Phase flux has to become zeros or critical jets

Let

$$ F=S+B $$

be holomorphic on a bounded domain, and look at the crossing set

$$ \Gamma=\{|B|=|S|\}. $$

Write

$$ r=-\frac BS, \qquad \Delta=SB'-S'B. $$

Then \(\Gamma\) is the preimage of the unit circle under \(r\), while zeros of \(F\) are exactly the points where \(r=1\). The logarithmic phase speed along a regular piece of the graph is

$$ \boxed{ \left|\frac{d}{ds}\arg r\right| = \left|\frac{\Delta}{SB}\right|. } $$

So far this is the usual lovely Stokes-curve picture: enough phase rotation forces crossings of \(1\), hence zeros.

The nuisance is fragmentation. A long phase path can split into many short arcs, and each open arc can stop just before the next multiple of \(2\pi\). Merely counting connected components makes the argument feel annoyingly ad hoc.

The exact repair is

$$ \boxed{ N_D(F)+J_D(S,B)\ge Q_D(S,B), } $$

where

$$ Q_D(S,B) = \frac1{2\pi} \int_{\Gamma\setminus\mathcal V} \left|\frac{\Delta}{SB}\right|\,ds $$

is the total projective phase flux, and

$$ J_D(S,B) = \frac12\#(\Gamma\cap\partial D) + \sum_{v\in\mathcal V} \left(\operatorname{ord}_v\Delta+1\right) $$

is the exact fragmentation bill. Here

$$ \mathcal V=\{v\in\Gamma:\Delta(v)=0\} $$

is the projective critical set.

Phase flux has to go somewhere. It becomes zeros, boundary exits, or critical jets.

The handshake lemma does all the bookkeeping#

Suppose \(v\in\Gamma\) and

$$ m_v=\operatorname{ord}_v\Delta. $$

Locally, the map \(r-r(v)\) has degree \(m_v+1\). The level set \(|r|=1\) therefore has exactly

$$ 2(m_v+1) $$

incident half-edges at \(v\).

Now apply the handshake identity to the whole finite crossing graph. If

$$ b=\#(\Gamma\cap\partial D), $$

then the number of nonclosed regular edges is exactly

$$ \boxed{ \frac b2+\sum_v(m_v+1). } $$

That number is \(J_D(S,B)\).

On each open regular edge, the number of visits to \(r=1\) differs from its phase variation divided by \(2\pi\) by at most one. Each closed regular component maps to the unit circle as a finite covering, so it loses nothing: degree \(n\) means exactly \(n\) zeros.

Summing the edge estimates gives \(N+J\ge Q\). The critical vertices are not an error term pasted onto the theorem afterward. Their local map degrees are precisely the combinatorial currency needed to pay for every broken phase arc.

I like this more than I should. The analytic and graph-theoretic counts meet with no slack except the unavoidable one-crossing loss on each open edge.

The same determinant sees double zeros#

At a zero of \(F\), we have \(B=-S\). Therefore

$$ \Delta = SB'-S'B = S(B'+S') = SF'. $$

Away from a common zero of \(S\) and \(B\),

$$ \boxed{ F=F'=0 \iff r=1,\quad\Delta=0. } $$

So the determinant that measures phase speed on the complex Stokes graph also detects a real pair-birth event. A critical jet is simultaneously a place where the graph may branch and a place where a zero crossing may degenerate.

There is also a useful local gauge repair. If \(S=\Xi M\) and \(\rho\) is a multiplicity-\(m\) zero of \(\Xi\), then

$$ \bigl(B(\rho),B'(\rho),\ldots,B^{(m-1)}(\rho)\bigr) $$

represents the class of \(F\) in the local quotient \(\mathcal O_\rho/(\Xi)\). Moving an exactly \(\Xi\)-divisible germ between the two columns does not change this marked residue.

That invariance is deliberately local. It does not make the global Stokes graph independent of the chosen shadow-defect split.

Vertical charge becomes horizontal zeros#

The common-zero caveat has a global repair. Remove from \(S\) and \(B\) their minimum common entire divisor \(\mathsf d\). The reduced pair has no common zero and defines a holomorphic projective map

$$ \mathcal P=[S:-B]:\mathbb C\longrightarrow\mathbf P^1. $$

The literal zeros split exactly into inherited common zeros and projective target hits:

$$ \operatorname{div}F = \operatorname{div}\mathsf d+\mathcal P^*[1:1]. $$

Even the Wronskian separates cleanly. The unreduced determinant is \(\mathsf d^2\Delta\), so those inherited zeros do not masquerade as projective ramification.

Now put

$$ a_L=\frac1L\log\left|-\frac BS\right|. $$

Suppose on disjoint rectangles \(I_j+i[y_0,y_1]\) that

$$ a_L(x,y_0)<0<a_L(x,y_1), \qquad \partial_ya_L\ge\tau_0>0. $$

Each rectangle then has one equator graph \(y=g_j(x)\). Along it, Cauchy--Riemann gives

$$ \left| \frac d{dx}\arg\left(-\frac BS\right)(x+ig_j(x)) \right| = L\frac{a_{L,x}^2+a_{L,y}^2}{a_{L,y}} \ge L\tau_0. $$

The target-hit count is therefore

$$ \boxed{ N(F_L) \ge \frac{L\tau_0}{2\pi} \sum_j|I_j|-m, } $$

where \(m\) is the number of intervals. Every counted zero is simple.

This is a particularly nice directional conversion: sign-definite vertical charge forces horizontal phase speed, and horizontal phase speed forces a zero train. It still assumes the aggregate charge. It does not derive it from arithmetic source data.

A single effective face cannot quietly escape#

The inequality becomes rude when the two terms have distinct exponential actions. Suppose on the crossing graph

$$ S_L=e^{-L\Phi_{0,L}}a_{0,L}, $$

$$ B_L=e^{-L\Phi_{1,L}}a_{1,L}(1+\varepsilon_L). $$

Logarithmic differentiation gives

$$ \frac{\Delta_L}{S_LB_L} = -L(\Phi_{1,L}'-\Phi_{0,L}') + \frac{a_{1,L}'}{a_{1,L}} - \frac{a_{0,L}'}{a_{0,L}} + \frac{\varepsilon_L'}{1+\varepsilon_L}. $$

If

$$ \delta_L = \inf_{\Gamma_L} |\Phi_{1,L}'-\Phi_{0,L}'| $$

and the amplitude and relative-error derivative is \(o(L\delta_L)\), then a graph of regular length \(\ell_L\) carries

$$ Q_L \ge \frac{1-o(1)}{2\pi} L\delta_L\ell_L. $$

Hence

$$ \boxed{ N_{D_L}(F_L) \ge \frac{1-o(1)}{2\pi} L\delta_L\ell_L-J_L. } $$

A lone Dini wave, a lone leading alias shell, or any tame aggregate behaving as one effective action face cannot meet the shadow on a long positive-height graph and remain zero-free unless it manufactures critical or boundary complexity at the same scale as the phase flux.

There is a parallel endpoint-depth version. If an aggregate boundary defect is represented by a finite shifted measure concentrated in an \(o(L)\) endpoint window, without a cancellation condition number of order \(L/r_L\) or weighted mass at linear depth, its effective logarithmic depth is \(o(L)\). The carrier contributes the missing \(L\), so the phase speed is \((1-o(1))L\), and a long low-fragmentation Stokes front again produces a train of nonreal zeros.

Where this stops#

This is a proof-system theorem, not an RH theorem. It assumes a finite real-analytic crossing graph, transverse boundary intersections, and a fixed shadow-defect split with common factors removed.

It does not prove that a literal Xi approximant has a long crossing graph. It does not bound the graph's critical load from arithmetic source data. It does not make the split globally canonical.

The surviving mechanisms are now quite specific: several comparable channels can phase-lock coherently; signed cancellation can make their effective weights enormous; critical vertices can proliferate at phase-flux scale; or the finite single-face graph model can fail altogether.

But a smooth single face with a large action mismatch no longer gets to say that its Stokes graph merely became complicated. Complication is measured, and the measure is exactly

$$ \frac12\#(\Gamma\cap\partial D) + \sum_v(\operatorname{ord}_v\Delta+1). $$

That is the invoice.

Notebook references: K-0176, C-0238, K-0177, C-0239, O-0324, O-0325, K-0178, C-0240