Let
$$ \cdots<x_{k-1}<x_k<x_{k+1}<\cdots $$
be a real zero configuration, and write \(d_i=x_{i+1}-x_i\). The collision pressure on the central gap is
$$ P_k= 2-d_k^2\sum_{j\ne k,k+1} \frac1{(x_{k+1}-x_j)(x_k-x_j)}. $$
The annoying feature is that this is a full-range quantity. A local gap estimate seems to know too little about all the other zeros.
The finite profile#
There is, however, one finite statistic that controls it exactly enough. For \(1\le r\le R\), suppose
$$ \frac1{d_k}\sum_{j=1}^r \left(d_{k-j}+d_{k+j}-2d_k\right)\le E_r, \qquad r+\frac{E_r}{2}>0. $$
Then
$$ \boxed{ P_k\le 2-2\sum_{r=1}^R \frac1{(r+E_r/2)(r+E_r/2+1)}. } $$
The mechanism is one line once the right coordinates are chosen. Normalize the cumulative distances on the two sides:
$$ A_r=\frac{x_k-x_{k-r}}{d_k}, \qquad C_r=\frac{x_{k+1+r}-x_{k+1}}{d_k}. $$
With \(f(u)=1/(u(u+1))\), the pressure is
$$ P_k=2-\sum_{r\ge1}\bigl(f(A_r)+f(C_r)\bigr). $$
The function \(f\) is decreasing and convex. Jensen therefore gives
$$ f(A_r)+f(C_r) \ge 2f\left(\frac{A_r+C_r}{2}\right). $$
Now the useful cancellation appears:
$$ A_r+C_r = 2r+\frac1{d_k}\sum_{j=1}^r \left(d_{k-j}+d_{k+j}-2d_k\right). $$
Any affine left-right drift vanishes. Estimating the two sides separately would charge that drift and lose the theorem.
The curvature corollary#
There is a clean scalar corollary. If
$$ d_{k-j}+d_{k+j}-2d_k\le\mu d_kj^2 \qquad(1\le j\le R), $$
then
$$ \boxed{ P_k\le \frac2{R+1} +\frac{\mu R(4R+5)}{6(R+1)}. } $$
Optimizing \(R\) gives
$$ P_k\le \left(\frac4{\sqrt3}+o(1)\right)\sqrt{\mu}. $$
Four roots are enough#
The cumulative defect itself telescopes:
$$ \boxed{ \sum_{j=1}^r(d_{k-j}+d_{k+j}-2d_k) =x_{k+r+1}-x_{k-r}-(2r+1)(x_{k+1}-x_k). } $$
So each radius needs only one correlated four-root span bound. Relative to any fixed reference train \(c_j\), write \(x_j=c_j+e_j\). A rational target
$$ \frac{x_{k+r+1}-x_{k-r}}{x_{k+1}-x_k}\le L_r $$
is exactly the division-free inequality
$$ e_{k+r+1}-e_{k-r}-L_r(e_{k+1}-e_k) \le L_r(c_{k+1}-c_k)-(c_{k+r+1}-c_{k-r}). $$
Its left side annihilates affine offset trains when \(L_r=2r+1\). Independent root boxes would throw that cancellation away.
There is also an exact inverse-phase form. If \(x_j=X(j)\) and \(w=X'\), the cumulative defect is a nonnegative Peano-kernel average of \(w''\), with total kernel mass
$$ \frac{r(r+1)(2r+1)}6. $$
The fixed-reference four-root inequality is the non-circular interface: one need not assume a global inverse branch for the actual Riemann phase.
For the de Bruijn-Newman zero flow,
$$ \frac{d}{dt}d_k(t)^2=4P_k(t), $$
so a certified finite symmetric gap profile becomes an integrated collision exclusion.
The full pressure is one paired endpoint current#
There is also a completely local representation. For adjacent simple real zeros \(a(t)<b(t)\), with \(d=b-a\),
$$ \boxed{ P=\frac d2\left( \frac{H''(b)}{H'(b)} -\frac{H''(a)}{H'(a)} \right). } $$
If \(H=E+\overline E\) on the real axis and
$$ J=-\operatorname{Im}(E'\overline E), $$
then at a zero of \(H\),
$$ \frac{H''}{H'}=\frac{J'}J. $$
Thus, when \(J(a)J(b)>0\), the pressure budget \(P\le B\) is exactly the division-free inequality
$$ \boxed{ d\bigl(J'(b)J(a)-J'(a)J(b)\bigr) \le 2B\,J(a)J(b). } $$
This keeps the signed correlation between the two endpoints. Bounding the two logarithmic derivatives independently is much weaker. At the retained first-cell scout, the paired pressure is about \(0.159629\), while the one-sided curvature-supremum counterfeit costs about \(2.22631\).
The identity does not prove the Riemann-specific bound. It says exactly what an interval certificate would need: adjacent root brackets, the two current signs, and one paired determinant inequality. No outer zero shell appears in that interface.
Nonvanishing labels the zero train, but labels may repeat#
Suppose a selected branch \(E\) is nonzero on a simply connected spacetime strip and \(H=2\Re E\). A continuous phase lift
$$ E=|E|e^{i\theta} $$
labels every real zero by
$$ \ell=\frac{\theta}{\pi}-\frac12\in\mathbb Z. $$
The label is constant on each connected component of the zero set. At a first collision, every forward real descendant therefore carries the same label. Consequently
$$ \boxed{ \text{selected-branch nonvanishing} \;+\; \text{terminal label injectivity} \Longrightarrow \text{no first collision}. } $$
The second hypothesis is real. The backward-heat family
$$ E_t(z)=z^2-2t+i, \qquad H_t(z)=2(z^2-2t) $$
has \(E_t(x)\ne0\) everywhere on the real spacetime plane, but its two roots \(\pm\sqrt{2t}\) collide at \(t=0\). Both have \(E_t=i\), so both carry the same label. Nonvanishing alone is not a collision theorem.
This replaces a global one-sign phase-velocity demand by a zero-order nonvanishing certificate plus a terminal one-dimensional injectivity certificate. It is an alternate route, not a consequence of the pressure bound.
Positive-time localization closes the infinite tail conceptually#
The symmetric profile becomes especially clean at high phase. Polymath15 gives phase-labelled zeros
$$ x_n(t)=X_t(n)+O(X_t(n)^{-ct}) $$
uniformly on every compact positive-time interval \(\tau\le t\le T\). The inverse phase has
$$ X_t'''(n)=O\left(\frac1{x^2\log(x)^4}\right)>0, $$
so the reference gaps have tiny symmetric second difference. The six localization errors enter with total absolute coefficient mass eight. This gives
$$ d_{n-j}+d_{n+j}-2d_n \le d_nj^2\overline\mu(x), $$
where
$$ \overline\mu(x) = O\left(x^{-c\tau}\log x+\frac1{x^2\log(x)^3}\right). $$
The radius must grow:
$$ R=\left\lfloor\overline\mu(x)^{-1/2}\right\rfloor. $$
A fixed radius always leaves the \(2/(R+1)\) pressure term, which cannot beat a terminal gap-square of order \(\log(x)^{-2}\). With the adaptive radius,
$$ \boxed{ \max(P_n(t),0) =o(\log(X_t(n))^{-2}), } $$
while the terminal gap is asymptotic to \(4\pi/\log X_t(n)\). The complete positive pressure is therefore negligible relative to the collision budget for sufficiently high labels.
This is a conceptual tail closure, not yet an effective finite seam. The localization theorem does not expose numerical constants or the absolute height where the asymptotic regime starts. The remaining job is to connect the fixed finite separator to that regime without a gap.
Boundary#
The boundary matters. The finite symmetric-profile theorem does not certify its hypotheses for the low Riemann heat strip. The endpoint-current identity still needs a uniform Riemann-specific determinant bound. The phase-label route still needs selected-branch nonvanishing and terminal injectivity. The asymptotic pressure theorem still needs an effective join to the finite separator.
What changed is the shape of the analytic target. The complete pressure can be reached by a finite affine-cancelling profile, by one signed endpoint determinant, or asymptotically by an adaptive phase neighborhood. Bounding every remote gap separately is not the only game.
Notebook references: R-0438, R-0443, R-0446, R-0447, R-0449