PBH collapse: independent verification and V0 / V1 / V1b implementation plan

Audit date: 23 September 2026.
Specification: the supplied preview.md, not an established result.
Status: literature audit, independently evaluated analytic/LTB diagnostics, and an implementation specification. No Einstein–Vlasov (EV), Einstein–Klein–Gordon (EKG), or production PIC collapse simulations were run. No production-code performance measurements were made.

Executive decision #

Proceed with a collisionless-first investigation, accompanied by a necessary scalar validation track. Do not adopt the brief's stronger conclusion that the EKG-to-EV collapse boundary is already known to an error below 10^-5. Nor should a spherical threshold above 0.33 automatically be labelled a code bug.

The three packages have distinct purposes:

The expensive nonspherical EKG campaign remains gated. If the original PIC branch is unavailable, V0 is a reconstruction project, not a cheap rerun. V1 can start independently with exact LTB and cosmological-geometry tests.

Evidence labels #

• VERIFIED: directly supported by a primary source checked for this audit.
• DERIVED: calculation reproduced here; assumptions are stated.
• WORKING HYPOTHESIS: scientifically plausible but not an acceptance criterion.
• UNVERIFIED / OVERSTATED: not established by the cited evidence.
• IMPLEMENTATION CHOICE: a proposed design, not a result or measured cost.

1. Verification ledger #

1.1 Relevant versions and scope #

The search did not identify a published calculation meeting all of the project's requirements: matched growing-mode data, directly perturbed real quadratic scalar, controlled collisionless comparison, post-caustic evolution, and converged finite-deadline PBH boundaries. This is a bounded literature-search conclusion, not proof that no such unpublished or differently indexed result exists.

1.2 Claims to retain, qualify, or remove #

2. Physical regime and effective-theory assumptions #

2.1 Correct mass scaling #

Use c=hbar=1 and the reduced Planck mass Mbar_P:

For M=10^19 g and gamma=1, m=1 GeV, 200 TeV and 10^13 GeV give q about 7.5×10^4, 1.5×10^10 and 7.5×10^17. The first is an illustrative weakly decaying scalar scale, not a complete model. At M=10^20 g multiply by ten. The lifetime and energy density needed for scalar domination remain independent assumptions.

If Gamma=c m^3/Mbar_P^2, the BBN floor produces a heavy-mass floor only after specifying c and the decay/thermalization history. For c=1/(8 pi), m=200 TeV and g*=10.75, the usual instantaneous-reheating estimate gives T_R about 11 MeV. Neither this decay law nor its coefficient follows from M_PBH.

In a matter-like era, collapse after D=t_c/t_k requires approximately Gamma < H_k/D. At D=200 and gamma=1, the corresponding temperature ceilings are about 2.2×10^5 GeV for 10^19 g and 6.9×10^4 GeV for 10^20 g, taking g*=106.75. Near reheating, evolve radiation production rather than replacing it with an arbitrary hard deadline.

2.2 Local validity, not background validity #

The proposed correspondence needs a hierarchy involving H/m, K/(am), local dynamical time, de Broglie wavelength, envelope gradients, sheet thickness, curvature, and self-interaction/decay scales. In particular,

Use the first-horizon mass here, not an accreted final mass. A tiny first seed can influence the ultimate PBH outcome by later accretion; its importance is not disproved by a small instantaneous mass fraction.

At a minimum record

and separately the local gradient/rest-energy ratio, the wavelength/sheet-width ratio, and the amplitude of anharmonic corrections. Relativistic streaming energy is not itself a failure of relativistic Vlasov. No universal Airy-caustic-width estimate translates directly into a PBH-boundary error.

The free real-scalar oscillaton maximum is about 0.607/(Gm) [R17]. Consequently M/Mmax≈0.824 gamma q. This identifies potentially core-dominated small-q runs, but is not an upper bound on every extended configuration or a theorem about collapse probabilities.

2.3 What the mathematical literature does and does not establish #

The relevant distinctions are:

• A single WKB phase gives a monokinetic dust description before caustics.
• A coarse phase-space description can admit multiple streams. Mocz et al. establish numerical Newtonian potential convergence in their tested configurations, not convergence of a black-hole indicator [R06].
• Widrow develops a relativistic scalar/Vlasov representation with a smoothing hierarchy; it is not a self-consistent cosmological horizon-convergence proof [R14].
• The one-dimensional pure-state semiclassical problem has delicate weak-solution/selection issues; the later selection work [R15] is a reason to control the cold regulator, not a general 3D strong-field theorem.
• Huneau–Luk's Burnett result concerns high-frequency vacuum metrics and an Einstein–massless Vlasov limit [R16]. It is not the massive matter limit in this project.
• Salvi's result is massive Klein–Gordon–Maxwell to relativistic Euler–Maxwell under its hypotheses [R18], not Einstein–Vlasov through cosmological caustics.

The working target remains EV, but the limits q→infinity, regulator width→0, amplitude→threshold, and deadline→infinity need not commute. A finite-q discrepancy over a limited scan can falsify a proposed convergence fit without falsifying all possible asymptotic correspondence.

2.4 Fragmentation before the target mode enters #

With an assumed small-scale density amplitude around 2×10^-5, linear matter-era growth needs about ln(1/delta)≈11 e-folds to reach unity. This conditional estimate is consistent with the mechanism investigated in inflaton-cluster and post-inflationary structure studies [R19,R20]. It does not establish that every model's target background has fragmented.

A fragmented medium is not automatically equivalent to smooth cold particles. Its clump masses, internal structure, tidal evolution, velocity dispersion, and relaxation times must justify that approximation. V1 should distinguish physical initial dispersion from an artificial numerical regulator.

3. Independent LTB and deadline checks #

3.1 Fixed conventions #

Let k_p denote the Gaussian profile parameter, not the characteristic horizon-entry wavenumber:

For reproducing the paper use a_i=1, H_i=5k_p and its constant-bang-time matter solution. Then t_i=2/(15 k_p) and t_k=100 sqrt(6)/k_p. All outputs must carry the convention explicitly. The spatial q(x) used in the paper's gradient expansion must be renamed in software to avoid confusion with m/H_k.

3.2 The proposed causal upper bound is promising, but conditional #

I reconstructed the exact LTB functions and integrated an approximation to the earliest outgoing central null ray, taking its launch radius successively smaller. The accompanying Python code reproduces these diagnostics.

For k_p=1, mu=0.33, a representative regular sphere has:

The ray endpoint changed by less than 10^-7 in k_p r between the two finest launch-radius/step choices. The mu=0.32 ray did not enter a trapped region on the computed interval to k_p r=4. This is consistent with the paper's LTB classification [R01].

Thus the geometrical ingredient of causal protection is supported: at mu=0.33, there is a strictly trapped sphere earlier than the central signal reaches it. Matter evolution modified only inside the central event's causal future cannot remove a trapped sphere outside that future.

What is not established here is the full transfer to a family of regularized EV solutions. That requires demonstrating convergence on the protected exterior region, controlling any earlier caustic or boundary influence, matching the initial geometry, and retaining a strict trapping margin. The launch calculation uses an asymptotic expansion and floating-point ODE integration, not interval arithmetic or a proof identifying every null generator.

Use mu=0.33 as a conditional cold-limit consistency check, not an unconditional bug detector. A finite-dispersion family has different stresses everywhere from the initial slice. The argument gives no lower bound near 0.2 and no theorem ordering spherical and triaxial thresholds.

3.3 Deadline estimate: correct diagnostic, not a collapse law #

For the linear matter growing mode,

Combining delta∝t^(2/3) with the spherical top-hat linear collapse value 1.686 yields

The proximity to the published particle transition motivates V0. It does not validate top-hat dynamics for the triaxial Gaussian peak.

A finite-time plateau does not prove permanent support. A decreasing boundary does not prove that no nonzero asymptotic floor exists. A floor plus a delay, critical slowing near an unstable state, and numerical artifacts can all produce intermediate behavior. Fit and publish the finite-deadline surface first; do not force a binary interpretation.

4. Common implementation contract #

4.1 Three outcomes, with finite-time meaning #

COLLAPSE: a future black-hole marginally outer-trapped surface is found, with outgoing expansion zero and ingoing expansion negative, and detection is stable under numerical checks. A falling lapse or 2M/R>=1 without checking the expansion signs is insufficient in an expanding cosmology.

RESOLVED NON-COLLAPSE BY D: no such surface forms by the specified exterior cosmological deadline, and the evolution to that deadline is resolved. It may still be infalling. A bounce, virial ratio near unity, or falling compaction is not a prerequisite.

UNRESOLVED: the calculation cannot support either classification because of numerical failure, insufficient resolution, uncontrolled constraints, an unvalidated regulator, or ambiguous horizon identification.

Give separate morphology tags: infall, rebound, multistream halo, long-lived core, dispersion. These tags are not substituted for horizon outcomes.

Do not require a newly formed horizon to survive for half the full collapse time as a generic detection rule. Use resolution-stable expansion signs, a short resolved continuation, or validated excision. Historical detection before D remains recorded even if a later optional continuation fails.

4.2 Shared geometry and matter interface #

Use a cosmological, horizon-capable spherical geometry formulation for V1 and V1b, for example reference-metric BSSN with regular origin treatment. A useful general line element is

An asymptotically flat polar-areal solver cannot simply be copied onto the entire initial superhorizon cosmological domain. Likewise, a dust-comoving coordinate system need not survive multistreaming. An independently formulated check can use another admissible gauge on overlapping regular regions.

Define a common matter-source interface returning E, J_i and S_ij, and independent diagnostic routines for the Hamiltonian/momentum constraints, Misner–Sharp mass, and both null expansions. Match scalars at equal exterior cosmological time and compare invariant profiles or proper/areal radii, not unmatched coordinate radii.

COSMOS is a useful public cosmological code base, but its release is advertised for a linear-EOS fluid plus a massless scalar; the code paper explicitly notes that publication-specific additions can be absent from the public release [R10]. COSMOS-S supplies a spherical starting point, not a verified ready-made EV or massive-EKG module. GRChombo/GRTresna provide alternatives, but a validated spherical/cartoon implementation must be demonstrated rather than assumed [R11,R12].

4.3 Record every run reproducibly #

Required metadata: source commit and patch digest; compiler/build flags; initial-data digest; amplitude/profile definition; scalar q and phase or kinetic regulator; metric gauge; spatial/phase-space grid; refinement/deposition rules; time-step history; domain and boundary treatment; exterior time mapping; resource usage; outcome by each deadline; and unresolved reason.

Required products: initial constraint residuals, run-time local and global constraints, horizon diagnostics, mass/flux accounting, density/stress profiles, field wavelength or particle sampling diagnostics, restart-validation logs, and threshold endpoint records.

Test monotonicity in amplitude before bisection. Keep interval-valued boundaries. An unresolved interior trial does not erase a valid outer bracket; an unresolved endpoint cannot establish that side of a bracket.

5. V0: reproduce and extend the published triaxial particle calculation #

Goal #

Separate a reproducible physical finite-deadline transition from a plotting deadline, numerical breakdown, and particle/discretization artifacts. V0 is not automatically a publication-quality EV continuum calculation.

V0.0 — provenance gate #

Obtain the original PIC module or an author-identified equivalent, its commit, parameter files, coordinate mapping, particle loading and deposition, gauge settings, horizon finder, domain, and actual termination reasons for the low-amplitude runs. Archive the exact paper version.

If the original branch is unavailable, label the work an independent reconstruction. Do not budget it as extending a checkpoint. No contact with the authors or acquisition of an unpublished branch has occurred in this audit.

V0.1 — reproduce controls before extending weak peaks #

Use the paper-matched e=0.2,p=0 profile and normalization, with

First run mu=0.045, 0.050 and 0.060 to D=250, keeping the published baseline numerical settings. Include a homogeneous cosmological particle test and a spherical pre-caustic dust/PIC comparison.

Validate the horizon itself, the timing convention, and the constraint histories; matching only a central lapse curve is insufficient. Differences between the reconstruction and the original setup must be recorded before claiming reproduction.

V0.2 — deadline screen #

Use mu={0.030,0.035,0.040,0.045,0.050,0.060}; evolve suitable cases to D=1000. Record outcomes at D={100,250,300,500,1000} from each trajectory. Include collapsing controls rather than testing only the previously non-collapsing interval.

Stop an individual run when its output becomes unresolved; never propagate its non-collapse label to a later deadline. A restart from D=250 must agree with an uninterrupted control evolution within the measured tolerance.

V0.3 — targeted controls, not an indiscriminate full factorial #

After screening, choose near-boundary pairs and test spatial grids N={80,120,160} per octant, physical central resolution held meaningful under the coordinate mapping. Vary particle sampling at fixed grid, then grid at controlled sampling. Candidate checks include 8,27,64 particles per cell with a quiet initial load; retain the original loading as a reproduction control. Change deposition order and check the resulting source conservation and force interpolation consistently.

These values are pilot choices. Increasing particles alone cannot cure a metric/deposition error. Increasing the grid while holding particles fixed can worsen sampling. Test both Hamiltonian and momentum constraints around crossings, not just a whole-domain mean.

V0.4 — boundary control #

For an ideal matter background the light-travel distance from entry to D is

At D=250 this is 10.60 r_=25.96/k_p, and at D=1000 it is 18 r_=44.09/k_p. The original initial slice adds an earlier travel distance of about 1.84 r_* before entry. Therefore simply increasing an outer face from 10/k_p to 25/k_p is not a general causal-isolation certificate.

Choose the boundary from the region being diagnosed, full start-to-deadline characteristic travel, and the actual boundary/gauge characteristics. Then demonstrate domain convergence; this may require a much larger coarse exterior. An absorbing prescription is not assumed perfect.

V0 acceptance and interpretation #

• Reproduction succeeds only if the original controls are recovered with their caveats.
• Evidence for delayed collapse requires newly formed resolved horizons at larger D, with the effect surviving numerical controls.
• Evidence for a persistent finite-time plateau requires converged neighboring brackets; it is not a proof about D→infinity.
• If particle constraints or outcomes do not converge, report V0 as unresolved and use V1 to distinguish physics from the 3D method. Do not relabel an affordable EKG run as a validated Vlasov replacement.

Deliverables: provenance report, an outcome-by-deadline table, convergence panels, and either controlled brackets or explicit one-sided/unresolved intervals. Allocation for subsequent V2 depends on these deliverables, not on a preferred threshold value.

6. V1: spherical Einstein–Vlasov implementation #

V1.0 — exact geometry tests #

Implement the constant-bang-time LTB solution from the supplied Gaussian curvature profile. Reproduce the small-radius limit, FLRW exterior, mass function, density, null expansions and causal diagnostic above. Compute R' and identify shell crossing independently of central focusing. Add automatic tests of positive density and coordinate regularity.

Use mu=1.3 only as a separate published-style strong spherical control when the code/data support its geometry; the small-radius diagnostic script supplied here is deliberately restricted to 0<mu<1. Do not force the same auxiliary coordinate assumptions onto every control.

V1.1 — kinetic matter rather than a moment closure #

Use a spherical distribution f(t,r,p_r,L), where L is the squared angular momentum per particle-rest-mass squared. For unit particle rest mass,

The characteristic equations are

Advect f along these characteristics. Obtain Einstein sources by invariant phase-space integration, or equivalently by orthonormal-frame integrals with a consistently normalized distribution:

Implement particle/characteristic shells for efficiency, with a conservative phase-space calculation for selected independent checks. Neither an isotropic-pressure closure nor a pressureless shell code continued through crossing is an EV calculation.

Test Minkowski free streaming, geodesics in a prescribed curved metric, expanding homogeneous kinetic matter, and cold pre-caustic agreement with LTB. Isolated EV critical-collapse cases [R13] are valuable additional tests, not cosmological initial data to transplant unchanged.

V1.2 — cold regulator and physical dispersion #

Do not assign the same nonzero angular momentum to every shell merely to keep it away from the origin. This creates a centrifugal barrier and can manufacture a threshold. A regular distribution has locally defined tangential velocities; L then scales as R^2 p_t^2 near the origin.

At exactly zero angular momentum, use an origin treatment that follows radial trajectories through the centre or reflects the signed radial momentum with the correct spherical symmetry. Validate this first on a non-self-gravitating test.

Separate two parameters:

1. numerical regularization of nominally cold data;
2. an actual cosmological velocity-dispersion model.

For the cold sequence, propose initial one-dimensional peculiar-speed widths 10^-3, 3×10^-4 and 10^-4 in c=1 units as pilot values, not established sufficiently small values. Resolve each smooth regulator at three numerical resolutions before reducing its width. Repeat selected cases with another positive distribution shape having matched low moments. If the regulators do not approach a common answer, the cold boundary is unresolved.

At finite width, construct the distribution's normalization and bulk velocity to match the intended E and J, then recheck the constraints. Its spatial stress is different from dust and must be included. A fixed physical dispersion is a new physical model; it must not be sent to zero as part of a numerical convergence test.

V1.3 — initial campaign #

Begin with the Gaussian Yoo profile only. A useful amplitude screen is

Record D={100,250,300,500,1000}. Refine amplitude intervals adaptively after testing monotonicity. Target 2% brackets in the pilot and 1% only after numerical and regulator errors support that precision. Apply the conditional 0.33 check only to a demonstrated cold limit on the protected region, not as a hard prior restricting the search.

V1.4 — two peak-theory profiles, explicitly defined #

After the first Gaussian boundary is controlled, add two conditional mean curvature profiles from a specified Gaussian field. A proposed reproducible choice is

with a stated Gaussian smoothing window and smoothing radius R_s=1/k_0. Define each mean profile at peak height nu=4, averaging the conditional field over the peak-weighted curvature distribution and the negative-definite Hessian condition. This is a Kac–Rice/peak-theory construction, not an arbitrary Gaussian or the covariance around an unconstrained point [R21]. Normalize the resulting shape to zeta(0)=1, then vary mu as a controlled amplitude multiplier. Preserve its outer oscillations/compensation and specify its characteristic entry radius.

These two spectra and nu are research choices, not facts supplied by the brief. They test profile sensitivity; they do not stand in for the entire abundance integral. Store the spectrum, smoothing, spectral moments, conditional weights, and generated profile arrays so the family is reproducible.

V1 deliverable and gate #

Publish

with finite-grid and regulator intervals, plus M_AH and formation times. A result above 0.33 triggers inspection of the causal-protection assumptions, phase-space regularization, initial geometry, boundaries and implementation; it is not automatically rejected.

V1 is successful if it gives a controlled answer or exposes that the proposed cold limit is not controlled. It is not required to find a floor near 0.3 or to contradict the published triaxial result.

7. V1b: spherical massive-scalar validation #

V1b.0 — equations and geometry #

Use V(phi)=m^2 phi^2/2, a minimally coupled real scalar and the same geometry/diagnostic interface as V1. With Pi=n^a grad_a phi and Phi_i=D_i phi,

Its sources are

A second independent formulation is needed for selected near-boundary results, not necessarily a duplicate full scan. A scalar implementation using a reference fluid must quantify the fluid's effect and its zero-density limit rather than silently changing the model [R02].

V1b.1 — initial data is the first science gate #

For a comoving constraint-satisfying dust slice, retaining gamma_ij and K_ij and setting phi=0, Pi=sqrt(2 rho_d) gives E=rho_d and J_i=0 exactly. This is a valid seed. It does not prove the absence of finite-q decaying or nonadiabatic contamination.

Construct the physical finite-q growing solution by evolving the homogeneous scalar/Friedmann background and the regular linear adiabatic perturbation equations, then matching a nonlinear gradient-expansion solution and solving the constraints. Tanaka–Sasaki provides relevant single-field gradient-expansion machinery [R22]. GRTresna solves constraints but does not choose the cosmological mode for the user [R11].

Compare initialization on different slices of the same background solution. Starting a fresh field from rest at each q_i is a change of cosmology, not a slice-convergence test. Phase shifts at fixed macroscopic data can likewise be physical at finite q; do not automatically count them as numerical error.

The small-amplitude tests must reproduce finite-q linear EKG, not only a fit to a+a^-3/2. Use the dust growing/decaying decomposition only after averaging and checking the scale lies in the relevant matter-like regime.

V1b.2 — early-time incompatibility #

In a dust-like background, with epsilon_i=K_*/(a_i H_i),

Copying the paper's early slice gives m/H_i≈q/1837. Thus q=100 would start at m/H_i≈0.054, not in an oscillating-matter regime. Starting around q_i=1 may be a valid physical choice, but the non-dust transition must be integrated. The dust power-law relations are not exact through that transition.

Do not impose a small-phase expansion when a phase shift q_i epsilon_i^2 is large; resum the phase where appropriate. More importantly, validate the resulting actual stress and growing mode. Arbitrary E,J fields cannot always be represented by one scalar: J_i=-Pi D_i phi imposes an integrability condition. The instantaneous oscillatory part of the stress is not generally small relative to its average merely because q is large.

V1b.3 — finite-q campaign #

Use q={50,100,200,400} as a conditional pilot, with 800 and 1600 added only when their predicted resource needs and likely information value are demonstrated. The number 50 is not asserted to be asymptotic; neither is 400. At each q record gamma_AH and local scale separation rather than accepting q alone as the regime classifier.

Begin with well-above and well-below candidate boundaries, then bracket mu once the data and wave resolution pass. Use D=300 for the first controlled comparisons, extracting D=100 and 250 at no extra evolution cost; extend selected cases toward D=1000 to distinguish scalar delays from EV delays. Match the EV regulator/coarse-graining assumptions rather than choosing a favorable comparison after seeing the result.

Repeat selected initialized perturbations at different starting slices, with an independently computed linear transfer, and at three spatial and time resolutions. The first gate is initialization consistency; if it fails, stop the scalar threshold scan while V0/V1 can continue.

V1b.4 — analysis and outcomes #

Measure the finite-q boundary difference, not central density alone:

Also compare time to AH, area mass, coarse density/stress, velocity dispersion, scalar/core mass, anisotropy where applicable, outgoing flux and constraints. Extract wave information with windowed spectra or a validated envelope reconstruction; a raw phase-gradient maximum diverging at a node is not by itself a reliable shortest-wavelength estimator.

Do not require alpha in [2/3,2], agreement by q=400, or convergence to a predetermined asymptote. Fit only when the sequence is demonstrably informative. With four points, a free three-parameter power law has little predictive leverage; require additional points or report unresolved extrapolation. An out-of-sample highest-q check is useful but not a substitute for a controlled asymptotic expansion.

A small-gap result supports using EV within the tested regime. It does not certify 10^-5 accuracy at q=10^9. A persistent discrepancy requires inspecting the cold limit, initialization, core scale and horizon mass before interpreting it as a failure of effective theory. Equality of two threshold numbers does not show that the scalar stayed single-stream.

8. Numerical error and compute corrections #

8.1 The RK4 carrier test #

For x=omega dt,

Over duration T, the leading amplitude loss exponent is omega T x^5/144, the energy-loss exponent is omega T x^5/72, and the phase lag is omega T x^4/120. At D=300, omega≈m and mT≈199.33q:

The supplied script also evaluates the exact RK4 oscillator modulus rather than only its leading expansion. Therefore the brief's chosen steps do not automatically satisfy an energy-drift target of 10^-3, even on the oscillator test.

For this diagnostic alone, require

and an independently chosen phase tolerance. In a cosmological run, test against the background continuity equation and a high-accuracy homogeneous solution; scalar energy itself is not constant in an expanding volume.

8.2 Convergence and honest uncertainties #

Keep amplitude-bracket width, spatial truncation, time integration, geometry refinement, source deposition, boundaries, initial data, horizon finding, and kinetic regulator effects separate. Add unknown systematic bounds conservatively. They are not automatically independent Gaussian errors suitable for quadrature or “3 sigma.”

At a smooth fixed kinetic regulator, demand local and integrated constraint convergence. In the cold limit a physical density caustic can make maximum-norm behavior singular, so also test weak/coarse-grained sources and geometry. A small box-averaged residual alone is inadequate. Choose normalizations that remain meaningful in low-density regions, using matter and curvature/background scales rather than dividing by nearly zero density.

A candidate physical EKG–EV difference must exceed the combined uncertainty interval and survive changes of grid, step, initialization, domain, and regulator. Do not treat AIC/BIC selection among four-point fits as proof of an asymptotic law.

8.3 Cost: profiling must precede allocation #

The public literature documents viable cosmological and AMR codes [R10–R12], but it does not calibrate the brief's per-run core-hour forecasts for this long-time problem. There is no measured basis here for promising V0 in 10^3–10^4 or the entire V1 in <=10^3 core-hours.

The proposed q^4.2 scaling is a conditional estimate: q^3 spatial work for a fixed wave-resolved volume, multiplied by q carrier cycles and an extra q^0.2 to control RK4 energy drift. Phase control can instead give q^4.25 under those same assumptions. Spatially localized refinement, a changing core mass, a different integrator and an increased collapse duration alter these exponents. Halo-radius estimates must retain the relevant mass fraction gamma and the chosen compaction convention.

Profile actual geometry updates, particle interpolation/deposition, kinetic quadrature, refinement and I/O. The work estimate is

with each tau measured on the target machine. Extrapolate only after the pilot reaches the expensive multistream/core regime.

A 32–64-core server with 128 GB RAM and roughly 1 TB initial scratch is a reasonable pilot platform, not a guarantee of campaign completion. A distribution grid 2048×512×64 already has about 67 million values, or 0.54 GB per double-precision array; eight such arrays exceed 4 GB before geometry and buffers. A 160^3 octant mesh with 64 particles/cell has 262 million particles, about 21 GB at 80 bytes/particle before grid storage. The brief's universal <1 GB spherical estimate is not justified for phase-space Vlasov.

Approve the next allocation from measured cost per resolved endpoint and the remaining run matrix. Parallelize independent runs where possible. Reserve distributed-memory HPC for configurations demonstrated to need it rather than assuming OpenMP-only public software will scale across nodes unchanged.

9. Abundance and observational conclusions that must not be imported #

A collapse boundary enters a mass-weighted joint distribution of peak scale, amplitude, shape, profile, environment and physical dispersion. Symbolically,

Only after checking monotonicity may the indicator be represented by a simple amplitude step. The initial spectrum controls more than amplitude: smoothing, spectral moments, profile/shape correlations, and small-scale dispersion also matter. The spectrum height is not equal to its integrated variance for arbitrary bandwidth.

Harada et al. derive 0.05556 sigma^5 using a particular shape distribution and collapse model [R23]. Ye et al.'s enhancement belongs to a particular peak-theory/spectrum calculation [R07]. Neither licenses a universal inverse fifth power of the single e=0.2 threshold. A uniform rescaling of an established small-e barrier can generate such a factor; the one existing triaxial point does not establish that rescaling.

The reheating conversion

is a useful estimate for all DM, assuming standard late entropy evolution, beta evaluated after relevant accretion, and the usual g*/g*s factors. It is not a formation-spectrum prediction.

Do not preserve the brief's blanket compatibility labels or its “Yoo particle slope” column. In particular:

• The asteroid window and its dependence on extended mass functions require a model-dependent observational integral [R24].
• Esser et al. v2 reports strong constraints with its preferred stellar-IMF priors, but substantially weaker constraints with uniform priors [R25].
• Gottlieb et al. argues that capture/delivery into stars and stellar demographics may weaken population limits; it does not publish a replacement dwarf-galaxy likelihood proving f=1 is allowed. Its stellar-evolution outcomes are not a blanket no-destruction result [R26].
• Use De la Torre Luque et al. v2, 25 March 2026, including its erratum, for the relevant evaporating tails [R27].
• The quoted ACT spectral index near 0.974 is a combined-data result, not a model-independent ACT-only exclusion of every long inflaton-dominated epoch [R28].
• Use the corrected gradual-transition and sudden-transition induced-GW calculations [R29,R30]. Radiation-era scaling does not by itself establish LISA detectability or exclusion in an early matter era.
• A thermal-history floor around a few MeV has assumptions about decay products and neutrino thermalization; use a specified reheating model [R31].

V0/V1/V1b can reduce collapse-model uncertainty. They cannot certify f_PBH=1 without a specified primordial spectrum, reheating history and mass function.

10. Dependency order and stop conditions #

1. Freeze the specification, conventions, versions and data contract. Keep the analytic diagnostic tests as continuous tests.
2. Run V0 provenance/reproduction and V1 geometry/free-streaming development in parallel. If the PIC source is missing, V1 remains executable as an independent project.
3. Obtain the first Gaussian V1 cold-regulator/deadline map. Do not impose the 0.33 value as a prior.
4. Validate V1b homogeneous and growing-mode data before the scalar threshold scan. Then compare finite-q outcomes with the controlled V1 reference.
5. Add the two explicitly defined peak profiles and physical dispersion only after the basic matched comparison is reliable.
6. Consider V2/V3 only after these outputs show what remains uncertain and whether a nonspherical scalar experiment would reduce that uncertainty.

Hard stops: no reproducible geometry/growing mode; nonconvergent horizons or constraints; noncontrolled cold regularization; unresolved wave/core scales at informative q; missing source provenance represented as a reproduction; or a specified cosmology independently excluded. None permits relabelling failure as non-collapse.

Scientific redirects: a robust Vlasov boundary plus small controlled scalar differences favors EV production calculations. A resolved discrepancy keeps scalar dynamics in the model. A deadline-dependent boundary requires a reheating-dependent prescription. An ambiguous finite-q trend requires more controlled analysis, not a predetermined theory verdict.

Decision: GO with V0 and V1; GO with V1b as a gated validation track. NO-GO for claiming a validated 10^-5 scalar-to-Vlasov threshold error or funding a production nonspherical EKG scan from the present brief.

References and retrieval scope #

The identifiers below are primary-source locators. Version dates above were checked against arXiv submission histories, not HTML-render dates. Public documentation establishes advertised capabilities; no production code was compiled or benchmarked in this audit.

• R01: Yoo, Escrivà, Harada & Kohri. Simulation of PBH formation in a matter-dominated universe. arXiv:2609.14218v1. Sections III–VI; equations 4.1–4.18; figures 1, 7 and 10. Relevant PDF figures were visually checked.
• R02: Milligan et al. Primordial Black Hole Formation in a Scalar Field Dominated Universe. arXiv:2504.02600v2. Sections II, IV–VI and appendix A. Quadratic results are bounds, not a resolved threshold series. I did not recover an unambiguous numerical m/H_k for the quadratic campaign from the inspected text; reproduction requires that configuration.
• R03: de Jong et al. Spinning primordial black holes formed during a matter-dominated era. arXiv:2306.11810v2.
• R04: de Jong, Aurrekoetxea & Lim. Primordial black hole formation with full numerical relativity. arXiv:2109.04896v2; JCAP 03 (2022) 029. Footnote 1.
• R05: Ebrahimian, Abolhasani & Mirbabayi. Primordial black hole formation in matter domination. arXiv:2507.18312v2. Analytic and Newtonian/shell evidence, not a full EV numerical horizon threshold.
• R06: Mocz et al. On the Schrödinger–Poisson–Vlasov–Poisson correspondence. arXiv:1801.03507v2.
• R07: Ye et al. Primordial Black Hole Formation and Spin in Matter Domination Revisited. arXiv:2508.10070v1. Peak-theory matter-era abundance and spin calculation.
• R08: Padilla et al. arXiv:2509.10431v1. Quartic scalar critical-collapse follow-up.
• R09: Escrivà et al. arXiv:2605.04487v1. Nonspherical collapse/GW calculation with Newtonian-force modelling.
• R10: Yoo et al. COSMOS: A numerical relativity code specialized for PBH formation. arXiv:2606.02053v1; JOSS DOI 10.21105/joss.09570. Public repositories cmyoo/cosmos and cmyoo/cosmos-s.
• R11: GRTresna: An open-source code to solve the initial data constraints in numerical relativity. arXiv:2501.13046. CTTK/CTTK-Hybrid and public capability documentation.
• R12: Clough et al. GRChombo: Numerical Relativity with Adaptive Mesh Refinement. arXiv:1503.03436. Capability/scaling evidence, not this project's measured throughput.
• R13: Rein, Rendall & Schaeffer, arXiv:gr-qc/9804040; Olabarrieta & Choptuik, arXiv:gr-qc/0107076, PRD 65 (2002) 024007; Akbarian & Choptuik, arXiv:1409.5176. Isolated spherical EV benchmark families.
• R14: Widrow. arXiv:astro-ph/9607124. Relativistic scalar/Vlasov representation.
• R15: Semiclassical regularization of Vlasov equations and wavepackets for nonlinear Schrödinger equations. arXiv:1505.04707. See also Zhang, Zheng & Mauser, CPAM 55 (2002) 582–632, DOI 10.1002/cpa.3017, for the earlier one-dimensional result; this audit does not assert a general 3D pure-state theorem from that reference.
• R16: Huneau & Luk. Burnett's conjecture in generalized wave coordinates. arXiv:2403.03470. Einstein–massless Vlasov high-frequency vacuum limit.
• R17: Alcubierre et al. Numerical studies of Phi^2-Oscillatons. arXiv:gr-qc/0301105. Maximum mass 0.607 m_Pl^2/m in the stated free-field family.
• R18: Salvi. Semi-classical limit of the massive Klein–Gordon–Maxwell system toward the relativistic Euler–Maxwell system via an adapted modulated energy method. arXiv:2502.06622.
• R19: Niemeyer & Easther. Inflaton Clusters and Inflaton Stars. arXiv:1911.01661.
• R20: Eggemeier et al. Post-inflationary structure formation boosted by parametric self-resonance. arXiv:2311.08780.
• R21: The peak-weighted construction must implement the full Gaussian peak/Kac–Rice measure, not merely set the central gradient to zero; Ye et al. [R07] provides a relevant primary implementation of peak theory in this application.
• R22: Tanaka & Sasaki. Gradient expansion approach to nonlinear superhorizon perturbations II — a single scalar field. arXiv:0706.0678.
• R23: Harada et al. arXiv:1609.01588v2; Kokubu et al. arXiv:1810.03490v1; Harada et al. arXiv:2211.13950v3. Distinct anisotropy, causal-inhomogeneity, and velocity-dispersion prescriptions; do not combine them as independently calibrated factors without checking their assumptions.
• R24: Gorton & Green. How open is the asteroid-mass primordial black hole window? arXiv:2403.03839.
• R25: Esser et al. arXiv:2503.03352v2, 25 June 2025. Section 5.2 and appendices.
• R26: Gottlieb et al. The Life and Death of Stars That Capture Primordial Black Holes. arXiv:2606.02700v1. Especially the discussion of capture rates and population constraints.
• R27: De la Torre Luque, Koechler & Balaji. Refining Galactic primordial black hole evaporation constraints. arXiv:2406.11949v2, 25 March 2026, with erratum.
• R28: Louis et al. The Atacama Cosmology Telescope: DR6 Power Spectra, Likelihoods and LCDM Parameters. arXiv:2503.14452.
• R29: Inomata et al. Gradual-transition induced GWs, arXiv:1904.12878v3, 28 February 2023; corrected numerical figures.
• R30: Inomata et al. Sudden-transition induced GWs, arXiv:1904.12879v4, 21 July 2023.
• R31: Bounds on very low reheating scenarios after Planck. arXiv:1511.00672. Model-dependent thermalization/BBN context.