For \(\xi>1\), let \(w\) be a nonnegative normalized \(C^1\) weight on \([1,\xi]\), and define the CHJ-II \(k=1\) source norm

$$ J_\xi(w) =\frac{w(\xi)}{\xi}+w(1) +\int_1^\xi\frac{w(u)}u\,du +\int_1^\xi\frac{|w'(u)|}u\,du. $$

There is one exact transition point

$$ \xi_c=2.110355059880694877\ldots $$

determined by

$$ 3+\xi_c^{-1}+\log\xi_c-2\xi_c=0. $$

For \(1<\xi\le\xi_c\), the unique minimizer is constant:

$$ w(u)=\frac1{\xi-1}. $$

For \(\xi>\xi_c\), no \(C^1\) minimizer exists. The infimum is approached by weights that are zero on an initial segment and constant on a right tail.

The optimizer does not gradually become lopsided. It stays completely flat, reaches one exact scalar equation, and then jumps to a boundary layer.

The fixed interval gives the coefficient 1.3962008#

At \(\xi=2\), the unique optimum is \(w=1\), and

$$ \inf 4N_{1,2} =\frac{3+2\log2}{\pi} =1.396200858856675\ldots $$

This replaces the published coefficient \(1.785\) within the identical nonnegative \(k=1\), \([T,2T]\) construction.

The dual certificate is a function

$$ \phi(u)=1+\log u-\left(\frac32+\log2\right)(u-1) $$

satisfying

$$ |\phi(u)|\le\frac1u. $$

Integration by parts converts the weighted variation into the exact lower bound. Equality forces \(w'=0\), so the constant is not merely an optimizer; it is the unique optimizer.

There is one important reality condition. The averaged theorem remains valid for arbitrary complex coefficients. Extracting one common point \(T^*\in[T,2T]\) from the average requires a real-valued remainder. A complex unit-circle function can average to zero while keeping modulus one everywhere, so the unrestricted complex pointwise inference is false.

Beyond the transition, the optimizer is a tail#

For \(\xi>\xi_c\), let \(a\in(1,\xi)\) solve

$$ 1+\frac2a+\log\frac{\xi}{a} -\frac{(a+1)\xi}{a^2}+\frac1\xi=0. $$

Then

$$ \inf J_\xi=\frac{a+1}{a^2}. $$

The bounded-variation optimizer is

$$ w(u)=\frac{\mathbf1_{[a,\xi]}(u)}{\xi-a}. $$

Smooth \(C^1\) ramps approximate it arbitrarily well, but continuity prevents exact attainment: a smooth weight that is zero before \(a\) and constant after \(a\) would have to be zero everywhere.

The transition is therefore also a transition from an attained interior optimum to a nonattained boundary infimum.

Every linear tradeoff reduces to one interval#

The same norm comes with a first moment

$$ \mu_\xi(w)=\int_1^\xi u\,w(u)\,du. $$

Weighted coarea decomposes the superlevel sets of \(w\) into interval components. After charging both endpoints and using layer cake, the pair \((J_\xi(w),\mu_\xi(w))\) becomes a barycenter of interval-step pairs

$$ \left( \frac{a^{-1}+b^{-1}+\log(b/a)}{b-a}, \frac{a+b}{2} \right). $$

Consequently, for every real \(\beta\),

$$ \inf_w(J_\xi(w)+\beta\mu_\xi(w)) $$

is exactly the minimum over \(1\le a

The reduction stops at linear tradeoffs. The complete explicit-formula remainder contains nonlinear cutoff terms, and their Hessian has negative determinant. Jensen cannot collapse that saddle-shaped objective to a single interval step. The norm and every support line are solved; the complete simultaneous optimization is not.

Notebook references: C-0101, C-0127, C-0128