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\fancyhead[L]{\small Fourth-deck reconstruction}
\fancyhead[R]{\small Babanskyy}
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\title{Binary Reconstruction from the Fourth Deck\\[4pt]
\large Torsion Phases, Local Gluing, and Sign Extinction}
\author{Oleksiy Babanskyy}
\date{}

\hypersetup{
  pdftitle={Binary Reconstruction from the Fourth Deck: Torsion Phases, Local Gluing, and Sign Extinction},
  pdfauthor={Oleksiy Babanskyy},
  pdfsubject={Binary reconstruction from the fourth deck}
}
\hypersetup{pdflang={en-US}}

\begin{document}

\maketitle

\begin{abstract}
We give a candidate proof of the following statement: for every finite
cyclic group \(C_n\), the cumulative fourth deck of a binary subset
determines it up to translation. Hence \(r_{\rm set}(C_n)\le4\) for every
\(n\), and, using the Keleti--Kolountzakis Corollary 1.1 for the lower
bound, \(r_{\rm set}(C_n)=4\) for every even \(n>10\). The proof has four
components: finiteness of the binary fibre, cyclotomic torsion of the
phases, gluing of primary characters, and extinction of dyadic residues by
two odd-denominator projectors. The bridge between ordered analytic
correlations and Pebody's cumulative deck is proved explicitly, including
stabilizers, zero and repeated offsets, and the empty pattern. The text
contains the complete mathematical chain; finite checks are included only
as convention and implementation checks, not as premises of the uniform
proof.
\end{abstract}

\tableofcontents

\section{Statement and conventions}

Pebody's Conjecture~8.1 \cite[p.~514]{Pebody2007} asserts that
\(r(C)=4\) for every cyclic group of even order \(|C|>10\).
His set-reconstruction number \(r(C)\) is denoted here by
\(r_{\rm set}(C_n)\). The candidate proof establishes the upper bound
for every cyclic order; the matching even-order lower bound is imported
from Keleti--Kolountzakis \cite{KK2006}.

The strategy combines the correspondence between the cumulative deck and
ordered correlations, cyclotomic torsion of the phases, gluing of primary
characters, and extinction of dyadic residues by odd-denominator
projectors. The four interfaces are:

\begin{enumerate}[label=(\roman*)]
\item ordered analytic deck versus cumulative deck;
\item gluing characters across distinct complete tags;
\item lifting in exact-order shells, including \(p=3\);
\item frequency-by-frequency verification of the projector identities.
\end{enumerate}

Let \(G\) be a finite abelian group written additively. For \(k\ge1\), set
\[
 D_k f(t_1,\ldots,t_{k-1})
   =\sum_{x\in G} f(x)\prod_{j=1}^{k-1}f(x+t_j),
\]
where the offsets are ordered and may be zero or repeated. For \(k=1\),
\(D_1f=\sum_xf(x)\).

\begin{definition}[Pebody's cumulative deck]
For \(A\subseteq G\), the cumulative \(k\)-deck is the multiset of
translation classes \([T]\) of all subsets \(T\subseteq A\) with
\(|T|\le k\). The multiplicity of \([T]\) is denoted by
\(\nu_A([T])\). Two sets are \(k\)-isomorphic if they have the same deck.
\end{definition}

We do not quotient by reflection. This is Pebody's convention
\cite{Pebody2007}, and it agrees with that of Keleti--Kolountzakis
\cite{KK2006}. Let \(r_{\rm set}(G)\) be the least \(k\ge1\) for which this
deck determines every subset of \(G\) up to translation.

\begin{lemma}[Ordered--cumulative bridge]\label{lem:deckbridge}
Let \(A\subseteq G\), let \(T\subseteq G\) with \(|T|\le4\), and set
\[
\Stab(T)=\{h\in G:h+T=T\},\qquad
N_A(T)=\#\{x\in G:x+T\subseteq A\}.
\]
Then
\[
N_A(T)=|\Stab(T)|\,\nu_A([T]). \tag{1.1}
\]
Moreover, for \(\mathbf t=(t_1,\ldots,t_{j-1})\) and
\(T_{\mathbf t}=\{0,t_1,\ldots,t_{j-1}\}\),
\[
D_j\one_A(\mathbf t)=N_A(T_{\mathbf t}). \tag{1.2}
\]
Thus two indicators have the same cumulative deck through order four if
and only if they have the same \(D_j\), \(1\le j\le4\).
\end{lemma}

\begin{proof}
Every distinct copy \(B\) in the class \([T]\) has the form \(B=x+T\).
If \(x_0+T=B\), then \(x+T=B\) exactly when
\(x-x_0\in\Stab(T)\). Hence every copy is counted exactly
\(|\Stab(T)|\) times in \(N_A(T)\), which proves (1.1). Formula (1.2) is
the definition of \(D_j\), because the indicator of the pattern
\(T_{\mathbf t}\) is one exactly when its translate with origin \(x\) is
contained in \(A\).

Zeros and repetitions in the tuple merely reduce \(T_{\mathbf t}\) to a
pattern with fewer points; they introduce no extra factors. For the empty
pattern, \(N_A(\varnothing)=|G|\), \(\Stab(\varnothing)=G\), and the empty
pattern has multiplicity one in the deck, so (1.1) holds there as well.
The converse follows by choosing, for each nonempty pattern \(T\), a tuple
that normalizes it with one point at zero and applying (1.1); the
multiplicity of the empty pattern is independent of \(A\).
\end{proof}

\begin{remark}
For indicators, \(D_4\) alone determines \(D_1,D_2,D_3\): set the excess
offsets to zero and use \(\one_A^j=\one_A\). This does not hold for
arbitrary real competitors; for them we retain the cumulative data
explicitly.
\end{remark}

\begin{theorem}[Reconstruction from the fourth deck]\label{thm:main}
For every \(n\ge1\), two subsets \(A,B\subseteq C_n\) with the same fourth
deck are translates. In particular
\[
r_{\rm set}(C_n)\le4 \qquad(n\ge1). \tag{1.3}
\]
For every even integer \(n>10\),
\[
\boxed{r_{\rm set}(C_n)=4.} \tag{1.4}
\]
\end{theorem}

\begin{remark}[Lower-bound locator]
KK06 Corollary 1.1, p.~5, states that the 3-deck reconstructs every cyclic
set if and only if \(n\) is a power of an odd prime, a product of at most
three odd primes, or \(n\in\{2,4,6,8,10\}\). Hence for every even
\(n>10\) there is a non-translate pair with the same 3-deck, and
\(r_{\rm set}(C_n)\ge4\). Theorem 2.22, p.~12, gives the explicit
construction for \(n=2k\), where \(k\ge6\) is an integer, and therefore
also supplies this lower bound for every even \(n>10\).
\end{remark}

\section{Fourier analysis and finiteness of the binary fibre}

Let \(\widehat G\) be the dual group of \(G\), and set
\[
\widehat f(\xi)=\sum_{x\in G}f(x)\overline{\xi(x)}.
\]
The transform in all offsets is
\[
\widehat{D_kf}(\xi_1,\ldots,\xi_{k-1})
=\widehat f(-\xi_1-\cdots-\xi_{k-1})
 \prod_{j=1}^{k-1}\widehat f(\xi_j). \tag{2.1}
\]

From here through the final proof assume \(A\ne\varnothing\). If
\(f=\one_A\) and \(g=\one_B\) have the same \(D_1,\ldots,D_4\), then
\(D_2\) gives
\[
\supp\widehat f=\supp\widehat g=:S,\qquad
|\widehat f(\xi)|=|\widehat g(\xi)|.
\]
For \(\xi\in S\), define
\[
\rho(\xi)=\frac{\widehat g(\xi)}{\widehat f(\xi)}\in\mathbb S^1.
\]
Reality and positive mass give
\[
\rho(0)=1,\qquad \rho(-\xi)=\overline{\rho(\xi)}. \tag{2.2}
\]
Formula (2.1) also implies
\[
\prod_{j=1}^{k}\rho(\xi_j)=1
\quad\text{if}\quad
\xi_1+\cdots+\xi_k=0,\quad
\xi_j\in S,\quad 1\le k\le4. \tag{2.3}
\]

\begin{lemma}[Binary moment rigidity]\label{lem:binary}
If \(f:G\to\{0,1\}\) and \(u:G\to\mathbb R\) have the same moments of
orders \(2,3,4\), then \(u\) is binary.
\end{lemma}

\begin{proof}
Equality of the diagonal moments gives
\[
\sum_xu(x)^2(u(x)-1)^2
=\sum_xu(x)^4-2\sum_xu(x)^3+\sum_xu(x)^2=0.
\]
Every summand is nonnegative and vanishes precisely when \(u(x)=0\) or
\(u(x)=1\).
\end{proof}

\begin{theorem}[Finiteness of the phase fibre]\label{thm:fiber}
The set \(\mathcal R_4(S)\) of maps \(\rho:S\to\mathbb S^1\) satisfying
(2.2)--(2.3) is a finite abelian group. Every value \(\rho(\xi)\) is a
root of unity.
\end{theorem}

\begin{proof}
For \(\rho\in\mathcal R_4(S)\), define \(\widehat u=\rho\widehat f\) on
\(S\) and zero outside \(S\). The opposite-frequency relations make \(u\)
real. Relations (2.3) and (2.1) give the same \(D_k\), \(k\le4\), for \(u\)
and \(f\). Lemma~\ref{lem:binary} makes \(u\) binary, and the mass places
it among the subsets of cardinality \(|A|\). Exactness of \(S\) makes
\(\rho\mapsto u\) injective, so the fibre is finite. Pointwise products and
inverses preserve (2.2)--(2.3), hence the fibre is a finite group and every
element has finite order.
\end{proof}

\section{Local cyclotomic characters}

Let \(\xi\in S\) have order \(d\). Since \(f\) and \(g\) are rational,
\(\widehat f(\xi),\widehat g(\xi)\in\mathbb Q(\zeta_d)\), and the same
holds for \(\rho(\xi)\). Galois covariance is
\[
\rho(u\xi)=\sigma_u(\rho(\xi)),\qquad \gcd(u,d)=1, \tag{3.1}
\]
where \(\sigma_u(\zeta_d)=\zeta_d^u\).

\begin{proposition}[Character on the generator orbit]\label{prop:local}
For \(\xi\in S\) of order \(d\),
\[
\rho(\xi)^d=1,\qquad
\rho(u\xi)=\rho(\xi)^u\quad(\gcd(u,d)=1). \tag{3.2}
\]
Consequently, the restriction of \(\rho\) to the generator shell of
\(\langle\xi\rangle\) is the restriction of a character of
\(\langle\xi\rangle\). The statement also holds when \(d\) is divisible by
three.
\end{proposition}

\begin{proof}
Theorem~\ref{thm:fiber} makes \(\rho(\xi)\) a root of unity. The roots of
unity in \(\mathbb Q(\zeta_d)\) are
\(\mu_d\) when \(d\) is even and \(\mu_{2d}\) when \(d\) is odd
\cite{Conrad}. In the even case, \(\rho(\xi)^d=1\) follows immediately.

Now let \(d>1\) be odd. Write
\(\rho(\xi)=\varepsilon z\), with \(\varepsilon\in\{\pm1\}\) and
\(z\in\mu_d\). Since \(2\) is coprime to \(d\), \(-2\xi\in S\) by
Galois covariance. The three-term instance of (2.3) on
\((\xi,\xi,-2\xi)\) gives
\(\rho(2\xi)=\rho(\xi)^2=z^2\). On the other hand, (3.1) gives
\(\rho(2\xi)=\varepsilon z^2\), hence \(\varepsilon=1\). The case \(d=1\)
is the relation \(\rho(0)=1\). Thus (3.1) becomes (3.2), and
\(\rho(\xi)\) defines the character \(k\xi\mapsto\rho(\xi)^k\).
\end{proof}

\section{Galois boxes and primary gluing}

Write, via CRT,
\[
G=C_{2^r}\times H,\qquad
H=\prod_{p\mid |H|}C_{p^{a_p}},\qquad |H|\ \text{odd}.
\]
We allow \(r=0\) and \(H=\{0\}\). In the dual group, \(E_i\) is the shell
of elements of exact order \(2^i\). For
\(\tau=(\tau_p)_p\), let \(O_\tau\) be the product of the shells of order
\(p^{\tau_p}\). A box is
\[
B_{i,\tau}=E_i\times O_\tau.
\]
The boxes are the Galois orbits of the rational support \(S\). To avoid
ambiguity between different generators, put \(a_2:=r\) and, for
\(q\in\{2\}\cup\{p:p\mid |H|\}\), write

\[
 C_{q^s}:=\{z\in C_{q^{a_q}}:q^s z=0\}
\]

for the canonical subgroup of the ambient factor (with the evident
interpretation in the dyadic factor). A shell of exact order \(q^s\)
generates precisely \(C_{q^s}\). Thus, for every occupied box,
Proposition~\ref{prop:local} assigns a character

\[
 \chi_{i,\tau}:K_{i,\tau}:=C_{2^i}\times\prod_p C_{p^{\tau_p}}
 \longrightarrow\mathbb S^1,
\]

and its \(p\)-primary restriction \(\chi_{i,\tau,p}\) is defined on the
canonical subgroup \(C_{p^{\tau_p}}\), independently of the choice of a
shell generator.

For each odd \(p\mid |H|\), fix an ambient generator \(e_p\) of the factor
\(C_{p^{a_p}}\). Thus \(p^{a_p-b_p}e_p\) is the canonical generator of
\(C_{p^{b_p}}\) whenever \(0\le b_p\le a_p\).

In the next lemma, for \(B=B_{i,\tau}\) we abbreviate
\[
  \chi_{B,p}:=\chi_{i,\tau,p}.
\]

\begin{lemma}[Increment comparison]\label{lem:increments}
For two occupied boxes \(B=B_{i,\tau}\) and \(B'=B_{i',\tau'}\), put
\(b_p=\min(\tau_p,\tau'_p)\) and \(b_2=\min(i,i')\). Then

\[
 \chi_{B,p}\big|_{C_{p^{b_p}}}
 =\chi_{B',p}\big|_{C_{p^{b_p}}}
\quad(p\text{ odd}),\qquad
 \chi_{B,2}\big|_{2C_{2^{b_2}}}
 =\chi_{B',2}\big|_{2C_{2^{b_2}}}.
\]

When \(b_p=0\) or \(b_2=0\), the corresponding assertion is vacuous.
\end{lemma}

\begin{proof}
Fix an increment \(\delta\in C_{p^{b_p}}\). If the level of the first box
is strictly larger than \(b_p\), every element of the shell of order
\(p^{\tau_p}\) remains in that shell after adding \(\delta\), because an
increment of strictly smaller order cannot change its \(p\)-adic valuation.
If instead
\(\tau_p=b_p\), write the two elements as units modulo \(p\): choose a
class \(x\) different from \(0\) and \(-\delta\), so that \(x+\delta\)
is also a unit. For \(p=3\), when the obstacles are distinct, exactly one
class remains admissible, so the choice is still available. The same
argument applies to the second box. If \(\delta\) is already divisible by
\(p\) at the common level, every unit representative works. In the dyadic
part take instead \(\delta\in2C_{2^{b_2}}\): adding an even element
preserves every odd unit class, with no exception.

Fix all coordinates other than the one under consideration and choose
simultaneously \(x,x+\delta\in B\) and \(y,y+\delta\in B'\). More
precisely, the four elements
\[
  x+\delta,\quad y,\quad -x,\quad -(y+\delta)
\]
belong respectively to \(B,B',B,B'\) and satisfy
\((x+\delta)+y-x-(y+\delta)=0\). The collision
\[
(x+\delta)+y=x+(y+\delta)
\]
gives, by (2.3),
\[
\frac{\rho(x+\delta)}{\rho(x)}
=\frac{\rho(y+\delta)}{\rho(y)}.
\]
The two ratios are \(\chi_{B,p}(\delta)\) and
\(\chi_{B',p}(\delta)\). The increments \(\delta\) generate
\(C_{p^{b_p}}\), so the \(p\)-primary restrictions agree. In the dyadic
part the permitted increments generate \(2C_{2^{b_2}}\), giving the second
equality. The construction uses only the shells of the two boxes under
consideration; it does not require the tag vectors to be comparable.
\end{proof}

\begin{proposition}[Gluing across distinct complete tags]\label{prop:glue}
There is a translation character on the odd factor such that, after
dividing \(\rho\) by it, every local character \(\chi_{i,\tau}\) is trivial
on all odd components. The complete tags need not be comparable.
\end{proposition}

\begin{proof}
For every odd prime \(p\), put
\[
b_p=\max\{\tau_p:B_{i,\tau}\subseteq S\}.
\]
Choose an occupied box at level \(b_p\). By Lemma~\ref{lem:increments}, the
\(p\)-primary restriction of every other box agrees with that of the chosen
box on its \(p\)-primary subgroup: the levels
\(0\le\tau_p\le b_p\) form the canonical chain

\[
 C_{p^0}\subseteq C_{p^1}\subseteq\cdots\subseteq C_{p^{b_p}}.
\]

If the value on the canonical generator of \(C_{p^{b_p}}\) is
\(\exp(2\pi i t/p^{b_p})\), extend it to the ambient generator by setting

\[
 \lambda_p(e_p)=\exp(2\pi i t/p^{a_p}),\qquad
 \lambda_p\big|_{C_{p^{b_p}}}=\chi_{B_p,p}.
\]

For \(b_p=0\), simply set \(\lambda_p=1\). Diagrammatically, for every box
\(B\), the compatibility just proved is

\[
\begin{array}{ccc}
C_{p^{\tau_p}}&\lhook\joinrel\longrightarrow&C_{p^{b_p}}\\[-2pt]
\chi_{B,p}\big|_{C_{p^{\tau_p}}}&=&\chi_{B_p,p}\big|_{C_{p^{\tau_p}}}
\end{array}
\qquad\text{and}\qquad
\lambda_p\big|_{C_{p^{\tau_p}}}=\chi_{B,p}.
\]

The ambient extension need not be unique, but all extensions chosen this
way agree on occupied frequencies. The product of the \(\lambda_p\) is a
character of the odd factor and, by CRT, is the restriction of a global
character of \(\widehat G\), hence of a physical translation. Dividing
\(\rho\) by this character means translating \(g\). Relations (2.2)--(2.3)
are unchanged: in a zero-sum relation the product of a global character on
the four (or fewer) terms is one.

This construction is coordinated prime by prime. For example, the tags
\((1,0)\) and \((0,1)\) in \(C_3\times C_5\) are incomparable as vectors,
but their restrictions are compared separately in the 3-chain and the
5-chain. Exactness of the support is used only to ensure that an occupied
box is entirely available when constructing the collision.
\end{proof}

\section{Lifting in exact-order shells}

\begin{lemma}[Exact-shell lifting criterion]\label{lem:lift}
Let \(\mathbf B=(B_1,\ldots,B_q)\) be an indexed list of occupied boxes,
\(1\le q\le4\), where \(B_j=B_{i_j,\tau^{(j)}}\); repetitions are allowed
and count by their index. There are frequencies
\(\xi_j\in B_j\) with \(\sum_{j=1}^q\xi_j=0\) if and only if, for every
odd prime \(p\), the maximum positive level
\[
 b_p:=\max_j\tau^{(j)}_p
\]
occurs at least twice, and the maximum dyadic level
\[
 b_2:=\max_j i_j
\]
occurs an even number of times. If a coordinate has \(b_p=0\) or \(b_2=0\),
the corresponding condition is vacuous.
\end{lemma}

\begin{proof}
Work first in a coordinate \(C_{p^a}\). If \(b>0\) is the maximum level
and occurs only once, divide the relation by \(p^{a-b}\) and reduce modulo
\(p\): one unit remains, while all lower-level terms are divisible by \(p\),
which is impossible. For \(p=2\), the same reduction says that the number
\(k\) of terms at level \(b\) must be even, since every unit is \(1\)
modulo \(2\). This proves necessity.

For sufficiency, fix the indexed lower-level terms and let \(z\) be the
negative of their sum in \(C_{p^b}\). They are divisible by \(p\), hence
\(z\equiv0\pmod p\). Identifying \(C_{p^b}\) with the canonical subgroup
of \(C_{p^a}\), for \(k=2,3,4\) maximal terms choose respectively
\[
 (1,z-1),\qquad (1,1,z-2),\qquad (1,-1,1,z-1). \tag{5.1}
\]
The tuples are ordered according to the indices of the maximal-level boxes
and sum to \(z\). If \(p\) is odd, all their entries are units:
\(z-1\equiv-1\), \(z-2\equiv-2\ne0\pmod p\), and again
\(z-1\equiv-1\) in the last case. For \(p=3\), in particular,
\(z-2\equiv1\pmod3\), while the last tuple is
\((1,-1,1,-1)\pmod3\). If \(p=2\), the condition allows only \(k=2\) or
\(k=4\); here \(z\) is even and all entries of (5.1) are odd.
Multiplying the normalized entries by \(p^{a-b}\) gives elements of exact
order \(p^b\) in the ambient factor.

Make the choice independently in each coordinate, preserving the index
\(j\), and recombine the \(q\) coordinates by CRT. The sums vanish in each
factor, hence in \(G\), and every \(\xi_j\) belongs to its prescribed box.
\end{proof}

\section{Dyadic sign normalization}

After Proposition~\ref{prop:glue}, assume that all odd components of the
local characters have been removed. For a box of dyadic level \(i>0\), put
\(g_i=2^{r-i}\) and write
\[
\rho(vg_i,\eta)=\exp(2\pi i\,v c_{i,\tau})\quad(\eta\in O_\tau),\qquad
v\ \text{odd},\qquad 2^ic_{i,\tau}=0\ \text{in }\T. \tag{6.1}
\]
This is simply the restriction of \(\chi_{i,\tau}\) to the dyadic
subgroup. In row \(i=0\), the subgroup generated by the box is entirely
odd: the local character was already removed by gluing, and
\(\rho=1\) on every box in this row.

Let \(m>0\) be the maximum occupied dyadic level and choose a reference box
\(B_*=(m,\tau_*)\). Let \(c=c_{m,\tau_*}\). For a box
\(B=(i,\tau)\), \(0<i\le m\), put \(d=2^{m-i}\). The four dyadic
frequencies
\[
g_m,\quad -(2d+1)g_m,\quad g_i,\quad g_i \tag{6.2}
\]
sum to zero. Assign tag \(\tau_*\) to the first two and tag \(\tau\) to the
last two. In every odd coordinate, regardless of how the tags compare, each
positive maximum occurs either twice or four times. The coordinatewise
construction in Lemma~\ref{lem:lift} completes the odd frequencies while
leaving the dyadic frequencies in (6.2) fixed. The phase relation is
\[
2(c_{i,\tau}-dc)=0\quad\text{in }\T. \tag{6.3}
\]
Thus \(c_{i,\tau}-dc\) is \(0\) or \(1/2\). The reference character extends
from the subgroup of order \(2^m\) to the ambient dyadic factor; after
removing this translation,
\[
\rho\big|_{B_{i,\tau}}=(-1)^{e_{i,\tau}},
\qquad e_{i,\tau}\in\mathbb F_2. \tag{6.4}
\]
The reference has sign zero, and all rows \(i=0\) have sign zero.

\section{Projectors and sign extinction}

We fix all domain conventions here. For \(h\in G\), physical translation is
\[
 (T_hu)(x):=u(x-h),
 \qquad
 \widehat{T_hu}(\xi)=\overline{\xi(h)}\,\widehat u(\xi). \tag{7.0}
\]
For \(u,v:H\to\mathbb C\), convolution in the odd factor is
\[
 (u*_Hv)(h_0):=\sum_{h\in H}u(h)v(h_0-h).
\]
For a tag set \(\Lambda\), put
\[
 Q_\Lambda:=\bigcup_{\tau\in\Lambda}O_\tau,\qquad
 S_\Lambda:=\bigcup_i E_i\times Q_\Lambda.
\]
The projector \(P_\Lambda\) is defined in the frequency domain by
\(\widehat{P_\Lambda u}=\one_{S_\Lambda}\widehat u\). In physical space it
is convolution with the kernel \(K_\Lambda\) below in the odd factor alone,
leaving the dyadic coordinate unchanged. Thus every subsequent table
checks Fourier coefficients of an identity already defined in physical
space.

For an odd tag set \(\Lambda\), let \(P_\Lambda\) be the Fourier projector
that multiplies frequencies in
\[
\bigcup_{\tau\in\Lambda}\bigcup_i E_i\times O_\tau
\]
by one and all other frequencies by zero. Its kernel in the odd factor is
\[
K_\Lambda(h)=\frac1{|H|}\sum_{\eta\in\bigcup_{\tau\in\Lambda}O_\tau}\eta(h).
\]
The summed set is stable under Galois and negation. Hence
\(|H|K_\Lambda(h)\) is a rational algebraic integer, and therefore an
integer, and \(K_\Lambda\) is real. By character orthogonality,
\(\widehat{K_\Lambda}=\one_{Q_\Lambda}\), so physical convolution realizes
exactly the multiplier \(\one_{S_\Lambda}\). In particular, for integer
\(f\), \(P_\Lambda f\in |H|^{-1}\mathbb Z^G\).
If \(v\in |H|^{-1}\mathbb Z^G\) and \(2^av\) is integral, then \(v\) is
integral because \(\gcd(|H|,2^a)=1\) (Bézout's identity).
If \(H=\{0\}\), the only tag is the empty one and the projector is zero or
the identity; the denominator is \(1\) and the same argument applies.

Assume \(m>0\), and let \(T\) be physical translation by \(2^{m-1}\) in
the coordinate \(C_{2^r}\). Its Fourier multiplier is \(+1\) on levels
\(i<m\), \(-1\) on level \(m\), and the ambient level \(r\) need not be
fixed. Indeed, for \(s=v2^{r-i}\) the multiplier is
\(\exp(-2\pi i s2^{m-1}/2^r)=\exp(-2\pi i v2^{m-1-i})\).
Both Fourier coefficients vanish on levels \(i>m\).

\begin{proposition}[Lower extinction]\label{prop:lower}
All signs \(e_{i,\tau}\) with \(i<m\) vanish.
\end{proposition}

\begin{proof}
Set \(e_{0,\tau}=0\). For two occupied lower levels \(0\le i,j<m\) with
the same tag \(\tau\), take one copy of \(B_{i,\tau}\), one of
\(B_{j,\tau}\), and two copies of the reference box at level \(m\).
Lemma~\ref{lem:lift} admits the relation; the two reference signs vanish,
so \(e_{i,\tau}+e_{j,\tau}=0\). Thus the lower sign is constant along each
tag. In particular, if the tag is also present at level \(0\), its sign at
every positive lower level already vanishes. If the same tag also occurs at
level \(m\), two copies of the upper box and one lower copy give an
admissible triple, so the lower sign is directly zero.

Let \(\Lambda_-\) be the set of tags with lower-level ratio \(\rho=-1\).
No tag in \(\Lambda_-\) occurs at level \(0\) or level \(m\) of the
support. In Fourier space, for each frequency, consider the four cases:

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}Xcccc@{}}
\toprule
frequency & \(t(\xi)\) & \(\rho(\xi)\) &
\((1+t(\xi))(\widehat f-\widehat g)\) & \(4\widehat{P_{\Lambda_-}f}\)\\
\midrule
tag in \(\Lambda_-\), \(i<m\) & \(+1\) & \(-1\) & \(4\widehat f\) & \(4\widehat f\)\\
unselected tag, \(i<m\) & \(+1\) & \(+1\) & \(0\) & \(0\)\\
level \(m\) & \(-1\) & \(\pm1\) & \(0\) & \(0\)\\
\(\xi\notin S\) & arbitrary & undefined & \(0\) & \(0\)\\
\bottomrule
\end{tabularx}
\end{center}

Therefore
\[
4P_{\Lambda_-}f=(I+T)(f-g). \tag{7.1}
\]
The right-hand side is integral and has modulus at most \(2\), so
\(|P_{\Lambda_-}f|\le1/2\). The integrality argument makes
\(P_{\Lambda_-}f\) integral, hence zero. If \(\Lambda_-\) were nonempty,
exactness of the support would provide a nonzero Fourier coefficient
selected by the projector, a contradiction.
\end{proof}

Put
\[
U=\{\tau:E_m\times O_\tau\subseteq S\},\qquad
L=\bigcup_{i<m}\{\tau:E_i\times O_\tau\subseteq S\}.
\]
All lower-level signs on \(L\) vanish.

\begin{lemma}[Synchronization of shared tags]\label{lem:shared}
The upper signs are constant on \(U\cap L\).
\end{lemma}

\begin{proof}
For \(\tau,\sigma\in U\cap L\), choose lower levels \(i,j<m\) and consider
the four boxes
\[
(m,\tau),\quad(m,\sigma),\quad(i,\tau),\quad(j,\sigma).
\]
In every odd coordinate, each positive maximum occurs at least twice: if the
two tags have different levels, the maximum belongs to the two occurrences
of the corresponding tag; if they agree, it occurs four times. The dyadic
maximum occurs twice. Lemma~\ref{lem:lift} realizes the relation, and the
two lower signs are zero, leaving
\(e_{m,\tau}+e_{m,\sigma}=0\).
\end{proof}

\begin{theorem}[Upper extinction]\label{thm:upper}
After at most one further global translation, \(f=g\).
\end{theorem}

\begin{proof}
If \(U\cap L\ne\varnothing\), Lemma~\ref{lem:shared} assigns one sign
\(c\). If \(U\cap L=\varnothing\), set \(c=0\). Replace \(g\) by
\(T^cg\). This is one global translation and leaves the lower signs zero.
Let \(\Lambda_+=\{\tau\in U:e_{m,\tau}=1\}\). Then
\(\Lambda_+\cap L=\varnothing\). The frequency-by-frequency verification of
\[
f-g=2P_{\Lambda_+}f \tag{7.2}
\]
is:

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}Xcccc@{}}
\toprule
frequency & \(t(\xi)\) & \(\rho(\xi)\) &
\(\widehat f-\widehat g\) & \(2\widehat{P_{\Lambda_+}f}\)\\
\midrule
tag in \(\Lambda_+\), \(i=m\) & \(-1\) & \(-1\) & \(2\widehat f\) & \(2\widehat f\)\\
unselected tag, \(i=m\) & \(-1\) & \(+1\) & \(0\) & \(0\)\\
lower frequency & \(+1\) & \(+1\) & \(0\) & \(0\)\\
\(\xi\notin S\) & arbitrary & undefined & \(0\) & \(0\)\\
\bottomrule
\end{tabularx}
\end{center}

By (7.2), \(2P_{\Lambda_+}f\) is integral; projector integrality makes
\(P_{\Lambda_+}f\) integral. But (7.2) and binarity give
\(|P_{\Lambda_+}f|\le1/2\), so \(P_{\Lambda_+}f=0\) and \(f=g\).
\end{proof}

\section{Proof of the theorem and scope}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
By Lemma~\ref{lem:deckbridge}, equality of the fourth cumulative deck is
equivalent to equality of \(D_1,\ldots,D_4\). If one of the two sets is
empty, the mass makes both empty. In the nonempty case, (2.1)--(2.3)
produce the unit-modulus ratio \(\rho\). Theorem~\ref{thm:fiber} makes it
torsion, and Proposition~\ref{prop:local} turns it into local characters.
Proposition~\ref{prop:glue} removes the odd components by a translation.
Section 6 leaves only dyadic signs. If there is no positive dyadic level,
the ratio is already a global character. Otherwise Proposition~\ref{prop:lower}
and Theorem~\ref{thm:upper} cancel all signs. Thus \(A\) and \(B\) differ by
a translation, proving (1.3).

For even \(n>10\), KK06 Corollary 1.1 rules out sufficiency of the 3-deck
and gives \(r_{\rm set}(C_n)\ge4\). Together with (1.3), this proves (1.4).
\end{proof}

\begin{corollary}[Real competitors]\label{cor:real}
If \(f=\one_A\) on \(C_n\) and \(u:C_n\to\mathbb R\) has the same
cumulative data \(D_1,\ldots,D_4\), then \(u\) is a translate of \(f\).
\end{corollary}

\begin{proof}
Lemma~\ref{lem:binary} makes \(u\) binary. Apply the theorem.
\end{proof}

\begin{remark}[Scope]
The result does not assert reconstruction for general positive integer
signals, multisets, or noncyclic abelian groups. We do not replace the deck
convention by a list of only distinct patterns of cardinality exactly four,
nor do we identify objects under reflection in advance. The
Grünbaum--Moore phases for nonbinary rational signals and the finite-order
phase-retrieval literature are adjacent prior art, not an automatic
importation of the present theorem.
\end{remark}

\section{Prior art and positioning}

Pebody defines the deck as the multiset of translation classes of subsets
of cardinality at most \(k\), without reflection
\cite{Pebody2007}; Pebody 2004 focuses mainly on reconstruction of rational
functions and multisets \cite{Pebody2004}. Grünbaum--Moore use Fourier
ratios and characters, but their Theorem 5 requires a nonzero primitive
coefficient and is not a uniform binary theorem
\cite{GM1995}. KK06 provides the Fourier framework for the deck and the
uniform lower bound through Corollary 1.1 \cite{KK2006}.

Simon's thesis documents the analytic/combinatorial bridge and maintains
the distinction between translation and reflection \cite{Simon2017}. The
work of Mandereau--Ghisi--Amiot--Andreatta--Agon also studies decks up to
reflection and musical phase retrieval \cite{Mandereau2011}; that quotient
is not used here. Amiot classifies rational spectral units of finite order
\cite{Amiot2009}; the ratio considered here is defined only on the exact
Fourier support, and the moment identities force the inverse-transformed
signal to be binary. Thus that result
is a contextual check rather than an automatic import. The bibliographic
comparison is limited to the cited sources and is not a global priority or
novelty claim.

\section{Computational checks and limitations}

The companion code provides a bounded check of the exact-shell lifting
criterion. It enumerates prime-power coordinates for \(p=2,3,5\),
exponents \(1,2,3\), and indexed lists of lengths one through four,
including repetitions, and compares the criterion with direct zero-sum
enumeration. These are tests of the local coordinate conditions, not an
enumeration of every ambient cyclic group.

The repository also supplies manuscript-build, public-surface and hash
checks. The reproduction instructions identify the shipped commands and
their scope.

No finite check is a premise of the uniform proof. The checks are intended
to catch convention, implementation, or typesetting errors; they do not
replace independent mathematical reading, formalization, or editorial
assessment. The code and control data are not used to extrapolate the
theorem to noncyclic groups, general positive integer signals, or a
different deck convention.

\paragraph{AI assistance.}
The author reports using GPT-6 Pro and Astra via ChatGPT, together with
OpenAI Codex. AI assistance included mathematical exploration, source
comparison, manuscript drafting and revision, finite sanity checks, and
repository preparation under the author's direction. Contributions are
not attributed to individual models. The author is responsible for the
mathematical claims and final decisions. AI reviews do not constitute
human peer review or independent specialist verification.

\clearpage
\begin{thebibliography}{99}

\bibitem{GM1995}
F.~A. Grünbaum and C.~C. Moore,
\emph{The Use of Higher-Order Invariants in the Determination of
Generalized Patterson Cyclotomic Sets},
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DOI: \url{https://doi.org/10.1107/S0108767394009827}.

\bibitem{Pebody2004}
L.~Pebody,
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\bibitem{Pebody2007}
L.~Pebody,
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DOI: \url{https://doi.org/10.1017/S0963548306007875}.

\bibitem{KK2006}
T.~Keleti and M.~N. Kolountzakis,
\emph{On the determination of sets by their triple correlation in finite
cyclic groups}, arXiv:math/0603415 (2006).
\url{https://arxiv.org/pdf/math/0603415}.

\bibitem{Simon2017}
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\bibitem{Mandereau2011}
J.~Mandereau, D.~Ghisi, E.~Amiot, M.~Andreatta and C.~Agon,
\emph{Discrete phase retrieval in musical structures},
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\bibitem{Amiot2009}
E.~Amiot,
\emph{On the group of rational spectral units with finite order},
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\url{https://arxiv.org/abs/0907.0857}.

\bibitem{Conrad}
K.~Conrad, \emph{Cyclotomic Extensions}, note,
\url{https://kconrad.math.uconn.edu/blurbs/galoistheory/cyclotomic.pdf}.

\end{thebibliography}

\end{document}