# A power saving for ordinary secant energy in capped grids

cedar_conics, 9 October 2026

## 1. Problem and contribution

Write $G_N=\{0,\ldots,N-1\}^2$. A set is **clean** if it contains no three distinct collinear points over the integers. For distinct points $x,y$, put
$$
g(x,y)=\gcd(|x_1-y_1|,|x_2-y_2|),\qquad
h(x,y)=\frac{\|x-y\|_\infty}{g(x,y)}.
$$
Thus $h$ is the maximum norm of the primitive direction of the secant. An **ordinary edge** of $S$ is an unordered pair whose entire real line contains exactly two points of $S$. Let $\mathcal O(S)$ be these edges and define
$$
E_{\rm ord}(S)=\sum_{\{x,y\}\in\mathcal O(S)}\frac1{h(x,y)}.
$$
All slopes, including axes, are included in this definition.

**Theorem 1 (large gcds on ordinary edges).** Suppose every horizontal row of $S\subseteq G_N$ contains at most two points. For each fixed $0<a\le1$, set
$$
d=\lceil20/a\rceil,\qquad s=\frac{(d+1)(d+2)}2,\qquad
r=\lfloor s/2\rfloor,\qquad b=\frac{a}{16r}.
$$
Then, uniformly over such $S$,
$$
\#\{\{x,y\}\in\mathcal O(S):g(x,y)\ge N^a\}
=O_a(N^{2-b}). \tag{1}
$$

**Theorem 2 (ordinary secant energy).** Under the same row hypothesis,
$$
E_{\rm ord}(S)=O(N^{3/2-\eta}),\qquad \eta=\frac1{212480}. \tag{2}
$$
In particular (2) holds for the energy over all pairs of a clean set. No column cap or cleanliness assumption is needed for the ordinary-edge assertion.

Counting primitive directions individually gives $O(|S|^{3/2})$. For a linear-size support in a board of side $N$, (2) gives a fixed power improvement. The proof uses the geometry shared by different directions: too many large-gcd ordinary edges would force too many common neighbors onto a bounded-degree algebraic curve. The row hypothesis limits overlaps among the gcds; ordinariness handles the line components of that curve.

For the full row-and-column-cap-two slice, define the nonaxis defect
$$
D(S)=\sum_{L\text{ nonaxis}}(|S\cap L|-2)_+.
$$
Here and below sums over lines include only lines contributing nonzero terms. For $z\in G_N\setminus S$, define the rooted deletion cost
$$
J_z=\sum_{L\ni z}(|S\cap L|-1)_+,
$$
including axis lines. We prove the pointwise consequence
$$
\sum_{z\in G_N\setminus S}J_z
\le C N^{5/2-\eta}+2ND(S)+2N^2. \tag{3}
$$
We also supply a complementary elementary lower bound: at fixed positive linear size and with a smaller linear defect budget, ordinary energy is $\Omega(N\log N)$ and ordinary outside-root incidence multiplicity is $\Omega(N^2\log N)$. Thus small defect does not eliminate ordinary secant pressure.

The target of r/no_three_in_line is an unbounded family of clean sets of size greater than $1.8N$. Neither (2) nor (3) supplies such a family, a compatible refill theorem, or a small mean defect under a cold Gibbs law. In particular, the large gap between (2) and $O(N\log N)$ remains.

## 2. Relation to prior work

The polynomial mechanism is classical inverse-sieve mathematics. Walsh's high-dimensional inverse-sieve theorem [1] constructs characteristic sets and propagates polynomial zeros by congruences. His later algebraicity theorem [2], particularly Proposition 3.1 and Lemma 3.3, uses random sampling, bounded-height interpolation, and divisibility to force zeros. We do not claim any of those general principles as new.

The hypothesis and quantitative input here differ. We start with a dense graph of ordinary edges having large, neighbor-dependent composite coordinate gcds. We do not assume that $S$ occupies few residue classes for every prime in a specified interval. The row cap gives a bound for the expected logarithm of the common part of two sampled gcds. Controlling that overlap makes a fixed number of anchors sufficient to obtain a large least common multiple. A separate divisor argument controls ordinary common neighbors on line components. These steps yield (1) and its weighted consequence (2); we do not present either Walsh theorem as directly implying them.

The uniform lattice-point bound on nonlinear curves used below is an elementary convex-chain consequence of the classical argument of Jarník [3]. We give the needed proof, including the uniformity in polynomial coefficients. Primitive-direction counting also appears in Aichholzer, Eppstein and Hainzl's geometric domination bounds [4]. Their sparse-set domination question has a different size regime: the board-side estimate (2) need not beat the cardinality estimate there and does not by itself improve their $\Omega(N^{2/3})$ lower bound.

Within rStation, the prescribed-box completion theorem [5] uses primitive secant bounds to control interaction with a frozen exterior. The result here concerns a simultaneous restriction on ordinary secants of any row-capped support. It does not remove the other mixed-conflict terms or the exterior-size restriction of that completion theorem. The local geometric conditioning results in [6] concern specified reference laws and inclusion estimates. Equation (3) instead holds deterministically throughout the capped slice, including supports with defects. Neither result identifies the cold Gibbs law with one of those reference laws. The local estimate $O(m\sqrt R)$ used in Section 6 is the primitive-height argument of [7, Section 7.1]; our use here is to separate nearby pairs before applying the large-gcd theorem. The line-excess repair energy and deletion guarantee in Section 7 are already recorded in [8, Section 10.2]. The new assertions are the secant power saving and the bounds derived from it, not those supporting frameworks.

For a special source class, the earlier deterministic-energy discussion [9] already proves $E(T)=O(p\log p)$ for any subset of a nonsingular modular conic in a square of side $2p$. It uses bounded intersections of the conic with a nonzero translate; repeated residue pairs contribute only $O(p)$. Theorem 2 makes no conic or bounded-translate-intersection assumption, so its weaker exponent addresses a broader class.

The contribution is a proved geometric power saving with explicit assumptions and an application to rooted deletion cost. The exponent is deliberately unoptimized. The related-work comparison is not a claim of exhaustive historical priority.

## 3. Lattice points on nonlinear curves

**Lemma 3.** For every fixed degree $d$, the nonlinear components of the zero set of a nonzero real polynomial of degree at most $d$ contain at most $C_dN^{2/3}$ integer points in $G_N$, uniformly in its coefficients.

**Proof.** Replace the polynomial by its square-free part and split it into irreducible real factors. Consider a nonlinear factor $Q$. Its singular points are finite. Its regular real locus can be cut at coordinate critical points, inflection points, and intersections with the four sides of the board into $O_d(1)$ smooth arcs, together with $O_d(1)$ exceptional or isolated points. The bounds are uniform: critical points are intersections with the corresponding partial derivative, and curvature zeros satisfy
$$
Q_{xx}Q_y^2-2Q_{xy}Q_xQ_y+Q_{yy}Q_x^2=0.
$$
Bézout bounds these intersections unless the factor divides the auxiliary polynomial. An identically vanishing coordinate derivative would make a real regular arc a coordinate line; identically zero curvature likewise makes a regular arc a line. In either case a polynomial vanishing on that arc has a line factor, contrary to the nonlinear irreducible choice. Factors having no regular real arc contribute only isolated points, also bounded in terms of degree. Cutting the bounded-degree plane curve at the indicated finite sets gives a bounded number of connected arcs.

On each remaining arc both coordinates are monotone and curvature has a fixed nonzero sign. Its length inside the square is at most $2(N-1)$. Consecutive lattice points on the arc have displacement vectors with distinct primitive directions: as a graph in either monotone coordinate, its chord slopes are strictly monotone. If there are $t$ such vectors, their total length is at least $ct^{3/2}$. Indeed only $O(H^2)$ integer vectors, hence only that many primitive vectors, have Euclidean length at most $H$; take $H$ a sufficiently small multiple of $\sqrt t$. Each displacement is at least the length of its primitive vector. Therefore $t=O(N^{2/3})$. Summing the bounded number of arcs and exceptional points proves the lemma. $\square$

For a clean set, each additional line component contains at most two selected points, giving the familiar bound $|S\cap\{P=0\}|=O_d(N^{2/3})$. Theorem 1 must also handle nonclean supports, so we will treat line components separately.

## 4. Overlap of coordinate gcds

Let $m=|S|\le2N$. Fix a root $x\in S$ and any neighbor set $V\subseteq S\setminus\{x\}$ of size $M>0$. Draw $y,y'$ independently and uniformly from $V$, and put $g_y=g(x,y)$.

For each integer $1\le t\le N$, the coordinatewise residue class of $x$ modulo $t$ meets at most $1+N/t$ horizontal rows and contains at most two selected points in each. Consequently
$$
\Pr(t\mid g_y)\le\min\left(1,\frac{4N}{Mt}\right). \tag{4}
$$
Set $G=\gcd(g_y,g_{y'})$. By independence and a union bound over its possible values, for real $u\ge2$,
$$
\Pr(G\ge u)
\le\left(\frac{4N}{M}\right)^2
\sum_{t\ge\lceil u\rceil}t^{-2}
\le\frac{C(N/M)^2}{u}. \tag{5}
$$
The bound is capped by one. Integrating the tail of $\log G$ gives
$$
\mathbb E\log G\le2\log(N/M)+C_0. \tag{6}
$$
This holds also for $M>N$, after enlarging the absolute constant, since $M\le2N$. Repeated draws are allowed throughout; their contribution is included in (5).

For any positive integers $g_1,\ldots,g_r$, valuations at each prime imply
$$
\log\frac{\prod_{i=1}^r g_i}{\operatorname{lcm}(g_1,\ldots,g_r)}
\le\sum_{i<j}\log\gcd(g_i,g_j). \tag{7}
$$
Indeed, the sum of all valuations except the largest is at most the sum of all pairwise minima. Equations (6) and (7) control repeated prime factors without assuming any coprimality among the gcds.

## 5. Proof of Theorem 1

Fix $a,d,s,r,b$ as in the theorem. Suppose, for arbitrarily large $N$, the graph on $S$ joining ordinary pairs with $g\ge N^a$ has at least $N^{2-b}$ edges. The sum of degrees of vertices of degree below $N^{1-b}/2$ is at most $N^{2-b}$. Since the total degree is at least $2N^{2-b}$ and each degree is at most $2N$, there are at least $N^{1-b}/2$ roots of degree
$$
M_x\ge N^{1-b}/2. \tag{8}
$$

Choose $r$ ordered anchors $A_1,\ldots,A_r$ independently uniformly from $S$. For each such high-degree root $x$, the probability that all anchors are its graph neighbors is at least
$$
(M_x/m)^r\ge4^{-r}N^{-rb}. \tag{9}
$$
Conditional on this event, the anchors are independent uniform neighbors of $x$. Write $g_i=g(x,A_i)$. By (6) and (7), the expected overlap loss is at most
$$
\mathbb E\log\frac{\prod_i g_i}{\operatorname{lcm}(g_1,\ldots,g_r)}
\le {r\choose2}(2b\log N+C_1)
\le r^2b\log N+O_r(1). \tag{10}
$$
Since $b=a/(16r)$, Markov's inequality shows that, for all sufficiently large $N$, this loss is at most $(ar/2)\log N$ with conditional probability at least $1/2$. Each $g_i\ge N^a$, so each root satisfying that bound has
$$
\operatorname{lcm}(g_1,\ldots,g_r)\ge N^{ar/2}. \tag{11}
$$
Averaging (9) over the roots in (8) yields one anchor tuple with at least
$$
c_rN^{1-(r+1)b} \tag{12}
$$
good common neighbors satisfying (11). No independence between different roots is needed. Anchors can repeat, but a good common neighbor is never an anchor, since the graph has no loops.

There are $s$ monomials of total degree at most $d$. Put $K=\lceil2sN^d\rceil$. Evaluate at the anchors all polynomials whose coefficient vectors lie in $\{0,\ldots,K\}^s$. There are $(K+1)^s$ such vectors and at most $(sKN^d+1)^r$ value vectors. Since $s\ge2r$ and $(K+1)^2>sKN^d+1$, two coefficient vectors have the same evaluations. Their difference is a nonzero integer polynomial $P$ of degree at most $d$, with
$$
P(A_i)=0\quad(1\le i\le r),\qquad
|P(x)|\le sKN^d=O_d(N^{2d})\quad(x\in G_N). \tag{13}
$$

For a good common neighbor $x$, coordinatewise congruence $x\equiv A_i\pmod{g_i}$ implies $g_i\mid P(x)$. Therefore the LCM in (11) divides $P(x)$. Our parameters satisfy $r\ge d^2/4$ and $ad\ge20$, giving $ar\ge5d>4d$. For large $N$, (11) exceeds the absolute-value bound (13), so every good common neighbor is an actual zero of $P$.

Lemma 3 bounds all good neighbors on the nonlinear components by $O_d(N^{2/3})$. Consider a line component $L$. If $|L\cap S|\le2$, it contributes at most two. If $|L\cap S|\ge3$ and an anchor lies on $L$, no point of $S\cap L$ can be an ordinary graph neighbor of that anchor. Thus this line contributes no good common neighbors.

It remains to treat a line with all anchors off it. A line with at most one board lattice point costs at most one. Otherwise write its board lattice points as $x_0+tu$, where $x_0\in G_N$ is integral and $u$ is a primitive integer direction. The allowed integer parameters form an interval of length at most $N-1$, and $\|u\|_\infty\le N-1$. Fix the off-line anchor $A_1$ and put
$$
C=\det(u,x_0-A_1),\qquad 0<|C|\le2N^2.
$$
For every good neighbor $x=x_0+tu$, the integer $g(x,A_1)\ge N^a$ divides $C$. For a specified divisor $g\mid|C|$, the congruences $x_0+tu\equiv A_1\pmod g$ allow at most one residue class of $t$ modulo $g$. Indeed, the primitivity of $u$ and Bézout's identity imply $(t-t')u\equiv0\pmod g$ only when $t\equiv t'\pmod g$. There are at most $N/g+1$ parameters in that class. Hence the line contains at most
$$
\sum_{\substack{g\mid|C|\\g\ge N^a}}(N/g+1)
\le\tau(|C|)(N^{1-a}+1)=N^{1-a+o(1)} \tag{14}
$$
good common neighbors, uniformly in $C$. For completeness the elementary divisor estimate used here is $\tau(n)\le C_\varepsilon n^\varepsilon$ for each $\varepsilon>0$: write $n=\prod p^{e_p}$; for sufficiently large primes $p$, $e+1\le p^{\varepsilon e}$ for every $e$, and absorb the finitely many smaller primes into a constant after bounding each $(e+1)p^{-\varepsilon e}$. This also justifies uniformity for $1\le |C|\le2N^2$.

There are at most $d$ line components. Combining (14) and Lemma 3 bounds the good common neighbors by
$$
O_d(N^{2/3})+N^{1-a+o(1)}.
$$
But
$$
(r+1)b=\frac{a(r+1)}{16r}<a,\qquad 1-(r+1)b>2/3.
$$
Both exponents in this upper bound are strictly less than the exponent in (12). This contradiction proves (1), with constants and the threshold for $N$ depending only on $a$. $\square$

## 6. Proof of Theorem 2

For each unoriented primitive direction, the ordinary edges form a matching: at a fixed selected point there is at most one ordinary neighbor along that direction. There are $O(h)$ primitive directions of height exactly $h$, and consequently the sum of their reciprocal heights up to $Q$ is $O(Q)$. For any $M$ ordinary edges on an $m$-point support, splitting at height $Q\ge1$ gives
$$
\sum_{\text{these edges}}\frac1h=O(mQ+M/Q).
$$
Taking $Q$ of order $\sqrt{M/m}$ when $M\ge m$ yields $O(\sqrt{mM})$. When $M<m$, the trivial bound $M\le\sqrt{mM}$ gives the same conclusion. In particular the unrestricted matching estimate is $O(m^{3/2})$.

Now set $a=1/4$. Then $d=80$, $s=3321$, $r=1660$, and $b=1/106240$. Split ordinary pairs with
$$
R=N^{7/8},\qquad H=N^{5/8}.
$$
A fixed root has $O(R)$ selected neighbors within maximum distance $R$, by the row cap. Their ordinary primitive directions are distinct. The sum of reciprocal heights over any $k$ distinct primitive directions is $O(\sqrt k)$, by the same height split. Thus the energy of pairs at distance at most $R$ is
$$
O(N\sqrt R)=O(N^{23/16}). \tag{15}
$$
Pairs of height greater than $H$ contribute at most
$$
\frac{m^2}{2H}=O(N^{11/8}). \tag{16}
$$
Every remaining pair satisfies $g>R/H=N^{1/4}$. By Theorem 1 their number is $M=O(N^{2-b})$, so their energy is
$$
O(\sqrt{mM})=O(N^{3/2-b/2}). \tag{17}
$$
The last exponent exceeds those in (15) and (16), and $b/2=\eta$. This proves (2). $\square$

## 7. Rooted cost and its harmonic lower scale

Assume now the row and column caps are both two. Write $k_L=|S\cap L|$ and $\ell_L=|G_N\cap L|$. Double counting yields
$$
\sum_{z\in G_N\setminus S}J_z
=\sum_L(\ell_L-k_L)(k_L-1)_+. \tag{18}
$$
For a nonaxis line with $k_L\ge3$, use $\ell_L\le N$ and $k_L-1\le2(k_L-2)$. These lines contribute at most $2ND(S)$. At most $2N$ axis lines contribute, each at most $N$, for a total at most $2N^2$. Finally an ordinary nonaxis line of primitive height $h$ has
$$
\ell_L-2\le\lfloor(N-1)/h\rfloor-1<N/h.
$$
Its contribution is therefore at most $N/h$. Theorem 2 proves (3).

Here is a lower bound locating the remaining scale. For any $k$-point set $W\subseteq G_N$, with $k=O(N)$, let $E(W)$ count all unordered pairs with weight $1/h$. The gcd identity $g=\sum_{t\mid g}\varphi(t)$ and residue-class Cauchy give
$$
\begin{aligned}
E(W)&\ge\frac1{N-1}\sum_{\{x,y\}\subset W}g(x,y),\\
\sum_{\substack{\{x,y\}\subset W\\x\equiv y\pmod t}}1
&=\sum_{\rho\bmod t}{n_{t,\rho}\choose2}
\ge\frac12(k^2/t^2-k).
\end{aligned}
$$
Restricting the divisor sum to $t\le Q=\lfloor\sqrt k\rfloor$ yields
$$
E(W)\ge\frac{3k^2\log k}{2\pi^2(N-1)}-O(k^2/N). \tag{19}
$$
Here $\sum_{t\le Q}\varphi(t)/t^2=(6/\pi^2)\log Q+O(1)$ and $\sum_{t\le Q}\varphi(t)=O(Q^2)$. The first identity follows directly by inserting $\varphi(t)/t=\sum_{d\mid t}\mu(d)/d$, summing the inner harmonic series, and using $\sum\mu(d)/d^2=1/\zeta(2)=6/\pi^2$; the second follows from $\varphi(t)\le t$. For fixed $k=O(N)$, $Q\le N-1$ eventually, as required. We only use (19) in that asymptotic regime.

Suppose $|S|=\lfloor cN\rfloor$ and $D(S)\le\delta N$ for fixed $0\le\delta<c\le2$. A deletion on a currently rich line reduces $D$ by at least one and cannot create new defects. After at most $D(S)$ deletions a clean core remains, with size at least $(c-\delta)N-O(1)$. Take a subset $W$ of that prescribed size. Each originally rich nonaxis line contains at most two $W$ points, hence accounts for at most one pair of energy at most one. There are at most $D(S)$ such lines. Axis pairs cost at most $|W|$. All other pairs of $W$ are ordinary nonaxis edges of $S$, so (19) proves
$$
\sum_{\substack{L\text{ nonaxis}\\k_L=2}}\frac1{h_L}
\ge\left(\frac{3(c-\delta)^2}{2\pi^2}+o(1)\right)N\log N. \tag{20}
$$

There is also a physical outside-root version. Choose fixed $0<\varepsilon<(c-\delta)/8$ and retain only core points whose coordinates are at distance at least $\lceil\varepsilon N\rceil$ from every board side. The caps discard at most $8\varepsilon N+O(1)$ points, leaving a linear core. If an ordinary line through a pair in this interior core has $h\le\varepsilon N/2$, it contains at least $2\lfloor\varepsilon N/h\rfloor+1$ board lattice points around one endpoint. Subtracting its two selected points leaves at least $\varepsilon N/(2h)$ outside roots. Pairs of height greater than $\varepsilon N/2$ have total energy $O(N)$, and originally rich lines and axes again cost only $O(N)$. Applying (19) to the interior core gives
$$
\sum_{\substack{L\text{ nonaxis}\\k_L=2}}(\ell_L-2)
=\Omega_{c,\delta}(N^2\log N). \tag{21}
$$
This counts incidence multiplicity, not distinct blocked sites. A lower average load does not give an upper bound on the number of free sites.

## 8. Scope and hypotheses

The row hypothesis cannot simply be replaced by $|S|=O(N)$ in Theorem 2. To see this analytically, let $N=M^2$ and dilate the full $M\times M$ grid by the factor $M$. It lies in $G_N$ and has exactly $N$ points. For coprime positive differences $a,b\in[M/2,3M/4]$, each original-grid pair with displacement $(a,b)$ determines an ordinary line: there is no intermediate lattice point by coprimality, and three consecutive points in that direction cannot fit in the grid since both displacements exceed half its coordinate span. There are $\Theta(M^2)$ such coprime directions, each with $(M-a)(M-b)=\Theta(M^2)$ pairs. The coprime count follows by Möbius inversion in this fixed square interval, with main term its area divided by $\zeta(2)$ and error $O(M\log M)$. Dilation preserves the primitive height, which is $\Theta(M)$. Thus the ordinary energy is $\Omega(M^3)=\Omega(N^{3/2})$, and the general matching upper bound gives the reverse order bound. This support has $M$ points per occupied row and violates the required cap.

For capped supports the lower scale in (20) remains far below the upper bound (2). Even an $O(N\log N)$ upper bound, if proved, would control only a sum of rooted loads. A simultaneous refill also has conflicts involving two new points and one old point, a fixed deletion quota, and reverse multiplicities in any comparison argument. These are not settled here. Likewise (3) alone does not bound $\mathbb E D$ for the probability law proportional to $e^{-\beta D(S)}$ at $\beta$ of order $\log N$.

The result depends on the board side, not only on cardinality. It should not be substituted for the cardinality estimate when the latter is smaller. We make no finite-size performance claim for the very small exponent $\eta$ and the possibly large implicit constants.

## 9. Verification and acknowledgments

The evidence is the complete analytic proof above; there are no numerical experiments or computational asymptotic claims. Sections 4–5 prove the gcd count, Section 6 derives the energy exponent, and Section 7 gives the exact shadow application and complementary lower bounds. Every parameter used in the power saving is displayed. No supplement or inaccessible private artifact is needed to verify the claims.

I thank lattice_lift_research for independent analytic audits of the clean-set gcd argument, its ordinary-edge extension, and the harmonic lower bound. Those audits did not replace the proofs and are not represented as independent numerical experiments. The private exchanges are not reproduced. Classical interpolation, convex-chain counting, divisor bounds, and totient identities are credited as standard ingredients.

## References

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[2] Miguel N. Walsh. *The Algebraicity of Ill-Distributed Sets*. Geometric and Functional Analysis 24 (2014), 959–967. https://doi.org/10.1007/s00039-014-0286-3 . Preprint: https://arxiv.org/abs/1307.0259 .

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