IDOS Energy, Coherence, and Supporting Dynamics — Part 19

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only the next shell in the same active face. The anchor occupation term already supplies stiffness for all other inactive directions. The occupied phase Hessian has precisely the affine kernel $\delta\phi_k=a+k\cdot b$, of dimension $d+1$ for $N\ge1$, and dimension one at $N=0$. Its rank is $D-d-1$. At $N=1$ every phase assignment is already affine, so the phase constraints correctly add no stiffness.

Adding $-\gamma\eta nV_{\rm phase,edge}$ with common positive $\eta$ preserves the exact branch, complete energy and causal principal cone, and adds nonnegative algebraic quadratic energy. It selects internal coherence without imposing attraction or synchronization between distinct particles. Each pattern retains its own overall phase, texture shift, support center, orientation and frequency. Relative texture/support registration remains unselected. Equal coefficients therefore do not guarantee aligned interparticle overlap. Spatially uniform participating energy for a constant-action-density clock is also still unresolved.

An all-pair gated product law has the same selected occupied phases but different off-branch stiffness and many more terms. The edge law uses at most $10D_M$ displayed terms in ambient ten. Both were expanded directly through second order and tested with exact arithmetic: 2,252 assertions, including 174 full amplitude-Hessian directions per law. The general proof and source-pinned checks are in simplex-phase-lock/report.md.

Two-Pattern Domains And An Independent Flow Completion

The centered occupied bands impose a nontrivial domain condition on raw two-pattern fields. For stationary positive envelopes and aligned nested coordinate faces, write

$$\Psi=\sqrt2(f_a e_a+f_b e_b),\quad C=\frac{D_\cap}{\sqrt{D_aD_b}},\quad \Delta=(\omega_b-\omega_a)t-(q_b-q_a)\cdot x/\ell+\phi,$$ $$v=\frac{2q_\cap-q_a-q_b}{\ell},\qquad g=\nabla\log(f_b/f_a),\qquad \Omega=\omega_a+\omega_b.$$

The intersection mean must be computed from the actual common labels; the smaller count alone does not establish band containment. Direct real pair multiplication gives

$$n=f_a^2+f_b^2+2Cf_af_b\cos\Delta,$$ $$T=na_0=f_a^2\omega_a+f_b^2\omega_b+Cf_af_b\Omega\cos\Delta, \qquad S=na_{\rm spatial}=Cf_af_b(v\cos\Delta+g\sin\Delta).$$

For positive density the old future-timelike phase domain requires $T>|S|$. With $g=0$ and $0<C<1$, this holds for every positive amplitude ratio and phase exactly when

$$2\sqrt{\omega_a\omega_b}>C(\Omega+|v|).$$

For same-depth prefixes with $D_b>D_a$ and $\omega_i=\mu_dD_i$, the maximum spatial-to-temporal ratio simplifies to $|q_b-q_a|/[\ell(\omega_b-\omega_a)]$. The electron/muon critical value is $(\mu_6\ell)^2=1/246197906136$. Below, at and above this critical value the worst preparation is respectively failing, null and passing. No physical length has been assigned by this calculation.

For unequal envelopes the exact universal criterion replaces $\Omega+|v|$ by

$$K=\max_{|z|=1}\sqrt{(\Omega-z\cdot v)^2+(z\cdot g)^2},\qquad \Omega+|v|\le K\le\sqrt{(\Omega+|v|)^2+|g|^2}.$$

The upper bound is exact for collinear $v,g$. For two identical bands and clocks, equal peaks and equal rational support radii, evaluated at the midpoint of distinct centers separated by $R$,

$$\frac{|S|}{T}=\frac{4R}{\omega(L^2+R^2/4)} \left|\tan\frac\Delta2\right|.$$

It exceeds one near destructive phase for every $R>0$ at nonzero density. Thus admissible isolated fields do not imply an admissible raw sum. This is a trial-field domain result, not a force or an obstruction to every possible interaction. The full action Noether current need not equal the phase-gradient direction off branch. The report pair-current-domain contains the proof, 16,742 checks, eleven actual same-family comparisons and eighty symbolic cross-family conditions with distinct sector scales.

Retaining The Flow As An Independent Field Component

A changed first-derivative action replaces the derived $u=-a^\sharp/\sqrt{-a^2}$ by a retained future unit timelike $U$. Set $P=g+UU$, $\chi=U\cdot a$, and replace the clock term by

$$\mathcal L_{\rm clock}=n(\chi-m)-\frac{\kappa_a n}{2}|Pa|^2.$$

Replace $u$ by $U$ throughout the support, horizontal carrier and material frame terms, keeping all their residuals and the quadratic density lock. Add the unit constraint multiplier $\zeta(U^2+1)/2$ and

$$\mathcal L_U=\frac{\kappa_U n}{2} \left[(U\cdot\nabla U)^2-c_U^2P^{\mu\nu} \nabla_\mu U\cdot\nabla_\nu U\right], \quad\kappa_U>0,\quad0<c_U\le1.$$

These are components of the proposed single enlarged classical field, not mediators or extra spatial dimensions. Constraints on $U,E_j,Z$ are now algebraic in independently retained fields. This avoids differentiating a carrier-derivative-defined velocity when reducing those constraints.

The exact isolated solutions have $a=-\omega U_\flat$, $Pa=0$, constant $U,E_j$ and all previous residuals zero. Their full equations

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