The load-bearing paper for Premise B of the LHC safety case: even if a TeV-scale black hole produced at the LHC were absolutely stable (Hawking radiation off), it would still pose no risk. The authors work through accretion mechanisms for a hypothetical stable BH trapped inside a massive body — subatomic-scale capture of nuclei/electrons (bounded by Schwinger pair production and electromagnetic binding-energy competition), transition to macroscopic Bondi accretion, and Eddington-limited radiative feedback — deriving growth timescales that are astronomically long (>10^11-10^12 years in most channels) compared to stellar/Earth lifetimes. They then invoke the survival of white dwarfs and neutron stars against cosmic-ray-produced stable BHs (which would be captured by these dense objects, unlike by Earth) as an independent empirical check: since these objects have survived for ~Gyr timescales, either stable BHs are not made by cosmic rays at the required rate, or their accretion is too slow to matter — either way, an LHC-made stable BH poses no danger on any timescale shorter than Earth’s natural lifetime. relevance_note: The single paper Premise B of the modern LHC safety case (LSAG 2008) rests on; establishes both the accretion-physics argument and the white-dwarf/neutron-star empirical bound.
Extracted summary
Abstract, §9 — thesis (Premise B)
H-22 - Even hypothetically stable LHC black holes pose no risk to Earth on timescales below the Earth's natural lifetime
This is “Premise B” of the LSAG safety case: the fallback that makes the safety conclusion independent of Hawking-radiation theory. Its logical structure is a dichotomy over the crossover radius R_C at which gravity becomes four-dimensional: (a) if R_C <~ 200 Angstrom, accretion in Earth is slower than the solar lifetime by direct calculation; (b) if R_C >~ 15-200 Angstrom, accretion would also destroy white dwarfs (and for D>=8, neutron stars) far faster than their observed >~ Gyr ages — contradicting observation — so such configurations cannot be realized. The paper’s supporting layers: charged stable holes are already excluded by Earth/Sun survival; neutral ones by the white-dwarf and neutron-star bounds.
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§1, §3 — framework hypotheses
H-23 - TeV-scale gravity with extra dimensions is realized, allowing black-hole production at the LHC
The enabling assumption of the whole black-hole-risk discussion; the paper itself calls it “a fascinating … but also a somewhat unlikely possibility”. Current lower bounds on M_D at writing were ~1 TeV. Black-hole formation threshold: the paper conservatively allows M_min = 3 M_D (entropy ~ 24 at M = 5 M_D is the usual semiclassicality criterion). Without this hypothesis no LHC black holes exist and the risk question is moot.
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H-24 - Micro black holes fail to Hawking-radiate yet still discharge, remaining stable and neutral
The hypothetical scenario the paper is built to bound — the only variant of “stable black hole” not already excluded by Earth and Sun survival (charged stable holes stop in Earth/Sun; see the charged-stopping argument). The paper notes there is no known consistent microphysics in which Schwinger discharge operates while Hawking radiation does not — both are quantum pair-creation effects, differing in that Schwinger discharge happens outside the horizon while Hawking radiation is trans-horizon — so the scenario is regarded as close to self-contradictory, but it is not strictly ruled out by that observation alone.
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§2.1 — why stability is already implausible
A-44 - Basic quantum principles imply LHC-scale black holes decay in about 1e-27 s - stability would need physics with no consistent framework
Reasoning. (1) Stability in quantum theory requires protection by a conservation law (baryon number, lepton number, etc.); a hole formed from two partons with mass <~ 10 M_D has no such protecting charge, so all decay channels to light matter are open, and generically an allowed decay occurs on the natural timescale — here 1/M_D ~ 10^-27 s. (2) Hawking’s original derivation has a known weak point (trans-Planckian modes), but the result has been rederived by routes that avoid or confront it: trace-anomaly stress-tensor constructions (Christensen-Fulling and higher-dimensional successors), modified short-distance dispersion, and condensed-matter analog systems exhibiting the effect (“despite the shaky derivation, the effect is almost certainly right”); the unresolved information-paradox issues concern subtle corrections to exact thermality, not whether decay occurs. (3) The only stable variant not already excluded by Earth/Sun survival requires the hole to be neutral, i.e. Schwinger (chromo-)electric discharge must operate; but Schwinger pair creation rests on the same quantum principles as Hawking emission — the one distinction is that discharge can be described as pair production outside the horizon while Hawking radiation is trans-horizon — so suppressing the latter but not the former requires artificial boundary conditions at the horizon with no known consistent microphysics. The paper nevertheless adopts the scenario as a working hypothesis precisely to test it against astrophysical data.
Step 6 — validity verdict
approved / checked. Reconstruction — premises: (i) quantum stability requires protection by a conservation law; (ii) a hole formed from two partons at <~10 M_D carries no conserved charge not carried by ordinary matter; (iii) the only dimensionful timescale is 1/M_D ~ 1e-27 s and no small parameter suppresses the open channels; (iv) Schwinger (chromo-)electric discharge and Hawking emission rest on the same vacuum-pair-creation principles, differing only in where the pair is created relative to the horizon. Load-bearing steps: (i)+(ii) open all decay channels to light matter, (iii) makes decay at ~1e-27 s the generic outcome; and (iv) means a scenario keeping discharge while deleting Hawking decay needs horizon boundary conditions with no known consistent microphysics. The hidden premise — “no known consistent framework” is not “logically impossible” — is honored by the hedged conclusion (“nearly self-contradictory”) and by the source itself adopting the scenario as a working hypothesis to test empirically; nothing stronger is claimed than the premises license. Traced; valid.
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§2.2 — cosmic-ray arguments on Earth and Sun
O-32 - Cosmic rays have produced of order 10^22 nucleon-nucleon collisions above LHC energy on Earth alone
The flux relation is a conservative lower bound to the measured spectra (Auger, HiRes) in the region 10^17 eV < E < E_GZK ~ 5 x 10^19 eV; the paper’s quantitative production-rate calculations (Appendix E) use the actual Auger spectrum up to 2 x 10^20 eV, interpolated linearly, with a nucleon-nucleon inelastic cross section of 100 mb. For nuclei only E/A per nucleon is available, so a pure-iron composition reduces the above-threshold rate by a large factor — which is why composition worst cases are carried through the paper’s bounds.
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A-45 - Charged stable black holes would stop in Earth or the Sun, so their multi-billion-year survival excludes that scenario
Reasoning. Electromagnetic energy loss is a long-range process independent of the particle’s internal microphysics, so a charged black hole’s stopping power equals a muon’s at equal Lorentz gamma. Bethe-Bloch regime (gamma < 10^3): ~2 MeV cm^2/g, i.e. ~11 MeV/cm at mean Earth density; above gamma ~ 10^3, pair production, bremsstrahlung and nuclear dissociation grow linearly, ~60 MeV/cm at gamma ~ 10^4. Production kinematics force the hole’s momentum above M^2/2m_p ~ 10^8 GeV (for M = 14 TeV), giving a slow-down distance to gamma ~ 10^3 of order M/(6 keV) cm — over 10^4 km at 14 TeV, more than Earth’s radius, with the remaining slow-down at ~1 GeV/m taking a comparable distance; a careful estimate still leaves Earth stopping unit-charge holes up to ~7 TeV. For heavier holes the Sun suffices: core density ~150 g/cm^3 over ~1.4 x 10^5 km stops well beyond 100 TeV. Magnetic monopoles stop ~100x more easily (magnetic charge >= 1/alpha times e). Any accreted extra charge only strengthens stopping. Given the ~10^22/A above-LHC-energy collisions over Earth’s history, many such objects would have stopped and accreted in Earth and Sun; their multi-Gyr health closes the charged branch — which is why the paper’s remaining work targets the neutral branch only.
Step 6 — validity verdict
approved / checked. Reconstruction — premises: (i) electromagnetic stopping depends only on charge and gamma, not internal microphysics, so a charged hole stops like a heavy muon; (ii) the stopping ranges: Earth column stops unit-charge holes up to ~7 TeV, the solar core (~150 g/cm^3 over 1.4e5 km, column ~2e12 g/cm^2) well beyond 100 TeV; (iii) ~1e22/A above-LHC-energy cosmic-ray collisions over Earth’s history imply many produced holes stopped and accreted, if stable charged holes exist at these masses; (iv) Earth and Sun persist after ~4.5 Gyr. Modus tollens closes the charged branch. The load-bearing hidden premise — the hole retains its charge while slowing — is the branch definition itself; a hole that neutralizes exits to the neutral branch, which the argument explicitly hands to the compact-star analysis (A-46 ff.), so the defeater “it might shed charge” changes the branch, not the validity. The range arithmetic is order-of-magnitude traceable (~2 MeV cm^2/g against the available columns). Checked.
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A-46 - Neutral cosmic-ray-produced black holes punch through Earth and Sun, voiding the naive cosmic-ray safety argument
Reasoning. Two gravitational slow-down channels exist for a relativistic neutral hole: capture of partons entering impact parameter b_min*R (accretion slow-down, with quantum capture cross section sigma_c = 2^((3D-13)/(D-3)) pi R^2, b_min ~ 1.4-2.4) and elastic gravitational scattering (Coulomb slow-down, subdominant: c_sc ~ 0.5, 0.25, 0.17 for D = 5, 6, 7). Integrating the coupled momentum-loss/mass-growth equations from initial boost gamma_i (production kinematics: gamma > M/2m_p, significant production up to gamma_i ~ 3M/m_p ~ 4.5 x 10^4 at 14 TeV) gives a required stopping column density of order M_0^3/pi R^2 ~ d_0 rho: about 4.6 x 10^12 g/cm^2, versus Earth’s actual column
1.1 x 10^10 g/cm^2 — a few GeV of accretion per transit. Hence Earth (and the Sun, and ordinary stars, and the interstellar medium — the hole even escapes the galaxy) neither stops nor keeps cosmic-ray-produced neutral holes, and the classic “cosmic rays hit Earth harder than the LHC ever will” argument fails exactly in the neutral-stable case that matters. (The same physics implies typical LHC-produced holes, born faster than Earth’s 11 km/s escape velocity, mostly leave: trapping probabilities are0.01 cm) are the only usable natural beam dumps — hence the paper’s compact-star strategy.10^-4-10^-3, and the expected number of trapped holes over the LHC lifetime falls below 1 for M >7 TeV.) White dwarfs (d_0 ~ 1.5 km vs radii of 10^3-10^4 km) and neutron stars (d_0 <Step 6 — validity verdict
approved / checked. Reconstruction — premises: a neutral hole interacts only gravitationally, with capture cross section ~pi R^2 (R ~ 1/M_D at TeV masses); production kinematics give boosts up to gamma_i ~ 4.5e4. Load-bearing step: required stopping column ~M_0^3/(pi R^2) ~ 4.6e12 g/cm^2 versus Earth’s actual ~1.1e10 g/cm^2, so the hole exits still relativistic and Earth/Sun survival carries no information about stable neutral holes. Independently spot-checked the core estimate: Earth column ~6.6e33 nucleons/cm^2 times pi R^2 ~ 1e-33 cm^2 gives only a handful of parton captures — a few GeV accreted against ~1e8 GeV of momentum — matching the body. The conclusion (the naive “cosmic rays hit Earth harder” argument fails exactly in the case that matters; only white dwarfs and neutron stars have d_0-scale columns) follows. This argument undercuts a safety argument; approving it records that the undercut is valid, which is what makes A-49-class compact-star bounds load-bearing. Checked.
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O-35 - Earth and Sun have existed for about 4.5 billion years and the Sun has about 5 billion years remaining
Firmly established, uncontested background fact used as the survival baseline in the cosmic-ray safety arguments (“the multi-billion year longevity of Earth and Sun”) and as the natural-timescale yardstick that accretion times are compared against.
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§4 — accretion inside Earth
A-47 - For crossover radius below about 200 Angstrom a stable hole in Earth accretes far slower than the solar lifetime
Reasoning. Subatomic phase: a slow hole captures a nucleus only when its D-dimensional gravity beats the electromagnetic restoring force K*d binding the nucleus to its site (K ~ 14 eV/Angstrom^2, from dipole estimates and Debye temperatures 300-600 K); this defines the capture radius R_EM = [(D-1)^(D-1) k_D M m / ((D-2)^(D-2) K M_D^(D-2))]^(1/(D-1)), well above the Schwarzschild radius. Integrating dM/dt = pi rho v R_EM^2 at escape velocity with mean Earth density (both conservative: real holes slow as they accrete and see higher central density but lower speed) gives growth-to-1-Angstrom times: negligible for D = 6, 7; but for D >= 8 the force law turns 4D at R_C (5-6 R_D, with R_D < 1 Angstrom), and the 4D phase takes t = 9.9 x 10^18/chi s ~ 3 x 10^11 yr — the weakness of 4D gravity is the bottleneck. Macroscopic phase: once R_EM > a ~ 1 Angstrom (mass
10^11 g in 4D), treat Earth’s interior as an inviscid compressible fluid (neglecting cohesion — again conservative) and apply Bondi accretion generalized to D dimensions: dM/dt = pi lambda_D c_s R_B^2 rho with R_B = ((D-3)/4c_s^2)^(1/(D-3)) R and 4 ⇐ lambda_4 ⇐ 18. With d_0 c_s = 1.33 x 10^-4 s (Birch’s law, tested for Fe to Earth-core densities), the 4D radius-doubling time from R_B ~ a is 1.2 x 10^12/lambda_4 yr. D = 7 (R_D > a) combines D-dimensional and 4D Bondi phases into 6.4-94 Gyr for M_D = 1-5 TeV. Cross-check: a captured minimum-mass (10^15 g) primordial black hole would consume Earth in >47 Myr — consistent with no PBH having been captured, and calibrating the formalism. Radiative (Eddington) feedback is argued NOT to slow this further (spherical BH accretion is radiatively inefficient), so the bound does not lean on it.Step 6 — validity verdict
approved / trusted. Reconstruction: trapped-hole growth is computed in two phases — subatomic capture only when D-dimensional gravity beats the electromagnetic restoring force (capture radius R_EM), then D-dimensional/4D Bondi accretion — under uniformly growth-maximizing choices (escape velocity with mean density, fluid Earth without cohesion, no Eddington throttling claimed as help); the resulting times (~3e11 yr for the D>=8 4D phase, ~1.2e12/lambda_4 yr for 4D Bondi, 6.4-94 Gyr for D=7) all vastly exceed the Sun’s
Link to original5 Gyr remainder, giving the conclusion for R_C <200 Angstrom. The logical frame — conservative lower bounds on accretion time exceeding the solar lifetime imply irrelevance — is traced and valid. Why trusted: the load-bearing numeric content (R_EM formula, D-dimensional Bondi generalization, phase-matching integrations across ~20 pages of section 4 and Appendix A) is a specialist derivation not retraceable at reasonable cost. Credibility: the internal calibration against a captured 1e15 g primordial hole (>=47 Myr to consume Earth, consistent with standard results) checks the formalism; adopted by the LSAG review; no published quantitative refutation.
A-48 - For crossover radius above about 200 Angstrom Earth accretion could be fast, so Earth-internal calculation alone cannot close the safety case
Reasoning. For D = 6, 7 the extra dimensions are larger than the atomic scale, so subatomic growth and the early Bondi phase proceed under the much stronger D-dimensional force law; the slow 4D bottleneck is entered only at R_B ~ R_C. Modeling the R_D-to-R_C transition conservatively (constant Bondi radius R_C), total times are ~10^5 (M_D/M_0)^2 yr for D = 6 but 6.4-94 Gyr for D = 7 — the D = 6 case is short on geologic timescales. The warped D = 5 extreme (R_C ~ 0.2 mm, just under lab bounds on deviations from Newtonian gravity) runs through 5D subnuclear/subatomic/Bondi phases in ~5 x 10^-3 s to 0.1 g, then a slow warped phase, then 4D evolution — each bounded below by ~3 x 10^5 yr. These are lower bounds under deliberately growth-maximizing assumptions (fluid Earth, no charge suppression, no Eddington limit), so they do not prove danger; but they mean the Earth-internal calculation cannot by itself exclude it. This honesty about the fast corner is what makes the white-dwarf/neutron-star empirical bounds load-bearing rather than decorative: the paper’s safety conclusion for exactly these scenarios rests on compact-star survival.
Step 6 — validity verdict
approved / checked. Reconstruction — premises: for unwarped D = 6 (R_D ~ 5e-2 cm) and warped D = 5 with R_C ~ 0.2 mm, the D-dimensional force law — far stronger than 4D gravity at short range — governs growth all the way to macroscopic radii, so the 4D-weakness bottleneck that produced A-47’s long times never operates below R_C; the same conservative formalism then returns lower bounds as short as ~1e4-1e5 yr (D=6) and ~3e5 yr (warped D=5). Load-bearing step: short lower bounds obtained under deliberately growth-maximizing assumptions do not demonstrate danger, but they do mean the Earth-internal calculation cannot exclude it — exclusion requires a long lower bound. That step is elementary and is the entire conclusion; it survives even if the specific ~1e5 yr figures shift, because the structural point (no 4D bottleneck when R_C is macroscopic) is qualitative and traceable. Hence the safety conclusion for these corners must come from compact-star bounds — exactly what the statement says, and what makes A-49 ff. load-bearing rather than decorative. Checked.
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§4.5, Appendix B — radiative feedback
A-53 - No Eddington limit throttles micro-black-hole accretion in Earth, white dwarfs, or neutron stars
Reasoning. An Eddington limit requires luminosity L = eta dM/dt whose radiation force etaMdotsigma/(4 pi m r^2) cancels gravity at the sonic radius; the modified Bernoulli equation then shuts off accretion when the Eddington term dominates, giving the 4D limit Mdot ⇐ 4 pi m G M/(eta sigma) and its higher-dimensional generalization Mdot ⇐ f(Gamma,D) 8 pi m R_B c_s^2/(eta sigma). Whether this is ever reached depends on how much radiation is generated and whether it can propagate outward. (1) Generation: for spherical (Bondi) inflow onto a bare horizon there is no hard surface or accretion disk; free-free emissivity integrated over the supersonic region is small — the well-known inefficiency of spherical accretion (Shapiro-Teukolsky; Frank-King-Raine). (2) Pressure comparison: ambient photon pressure over fluid pressure is T^4/p ~ 10^-6 (young white dwarf, T ~ 10^8 K vs degenerate pressure),
4 x 10^-12 (Earth center), <10^-11 (neutron star), and the ratio p_gamma/p scales as rho^(3Gamma-4), staying small inward. (3) Transport: in white dwarfs the ambient photon mean free path is l_0 ~ 5 x 10^-8 cm (opacity calculations, falling100x by 1 Gyr age), so beyond tiny radii the medium is optically thick; the photon-trapping condition r^2 < R R_B^3/l_0^2 holds out to the sonic radius once R_B >l_0/c_s — outward diffusion loses to inward advection even for very small 4D holes. Earth is the least opaque case (l_0 ~ 0.5 cm, an optically-thin 4D window between R_C and l_0): here luminosity can escape, but attaining Eddington with efficiency eta forces the thermalization radius r_T ~ R/eta^2 to be large, which caps the last-scattering temperature and undercuts the required luminosity — a self-consistency failure matching the known difficulty of radiation-limited spherical accretion; and any melting merely reproduces the fluid treatment already assumed. (4) Observable cross-check: were white-dwarf holes Eddington-limited radiators at L_Edd ~ 8 pi m R_B c_s^2/sigma, the thousands of accumulated holes would exceed 10^-2 L_sun once N R_B/cm >~ 60, interfering with measured cooling rates (10^-1-10^-3 L_sun) — not observed. Hence Earth safety cannot be rescued from the fast-accretion corners by radiative feedback, and (conversely) the white-dwarf/neutron-star destruction times are not lengthened by it.Step 6 verdict
Verdict: approved (checked). Load-bearing step: for Bondi inflow onto a bare horizon, radiation cannot both be generated efficiently and propagate outward against the flow - (1) spherical-accretion radiative inefficiency (standard result), (2) p_gamma/p ⇐ 1e-6 in all three bodies, (3) photon trapping/advection once optically thick; Earth’s optically-thin 4D window is closed by the eta-vs-thermalization-radius self-consistency argument. Plus an independent observational modus tollens: Eddington-limited WD holes would exceed 1e-2 L_sun, interfering with measured cooling - not observed. Each sub-step traced at the physics-argument level; no defeater found conditional on the no-disk spherical premise (reasonable for a hole at rest in pressure-supported matter). Note: A-7 (Plaga) contests the premise set for the metastable Hawking-radiating case; that is a premise dispute for steps 7-8, not an inference failure here.
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§5-7 — the white-dwarf bound
A-49 - Solar-mass white dwarfs provide enough column density to stop cosmic-ray-produced neutral black holes for D up to 7
Reasoning. The relativistic phase (boost gamma_i up to ~3M_i/m_p ~ 4.5 x 10^4) is governed by parton capture within b_min*R plus gravitational Coulomb scattering; the required column density (5.25) scales as (M_D/M_0)^3 (gamma_i M_i/M_D)^((D-5)/(D-3)) M_0^3 with M_0^3 = 4.6 x 10^12 g/cm^2. The nonrelativistic phase (from p ~ M down to below the escape velocity v_WD ~ 0.02) proceeds by sub-nuclear “bag-model” accretion of nucleons (checked in Appendix D: captured mass per nucleon crossing < m_p, capture radius stays under 1 fm); its column density (5.32) is of the same order. Parameter robustness: the stopping distance is maximized at c_ac,M = c_ac,p (perfect accretion) and small c_sc; varying c_ac,p from 1 to 1/4 changes the bound < 25% provided c_sc > (0.12, 0.07) for D = 6, 7 — and the actual quantum-scattering estimates give c_sc = (0.25, 0.17); the final numbers take the worst case (c_sc = 0, c_ac,p = 1, M_D = M_i/3). Comparison against realistic density profiles (Shen models; column densities 13/21/38 x 10^15 g/cm^2 for 1.0/1.1/1.2 Msun, ~10% systematic from temperature and composition) yields the stopping verdicts; off-zenith incidence reduces available column (Fig. 1 right), which is priced into the production-rate acceptances (10%, 10%, 2% of flux kept for D = 5, 6, 7). Note the D = 7 gap is harmless: D = 7 holes above ~6 TeV already have Earth-accretion times > 20 Gyr, and the sub-6-TeV ones stop in a solar-mass star.
Step 6 — validity verdict
approved / trusted. Reconstruction: the column density required to stop a 7-14 TeV neutral hole — relativistic parton-capture-plus-scattering phase (Eq. 5.25, scaling as (M_D/M_0)^3 gamma_i^((D-5)/(D-3))) followed by nonrelativistic bag-model nucleon accretion (Eq. 5.32, same order) — is compared with available white-dwarf diameters (13/21/38 e15 g/cm^2 for 1.0/1.1/1.2 Msun, +/-10% systematic): D = 5, 6 stop in a solar-mass star, D = 7 needs >~1.1 Msun, D >= 8 generally exceeds WD stopping power and is handed to neutron stars. The comparison logic, worst-case parameter bookkeeping (c_sc = 0, c_ac,p = 1, M_D = M_i/3; <25% sensitivity across the allowed range), off-zenith acceptance haircuts, and the closure of the D = 7 gap via A-47’s >20 Gyr Earth times are traced and internally consistent. Why trusted: the load-bearing numbers come from integrating the coupled slow-down/mass-growth equations through realistic WD profiles (section 5, Appendices C-D) — a specialist computation not retraceable here. Credibility: same internally cross-checked formalism as A-47, adopted by LSAG, no published refutation.
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A-50 - Cosmic rays produce thousands of black holes on low-field white dwarfs within tens of millions of years under any composition
Reasoning. Cross section: sigma ~ pi R^2(sqrt(s-hat))/4, black hole formed only for b < 0.5 R with inelasticity y (fraction of parton CM energy trapped); conservatively y = M_min/14 TeV — the smallest value still compatible with producing mass M_min at the LHC (larger y only raises cosmic-ray rates), and M_min = 3 M_D (lower than the usual 5 M_D threshold, again lowering rates only through larger allowed M_D). Convolution: CTEQ6M PDFs at scale 1/R; bulk of production at parton x <~ 0.6 where PDFs are known to <10%; Auger spectrum (linearly interpolated, up to 2 x 10^20 eV) with sigma_NN = 100 mb; for mass-A primaries only E/A per nucleon. Magnetic screening (Appendix G): synchrotron loss in the star’s dipole field caps the surface-reaching energy at E_max = 3.6 x 10^18 eV (5000 km / R_0)(10^6 G / B_p)^2 for protons at theta = pi/2, so fields <~ few x 10^5 G pass the needed ~10^18-10^20 eV flux — and such low-field massive white dwarfs are observed. Robustness checks in the paper: a hypothesized 20% Auger energy-scale overestimate roughly halves rates; pure-Fe composition (inconsistent with shower-depth data showing mixed composition, ~ 5, and with a >~10% proton fraction indicated near the GZK energy) still yields thousands of holes per few thousand years in the main cases; the interstellar-medium channel (production probability ~1.4 x 10^-4 en route) provides ~1/Myr even for pure Fe onto white dwarfs of ANY magnetic field — enough to doom a 120-Myr-old Sirius B in the dangerous scenarios. Stopping-acceptance reductions (10%/10%/2% for D = 5/6/7) are already included in the >5000-per-10-Myr statement.
Step 6 verdict
Verdict: approved (checked). Reconstruction: measured Auger flux x trapped-surface geometric cross section x PDF convolution x magnetic-screening cap, with every disputable choice (y = M_min/14 TeV, M_min = 3 M_D, pure-Fe composition, stopping-geometry acceptance) pushed in the rate-lowering direction, so the accumulated-hole counts are lower bounds and “production is never the bottleneck” follows. Load-bearing step: the direction-of-conservatism plus orders-of-magnitude margin. Traced with an independent order-of-magnitude cross-check: flux above 1e19 eV ~ 1/km^2/yr on a pi(5400 km)^2 disc gives ~1e8 primaries/yr; per-impact BH probability 1e-8..1e-10 (sigma_BH ~ pb-nb vs sigma_NN ~ 100 mb) yields 1-100/yr, matching the quoted per-Myr ranges. Hidden premise surfaced: trapped-surface production estimates remain valid at these parton energies - a premise priced downstream, not an inference gap. No undercutting defeater survives conditional on the premises.
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A-51 - White-dwarf survival excludes every scenario in which Earth accretion would be dangerous - the accretion-time ratio is about 2e4
Reasoning. Setting: the stopped hole sits in degenerate-electron fluid with embedded C/O nuclei, rho ~ 10^7 g/cm^3, a_WD ~ 10^-10 cm, thermal velocities v_T >~ 3 x 10^-4; cores crystallize only at
0.6 Gyr, after the relevant times. Subatomic phase: same electromagnetic-competition formalism as Earth with K/(m M_0^2) ~ 1.2 x 10^-18 gives t_EM = 1.5 x 10^-7 s (D=5), 0.09 s (D=6), 6 x 10^4 s (D=7) to reach a_WD — negligible. Bondi phase: white-dwarf parameters d_0 = 1.4 x 10^-6 s, c_s = 1.4 x 10^-2 make d_0 c_s ~ 1.5 x 10^-4 of Earth’s value, so all Earth Bondi timescales carry over divided by5 x 10^6 yr, and even the unrealistic constant-R_D variant only multiplies this by (R_C/R_D)^2 ~ 400, still <~ Gyr for R_C >~ 30 Angstrom. The clean general relation: t_Earth/t_WD ~ [d_0 c_s (Earth)/d_0 c_s (WD)] x [lambda_4(WD)/lambda_4(Earth)] ~ 1.9 x 10^4 (Gamma = 5/3 Earth, 4/3 WD). Direction of all approximations: slowest-possible star accretion (constant-R_D transition modeling, no Eddington limit assumed for the star, liquid-phase medium) versus fastest-possible Earth accretion — so the ratio, and hence the exclusion, is conservative. Physical reinforcements: white-dwarf matter is liquid (Earth’s is partly solid, cohesion neglected only for Earth) and vastly denser with higher internal pressure. Endgame: once the hole exerts large-scale influence the evolution may deviate from Bondi but in any case disrupts the star — the observable that Gyr-old massive white dwarfs contradict. A captured minimum-mass primordial black hole would eat a white dwarf in ~1.8 Gyr, a side-prediction consistent with their observed abundance.10^4: D = 6 phases of 8/lambda_6, 4 x 10^2/lambda_4, 15/lambda_4 yr; D = 7 phases up to 6 x 10^7/lambda_4 yr — under 80 Myr even at the largest relevant M_D ~ 4.7 TeV, and 2-4x faster in 1.1-1.2 Msun stars (higher central density). Warped/general case: growth is dominated by the crossover radius, t_w ~ (16 pi c_s d_0 / lambda_4)(M_4/M_0)^2 / (M_0 R_C); R_C >15 Angstrom gives t_WD <Step 6 verdict
Verdict: approved (checked). Load-bearing step: the dichotomy - any (D, M_D, R_C) configuration whose Earth accretion completes within the solar lifetime has white-dwarf accretion faster by t_Earth/t_WD ~ 1.9e4, contradicting observed >~Gyr white-dwarf ages. Traced: the Bondi time scaling t ~ d_0 c_s / lambda carries over because the identical 4D/warped crossover formalism governs both bodies, and the stated approximation directions (slowest-possible star, fastest-possible Earth) make the ratio conservative. Conditional premises - holes are produced and stopped in WDs (A-49/A-50), no radiative throttling (A-53), Bondi applicability in the degenerate liquid - are premises priced in steps 7-8. Undercutting candidates (Eddington limit, Earth-solid vs WD-liquid asymmetry) attack premises, not the inference, and are addressed by companion arguments. Valid.
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O-33 - Massive white dwarfs with very low magnetic fields and ages from 100 Myr to gigayears are observed
These specific stars are the empirical anchor of the white-dwarf bound: each is simultaneously (i) massive enough to stop cosmic-ray-produced neutral black holes, (ii) weakly magnetized enough that cosmic rays reach its surface unscreened, and (iii) old enough that catalyzed decay on <~100 Myr timescales would already have destroyed it. Sirius B (1.0 Msun, ~120 Myr) plays the same role for the interstellar-medium production channel, which is magnetic-field-independent. Quoted mass/age/field values carry ~10% column-density systematics (temperature, composition).
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§8 — the neutron-star bound
A-52 - Neutron-star survival bounds the D of 8 and higher scenarios via binary beam dumps and cosmogenic neutrinos
Reasoning. Stopping: at nuclear densities (>~ 2 x 10^14 g/cm^3) the gravitational stopping scale d_0 = M_0^3/pi rho <~ 0.01 cm << 10 km, so even the most boosted holes stop promptly and fall below the ~c escape velocity. Production route 1 — binaries: cosmic rays aimed at the neutron star but intercepted by its companion convert to black holes on the companion (no magnetic obstacle for the neutral product); the FCE = integral of the companion’s solid-angle fraction over the system lifetime is 2-30 Myr across known X-ray-binary classes (2-6 Myr very confident), and with even a 10% proton fraction the 14-TeV worst-case rate
5/Myr x 2 Myr FCE suffices; smaller M_min or larger proton fraction gives orders of magnitude more. Route 2 — cosmogenic neutrinos: protons above the GZK energy photo-produce pions on the CMB whose decays guarantee a neutrino flux; neutrinos evade synchrotron losses entirely, and their entire energy is available (no 1/A or parton-x penalty), with black-hole production competing only against the electroweak cross section; using a parameterization at the lower edge of worst-case (Fe-only-injection) cosmogenic-flux fits, rates are >= 4.5 x 10^3/Myr (D = 5) to 6.2 x 10^4/Myr (D = 11) at M_min = 14 TeV, y = 0.5. Interior evolution: a hole temporarily color-bound in the crust absorbs the rest of the nucleon in t_abs ~ 10^7 fm, re-neutralizes, and drifts down with average velocity sqrt(g_NS d_free/2); crust transit <10 s x r_c[1/TeV]. In the core, subnuclear geometric accretion dM/dt = pi rho R^2 reaches r_N within a fraction of a second to a few weeks (6 ⇐ D ⇐ 11); thereafter Bondi accretion with d_0 = 7 x 10^-3 s, c_s ~ 0.1 (10^-6 of white-dwarf times) completes in ⇐ ~10 Myr even for the slowest (D = 11, 4D phase) case, ~20 yr in warped cases with R_C >= 5 Angstrom. Caveats the paper itself flags: sub-GZK composition may be heavy (though shower data indicate mixed, ~ 5, and a proton component at the GZK edge); the cosmogenic neutrino flux, while theoretically guaranteed by observed super-GZK cosmic rays plus any nonzero proton fraction, was not yet directly measured, and brane-world models with quarks and neutrinos on different branes could in principle suppress neutrino-parton gravity. Hence the bound is presented as very strong but not absolutely certain — the escape hatch requires two independent unlikely failures simultaneously.Step 6 verdict
Verdict: approved (checked). The conclusion is properly conditionalized inside the statement (“unless cosmic rays are almost purely heavy AND the cosmogenic neutrino flux is absent or non-reactive”) - the two escape hatches are internalized, so no undercutting defeater survives. Traced: stopping (d_0 = M_0^3/pi rho <~ 0.01 cm << 10 km at nuclear density), the FCE arithmetic for binary beam dumps, the neutrino route’s freedom from 1/A and synchrotron penalties, and interior consumption ⇐ ~10 Myr << Gyr ages. Rate inputs (Auger flux, cosmogenic-flux parameterization, binary demographics) are premises priced downstream. Valid conditional on them.
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O-34 - Neutron stars with gigayear lifetimes, fields of 10^8 G and up, and long-lived binary companions are observed
The neutron-star side of the empirical anchor: their extreme density (>
Link to original2 x 10^14 g/cm^3, d_0 <0.01 cm stopping scale) makes them stop even the black holes that white dwarfs cannot (D >= 8), and their known ages bound how long a captured stable black hole can have been accreting. The strong magnetic fields are a complication (synchrotron screening of charged cosmic rays), which is why the binary-companion beam-dump geometry and the cosmogenic-neutrino flux carry the production side of the neutron-star bound.