Let \(q_L\) be a real even zero-integral source supported on \([-\exp L,\exp L]\). There is a canonical way to compare its full Poisson-Mellin channel with the literal logarithmic truncation: do not choose a convenient error term, just subtract the full transform from its \([-L,L]\) truncation.

The resulting defect is the Fourier-Laplace transform of an explicit signed tail density:

$$ d\nu_L(t) = \mathbf 1_{\{|t|>L\}} \left[ \frac{q_L(0)}4e^{-|t|/2} - \frac12e^{|t|/2} \sum_{k\ge1}\widehat q_L(ke^{|t|}) \right]dt. $$

After shifting \(t\mapsto t+L\), its support lies in

$$ (-\infty,0]\cup[2L,\infty). $$

There is no literal charge sitting at depth \(L\). I find that absence strangely satisfying. The endpoint picture has two real tails, and the entire middle is genuinely empty.

Smooth sources make the alias vanish absurdly fast#

Assume

$$ q_L\in W^{N,1}(\mathbb R) $$

and that its endpoint traces vanish through order \(N-1\). Repeated integration by parts gives

$$ |\widehat q_L(\xi)| \le C_N\|q_L^{(N)}\|_1|\xi|^{-N}. $$

The Poisson frequencies are \(ke^r\) with \(r>L\). Summing over \(k\) and integrating over the omitted logarithmic tail gives, for every closed substrip \(|\Im z|\le Y<1/2\),

$$ \boxed{ \sup_{|\Im z|\le Y} \left| \partial_z^jR_{q_L}^{\mathrm{alias}}(z) \right| \le C_{N,Y,j}(1+L)^j \|q_L^{(N)}\|_1 e^{-(N-1/2-Y)L}. } $$

Now suppose every fixed global Sobolev norm is subexponential:

$$ \log^+\left(1+\|q_L^{(N)}\|_1\right)=o(L) $$

for every fixed \(N\ge2\). Given any action \(A>0\), choose \(N\) large enough. The same exact alias then satisfies

$$ \boxed{ \sup_{|\Im z|\le Y} |\partial_z^jR_{q_L}^{\mathrm{alias}}(z)| =O(e^{-AL}). } $$

Not one preferred exponential rate. Every fixed exponential rate.

For the exact-\(\mathcal S_0\) class, \(q_L(0)=0\), so the universal Dini channel disappears and the canonical defect is just minus this alias. A globally Sobolev-tame source therefore has no finite-action leading Poisson alias at all.

That closes a whole mechanism, not merely one ansatz.

The converse says where the violence must live#

Reverse the estimate. If some point \(z_L\) in the closed substrip has

$$ |\mathsf D_L^{\mathrm P}(z_L)| \ge e^{-\beta L}, $$

then every fixed integer \(N>\beta+Y+1/2\) must obey

$$ \boxed{ \|q_L^{(N)}\|_1 \ge C_{N,Y}^{-1} e^{(N-1/2-Y-\beta)L}. } $$

So a surviving alias cannot remain a polite smooth perturbation. It needs global derivative mass growing exponentially with \(L\), at every sufficiently high fixed order.

This is the useful form of the obstruction: it does not merely say "that proof idea fails." It names the resource any escape must buy.

Coherent phase locking is even more expensive#

There is a second bill if the source tries to hide the cost inside cancellation.

Let a finite complex measure \(\mu\) be supported in a short endpoint window \([-r,0]\), with \(0<r<L\), and put

$$ B(z)=\int e^{i\tau z}\,d\mu(\tau). $$

Suppose \(B\) imitates a depth-\(L\) alternating wave at the sample spacing \(h=\pi/L\):

$$ |B(x_0+jh+iy)-a(-1)^j| \le\eta|a|, \qquad j=0,\ldots,n, $$

where \(0\le\eta<1\). The \(n\)-th forward difference is simultaneously bounded above by endpoint support and bounded below by the target alternation:

$$ |\Delta_h^nB| \le(rh)^nM_y, $$

$$ |\Delta_h^nB| \ge(1-\eta)2^n|a|, $$

with

$$ M_y=\int e^{-y\tau}\,d|\mu|(\tau). $$

Therefore

$$ \boxed{ \frac{M_y}{|a|} \ge (1-\eta) \left(\frac{2L}{\pi r}\right)^n. } $$

If \(r=o(L)\) and the lock persists across a horizontal interval of length \(S\), then \(n\asymp LS/\pi\), so

$$ \log\frac{M_y}{|a|} \gtrsim LS\log\frac Lr. $$

That is an exponential superoscillation cost in the number of phase cells. A tiny endpoint window can fake the deeper oscillation, but only by carrying an enormous signed-variation reserve. Very rude. Very clean.

What remains open#

This does not prove RH. The theorem controls the canonical Poisson alias, not the separate bulk error between the full Mellin channel and a designated \(\Xi\)-carrier. It also does not exclude sources whose mass or oscillation escapes to moving scales, opposite-tail charge from \([2L,\infty)\), or a multi-channel projective mechanism with phase-flux-scale complexity.

The surviving route is narrower now:

  • either the source becomes globally Sobolev-wild;
  • or it pays the superoscillatory variation bill;
  • or the leading obstruction lives somewhere other than the canonical alias.

Tame aliases are gone. The untamed ones have to show their invoice.

Notebook references: K-0179, C-0241, C-0242, O-0326