Enumeration and Sparsity of Growing-$k$ SAT Functions

Research-attempt provenance

Definitions and mathematical statement

Shared notation.

Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

A $k$-CNF function is the Boolean function computed by a conjunction of clauses of width at most $k$. Distinct functions are counted separately from distinct formulas; a unate function becomes monotone after fixed negations of selected input coordinates.

Record scope and qualifications.

The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. This overlaps the growing-width source question under ID 30005109. The additional sparse-formula thresholds are not explicitly specified.

Source-record limitation: The growing-width function-count target is already retained under OWR ID 30005109; extra sparse-formula thresholds are unnamed and lack a supplied target.

Related catalogue records: 30005109. Separate source IDs are preserved for their provenance.

Statement recorded in the supplied source.

Determine the asymptotic number of distinct $k$-SAT Boolean functions when $k$ grows with the number $n$ of variables. For which growth rates of $k$ are almost all such functions unate, and what are the corresponding sparse-formula thresholds? [S2]

Source status.

The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Count growing-width satisfiability functions and understand their typical structure.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30005341 (OWR-12697684-018). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Combinatorics $2023$, Oberwolfach Reports, DOI 10.4171/owr/2023/1, pp. 5--89 https://doi.org/10.4171/owr/2023/1.
• S3 Dong--Mani--Zhao, Enumerating k-SAT Functions https://arxiv.org/abs/2107.09233.
• Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.

Preparation and review.

The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30005341 and code OWR-12697684-018 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Related canonical identities: 30005109. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Literature and context check

I read the primary 2022 report's talk on pages 1222--1224 and the later 2023 report's talk on pages 49--50. The latter explicitly states the full fixed-$k$ theorem and distinguishes the still-discussed growing-width and sparse-formula directions. I also read the abstracts of Dong--Mani--Zhao, arXiv:2107.09233, and Balogh--Dong--Lidický--Mani--Zhao, arXiv:2209.04894v2. The second announces resolution of the fixed-$k$ unateness conjecture. These source results are context, not newly proved here; their container and extremal-hypergraph arguments were not independently re-derived. In particular the original partially-solved catalogue label is historical metadata, not a reason to describe the fixed-width conjecture as currently open.

Research attempt: A very sparse unate regime and the limits of density extrapolation

A formula family that can be counted without collisions.

Use the source's equivalent DNF formulation. For $1\leq k\leq n$ and a family $\mathcal H\subseteq\binom{[n]}k$, define \[ f_{\mathcal H}(x)=\bigvee_{S\in\mathcal H}\ \bigwedge_{i\in S}x_i. \] The empty disjunction is zero. On the input $1_T$ with exactly $k$ ones, $f_{\mathcal H}(1_T)=1$ if and only if $T\in\mathcal H$, because a $k$-set $S$ is contained in the $k$-set $T$ only when $S=T$. Thus these $2^{\binom nk}$ formulas represent distinct functions. Every clause is indispensable: $1_S$ is a private satisfying input for its own clause. A variable $i$ is essential exactly when some $S\in\mathcal H$ contains it. Indeed $1_S$ and $1_{S\setminus\{i\}}$ distinguish its two values; if no clause contains $i$, independence is immediate. With each possible clause independently included with probability $1/2$, the probability that a specified variable is absent is $2^{-\binom{n-1}{k-1}}$. A union bound shows that at least \[ \left(1-n2^{-\binom{n-1}{k-1}}\right)2^{\binom nk} \] functions in this family depend on all variables, whenever the displayed lower bound is positive. Each has $2^n$ different input-coordinate sign orientations, and the resulting functions over all such families are distinct. To prove this, a function monotone in both directions in a coordinate must be independent of that coordinate: with all other inputs fixed, its two values satisfy both opposite inequalities. An essential coordinate therefore has a unique monotonic direction. Equality of two oriented functions forces their orientations to agree, after which the $k$-weight evaluations identify $\mathcal H$. Consequently the total number $N(n,k)$ of source functions satisfies \[ N(n,k)\geq \left(1-n2^{-\binom{n-1}{k-1}}\right)2^{\binom nk+n}. \] Complementing the output gives the corresponding CNF count if that convention is preferred. This is a lower bound for a specifically constructed subclass; it does not count all possible prime implicants of a general unate function.

A concrete sparse-formula calculation.

Now take the auxiliary, explicitly specified model of $m$ independent uniform signed clauses, each with $k$ distinct variable indices, where $1\leq k\leq n$. If their supports are pairwise disjoint, the formula is syntactically unate and minimal: for any clause, make all its literals true and make at least one literal false in each other clause. Disjoint supports allow these choices simultaneously and give a private witness. For two independent uniform $k$-subsets $S,T$, a union bound over $i\in S$ gives $\Pr(S\cap T\ne\varnothing)\leq k^2/n$, since $\Pr(i\in T)=k/n$. Therefore disjointness has probability at least $1-\delta$, where \[ \delta=\binom m2\frac{k^2}{n}. \] Let $\mathcal M$ be the event that the DNF is minimal, treating repeated clauses as separate clauses (so any repetition fails minimality). For $\delta<1$, \[ \Pr(\hbox{not syntactically unate}\mid\mathcal M) \leq\frac{\delta}{1-\delta}. \] The numerator is at most the probability of nondisjoint supports; the denominator is at least $1-\delta$. Conditional on minimality, the unordered clause set is uniform among minimal $m$-clause sets: each has precisely $m!$ ordered samples of identical probability. Thus the same estimate applies to that source-like uniform-minimal-formula model. In particular, if $mk/\sqrt n\to0$, almost all such minimal formulas are unate. The bound is about formulas, and it does not induce a uniform distribution on the Boolean functions they represent.

An explicit obstruction once supports interact.

The two-clause minimal DNF \[ f=(x_1\wedge x_2)\vee(\neg x_1\wedge x_3) \] is not unate. With $(x_2,x_3)=(1,0)$ it increases in $x_1$, while with $(x_2,x_3)=(0,1)$ it decreases. Each clause has a private witness, so conditioning on minimality does not erase this interaction. Multiplying both clauses by $x_4\cdots x_{k+1}$ gives the same example at width $k\geq2$, with those common variables fixed to one in the witnesses. Consequently minimality alone is not the structural substitute needed between the very sparse disjoint-support regime and the source's dense typical-unateness theorem. The fact that both extremely sparse formulas and the dense setting can be unate also cautions against presuming a single monotone threshold solely from the word "sparse." No intermediate threshold is derived here.

Verification and adversarial audit

The sampling model is stated before conditioning. The transfer to uniform minimal clause sets counts the exact $m!$ multiplicity, and explicitly avoids transferring to uniformly random functions. The probability inequality is used only for $\delta<1$ and its asymptotic consequence only for $\delta\to0$. For $m=0$ the zero DNF is unate and minimal vacuously; for $m=1$ the bound is exact. The mixed-sign example has $k=2$ (or $k+1\leq n$ for its extension), not the unit-clause boundary.

Audit corrections.

Separated uniform formulas from uniform functions, justified conditioning on minimality, and kept the small-density result from being presented as the source's full sparse-threshold classification.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The independent-support estimate gives a rigorously defined very sparse auxiliary regime and the common lower count handles growing width. It does not determine the requested growing-$k$ function counts, intermediate sparse-formula thresholds, or typical nonunate structure. The record overlaps 30005108 and 30005109; the repeated lower-bound argument is declared, not counted as an independent discovery.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://ems.press/content/serial-article-files/46961 (consulted 2026-10-11).
• https://ems.press/content/serial-article-files/46994 (consulted 2026-10-11).
• https://arxiv.org/abs/2107.09233 (consulted 2026-10-11).
• https://arxiv.org/abs/2209.04894 (consulted 2026-10-11).

Model and contributor disclosure

Codex GPT-6 completed written-proof review of this attempt at its declared scope; this is model-assisted review, not independent human proof checking. OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration and repository-format review; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.