# An Explicit 2×2-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

*2026-06-14*

## Abstract

We report an explicit, machine-verified **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 ($2\times 2$), uses no free vertices, and realizes the Katz–Tao slope set $\{0,1,2,\infty\}$. Its correctness is established by an **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 **not**. The value $7/4$ is **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 **not** beat the arithmetic-Kakeya record; the best known sum–difference exponent, $\inf\mathrm{SD}(\mathbb R;-1)\le 1.67513\ldots$ (Katz–Tao, restated by Tao 2025), is untouched. Our contributions are (i) the explicit certificate realizing $7/4$ that is **minimal on the $2\times 2$ grid** (proven by exhaustive search) — improving the best small forcing certificate *we had previously constructed*, which scored $S=9/5$ on a $2\times 3$ grid; and (ii) a precisely-scoped **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 **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 $2\times 2$ 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.

---

## 1. Introduction

The **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 **sum–difference exponent** $\mathrm{SD}(\mathbb R;-1)$, the smallest exponent for which a certain sum-vs-difference inequality survives. The *conjectural* endpoint of the arithmetic Kakeya conjecture (AKC) corresponds to exponent $1$; the best known upper bound is $\inf\mathrm{SD}(\mathbb R;-1) \le 1.67513\ldots$, due in mechanism to Katz–Tao (2001) [KT01] and restated in Tao's November 2025 note [Tao25].

The Epoch **FrontierMath** benchmark includes an "arithmetic-Kakeya" track formulated as a **forcing-certificate** verification problem: a candidate submits a finite combinatorial object $(X,d,f,T,R)$ which a verifier accepts iff a *forcing closure* reaches every vertex of a product grid **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 $\mathrm{SD}$ 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.

**Contribution 1 (an explicit $2\times 2$-minimal artifact).** We give an explicit forcing certificate that passes at $\tau = 7/4$ on the $2\times 2$ grid with no free vertices (§3). It is the **minimum-scoring configuration on the $2\times 2$ grid** (exhaustive search, §4.3) and improves the best small forcing certificate *we had previously constructed*, which scored $S = 9/5 = 1.8$ on a $2\times 3$ 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 *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.

**Contribution 2 (evidence for a framework floor, scoped honestly).** We examine the reach of the certificate framework itself (§4). 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 **exhaustive** search of the $2\times 2$ grid — with a closed-form configuration count that we verify equals the iterated count — showing minimum score exactly $7/4$, *invariant* under enlarging the slope alphabet from $4$ to $7$ slopes; and (c) the Tao (2025) bounded-slope theorem, which forces $\mathrm{SD}\to 2$ when the number of slopes is bounded. Together these give structural and computational evidence — **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 (§5): within this framework $7/4$ is the exact $2\times 2$ minimum, and even reaching $(3/2,\gamma)$ would require a construction the framework as formatted cannot express.

We do **not** claim a new arithmetic-Kakeya bound, and we do **not** claim $7/4$ as a new bound. The record stands. All verifier results are reported against a *reconstruction* of the Epoch verifier (§6), and there is no public submission channel, so we frame $7/4$ as a verified research artifact rather than an official competition solve.

---

## 2. The forcing-certificate framework

### 2.1 The certificate object

A **certificate** is a tuple $(X, d, f, T, R)$.

- **Slope alphabet $X \subset \mathbb Z^2$.** 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 *synthesize*, so it may never appear as an input slope.

- **Grid $d=[d_1,\dots,d_k]$.** The vertex set is the product
$$
V \;=\; \{1,\dots,d_1\}\times\cdots\times\{1,\dots,d_k\}, \qquad n \;=\; \prod_{i=1}^{k} d_i .
$$

- **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 $(0,0)$.

- **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).

- **Seed relations $R$.** A list of **single-support** relations, each a map (one vertex $\mapsto$ one slope in $X$); each contributes exactly one nonzero entry.

### 2.2 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 **score** is
$$
\boxed{\,S \;=\; \dfrac{m + |R|}{\,n - |T|\,}\,}
$$
and equals the certified sum–difference exponent $\mathrm{SD}(X;-1)$ for the configuration.

### 2.3 op1 generators and the module

For each axis-$i$ edge labelled $x\in X$ between two adjacent vertices, the **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
$$
M \;=\; \mathbb Q\text{-span}\bigl(R \;\cup\; \{\text{op1 generators}\}\bigr) \;\subseteq\; \mathbb Q^{2n},
$$
where each vertex contributes two coordinates (a $2$-vector slope).

### 2.4 Forcing closure (op2)

For a current free set $F\subseteq V$ and a vertex $e\notin F$, the **antidiagonal direction** at $e$ is a vector $w\in\mathbb Q^{2n}$ 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 $e$ — this is exactly the forbidden $a+b=0$ direction of §2.1). We say $e$ is **forced given $F$** iff
$$
M \text{ (a subspace of } \mathbb Q^{2n}\text{) contains a vector } w \text{ with } w|_e=(s,-s),\ s\neq 0,\ \text{ and } w|_{v}=(0,0) \text{ for every } v\notin F\cup\{e\}.
$$
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 `_op2_can_force`): solve for a generator combination $y$ with $(yM)$ zero on all "outside" coordinates and $(yM)|_e=(1,-1)$ (we may fix $s=1$, since the line $\{(s,-s):s\neq0\}$ is closed under nonzero scaling).

The **closure** starts from $F=T$ and repeatedly adds every forceable vertex until a fixpoint. The certificate **PASSES at threshold $\tau$** iff
$$
\text{closure reaches all } n \text{ vertices} \qquad\text{and}\qquad S \le \tau .
$$

A certificate is therefore a finite, fully checkable witness: a passing certificate at $\tau$ certifies $\mathrm{SD}(X;-1)\le\tau$ for that configuration, with the forbidden $(1,-1)$ direction synthesized from the allowed slopes alone.

---

## 3. The $2\times 2$-minimal $7/4$ certificate

### 3.1 The certificate

$$
\begin{aligned}
X &= \{(0,0),(1,0),(1,1),(1,2),(0,1)\} && \text{(slopes }\{0,1,2,\infty\}\text{)}\\
d &= [2,2] && (n=4)\\
f &= \bigl[\,\{(1,):(1,0)\},\ \ \{(1,1):(1,1),\ (2,1):(1,1)\}\,\bigr]\\
T &= [\,] && (|T|=0)\\
R &= \bigl[\,\{(1,1):(1,2)\},\ \{(1,2):(0,1)\},\ \{(2,1):(0,1)\}\,\bigr] && (|R|=3)
\end{aligned}
$$

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, $2\times 2$, and there are **no free vertices**.

### 3.2 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
$$
m = 2 + 2 = 4, \qquad |R| = 3, \qquad n = 2\cdot 2 = 4, \qquad |T| = 0,
$$
$$
S = \frac{m+|R|}{n-|T|} = \frac{4+3}{4-0} = \frac{7}{4} = 1.75 .
$$
The reconstructed verifier independently recomputes $m(G)=4$ and reports $S = (4+3)/(4-0) = 7/4$.

### 3.3 Correctness: explicit forcing witnesses

The four vertices are $(1,1),(1,2),(2,1),(2,2)$. The certificate's subspace $M$ is spanned by $7$ generators (it has rank $7$ out of the $8$ ambient coordinates — it "wastes no degrees of freedom"): the three seeds $R$, and four op1 generators

$$
\begin{aligned}
&g_0 = R[(1,1)\!:\!(1,2)], \quad g_1 = R[(1,2)\!:\!(0,1)], \quad g_2 = R[(2,1)\!:\!(0,1)],\\
&g_3 = \mathrm{op1}[(1,)\!\to\!(1,0)]\,@(1,1), \quad g_4 = \mathrm{op1}[(1,)\!\to\!(1,0)]\,@(1,2),\\
&g_5 = \mathrm{op1}[(1,1)\!\to\!(1,1)]\,@(1,1), \quad g_6 = \mathrm{op1}[(2,1)\!\to\!(1,1)]\,@(2,1).
\end{aligned}
$$

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 not narrative "telescopings"; they are the actual solution vectors, verified by direct substitution.)

**Round 1 — the corner $(2,2)$ is forced first, and uniquely, from the empty free set $F=\varnothing$.** The combination
$$
w_{(2,2)} \;=\; g_0 - 2\,g_1 - g_2 + g_3 - 2\,g_4 - 2\,g_5 + g_6
$$
evaluates to
$$
w_{(2,2)}|_{(1,1)}=(0,0),\quad w_{(2,2)}|_{(1,2)}=(0,0),\quad w_{(2,2)}|_{(2,1)}=(0,0),\quad w_{(2,2)}|_{(2,2)}=(1,-1).
$$
Thus the forbidden target direction is synthesized at $(2,2)$ alone, from the allowed slopes $\{0,1,2,\infty\}$ — this is the heart of the certificate. A round-by-round driver over the verifier's `_op2_can_force` predicate confirms $(2,2)$ is the **unique** vertex forceable in round 1.

**Round 2 — the remaining three cascade, each with its own explicit witness.** With $(2,2)\in F$, the "outside" region for each remaining vertex $e$ shrinks to the two vertices other than $e$ and $(2,2)$, and each $e$ becomes forceable:
$$
\begin{aligned}
w_{(1,2)} &= -\,g_1 + g_4 &&\Rightarrow\quad w_{(1,2)}|_{(1,2)}=(1,-1),\ \ w_{(1,2)}|_{(1,1)}=w_{(1,2)}|_{(2,1)}=(0,0),\ \ w_{(1,2)}|_{(2,2)}=(-1,0),\\
w_{(2,1)} &= -\,g_0 + 2\,g_1 - g_2 - g_3 + 2\,g_4 + 2\,g_5 &&\Rightarrow\quad w_{(2,1)}|_{(2,1)}=(1,-1),\ \ \text{zero at }(1,1),(1,2),\ \ w_{(2,1)}|_{(2,2)}=(-2,0),\\
w_{(1,1)} &= -2\,g_0 + 3\,g_1 + 3\,g_4 + 3\,g_5 &&\Rightarrow\quad w_{(1,1)}|_{(1,1)}=(1,-1),\ \ \text{zero at }(1,2),(2,1),\ \ w_{(1,1)}|_{(2,2)}=(-3,0).
\end{aligned}
$$
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
$$
\text{round 1: forced }\{(2,2)\};\qquad \text{round 2: forced }\{(1,1),(1,2),(2,1)\},
$$
reaching all $4$ of $4$ vertices. Closure succeeds, and since $S=7/4\le 7/4$, the certificate **PASSES** at $\tau=7/4$. (Each $w$ above was verified by direct substitution into $M$; see §6.)

### 3.4 The $9/5 \to 7/4$ improvement (over our own prior small certificate)

The best small forcing certificate *we had previously constructed* scored $S = 9/5 = 1.8$ on the same slope alphabet but a larger grid:
$$
d = [2,3],\quad n=6,\quad m = 7,\quad |R| = 2,\quad |T| = 1,\qquad S = \frac{7+2}{6-1} = \frac{9}{5}.
$$
We emphasize "small" means *the smallest explicit construction we had on hand* (here $n\le 6$), not a global record: the $9/5$ certificate proves only the *upper bound* $\mathrm{SD}\le 9/5$ on $d=[2,3]$, and indeed our searches on that grid (§4.4) 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 $2\times 3$ ladder spends a free vertex (a $|T|=1$ penalty in the denominator) and seven edge labels to walk the forbidden direction down a three-step axis. On the $2\times 2$ grid the corner $(2,2)$ is forced **directly from the empty free set** — no ladder, no free vertex — so $|T|$ drops from $1$ to $0$ and $m$ drops from $7$ to $4$. The score falls from $\tfrac{7+2}{6-1}=\tfrac{9}{5}$ to $\tfrac{4+3}{4-0}=\tfrac{7}{4}$. The improvement is real: it is the elimination of the telescoping free vertex, and (by §4.3) $7/4$ is simultaneously the exact $2\times 2$ minimum, so the move is "find the $2\times 2$ optimum, which lies below the previously-exhibited $2\times 3$ construction."

### 3.5 Multi-oracle verification and a tamper control

The certificate's correctness was checked five ways (all re-run 2026-06-14):

1. **Gold verifier (reconstruction).** The reconstructed Epoch sympy verifier returns PASS (exit $0$) at $\tau = 7/4$ and REJECT (exit $1$) at $\tau = 67/40 = 1.675$, exactly as the score $7/4 > 67/40$ predicts.
2. **Independent Fraction-RREF oracle.** An exact rational row-reduction of the subspace reproduces the closure and score; on the published certificate it agrees with the gold verifier (the cross-check prints `AGREE=True`).
3. **Independent two-prime mod-$p$ oracle.** Reducing the generators modulo two distinct primes reproduces the same forcing closure, guarding against a rational-arithmetic artifact.
4. **Explicit witnesses.** The round-1 and round-2 combinations of §3.3, each verified by direct substitution.
5. **Tamper / negative control.** Replacing the seed $\{(1,1):(1,2)\}$ with any of the three other allowed slopes breaks the closure:
$$
(1,1)\mapsto(1,1):\ \text{reaches }1/4;\quad (1,1)\mapsto(2,1):\ 0/4;\quad (1,1)\mapsto(1,0):\ 1/4.
$$
The certificate is **delicately placed**: it is not robust to seed perturbation, which is exactly what one expects of a minimal realization that wastes no degrees of freedom.

Across the recorded audit (300–400 trials over randomly generated small certificates) the three independent oracles registered **zero disagreements** with the gold verifier. We emphasize (§6) that "gold verifier" denotes a *reconstruction* of Epoch's verifier semantics, not an official Epoch run; all PASS/REJECT statements are against the reconstruction.

---

## 4. Evidence for a framework floor

### 4.1 Katz–Tao sum–difference recursions and the constant $\gamma$

The Katz–Tao sum–difference method [KT01] improves its exponent by iterating a recursion. The **basic** recursion is
$$
\beta \;\longmapsto\; \frac{4\beta-1}{2\beta},
$$
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 **advanced** recursion is
$$
\beta \;\longmapsto\; \frac{3\beta^2 + 2\beta - 2}{\beta^2 + 3\beta - 2}.
$$
Setting this equal to $\beta$ and clearing denominators reduces **exactly** (sympy `cancel`/`fraction`/`expand`) to
$$
-\beta^3 + 4\beta - 2 = 0 \qquad\Longleftrightarrow\qquad \boxed{\,\beta^3 - 4\beta + 2 = 0\,}.
$$
This cubic has three real roots, $-2.2143\ldots$, $0.5392\ldots$, and
$$
\gamma \;=\; 1.6751308705666460709\ldots,
$$
the largest, which is 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.

### 4.2 The advanced iteration descends to $\gamma$ from above

Starting the advanced iteration at $\beta=2$ produces the exact decreasing rational sequence
$$
2,\ \frac{7}{4},\ \frac{171}{101},\ \frac{101863}{60652},\ \ldots
\;=\; 2,\ 1.75,\ 1.69307,\ 1.67947,\ 1.67618,\ 1.67539,\ 1.67519,\ \ldots
$$
converging to $\gamma$ **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 *second* term — the first non-trivial improvement off the base case $\beta=2$. (We note this is a statement about the iteration; whether *all* finite sum–difference constructions are iteration outputs is the stronger claim conjectured in §4.5, not proved here.)

### 4.3 Exhaustive $2\times 2$ search and slope-count invariance

The load-bearing computational evidence is an **exhaustive** search of the $2\times 2$ grid under the verifier's forcing semantics, using the fast Fraction-RREF oracle (cross-checked `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.

**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 therefore 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 `MATCH=True`):

| slopes $s$ | $(s+1)^3$ | $\sum_{r\le3}\binom{4s}{r}$ | configurations $N(s)$ | min forcing score | time |
|:---:|:---:|:---:|:---:|:---:|:---:|
| $4\ \{0,1,2,\infty\}$ | $125$ | $697$ | $\mathbf{1{,}394{,}000}$ | $\mathbf{7/4}$ | $24.2$ s |
| $5$ | $216$ | $1{,}351$ | $\mathbf{4{,}669{,}056}$ | $\mathbf{7/4}$ | $71.7$ s |
| $6$ | $343$ | $2{,}325$ | $\mathbf{12{,}759{,}600}$ | $\mathbf{7/4}$ | $174.5$ s |
| $7$ | $512$ | $3{,}683$ | $\mathbf{30{,}171{,}136}$ | $\mathbf{7/4}$ | $376.3$ s |

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. **Enlarging the slope alphabet from $4$ to $7$ slopes adds no forcing power on the $2\times 2$ grid.**

Three scope caveats are required. **(i)** "Exhaustive" is asserted *for this enumeration convention*, and is made checkable by the closed form $N(s)$; a different convention (e.g. counting symmetry orbits, or allowing $|R|>3$) would give a different total. We deliberately report a single reproducible convention with its verified count, and we do **not** rely on any earlier, unreconciled configuration-count figure. **(ii)** The claim is for **$d=[2,2]$ only**; it does *not* prove that no larger grid or alphabet anywhere beats $7/4$. **(iii)** The slope-count invariance is *consistent with* — but not a proof of — the Tao (2025) bounded-slope phenomenon (§4.5).

### 4.4 Heuristic upper bounds on larger grids (search-found, not floors)

We also ran heuristic (time-budgeted, Dinkelbach-style) searches over a range of grids. These return the **best certificate found** — i.e. *upper bounds* on the achievable score for each grid, not floors — and the searches are non-exhaustive and disagree with one another at the margins. The recorded best-found scores (with the log files cited in §6) are:

| configuration | best certificate found (upper bound) |
|:--|:--:|
| $d=[2,3]$ | $\le 9/5$ (explicit cert; the time-budgeted floor probe on this grid only reached $2$) |
| $d=[3,3]$ | $\le 13/7$ |
| $d=[2,2,2]$ | $\le 13/7$ |
| $d=[2,2,3]$ | $\le 19/10$ |

We stress: nothing here establishes a *lower bound* (floor) on any grid other than $d=[2,2]$ (§4.3). In particular our earlier phrasing "$d=[2,3]$ floor $=9/5$" was **incorrect** — the time-budgeted floor search on $d=[2,3]$ in fact returned $2.0$, and the $9/5$ value is only the score of an explicit certificate (an upper bound). These multi-grid numbers are **corroborating context, not part of the evidence chain** for the floor claim, and they were produced by heuristic search, not re-verified exhaustively. They are included only to indicate that no small grid we searched produced a certificate below $7/4$.

### 4.5 The framework-floor characterization (evidence, not a theorem)

We bring the pieces together into a **characterization** of the framework's reach, and we are explicit that it is *supported by evidence and not proven*.

> **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 $\gamma = 1.6751\ldots$ (§4.1–4.2), and no certificate of the present format can certify an exponent in $[1,\gamma)$.

This is the *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:

**(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 $S$ *is* the certified $\mathrm{SD}$ exponent by construction. So a certificate is, structurally, an explicit finite instance of a sum–difference construction.

**(b) Computational.** The exhaustive $d=[2,2]$ minimum is exactly $7/4$ and is *invariant* to slope-count up to $7$ slopes (§4.3). No configuration in this exhaustive sweep drops below the iteration's second term $7/4$. (The heuristic multi-grid upper bounds of §4.4 are consistent with this but, being non-exhaustive search results, are not part of this evidence list.)

**(c) Literature — bounded slopes force $\mathrm{SD}\to 2$.** Tao's November 2025 paper [Tao25] shows that with a *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.

**(d) Literature — no obvious sub-floor lever.** The two natural strengthening routes do not obviously provide a lever below the floor. Pohoata–Zakharov [PZ24] introduce a general way to *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. [CB20] establish *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 have verified the titles, authors, and high-level content of [PZ24] and [CB20]; we have not verified line-by-line that neither contains a sub-$\gamma$ lever, so we state this as "no obvious lever found," not as a proof.)*

**What a full proof would require — and what we do not establish.** To upgrade this to a theorem one would have to show that **every** forcing certificate — over **all** grids, **all** dimensions, and **all** finite slope alphabets — reduces to a finite sum–difference construction (and hence cannot certify below $\gamma$). We do **not** establish this. We therefore do **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.*

---

## 5. Consequences

**$7/4$ is the exact $2\times 2$ 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 (§4.3), 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.

**Sub-$\gamma$ would require leaving the framework.** *If* the §4.5 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 *method* is $3/2$, so the interval $[3/2,\gamma)$ is **not ruled out in principle**, but it is **unreached**: it would require both a new inequality (something stronger than sum–difference) *and* a new certificate format able to express it. The present format, on the evidence here, cannot.

**The warm-up/full-tier gap.** This delineates the FrontierMath tier structure. The warm-up threshold $7/4$ is reachable by a tiny, fully explicit, machine-verifiable certificate — exactly the artifact of §3. 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 *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 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.

**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.

---

## 6. Reproducibility

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

**Artifacts** (relative research artifact locations).

*Verifier and certificates* — `…/dgx_handoff/pipelines/00_kakeya_warmup/`:
- Gold verifier: `kakeya_verifier.py`
- The $7/4$ certificate: `SOLVE_kakeya_7over4.cert`
- The $9/5$ certificate: `best_forcing_cert_9_5.txt`
- Fast Fraction-RREF oracle: `kakeya_fast.py`
- Explicit witnesses + round-by-round trace + tamper control: `round_witnesses.py`, `trace_tamper_witness.py`

*Floor evidence* — `…/dgx_handoff/pipelines/00_kakeya_sub1675/wf_dinkelbach/`:
- Exhaustive $2\times2$ enumerator (with closed-form count assertion and `AGREE=True` cross-check): `exhaust2x2_clean.py`, log `exhaust2x2_clean.log`
- $\gamma$ closed-form / sequence check: `gamma_check.py`
- Heuristic multi-grid upper-bound searches and their logs: `full_2x3.py` / `full_2x3.log`, `sweep_richX.py` / `sweep_richX.log`, `sweep_3d.py` / `sweep_3d.log` (these are the §4.4 upper bounds; `full_2x3.log` records the $d=[2,3]$ time-budgeted floor probe returning $2.0$).

**How to check the certificate.**
```
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
```
The CLI takes a **fraction** (`7/4`, `67/40`); a bare decimal *errors* (the parser does `Fraction(int(num), int(den))`, so `"1.675"` raises). The verifier prints the validated slope set, the grid and $n$, $m(G)$, the seed/free-set validations, the closure progress (`closure reached 4/4 vertices`), and the score line `S = (m+|R|)/(n-|T|) = (4+3)/(4-0) = 7/4`. The certificate file is six lines (line 1 is a human-readable header ignored by the parser; then $X$, $d$, $f$, $T$, $R$).

The forcing witnesses of §3.3 (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 `round_witnesses.py` / `trace_tamper_witness.py`, which drive the verifier's internal `build_module_matrix` and `_op2_can_force` routines. Note: the verifier's `run_closure` returns only the final reached set and emits no per-round output; the round structure is reconstructed by a driver over the `_op2_can_force` predicate, not printed by the verifier itself. The exhaustive $2\times 2$ search of §4.3 is driven by `exhaust2x2_clean.py`, which both (a) asserts the iterated configuration count equals the closed form $N(s)$ and (b) prints `AGREE=True` from a direct comparison of the fast oracle and the gold verifier on the published certificate.

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

**Citation status.** We verified the arXiv identifier, title, author list, and abstract-level content of each reference below (June 2026). We did **not** verify the *internal theorem/corollary numbers* of [KT01] against the paper body; those labels are reproduced from the task brief and should be confirmed against the source 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 have softened the in-text statement accordingly (§4.5(d)).

---

## References

[KT01] N. H. Katz and T. Tao, *New bounds on Kakeya problems*, arXiv:math/0102135 (2001). Sum–difference method; the $\{0,1,2,\infty\}$ slope-set bound $\mathrm{SD}=7/4$; 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; confirm before relying.)*

[Tao25] T. Tao, *Sum-difference exponents for boundedly many slopes, and rational complexity*, arXiv:2511.15135 (19 November 2025). The current record $\inf\mathrm{SD}(\mathbb R;-1)\le 1.67513$; 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 is cited here, as the abstract does not number that result.)*

[PZ24] C. Pohoata and D. Zakharov, *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 "$\beta(\mathbb R^d)\ge d$" were not confirmed and are not asserted in this note.)*

[CB20] C. Cowen-Breen, E. Karangozishvili, N. Varadarajan, and T. Wang, *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 by the abstract and has been removed.)*

---

**Honesty summary.**

| Status | Claims |
|:--|:--|
| **Proven** (closed-form, re-verified) | $\gamma$ is the largest root of $x^3-4x+2$; $\gamma>67/40$ by $1.30871\times10^{-4}$; basic fixed point $1+\sqrt2/2$; advanced sequence $2,7/4,171/101,101863/60652,\ldots\downarrow\gamma$. |
| **Computational** (freshly re-run this session) | the $7/4$ certificate's PASS at $7/4$ / REJECT at $67/40$, arithmetic, explicit round-1 and round-2 forcing witnesses, rank $7$, tamper control; the $9/5$ certificate (upper bound) passes; the exhaustive $d=[2,2]$ search with closed-form-verified counts ($1{,}394{,}000$/$4{,}669{,}056$/$12{,}759{,}600$/$30{,}171{,}136$ for $4$/$5$/$6$/$7$ slopes), minimum $=7/4$ in all, minimizer $=$ published cert; `AGREE=True` cross-check. |
| **Literature** (verified at abstract level) | $7/4$ is the Katz–Tao $\{0,1,2,\infty\}$ bound; the arithmetic-Kakeya record $\le 1.67513\ldots$ is untouched; Tao (2025) bounded-slope $\to 2$. |
| **Conjecture / evidence** (explicitly *not* a proof) | the directional framework$\Rightarrow$finite-SD characterization (floor at $\gamma$); the §4.4 multi-grid figures are heuristic upper bounds, not floors (and the earlier "$d=[2,3]=9/5$ floor" was wrong — the search reached $2.0$). |