen-source gaps versus reciprocal response frequencies

The preceding spectra hold the source fixed during a spinor variation. They are not the normal frequencies of the entire coupled source-spinor system. The flat minimum family supplies six zero modes in that larger system. This distinction can be followed through exactly.

For one complex tangent $u$, choose the corresponding normalized complex source displacement $b$. If the squared moment-map derivative weight is $w$, then $\Delta=g^2w/(4\mu)$ and the linearized equations are

$$M\ddot b=-2\mu b+g\sqrt w\,u, \qquad i\hbar\dot u=\Delta u-\frac{g\sqrt w}{2}b.$$

With time dependence $e^{-i\omega t}$ the characteristic polynomial is

$$\omega\left(M\hbar\omega^2-M\Delta\omega-2\mu\hbar\right)=0.$$

Apart from the zero mode, the positive frequency pair is

$$\omega_{\rm high,low} =\frac{\sqrt{\Delta^2+8\mu\hbar^2/M}\ \mathord{\pm}\ \Delta}{2\hbar}.$$

In particular

$$\omega_{\rm high}-\omega_{\rm low}=\frac{\Delta}{\hbar}, \qquad\omega_{\rm high}\omega_{\rm low}=\frac{2\mu}{M}.$$

Thus reciprocal source motion changes the frequencies but preserves the gap as a pair separation in this quadratic, isotropic-inertia model. Five dimensions have two identical pairs and one different pair; six dimensions have three identical pairs. The remaining source-normal modes have frequency $\sqrt{2\mu/M}$, with multiplicity four and nine respectively. These are additional source degrees of freedom, not spatial dimensions.

At $\mu=1.3,g=0.7,M=0.4,\hbar=2$, the five-dimensional pairs are $(2.45701979,2.64548133)$ twice and $(2.36800433,2.74492741)$ once. Six dimensions give the latter pair three times. Direct full nonlinear projective evolution matches the analytic tangent response: reducing perturbations from $10^{-4}$ to $10^{-5}$ reduces the scaled prediction error by a factor of ten in both models.

This is a calculable response target, not a demonstrated experimental access route. Other source inertias, extra couplings, damping, nuclear interaction, and detector selection require their own calculation.

For the explicitly specified drive “force on the source tangent” and readout “the same source displacement,” the complex-helicity susceptibility is

$$\chi_b(\omega)=\frac{\Delta-\hbar\omega} {\omega(M\hbar\omega^2-M\Delta\omega-2\mu\hbar)}.$$

Its nonzero mode poles have nonvanishing numerators, so they are not dark to this input/output pair. The full linear-system resolvent reproduces this formula at fifteen off-resonant probes in each dimension to below $1.8\times10^{-15}$. The frozen-source value $\omega=\Delta/\hbar$ is an antiresonance of this helicity response, not a coupled-system pole. These are ideal model controls and observations; their laboratory access has not been derived.

Operator-class escape route: why the hypotheses matter

The same five-dimensional matrices obey

$$S(b)+\sum_a n_a\gamma_a=4zz^\dagger-I_4.$$

Allowing an additional vector source therefore reproduces the singlet-triplet operator exactly in five dimensions. The no-go theorem is a theorem about linear bivector coupling, not arbitrary five-dimensional interactions. Scalar nonlinearities in the bivector operator could also alter its spectrum. A microscopic derivation must justify the allowed operator class before this discriminator can decide spatial dimension.

None of these source-polarization energy gaps have yet been identified with measured atomic excitation energies. Their multiplicities should not be relabelled p/d/f shell counts.

Verification

spin5_selection_checks.py checks universal Clifford polynomial identities, moment-map injectivity, the completed-square minimum, all six projective tangent directions, the selected spectrum, normal Hessian, induced metric, Berry curvature, and reciprocal conservation. The first run completed 501 checks. Later spectral-distance and vector-channel checks extend that run; saved reports retain their separate counts.

spinor_response_poles_checks.py checks the reciprocal frequency calculation against the full variational matrix and nonlinear evolution. Evidence: extended selection checks and reciprocal response checks. The forced-response extension completes 93 runtime checks, including source-pole observability.

Related

• Dimensional magnetic kernel comparison — applies the same spinors to the existing constrained magnetic kernel and checks the scalar-exchange escape channel.
• Chiral CP3 selection — six-dimensional construction compared under the same type of source energy.
• Dimensional necessity from response — independent rank theorem and dynamical five-dimensional competitor.