Theory predicts that metal-free first-generation (Population III) stars with initial masses of roughly 140–260 solar masses end their lives as pair-instability supernovae (PISNe), whose ejecta carry a distinctive nucleosynthetic fingerprint — strong contrasts between even-Z and odd-Z elements and low [Zn/Fe] — that should be preserved in the photospheres of second-generation low-mass stars. Despite dedicated searches across tens of thousands of extremely metal-poor (EMP) halo stars, no unambiguous PISN-enriched star has ever been identified: the two historical candidates were both reinterpreted by later independent analyses, and no PISN has been directly observed. The open problem is to explain this absence quantitatively — disentangling whether it reflects a Population III initial mass function (IMF) with few or no PISN-progenitor masses, dilution and metal mixing of the large PISN ejecta, modification of the fingerprint at finite metallicity, or survey selection and rarity — and thereby either identify a bona fide mono-enriched PISN descendant or convert the non-detection into a rigorous constraint on the frequency of very massive first stars. The problem remains open: recent cosmological-enrichment simulations, updated finite-metallicity PISN yield calculations, and statistical IMF inferences each cover part of the question, but no published work provides a validated quantitative reconciliation.

Background

Population III (Pop III) stars are the first, metal-free generation of stars, formed from primordial hydrogen and helium. Stellar-archaeology 'fossil' searches look for their nucleosynthetic products preserved in the atmospheres of long-lived, low-mass, extremely metal-poor (EMP) stars, conventionally those with [Fe/H] < -3, where the bracket notation [A/B] denotes the logarithm of an element ratio relative to the solar value. Very massive Pop III stars (initial masses ≈ 140–260 solar masses, corresponding to helium-core masses ≈ 64–133 solar masses) encounter the pair-instability: production of electron–positron pairs softens the equation of state, the oxygen core contracts and ignites explosively, and the star is completely disrupted in a thermonuclear runaway (a pair-instability supernova, PISN) leaving no compact remnant. Heger and Woosley (2002) computed the nucleosynthetic yields of Pop III PISNe and identified a distinctive fingerprint: large production of even-Z alpha-elements (elements whose nuclei are built mainly from helium nuclei, such as Mg, Si, S, Ca) with strong deficiency of odd-Z elements (the 'odd–even effect', e.g., low Na, Al, P, Sc) together with low [Zn/Fe] — a pattern so extreme that even partial enrichment of a second-generation star should be recognizable. Yet abundance surveys of EMP halo stars have never shown this pattern. Nomoto et al. (2006) emphasized that observed iron-peak and CNO (carbon, nitrogen, oxygen) abundance patterns in EMP stars are instead well fit by energetic core-collapse supernovae ('hypernovae') with mixing and fallback (ejecta only partially escaping the collapsing star), implying progenitor masses below the PISN regime. This tension motivated physical explanations for the absence: Hosokawa et al. (2011) showed that radiative feedback from a growing protostar halts accretion, limiting typical first-star masses to a few tens of solar masses — below the PISN range. Dedicated observational searches (e.g., Ren, Christlieb and Zhao 2012, selecting candidates via the PISN calcium excess) found no relics. Two stars were subsequently proposed as candidate PISN descendants — SDSS J0018-0939 (Aoki et al. 2014, tentatively) and LAMOST J1010+2358 (Xing et al. 2023, Nature) — and both were later disputed or reinterpreted by independent high-resolution spectroscopic and nucleosynthetic analyses. On the theory side, the predicted fingerprint itself has caveats: at finite metallicity the odd–even effect is weaker (Kozyreva et al. 2014), and PISNe arising in gas already enriched by earlier PISNe show further weakened odd–even contrasts and enhanced Zn–Ge production (Xin et al. 2026). Cosmological simulations of PISN enrichment and turbulent metal mixing (Karlsson, Bromm and Bland-Hawthorn 2013; Magg et al. 2022; Kordt, Glover and Klessen 2026) show that the observed metallicity distribution of PISN descendants depends strongly on the PISN mass and explosion energy, while statistical analyses (Koutsouridou, Salvadori and Skúladóttir 2024) have begun converting the non-detection into constraints on the initial mass function (IMF), excluding flat Pop III IMFs. In parallel, no PISN has ever been unambiguously observed as a transient, either locally (the SN 2007bi claim was reinterpreted) or at high redshift with JWST, where several tentative abundance-based and photometric claims remain unsettled.

Problem Statement

Explain why the distinctive nucleosynthetic signature of pair-instability supernovae — strong odd-even abundance contrasts and low [Zn/Fe], as predicted for 140-260 solar-mass metal-free stars — is absent from the observed population of extremely metal-poor stars, and determine what that absence implies for the Population III initial mass function. The absence could reflect (a) a bona fide second-generation star whose detailed abundance pattern, including the discriminating elements (C, Al, odd-Z elements such as Na and Sc, and Zn), can only be reproduced by PISN ejecta, with the signature then hidden by rarity and selection rather than absent; (b) the expected frequency and metallicity distribution of mono-enriched PISN descendants (stars whose metals derive from a single earlier supernova) as a function of the Pop III IMF, which follows from PISN yield predictions (zero-metallicity and finite-metallicity) together with modeling of enrichment, dilution, and metal mixing and the selection functions of existing EMP-star surveys, and which would convert the observed non-detection into a rigorous bound on the frequency with which the first stellar generation produced stars in the PISN mass range; or (c) the impossibility in principle of PISN descendants appearing in the observable EMP population across the full PISN mass and energy range, which itself fixes the IMF implication. The answer must discriminate among the competing suppression mechanisms — IMF shape, dilution and mixing, finite-metallicity modification of the yields, and observational selection — rather than assuming one, and its conclusions must be robust across, rather than artifacts of, any single choice of survey, yield model, or enrichment code.

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