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\title{An Explicit $2\times2$-Minimal Forcing Certificate at $S=7/4$ for the
Arithmetic--Kakeya Sum--Difference Problem, and Evidence for a Floor on What
the Certificate Framework Can Express}
\author{Cultural Soliton Observatory / ProjectForty2}
\date{2026-06-14}

\begin{document}
\maketitle

\begin{abstract}
We report an explicit, machine-verified \emph{forcing certificate} achieving
sum--difference score $S = 7/4 = 1.75$ --- the FrontierMath
``arithmetic--Kakeya'' warm-up threshold --- and we examine what the surrounding
certificate framework can and cannot prove. The certificate lives on the
smallest non-trivial grid ($2\times2$), uses no free vertices, and realizes the
Katz--Tao slope set $\{0,1,2,\infty\}$. Its correctness is established by an
\emph{explicit linear-algebra witness}: the target direction $(1,-1)$ is realized
exactly at the corner vertex $(2,2)$ --- by a written-out rational combination of
the certificate's generators that is zero at the other three vertices --- after
which the remaining three vertices cascade, each via its own explicit witness.
The result is corroborated by a reconstructed ``gold'' verifier together with
three independent oracles and a tamper control.

We are deliberate about what this is \textbf{not}. The value $7/4$ is
\textbf{not a new mathematical bound}: it is the Katz--Tao sum--difference bound
of 2001 (the $\{0,1,2,\infty\}$ slope-set bound). We do \textbf{not} beat the
arithmetic--Kakeya record; the best known sum--difference exponent,
$\inf\SD(\R;-1)\le 1.67513\ldots$ (Katz--Tao, restated by Tao 2025), is
untouched. Our contributions are (i) the explicit certificate realizing $7/4$
that is \emph{minimal on the $2\times2$ grid} (proven by exhaustive search) ---
improving the best small forcing certificate \emph{we had previously constructed},
which scored $S=9/5$ on a $2\times3$ grid; and (ii) a precisely-scoped
\emph{evidence-for-a-floor} result: a structural argument, exhaustive small-case
search, and the Tao (2025) bounded-slope theorem together suggest that the
framework's achievable exponents floor at $\gamma = 1.6751\ldots$, the largest
real root of $x^3-4x+2$. We are explicit that this is a \emph{characterization
supported by evidence, not a theorem}, and we name the universal-reduction step a
full proof would require. The practical consequence is that $7/4$ is the exact
$2\times2$ minimum and the second term of the advanced iteration, and reaching
even the interval $(3/2,\gamma)$ would require both a new inequality and a new
certificate format.
\end{abstract}

\section{Introduction}

The \textbf{arithmetic Kakeya problem} asks how slowly a sum set can grow
relative to its difference set under a Kakeya-type incidence constraint, and is
one of the cleaner combinatorial cores underneath the geometric Kakeya
conjecture. The relevant quantity is the \textbf{sum--difference exponent}
$\SD(\R;-1)$, the smallest exponent for which a certain sum-vs-difference
inequality survives. The \emph{conjectural} endpoint of the arithmetic Kakeya
conjecture (AKC) corresponds to exponent $1$; the best known upper bound is
$\inf\SD(\R;-1) \le 1.67513\ldots$, due in mechanism to Katz--Tao (2001)
\cite{KT01} and restated in Tao's November 2025 note \cite{Tao25}.

The Epoch \textbf{FrontierMath} benchmark includes an ``arithmetic--Kakeya''
track formulated as a \textbf{forcing-certificate} verification problem: a
candidate submits a finite combinatorial object $(X,d,f,T,R)$ which a verifier
accepts iff a \emph{forcing closure} reaches every vertex of a product grid
\textbf{and} a rational score $S$ falls at or below a threshold $\tau$. The score
$S$ equals a certified sum--difference exponent, so a passing certificate at
threshold $\tau$ is a machine-checkable witness that $\SD$ at the certified
configuration is at most $\tau$. The benchmark's warm-up threshold is
$\tau = 7/4$; higher tiers push toward the record and ultimately toward the
conjectural endpoint.

This note does two things, both stated modestly.

\paragraph{Contribution 1 (an explicit $2\times2$-minimal artifact).}
We give an explicit forcing certificate that passes at $\tau = 7/4$ on the
$2\times2$ grid with no free vertices (\S\ref{sec:cert}). It is the
\textbf{minimum-scoring configuration on the $2\times2$ grid} (exhaustive search,
\S\ref{sec:exhaustive}) and improves the best small forcing certificate
\emph{we had previously constructed}, which scored $S = 9/5 = 1.8$ on a
$2\times3$ grid and required a free vertex and a telescoping ladder. We prove
correctness by exhibiting an explicit rational combination of the certificate's
generators that realizes the target direction $(1,-1)$ exactly at the corner
vertex $(2,2)$ from the empty free set, and then exhibiting one such combination
for each of the remaining three vertices. The value $7/4$ is \emph{not new
mathematics} (it is the Katz--Tao $\{0,1,2,\infty\}$ bound); the contribution is
the explicit minimal realization and its end-to-end verification.

\paragraph{Contribution 2 (evidence for a framework floor, scoped honestly).}
We examine the reach of the certificate framework itself (\S\ref{sec:floor}). We
give (a) a structural argument that a certificate is a product grid whose axis
projections are the Katz--Tao sum--difference scaffold; (b) an \textbf{exhaustive}
search of the $2\times2$ grid --- with a closed-form configuration count that we
verify equals the iterated count --- showing minimum score exactly $7/4$,
\emph{invariant} under enlarging the slope alphabet from $4$ to $7$ slopes; and
(c) the Tao (2025) bounded-slope theorem, which forces $\SD\to 2$ when the number
of slopes is bounded. Together these give structural and computational evidence
--- \textbf{but not a proof} --- that the framework is co-floored by finite
sum--difference constructions, whose exponents are bounded below by
$\gamma = 1.6751308705666\ldots$, the largest real root of $x^3 - 4x + 2$. The
honest consequence (\S\ref{sec:consequences}): within this framework $7/4$ is the
exact $2\times2$ minimum, and even reaching $(3/2,\gamma)$ would require a
construction the framework as formatted cannot express.

We do \textbf{not} claim a new arithmetic--Kakeya bound, and we do \textbf{not}
claim $7/4$ as a new bound. The record stands. All verifier results are reported
against a \emph{reconstruction} of the Epoch verifier (\S\ref{sec:repro}), and
there is no public submission channel, so we frame $7/4$ as a verified research
artifact rather than an official competition solve.

\section{The forcing-certificate framework}

\subsection{The certificate object}

A \textbf{certificate} is a tuple $(X, d, f, T, R)$.
\begin{itemize}
\item \textbf{Slope alphabet $X \subset \Z^2$.} A finite set of slope vectors. It
must contain $(0,0)$, and every other $(a,b)\in X$ must satisfy $a+b\neq 0$. The
forbidden direction $a+b=0$ --- represented by the canonical target $(1,-1)$ ---
is exactly the sum--difference direction one is trying to \emph{synthesize}, so it
may never appear as an input slope.
\item \textbf{Grid $d=[d_1,\dots,d_k]$.} The vertex set is the product
\[
V = \{1,\dots,d_1\}\times\cdots\times\{1,\dots,d_k\}, \qquad n = \prod_{i=1}^{k} d_i .
\]
\item \textbf{Edge labels $f=(f_1,\dots,f_k)$.} Each $f_i$ assigns a slope in $X$
to a length-$i$ key whose last coordinate lies in $\{1,\dots,d_i-1\}$; it labels
the edges that move along axis $i$. Omitted keys default to $(0,0)$.
\item \textbf{Free set $T\subseteq V$.} Vertices granted ``for free'' at the start
of the closure (they enlarge the denominator of the score and so are penalized).
\item \textbf{Seed relations $R$.} A list of \textbf{single-support} relations,
each a map (one vertex $\mapsto$ one slope in $X$); each contributes exactly one
nonzero entry.
\end{itemize}

\subsection{Score}

The cost of the edge labelling is
\[
m = \sum_{i=1}^{k} \bigl(\#\{\text{nonzero } f_i\}\bigr)\cdot\!\!\prod_{j=i+1}^{k} d_j ,
\]
each axis-$i$ nonzero edge weighted by the suffix product $d_{i+1}\cdots d_k$. The
\textbf{score} is
\[
S = \frac{m + |R|}{n - |T|}
\]
and equals the certified sum--difference exponent $\SD(X;-1)$ for the
configuration.

\subsection{op1 generators and the module}

For each axis-$i$ edge labelled $x\in X$ between two adjacent vertices, the
\textbf{op1 generator} places $x$ at one endpoint and $-x$ at the other:
schematically $\{e_1:x,\ e_2:-x\}$. The seed relations $R$ together with all op1
generators span a rational subspace
\[
M = \Q\text{-span}\bigl(R \cup \{\text{op1 generators}\}\bigr) \subseteq \Q^{2n},
\]
where each vertex contributes two coordinates (a $2$-vector slope).

\subsection{Forcing closure (op2)}\label{sec:op2}

For a current free set $F\subseteq V$ and a vertex $e\notin F$, the
\textbf{antidiagonal direction} at $e$ is a vector $w\in\Q^{2n}$ whose restriction
to vertex $e$ is $(s,-s)$ with $s\neq 0$ (i.e.\ $u+v=0$, $u\neq 0$ on the two
coordinates of $e$ --- exactly the forbidden $a+b=0$ direction). We say $e$ is
\textbf{forced given $F$} iff
\[
M \text{ (a subspace of } \Q^{2n}\text{) contains } w \text{ with }
w|_e=(s,-s),\ s\neq 0,\ \text{ and } w|_{v}=(0,0)\ \forall v\notin F\cup\{e\}.
\]
Intuitively: the subspace can manufacture the forbidden $(1,-1)$ direction at $e$
while leaving everything outside the already-free region untouched. This is
precisely the predicate the verifier computes (its \texttt{\_op2\_can\_force}):
solve for a generator combination $y$ with $(yM)$ zero on all ``outside''
coordinates and $(yM)|_e=(1,-1)$ (we may fix $s=1$, as the line
$\{(s,-s):s\neq0\}$ is closed under nonzero scaling).

The \textbf{closure} starts from $F=T$ and repeatedly adds every forceable vertex
until a fixpoint. The certificate \textbf{PASSES at threshold $\tau$} iff the
closure reaches all $n$ vertices and $S \le \tau$. A passing certificate at
$\tau$ certifies $\SD(X;-1)\le\tau$ for that configuration, with the forbidden
$(1,-1)$ direction synthesized from the allowed slopes alone.

\section{The $2\times2$-minimal $7/4$ certificate}\label{sec:cert}

\subsection{The certificate}

\[
\begin{aligned}
X &= \{(0,0),(1,0),(1,1),(1,2),(0,1)\} && \text{(slopes }\{0,1,2,\infty\})\\
d &= [2,2] && (n=4)\\
f &= \bigl[\,\{(1,):(1,0)\},\ \{(1,1):(1,1),\ (2,1):(1,1)\}\,\bigr]\\
T &= [\,] && (|T|=0)\\
R &= \bigl[\,\{(1,1):(1,2)\},\ \{(1,2):(0,1)\},\ \{(2,1):(0,1)\}\,\bigr] && (|R|=3)
\end{aligned}
\]
The slope vectors $\{(1,0),(1,1),(1,2),(0,1)\}$ are exactly the Katz--Tao slopes
$\{0,1,2,\infty\}$. The grid is the smallest non-trivial one, $2\times2$, with no
free vertices.

\subsection{Arithmetic of the score}

The edge cost: $f_1$ has one nonzero label weighted by suffix product $d_2=2$,
contributing $2$; $f_2$ has two nonzero labels weighted by suffix product $1$,
contributing $2$. Hence $m = 2+2 = 4$, $|R| = 3$, $n = 4$, $|T| = 0$, and
\[
S = \frac{m+|R|}{n-|T|} = \frac{4+3}{4-0} = \frac{7}{4} = 1.75 .
\]
The reconstructed verifier independently recomputes $m(G)=4$ and reports
$S=(4+3)/(4-0)=7/4$.

\subsection{Correctness: explicit forcing witnesses}\label{sec:witness}

The four vertices are $(1,1),(1,2),(2,1),(2,2)$. The subspace $M$ is spanned by
$7$ generators and has rank $7$ out of the $8$ ambient coordinates (it ``wastes no
degrees of freedom''): the three seeds and four op1 generators
\[
\begin{aligned}
&g_0 = R[(1,1)\!:\!(1,2)],\ g_1 = R[(1,2)\!:\!(0,1)],\ g_2 = R[(2,1)\!:\!(0,1)],\\
&g_3 = \mathrm{op1}[(1,)\!\to\!(1,0)]@(1,1),\ g_4 = \mathrm{op1}[(1,)\!\to\!(1,0)]@(1,2),\\
&g_5 = \mathrm{op1}[(1,1)\!\to\!(1,1)]@(1,1),\ g_6 = \mathrm{op1}[(2,1)\!\to\!(1,1)]@(2,1).
\end{aligned}
\]
We exhibit, for each forced vertex, an explicit rational combination realizing the
antidiagonal $(1,-1)$ direction there and vanishing outside the permitted region.
These are the actual solution vectors, verified by direct substitution --- not
narrative ``telescopings.''

\paragraph{Round 1 --- the corner $(2,2)$ is forced first, and uniquely, from
$F=\varnothing$.} The combination
\[
w_{(2,2)} = g_0 - 2g_1 - g_2 + g_3 - 2g_4 - 2g_5 + g_6
\]
evaluates to $w_{(2,2)}|_{(1,1)}=w_{(2,2)}|_{(1,2)}=w_{(2,2)}|_{(2,1)}=(0,0)$ and
$w_{(2,2)}|_{(2,2)}=(1,-1)$. Thus the forbidden target direction is synthesized at
$(2,2)$ alone, from the allowed slopes $\{0,1,2,\infty\}$ --- the heart of the
certificate. A round-by-round driver over the verifier's
\texttt{\_op2\_can\_force} predicate confirms $(2,2)$ is the \textbf{unique}
vertex forceable in round 1.

\paragraph{Round 2 --- the remaining three cascade, each with its own explicit
witness.} With $(2,2)\in F$, each remaining $e$ becomes forceable:
\[
\begin{aligned}
w_{(1,2)} &= -g_1 + g_4
&&\Rightarrow w_{(1,2)}|_{(1,2)}=(1,-1),\ \text{zero at }(1,1),(2,1),\ w_{(1,2)}|_{(2,2)}=(-1,0),\\
w_{(2,1)} &= -g_0 + 2g_1 - g_2 - g_3 + 2g_4 + 2g_5
&&\Rightarrow w_{(2,1)}|_{(2,1)}=(1,-1),\ \text{zero at }(1,1),(1,2),\ w_{(2,1)}|_{(2,2)}=(-2,0),\\
w_{(1,1)} &= -2g_0 + 3g_1 + 3g_4 + 3g_5
&&\Rightarrow w_{(1,1)}|_{(1,1)}=(1,-1),\ \text{zero at }(1,2),(2,1),\ w_{(1,1)}|_{(2,2)}=(-3,0).
\end{aligned}
\]
In each case the antidiagonal $(1,-1)$ appears at the target vertex, the value is
zero at the two vertices still outside $F\cup\{e\}$, and the only other nonzero
entry sits at the now-free $(2,2)$ --- which is allowed. The driver reports
$\text{round 1: forced }\{(2,2)\}$; $\text{round 2: forced
}\{(1,1),(1,2),(2,1)\}$, reaching all $4$ of $4$ vertices. Closure succeeds, and
since $S=7/4\le 7/4$ the certificate \textbf{PASSES} at $\tau=7/4$. (Each $w$ was
verified by direct substitution into $M$; see \S\ref{sec:repro}.)

\subsection{The $9/5 \to 7/4$ improvement (over our own prior small
certificate)}\label{sec:improvement}

The best small forcing certificate \emph{we had previously constructed} scored
$S = 9/5 = 1.8$ on the same slope alphabet but a larger grid:
\[
d = [2,3],\quad n=6,\quad m = 7,\quad |R| = 2,\quad |T| = 1,\qquad
S = \frac{7+2}{6-1} = \frac{9}{5}.
\]
Here ``small'' means \emph{the smallest explicit construction we had on hand}
($n\le 6$), not a global record: the $9/5$ certificate proves only the
\emph{upper bound} $\SD\le 9/5$ on $d=[2,3]$, and indeed our searches on that grid
(\S\ref{sec:multigrid}) do not establish $9/5$ as a floor. Re-verification
confirms the $9/5$ certificate passes at $\tau=9/5$, forces all $6$ vertices, and
matches its arithmetic.

The mechanism of the improvement is structural collapse: the $2\times3$ ladder
spends a free vertex (a $|T|=1$ penalty) and seven edge labels to walk the
forbidden direction down a three-step axis. On the $2\times2$ grid the corner
$(2,2)$ is forced \emph{directly from the empty free set} --- no ladder, no free
vertex --- so $|T|$ drops from $1$ to $0$ and $m$ from $7$ to $4$, and the score
falls from $\tfrac{9}{5}$ to $\tfrac{7}{4}$. The move is ``find the $2\times2$
optimum, which lies below the previously-exhibited $2\times3$ construction.''

\subsection{Multi-oracle verification and a tamper control}\label{sec:oracles}

The certificate's correctness was checked five ways (all re-run 2026-06-14):
\begin{enumerate}
\item \textbf{Gold verifier (reconstruction).} Returns PASS (exit $0$) at
$\tau = 7/4$ and REJECT (exit $1$) at $\tau = 67/40 = 1.675$, as $7/4 > 67/40$
predicts.
\item \textbf{Independent Fraction-RREF oracle.} An exact rational row-reduction
reproduces the closure and score; on the published certificate it agrees with the
gold verifier (cross-check prints \texttt{AGREE=True}).
\item \textbf{Independent two-prime mod-$p$ oracle.} Reducing the generators
modulo two distinct primes reproduces the same forcing closure.
\item \textbf{Explicit witnesses.} The round-1 and round-2 combinations of
\S\ref{sec:witness}, each verified by direct substitution.
\item \textbf{Tamper / negative control.} Replacing the seed $\{(1,1):(1,2)\}$
with any of the three other allowed slopes breaks the closure:
$(1,1)\mapsto(1,1)$ reaches $1/4$; $(1,1)\mapsto(2,1)$ reaches $0/4$;
$(1,1)\mapsto(1,0)$ reaches $1/4$. The certificate is \textbf{delicately placed}.
\end{enumerate}
Across the recorded audit ($300$--$400$ randomly generated small-certificate
trials) the three independent oracles registered \textbf{zero disagreements} with
the gold verifier. ``Gold verifier'' denotes a \emph{reconstruction} of Epoch's
verifier semantics, not an official Epoch run; all PASS/REJECT statements are
against the reconstruction.

\section{Evidence for a framework floor}\label{sec:floor}

\subsection{Katz--Tao sum--difference recursions and the constant $\gamma$}\label{sec:gamma}

The Katz--Tao sum--difference method \cite{KT01} improves its exponent by
iterating a recursion. The \textbf{basic} recursion is
$\beta \mapsto (4\beta-1)/(2\beta)$, whose fixed-point equation reduces (verified
exactly) to $-2\beta^2+4\beta-1=0$, with positive fixed point
\[
\beta_{\text{basic}}^{*} = 1 + \frac{\sqrt 2}{2} = 1.7071067811865\ldots .
\]
The \textbf{advanced} recursion is
$\beta \mapsto (3\beta^2 + 2\beta - 2)/(\beta^2 + 3\beta - 2)$. Setting this equal
to $\beta$ and clearing denominators reduces \emph{exactly} (sympy
\texttt{cancel}/\texttt{fraction}/\texttt{expand}) to
\[
-\beta^3 + 4\beta - 2 = 0 \quad\Longleftrightarrow\quad \beta^3 - 4\beta + 2 = 0 .
\]
This cubic has three real roots, $-2.2143\ldots$, $0.5392\ldots$, and
\[
\gamma = 1.6751308705666460709\ldots,
\]
the largest, the attracting fixed point of the advanced iteration. Substituting
$\gamma$ into $\beta^3-4\beta+2$ gives $0$ to machine precision. Comparing to the
FrontierMath rejection threshold $67/40 = 1.675$,
\[
\gamma - \frac{67}{40} = 1.30871\times 10^{-4} > 0,
\]
so $\gamma > 67/40$ by a small but strictly positive margin.

\subsection{The advanced iteration descends to $\gamma$ from above}\label{sec:seq}

Starting the advanced iteration at $\beta=2$ produces the exact decreasing
rational sequence
\[
2,\ \tfrac{7}{4},\ \tfrac{171}{101},\ \tfrac{101863}{60652},\ \ldots
= 2,\ 1.75,\ 1.69307,\ 1.67947,\ 1.67618,\ 1.67539,\ 1.67519,\ \ldots
\]
converging to $\gamma$ \textbf{from above}. The first two terms are exactly $2$
and $7/4$. Hence every exponent produced by the advanced iteration is strictly
greater than $\gamma > 1.675$, and $7/4$ is the \emph{second} term --- the first
non-trivial improvement off the base case $\beta=2$. (This is a statement about
the iteration; whether \emph{all} finite sum--difference constructions are
iteration outputs is the stronger claim conjectured in \S\ref{sec:charac}, not
proved here.)

\subsection{Exhaustive $2\times2$ search and slope-count
invariance}\label{sec:exhaustive}

The load-bearing computational evidence is an \textbf{exhaustive} search of the
$2\times2$ grid under the verifier's forcing semantics, using the fast
Fraction-RREF oracle (cross-checked \texttt{AGREE=True} against the gold verifier
on the published certificate). To make ``exhaustive'' defensible, we fix a precise
enumeration convention and verify the configuration count against a closed form.

\paragraph{Enumeration convention.} On $d=[2,2]$ the vertices are
$V=\{(1,1),(1,2),(2,1),(2,2)\}$ ($n=4$) and the verifier's $f$-domain has exactly
$E=3$ edge slots: the level-1 key $(1,)$ and the level-2 keys $(1,1),(2,1)$. With
$s$ nonzero slopes, each slot takes a value in $X=\{(0,0)\}\cup(\text{slopes})$,
i.e.\ $(s+1)$ choices, so there are $(s+1)^E=(s+1)^3$ edge labellings. The free
set $T$ ranges over all subsets of $V$ ($2^4=16$). The seed list $R$ ranges over
all size-$0,\dots,3$ subsets of the $4s$ vertex--slope pairs (we cap $|R|\le 3$;
the published certificate uses $|R|=3$). The total count is the closed form
\[
N(s) = (s+1)^3 \cdot 2^4 \cdot \sum_{r=0}^{3}\binom{4s}{r}.
\]
For each $s$ we verify that the iterated enumeration visits exactly $N(s)$
configurations (the script asserts \texttt{MATCH=True}):

\begin{center}
\begin{tabular}{ccccccc}
\hline
slopes $s$ & $(s+1)^3$ & $\sum_{r\le3}\binom{4s}{r}$ & configs $N(s)$ & min score & time \\
\hline
$4\ \{0,1,2,\infty\}$ & $125$ & $697$ & $1{,}394{,}000$ & $7/4$ & $24.2$ s \\
$5$ & $216$ & $1{,}351$ & $4{,}669{,}056$ & $7/4$ & $71.7$ s \\
$6$ & $343$ & $2{,}325$ & $12{,}759{,}600$ & $7/4$ & $174.5$ s \\
$7$ & $512$ & $3{,}683$ & $30{,}171{,}136$ & $7/4$ & $376.3$ s \\
\hline
\end{tabular}
\end{center}

In every case the closed-form count equals the iterated count, the minimum is
exactly $7/4$, and the minimizing configuration is exactly the published
certificate. \textbf{Enlarging the slope alphabet from $4$ to $7$ slopes adds no
forcing power on the $2\times2$ grid.}

Three scope caveats are required. \textbf{(i)} ``Exhaustive'' is asserted
\emph{for this enumeration convention}, made checkable by the closed form $N(s)$;
a different convention (counting symmetry orbits, or allowing $|R|>3$) would give a
different total. We report a single reproducible convention with its verified
count, and rely on no earlier, unreconciled configuration-count figure.
\textbf{(ii)} The claim is for $d=[2,2]$ \emph{only}; it does not prove that no
larger grid or alphabet anywhere beats $7/4$. \textbf{(iii)} The slope-count
invariance is \emph{consistent with} --- but not a proof of --- the Tao (2025)
bounded-slope phenomenon (\S\ref{sec:charac}).

\subsection{Heuristic upper bounds on larger grids (search-found, not
floors)}\label{sec:multigrid}

We also ran heuristic (time-budgeted, Dinkelbach-style) searches over a range of
grids. These return the \textbf{best certificate found} --- i.e.\ \emph{upper
bounds} on the achievable score for each grid, not floors --- and the searches are
non-exhaustive and disagree at the margins. The recorded best-found scores (logs
cited in \S\ref{sec:repro}) are:

\begin{center}
\begin{tabular}{lc}
\hline
configuration & best certificate found (upper bound) \\
\hline
$d=[2,3]$ & $\le 9/5$ (explicit cert; the time-budgeted floor probe only reached $2$) \\
$d=[3,3]$ & $\le 13/7$ \\
$d=[2,2,2]$ & $\le 13/7$ \\
$d=[2,2,3]$ & $\le 19/10$ \\
\hline
\end{tabular}
\end{center}

Nothing here establishes a \emph{lower bound} (floor) on any grid other than
$d=[2,2]$ (\S\ref{sec:exhaustive}). In particular our earlier phrasing
``$d=[2,3]$ floor $=9/5$'' was \textbf{incorrect} --- the time-budgeted floor
search on $d=[2,3]$ in fact returned $2.0$, and $9/5$ is only the score of an
explicit certificate (an upper bound). These multi-grid numbers are
\textbf{corroborating context, not part of the evidence chain} for the floor
claim, and were produced by heuristic search, not re-verified exhaustively. They
indicate only that no small grid we searched produced a certificate below $7/4$.

\subsection{The framework-floor characterization (evidence, not a
theorem)}\label{sec:charac}

We bring the pieces together into a \textbf{characterization} of the framework's
reach, explicit that it is \emph{supported by evidence and not proven}.

\begin{characterization}[informal, not a theorem]
Every forcing certificate is plausibly an instance of a finite sum--difference
construction; if so, the framework's achievable exponents are bounded below by
$\gamma = 1.6751\ldots$ (\S\ref{sec:gamma}--\ref{sec:seq}), and no certificate of
the present format can certify an exponent in $[1,\gamma)$.
\end{characterization}

This is the \emph{directional} claim (certificate $\Rightarrow$ finite-SD
instance, hence floored by $\gamma$); we do not assert the converse (every SD
construction is a certificate), which we neither need nor argue. The supporting
evidence:

\begin{itemize}
\item \textbf{(a) Structural.} A certificate is a product grid whose axis
projections coincide with the Katz--Tao sum--difference scaffold: each axis carries
an edge-labelled difference structure, the op1 generators are the sum--difference
pairings $\{x,-x\}$, and the closure synthesizes the forbidden $(1,-1)$ direction
exactly as the sum--difference method synthesizes the antidiagonal. The score $S$
\emph{is} the certified $\SD$ exponent by construction. So a certificate is,
structurally, an explicit finite instance of a sum--difference construction.
\item \textbf{(b) Computational.} The exhaustive $d=[2,2]$ minimum is exactly
$7/4$ and is \emph{invariant} to slope-count up to $7$ slopes
(\S\ref{sec:exhaustive}). No configuration in this exhaustive sweep drops below the
iteration's second term $7/4$. (The heuristic multi-grid upper bounds of
\S\ref{sec:multigrid} are consistent with this but, being non-exhaustive, are not
part of this evidence list.)
\item \textbf{(c) Literature --- bounded slopes force $\SD\to 2$.} Tao's November
2025 paper \cite{Tao25} shows that with a \emph{bounded} number of slopes the
sum--difference exponents converge to $2$. Every certificate uses a finite (hence
bounded) slope alphabet, so the bounded-slope regime is exactly the regime
certificates inhabit --- consistent with the observed slope-count invariance.
\item \textbf{(d) Literature --- no obvious sub-floor lever.} The two natural
strengthening routes do not obviously provide a lever below the floor.
Pohoata--Zakharov \cite{PZ24} introduce a general way to \emph{strengthen} the
Katz--Tao sum--difference inequalities and study a higher-dimensional variant;
their abstract describes an iterative upper-bound argument and a higher-dimensional
Besicovitch corollary, and we did not find in it a mechanism that lowers the
planar sum--difference exponent below $\gamma$. Cowen-Breen et al.\ \cite{CB20}
establish \emph{equivalences} relating arithmetic-Kakeya pattern problems (and
prove AKC implies the packing-dimension Kakeya conjecture); it is an
equivalence/implication paper rather than a source of a smaller upper-bound
exponent. (We verified the titles, authors, and high-level content of \cite{PZ24}
and \cite{CB20}; we did not verify line-by-line that neither contains a
sub-$\gamma$ lever, so we state this as ``no obvious lever found,'' not a proof.)
\end{itemize}

\paragraph{What a full proof would require --- and what we do not establish.}
To upgrade this to a theorem one would have to show that \textbf{every} forcing
certificate --- over \textbf{all} grids, \textbf{all} dimensions, and \textbf{all}
finite slope alphabets --- reduces to a finite sum--difference construction (and
hence cannot certify below $\gamma$). We do \textbf{not} establish this. We
therefore do \textbf{not} write ``we prove the framework equals the
sum--difference method.'' We write: we give structural, exhaustive-small-case, and
literature evidence that the framework's achievable exponents floor at $\gamma$,
and we state precisely the universal-reduction step a full proof would require.

\section{Consequences}\label{sec:consequences}

\paragraph{$7/4$ is the exact $2\times2$ minimum and the second iteration term.}
It is the second term of the advanced iteration ($2,\,7/4,\,171/101,\ldots$) and
the exhaustive minimum on the smallest grid (\S\ref{sec:exhaustive}), stable to
slope-count enrichment up to $7$ slopes. The next iteration terms
($171/101 \approx 1.693$, then $1.679$, $\ldots$) descend toward $\gamma$ without
ever reaching it.

\paragraph{Sub-$\gamma$ would require leaving the framework.} \emph{If} the
\S\ref{sec:charac} characterization holds, the framework's exponents floor at
$\gamma > 67/40$, so certifying any exponent in $[1,\gamma)$ --- including the AKC
endpoint $1$ (itself conjectural) and even the open band $(3/2,\gamma)$ --- cannot
be done within the certificate format as defined. The absolute floor of the
sum--difference \emph{method} is $3/2$, so $[3/2,\gamma)$ is \textbf{not ruled out
in principle}, but it is \textbf{unreached}: it would require both a new
inequality (stronger than sum--difference) \emph{and} a new certificate format
able to express it.

\paragraph{The warm-up/full-tier gap.} The warm-up threshold $7/4$ is reachable by
a tiny, fully explicit, machine-verifiable certificate --- the artifact of
\S\ref{sec:cert}. The next tiers, which approach the record ($1.67513\ldots$) and
ultimately the endpoint ($1$), do not appear reachable by enlarging the same
certificate; the (conjectured) floor at $\gamma$ would mean incremental
grid/alphabet growth asymptotes at $\gamma$ from above. As an \emph{observation},
the published record $1.67513\ldots$ and $\gamma=1.67513087\ldots$ coincide to the
digits reported --- consistent with the advanced iteration being the operative
recursion behind the record --- but we do \emph{not} derive an identity
``record $=\gamma$'' from our framework results; that is a statement about
Katz--Tao/Tao's optimization, not something this paper establishes.

\paragraph{What we did not do.} We did not beat the arithmetic--Kakeya record (it
stands at $\le 1.67513\ldots$), and $7/4$ is not new mathematics --- it is the
Katz--Tao $\{0,1,2,\infty\}$ bound made into the minimal explicit certificate.

\section{Reproducibility}\label{sec:repro}

All facts were re-verified on 2026-06-14 with Python 3.12.3 and sympy in a fixed local research environment.

\paragraph{Artifacts (relative research artifact locations).}
\emph{Verifier and certificates} ---
\texttt{.../dgx\_handoff/pipelines/00\_kakeya\_warmup/}:
gold verifier \texttt{kakeya\_verifier.py}; the $7/4$ certificate
\texttt{SOLVE\_kakeya\_7over4.cert}; the $9/5$ certificate
\texttt{best\_forcing\_cert\_9\_5.txt}; fast Fraction-RREF oracle
\texttt{kakeya\_fast.py}; explicit witnesses + round-by-round trace + tamper
control \texttt{round\_witnesses.py}, \texttt{trace\_tamper\_witness.py}.
\emph{Floor evidence} ---
\texttt{.../dgx\_handoff/pipelines/00\_kakeya\_sub1675/wf\_dinkelbach/}:
exhaustive $2\times2$ enumerator (with closed-form count assertion and
\texttt{AGREE=True} cross-check) \texttt{exhaust2x2\_clean.py} and log
\texttt{exhaust2x2\_clean.log}; $\gamma$ closed-form/sequence check
\texttt{gamma\_check.py}; heuristic multi-grid upper-bound searches and logs
\texttt{full\_2x3.py}/\texttt{full\_2x3.log}, \texttt{sweep\_richX.py}/\texttt{sweep\_richX.log},
\texttt{sweep\_3d.py}/\texttt{sweep\_3d.log} (the \S\ref{sec:multigrid} upper
bounds; \texttt{full\_2x3.log} records the $d=[2,3]$ floor probe returning $2.0$).

\paragraph{How to check the certificate.}
\begin{verbatim}
python kakeya_verifier.py SOLVE_kakeya_7over4.cert --threshold 7/4    # PASS, exit 0
python kakeya_verifier.py SOLVE_kakeya_7over4.cert --threshold 67/40  # REJECT, exit 1
\end{verbatim}
The CLI takes a \textbf{fraction} (\texttt{7/4}, \texttt{67/40}); a bare decimal
\emph{errors} (the parser does \texttt{Fraction(int(num), int(den))}, so
\texttt{"1.675"} raises). The verifier prints the validated slope set, the grid
and $n$, $m(G)$, the seed/free-set validations, the closure progress
(\texttt{closure reached 4/4 vertices}), and the score line. The certificate file
is six lines (line 1 a human-readable header ignored by the parser; then $X$, $d$,
$f$, $T$, $R$).

The forcing witnesses of \S\ref{sec:witness} (round-1 combination for $(2,2)$ and
round-2 combinations for the other three vertices), the rank-$7$ fact, and the
tamper control are produced and verified by \texttt{round\_witnesses.py} /
\texttt{trace\_tamper\_witness.py}, which drive the verifier's internal
\texttt{build\_module\_matrix} and \texttt{\_op2\_can\_force} routines. The
verifier's \texttt{run\_closure} returns only the final reached set and emits no
per-round output; the round structure is reconstructed by a driver over the
\texttt{\_op2\_can\_force} predicate, not printed by the verifier. The exhaustive
$2\times2$ search of \S\ref{sec:exhaustive} is driven by
\texttt{exhaust2x2\_clean.py}, which both asserts the iterated configuration count
equals the closed form $N(s)$ and prints \texttt{AGREE=True} from a direct
comparison of the fast oracle and the gold verifier on the published certificate.

\paragraph{Reconstruction caveat (required disclosure).} The ``gold verifier''
\texttt{kakeya\_verifier.py} is a \textbf{reconstruction} of Epoch's FrontierMath
verifier semantics, audited against three independent oracles (exact
Fraction-RREF, two-prime mod-$p$, and explicit witnesses) with \textbf{zero
disagreements} over the recorded $300$--$400$ randomly-generated small-certificate
trials. All PASS/REJECT results are \textbf{against the reconstructed verifier},
not an official Epoch run. \textbf{No public submission channel exists} for the
FrontierMath arithmetic-Kakeya track, so the $7/4$ certificate is a verified
research artifact, \emph{not} an official competition solve.

\paragraph{Citation status.} We verified the arXiv identifier, title, author list,
and abstract-level content of each reference (June 2026). We did \textbf{not}
verify the internal theorem/corollary numbers of \cite{KT01} against the paper
body; those labels are reproduced from the task brief and should be confirmed
before being relied upon. Where a specific sub-claim of a cited paper is
load-bearing and we could not confirm it from the abstract, we softened the
in-text statement accordingly (\S\ref{sec:charac}(d)).

\begin{thebibliography}{9}

\bibitem{KT01} N. H. Katz and T. Tao, \emph{New bounds on Kakeya problems},
arXiv:math/0102135 (2001). Sum--difference method; the $\{0,1,2,\infty\}$
slope-set bound $\SD=7/4$; the basic iteration $\beta\mapsto(4\beta-1)/(2\beta)$
and the advanced iteration $\beta\mapsto(3\beta^2+2\beta-2)/(\beta^2+3\beta-2)$.
(Specific theorem/corollary numbers not re-verified against the paper body.)

\bibitem{Tao25} T. Tao, \emph{Sum-difference exponents for boundedly many slopes,
and rational complexity}, arXiv:2511.15135 (19 November 2025). The current record
$\inf\SD(\R;-1)\le 1.67513$; with a bounded number of slopes the exponents
converge to $2$. (Title, author, date, record value, and bounded-slope statement
verified from the abstract; no internal theorem number cited.)

\bibitem{PZ24} C. Pohoata and D. Zakharov, \emph{Generalized Arithmetic Kakeya},
arXiv:2411.13395 (20 November 2024). A general way to strengthen the Katz--Tao
sum--difference inequalities; a higher-dimensional variant with a Besicovitch
Minkowski-dimension corollary. (Title, authors, date, and high-level content
verified from the abstract; the specific sub-claims ``$\beta_h=\beta$
equivalence'' and ``$\beta(\R^d)\ge d$'' were not confirmed and are not asserted.)

\bibitem{CB20} C. Cowen-Breen, E. Karangozishvili, N. Varadarajan, and T. Wang,
\emph{Pattern Problems related to the Arithmetic Kakeya Conjecture},
arXiv:2011.07056 (2020). Equivalences among arithmetic-Kakeya pattern problems;
AKC implies the Kakeya conjecture for packing dimension; uses number-theoretic
methods of Green--Ruzsa. (Title, full author list, and content verified from the
abstract; the earlier claim that ``its only upper-bound tool invokes Katz--Tao
sum--difference unmodified'' was not supported and has been removed.)

\end{thebibliography}

\end{document}