0.1 — Pre-Coherence Continuum

Continuous scalar fabrics before a shared recurrence or lattice exists.

Independent Oscillatory Fabrics

Before coherence, the EOTU consists of independent scalar fabrics with differing amplitudes, recurrence rates, and continuous phases. No universal phase reference, King Frequency, spatial lattice, or CPP identity exists at this stage. A pre-coherent fabric is represented by

\[ \Psi_i(t)=A_i\sin(\omega_i t+\phi_i), \qquad 0\leq\phi_i<2\pi. \]

Each oscillator has both an instantaneous response and the normalized response of its recurrence rate:

\[ Q_i = \frac{\dot{\Psi}_i}{\omega_i} = A_i\cos(\omega_i t+\phi_i). \]

The pair \((\Psi_i,Q_i)\) comprises two conjugate quadratures of the same scalar oscillation. The quantity \(Q_i\) is not a second field, a second fabric, or an independently assigned phase state. The two responses satisfy

\[ \Psi_i^2+Q_i^2=A_i^2. \]

Relative Alignment and Mismatch

For two fabrics, the interaction depends on their relative phase \(\delta_{ij}=\theta_j-\theta_i\), not on an absolute phase reference. The normalized alignment and orthogonal-mismatch responses are

\[ D_{ij} = \frac{\Psi_i\Psi_j+Q_iQ_j}{A_iA_j} = \cos\delta_{ij}, \qquad M_{ij} = \frac{\Psi_iQ_j-Q_i\Psi_j}{A_iA_j} = \sin\delta_{ij}. \]

These conjugate relative responses obey \(D_{ij}^2+M_{ij}^2=1\). The Fabric interaction is quadrature-degenerate: the same response function applies to both squared channels,

\[ V_{ij}=G\!\left(D_{ij}^2\right)+G\!\left(M_{ij}^2\right). \]

The complete lowest-order contribution is therefore independent of relative phase. This cancels the second-harmonic phase preference rather than selecting two preferred orientations.

Fourth-Order Phase Selection

For the saturating nonlinear Fabric response, the first surviving phase-dependent contribution is, apart from a phase-independent constant,

\[ V_{\mathrm{phase}} = \frac{\kappa}{4} \left[1-\cos\left(4\delta_{ij}\right)\right], \qquad \kappa>0. \]

Its stable minima occur at \(\delta_{ij}=n\pi/2\), giving four relative-phase families within one cycle. The four-family structure is therefore not inserted as an initial condition or obtained by assuming a fourth harmonic. It follows, within the framework, from scalar recurrence, equal coupling through the conjugate quadratures, and a saturating nonlinear response.

As these stable relationships spread through the fabric population, recurrence rates converge toward a common rhythm. That shared recurrence defines the King Frequency and supplies the reference against which the four primordial CPP phases are later fixed.

Summary

  • Pre-coherent fabrics have continuous phase and no universal King reference.
  • \(\Psi_i\) and \(Q_i\) are conjugate responses of one scalar oscillator.
  • Equal nonlinear coupling through alignment and mismatch cancels the lower phase preference.
  • The first surviving phase-selective term is fourth order and has quarter-cycle minima.
  • The King Frequency emerges as compatible recurrence rates converge.