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\title{Arithmetic progressions with prescribed $\ell^p$ gaps}

\author{Generated by Vjekoslav Kova\v{c} using ChatGPT-5.6 Sol and ChatGPT-6 Astra}

\address{Vjekoslav Kova\v{c}, University of Zagreb Faculty of Science, Department of Mathematics, Bijeni\v{c}ka cesta 30, 10000 Zagreb, Croatia}
\email{vjekovac@math.hr}

\date{September 9, 2026}

\begin{document}

\begin{abstract}
For every $n\ge3$ and $p\in[1,\infty)\setminus\{1,2,\ldots,n-1\}$, we prove that every measurable subset of $\mathbb{R}^d$ of positive upper Banach density, with $d\ge2^{n-2}+1$, contains an $n$-term arithmetic progression with every sufficiently large prescribed $\ell^p$ gap.
\end{abstract}

\maketitle

\section{The result and its ingredients}
For a measurable set $A\subset\mathbb R^d$, define its upper Banach
density by
\[
 \overline\delta(A)=\limsup_{R\to\infty}\sup_{z\in\mathbb R^d}
 R^{-d}|A\cap(z+[0,R]^d)|.
\]
\begin{theorem}\label{thm:power-main}
Let $n\ge3$ be an integer, let
$p\in(1,\infty)\setminus\{2,\ldots,n-1\}$, and let
$d\ge2^{n-2}+1$. Every measurable $A\subset\mathbb R^d$ with
$\overline\delta(A)>0$ admits $\lambda_0>0$ such that, for every
$\lambda\ge\lambda_0$, there exist $x,y\in\mathbb R^d$ with
\[
 x,x+y,\ldots,x+(n-1)y\in A,
 \qquad \|y\|_{\ell^p}=\lambda.
\]
\end{theorem}
For $n\ge4$ this answers Problem~4 by Durcik and Kova\v{c} \cite{DurcikKovac2020}, implicitly asked earlier by Cook, Magyar, and Pramanik \cite{CookMagyarPramanik2017}.
The sufficient dimensions for $n=3,4,5,6$ are, respectively,
$3,5,9,17$; the same proof gives $2^{n-2}+1$ for every larger $n$.
These bounds are independent of $p$, although constants may degenerate
near the excluded exponents. We do not establish optimality or settle
the planar case (i.e., the case $d=2$, which is the smallest dimension in which the problem makes sense).

The distance theorems of Furstenberg, Katznelson, and Weiss
\cite{FurstenbergKatznelsonWeiss1990} and of Falconer and Marstrand
\cite{FalconerMarstrand1986} initiated the study of prescribed large
sizes of configurations in positive-density subsets of Euclidean space.
Bourgain \cite{Bourgain1986} treated nondegenerate simplices and also
exhibited an obstruction for three-term progressions with prescribed
Euclidean gaps. Cook, Magyar, and Pramanik \cite{CookMagyarPramanik2017}
proved the conclusion for every sufficiently large gap for three-term
progressions in the $\ell^p$ metric, for every
$p\in(1,\infty)\setminus\{2\}$ and sufficiently large dimension
depending on $p$.

For longer progressions, Durcik and Kova\v{c} \cite{DurcikKovac2020}
proved that positive-measure subsets of the unit cube determine an
interval of $\ell^p$ gap lengths whose size has a positive lower bound
depending only on the density and the fixed parameters. Their
Problem~4 asks for every sufficiently large gap in positive-density
subsets of the whole space. Theorem~\ref{thm:power-main} supplies this
conclusion with a dimension bound independent of $p$. Related results
for corners and extended boxes appear in
\cite{DurcikKovacRimanic2018,DurcikKovac2021}; the question for longer
progressions was already discussed in
\cite[Section~4]{DurcikKovacRimanic2018}.

Put $k=n-2$ and $Q=2^k$. Bilinear stopping reduces uniform shell
comparison to Fourier convergence of $Q$-fold shell intersections.
In dimension $Q+1$, these intersections are generically curves.
Gowers convergence first gives well-defined positive cube measures;
recursive disintegration identifies their full mass with ordinary
coarea measure on regular configurations. A polynomial continuation
argument then transfers one nonzero-torsion witness to almost every
such configuration. We construct that witness for every admissible
exponent and obtain the required Fourier convergence by localization.

The final density argument uses two external inputs: the continuous
Varnavides estimate \cite[Proposition~1]{DurcikKovac2020} and
norm variation of multiple ergodic averages
\cite[Theorem~1.2]{VanDoornEtAl2026}. Their precise forms are stated
in Section~\ref{sec:radial-bridge}, including the passage from discrete
to continuous translations. The latter theorem strengthens Tao's
norm-convergence theorem \cite{Tao2008} and extends earlier variation
estimates for two and three commuting transformations
\cite{DurcikKovacSkrebThiele2019,DurcikSlavikovaThiele2026}.

The use of shell approximations and uniformity norms has precedents in
\cite{CookMagyarPramanik2017,DurcikKovac2020}. Here we stop the
Cauchy--Schwarz iteration at a bilinear form and use the geometry of
shell intersections to control the resulting Fourier supremum.
We also use the coarea formula \cite{Federer1969}, the analytic
implicit-function theorem, one-dimensional van der Corput estimates
\cite[Chapter~VIII, Section~1]{Stein1993}, and dimension theory for
locally subanalytic sets \cite{BierstoneMilman1988}. All dimension
arguments are made on relatively compact positive-coordinate charts
and then extended by countable exhaustion.

\section{Local shells and counting forms}%\label{sec:counting}
Until the final dimension-lifting argument, fix
\[
 k=n-2\ge1,\qquad Q=2^k,\qquad d=Q+1,
 \qquad p\in(1,\infty)\setminus\{2,\ldots,k+1\}.
\]
We call such an exponent $p$ admissible. On the positive orthant, set
$P(y)=\sum_{i=1}^d y_i^p$. Choose nonnegative
$\chi_i\in C_c^\infty((0,\infty))$, each positive at $d^{-1/p}$,
and write $\chi(y)=\prod_i\chi_i(y_i)$. Define the finite positive
measure
\[
 d\sigma(y)=\frac{\chi(y)}{|\nabla P(y)|}\,
 d\mathcal H^{d-1}(y)\big|_{\{P=1\}}.
\]
Its mass is strictly positive. Fix nonnegative probability densities
$\psi\in C_c^\infty((-1,1))$ and
$\varphi\in C_c^\infty(\mathbb R^d)$, and put
\[
 \psi_\eta(s)=\eta^{-1}\psi(s/\eta),\qquad
 S_\eta(y)=\chi(y)\psi_\eta(P(y)-1).
\]
The functions are extended by zero away from the positive orthant.
Coarea gives $S_\eta(y)\,dy\rightharpoonup\sigma$, with convergence
of masses. All shell supports lie in the same compact positive box.

For a finite measure $\mu$, let $\mu_\lambda$ be its pushforward by
$y\mapsto\lambda y$. For functions we use the corresponding
mass-preserving dilation $g_\lambda(y)=\lambda^{-d}g(y/\lambda)$.
For bounded, compactly supported inputs, define
\[
 \Lambda_m(\mathbf f;\mu)=
 \int_{\mathbb R^d}\int_{\mathbb R^d}
       \prod_{j=0}^{m-1}f_j(x+jy)\,dx\,d\mu(y).
\]
We write $\Lambda_m(\mathbf f;g)$ when $d\mu=g(y)\,dy$.
The forms depend only on the almost-everywhere classes of the inputs:
a null-set change contributes zero after integration in $x$, by
Fubini. The Fourier transform is
$\widehat g(\xi)=\int g(y)e^{-2\pi i y\cdot\xi}\,dy$.
We write $(q)_s=q(q-1)\cdots(q-s+1)$ for falling factorials, with
$(q)_0=1$. Constants may depend on the fixed parameters and cutoffs.

\section{Bilinear stopping and the shell criterion}%\label{sec:uniform}

For an integer $m\geq2$, put
\[
 Q_m=2^{m-2},\qquad b_m=1-\frac{m-1}{Q_m}.
\]
For $h\in\mathbb R^d$ define
$\Delta_hg(y)=g(y)\overline{g(y+h)}$, and let
\begin{equation}\label{eq:B-seminorm}
 \mathfrak B_m(g)^{Q_m}
 =\int_{(\mathbb R^d)^{m-2}}
 \big\|\widehat{\Delta_{h_{m-2}}\cdots\Delta_{h_1}g}\big\|_\infty
 \,dh_1\cdots dh_{m-2}.
\end{equation}
For $m=2$ this means $\mathfrak B_2(g)=\|\widehat g\|_\infty$.
All kernels in the next lemma are bounded and compactly supported, so
Fubini's theorem applies to the integrals used in its proof. The
suprema in the definition are measurable: each Fourier transform is
continuous in $\xi$, so its supremum equals the supremum over
$\xi\in\mathbb Q^d$.

\begin{lemma}[Explicit localized stopping estimate]
\label{lem:bs-explicit}
Let $\Omega\subset\mathbb R^d$ be a bounded measurable set of measure
$M>0$, and let $K\subset\mathbb R^d$ be a compact set with
$H:=|K-K|>0$. Suppose that $|f_j|\leq1$ and
$\operatorname{supp}f_j\subset\Omega$ for $0\leq j<m$, and that
$\operatorname{supp}g\subset\lambda K$ for some $\lambda>0$. Then
\begin{equation}\label{eq:bs-explicit}
 \left|\int_{\mathbb R^{2d}}
   \prod_{j=0}^{m-1}f_j(x+jy)g(y)\,dx\,dy\right|
 \leq M H^{b_m}\lambda^{db_m}\mathfrak B_m(g).
\end{equation}
For the mass-preserving dilation
$g_\lambda(y)=\lambda^{-d}g(y/\lambda)$ one has
\begin{equation}\label{eq:bs-exact-scaling}
 \mathfrak B_m(g_\lambda)=\lambda^{-db_m}\mathfrak B_m(g).
\end{equation}
In particular, if $\operatorname{supp}g\subset K$, then
\begin{equation}\label{eq:bs-dilation-cancelled}
 |\Lambda_m(\mathbf f;g_\lambda)|
 \leq M H^{b_m}\mathfrak B_m(g)
 \qquad(\lambda>0).
\end{equation}
\end{lemma}

\begin{proof}
For $m=2$, write $\check g(y)=g(-y)$. The inner integral in $y$ is
$(f_1*\check g)(x)$. Cauchy--Schwarz in $x$ and Plancherel therefore give
\[
 |\Lambda_2(f_0,f_1;g)|
 \leq\|f_0\|_2\|f_1*\check g\|_2
 \leq\|f_0\|_2\|f_1\|_2\|\widehat g\|_\infty
 \leq M\mathfrak B_2(g),
\]
which is \eqref{eq:bs-explicit}, since $b_2=0$.

Suppose the estimate has been proved for $m-1$, where $m\geq3$.
Removing $f_0$ by Cauchy--Schwarz gives
\[
 |\Lambda_m(\mathbf f;g)|^2
 \leq M\int_{\mathbb R^d}
  \left|\int_{\mathbb R^d}
      \prod_{j=1}^{m-1}f_j(x+jy)g(y)\,dy\right|^2 dx.
\]
Expand the square, put the second integration variable equal to $y+h$,
and then make the measure-preserving substitution $z=x+y$.
With
\[
 F_{j,h}(z)=f_j(z)\overline{f_j(z+jh)}
 \qquad(1\leq j\leq m-1)
\]
this yields
\begin{equation}\label{eq:bs-one-step}
 |\Lambda_m(\mathbf f;g)|^2
 \leq M\int_{\lambda(K-K)}
  |\Lambda_{m-1}(F_{1,h},\ldots,F_{m-1,h};\Delta_hg)|\,dh.
\end{equation}
Indeed, the integral over $h$ before taking absolute values equals
the nonnegative square integral above; taking the absolute value inside can only
increase it. The restriction on $h$ follows from
$g(y)\overline{g(y+h)}\ne0$ only when $h\in\lambda(K-K)$.
Every $F_{j,h}$ is bounded by one and supported in $\Omega$, while
$\Delta_hg$ is supported in $\lambda K$. The induction hypothesis
applied for each $h$ therefore bounds the right side of
\eqref{eq:bs-one-step} by
\[
 M^2H^{b_{m-1}}\lambda^{db_{m-1}}
 \int_{\lambda(K-K)}\mathfrak B_{m-1}(\Delta_hg)\,dh.
\]
H\"older's inequality, with exponent $Q_{m-1}$, gives
\begin{align*}
 \int_{\lambda(K-K)}\mathfrak B_{m-1}(\Delta_hg)\,dh
 &\leq(\lambda^dH)^{1-1/Q_{m-1}}
 \left(\int_{\mathbb R^d}
      \mathfrak B_{m-1}(\Delta_hg)^{Q_{m-1}}\,dh
 \right)^{1/Q_{m-1}}\\
 &= (\lambda^dH)^{1-1/Q_{m-1}}\mathfrak B_m(g)^2.
\end{align*}
The last equality follows directly from the definition, the commuting
identity $\Delta_h\Delta_kg=\Delta_k\Delta_hg$, and
$Q_m=2Q_{m-1}$. For $m=3$, the exponent in H\"older's inequality is one, and the
same calculation is simply an equality. Finally,
\[
 b_m=\frac12\left(b_{m-1}+1-\frac1{Q_{m-1}}\right).
\]
Taking square roots proves the induction claim, including its stated
constant.

For the dilation identity, write $r=m-2$ and use
\[
 \Delta_{\lambda h_r}\cdots\Delta_{\lambda h_1}g_\lambda(y)
 =\lambda^{-dQ_m}
   (\Delta_{h_r}\cdots\Delta_{h_1}g)(y/\lambda).
\]
Consequently,
\[
 \widehat{\Delta_{\lambda h_r}\cdots\Delta_{\lambda h_1}g_\lambda}(\xi)
 =\lambda^{-d(Q_m-1)}
 \widehat{\Delta_{h_r}\cdots\Delta_{h_1}g}(\lambda\xi).
\]
Changing all $r$ increment variables in the defining integral gives
\[
 \mathfrak B_m(g_\lambda)^{Q_m}
 =\lambda^{-d(Q_m-1-r)}\mathfrak B_m(g)^{Q_m}
 =\lambda^{-dQ_mb_m}\mathfrak B_m(g)^{Q_m}.
\]
This proves \eqref{eq:bs-exact-scaling}; inserting it into
\eqref{eq:bs-explicit} proves \eqref{eq:bs-dilation-cancelled}.
\end{proof}

\begin{remark}
A compact box containing the support may always be used for $K$.
The precise constants above are included to exhibit every localization
factor. No triangle inequality for $\mathfrak B_m$ is required anywhere
in this argument.
\end{remark}

\subsection{From shell comparison to convolution comparison}

\begin{lemma}[Direct shell-to-convolution comparison]
\label{lem:bs-shell-to-convolution}
Fix $m\geq2$ and a bounded input-support set $\Omega$. Let
$L>0$ satisfy $\operatorname{supp}\varphi\subset B(0,L)$, and set
\[
 \Omega^*=\Omega+B(0,(m-1)L).
\]
Suppose that for some $a_0\in(0,1]$ there is a nondecreasing function $v:(0,a_0)\to[0,\infty)$,
with $v(a)\to0$ as $a\to0$, such that
\begin{equation}\label{eq:bs-shell-hypothesis}
 |\Lambda_m(\mathbf f;(S_\eta-S_\tau)_\lambda)|\leq v(a)
\end{equation}
whenever $0<\eta<\tau\leq a<a_0$, $0<\lambda\leq1$, and all inputs
are bounded by one and supported in $\Omega^*$.
Then, for $0<\varepsilon<\tau<\tau_0$ and $0<\lambda\leq1$,
\begin{equation}\label{eq:bs-surface-convolution}
 |\Lambda_m(\mathbf f;\sigma_\lambda-
             \sigma_\lambda*\varphi_{\varepsilon\lambda})|
 \leq 2v(\tau)+C|\Omega|\frac\varepsilon\tau
\end{equation}
for all inputs bounded by one and supported in $\Omega$.
Here $0<\tau_0<a_0$ is sufficiently small, and $C$ depends only
on the fixed defining function and cutoffs. In particular, taking $\tau=\sqrt\varepsilon$ makes the right-hand
side tend to zero uniformly in the inputs and in $\lambda$.
\end{lemma}

\begin{proof}
First, the coarea formula in finitely many coordinate neighborhoods
where a partial derivative of $P$ is bounded away from zero gives
\[
 S_\eta(y)\,dy\ \longrightarrow\ \sigma
\]
weakly as finite measures. Their supports lie in the fixed compact set
$\operatorname{supp}\chi$, and their masses are uniformly bounded.
For compactly supported inputs bounded by one, the correlation
\[
 F_{\mathbf f}(y)=\int_{\mathbb R^d}
                  \prod_{j=0}^{m-1}f_j(x+jy)\,dx
\]
is continuous and compactly supported. To see continuity, expand the
difference of the two products by telescoping and use
\[
 |F_{\mathbf f}(y+h)-F_{\mathbf f}(y)|
 \leq\sum_{j=1}^{m-1}
       \|f_j(\,\cdot+jh)-f_j\|_1\longrightarrow0.
\]
The compact support follows from the first two inputs: a nonzero
product requires $x\in\operatorname{supp}f_0$ and
$x+y\in\operatorname{supp}f_1$.
Thus we may pass $\eta\to0$ in \eqref{eq:bs-shell-hypothesis}, for
fixed inputs, $\tau$, and $\lambda$, obtaining
\begin{equation}\label{eq:bs-exact-thick}
 |\Lambda_m(\mathbf f;\sigma_\lambda-(S_\tau)_\lambda)|
 \leq v(\tau).
\end{equation}
Although the weak convergence itself is only pointwise in the inputs,
the bound obtained after taking the limit is uniform, since the
prelimit bound is uniform.

There is also the elementary estimate
\begin{equation}\label{eq:bs-shell-gradient}
 \|\nabla S_\tau\|_1\leq C\tau^{-1}
 \qquad(0<\tau<\tau_0).
\end{equation}
Indeed,
\[
 \nabla S_\tau
 = (\nabla\chi)\psi_\tau(P-1)
    +\chi\psi'_\tau(P-1)\nabla P.
\]
The coordinate neighborhoods above give
$|\operatorname{supp}\chi\cap\{|P-1|\leq C_\psi\tau\}|=O(\tau)$.
The two terms have heights $O(\tau^{-1})$ and $O(\tau^{-2})$,
respectively, which proves \eqref{eq:bs-shell-gradient}.
By the fundamental theorem of calculus and translation invariance,
\[
 \|S_\tau(\,\cdot-z)-S_\tau\|_1
 \leq |z|\|\nabla S_\tau\|_1.
\]
It follows that
\begin{equation}\label{eq:bs-coarse-convolution}
 \|S_\tau-S_\tau*\varphi_\varepsilon\|_1
 \leq\varepsilon\|\nabla S_\tau\|_1
             \int |z|\varphi(z)\,dz
 \leq C\varepsilon/\tau.
\end{equation}

Use the exact decomposition
\begin{align*}
 \sigma_\lambda-\sigma_\lambda*\varphi_{\varepsilon\lambda}
 ={}&[\sigma_\lambda-(S_\tau)_\lambda]\\
 &+[(S_\tau)_\lambda-(S_\tau)_\lambda*
                                  \varphi_{\varepsilon\lambda}]\\
 &+[(S_\tau)_\lambda-\sigma_\lambda]*
                                  \varphi_{\varepsilon\lambda}.
\end{align*}
The first term contributes at most $v(\tau)$ by
\eqref{eq:bs-exact-thick}. Since
$|F_{\mathbf f}(y)|\leq|\Omega|$, the second contributes at most
$C|\Omega|\varepsilon/\tau$ by
\eqref{eq:bs-coarse-convolution} and invariance of the $L^1$ norm
under mass-preserving dilation.

For the third, expand the convolution and change the gap variable.
For $z\in\operatorname{supp}\varphi_{\varepsilon\lambda}$ put
$f_{j,z}(x)=f_j(x+jz)$. Fubini's theorem gives
\begin{align*}
 &\Lambda_m(\mathbf f;
       [(S_\tau)_\lambda-\sigma_\lambda]*
                                    \varphi_{\varepsilon\lambda})\\
 &\qquad=\int\varphi_{\varepsilon\lambda}(z)
       \Lambda_m(\mathbf f_z;(S_\tau)_\lambda-\sigma_\lambda)\,dz.
\end{align*}
Because $\varepsilon\lambda\leq1$, all $f_{j,z}$ are supported in
$\Omega^*$ and bounded by one. Hence \eqref{eq:bs-exact-thick} and
$\|\varphi_{\varepsilon\lambda}\|_1=1$ bound this contribution by
$v(\tau)$. Adding the three bounds proves
\eqref{eq:bs-surface-convolution} and the convergence assertion.
\end{proof}

\begin{corollary}[A sufficient shell criterion]
\label{cor:bs-shell-criterion}
If
\[
 \omega_m(a):=\sup_{0<\eta<\tau\leq a}
                 \mathfrak B_m(S_\eta-S_\tau)\longrightarrow0
 \quad(a\to0),
\]
then the exact-to-convolution comparison in
\eqref{eq:bs-surface-convolution} holds with a modulus tending to zero,
uniformly for inputs bounded by one and supported in a fixed bounded
set, and for $0<\lambda\leq1$.
\end{corollary}
\begin{proof}
Choose $a_0>0$ small enough that $\omega_m(a)<\infty$ for $a<a_0$.
All shell differences are supported in the fixed compact set
$\operatorname{supp}\chi$. Lemma~\ref{lem:bs-explicit}, applied with
$\Omega^*$ in place of $\Omega$, proves
\eqref{eq:bs-shell-hypothesis} with $v(a)=C\omega_m(a)$.
Apply Lemma~\ref{lem:bs-shell-to-convolution}.
\end{proof}

\section{Gowers convergence of the shells}
%\label{subsec:gs-general}

Gowers introduced uniformity norms in his work on arithmetic
progressions \cite{Gowers2001}. We use their Euclidean version;
see \cite{EisnerTao2012} for the locally compact abelian group setting.
For an integer $k\geq2$, the unnormalized Gowers norm on
$\mathbb R^d$ is
\begin{equation}\label{eq:gs-definition}
 \|f\|_{U^k(\mathbb R^d)}^{2^k}
 =\int_{\mathbb R^{d(k+1)}}
     \prod_{\omega\in\{0,1\}^k}
       \mathcal C^{|\omega|}f(x+\omega\cdot h)\,dx\,dh,
 \qquad \omega\cdot h=\sum_{j=1}^k\omega_jh_j.
\end{equation}
Here $\mathcal C$ is complex conjugation. The integral on the right
is nonnegative. For example, separating the last increment writes it
as the integral, over $(h_1,\ldots,h_{k-1})$, of
\[
 \left|\int_{\mathbb R^d}
       \prod_{\omega\in\{0,1\}^{k-1}}
          \mathcal C^{|\omega|}f(x+\omega\cdot h)\,dx\right|^2.
\]
We use the usual Gowers--Cauchy--Schwarz inequality
\begin{equation}\label{eq:gs-GCS}
 \left|\int
     \prod_{\omega\in\{0,1\}^k}
          \mathcal C^{|\omega|}f_\omega(x+\omega\cdot h)
       \,dx\,dh\right|
 \leq\prod_{\omega\in\{0,1\}^k}\|f_\omega\|_{U^k}.
\end{equation}
The triangle inequality for $U^k$ follows by expanding the $2^k$
factors for $f+g$, applying \eqref{eq:gs-GCS} to each of the
$2^{2^k}$ terms, and summing:
\[
 \|f+g\|_{U^k}^{2^k}
 \leq(\|f\|_{U^k}+\|g\|_{U^k})^{2^k}.
\]
Only bounded compactly supported functions and limits controlled by
these inequalities are used below.

\begin{lemma}[The one-dimensional phase estimate]
\label{lem:gs-scalar-phase}
Let $k\geq2$, let $b\in C_c^\infty((0,\infty))$, and suppose
\begin{equation}\label{eq:gs-exponent-condition}
 p>1,\qquad p(p-1)\cdots(p-k+1)\ne0.
\end{equation}
Then
\begin{equation}\label{eq:gs-scalar-bound}
 \|b(x)e^{2\pi iux^p}\|_{U^k(\mathbb R)}
 \leq C_{k,p,b}(1+|u|)^{-1/2^k}
       \bigl(1+\log(2+|u|)\bigr)^{(k-2)/2^k}.
\end{equation}
For $k=2$, condition \eqref{eq:gs-exponent-condition} holds for every
$p>1$, and the logarithmic exponent is zero.
\end{lemma}

\begin{proof}
Choose $0<c<C<\infty$ such that
$\operatorname{supp}b\Subset(c,C)$. Put $f_u(x)=b(x)e^{2\pi iux^p}$.
Its cube product has amplitude and phase
\[
 B(x,h)=\prod_{\omega\in\{0,1\}^k}
          \mathcal C^{|\omega|}b(x+\omega\cdot h),
 \qquad
 \Phi(x,h)=\sum_{\omega\in\{0,1\}^k}
                    (-1)^{|\omega|}(x+\omega\cdot h)^p.
\]
The phase is only needed where its arguments are positive. On the
amplitude support every vertex belongs to $[c,C]$. In particular,
$x\in[c,C]$ and $|h_j|\leq C-c$ for every $j$.

Fix $x$ and $h'=(h_1,\ldots,h_{k-1})$ for which the factors with
$\omega_k=0$ do not vanish. Integrate first in $h_k$.
On the interval on which all the remaining vertices belong to
$[c,C]$, the fundamental theorem of calculus, applied $k-1$ times,
gives
\begin{align}
 \partial_{h_k}\Phi(x,h)
 &=(-1)^k p(p-1)\cdots(p-k+1)
        \left(\prod_{j=1}^{k-1}h_j\right)
        A(x,h),\label{eq:gs-first-increment-derivative}\\
 A(x,h)
 &=\int_{[0,1]^{k-1}}
       \left(x+h_k+\sum_{j=1}^{k-1}t_jh_j\right)^{p-k}\,dt.
       \notag
\end{align}
Every point in the last integral lies in $[c,C]$: it is a convex
combination of the vertices with $\omega_k=1$. Consequently $A$ is
bounded above and below by positive constants. Every derivative of
$A$ in $h_k$ is uniformly bounded, and the same holds for every
fixed-order derivative of $A^{-1}$. These bounds are independent of
$x,h'$ on the active set.

For $\prod_{j<k}h_j\ne0$, integrate by parts using
\[
 e^{2\pi iu\Phi}
 =\frac{1}{2\pi iu\,\partial_{h_k}\Phi}
       \partial_{h_k}e^{2\pi iu\Phi}.
\]
The amplitude and each of its derivatives are uniformly bounded and
supported in an interval of uniformly bounded length. They vanish
near the endpoints of the interval where any vertex first reaches
$c$ or $C$, so no boundary terms occur. Equation
\eqref{eq:gs-first-increment-derivative} shows that each integration
by parts contributes one factor
$(|u|\prod_{j<k}|h_j|)^{-1}$. After $N$ repetitions,
\[
 \left|\int_{\mathbb R}B(x,h)e^{2\pi iu\Phi(x,h)}\,dh_k\right|
 \leq C_N\min\left\{1,
       \left(|u|\prod_{j=1}^{k-1}|h_j|\right)^{-N}\right\}.
\]
The bound by a constant is the trivial estimate; it covers zero
increments as well. Combining the nonnegativity in
\eqref{eq:gs-definition} with Fubini and the triangle inequality
therefore yields
\begin{equation}\label{eq:gs-reduced-integral}
 \|f_u\|_{U^k}^{2^k}
 \leq C_N\int_{[-(C-c),C-c]^{k-1}}
       \min\left\{1,
          \left(|u|\prod_{j=1}^{k-1}|h_j|\right)^{-N}\right\}
       \,dh'.
\end{equation}
The bounded $x$ integral has been absorbed into $C_N$.

For completeness, if $r\geq1$, $N>1$, and $U\geq2$, then
\begin{equation}\label{eq:gs-product-integral}
 \int_{[0,1]^r}\min\{1,(Us_1\cdots s_r)^{-N}\}\,ds
 \leq C_{r,N}U^{-1}(1+\log U)^{r-1}.
\end{equation}
To prove this, set $s_j=e^{-v_j}$. Integration over the simplex of
points with $v_1+\cdots+v_r=v$ turns the left side into
\[
 \frac1{(r-1)!}\int_0^\infty
      \min\{1,U^{-N}e^{Nv}\}e^{-v}v^{r-1}\,dv.
\]
On $0\leq v\leq\log U$ it is at most
\[
 \frac{U^{-N}(1+\log U)^{r-1}}{(r-1)!}
       \int_0^{\log U}e^{(N-1)v}\,dv
 \leq C_{r,N}U^{-1}(1+\log U)^{r-1}.
\]
On $v\geq\log U$, put $v=\log U+w$ and use
$(\log U+w)^{r-1}\leq C_r(1+\log U)^{r-1}(1+w)^{r-1}$.
The remaining integral of $e^{-w}(1+w)^{r-1}$ is finite and gives
the same bound. This also proves the case $r=1$.

Apply \eqref{eq:gs-product-integral} with $r=k-1$, after rescaling
and using symmetry in the increments in \eqref{eq:gs-reduced-integral}.
For $|u|$ sufficiently large this gives
\[
 \|f_u\|_{U^k}^{2^k}
 \leq C|u|^{-1}(1+\log|u|)^{k-2}.
\]
The remaining bounded range of $u$ follows from the trivial cube
integral estimate. Taking the $2^k$-th root proves
\eqref{eq:gs-scalar-bound}.
\end{proof}

\begin{theorem}[Gowers Cauchy convergence of shells]
\label{thm:gs-shell-Cauchy}
Suppose that $k\geq2$, $d>2^k$, and $p$ satisfies
\eqref{eq:gs-exponent-condition}. Let
\[
 \chi(y)=\prod_{\ell=1}^d b_\ell(y_\ell),
 \qquad b_\ell\in C_c^\infty((0,\infty)),
 \qquad P_p(y)=\sum_{\ell=1}^dy_\ell^p.
\]
Fix $\psi\in C_c^\infty(\mathbb R)$ with $\int\psi=1$, and put
\[
 S_\eta(y)=\chi(y)\eta^{-1}\psi((P_p(y)-1)/\eta).
\]
Then
\[
 \|S_\eta-S_\tau\|_{U^k(\mathbb R^d)}\longrightarrow0
 \qquad(\eta,\tau\downarrow0),
\]
and the shell $U^k$ norms are uniformly bounded for $0<\eta\le1$.
\end{theorem}

\begin{proof}
For $G_u(y)=\chi(y)e^{2\pi iuP_p(y)}$, the defining cube integral
factors over the $d$ scalar coordinates. Thus the tensor identity is
exact:
\[
 \|G_u\|_{U^k(\mathbb R^d)}
 =\prod_{\ell=1}^d
       \|b_\ell(x)e^{2\pi iux^p}\|_{U^k(\mathbb R)}.
\]
By Lemma~\ref{lem:gs-scalar-phase}, with
$\gamma=d/2^k$ and $L=d(k-2)/2^k$,
\begin{equation}\label{eq:gs-tensor-phase}
 \|G_u\|_{U^k}
 \leq C(1+|u|)^{-\gamma}
                \bigl(1+\log(2+|u|)\bigr)^L.
\end{equation}
The right side is integrable on $\mathbb R$, since $\gamma>1$.

Use the scalar Fourier-transform convention
$\widehat\psi(u)=\int\psi(s)e^{-2\pi ius}\,ds$.
Fourier inversion gives the exact formula
\[
 S_\eta-S_\tau
 =\int_{\mathbb R} e^{-2\pi iu}
       [\widehat\psi(\eta u)-\widehat\psi(\tau u)]G_u\,du.
\]
For fixed positive $\eta,\tau$, the scalar coefficient is integrable,
so this integral converges uniformly in $y$ and has fixed compact
support. The triangle inequality for $U^k$ yields
\begin{equation}\label{eq:gs-Minkowski}
 \|S_\eta-S_\tau\|_{U^k}
 \leq\int_{\mathbb R}
   |\widehat\psi(\eta u)-\widehat\psi(\tau u)|\,
                                      \|G_u\|_{U^k}\,du.
\end{equation}
One may justify the integral triangle inequality here without
invoking a completed Gowers space. On a bounded $u$ interval,
Riemann sums converge uniformly in $y$, and hence in $U^k$: for
functions supported in a fixed compact set $E$,
\[
 \|f\|_{U^k}^{2^k}
 \leq |E|\,|E-E|^k\|f\|_\infty^{2^k}.
\]
Apply the finite-sum triangle inequality, pass to the Riemann limit,
and then let the interval increase to $\mathbb R$. The integrability
of the Fourier coefficient gives uniform convergence of the tails;
\eqref{eq:gs-tensor-phase} controls the right side.

The Fourier coefficient in \eqref{eq:gs-Minkowski} tends to zero
pointwise as $\eta,\tau\downarrow0$ and is bounded by
$2\|\psi\|_1$. Dominated convergence using
\eqref{eq:gs-tensor-phase} proves the Cauchy assertion. Applying the
same integral estimate to $S_\eta$ itself, with coefficient
$e^{-2\pi iu}\widehat\psi(\eta u)$, gives uniform boundedness.
\end{proof}

\section{Cube measures and exclusion of singular concentration}
%\label{sec:recursive-cube-measures}

We first record the measure-theoretic consequence of Gowers convergence.
Throughout the next lemma, let
\[
 S_\epsilon(y)=\chi(y)\psi_\epsilon(P(y)-1),
 \qquad \psi_\epsilon(s)=\epsilon^{-1}\psi(s/\epsilon),
\]
where $\chi\geq0$ is smooth and compactly supported, $P$ is smooth on a
neighborhood of its support, and $\psi\geq0$ is a smooth compactly
supported probability density. Write $E=\operatorname{supp}\chi$.
All the configuration measures below have common compact support: their
base vertex belongs to $E$, and each increment belongs to $E-E$.

\begin{lemma}[Mixed cube limits and kernel factorization]
\label{lem:cube-kernel-disintegration}
Let $r\geq2$, and suppose $S_\epsilon$ is Cauchy in $U^r(\mathbb R^d)$.
Then the positive measures
\[
 dM_{\boldsymbol\epsilon}(y,h)
 =\prod_{\omega\in\{0,1\}^r}
 S_{\epsilon_\omega}(y+\omega\cdot h)\,dy\,dh
\]
converge weakly and in total mass to a common finite measure $M_r$ as
$\max_\omega\epsilon_\omega\to0$, independently of the relative
widths. This limit is supported on the exact incidence
\[
 P(y+\omega\cdot h)=1\qquad(\omega\in\{0,1\}^r).
\]
Put $\ell=(h_1,\ldots,h_{r-1})$, and define
\[
 d\nu_{\ell,\epsilon}(x)
 =\prod_{\omega\in\{0,1\}^{r-1}}
 S_\epsilon(x+\omega\cdot\ell)\,dx,
 \qquad m_\epsilon(\ell)=\nu_{\ell,\epsilon}(\mathbb R^d).
\]
There is a measurable kernel of finite positive measures $\nu_\ell$
such that $m_\epsilon\to m:=\nu_\ell(\mathbb R^d)$ in $L^2(d\ell)$,
the lower-cube measures converge weakly to
\[
 L=\int\nu_\ell\,d\ell,
\]
and
\begin{equation}\label{eq:recursive-cube-disintegration}
 \int\Phi\,dM_r
 =\int d\ell\iint
 \Phi(x,\ell,z-x)\,d\nu_\ell(x)\,d\nu_\ell(z)
\end{equation}
for every continuous compactly supported $\Phi$.
The kernel is uniquely determined for almost every $\ell$ by $L$.
It is supported on the exact lower-cube fiber for almost every $\ell$.

If $L$ is concentrated where the lower-cube derivative in $x$ has
rank $2^{r-1}$ and there equals ordinary weighted coarea measure,
then $\nu_\ell$ is the corresponding fiber coarea measure for almost
every $\ell$. In that case $M_r$ is concentrated on the regular locus
of the full $2^r$-level map and equals ordinary weighted coarea
measure there.
\end{lemma}

\begin{proof}
Let $C$ bound the shell $U^r$ norms. Gowers--Cauchy--Schwarz,
\eqref{eq:gs-GCS}, bounds every mixed $r$-cube integral by the product
of its input $U^r$ norms. The same bound is available with a smooth compactly supported
configuration test $\Phi$, at the cost of $\|\widehat\Phi\|_1$.
Indeed, every Fourier character factors into characters of the base
and singleton vertices by
\[
 \xi\cdot y+\sum_{j=1}^r\eta_j\cdot h_j
 =\left(\xi-\sum_{j=1}^r\eta_j\right)\cdot y
       +\sum_{j=1}^r\eta_j\cdot(y+h_j).
\]
Linear modulation preserves $U^r$ for $r\geq2$. Absorbing the
conjugations at odd vertices into the inputs and using Fourier
inversion therefore gives
\[
 \left|\int\Phi(y,h)\prod_\omega
 f_\omega(y+\omega\cdot h)\,dy\,dh\right|
 \leq\|\widehat\Phi\|_1\prod_\omega\|f_\omega\|_{U^r}.
\]
Telescope one input at a time when comparing two systems of widths.
Each summand has one $U^r$-small difference and uniformly bounded
remaining inputs. Thus all smooth tested integrals are Cauchy, with
the same limit for all independent widths. The untested inequality
also bounds the masses uniformly. Positivity and common compact
support give weakly convergent subsequences, while the Cauchy
property on smooth tests makes their limit unique. Uniform
approximation by smooth tests on the common compact support proves
weak convergence on continuous tests. A smooth test equal to one
there proves convergence of total masses.

For $2^{r-1}$ functions define
\[
 m_{\mathbf f}(\ell)
 =\int\prod_{\omega\in\{0,1\}^{r-1}}
 f_\omega(x+\omega\cdot\ell)\,dx.
\]
Expanding its squared $L^2$ norm and substituting $z=x+h_r$ produces
an $r$-cube integral. Gowers--Cauchy--Schwarz gives
\begin{equation}\label{eq:cube-lower-mass-bound}
 \|m_{\mathbf f}\|_2
 \leq\prod_{\omega\in\{0,1\}^{r-1}}\|f_\omega\|_{U^r}.
\end{equation}
Consequently
\[
 \|m_\epsilon-m_\tau\|_2
 \leq 2^{r-1}C^{2^{r-1}-1}
                  \|S_\epsilon-S_\tau\|_{U^r}\longrightarrow0.
\]
For a smooth compactly supported $\phi(x)$, write
\[
 m_\epsilon^\phi(\ell)=\int\phi(x)\,d\nu_{\ell,\epsilon}(x).
\]
Fourier inversion in $x$, Minkowski's inequality, and
\eqref{eq:cube-lower-mass-bound} yield the analogous tested estimate
\[
 \left\|\int\phi(x)\prod_\omega
 f_\omega(x+\omega\cdot\ell)\,dx\right\|_2
 \leq\|\widehat\phi\|_1\prod_\omega\|f_\omega\|_{U^r}.
\]
Here the Fourier character modulates only the base-vertex input.
Telescoping proves that $m_\epsilon^\phi$ is Cauchy in $L^2$ for
every such $\phi$.

Choose countably many smooth tests whose restrictions are uniformly
dense in $C(E)$, including a test equal to one on $E$. A diagonal
subsequence $\epsilon_j\to0$ has almost-everywhere convergence of
all their tested masses and the total masses. For almost every
$\ell$ the latter are bounded. Weak compactness of positive measures
on $E$, followed by uniqueness from the determining tests, therefore
gives a finite measure $\nu_\ell$ with
\[
 \nu_{\ell,\epsilon_j}\longrightarrow\nu_\ell
 \quad\hbox{weakly for almost every }\ell.
\]
Its total mass is the $L^2$ limit $m(\ell)$ of $m_\epsilon$.
The countable determining family also proves measurability of the
kernel $\ell\mapsto\nu_\ell$.

The lower-cube measures have a unique weak limit. For tests
$\phi(x)a(\ell)$ this follows from the preceding $L^2$ convergence
and the bounded support in $\ell$. Finite sums of such product tests
are uniformly dense on the common compact support, and the masses
are uniformly bounded by \eqref{eq:cube-lower-mass-bound} and
Cauchy--Schwarz in $\ell$. It follows that the limit is
$L=\int\nu_\ell\,d\ell$. Uniqueness of disintegration determines
$\nu_\ell$ almost everywhere.

For equal widths, the substitution $h_r=z-x$ gives
\eqref{eq:recursive-cube-disintegration} with
$\nu_{\ell,\epsilon}$ in place of $\nu_\ell$. Along the chosen
subsequence the inner integrals converge for almost every $\ell$,
by weak convergence of products of finite measures on a compact set.
Their absolute values are bounded by
$\|\Phi\|_\infty m_\epsilon(\ell)^2$. Since
\[
 \|m_\epsilon^2-m^2\|_1
 \leq\|m_\epsilon-m\|_2(\|m_\epsilon\|_2+\|m\|_2)
 \longrightarrow0,
\]
these bounds are uniformly integrable. Vitali's theorem proves
\eqref{eq:recursive-cube-disintegration}. The already established
uniqueness of the full-cube weak limit removes any subsequence
restriction from that identity.

Both $M_r$ and $L$ are supported on their exact incidences. To see
this, any point off the exact incidence has a neighborhood on which
at least one level differs from one by a fixed positive amount;
the associated shell factor vanishes there for all sufficiently
small widths. The same conclusion for almost every conditional
measure follows from $L=\int\nu_\ell\,d\ell$ and Tonelli's theorem.
More generally, concentration of $L$ on any measurable set implies
concentration of its conditional measures on the corresponding
sections of that set.

On the full-rank part of the lower-cube derivative in $x$, ordinary
coarea charts disintegrate the lower-cube level measure into its
fiber coarea measures with respect to $d\ell$. Uniqueness of the
kernel proves the stated identification of $\nu_\ell$ there. In
coordinates $(x,z,\ell)$, the two groups of $2^{r-1}$ levels in
\eqref{eq:recursive-cube-disintegration} have independent derivative
blocks in $x$ and $z$. Thus the full derivative has rank $2^r$
almost everywhere for the product kernel.
Finally, on any relatively compact submersion chart for the full
level map, the shell integrals are products of one-dimensional
approximate identities against a smooth density. They converge to
ordinary weighted coarea measure, even with independent widths.
This identifies $M_r$ on its regular locus and completes the proof.
\end{proof}

\begin{proposition}[Regular mass at the target dimension]
\label{prop:regular-cube-mass}
Let $k\geq1$, $d=2^k+1$, and
\[
 p>1,\qquad p\notin\{2,\ldots,k+1\}.
\]
Let $P(y)=\sum_{i=1}^d y_i^p$ on the positive orthant, and choose a
nonnegative smooth product cutoff $\chi$ supported compactly there.
For every $1\leq j\leq k$, the positive mixed $j$-cube shell measures
have a common finite weak limit $M_j$, with convergence of total
masses. This measure is supported on the exact incidence
\[
 \Gamma_j=\{(y,h):P(y+\omega\cdot h)=1
                         \text{ for every }\omega\in\{0,1\}^j\},
\]
is concentrated on the regular locus of its full $2^j$-level map,
and equals ordinary weighted coarea measure there.
\end{proposition}

\begin{proof}
The single shells converge weakly and in mass to
\[
 d\sigma(y)=\frac{\chi(y)}{|\nabla P(y)|}\,d\mathcal H^{d-1}(y)
 \quad\hbox{on }\{P=1\},
\]
because $\nabla P$ does not vanish on the active positive patch.
Under $z=y+h$, the mixed one-cube measure becomes
$S_\eta(y)\,dy\otimes S_\tau(z)\,dz$. Therefore $M_1$ is the image
of $\sigma\otimes\sigma$, with weak and mass convergence for
independent widths. The map $(y,z)\mapsto(P(y),P(z))$ is a
submersion everywhere on the active patch, so $M_1$ is ordinary
coarea measure on the full two-level incidence.

For $2\leq j\leq k$, Theorem~\ref{thm:gs-shell-Cauchy} applies in
$U^j$, since $d>2^j$ and the required falling factorials of $p$ are
nonzero. Lemma~\ref{lem:cube-kernel-disintegration} supplies the
mixed weak limits $M_j$. If $k=1$, the preceding paragraph already
proves the proposition. Otherwise proceed by induction from $M_1$.

Suppose $1\leq j<k$ and that $M_j$ is ordinary coarea measure on
the regular full incidence. We show that the derivative in $y$
alone has rank $2^j$ at $M_j$-almost every point. Rank deficiency
implies the existence of
$[c]\in\mathbb P^{2^j-1}(\mathbb R)$ such that
\begin{equation}\label{eq:regular-mass-rank-equations}
 \sum_{\omega\in\{0,1\}^j}
 c_\omega(y_i+\sum_{a=1}^j\omega_a h_{ai})^{p-1}=0
 \qquad(1\leq i\leq d).
\end{equation}
For every nonzero $c$, the scalar function in one coordinate block
of $j+1$ variables is nonzero analytic on its connected positive
cone. Indeed, set its base variable equal to one and differentiate
at zero increments in a square-free subset $S\subseteq[j]$.
An identically zero function would give
\[
 (p-1)_{|S|}\sum_{\omega:\,\omega_a=1\ (a\in S)}c_\omega=0
 \qquad(S\subseteq[j]).
\]
All displayed falling factorials are nonzero. Starting with
$S=[j]$ and proceeding in decreasing subset size forces every
$c_\omega=0$, a contradiction.

Hence, for fixed $[c]$, each coordinate zero set in
\eqref{eq:regular-mass-rank-equations} has dimension at most $j$.
The $d$ coordinate blocks are independent. Including the projective
coefficient parameter and projecting shows that the rank-defect
set has dimension at most
\[
 jd+(2^j-1).
\]
These dimension statements can be made on countably many relatively
compact positive charts using analytic stratification; projection
in the compact projective coefficient variable does not increase
dimension. The regular exact incidence has dimension
$(j+1)d-2^j$, and
\[
 jd+(2^j-1)<(j+1)d-2^j,
 \qquad\text{since }d=2^k+1>2^{j+1}-1.
\]
Thus its rank-defect subset is null for ordinary coarea measure,
and consequently for $M_j$.

Apply Lemma~\ref{lem:cube-kernel-disintegration} at order $r=j+1$.
The lower-cube weak limit in that lemma is $M_j$, by uniqueness of
the already established weak limit. We have just verified its
concentration on full-rank $y$-fibers and its coarea description.
The final assertion of the lemma therefore identifies $M_{j+1}$
as ordinary coarea measure on its regular full incidence, with no
mass on its singular complement. This completes the induction.
\end{proof}

\section{Polynomial transfer from one signed witness}
%\label{sec:weight-transfer}

The geometry of exact intersections can have several connected
components. The following argument transfers nonzero torsion through a
connected space of coordinate ratios instead. Allowing signed weights
in the initial witness will be useful for integer exponents.

For a regular curve $y=Y(t)$ in $\mathbb R^d$, nonzero
\emph{affine torsion} at a point means
\[
 \det(Y'(t),Y''(t),\ldots,Y^{(d)}(t))\ne0.
\]
This condition is independent of the regular parameterization and
is preserved by invertible affine maps. On a regular family of
curves, it is an open condition.

\begin{proposition}[Single-witness transfer]
\label{prop:single-witness-transfer}
Let $k\ge1$, $Q=2^k$, $d=Q+1$, and
$p\in(1,\infty)\setminus\{2,\ldots,k+1\}$. Suppose there are
nonzero real weights $c_1,\ldots,c_d$, a positive vector $y$, and
increments $h_1,\ldots,h_k$ for which every $y+\omega\cdot h$ is
positive and
\[
 \sum_{i=1}^d c_i\Big(y_i+\sum_{j=1}^k\omega_jh_{ji}\Big)^p=1
 \qquad(\omega\in\{0,1\}^k).
\]
Suppose the fixed-increment intersection through $y$ is a regular
curve with nonzero affine torsion at $y$.

Then almost every regular exact configuration on the ordinary
positive unit $\ell^p$ sphere has a regular fixed-increment curve
with nonzero affine torsion. Here regularity of the exact
configuration means rank $Q$ for the full level derivative in
$(y,h_1,\ldots,h_k)$, and almost everywhere refers to its ordinary
coarea level measure. In particular, every nonempty relatively open
part of that regular incidence contains an ordinary positive witness.
\end{proposition}

\begin{proof}
Put
\[
 \mathcal U_k=
 \{r\in\mathbb R^k:1+\omega\cdot r>0
                         \text{ for all }\omega\in\{0,1\}^k\},
 \qquad A(r)_{\omega i}=(1+\omega\cdot r_i)^p.
\]
The domain $\mathcal U_k$ is open and convex and contains the origin.
Normalize a configuration with the fixed weights $c_i$ by
\[
 b_i=c_i y_i^p,\qquad
 r_i=(h_{1i},\ldots,h_{ki})/y_i.
\]
Its exact equations become
\begin{equation}\label{eq:transfer-linear-incidence}
 A(r)b=\mathbf1.
\end{equation}
On the chamber $\operatorname{sgn}b_i=\operatorname{sgn}c_i$, this
normalization is a real-analytic diffeomorphism. Its inverse is
\[
 y_i=(b_i/c_i)^{1/p},\qquad h_{ji}=y_i(r_i)_j.
\]
For the ordinary sphere the same assertion holds with all $c_i=1$
and all $b_i>0$. Consequently these maps preserve regularity of the
universal incidence and its surface-null sets. On any regular level
chart, surface measure and coarea level measure have the same null
sets, since their density ratio is smooth and strictly positive.

We first show that $A$ has rank $Q$ almost everywhere on every
regular incidence in question, irrespective of the signs of $b$.
The feature map
\[
 a(r)=\big((1+\omega\cdot r)^p\big)_{\omega\in\{0,1\}^k}
\]
spans $\mathbb R^Q$ as $r$ ranges over $\mathcal U_k$. Indeed, if
$\sum_\omega u_\omega(1+\omega\cdot r)^p=0$ identically, then
for each $S\subseteq[k]$, square-free differentiation at $r=0$ gives
\[
 (p)_{|S|}\sum_{\omega:\,\omega_j=1\ (j\in S)}u_\omega=0.
\]
All displayed factors are nonzero. Starting with $S=[k]$ and
proceeding in decreasing subset size, Boolean inversion gives
$u_\omega=0$ for every $\omega$. Thus the analytic preimage of
any proper linear subspace of $\mathbb R^Q$ under $a$ has dimension
at most $k-1$: it lies in the zero set of a nonzero real-analytic
linear combination of the features.

On a rank-$s$ stratum of $A$, $s<Q$, choose $s$ independent columns.
Their ratios contribute at most $ks$ dimensions. Once these ratios
are fixed, each remaining ratio must map into their proper linear
span, and hence contributes at most $k-1$ dimensions. The ratio
stratum therefore has dimension at most
\[
 ks+(k-1)(d-s).
\]
For each such ratio tuple the solutions of
\eqref{eq:transfer-linear-incidence}, if any, form an affine space
of dimension $d-s$. It follows that the exact rank-deficient set
has dimension at most
\begin{equation}\label{eq:transfer-rank-defect-dimension}
 ks+(k-1)(d-s)+(d-s)=kd.
\end{equation}
These dimension statements are local: on relatively compact analytic
charts, rank conditions and the incidence equations define
subanalytic sets, and the dimension bound for a family is the
dimension of its parameter set plus the largest fiber dimension.
Apply this fact first to the remaining ratios and then to the
weights, and take the finite union over the choices of independent
columns. A countable exhaustion handles the unrestricted domains.
The regular exact incidence has dimension
\[
 (k+1)d-Q=kd+1.
\]
Thus \eqref{eq:transfer-rank-defect-dimension} makes its rank-defect
subset surface-null. The argument imposes no sign condition on the
weights.

We next express torsion polynomially in the normalized weights. For
fixed $b,r$ let
\[
 F_\omega(Y;b,r)=\sum_{i=1}^d b_i(Y_i+\omega\cdot r_i)^p,
\]
defined near $Y=\mathbf1$. Let $T(Y;b,r)$ be the signed cofactor vector of the $Q\times d$
matrix $D_YF$, so
\[
 T_i=(-1)^{i+1}\det\bigl((D_YF)_{\widehat i}\bigr),
 \qquad 1\le i\le d,
\]
where the subscript $\widehat i$ denotes deletion of column $i$.
Then $D_YF\,T=0$. Write $\mathcal D=T\cdot\nabla_Y$,
differentiating with $b,r$ held fixed. Set
\[
 \mathscr P(b,r)=
 \left.\det(T,\mathcal DT,\ldots,\mathcal D^QT)
                                              \right|_{Y=\mathbf1}.
\]
The vector $T$ has homogeneous degree $Q$ in $b$.
Consequently $\mathcal D^jT$ has homogeneous degree $Q(j+1)$,
and $\mathscr P$ is a homogeneous polynomial in $b$ of degree
\[
 L=Q\sum_{j=1}^d j=\frac{Qd(d+1)}2.
\]
Its coefficients are real analytic on all of $\mathcal U_k^d$.

If $D_YF$ has rank $Q$, then $T$ is a nonzero tangent field to the
one-dimensional level fiber. Its integral curves have successive
derivatives $T,\mathcal DT,\ldots$. Reparameterizing a regular curve
multiplies its torsion determinant by a nonzero factor. Hence
$\mathscr P\ne0$ is exactly ordinary affine torsion nonvanishing
on a regular fiber; it also implies the required rank by itself.
At a fixed original configuration, the linear change
$x_i=y_iY_i$ takes this normalized weighted fiber to the original
fiber with weights $c_i$ and fixed increments $h_j$.
It preserves regularity and torsion nonvanishing. In particular,
$\mathscr P$ is nonzero at the normalized signed witness.

The fixed-increment rank at the witness implies full universal rank,
so its exact incidence is a manifold of dimension $kd+1$ near that
point. By \eqref{eq:transfer-rank-defect-dimension}, we may perturb
the witness within this manifold, retaining its sign chamber and
nonzero torsion, until $A$ has rank $Q$. Choose a $Q$-column minor
$\Delta(r)$ that is nonzero there, and let $t$ be the omitted
weight. With $I$ the chosen columns and $j$ the omitted column,
Cramer's rule gives
\[
 b_I(r,t)=\frac{\operatorname{adj}(A_I(r))
                           (\mathbf1-tA_j(r))}{\Delta(r)},
 \qquad b_j(r,t)=t.
\]
Write the displayed numerator as $B_I(r,t)$ and put
$B_j(r,t)=t\Delta(r)$. All components of $B$ are analytic on
$\mathcal U_k^d\times\mathbb R$. Homogeneity now shows that
\begin{equation}\label{eq:general-cleared-torsion}
 \mathscr Q(r,t):=\mathscr P(B(r,t),r)
       =\Delta(r)^L\mathscr P(b(r,t),r)
       \qquad(\Delta(r)\ne0).
\end{equation}
This explicitly supplies an analytic extension to the whole
connected domain $\mathcal U_k^d\times\mathbb R$. It is nonzero
at the perturbed signed witness. Therefore its zero set has
Lebesgue measure zero in that domain.

The fixed minor $\Delta$ is also a nonzero analytic function on
the connected domain $\mathcal U_k^d$. On every full-rank incidence
chart, including a chart parameterized using another minor, all
$kd$ ratios and one omitted weight are free coordinates. Hence
$\{\Delta=0\}$ is surface-null on every such chart, by the
analytic zero-set theorem and Fubini's theorem. Outside that null
set, the fixed Cramer parameterization above applies. Equation
\eqref{eq:general-cleared-torsion} then makes $\{\mathscr P=0\}$
null there as well. Together with the rank-defect estimate, this
proves $\mathscr P\ne0$ almost everywhere on the regular
positive-weight incidence.

Finally, when $b_i>0$, the fixed diagonal map
$Y_i\mapsto b_i^{1/p}Y_i$ converts the normalized weighted equations
into ordinary unit-sphere equations, with increments scaled by the
same factors. Equivalently, it is the fiber version of the inverse
normalization with $c_i=1$. It preserves regularity and nonzero
torsion, and the normalization preserves null sets on the universal
incidence. This proves the asserted almost-everywhere conclusion
for the ordinary sphere. Every nonempty relatively open subset of
its regular incidence has positive local surface measure, giving
the final existence assertion. No connectedness of that incidence
has been used.
\end{proof}

\section{Construction of the witnesses}%\label{sec:witnesses}
\subsection{Noninteger exponents}
%\label{sec:general-walsh-witness}

\begin{lemma}[Walsh witness]
\label{lem:general-walsh-witness}
Let $k\geq1$, $Q=2^k$, and let $p>1$ be noninteger.
There is a regular positive exact $k$-cube on the $\ell^p$ sphere in
$\mathbb R^{Q+1}$ whose fixed-increment curve has nonzero affine torsion.
\end{lemma}
\begin{proof}

Here is a direct construction. Index the first $Q$ coordinates by
$\epsilon\in\{-1,1\}^k$ and the vertices by
$\sigma\in\{-1,1\}^k$. Choose $a>0$ and $\delta>0$ small. At the
vertex $\sigma$ prescribe the first $Q$ coordinates to be
\[
 a+\delta\sum_{j=1}^k\epsilon_j\sigma_j.
\]
The sum of their $p$th powers is independent of $\sigma$, by
permuting the signs $\epsilon$; denote it by $C$. Writing
$\sigma_j=2\omega_j-1$, these are the vertices of an affine cube
with base coordinates $a-\delta\sum_j\epsilon_j$ and increments
$h_{j,\epsilon}=2\delta\epsilon_j$.

The derivative matrix in the base coordinates is
\[
 M_{\sigma,\epsilon}=p\left(a+\delta
       \sum_j\epsilon_j\sigma_j\right)^{p-1}.
\]
The Walsh character of a subset $S$ is an eigenvector, with eigenvalue
\[
 \lambda_S=p\sum_{\epsilon\in\{-1,1\}^k}
 \left(\prod_{j\in S}\epsilon_j\right)
 \left(a+\delta\sum_j\epsilon_j\right)^{p-1}.
\]
For $s=|S|$ its leading term as $\delta\to0$ is
\[
 \lambda_S=2^k p(p-1)\cdots(p-s)a^{p-1-s}\delta^s
                 +O(\delta^{s+1}).
\]
All these leading coefficients are nonzero under the stated
assumption. Thus $M$ is invertible for sufficiently small fixed
$\delta>0$.

Choose positive numbers $c_1,\ldots,c_k$ whose subset sums
$s_\omega=\sum_j\omega_jc_j$ are pairwise distinct, for example
$c_j=2^{j-1}$. Introduce a small positive weight $\beta$ in a final
coordinate, whose base coordinate is $t>0$ and whose increments are
$c_j$. The implicit function theorem solves the $Q$ equations
\[
 \sum_\epsilon\left(Y_\epsilon(t,\beta)
                  +2\delta\sum_j\omega_j\epsilon_j\right)^p
                  +\beta(t+s_\omega)^p=C
\]
near any fixed $t_0>0$, with
\[
 Y_\epsilon(t,0)=a-\delta\sum_j\epsilon_j.
\]
The solution is jointly analytic in $(t,\beta)$ and
\[
 Y(t,\beta)=Y(t,0)-\beta M^{-1} f(t)+O(\beta^2),\qquad
 f_\omega(t)=(t+s_\omega)^p,
\]
with the vertex rows permuted if necessary. The error is analytic,
so it can be differentiated any fixed number of times.
For the curve $\gamma_\beta(t)=(Y(t,\beta),t)$, expansion along the
last coordinate gives
\[
 \det(\gamma_\beta',\ldots,\gamma_\beta^{(Q+1)})
 =\pm(-\beta)^Q(\det M)^{-1}
   \det\bigl(f^{(2)}(t),\ldots,f^{(Q+1)}(t)\bigr)
   +O(\beta^{Q+1}).
\]
Put $x_\omega=t+s_\omega$. The determinant on the right equals,
up to the harmless sign from reversing columns,
\[
 \left(\prod_{m=2}^{Q+1}p(p-1)\cdots(p-m+1)\right)
 \left(\prod_\omega x_\omega^{p-Q-1}\right)
 \prod_{\omega<\omega'}(x_{\omega'}-x_\omega).
\]
It is nonzero. Consequently the curve has nonzero torsion for every
sufficiently small $\beta>0$. Its defining gradients are independent
because the first $Q$ columns remain invertible. All its cube
vertices are positive.

Finally apply the fixed invertible diagonal map multiplying the
last coordinate by $\beta^{1/p}$, then scale every coordinate by
$C^{-1/p}$. This converts the weighted equal-level equations to
ordinary unit $\ell^p$ sphere equations. Nonzero affine torsion and
regularity are preserved. This proves the lemma.
\end{proof}

\begin{remark}[A positive regular incidence exists for all admissible exponents]
The Walsh eigenvalue calculation needs only
$p(p-1)\cdots(p-k)\ne0$, which holds for every admissible $p$.
For any such $p$, add a last coordinate equal to one, with zero increments and unit
weight, and scale the common level to one. The first $Q$
normal columns are still independent. Thus the regular positive exact
incidence is nonempty for every exponent used below, including the
integers where the displayed leading torsion coefficient vanishes.
Every such regular point has a neighborhood of dimension $kd+1$;
the rank-defect bound ensures that this neighborhood contains a
positive full-rank Cramer chart.
\end{remark}

\subsection{Integer exponents}%\label{sec:integer-triangular-witness}

\begin{lemma}[Integer exponents]
\label{lem:integer-triangular-witness}
For every $k\ge1$ and every integer $p\ge k+2$, there is a signed
witness satisfying Proposition~\ref{prop:single-witness-transfer} in
$d=2^k+1$ coordinates.
\end{lemma}

\begin{proof}
Index $Q-1$ coordinates by the nonempty subsets $S\subseteq[k]$,
and use two more coordinates denoted by $u,v$. Set
$d_S=p-|S|\ge2$. Choose positive numbers $\beta_1,\ldots,\beta_k$
so that
\[
 R_S:=\Big(\prod_{j\in S}\beta_j\Big)^{1/d_S}>3
 \quad(S\ne\emptyset)
\]
are pairwise distinct. For an entirely explicit choice, put
$M=2p+1$, $B=3^p$, and
$\beta_j=B^{M^{j-1}}$. If $R_S=R_T$, then
\[
 d_T\sum_{j\in S}M^{j-1}
   -d_S\sum_{j\in T}M^{j-1}=0.
\]
The coefficients in this integer relation have absolute value at most
$p$, and the largest nonzero term exceeds the sum of all smaller
terms because $M-1>p$. Hence all coefficients vanish, forcing
$S=T$. The lower bound $R_S>3$ is immediate.
Write $\beta_S=\prod_{j\in S}\beta_j$.

Define $k$ increment-direction vectors by
\[
 a^{(j)}_S=\mathbf1_{j\in S},\qquad
 a^{(j)}_u=1,\qquad a^{(j)}_v=\beta_j.
\]
For a parameter $\varepsilon>0$, give the coordinates the weights
\[
 b_S=\varepsilon^{-d_S},\qquad b_u=1,\qquad b_v=-1.
\]
First consider the homogeneous finite-difference limit equations
\begin{equation}\label{eq:integer-triangular-model}
 H_R(y):=\sum_i b_i\Big(\prod_{j\in R}a^{(j)}_i\Big)
                 y_i^{p-|R|}=0
       \quad(\emptyset\ne R\subseteq[k]),
 \qquad P_b(y):=\sum_i b_i y_i^p=1.
\end{equation}
Before imposing the last normalization, set
$z_S=\varepsilon X_S$, $z_u=t$, and $z_v=1$. The nonempty equations
become the triangular system
\[
 X_R^{d_R}+
 \sum_{S\supsetneq R}\varepsilon^{|S|-|R|}X_S^{d_R}
 +t^{d_R}-\beta_R=0.
\]
At $\varepsilon=0$ its positive solution on $3/2<t<5/2$ is
\[
 X_R(t,0)=(\beta_R-t^{d_R})^{1/d_R}.
\]
The derivative with respect to the $X_R$ variables is diagonal and
invertible there. The analytic implicit-function theorem therefore
supplies positive real-analytic solutions $X_R(t,\varepsilon)$ near
any compact subinterval when $\varepsilon$ is sufficiently small.
Put
\[
 K_\varepsilon(t)=
 \left(t^p-1+\sum_{S\ne\emptyset}
                  \varepsilon^{|S|}X_S(t,\varepsilon)^p\right)^{1/p}.
\]
This is positive on the same interval for sufficiently small
$\varepsilon$. By homogeneity, $y=z/K_\varepsilon$ solves
\eqref{eq:integer-triangular-model}. After the invertible diagonal
change of coordinates that divides the subset coordinates by
$\varepsilon$, the curve is
\[
 \gamma_\varepsilon(t)=
 \frac{((X_S(t,\varepsilon))_{S\ne\emptyset},t,1)}
      {K_\varepsilon(t)}.
\]
As $\varepsilon\to0$, it converges in $C^m$ on compact subintervals
for every fixed nonnegative integer $m$ to
\[
 \gamma_0(t)=
 \frac{(((\beta_S-t^{d_S})^{1/d_S})_{S\ne\emptyset},t,1)}
      {(t^p-1)^{1/p}}.
\]

We claim that this limit curve lies in no affine hyperplane. It is
enough to prove linear independence, over $\mathbb C$, of the analytic
germs
\[
 1,\quad t,\quad K_0(t)=(t^p-1)^{1/p},\quad
 X_S(t,0)=(\beta_S-t^{d_S})^{1/d_S}
      \quad(S\ne\emptyset).
\]
The branch points of $K_0$ lie on the unit circle. Those of $X_S$
lie on $|t|=R_S$, and these circles are pairwise distinct. Suppose a
linear relation of these germs holds. Continue it from the real
interval to a neighborhood of a zero of $\beta_S-t^{d_S}$, avoiding
all the finitely many other branch points, and continue once around
a small loop about that zero. This multiplies $X_S$ by
$e^{2\pi i/d_S}\ne1$ and fixes all other germs. Subtracting the
original relation forces the coefficient of $X_S$ to vanish. Doing
this for every $S$, and then around one zero of $t^p-1$, forces the
coefficient of $K_0$ to vanish. The remaining affine polynomial in
$t$ has both coefficients zero. This proves the claim.

For real-analytic functions, linear independence implies that their
Wronskian is not identically zero; see \cite{BostanDumas2010}. Applied to the constant function
and the $d$ coordinates of $\gamma_0$, it follows that
\[
 \det(\gamma_0'(t_*),\ldots,\gamma_0^{(d)}(t_*))\ne0
\]
for some real $t_*\in(3/2,5/2)$. Here is an elementary justification of the analytic Wronskian fact.
In any finite-dimensional space of independent analytic germs, constant
linear changes of basis eliminate successive leading Taylor terms
and give a basis $g_j(t)=c_j(t-t_0)^{m_j}+\cdots$, where
$c_j\ne0$ and the nonnegative integers $m_j$ are distinct.
The leading Wronskian coefficient is the product of the $c_j$ times
the determinant of the falling factorials $(m_j)_r$,
$r=0,\ldots,d$. That determinant is the Vandermonde product in the
distinct $m_j$ and is nonzero. Hence the Wronskian cannot vanish
identically. The Wronskian of $1$ and the coordinates of $\gamma_0$
expands along its constant column to precisely the displayed torsion. The real-analytic identity
theorem ensures that a nonzero Wronskian cannot vanish on the entire
real interval. The determinant remains nonzero for
$\gamma_\varepsilon$ when $\varepsilon>0$ is sufficiently small.
Thus the original curve $y=z/K_\varepsilon$ has nonzero affine torsion.

The nonempty gradients in \eqref{eq:integer-triangular-model} have
rank $Q-1$: the submatrix using the subset coordinates is triangular
in decreasing subset size, with nonzero diagonal entries.
Euler's identity gives $\nabla H_R(y)\cdot y=0$ at a solution,
whereas $\nabla P_b(y)\cdot y=p$. Thus all $Q$ defining gradients
are independent, and this is a regular curve.

Finally, fix such a sufficiently small $\varepsilon>0$ and deform
from the limit equations to genuine translated shells. For nonempty
$R$, replace $H_R$ by
\[
 H_{R,\delta}(y)=
 \frac{\displaystyle\sum_{T\subseteq R}(-1)^{|R|-|T|}
    P_b\Big(y+\delta\sum_{j\in T}a^{(j)}\Big)}
      {(p)_{|R|}\delta^{|R|}},
 \qquad (p)_m=p(p-1)\cdots(p-m+1).
\]
These are polynomial functions, hence analytic, in $y$ and $\delta$ after
removing the displayed removable singularity, and
$H_{R,0}=H_R$. Keep the equation $P_b(y)=1$. Choose a linear functional $\ell$
with $\ell(y'(t_*))\ne0$ and adjoin the equation $\ell(y)=s$.
The analytic implicit-function theorem gives a jointly analytic
solution $y=y(s,\delta)$ near
$(s,\delta)=(\ell(y(t_*)),0)$. At $\delta=0$, this is a regular
reparameterization of the curve above. Thus the nearby regular
curves converge in $C^d$ as $\delta\to0$, and their torsion remains
nonzero at the corresponding point for sufficiently small $\delta$. Vanishing of all the finite differences is
equivalent, by Boolean inversion, to
\[
 P_b\Big(y+\delta\sum_j\omega_j a^{(j)}\Big)=1
          \quad(\omega\in\{0,1\}^k).
\]
All vertex coordinates are positive if $\delta$ is sufficiently
small. This is the required signed witness.
\end{proof}

\begin{theorem}[Witnesses in every progression length]
\label{thm:all-lengths-witness}
For every $k\ge1$ and every
$p\in(1,\infty)\setminus\{2,\ldots,k+1\}$, the positive unit
$\ell^p$ sphere in $\mathbb R^{2^k+1}$ has a regular exact $k$-cube
whose fixed-increment intersection has nonzero affine torsion.
Moreover, this property holds at almost every regular positive exact
configuration, with respect to ordinary coarea level measure.
\end{theorem}
\begin{proof}
If $p$ is noninteger, Lemma~\ref{lem:general-walsh-witness} gives a
positive witness directly. If $p$ is an integer, admissibility says
$p\ge k+2$; Lemma~\ref{lem:integer-triangular-witness} and
Proposition~\ref{prop:single-witness-transfer} give the almost-everywhere
positive conclusion. The regular positive incidence is nonempty by
the remark following the Walsh construction, so this also gives an
actual positive witness. In both cases the polynomial transfer in
Proposition~\ref{prop:single-witness-transfer} gives the asserted
almost-everywhere statement.
\end{proof}

\section{Fourier localization and uniform shell comparison}
%\label{sec:fourier-localization}

\begin{lemma}[Fourier convergence on almost-everywhere good curves]
%\label{lem:weak-cube-Fourier-localization}
Let $k\ge1$, $Q=2^k$, and $d=Q+1$. Suppose the positive mixed
$k$-cube shell measures converge weakly and in mass to $M$.
Suppose $M$-almost every exact configuration has a regular
fixed-increment fiber with nonzero affine torsion. Then
\[
 \int\sup_{\xi\in\mathbb R^d}
 \left|\int e^{-2\pi i\xi\cdot y}
 \prod_{\omega\in\{0,1\}^k}(S_\eta-S_\tau)(y+\omega\cdot h)
 \,dy\right|\,dh\longrightarrow0
 \qquad(0<\eta<\tau\downarrow0).
\]
\end{lemma}
\begin{proof}
Let $\gamma>0$. By finiteness and inner regularity of $M$, choose a smooth
$0\le\rho\le1$, supported in finitely many regular nonzero-torsion
charts, with $\int(1-\rho)\,dM<\gamma$. This is possible by first
taking a compact subset of the good locus of complement mass below
$\gamma$, then a smooth cutoff supported in the good open set.
A partition of unity decomposes $\rho$ into smooth amplitudes
compactly supported in these charts. Shrink the charts so torsion
stays bounded away from zero on their closures and on nearby levels.

In one chart the submersion coordinates are $y=Y(t,s,h)$, with
$t\in\mathbb R$ and $s=(P(y+\omega\cdot h)-1)_\omega\in\mathbb R^Q$.
The Fourier transform of its localized level measure is
\[
 J(s,h,\xi)=\int e^{-2\pi i\xi\cdot Y(t,s,h)}a(t,s,h)\,dt,
\]
where $a$ is smooth and compactly supported. It is jointly continuous.
The vectors $\partial_t^jY$, $1\le j\le d$, span $\mathbb R^d$
uniformly on the compact chart. Therefore, for every unit direction
$\theta$, at least one $|\theta\cdot\partial_t^jY|$ has a fixed
positive lower bound. A finite smooth partition in the chart variables and
in $\theta$, using sufficiently small $t$-intervals, lets us fix such
a derivative throughout each integration interval. For $j\ge2$,
apply the one-dimensional van der Corput estimate
\cite[Chapter~VIII, Section~1]{Stein1993}; for $j=1$ integrate by parts,
using the uniformly bounded second derivative and amplitude derivatives.
It follows that
\[
 |J(s,h,\xi)|\le C(1+|\xi|)^{-1/d}
\]
uniformly in the relevant nearby levels and increments.

Set $e_{\eta,\tau}=\psi_\eta-\psi_\tau$. The localized signed-shell
Fourier transform is
\[
 \int J(s,h,\xi)\prod_\omega e_{\eta,\tau}(s_\omega)\,ds.
\]
Each factor has integral zero, $L^1$ norm at most two, and shrinking
support. The displayed decay controls large frequencies uniformly.
For bounded frequencies, subtract $J(0,h,\xi)$ and use uniform
continuity and cancellation. The resulting Fourier supremum tends
to zero uniformly in $h$, whose support is fixed and compact.
Thus the integral tends to zero on the part multiplied by $\rho$.

For the remaining part, discard the Fourier phase and expand the
absolute product. Its integrated Fourier supremum is at most
\[
 \sum_{\boldsymbol\epsilon\in\{\eta,\tau\}^{Q}}
       \int(1-\rho)\,dM_{\boldsymbol\epsilon}.
\]
Weak convergence and common compact support bound its limsup by
$2^Q\int(1-\rho)\,dM<2^Q\gamma$. Let $\gamma\downarrow0$.
\end{proof}

\begin{corollary}[The single-witness shell criterion]
\label{thm:general-single-witness}
For admissible $p$ and $d=2^{n-2}+1$, the existence of a signed
witness as in Proposition~\ref{prop:single-witness-transfer} implies
\[
 \mathfrak B_n(S_\eta-S_\tau)\longrightarrow0
 \qquad(0<\eta<\tau\downarrow0).
\]
Consequently the exact-to-convolution comparison holds uniformly for
unit-bounded inputs on a fixed bounded set and $0<\lambda\le1$.
\end{corollary}
\begin{proof}
Proposition~\ref{prop:regular-cube-mass} identifies the limiting
cube measure with ordinary coarea measure on the full regular
incidence. Proposition~\ref{prop:single-witness-transfer} makes its
fixed-increment fibers regular with nonzero torsion almost everywhere.
The preceding lemma is exactly the convergence of the $Q$th power
of \eqref{eq:B-seminorm}, since the shells are real. Apply
Corollary~\ref{cor:bs-shell-criterion} for the final assertion.
\end{proof}

\section{A finite-box bridge using radial regularization}
\label{sec:radial-bridge}

This section gives the passage from a uniform approximation
estimate for the exact spherical count to the conclusion for every
sufficiently large radius.  It applies to a measure supported on an arbitrarily small open cap.

Write $\rho(y)=\|y\|_{\ell^p}$, $V_p=|\{y:\rho(y)<1\}|$, and
$S_p=\{y:\rho(y)=1\}$.  Let $\kappa_p$ be the probability cone measure,
normalized by the polar-coordinate formula
\[
 \int_{\mathbb R^d}F(y)\,dy
 =dV_p\int_{S_p}\int_0^\infty
       F(s\omega)s^{d-1}\,ds\,d\kappa_p(\omega).
\]
Throughout this section let $\mu=a\,\kappa_p$ be a probability measure,
where $a$ is nonnegative and continuous and is strictly positive on
some relatively open subset of $S_p$.  A smooth, nonzero, nonnegative
cap cutoff satisfies this hypothesis.  For a measurable set $B$ of
finite measure define
\[
 C_\lambda(B)=\int_{S_p}\int_{\mathbb R^d}
       \prod_{i=0}^{n-1}1_B(x+i\lambda\omega)\,dx\,d\mu(\omega).
\]
The continuity of translations in $L^1$ shows that the integrated
correlation is continuous in its gap vector; hence the spherical and
radial integrals below are well defined.

We use the following uniform approximation hypothesis.  For a fixed
smooth compactly supported probability density $\varphi$, there is a
function $u(\varepsilon)\to0$ as $\varepsilon\downarrow0$ such that,
for every $R>0$, every translate $Q$ of $[0,R]^d$, every measurable
$B\subset Q$, and every $0<\lambda\le R$,
\begin{equation}
 \left|C_\lambda(B)-
 \int_{\mathbb R^d}\int_{\mathbb R^d}
 \prod_{i=0}^{n-1}1_B(x+iy)
 (\mu_\lambda*\varphi_{\varepsilon\lambda})(y)\,dy\,dx\right|
 \le R^d u(\varepsilon).
 \label{eq:radial-uniform-input}
\end{equation}
\begin{lemma}[Uniform continuity in relative radius]
\label{lem:radial-continuity}
Under \eqref{eq:radial-uniform-input}, for every $\tau>0$ there is an
$\eta\in(0,1/4)$ such that
\[
 |C_{s\lambda}(B)-C_\lambda(B)|\le\tau R^d
 \qquad(1-\eta\le s\le1+\eta)
\]
whenever $B\subset Q$, $Q$ is a cube of side $R$, and
$(1+\eta)\lambda\le R$.
\end{lemma}

\begin{proof}
Set $K_\varepsilon=\mu*\varphi_\varepsilon$ and use mass-preserving
dilations $(K_\varepsilon)_s(y)=s^{-d}K_\varepsilon(y/s)$.
Applying \eqref{eq:radial-uniform-input} at $\lambda$ and $s\lambda$
and observing that the integrated correlation is bounded by $|B|$ gives
\[
 |C_{s\lambda}(B)-C_\lambda(B)|
 \le 2R^d u(\varepsilon)
       +|B|\,\|(K_\varepsilon)_s-K_\varepsilon\|_1.
\]
Choose $\varepsilon$ so that $2u(\varepsilon)<\tau/2$.
Continuity of dilation in $L^1$, with this $\varepsilon$ fixed, then
gives the assertion, since $|B|\le R^d$.
\end{proof}

\begin{lemma}[A coarse count in an angular cap]
\label{lem:radial-cone-varnavides}
For every $\delta\in(0,1]$ there are constants $c>0$ and $L_0\ge1$,
depending only on $n,d,p,\mu,\delta$, such that
\begin{equation}
 P_\lambda(B):=\int_0^1d s^{d-1}C_{s\lambda}(B)\,ds
 \ge cR^d
 \label{eq:radial-coarse-positive}
\end{equation}
for every cube $Q$ of side $R$, every measurable $B\subset Q$ with
$|B|\ge\delta R^d$, and every $0<\lambda\le R/L_0$.
\end{lemma}

\begin{proof}
Polar coordinates identify $P_\lambda(B)$ with the progression count
whose gap kernel is the dilation of
\[
 K_0(y)=V_p^{-1}a(y/\rho(y))1_{\{0<\rho(y)<1\}}.
\]
Choose a point in the open set where $a>0$.  By taking a sufficiently
small closed cone around the corresponding ray, there exist an
invertible real matrix $T$ and a number $\gamma>0$ such that
\begin{equation}
 a(Tv/\rho(Tv))\ge\gamma
 \quad\text{for every }v\in[0,\infty)^d\setminus\{0\}.
 \label{eq:radial-positive-cone}
\end{equation}
Indeed, choose the $d$ linearly independent columns of $T$ sufficiently
close to a common nonzero vector on the selected ray.  Every nonzero
nonnegative linear combination, after division by the sum of its
coefficients, lies in the convex hull of these columns.  Radial
projection of that convex hull stays inside the set where
$a\ge\gamma$ if the columns were chosen sufficiently close.

Put
\[
 M=\max_{v\in[0,1]^d}\rho(Tv),\qquad
 D=\operatorname{diam}(T[0,1]^d),\qquad
 \ell=\lambda/M,\qquad v_\lambda=\ell^d|\det T|.
\]
Tile $\mathbb R^d$ by the half-open parallelepipeds
$P_k=\ell T(k+[0,1)^d)$, $k\in\mathbb Z^d$.  Each has measure
$v_\lambda$ and Euclidean diameter $D\ell$.
Retain those cells that are contained in $Q$, disregarding boundaries
of measure zero.  Their union contains the points of $Q$ at distance
greater than $D\ell$ from its boundary.  Consequently the measure
discarded from $Q$ is at most
\[
 2dD\ell R^{d-1}
\]
when $R\ge2D\ell$.  Taking, for example,
\[
 L_0=\max\{1,4dD/(M\delta)\}
\]
ensures that the discarded measure is at most $\delta R^d/2$.
The mass of $B$ in retained cells is therefore at least
$\delta R^d/2$.

Call a retained cell dense when $|B\cap P_k|\ge(\delta/4)|P_k|$.
The other retained cells contain at most $\delta R^d/4$ of $B$.
Hence the union of the dense cells has measure at least
$\delta R^d/4$.

Let $q=q(n,d,\delta/4)>0$ be the continuous Varnavides lower bound
on the unit cube: every $E\subset[0,1]^d$ of measure at least
$\delta/4$ satisfies
\[
 \int_{[0,1]^d}\int_{[0,1]^d}
 \prod_{i=0}^{n-1}1_E(u+iv)\,dv\,du\ge q,
\]
where $1_E$ is extended by zero. This is
\cite[Proposition~1]{DurcikKovac2020}, a continuous form of
Szemer\'edi's theorem \cite{Szemeredi1975} obtained by the averaging
argument of Varnavides \cite{Varnavides1959}.
Apply it on a dense cell using
$u\mapsto\ell T(k+u)$.  The resulting contribution, with starting
point in that cell and gap in $\ell T[0,1]^d$, is at least
$qv_\lambda^2$.  The Jacobian is $\ell^d|\det T|$ in each of
the starting-point and gap variables.

For almost every $y\in\ell T[0,1]^d$ one has $\rho(y)<\lambda$,
and \eqref{eq:radial-positive-cone} gives
$(K_0)_\lambda(y)\ge\gamma/(V_p\lambda^d)$.
The exceptional boundary is immaterial.  Summing the preceding
Varnavides contributions is legitimate because their starting-point
cells are disjoint and each restricted indicator is bounded by $1_B$.
It follows that
\[
 P_\lambda(B)
 \ge\frac{\gamma q}{V_p\lambda^d}
        v_\lambda\frac{\delta R^d}{4}
 =\frac{\gamma q|\det T|\delta}{4V_pM^d}R^d.
\]
This proves \eqref{eq:radial-coarse-positive}.
\end{proof}

We next record exactly the consequence of ray variation used in the
finite-box argument.  Set $m=n-1$ and
\[
 M_t^\omega(f_1,\ldots,f_m)(x)
   =\frac1t\int_0^t\prod_{i=1}^m f_i(x+iu\omega)\,du.
\]
For $f_i\in L^{2(n-1)}(\mathbb R^d)$, $1\le i\le n-1$,
the continuous ray estimate is
\begin{equation}
 \left(\sum_{k=0}^{N-1}
 \|M_{t_{k+1}}^\omega-M_{t_k}^\omega\|_2^r\right)^{1/r}
 \le C_{n,r}\prod_{i=1}^{n-1}\|f_i\|_{2(n-1)}
 \quad(0<t_0<\cdots<t_N).
 \label{eq:radial-ray-variation}
\end{equation}
Here one may take $r=2$ when $n=3$ and any $r>2^{n-2}$ when
$n\ge4$.  These are the exponents for $n-1$ moving factors; the
unshifted factor will be paired with the output in $L^2$.
Here is the complete passage from discrete to continuous time.
Fix $h>0$ and use the commuting transformations
$T_i x=x+ih\omega$, $1\le i\le n-1$, on the $\sigma$-finite
space $(\mathbb R^d,dx)$. Theorem~1.2 of
\cite{VanDoornEtAl2026} gives the same bound for the averages
$N^{-1}\sum_{j=0}^{N-1}\prod_i f_i(\cdot+ijh\omega)$,
uniformly in $h$ and in the direction.
Put $H_s=\prod_{i=1}^{n-1}f_i(\cdot+is\omega)$. Telescoping and
H\"older give
\[
 \|H_s-H_t\|_2\le\sum_{i=1}^{n-1}
 \|f_i(\cdot+is\omega)-f_i(\cdot+it\omega)\|_{2(n-1)}
 \prod_{j\ne i}\|f_j\|_{2(n-1)}.
\]
Strong continuity of translations makes $s\mapsto H_s$ continuous
in $L^2$, uniformly on compact time intervals. For any finite list
$0<t_0<\cdots<t_N$, the integers $N_j(h)=\lfloor t_j/h\rfloor$
are positive and strictly increasing for sufficiently small $h$.
The difference between the discrete average of length $N_j(h)$ and
$(hN_j(h))^{-1}\int_0^{hN_j(h)}H_s\,ds$ is bounded in $L^2$ by
\[
 \sup\{\|H_s-H_t\|_2:0\le s,t\le t_N+1,\ |s-t|\le h\},
\]
which tends to zero. Since $hN_j(h)\to t_j$, the integral averages
also tend to $M_{t_j}^\omega$ in $L^2$. Apply the discrete variation
bound to this finite list and pass to the limit in its finite sum.
This proves \eqref{eq:radial-ray-variation} with precisely the stated
exponents and constants.

\begin{lemma}[Radial profiles on separated scales]
\label{lem:radial-profile-error}
Let $0<\alpha<\beta<\infty$, and let $w$ be a real function of
bounded variation on $\mathbb R$, supported in $[\alpha,\beta]$,
with $\int w(s)\,ds=0$.  Define the finite signed measure
$d\nu(s)=-s\,dw(s)$, with the derivative taken on all of
$\mathbb R$, including endpoint jumps.  If
\[
 \beta\lambda_j<\alpha\lambda_{j+1}\qquad(1\le j<J),
\]
then, for every measurable $B$ of finite measure,
\begin{equation}
 \left(\sum_{j=1}^J
 \left|\int w(s)C_{s\lambda_j}(B)\,ds\right|^r\right)^{1/r}
 \le C_{n,r}\|\nu\|_{\mathrm{TV}}|B|.
 \label{eq:radial-profile-bound}
\end{equation}
\end{lemma}

\begin{proof}
Integration by parts gives $\nu(\mathbb R)=\int w=0$.  Apply
Bochner integration by parts to the locally absolutely continuous
$L^2$-valued function $s\mapsto sM_{s\lambda}^\omega$; its
derivative is $\prod_{i=1}^{n-1}1_B(\cdot+i\lambda s\omega)$.
It follows that
\[
 \int w(s)C_{s\lambda}(B)\,ds
 =\int_{S_p}\int
 \langle1_B,M_{s\lambda}^\omega-M_{\alpha\lambda}^\omega\rangle
 \,d\nu(s)\,d\mu(\omega).
\]
For every fixed $s\in[\alpha,\beta]$, the intervals
$[\alpha\lambda_j,s\lambda_j]$ are ordered and disjoint.  Include
all their endpoints in the partition in
\eqref{eq:radial-ray-variation}; the intervening increments only add
nonnegative summands.  Zero-length intervals may be omitted.
Minkowski's inequality in $\ell^r$, followed by the $L^2$ pairing
and \eqref{eq:radial-ray-variation}, proves
\[
 \left(\sum_j\left|\int w(s)C_{s\lambda_j}(B)\,ds\right|^r
 \right)^{1/r}
 \le C_{n,r}\|\nu\|_{\mathrm{TV}}
       \|1_B\|_2\prod_{i=1}^{n-1}\|1_B\|_{2(n-1)}.
\]
The product of norms on the right equals $|B|$.
\end{proof}

\begin{proposition}[Finite-list lemma with exact quantifiers]
\label{prop:radial-finite-list}
Assume \eqref{eq:radial-uniform-input} and
\eqref{eq:radial-ray-variation}.  For every $\delta\in(0,1]$ there
exist numbers $\Lambda>1$, $L\ge1$, and an integer $J_\delta\ge1$
with the following property.  For every list of $J_\delta$ positive
radii satisfying
\[
 \lambda_{j+1}>\Lambda\lambda_j\qquad(1\le j<J_\delta),
\]
every $R\ge L\lambda_{J_\delta}$, every translate $Q$ of
$[0,R]^d$, and every measurable $B\subset Q$ with
$|B|\ge\delta R^d$, at least one of the counts
$C_{\lambda_j}(B)$ is strictly positive.
All three constants may depend on $n,d,p,\mu,\varphi,\delta$, the
uniform modulus in \eqref{eq:radial-uniform-input}, and the fixed
admissible exponent $r$; they do not depend on the radii, $Q$, or $B$.
\end{proposition}

\begin{proof}
Take $c,L_0$ from Lemma~\ref{lem:radial-cone-varnavides}.
Choose $\alpha\in(0,1/4)$ with $2\alpha^d<c/4$, and put
\[
 w_\alpha(s)=\frac{d s^{d-1}}{1-\alpha^d}1_{[\alpha,1]}(s).
\]
This is a probability density, and its $L^1$ distance from
$d s^{d-1}1_{(0,1)}(s)$ equals $2\alpha^d$.
Since $0\le C_t(B)\le|B|\le R^d$, we obtain
\begin{equation}
 \int w_\alpha(s)C_{s\lambda}(B)\,ds\ge3cR^d/4
 \qquad(\lambda\le R/L_0).
 \label{eq:radial-truncated-lower}
\end{equation}
Choose $\eta\in(0,1/4)$ from
Lemma~\ref{lem:radial-continuity} with $\tau=c/4$, and choose a
smooth nonnegative probability density $w_\eta$ supported in
$[1-\eta,1+\eta]$.  Then
\begin{equation}
 \left|\int w_\eta(s)C_{s\lambda}(B)\,ds-C_\lambda(B)\right|
 \le cR^d/4
 \qquad((1+\eta)\lambda\le R).
 \label{eq:radial-thin-approximation}
\end{equation}
Set $\beta=1+\eta$, $\Lambda=\beta/\alpha$,
$L=\max\{L_0,\beta\}$, and $w=w_\eta-w_\alpha$.
The hypotheses of Lemma~\ref{lem:radial-profile-error} hold, with
$\nu=-s\,dw$.  Define
\[
 J_\delta=
 1+\left\lfloor
       \left(\frac{2C_{n,r}\|\nu\|_{\mathrm{TV}}}{c}\right)^r
       \right\rfloor.
\]
If all $C_{\lambda_j}(B)$ vanished, then
\eqref{eq:radial-truncated-lower} and
\eqref{eq:radial-thin-approximation} would imply
\[
 \left|\int w(s)C_{s\lambda_j}(B)\,ds\right|\ge cR^d/2
 \quad\text{for every }j.
\]
Equation~\eqref{eq:radial-profile-bound} would consequently give
$J_\delta^{1/r}cR^d/2
 \le C_{n,r}\|\nu\|_{\mathrm{TV}}R^d$, contradicting the definition
of $J_\delta$.
\end{proof}

\subsection{Completion of the proof}
\begin{proof}[Proof of Theorem~\ref{thm:power-main}]
Let $k=n-2$ and $d_0=2^k+1$. In dimension $d_0$,
Theorem~\ref{thm:all-lengths-witness} supplies the witness required
by Corollary~\ref{thm:general-single-witness}. Thus the
exact-to-convolution comparison has a modulus tending to zero,
uniformly on a unit input cube and for $0<\lambda\le1$.

Normalize the cap measure by $\mu=\sigma/\sigma(\mathbb R^{d_0})$.
Homogeneity and the cone-measure formula give
$d\sigma=(d_0V_p/p)\chi\,d\kappa_p$, so this $\mu$ has the
continuous nonnegative angular density required above, positive on
a nonempty open patch. Translating and rescaling an input cube of
side $R$ multiplies the counting forms by $R^{d_0}$ and changes
$\lambda$ to $\lambda/R$. Hence \eqref{eq:radial-uniform-input}
holds in dimension $d_0$, and the finite-list proposition applies.

To prove the result in any $d\ge d_0$, fix
$0<\delta<\overline\delta(A)$ and take the constants
$J_\delta,\Lambda,L$ of Proposition~\ref{prop:radial-finite-list}
in dimension $d_0$. If missing radii were unbounded, choose
$J_\delta$ of them with successive ratios larger than $\Lambda$.
Then choose a $d$-dimensional cube of side
$R\ge L\lambda_{J_\delta}$ whose relative density in $A$ exceeds
$\delta$. If $d>d_0$, Fubini supplies a measurable
$d_0$-dimensional coordinate slice of relative density at least
$\delta$; if $d=d_0$, use the cube itself.
The finite-list proposition gives a positive exact count at a listed
missing radius, a contradiction. Appending the fixed slice coordinates
to the progression points and zero coordinates to the gap preserves
membership in $A$ and the $\ell^p$ gap length. No fixed slice of
positive upper Banach density is required.
\end{proof}

\section*{AI disclosure}
The manuscript was generated gradually in the following three chats with OpenAI's models:
\begin{itemize}
\item \href{https://chatgpt.com/share/6aca569a-f04c-83eb-833c-81ab5822a32f}{ChatGPT-5.6 Sol, August 28 -- September 3, 2026},
\item \href{https://chatgpt.com/share/6aa15da8-6750-83ed-82e0-2f5418e7df7c}{ChatGPT-6 Astra, September 5--9, 2026},
\item \href{https://chatgpt.com/share/6aa15df7-db54-83ed-8f32-d0f944296ac0}{ChatGPT-6 Astra, September 9, 2026}.
\end{itemize}
The chats were mutually dependent via ChatGPT's internal library. They were initially directed towards the papers \cite{DurcikKovac2020} and \cite{VanDoornEtAl2026}, and then guided by generic prompts.
My intention is to digest and address the proof in a cooperative, non-competitive, time-insensitive environment.

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\end{document}