Completing an Orientation Defect with Its Phase and Amplitude

One Ray and an Independent Source Carrier Are Different Reductions

The canonical-source comparison retains orientation J and independent normal coefficients chi. That is a conditional multiple-mode reduction. It must not be presented as the polar decomposition of one complete chiral spinor. At q=3 a nonzero chiral spinor has eight real components: one amplitude, one phase and six projective coordinates already determined by J. Adding an independent four-complex-component chi to J would supply fourteen real coordinates instead.

For a single pure ray, the established metric identity gives the complete second-order comparison norm

$$\Psi=R e^{i\theta}\psi(J),\quad \psi^\dagger\psi=1, \qquad \|d\Psi\|^2=dR^2+R^2\left[(d\theta+a)^2+\frac1{16}\|dJ\|_F^2\right], \quad a=-i\psi^\dagger d\psi.$$

The formula is exact on the pure orbit. At q=2,3 it covers the complete chiral ray; q=4 needs its additional nonpure directions unless purity is imposed. The canonical Spin connection acts on this defining vacuum line through minus one half of its normal complex trace. Its traceless part annihilates the vacuum. Thus a source in this same line is central; it is not an independently occupied triplet source. Spatial derivatives, other retained modes and another underlying field remain possible sources but require their actual action and reduction.

The Phase Energy of the Exact Root Defect

The root-sphere orientation from the determinant calculation corresponds locally to

$$\psi=\frac{(a,z)}{\sqrt{a^2+|z|^2}},\qquad z=x+iy, \quad a>0.$$

The other chiral components vanish. On the transverse plane in polar coordinates r,phi,

$$ds_{\rm FS}^2=\frac{a^2}{(a^2+r^2)^2}(dr^2+r^2d\varphi^2), \qquad a_{\rm B}=\frac{r^2}{a^2+r^2}d\varphi.$$

Here a_B denotes the real phase connection; it is not the width parameter a. Its curvature integrates to 2 pi, while the anti-Hermitian line connection i a_B has Chern number minus one in the earlier i F/(2 pi) convention. This is the same vacuum-line class derived from the normal determinant connection.

The normalized spinor is smooth and nonzero on the plane, but its spinor boundary value winds once at large r. Its orientation has a constant limiting ray. The complete spinor gradient norm is

$$\|\nabla\psi\|^2=\frac{r^2+2a^2}{(a^2+r^2)^2}, \qquad \int_{r<L}\|\nabla\psi\|^2d^2x =\pi\left[\log\left(1+\frac{L^2}{a^2}\right)+1-\frac{a^2}{a^2+L^2}\right].$$

The projective part has integral 2 pi; the remaining phase part diverges logarithmically. A fixed-amplitude orientation defect with finite orientation energy therefore need not have finite complete spinor energy.

The obstruction does not depend on this phase choice. For a single-valued smooth phase on a disk, its loop integral vanishes. The circulation of d theta + a_B tends to 2 pi on large circles. Cauchy–Schwarz on each circle gives an angular-energy lower bound proportional to 1/r, whose radial integral diverges. Equivalently, a nonzero spinor extending continuously to the one-point closure of the plane would trivialize its ray line, whereas this map has nonzero first Chern class. This closure states a constant boundary condition and is not a physical compactification. A phase singularity, amplitude zero, changed boundary or an additional connection changes the assumptions and must be included in the energy.

A Smooth Amplitude-Zero Lift

There is a constructive lift that uses the amplitude instead of discarding the phase contribution. For the same ray projector P=psi psi-dagger and fixed reference vector e=(0,1), take

$$\Psi=P e=\frac{(a\bar z,|z|^2)}{a^2+|z|^2}.$$

It is smooth on the entire plane, vanishes at the core and tends to the constant vector e at infinity. For r>0 it has the same projective ray as psi. Its amplitude and local phase are R=r/sqrt(a squared+r squared) and theta=−phi; their apparently singular polar description reconstructs a smooth Cartesian field. Direct differentiation gives

$$\|d\Psi\|^2=ds_{\rm FS}^2,\qquad \int_{\mathbb R^2}\|\nabla\Psi\|^2d^2x=2\pi.$$

The radial, phase and orientation terms all contribute. Finite complete gradient energy is achieved without setting the phase energy to zero by hand. This is a kinematic construction, not yet a spatial solution of a chosen unrestricted field action.

The image lies on the ordinary field-value sphere

$$|\Psi_1|^2+(\Psi_2-1/2)^2=1/4,\qquad \operatorname{Im}\Psi_2=0.$$

Its radius-one-half induced metric is precisely the unit-vector Fubini–Study metric on this CP1. A field constrained to this sphere therefore has the harmonic-map realization of the root defect. Such a target constraint and selected reference vector are extra constitutive structure, not consequences of a free spinor equation.

In fact the unrestricted flat-complex-field residual is nonzero:

$$\Delta\Psi=-\frac{8a^2}{(a^2+r^2)^2}(\Psi-e/2).$$

No unitary-invariant potential V(Psi-dagger Psi) supplies the missing force for this exact profile: its derivative is parallel to Psi, whereas this residual has a nonparallel e contribution wherever the first component is nonzero. A determinant action built from the complete flat coefficient metric also reduces to the flat harmonic equation on this conformal map, so the same residual remains. The profile is an explicit finite-energy preparation or constrained-target solution, not an unrestricted self-supporting particle.

Why One Projector Column Is Special to CP1

For a normalized ray in C^s, let P=psi psi-dagger, sigma=P e, t=|psi-dagger e| squared and let h be a horizontal ray tangent, psi-dagger h=0. Then

$$\|d\sigma(h)\|^2=t\|h\|^2+|h^\dagger e_\perp|^2, \qquad e_\perp=(I-P)e.$$

For 0<t<1 the real metric eigenvalues relative to Fubini–Study are 1 twice and t with multiplicity 2(s−2). For s=2 they are all one, explaining the exact root-sphere completion. For s=4, which is the complete q3 chiral carrier, four real directions have coefficient t and collapse as t tends to zero. The whole CP2 hyperplane perpendicular to e maps to sigma=0. This column prescription therefore does not globally complete the full six-dimensional q3 orientation metric, despite working exactly on each chosen root sphere.

The full projector instead obeys norm(dP) squared = 2 ds_FS squared at every ray and is a global representation of the projective metric. It uses a constrained matrix-valued field rather than a fundamental spinor; that alternative is allowed as a research hypothesis but has its own dynamics and spin question. Neither description introduces physical spatial dimensions through the matrix rank.

Sector Scope and Consequences

Every occupied nonempty q=2,3,4 carrier contains this same root CP1. Its line phase obstruction and amplitude-zero construction therefore apply at depths two, three, four, five and six. Above depth two the associated ideal defect still extends along d−2 ordinary directions; transverse finiteness is not finite total particle energy. Depth ten has no normal-orientation root under the present selected-parent prescription.

The calculation establishes a complete phase-and-amplitude test that an orientation-only defect must pass. It also gives an exact smooth finite-gradient lift and identifies its missing dynamical force. It does not derive color, electric charge, a preferred particle width, a decay rate or the simplex frequency operator. The next useful construction is a common complete-field action that supplies the needed target or supporting force while retaining the causal kinetic metric and all occupied-sector controls.

Reproduction

The standalone checker and recorded report preserve 516 exact conditions. They calculate the phase curvature and line class, full fixed-amplitude and amplitude-zero gradient integrals, all three polar contributions, the unrestricted field residual and potential obstruction, complete real differential Gram matrices for projector columns and full projectors in complex ranks two, four and eight, and the six occupied-sector controls. A Python interpreter with SymPy creates a dated report alongside the executed source. No spatial evolution or particle simulation enters these exact checks.