Monochromatic finite sums and products

Source and verification

This article explains family 164 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Finite sums and finite products

For a finite set $A=\{a_1,\dots ,a_m\}$ of distinct positive integers, write $[m]=\{1,\dots ,m\}$ and define

$$\begin{array}{c}\mathrm{FS}(A)=\left\{\sum _{i\in I}{a}_i:\varnothing \ne I\subseteq [m]\right\},\\ \mathrm{FP}(A)=\left\{\prod _{i\in I}{a}_i:\varnothing \ne I\subseteq [m]\right\}.\end{array}$$

Each selected term appears once. Empty sums and empty products are excluded, while singleton subsets are included in both sets. A finite coloring of $\mathbb{N}$ is an arbitrary map to a finite set of colors, with no compatibility assumed between the coloring and either arithmetic operation.

Manuscript claim

For every finite coloring and every positive integer $m$, there are distinct positive integers $a_1,\dots ,a_m$ such that all of $\mathrm{FS}(A)\cup \mathrm{FP}(A)$ has one color. The terms can be required to grow arbitrarily rapidly in the following precise sense: for any $R\ge 2$ and $D\ge 1$, one can impose

$$\begin{array}{c}{a}_1>R,\\ a_j>R{\left(\sum _{i<j}{a}_i+\prod _{i<j}{a}_i\right)}^D.\end{array}$$

The second inequality applies for $2\le j\le m$.

(OpenAI, 2026)

For $m=2$, the configuration is $\{a,b,a+b,ab\}$. The same color must work for the sums and products together; two separate monochromatic conclusions would be weaker.

🏷️ Finite size and distinctness

The theorem is finite: the sequence can depend on the requested length $m$. It does not produce one infinite sequence whose entire finite-sums and finite-products sets are monochromatic. The source explicitly distinguishes that stronger statement from its conclusion.

The rapid-growth conditions also allow collisions to be avoided. The manuscript obtains choices for which every nonempty subset sum is distinct, every nonempty subset product is distinct, and the only values common to the two families are the original terms. In that case

$$\begin{array}{c}|\mathrm{FS}(A)|=|\mathrm{FP}(A)|=2^m-1,\\ |\mathrm{FS}(A)\cup \mathrm{FP}(A)|\\ =2(2^m-1)-m.\end{array}$$

This is a count of a specially chosen collision-free configuration, not a formula for arbitrary $A$.

🏷️ Why separate recurrence arguments do not combine automatically

Additive and multiplicative recurrence each supply large structured configurations. Their favorable sets, however, need not agree. A proof must arrange that the same chosen integers work for both operations and land in the same color class.

Passing to rational numbers does not remove this compatibility problem. Suppose a rational configuration has a desired property and its denominators are cleared by multiplication by $T$. A sum scales by $T$, but a product of $j$ terms scales by $T^j$. An arbitrary coloring need not assign related colors to those different scales. Thus denominator clearing is not a valid shortcut.

The manuscript begins with ordered blocks and multiplicative monochromaticity, then uses a finite Ramsey selection to organize the sums. This leaves a prediction-and-alignment problem: one must understand enough of the coloring on carefully chosen shifts and dilations to make the remaining additive patterns agree with the multiplicative ones.

🏷️ Prediction with multiplicative constraints

The prediction step approximates the relevant color indicators by structured sequences, with multiplicative masks retained. These masks restrict which substitutions are admissible. Divisor weights and prime substitutions make it possible to apply repeated Cauchy–Schwarz arguments without losing those restrictions.

The resulting structured models are piecewise nilsequences: sequences obtained from polynomial motion on compact spaces with a finite hierarchy of commutators, restricted to suitable pieces. Their role is to provide controlled recurrence and predict correlations. The post uses that role as a stated input rather than developing the algebraic theory of those spaces.

Different tasks require different approximation scales. Counting short shifts and aligning much larger dilations cannot generally use one error tolerance chosen at the outset. The source builds nested models, with each later choice respecting the data already fixed at earlier stages.

The alignment estimate handles finitely many rational scales simultaneously. A central feature is a positive success probability that does not collapse as the model complexity increases. This permits the proof to choose a useful alignment mass first and only then make approximation errors small relative to it. Reversing that order could produce a circular dependence in the parameters.

🏷️ Returning to integers

After the alignment stage, integer residue corrections make the proposed parameters compatible with the original arithmetic problem. The models are then lifted jointly into a common dynamical state so that their shifts can recur simultaneously. The structural depth needed for this joint state is bounded in terms of the requested finite size $m$.

This joint lifting is the closure step: recurrence for each model separately would provide different shifts and would not complete the construction. Once the shared recurrence is available, the proof chooses the integers in an order that also meets the prescribed growth conditions.

The result is qualitative in the size of the ambient search. These notes do not attach an explicit usable upper bound to the largest integer in the configuration. The elementary example and cardinality calculation explain the target; the simultaneous prediction, alignment, and integer-return estimates carry the new theorem.

🏷️ Key steps and proof outlines

Choose the success mass before choosing approximation errors

The “Framework” section reduces the theorem to prediction and alignment principles with a crucial order of quantifiers. Fix the desired length $m$, the color count $r$, and the structural depth required by prediction. Alignment then supplies finite scale lists and a success probability $\delta >0$, depending on these fixed data but independent of the complexity of the models later used. (OpenAI, 2026)

Let $C$ count the finitely many calibration events, and let $K$ count the possible scale-vector, ordered-block-chain, and color choices. Select a threshold $\tau \in (0,1/4)$ and then a prediction error $\eta >0$ satisfying

$$3C\tau <\delta /4,C\eta <\delta /4,K\eta <\delta {\tau }^{2^m}/4.$$

These inequalities state exactly what the approximations must accomplish. Calibration failures cost less than half the alignment mass. The remaining counting error is smaller than the positive product of model values on that intersection. A success constant chosen only after model complexity could make this parameter choice circular.

Obtain positive mass from finitely many recurrence options

The “Alignment” section first constructs, at each pivot $i$, at most $Q_i$ options whose success sets cover every starting state after suitable transformations. The individual transformations need not be identical. Each transformed state has the same limiting point law, and the relevant comparison sets are closed. Applying the closed-set direction of weak convergence to the finite covering inequality gives

$$1\le \sum _o\mu (F_o),\mu \left(\bigcup _o{F}_o\right)\ge 1/Q_i.$$

Here $F_o$ is the success set for option $o$ and $\mu$ is the limiting law. The comparisons use threshold $3\tau /2$. Enlarge them to the open condition “center $<7\tau /4$ or translated value $>5\tau /4$.” The open-set direction of weak convergence then transfers positive mass back to actual finite models. Uniform approximation error below $\tau /4$ makes an actual center exceeding $2\tau$ force an actual translated reading exceeding $\tau$.

This closed-to-open maneuver is a proof step, not cosmetic threshold slack: weak convergence gives upper bounds on closed sets and lower bounds on open sets in opposite directions.

Remove pivot shifts without conditioning on the wrong event

The local argument allows a bounded list of shifted pivots $p+s(v)$, with at most $V_i$ slots. For the raw pivot law ${\mu }_i$, the source proves

$$\Vert T_a{\mu }_i-{\mu }_i{\Vert }_{\mathrm{TV}}\le \frac{|a|}{X_i(\log X_i-W/X_i)}=o(1)$$

for its admissible shifts $a\in W\mathbb{Z}$, uniformly over earlier raw variables. Here $T_a$ is translation and $X_i$ is the pivot’s size scale. The shifts are negligible compared with $X_i$.

If $E$ is any event determined by earlier raw variables, the shifted success estimate therefore becomes

$$\mathbb{P}(E,\text{success at pivot }i)\ge \frac{\mathbb{P}(E)}{3Q_i{V}_i}-o(1).$$

One must use the raw law at this point: translation invariance inside a previously fixed small conditioning cell is not asserted. Choose the first successful scale option in a fixed finite enumeration and partition into the resulting path events. Iterating gives total mass at least

$$\delta -o(1),\delta =\prod _{i\text{ with targets}}(3Q_i{V}_i{)}^{-1}>0.$$

Earlier requirements survive because each pivot was prepared simultaneously for the finite list of all possible future multiplier products. Later pivots change the scale, but not the earlier raw variables.

Convert model positivity into actual integers

On the good alignment and calibration event, finite Ramsey selection provides an ordered block chain whose multiplicative mask has one color. Every singleton model value exceeds $2\tau$; alignment makes each nonsingleton sum model value exceed $\tau$. There are $2^m-m-1$ such sum factors. Summing over the $K$ possible witnesses gives model count at least $\delta {\tau }^{2^m}/2$ in the limiting argument. Prediction loses less than half of this by the chosen bound on $K\eta$.

The original count is a finite nonnegative average. A positive count therefore yields one actual tuple for which all product masks and all sum indicators equal one. Extra divisor weights only restrict admissibility and cannot create a false positive. Finally, the raw scales were chosen to dominate every fixed power of previous scales, so every sufficiently late support tuple already satisfies the requested growth inequalities. Positivity and growth thus hold for the same tuple. The deep prediction and joint-state construction remain manuscript inputs; this outlines how their quantitative outputs compose into the integer theorem.

See Also

• Arithmetic progression bounds also uses structured sequences and a return step to preserve previously imposed constraints.
• Pointwise multiple averages explains another setting where simultaneous recurrence requires more than separate one-parameter information.

🏷️ Manuscripts and verification scope

• Monochromatic finite sums and products in the positive integers — pinned manuscript and companion source.

No family scope page was present at lean/docs/164.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. Monochromatic finite sums and products in the positive integers.