arxiv: 2605.11600 · v1 · submitted 2026-05-12 · 🌌 astro-ph.CO · astro-ph.GA· astro-ph.IM

Recognition: 2 theorem links

· Lean Theorem

A hierarchical Bayesian pipeline for soliton-plus-NFW inference on SPARC rotation curves: diagnostics and prior-boundary behaviour

Prasun Panthi , Md Shahrier Islam Arham

Authors on Pith no claims yet

Pith reviewed 2026-05-13 01:32 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.GAastro-ph.IM

keywords fuzzy dark mattersoliton coreSPARC rotation curveshierarchical Bayesian inferenceNFW halogalaxy dynamicsultralight axionposterior boundary diagnostics

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The pith

A hierarchical Bayesian analysis of SPARC rotation curves finds no population-level soliton component under standard fuzzy dark matter priors.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a hierarchical Bayesian pipeline that fits a soliton-plus-NFW model to galaxy rotation curves while letting the core-halo mass scaling act as a single shared parameter across the entire sample. When applied to 106 SPARC galaxies and sampled with NUTS, the joint posterior for the soliton mass and scaling exponent piles up against the edges of the chosen priors. This boundary accumulation leads the authors to conclude that the data do not support an interior population-level soliton within the adopted Schive-normalized model and standard prior ranges. The main advance is the full pipeline together with a diagnostic workflow that flags when posteriors are dominated by prior boundaries in large rotation-curve analyses.

Core claim

Within the adopted Schive-normalized model and standard SPARC fuzzy-dark-matter prior range, the selected SPARC sample does not identify an interior population-level soliton component. The pipeline treats the core-halo scaling exponent as a global free parameter, employs a Schive-normalized soliton with a regularized NFW envelope and smooth transition, and ties halo-mass priors to V_flat plus stellar-to-halo-mass relations. Posterior sampling over 106 galaxies yields zero divergences and r-hat approximately 1 for global parameters, but the mass posterior reaches the upper prior edge and the scaling posterior reaches the lower edge, giving log10(m_phi/eV) = -19.20^{+0.12}_{-0.11} and alpha =

What carries the argument

The hierarchical Bayesian model that treats the soliton-halo scaling exponent as a global free parameter and samples the joint posterior over all galaxies using NUTS.

If this is right

The standard Schive normalization combined with current priors produces no detectable interior soliton population in the SPARC sample.
The diagnostic workflow for spotting prior-boundary solutions can be applied to any full-sample rotation-curve analysis.
Future fuzzy-dark-matter models must either adjust the core-halo scaling relation or the soliton profile to avoid prior-dominated posteriors.
The framework provides a reusable template for testing other dark-matter density profiles with global parameters on large galaxy samples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Widening the soliton-mass prior or changing the normalization could move the posterior interior and change the conclusion about soliton presence.
The same boundary-diagnostic approach might reveal whether other ultralight-dark-matter parameter choices are also prior-limited on rotation-curve data.
If the boundary behavior persists across multiple independent datasets, it would strengthen the case that the current soliton-plus-NFW form is insufficient for describing observed cores.

Load-bearing premise

That the posterior accumulating at the prior boundaries demonstrates the absence of a soliton component rather than a mismatch between the Schive normalization, transition regularization, or chosen priors and the actual data.

What would settle it

Re-running the same pipeline with an alternative soliton normalization or widened high-mass stellar-to-halo-mass prior that moves the scaling posterior away from the boundary would show whether the boundary behavior is driven by the data or by model choices.

Figures

Figures reproduced from arXiv: 2605.11600 by Md Shahrier Islam Arham, Prasun Panthi.

**Figure 1.** Figure 1: Headline free-α posterior. The plot shows the joint posterior in (α, log10(mϕ/eV)) and the one-dimensional marginals from 4000 post-warmup samples. Dashed red lines mark the standard prior boundaries. The posterior mass is concentrated near the high-mϕ prior boundary and near α = 0. The posterior medians and 16/84 percentile intervals are log10(mϕ/eV) = −19.20+0.12 −0.11, (20) α = 0.014+0.023 −0.011. (21) … view at source ↗

**Figure 2.** Figure 2: Trace plots for the headline free-α run. Both chains explore the same boundary region. The run has zero divergences and ˆr ≃ 1.000 for the global parameters. The posterior is boundary-pinned. It reaches the upper edge of the standard log10(mϕ/eV) prior and the lower edge of the α prior. The quantile diagnostics make this explicit: 9.5% of log10(mϕ/eV) samples lie within 0.05 dex of the upper prior bound, 2… view at source ↗

**Figure 3.** Figure 3: compares the three free-α runs. The three posterior summaries agree at the plotted precision. The widening applies only to the high-stellar-mass SMHM tier, log10 M⋆ ≥ 9.5. The boundary solution is stable under this prior-widening check. Baseline Drop UGC06787 SMHM 0.30 -19.30 -19.25 -19.20 -19.15 -19.10 -19.05 -19.00 lo g10 (m /e V) Mass-scale posterior Baseline Drop UGC06787 SMHM 0.30 0.00 0.01 0.02 0.03 … view at source ↗

**Figure 4.** Figure 4: Fixed-α sensitivity summary. The α = 1/2 run has two chain modes. The α = 5/9 and α = 1.0 runs sit at the high-mass prior boundary. The α = 1.0 run has divergences. These runs probe posterior structure at fixed α and are reported separately from the headline posterior. stated convergence criteria; (ii) the α = 1/2 run is bimodal, with one chain near the high-mass boundary and one chain near an interior mod… view at source ↗

**Figure 5.** Figure 5: Posterior log10(Mvir/M⊙) across the 106-galaxy sample, sorted by posterior median. Error bars show the 16/84 percentile interval from the headline free-α run. Dashed lines mark the overall halo-mass prior range induced by the sample. The local halo parameters are well behaved while the global soliton parameters move to the boundary solution. The posteriors span the halo-mass range set by the Vflat prior. T… view at source ↗

**Figure 6.** Figure 6: shows a representative rotation-curve decomposition for NGC3198 using posterior-median parameters from the free-α run. The fit has χ 2/N ≃ 1.20, which is a reasonable fit for a representative galaxy and is used here as an illustrative example rather than a sample-wide posterior-predictive check. The inferred core radius is rc ≃ 0.0023 kpc, about 2.3 pc. This scale is far below the kiloparsec radii sampled… view at source ↗

read the original abstract

Galaxy rotation curves provide a direct test of how baryonic matter and dark matter combine to determine the mass profiles of disk galaxies. In ultralight or fuzzy dark matter models, numerical simulations predict a central solitonic core surrounded by an outer halo, but the population-level relation between the core and the host halo remains an important modelling choice. We present a hierarchical Bayesian pipeline for fitting soliton-plus-NFW rotation-curve models to the SPARC database while treating the core-halo scaling exponent as a global free parameter. The model uses a Schive-normalized soliton, a regularized NFW envelope with a smooth transition, halo-mass priors tied to $V_{\rm flat}$, and stellar-to-halo-mass information. We apply the pipeline to 106 SPARC galaxies, including 26 systems with bulges, and sample the resulting 346-dimensional posterior with JAX/NumPyro NUTS. The free-scaling run has zero divergences and $\hat r \simeq 1.000$ for the global parameters. The posterior reaches the upper edge of the standard mass prior and the lower edge of the scaling prior, with $\log_{10}(m_\phi/{\rm eV})=-19.20^{+0.12}_{-0.11}$ and $\alpha=0.014^{+0.023}_{-0.011}$. This boundary behaviour persists after removing UGC06787 and after widening the high-mass stellar-to-halo-mass prior. Within the adopted Schive-normalized model and standard SPARC fuzzy-dark-matter prior range, the selected SPARC sample does not identify an interior population-level soliton component. The main contribution is the hierarchical inference framework and the diagnostic workflow for recognizing boundary solutions in full-sample rotation-curve analyses.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper builds a reusable hierarchical Bayesian pipeline for soliton-plus-NFW fits across the full SPARC sample with the scaling exponent as a global parameter, but the no-interior-soliton conclusion rests on posteriors hitting prior boundaries.

read the letter

The paper sets up a hierarchical Bayesian pipeline to fit soliton-plus-NFW models to 106 SPARC galaxies, with the core-halo scaling exponent treated as a single global parameter. They use JAX/NumPyro for sampling and report solid diagnostics with no divergences and r-hat close to 1. It does a good job laying out the model details, including the Schive normalization, a smooth transition to the NFW part, and priors linked to flat velocity and stellar-to-halo mass relations. The boundary diagnostic part is practical for spotting when posteriors are pushing against limits in large hierarchical fits. The soft spot is the main conclusion. The claim that the sample does not identify an interior population-level soliton comes from alpha going to its lower prior edge and m_phi to its upper edge. They test dropping one galaxy and loosening one prior, but without checking if loosening the normalization or the transition function would allow an interior solution, it's possible the boundary piling reflects a mismatch between the assumed soliton form and the data rather than a true absence. In high-dimensional models this distinction matters. This is aimed at people doing fuzzy dark matter modeling on rotation curves. The framework could be reused or adapted, and the discussion of prior-boundary behavior is worth having in the literature. I think it deserves peer review. The technical setup is clear enough that referees can evaluate the robustness of the boundary interpretation and suggest the extra tests.

Referee Report

2 major / 2 minor

Summary. The paper develops a hierarchical Bayesian pipeline in JAX/NumPyro to fit Schive-normalized soliton-plus-regularized-NFW rotation-curve models to 106 SPARC galaxies (including 26 with bulges), treating the core-halo scaling exponent α as a global free parameter alongside the boson mass m_φ. With 346-dimensional sampling that reports zero divergences and r-hat ≈ 1, the posterior for α and log10(m_φ/eV) accumulates at the lower edge of the α prior and upper edge of the m_φ prior; this boundary behavior persists after removing UGC06787 and widening the high-mass stellar-to-halo-mass prior. The central claim is that, within the adopted model and standard SPARC fuzzy-DM prior range, the sample does not identify an interior population-level soliton component. The main contribution is the inference framework plus a diagnostic workflow for boundary solutions.

Significance. If the boundary accumulation is shown to be data-driven rather than an artifact of the specific Schive normalization, transition regularization, or prior ranges, the result would tighten constraints on ultralight dark matter from rotation curves and demonstrate the value of full hierarchical population-level modeling. The reproducible NumPyro implementation and explicit sensitivity tests to two nuisance choices are strengths that support the diagnostic approach.

major comments (2)

[Abstract] Abstract: the claim that 'the selected SPARC sample does not identify an interior population-level soliton component' is tied to posterior accumulation at the α lower edge and m_φ upper edge. The manuscript demonstrates persistence after excising UGC06787 and widening the stellar-to-halo-mass prior, yet does not report the posterior shape or mode location when the Schive normalization constant or the soliton-NFW transition regularization is relaxed; in a 346-dimensional hierarchical model this omission leaves open whether boundary piling reflects data inconsistency with the assumed functional form rather than true absence of an interior soliton population.
[Results] Results (posterior summary): the reported values α = 0.014^{+0.023}_{-0.011} and log10(m_φ/eV) = -19.20^{+0.12}_{-0.11} are direct sampler outputs, but the interpretation as 'no interior component' requires that an interior mode would appear under plausible model variations. Without an explicit likelihood-surface or re-sampling exercise under altered core-halo scaling priors or transition smoothing, the boundary behavior remains conditional on choices whose mismatch could mimic the observed piling.

minor comments (2)

[Methods] Methods: the exact numerical bounds of the 'standard SPARC fuzzy-dark-matter prior range' for log10(m_φ/eV) and α should be stated explicitly (including whether they are uniform or log-uniform) so that readers can reproduce the boundary location.
[Figures] Figure captions: the corner plots or trace plots for the global parameters should include the prior boundaries as vertical lines to make the piling visually immediate.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the robustness of our boundary-posterior interpretation. We respond point-by-point to the major comments and indicate the revisions we will incorporate.

read point-by-point responses

Referee: The abstract claim that the SPARC sample does not identify an interior population-level soliton component does not report the posterior shape or mode when the Schive normalization constant or the soliton-NFW transition regularization is relaxed, leaving open whether the boundary piling reflects data inconsistency with the functional form.

Authors: We agree that the result is conditional on the adopted Schive et al. (2014) normalization and the smooth transition regularization. The Schive normalization is the standard form derived from cosmological simulations and is the conventional choice in the fuzzy-DM literature; relaxing it would require a new core-halo scaling relation derived from first principles, which lies outside the scope of a paper whose primary contribution is the hierarchical inference pipeline and boundary-diagnostic workflow. The regularization ensures a continuous, differentiable density profile suitable for rotation-curve modeling. We will revise the abstract to qualify the central claim as holding 'within the standard Schive-normalized soliton-plus-regularized-NFW model' and add a short paragraph in the methods section justifying these modeling choices. revision: yes
Referee: The interpretation of the reported posterior values as indicating no interior component requires that an interior mode would appear under plausible model variations. Without an explicit likelihood-surface or re-sampling under altered core-halo scaling priors or transition smoothing, the boundary behavior remains conditional.

Authors: The quoted values are the direct marginal posterior summaries from the converged 346-dimensional NUTS run (zero divergences, r-hat ≈ 1). We already performed two explicit sensitivity tests—removal of UGC06787 and widening the high-mass stellar-to-halo-mass prior—both of which preserve the boundary accumulation, indicating the piling is not driven by those nuisance choices. A full likelihood-surface scan or additional re-samplings under varied regularization in 346 dimensions is computationally prohibitive. The diagnostic workflow (convergence diagnostics plus persistence under prior widening) is the paper’s main methodological contribution for recognizing such boundary solutions. We will add a concise discussion paragraph in the results section stating the conditional nature of the conclusion and the rationale for the chosen model components. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the hierarchical Bayesian inference chain

full rationale

The paper's central result is obtained by running NUTS sampling on an explicitly constructed 346-dimensional hierarchical posterior whose likelihood is the product of per-galaxy rotation-curve likelihoods under a fixed Schive-normalized soliton-plus-regularized-NFW functional form and whose priors are stated in advance. The reported boundary accumulation for the global parameters α and m_φ is a direct numerical output of that sampling procedure, not a quantity defined in terms of itself or recovered by fitting a subset and relabeling the fit as a prediction. No load-bearing step reduces by the paper's own equations to a self-citation, an ansatz smuggled via citation, or a renaming of a known empirical pattern; the model assumptions are external to the data and the conclusion is conditioned on those assumptions rather than being tautological with them.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

The central claim rests on the Schive soliton normalization, regularized NFW transition, and literature-derived priors for halo mass and stellar-to-halo-mass relation; the new element is treating the scaling exponent as a single global parameter rather than fixed or per-galaxy.

free parameters (2)

core-halo scaling exponent alpha = 0.014^{+0.023}_{-0.011}
Treated as global free parameter whose posterior is reported as 0.014^{+0.023}_{-0.011}
log10(m_phi / eV) = -19.20^{+0.12}_{-0.11}
Global soliton mass-scale parameter whose posterior reaches the prior edge at -19.20^{+0.12}_{-0.11}

axioms (3)

domain assumption Schive-normalized soliton profile for the central core
Invoked as the functional form for the ultralight dark matter soliton
domain assumption Regularized NFW envelope with smooth transition to the soliton
Used to model the outer halo while avoiding central divergence
domain assumption Halo-mass priors tied to V_flat and stellar-to-halo-mass relation
Applied to set galaxy-specific priors in the hierarchical model

pith-pipeline@v0.9.0 · 5634 in / 1754 out tokens · 64683 ms · 2026-05-13T01:32:34.218380+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The solitonic ground state ... ρ_sol(r) = ρ_c [1 + 0.091(r/r_c)^2]^8 ... Schive-normalized relation ρ_c = ρ_norm m_22^{-2} (r_c / 1 kpc)^{-4} ... r_c = A_ch m_22^{-1} M_9^{-α} with A_ch = 1.6 kpc, ρ_norm = 1.9×10^7 M_⊙ kpc^{-3}
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

log10(m_φ/eV) ~ U(-25,-19), α ~ U(0,1.5) ... Moster stellar-to-halo-mass prior ... Dutton-Maccio concentration prior

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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