From integer counts to a two-component readout

Section 3.48 assigns one complex amplitude to each of two four-cell cycles. The real part is the first count minus the third, and the imaginary part is the second count minus the fourth. Both amplitudes share a positive normalization. Starting with the allocations in Table 1 gives real and imaginary numerators of four and two for the first component and four and zero for the second. The normalization is six, so the squared component magnitudes sum to one. [1]

The source applies a chosen rotation about the spin y axis. Its real matrix has diagonal entries three fifths, upper right entry minus four fifths and lower left entry four fifths. Acting on the starting column gives first-component real and imaginary numerators minus two and three over fifteen. The second component has numerators fourteen and four over fifteen. The exact integer output in Table 1 uses the source’s fivefold refinement of the counts and normalization. [1]

The positive-spin probability is the squared magnitude of the first component when the detector is fixed along z. It changes from twenty over thirty-six to thirteen over two hundred twenty-five. The negative outcome carries the complementary probability. The density matrix remains a pure state of unit trace. If the detector projector rotates with the spin state, the trace probability stays at five ninths because the common unitary cancels from the trace. [1]

Before rotation · 5/9

After rotation · 13/225

Figure 1 describes a probability assigned to the fixed spin projector. With the orbital factor held common, the scalar spatial density remains the same before and after rotation. The detector interaction would need to be modeled to predict spatial separation of outcomes. [1] [3]

Which current remains fixed

For a normalized spin column multiplying a common orbital field, the local spin matrix density is the scalar orbital density times the spin density matrix. The scalar convective current is the trace of the convective matrix current. A spatially uniform spin rotation preserves that trace. In the Pauli balance law, the spin torque enters through a commutator, whose trace is zero. The matrix can therefore precess while scalar number continuity is preserved. [1]

This comparison assumes a common orbital factor and a spatially uniform spin rotation. In a magnetic realization, the orbital equation retains its vector potential. A spatially varying spin column introduces a phase-derivative contribution to convective current and an orientation-gradient contribution to kinetic energy. Both enter the action of the same field. Comparing spin probabilities alone leaves those spatial data unspecified. [1]

A Dirac-compatible current comparison also retains the magnetization contribution. For a spatially uniform polarization and varying orbital density, its change is proportional to the cross product of the density gradient and the polarization change. In the source’s periodic cosine example, the density gradient along x and a change in z polarization produce a transverse current along y. The curl has zero divergence and the total flux through a closed surface vanishes, while flux through a selected open face can be nonzero. Scalar continuity therefore leaves the full local current and boundary flux underdetermined. [1]

What the action test adds

An exact endpoint readout can be produced by more than one integer history. The opposite-cell differences in this example are unchanged by a suitable alternating adjustment of counts within a cycle. The totals and the projected column stay fixed. Distinguishing the histories requires the physical rule that selects admissible transfers and supplies the action along those histories. [1]

Varying only along one prescribed rotation gives a particularly weak action test. In Section 3.51, a rotation at twice the required half-angle rate leaves a nonzero field-equation residual whose real pairing with the sole rotation tangent vanishes. The resulting restricted variation misses the error. Section 3.52 supplies the stronger comparison. A complete two-direction normalized variation can determine the isolated spin ray while leaving its representative common phase free. [1]

Sections 3.62–3.64 test the kinetic term with calibrated spin rotations. If the three physical rotations are generated by the calibrated spin expectations and their count lifts are admissible, their brackets force the local first-order kinetic curvature coefficient to equal the source reduced action scale. The three expectations recover the pure spin ray, with transverse components carrying relative-phase information that the z probability alone lacks. This result fixes a local response for the normalized spin state under its premises. Energy, physical duration and the action of unresolved count variables require further input. [1]

Table 2 separates the narrower checks from the joint claim. In the later sixteen-cell example of Section 3.85, the kinetic skew form has rank ten, with rank two on the balanced section and rank eight on the twelve hidden directions. The candidate also gives positive hidden energy. These results identify physical action dependence that must be addressed before eliminating those directions. Readout agreement alone gives no justification for removing them. [1]

Scope and source date

October 23, 2026 records the manuscript version used by the archive. The exact probabilities, conditional current identities and kinetic tests are mathematical examples. Extending them to a physical matter theory requires a source-selected count history and action that also determine clock normalization and stress under metric variations. The full calculations are identified below. This Brief was prepared November 3, 2026. [1] [4]