Common-Fundamental Portions and Rich Spatial Polarization

A field pattern can carry much more polarization information than a single rotation plane. Its component directions, relative phases, handedness and variation with position all matter. The same count of portions can therefore support different spatial textures without introducing portions that resonate at separate physical frequencies.

The constructions below describe one field on infinitely many ordinary spatial dimensions plus time. The displayed finite-dimensional coordinates describe the matter domain, involving up to ten spatial directions. Complex notation records real oscillation quadratures; it does not introduce a separate physical space. The electron's centre trajectory remains distinct from its internal texture.

1. A Portion Count and a Shape Are Different Data

Use dimensionless time measured in fundamental periods, and set

$$s(t)=\{t\}-\frac12,\qquad E(t)=\sum_{j=0}^{C-1}u_j s(t+j/C).$$

The vectors $u_j$ specify the spatial contribution of each portion. Between resets the derivative is $\sum_j u_j$; resetting portion $j$ changes the vector by $-u_j$. If the vectors sum to zero, the result is a periodic staircase through a closed list of vertices. Choosing the vectors as cyclic vertex differences realizes any centred list of $C$ polygon vertices.

This is an exact construction of vertex data, rather than continuous motion along polygon edges. A continuous physical profile could smooth the staircase, but its smoothing law must be stated. Its Fourier coefficients would acquire the smoothing kernel's Fourier factor. A several-plane staircase similarly samples a toroidal figure; continuity and a knot interpretation require further construction.

The portion schedule has $C$ reset slots per period. Reading each portion's reset additively retains the proposed count rate $C/T$. A particular vector observable can suppress a reset or combine unresolved ones, so its shortest period is a separate question. Uneven phase offsets also preserve that mean event count when every portion still contributes one registered event per fundamental cycle. This statement concerns the stipulated event readout, not a claim that every vector signal oscillates at $C/T$.

2. Exact Spectra and a Controlled Overlap Approximation

For a regular once-winding staircase in an oriented plane, encode its two coordinates as

$$z_C(t)=a\exp\!\left(2\pi i\frac{\lfloor Ct\rfloor}{C}\right).$$

Integrating over each of its constant intervals gives

$$\widehat z_C(n)= \begin{cases} a e^{-\pi i n/C}\operatorname{sinc}(n/C),&n\equiv1\pmod C,\\ 0,&\text{otherwise}, \end{cases} \qquad \operatorname{sinc}(x)=\frac{\sin(\pi x)}{\pi x}.$$

These are Fourier descriptions of the common-fundamental waveform. They are not additional independently resonating objects. A real coordinate also has the conjugate frequency set. At fixed radius, the fundamental coefficient is $a\operatorname{sinc}(1/C)$, differing from $a$ first by $a\pi^2/(6C^2)$. This supplies a useful example in which increasing the count does not proportionally increase a low-frequency response coefficient.

For counts $A,B$, equal unit radii, aligned planes and the displayed phase convention, the exact real dot-product mean includes every shared alias:

$$O(A,B)=\operatorname{Re} \sum_{n\equiv1\;({\rm mod}\;\operatorname{lcm}(A,B))} e^{-\pi i n(1/A-1/B)} \operatorname{sinc}(n/A)\operatorname{sinc}(n/B).$$

Keeping only $n=1$ is an approximation. For example, direct averaging of the four intervals at $A=2,B=4$ gives $O=1/2$; the fundamental-only expression is $4/\pi^2$. Parseval's identity and Cauchy--Schwarz give the useful bound

$$|O-O_{\rm fundamental}| \leq \sqrt{\bigl[1-\operatorname{sinc}^2(1/A)\bigr] \bigl[1-\operatorname{sinc}^2(1/B)\bigr]}.$$

The bound decreases as $1/(AB)$ at large counts. A physical interaction can still sample other waveform information, so this does not derive equal electron and muon electric coupling. General plane orientations and phases must also be retained; sharing one direction does not alone fix an overlap.

3. Polarization Freedom and Spatial Circulation

A nonzero complex $N$-component amplitude has $2N-2$ real shape coordinates after removing overall magnitude and common phase. This is a continuous shape count, not the number of independent basis polarizations. For example, two complex light components give two real normalized-shape coordinates, while three complex components would give four. Comparing either number with a basis-state count without that distinction is inappropriate.

Spinor counts must specify their representation. A doubled odd-dimensional module has twice the complex dimension of the minimal odd-dimensional Clifford module. Its commuting plane-sign labels therefore occur with a different multiplicity. Those mathematical conventions cannot silently be interchanged when assigning physical degrees of freedom.

The existing radial field already supplies a spatial texture. Let $\phi=F s+iG\gamma(\widehat x)s$, with real radial profiles, a normalized $\beta$-eigenvector $s$, Hermitian Clifford generators and

$$s^\dagger\gamma_a\gamma_b s=\delta_{ab}+iJ_{ab}.$$

Direct expansion gives

$$V_a=\phi^\dagger\gamma_a\phi=-2FGJ_{ab}\widehat x_b.$$

The antisymmetry of $J$ makes this current tangent to a sphere. A pure even-depth seed with $J^2=-I$ gives circular angular curves wherever $FG\ne0$. An odd-dimensional antisymmetric matrix has a kernel, giving an axis or a larger set of angular zeros. These are current curves of the internal field, not additional clocks or electron-centre trajectories.

For the pure seed the radial quantities

$$c=\frac{F^2-G^2}{F^2+G^2},\qquad v=\frac{2FG}{F^2+G^2},\qquad c^2+v^2=1$$

define a useful texture angle $\psi=2\operatorname{atan2}(G,F)$. The magnitude $|v|$ is a radial circulation envelope. The actual speed also contains $|J\widehat x|$. For the specified odd-depth seed, angular averaging multiplies the envelope by $\pi/4$ at depth three and $9\pi/32$ at depth five. A reported radial mean must retain that label.

4. What Can Produce a Directional Hook?

For identical scalar waveforms with a symmetric local sum energy,

$$U(\delta)=\int_0^1 f\bigl(a(t)+a(t+\delta)\bigr)\,dt =U(-\delta).$$

Changing the integration variable proves the equality. A smooth conservative phase force $-U'$ is consequently odd. Rich spatial orientation, unlike profiles or a contact-ended preparation can change the assumptions of this control and must be calculated explicitly.

The sawtooth correlation itself is

$$\mathcal C(\delta)=\frac1{12}-\frac{\delta(1-\delta)}2, \qquad0\leq\delta\leq1.$$

In a supplied scalar phase equation $\dot\delta=\Delta-K\mathcal C(\delta)$, its unequal extrema give stationary roots for $\Delta/K\in[-1/24,1/12]$. The endpoints are marginal. This is a conditional asymmetric locking example, not yet the relative-phase equation of two identical reciprocal particles: equal reciprocal responses to an even correlation cancel from their phase difference. A viable reduction must derive both responses, their mechanical effects and the field's energy flow.

The earlier numerical hook example used $\rho^{5/9}$, whereas the maintained density energy uses $\rho^{10/9}$. The corrected author calculation uses the maintained exponent and gives threshold ratios approximately 1.0324, 1.0844 and 1.0686 for the three tested unequal reset-width pairs; the old values 1.226, 1.557 and 1.414 do not apply to that energy. Identical waveforms retain the evenness identity at either exponent. These are prescribed discrete-waveform controls with the supplied background offset, not a solved localized pair or an error-bounded continuum interaction threshold. Independent assessment checked the corrected source and saved outputs without replaying the run.

5. Connection to Charge and Contact

For three equally weighted orthogonal planes in the stated aligned preparations, an agreeing-plane fraction is $h/3$. The existing charge table can be rewritten as the positive agreeing fraction or the negative disagreeing fraction, with the branch parity choosing between them. This is an exact rewrite of that table. Identifying its exterior degree with a count of agreeing physical planes is an additional interpretation to test through the actual reciprocal response.

Finite contact-ended spans also retain phase-dependent residual overlaps that an infinite or full-period average can remove. Thus handedness and shape provide concrete interaction data, while a zero under a special averaging convention is not a universal absence of interaction.

The constructive result is a rich common-fundamental texture language with explicit count, spectrum, current and matching identities. Sustaining a particular texture, selecting its count and deriving its physical force remain dynamical calculations. The sector-structure case and polarization account explain the existing geometric property constructions to which it connects.