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A Cap-Move Reformulation of Ling's Proof of Samuels' Conjecture

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Pith's one-line read A cap-move proof establishes Samuels' conjecture for finitely supported distributions, giving an alternative to Ling's Bernoulli-based proof.

arxiv 2608.22816 v1 pith:VTMXEHKT submitted 2026-08-24 math.PR math.CO

Yanjun Han This is my paper · ORCID
classification math.PRmath.CO
keywords conjecturedeltalingproofleftmathbbrightsamuels
verification ladder T0 review T1 audit T2 compute T3 formal

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The reading

Samuels' conjecture says that if you add independent nonnegative random variables with given averages, the chance the sum stays below a fixed level is at least a certain product of fractions. The inequality had been open for decades and implies other results, such as Feige's conjecture about sums of independent variables. Zhi Ling recently proved it in 2026. This note gives a second, self-contained proof.

The new proof starts by assuming each variable takes only finitely many values, which is a standard approximation step. It then applies a 'cap move': pick one variable and replace it by the smaller of itself and a number a. This only makes the sum smaller, so the probability of staying below the target cannot go down. The proof shows that a cap move is 'safe' whenever it does not lower a certain conditional-gap quantity r, and that at least one safe move always exists until all variables become deterministic. Repeating safe moves, each step removes at least one possible value, so the process ends at a deterministic sum for which the inequality is obvious. A telescoping product then bounds the original probability by the conjectured product. The last step is a calculus lemma showing the resulting product is at least the minimal value in Samuels' formula.

Extended reading notes

Core claim

The cap-move argument proves Samuels' conjecture: for independent nonnegative Xi with E[Xi] = μ_i, P(sum Xi < δ + sum μ_i) is at least min_i product_{j=i}^n (1 - μ_j / (δ + sum_{k=i}^n μ_k)). If the proof is correct, the conjecture holds for all finitely supported distributions and, by the asserted approximation, for all independent nonnegative random variables.

Load-bearing premise

The paper assumes without proof that proving the inequality for finitely supported discrete random variables suffices for arbitrary independent nonnegative random variables. Section 2 opens with 'By approximating a general random variable with discrete ones, it suffices to consider the case where each Xi is a discrete random variable with finite support.' The entire cap-move argument operates on finite support, so the unrestricted form of Samuels' conjecture depends on this unstated limit argument holding, including the behavior at the threshold lambda.

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Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The proof introduces no fitted parameters and no new physical or mathematical entities. It uses standard approximation and tilting arguments; the finite-support approximation is a domain assumption that is asserted rather than proved.

assumptions (2)
  • domain assumption Arbitrary nonnegative independent random variables can be approximated by finitely supported discrete random variables with the same means, preserving the inequality in the limit.
    Invoked at the beginning of Section 2 without proof; the cap-move proof only treats finite-support distributions.
  • standard math Standard properties of exponential tilts, including the derivative identities used in Lemma 2.3.
    Used to derive the covariance identity and the measure decomposition; standard but not proven in the note.

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Pith. "Pith review of A Cap-Move Reformulation of Ling's Proof of Samuels' Conjecture." pith.science (2026). https://pith.science/paper/VTMXEHKT

@misc{pith2026260822816,
  author       = {Pith},
  title        = {Pith review of: A Cap-Move Reformulation of Ling's Proof of Samuels' Conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VTMXEHKT}},
  note         = {Machine review of arXiv:2608.22816}
}
abstract

Let $0\le \mu_1\le \mu_2 \le \cdots \le \mu_n$ and $\delta > 0$. Samuels' conjecture claims that if $X_1,\dots,X_n$ are independent non-negative random variables with $\mathbb{E}[X_i] = \mu_i$, then $$ \mathbb{P}\left( \sum_{i=1}^n X_i < \delta + \sum_{i=1}^n \mu_i \right) \ge \min_{1\le i\le n} \prod_{j=i}^n \left(1-\frac{\mu_j}{\delta + \sum_{k=i}^n \mu_k}\right).$$ This conjecture was recently proved by Ling. In this note, we provide an alternative presentation of Ling's proof via an operation called the \emph{cap move}. This proof works directly with finitely supported distributions and avoids the reduction to the Bernoulli case.

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Works this paper leans on

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Reviewed August 28, 2026 · model on record in the stance chip above.