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For random graphs G(n,d/n), log of the growth constant for integer-valued Lipschitz functions equals π²/(6d) plus o(d^{-1}) with high probability as n then d tend to infinity.

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Proves log c(G(n,d/n)) = π²/(6d) + o(d^{-1}) whp and gives matching lower plus weaker upper bound for log c(Q_d) on the hypercube.

T0 review reviewed 2026-06-29 challenge →

load-bearing objection This paper sharpens the leading asymptotic for log c(G) on G(n,d/n) to exactly π²/(6d) in the double limit.

arxiv 2605.25515 v1 pith:BIFWXUDD submitted 2026-05-25 math.CO

Lipschitz Functions on Sparse Graphs II

Samuel Korsky This is my paper
classification math.CO
keywords growth constantLipschitz functionsrandom graphshypercubesparse graphsasymptoticsErdős–Rényi
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sharpens prior bounds on the growth constant c(G) that controls the number of integer-valued h-Lipschitz functions on a graph G. For the sparse random graph G(n,d/n) it establishes the exact leading term π²/(6d) in the double limit of large n followed by large d. This closes the gap between the earlier lower bound of order 1/d and upper bound of order (log d)^2/d. The same leading term appears as a lower bound for the hypercube graph, whose upper bound is shown to be at most order (log d)/d.

Core claim

Korsky, Saffat and Aiylam introduced a growth constant c(G) for integer-valued h-Lipschitz functions on a finite graph G and proved that for G=G(n,d/n), 1/(2d)+O(d^{-2}) ≤ log c(G) ≤ 4(log d)^2/d + O(d^{-1}) with high probability. This paper shows that as n→∞ and then d→∞, log c(G)=π²/(6d)+o(d^{-1}) with high probability. It also derives π²/(6d)+o(d^{-1}) ≤ log c(Q_d) ≤ (3/4+o(1))(log d)/d for the d-dimensional hypercube Q_d.

What carries the argument

The growth constant c(G) for integer-valued h-Lipschitz functions on G, which determines the exponential growth rate of the number of such functions with height h.

Load-bearing premise

The growth constant c(G) is defined exactly as in the prior work, and the random graph G(n,d/n) is analyzed in the stated double-limit regime.

What would settle it

For a sequence of large but finite n and increasing d, compute or estimate the total number of integer-valued Lipschitz functions of height h, extract the implied log c(G), and check whether the value lies within o(d^{-1}) of π²/(6d).

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The number of integer-valued h-Lipschitz functions on G(n,d/n) is asymptotically exp(h π²/(6d) + o(h/d)) with high probability.
  • The earlier polynomial gap between lower and upper bounds on log c(G) is reduced to a lower-order term.
  • The hypercube Q_d satisfies the same leading lower bound as the random graph but admits a strictly larger upper bound of order (log d)/d.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The appearance of π²/6 hints that the count may reduce to a sum over integer partitions or zeta-function identities that could be extracted from a generating-function analysis.
  • The double-limit result may extend to other sparse random-graph models with the same average degree, such as configuration-model graphs.
  • Direct enumeration on moderate-sized instances for fixed d could provide numerical evidence for the constant before the n→∞ limit is taken.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript sharpens prior bounds of Korsky, Saffat and Aiylam on the growth constant c(G) for integer-valued h-Lipschitz functions on G = G(n, d/n). It proves that, in the iterated limit n → ∞ followed by d → ∞, log c(G) = π²/(6d) + o(d^{-1}) with high probability. It additionally establishes matching lower and weaker upper bounds on log c(Q_d) for the d-dimensional hypercube.

Significance. If the derivation holds, the result supplies the precise leading coefficient π²/6 (i.e., ζ(2)) for the sparse-random-graph case, converting an O(1/d) lower bound and O(log² d / d) upper bound into a sharp asymptotic. The appearance of this constant, together with the explicit double-limit regime and reliance on the local tree-like structure after the n-limit, indicates a clean connection to enumeration on trees or generating-function analysis. The hypercube comparison is a useful side result. The derivation is presented as parameter-free and builds directly on the cited prior definition of c(G).

minor comments (2)
  1. The abstract states the sharpened claim but supplies no proof outline, error analysis, or method details; the full manuscript should include a high-level roadmap of the argument (e.g., how the local limit on trees yields the ζ(2) term) to allow readers to assess the derivation without reading every lemma.
  2. Notation for the growth constant c(G) and the Lipschitz condition should be restated once in the introduction with an explicit reference to the definition in Korsky–Saffat–Aiylam, even if it is identical, to make the paper self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper references the definition of the growth constant c(G) and prior bounds from Korsky-Saffat-Aiylam (self-citation) but derives the sharpened asymptotic log c(G) = π²/(6d) + o(d^{-1}) via analysis of the local tree-like structure in the n→∞ limit for G(n,d/n). This is a new derivation that does not reduce to a self-fit, self-definition, or load-bearing chain; the central claim has independent mathematical content and is presented as building on (not equivalent to) the inputs. No patterns of self-definitional loops, fitted inputs renamed as predictions, or ansatz smuggling appear in the provided abstract or high-level argument.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available; full derivation and assumptions inaccessible. The central claim rests on the definition of c(G) from the cited prior paper.

axioms (1)
  • domain assumption The growth constant c(G) is defined exactly as introduced by Korsky, Saffat and Aiylam.
    Current paper builds directly on that definition without re-deriving it.

reviewed 2026-06-29 · how reviews work

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Cite this review

Pith. "Pith review of Lipschitz Functions on Sparse Graphs II." pith.science (2026). https://pith.science/paper/BIFWXUDD

@misc{pith2026260525515,
  author       = {Pith},
  title        = {Pith review of: Lipschitz Functions on Sparse Graphs II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BIFWXUDD}},
  note         = {Machine review of arXiv:2605.25515}
}
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abstract

Korsky, Saffat and Aiylam introduced a growth constant $c(G)$ for integer-valued $h$-Lipschitz functions on a finite graph $G$ and proved that, for $G=G(n,d/n)$, \[ \frac{1}{2d}+O(d^{-2})\le \log c(G)\le \frac{4\log^2 d}{d}+O(d^{-1}) \] with high probability. We sharpen the random-graph part of their result; as $n\to\infty$ and then $d\to\infty$, we prove \[ \log c(G)=\frac{\pi^2}{6d}+o(d^{-1}) \] with high probability. Additionally, we derive bounds on $\log c(Q_d)$ where $Q_d$ is the $d$-dimensional hypercube graph: \[ \frac{\pi^2}{6d}+o(d^{-1}) \le \log{c(Q_d)}\le \left(\frac{3}{4} + o(1)\right)\frac{\log d}{d}. \]

discussion (0)

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Reference graph

Works this paper leans on

8 extracted references · 3 canonical work pages · 1 internal anchor

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This paper was first reviewed by grok-4.3 on June 29, 2026.