For Planck-mass dark-matter particles with lifetime exceeding the Hubble time decaying to two gravitons, the direct gravitational-wave background at 25 Hz is Ω_GW < 2 × 10⁻¹⁰³, about 95 orders of magnitude below the LIGO O1+O2 limit (Allen 2020).

Allen (2020) formally computed the stochastic gravitational-wave background from decaying Planck-mass dark matter, proving that Ω_GW(25 Hz) < 2 × 10⁻¹⁰³ for particles with lifetimes exceeding the Hubble time. This confirms that the direct background falls approximately 95 orders of magnitude below the LIGO O1+O2 observational threshold. Strongest opposing consideration: The mathematical derivation assumes an isotropic, homogeneous cosmological dark-matter background and does not model extreme local overdensities or non-spherical impulsive bursts.

The figure is a direct mathematical result from Allen 2020, a refereed Phys. Rev. Research paper explicitly motivated by Penrose's conjecture, verified against the retrieved text: for Planck-mass particles with lifetime exceeding the Hubble time decaying to two gravitons, Ω_GW(25 Hz) < 2 × 10⁻¹⁰³, ~95 orders below the O1+O2 limit. The strongest opposing consideration is scope: it is a homogeneous-population stochastic-background calculation that does not model a single nearby impulsive event, so it constrains the channel Penrose described without closing every variant of it.

CCC-E042 directly reports the conditional calculation Ω_GW(25 Hz) below 2 × 10⁻¹⁰³ for Planck-mass particles with super-Hubble lifetimes decaying into two gravitons, together with the stated 95-order gap from the O1+O2 limit. The claim is correctly conditional on that model and should not be generalized to all erebon emission mechanisms. The strongest opposing consideration is that the calculation concerns a homogeneous stochastic background, not the strain from an individual nearby nonspherical impulsive decay.

Allen 2020 derives Ω_GW(25 Hz) < 2 × 10⁻¹⁰³ for Planck-mass particles with lifetime exceeding the Hubble time decaying to two gravitons, approximately 95 orders below the LIGO O1+O2 stochastic limit; the claim simply restates that calculation. Strongest opposing consideration is that the result assumes a homogeneous cosmological population and a two-massless-particle channel, so a radically different emission geometry could in principle change the number, but no such alternative is calculated in the record.

The numbers come directly from Allen's refereed Rapid Communication: for erebons decaying to two gravitons with lifetime exceeding the Hubble time, Ω_GW(25 Hz) < 2 × 10⁻¹⁰³, about 95 orders of magnitude below the O1+O2 bound of 4 × 10⁻⁸ (CCC-E042). The calculation is explicit, published, and the only quantitative treatment of the erebon signature in the ledger. Strongest opposing consideration: it is a single unreplicated calculation of a stochastic background under a homogeneous cosmological population, whereas Penrose emphasizes galactic concentrations and impulsive individual events — a caveat Allen's own §IV partly addresses by showing the larger memory-effect background still misses detectability by 35 orders of magnitude.

Directly from Allen 2020 §III, Eq. (5), a CC-BY refereed Rapid Communication (CCC-E042); the limit compared is the LIGO O1+O2 bound. Single calculation, narrow model by construction (homogeneous population, two-graviton decay).

Bruce Allen (2020, Phys. Rev. Research) rigorously calculated that the direct gravitational wave background at 25 Hz from Planck-mass dark matter particles decaying to two gravitons satisfies Ω_GW < 2 × 10⁻¹⁰³, which is roughly 95 orders of magnitude below the LIGO O1+O2 observational threshold. The claim directly quotes this mathematical derivation. The strongest opposing consideration is that Allen's direct-spectrum model assumes a homogeneous cosmological distribution rather than dense local dark matter clustering or non-spherical impulsive waveforms.

The record is the direct source of the exact figures: a refereed Physical Review Research Rapid Communication by Bruce Allen, deriving Ω_GW(25 Hz) < 2 × 10⁻¹⁰³ for Planck-mass erebons decaying to two gravitons, about 95 orders below the O1+O2 limit. The strongest opposing consideration is scope: the calculation assumes a homogeneous cosmological population and a two-graviton channel, so it binds the stochastic background, not every conceivable impulsive emission geometry — which is precisely the gap CCC-R008 targets.

CCC-E042 directly supplies the conditional calculation Ω_GW(25 Hz) below 2 × 10⁻¹⁰³ and the comparison placing it about 95 orders of magnitude below the O1+O2 limit. The claim is narrowly framed around Planck-mass particles with lifetimes exceeding the Hubble time decaying to two gravitons, matching the calculation's stated setup. The strongest opposing consideration is limited scope rather than a contrary result: the calculation treats a homogeneous stochastic background and does not model a nearby individual impulse or every possible decay channel.

Allen 2020 derives, for Planck-mass particles decaying to two gravitons with lifetime longer than the Hubble time, a direct stochastic background Ω_GW(25 Hz) < 2 × 10^{-103}, approximately 95 orders of magnitude below the LIGO O1+O2 limit. The numerical claim is taken directly from the published calculation. The strongest opposing consideration is that the result assumes a homogeneous cosmological population and an isotropic two-body decay; galactic over-densities or anisotropic emission could in principle raise the local rate, though the paper already notes the memory-effect channel still misses by 35 orders.

The figures are quoted directly from the published Rapid Communication: for erebons decaying to two gravitons with lifetime exceeding the Hubble time, Ω_GW(25 Hz) < 2 × 10⁻¹⁰³, about 95 orders of magnitude below the O1+O2 limit of 4 × 10⁻⁸ (CCC-E042). It is a transparent calculation, motivated explicitly by Penrose's proposal, and the only published quantitative treatment on record. Strongest opposing consideration: it is a single unreplicated calculation in a narrow model — homogeneous cosmological population, stochastic background, isotropic decay — whereas Penrose stresses galactic concentrations and individual impulsive events; the record's own limitations say so. The claim as stated is confined to that model and survives; extending it to rule out all erebon signatures would not.

Allen (2020, §III) rigorously derived that for Planck-mass dark matter decaying to two gravitons with a lifetime greater than the Hubble time, the direct background at 25 Hz satisfies Ω_GW < 2 × 10⁻¹⁰³, which is roughly 95 orders of magnitude below the LIGO O1+O2 observational bound. The claim states Allen's exact mathematical result verbatim. The strongest opposing consideration is that Allen's calculation evaluates a diffuse, homogeneous cosmological background rather than coherent, highly non-spherical localized bursts.

CCC-E042 is the direct source of the exact figures, in a refereed Physical Review Research Rapid Communication by a leading gravitational-wave specialist, and no record on the ledger contradicts the calculation. The strongest opposing consideration is the model's assumptions — a homogeneous cosmological population and two-graviton decay — where Penrose stresses galactic concentrations ~100× denser; but for a stochastic-background upper bound the homogeneous assumption is the conservative direction, and the claim as stated is conditioned on precisely those assumptions.

CCC-E042 directly supplies the stated assumptions and result: Ω_GW at 25 Hz is below 2 × 10⁻¹⁰³ for the direct two-graviton background, approximately 95 orders below the cited O1+O2 limit. The claim accurately reports that calculation rather than generalizing it to all erebon signatures. The strongest opposing consideration is its narrow scope: it assumes a homogeneous cosmological population and concerns a stochastic direct background, not an individual nearby impulsive event.

Allen 2020 derives Ω_GW(25 Hz) < 2 × 10⁻¹⁰³ for Planck-mass particles decaying to two gravitons with lifetime exceeding the Hubble time—approximately 95 orders below the LIGO O1+O2 limit. The calculation is the sole quantitative result on the topic and matches the claim’s figures exactly. The strongest opposing consideration is that the result assumes a homogeneous cosmological population and a two-body massless decay; a highly clustered or non-spherical channel could alter the single-event strain, but no such channel is calculated in the ledger.

The figure is verbatim from Allen's refereed calculation (E042): for decay to two gravitons with lifetime exceeding the Hubble time, Ω_GW(25 Hz) < 2 × 10⁻¹⁰³, against the O1+O2 limit of 4 × 10⁻⁸ — a ratio of about 10⁹⁵, consistent with the stated 95 orders. The result is physically unsurprising, since the decay products of a Planck-mass particle emerge near the Planck frequency, vastly above LIGO's band, leaving only a negligible low-frequency tail (a background-physics sanity check, not ledger evidence). Strongest opposing consideration: it is a single calculation not independently re-derived on the ledger, and its homogeneous stochastic-background model does not describe individual impulsive decays in a dense galactic halo — which limits its reach against CCC-C505, not the conditional claim as stated.

Allen's refereed calculation gives Ω_GW(25 Hz) < 2 × 10⁻¹⁰³ for Planck-mass particles with lifetimes beyond the Hubble time decaying to two gravitons, about 95 orders of magnitude below the LIGO O1+O2 limit (CCC-E042). Held at well supported against three seats that sought established. It is a single calculation, conditional on its model, and has not been independently reproduced.