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\section{Synthesis and open problems}\label{sec:synthesis}
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\Cref{tab:atlas} collects the calibrations discussed in this paper in one place.

\input{atlas}

\subsection{Five principles, revisited}

\paragraph{P1. Convexity and tameness substitute for compactness.}
Compactness is what makes general existence theorems cost $\WKL$: the extreme value theorem, Brouwer's fixed point
theorem, the Hahn--Banach theorem. Convexity makes existence cheap again (\cref{thm:convex-rca}), for a reason
that is visible in the proof: a convex function cannot hide a minimizer behind a local maximum, so a
one-dimensional search never has to backtrack. In dimension one this is a complete answer; in higher dimensions
we need a modulus of uniform continuity, and whether convexity provides it in $\RCA$ is open
(\cref{prob:uniform-n}). Tameness makes it cheap for a different reason: fixed-format
semialgebraic statements follow from finitely many algebraic axioms (\cref{prop:tarski}). The same phenomena appear
in Bishop's constructive mathematics, where convexity replaces the fan theorem
\cite{BergerSvindland2016Infima,BergerSvindland2018Fan,BergerSvindland2019UniqueMin}, in Weihrauch complexity, where
convex choice sits strictly below closed choice \cite{LeRouxPauly2015LMCS,KiharaPauly2019TAMC}, and in combinatorial
optimization, where the greedy algorithm makes matroid bases free \cite{HirstMummert2017}.

\paragraph{P2. Selection is the price of convergence.}
\Cref{thm:A,thm:D} show that what costs $\ACA$ is not existence but commitment. The limit of a convergent method
is a realizer of convex minimization, a function of the data and the starting point; finding \emph{some} minimizer
is non-uniformly computable, but the realizers computed by gradient descent in dimension one and by anchored methods
in every dimension select the projection of the starting point or anchor onto the solution set, and this selection is
$\lm$-hard (\cref{thm:A,thm:D}; Neumann \cite{Neumann2015LMCS} proved this for projections and for a large class of
fixed-point iterations). This explains why the strength of convergence theorems does not depend on whether the
algorithm is a descent method, a fixed-point iteration or an anchored scheme: each of them makes a choice of this
kind, at least on one-dimensional instances. The same pattern
appears in existence theorems that ask for an extremal solution: Peano's theorem is $\WKL$, but Osgood's maximal
solution is $\ACA$ \cite{Simpson1984}.

\paragraph{P3. Uniqueness and regularity are computational resources.}
A unique minimizer is computable, and so is a rate (\cref{thm:B}). A modulus of uniqueness or regularity is a
uniform rate. This is Kohlenbach's observation for best approximation \cite{Kohlenbach1993Chebycheff}, and it is
the logical content of the many ``error bound implies convergence rate'' theorems of optimization: the error bound
is a modulus of regularity, and the theorem is its quantitative unwinding. (\Cref{thm:B}(b) applies the modulus once;
the familiar geometric rates come from applying a linear modulus repeatedly along the iteration.) Conversely, a uniform
rate of convergence yields a modulus of regularity \cite{KohlenbachLopezAcedoNicolae2019}, and the existence of such
moduli is equivalent to $\ACA$ \cite{Kohlenbach2019Moduli}; together these show that uniform rates for the iterates
exist exactly under a regularity assumption, which is not available in general. O-minimality explains why
regularity is so often available, though not why it should be computable from the data.

\paragraph{P4. Bounds on values are cheap, convergence of iterates is not.}
Every uniform rate in \cref{lem:basic} follows from finitely many instances of \eqref{eq:pair}: it has a proof that
uses no compactness and no induction beyond $\Pi^0_1$, and the metatheorems guarantee that such a proof yields bounds
depending only on $L$, $D$ and $k$. Convergence of iterates is a $\Pi^0_3$ statement whose proofs use compactness;
metastability is what is left after Kreisel's no-counterexample interpretation, and fluctuation bounds are what is
left when, in addition, the geometry cooperates (\cref{thm:E}). The principle is a heuristic: what makes the values
uniform is the existence of a simple proof of a bound, not the logical form of the convergence statement as such.

\paragraph{P5. Dimension is the price of compactness.}
The rates of \cref{lem:basic} hold in every Hilbert space; the fluctuation and metastability bounds of \cref{thm:E}
depend on $d$ and cannot be made dimension-free. The dividing line is the use of compactness (Bolzano--Weierstrass, or
the length of self-contracted curves) in the convergence proof, which proof mining detects as a dependence on a
modulus of total boundedness. The Halpern iteration converges strongly in Hilbert space by a proof that uses no norm
compactness (only weak sequential compactness or projections, available uniformly in every Hilbert space), and
accordingly it has dimension-free metastability rates
\cite{Kohlenbach2011AdvMath,KohlenbachLeustean2012AdvMath}.

\subsection{Open problems}

\begin{problem}\label{prob:uniform-n}
Does $\RCA$ prove that for every $n$, every convex function on $[0,1]^n$ with a modulus of uniform continuity attains
its minimum? The proof of \cref{thm:convex-rca}(b) is an external induction on $n$. Does $\RCA$ prove that every
continuous convex function on $[0,1]^2$ has a modulus of uniform continuity? Similarly, does $\RCA$ prove, with $n$ and
$d$ as variables, that polynomials of degree $d$ attain their minima on $[0,1]^n$, or that finite games of every size
have Nash equilibria (\cref{prop:tarski})? Does the method of \cref{thm:lp-rca}, applied to Lemke--Howson pivoting,
give bimatrix Nash equilibria of every size in $\RCA$?
\end{problem}

\begin{problem}\label{prob:realizers}
Is every limit-computable single-valued realizer of $\CMin_1$ Weihrauch equivalent to $\lm$? \Cref{thm:A,thm:D}
show this for the projection-type selections computed by gradient descent and anchored methods, while the
multivalued problem $\CMin_1$ itself is only $\IVT$.
\end{problem}

\begin{problem}\label{prob:kkt}
Determine the Weihrauch degree of $\KKT_n$. By \cref{prop:kkt1}, $\KKT_1\seqv\IVT$ and $\KKT_n\lev\BFT_n\seqv\WKLp$
for $n\ge2$. Does every computable $C^1$ function on $[0,1]^2$ have a computable KKT point? A positive answer would make
$\KKT_n$ a natural continuous counterpart of $\CLS=\PPAD\cap\PLS$, sitting strictly below the Brouwer problem. We
found no example of a computable $C^1$ function on the square without a computable KKT point. A counterexample must
be degenerate at its interior KKT points, since non-degenerate zeros of computable vector fields are computable
\cite{GracaZhong2021HyperbolicEquilibria};
co-c.e.\ planar continua without computable points \cite{Kihara2012PlanarContinua} are a possible starting point.
\end{problem}

\begin{problem}\label{prob:threshold}
Let $M_d(R)$ be the largest number of $\eps$-fluctuations of gradient descent on convex $1$-smooth functions on $\R^d$
with $\dist(x_0,\argmin f)\le R\eps$. \Cref{thm:E} and \cref{prop:staircase} give $M_d(2\sqrt d)\ge d$,
$M_d(R)\le2C_dR$ and $\sup_dM_d(R_0)=\infty$ for some $R_0$. Let $c_d^{\mathrm{GD}}$ be the best constant $c$ such that
every gradient-descent orbit (step $1/L$) of a convex $L$-smooth function on $\R^d$ has length at most $c\,D$.
Determine the growth of $c_d^{\mathrm{GD}}$, which lies between $\sqrt d/2$ (\cref{prop:staircase}) and
$2^{O(d^2)}$
\cite{GuptaBalakrishnanRamdas2021}, and the threshold $R^\ast=\inf\{R:\sup_dM_d(R)=\infty\}$.
\end{problem}

\begin{problem}\label{prob:hilbert-limits}
Which reals are coordinates of limits of gradient descent on computable convex $L$-smooth functions on $\ell^2$ when the
iterates converge in norm? Are all limit-computable reals possible? Over $\RCA$, is weak convergence of gradient
descent in separable Hilbert spaces equivalent to $\ACA$? Under the rate-free coding of reals of
\cite{FreundPischkeUftring2026Avoiding}, does $\RCA$ prove convergence of gradient descent in $\R^d$, as the finite-length argument behind
\cref{thm:F}(b) suggests (\cref{rem:A}(c) discusses dimension one)?
\end{problem}

\begin{problem}\label{prob:duality}
Calibrate the duality theorems of convex optimization in separable Banach spaces: Fenchel--Rockafellar duality,
Lagrange duality under Slater's condition, and the existence of subgradients. The separation theorems suggest $\WKL$
when an interior point is available and $\ACA$ otherwise. In the finite world, is linear programming duality provable
in a bounded-arithmetic theory of linear algebra?
\end{problem}

\begin{problem}\label{prob:mining-length}
The generic proof-mined bounds for Fej\'er monotone sequences in $\R^d$ are of order $(\sqrt d\,D/\eps)^d$
\cite{KohlenbachLeusteanNicolae2018}; \cref{thm:E} gives $O_d(D/\eps)$. Is there a logical metatheorem that explains
the improvement, for instance one that treats self-contractedness as a monotonicity principle with linear fluctuation
bounds?
\end{problem}

\begin{problem}\label{prob:menger}
Determine the exact strength of Menger's theorem for countable graphs (between $\ATR$ and $\mathsf{TLPP}_0$
\cite{Shafer2012,Towsner2013}), of max-flow/min-cut for countable networks
\cite{AharoniBergerGeorgakopoulosPerlsteinSpruessel2011} and of matroid intersection for countable matroids
\cite{Joo2021}.
\end{problem}

\begin{problem}\label{prob:stochastic}
Calibrate the almost-sure convergence theorems of stochastic approximation (Robbins--Siegmund, stochastic gradient
descent) in reverse mathematics and Weihrauch complexity, in the spirit of the quantitative results of Neri, Pischke and
Powell \cite{NeriPowell2026AAP,NeriPischkePowell2025abstract}.
\end{problem}