AMR Reproduction — Lab Log

Record the methodological reset that opens the AMR programme: why we stopped the uniform-grid approach at scale, how the project is now organised, and the rules under which all subsequent science will be logged. This entry is the lab log's own first page — it documents the frame, not yet any physics result.

Expectation going in: a uniform spectral grid cannot reach the published soliton-core resolution at a defensible cost, because it must refine the whole box to resolve a ~pc core. I expect AMR (GAMER, the original method) to reach converged cores on far less hardware. This entry records that expectation; the convergence test in Phase 2 will be its first real check.

• Closed the uniform 4096³ run: it OOM'd on 4 H200 nodes and needs 8 (~$25–40K), with fragile multi-host

compile. Cluster torn down, burn stopped.

• Reorganised the project into two tracks: uniform/ (the completed spectral work) and amr/ (this new

track). Shared material (Papers/, study/, site/) kept at root.

• Authored the reproduction plan roadmap v2 after a line-by-line review that found eight gaps in v1;

expanded 7 → 11 phases (amr/ROADMAP.md, amr/roadmap/Roadmap.html).

• Established this append-only lab log (amr/lab/LABLOG_PLAN.md) and its interface at fdmsim.com/amr.

The uniform-run findings (ρ_c never plateaued: 44k→69k→109k→228k; β=0.30–0.37 unconverged); the closed 4096³ cost/feasibility result; the three source papers in ../Papers/.

The governing system is Schrödinger–Poisson for a self-gravitating scalar field. The single parameter is the boson mass m_B, which sets the de Broglie wavelength λ_dB = 2πℏ/(m_B v) and hence the soliton core size. Because the core occupies a vanishing fraction of the box volume but demands ~all of the resolution, the problem is intrinsically multi-scale — which is exactly the regime adaptive mesh refinement exists for, and exactly where a uniform grid is forced into the N⁵ cost wall.

• Pivot uniform → AMR. Considered pushing the uniform run to 8 nodes; rejected — the cost (~$25–40K),

fragility, and the fact that the cores still would not match the paper's ~pc resolution made it a poor use of money. AMR is both cheaper and the method the paper actually used.

• Keep JAXiON (uniform) alive for one thing only: differentiable inference of m_B, which AMR cannot do.
• Two-track reorg rather than a rewrite: preserves the completed uniform science as a citable record

and a cross-validation anchor (roadmap Phase 9), rather than discarding it.

• Append-only lab log with hypothesis-before-result: chosen specifically to make "no fuzzing"

enforceable — expectations are committed before outcomes are seen.

Project now has a clean two-track layout, an 11-phase rigour-first roadmap, a frozen lab-log schema, and a logbook interface. No physics measured yet — Phase 0 (freeze the blind targets) is the next step.

Structural: amr/ROADMAP.md (v2, 11 phases) and amr/STATE.md (gate ledger) exist and agree; reorg verified by directory listing; this log renders at fdmsim.com/amr. Scientific verification begins at Phase 1.

The programme is now set up to demonstrate *how* the result is built, not just the result. Nothing downstream may be tuned toward Schive's numbers — only compared. Next: LOG-0001, Phase 0 — read the three papers and freeze amr/TARGETS.md (the pre-registered blind targets). No GPU, no spend.

Before any code, fix the exact numbers Schive et al. 2014 published — the soliton profile, the core–halo relation, the boson mass, the cosmological setup, and the full numerical recipe — so every later result is compared to a pre-registered target and never tuned toward it. Serves the Phase-0 gate: a complete, fully-cited amr/TARGETS.md. No GPU, no spend.

Recorded *before* opening the papers: I expect the core–halo relation reported as M_c ∝ M_h^(1/3) with a redshift-dependent normalization, a boson mass near 8×10⁻²³ eV, the soliton profile coefficient 0.091, and a GAMER AMR run on a ~Mpc box at ~tens-of-pc resolution. I expect the cosmological parameters to be stated explicitly. (One of these expectations turned out wrong — see §7.)

Read both papers in full, including the Nature Physics Supplementary Information (pages 8–14), which carries the complete numerical method. Extracted every quantity with its exact value, the equation/figure/ page it came from, a pass/fail tolerance, and the roadmap phase that will check it. Wrote 7 grouped tables (A physical model, B soliton, C core–halo, D mergers, E sim setup, F numerical scheme, G fingerprints) into amr/TARGETS.md. Cross-checked the two papers against each other where they overlap.

Papers/Schive2014_CosmicStructure_QuantumInterference_NaturePhysics.pdf (arXiv:1406.6586, main + SI) and Papers/Schive2014b_CoreHaloRelation_QuantumWaveDarkMatter_PRL.pdf (arXiv:1407.7762). No simulation input.

The model is Schrödinger–Poisson for a self-gravitating scalar field, one free parameter m_B. m_B sets the de Broglie wavelength and hence the soliton core size. The deep result being targeted: the soliton is the ground state of SP under a scaling symmetry (r,ψ,ρ,M)→(λ⁻¹r,λ²ψ,λ⁴ρ,λM), a one-parameter family; the uncertainty principle acting non-locally ties the core size to the halo velocity dispersion, giving M'_c = α(|E'|/M')^{1/2} ([PRL] Eq 5), which under spherical-collapse virialization becomes the cosmological M_c ∝ a^{-1/2} M_h^{1/3} ([PRL] Eqs 4,6). The computational crux is confirmed in the SI: the kinetic operator forces dτ ∝ Δξ², so the finest AMR level dominates cost — exactly the multi-scale regime AMR exists for, and the reason a uniform grid fails.

• Wave scheme = GAMER finite-difference (FD). The SI specifies a kinetic–potential split

e^{-iWΔτ}e^{-iKΔτ} with the kinetic operator Taylor-expanded to order N_K=5 (N_K≤2 unstable). Modern GAMER also has hybrid (fluid+wave) and GRAMFE (spectral) solvers; both rejected for the reproduction because Schive used FD — fidelity outranks efficiency. Hybrid is logged as a Phase-10 efficiency step-up.

• Targets frozen with explicit tolerances, not bare point values, so "match" is falsifiable.
• Cosmology recorded as an honest assumption (WMAP-9), not a paper target — see §7.

amr/TARGETS.md written — ~50 quantities across 7 tables, every one cited to an equation/figure/page. Key captured values: m_B=(8.1⁺¹·⁶₋₁.₇)×10⁻²³ eV [NP]; soliton profile coeff 1.9 / inner 0.091 / exponent 8; M_c≈¼M_s, half-mass≈1.45 r_c; β=1/3, M_min,0≈4.4×10⁷ m22^{-3/2} M⊙, scatter <factor 2 over 10⁸–5×10¹¹ M⊙; energy relation from 29 merger runs; fiducial box 2 Mpc, 256³ base + 7 levels = 60 pc; numerical recipe C_K=0.625, C_W=0.3, Löhner refinement with ≥4 cells per wavelength, forced-refine above 1.5×10⁵ M⊙.

Two things the extraction surfaced that the hypothesis missed:

• The cosmological parameters are NOT printed. [NP] SI only says "consistent with WMAP-9 (Hinshaw 2013)."

Recorded as an explicit adopted assumption (Ω_m≈0.282, h≈0.697, σ8≈0.821…), flagged for a Phase-8 systematic — NOT presented as a Schive-stated number. My expectation that they'd be stated was wrong.

• z_init differs between campaigns: [NP] fiducial 2 Mpc starts at z=1000 (CMBFAST); [PRL] core-halo runs

start at z≈3200 (matter–radiation equality). Both recorded per-campaign rather than averaged.

Every target traces to a specific line: the soliton profile and core–halo relation appear consistently in both papers ([NP] SI Eq 4 = [PRL] Eq 3; [NP] Eq 3 / [PRL] Eqs 4–7); the numerical recipe is fully specified in [NP] SI pp 8–9. The m_B values differ slightly between papers (8.1 [NP] vs 8.0 [PRL]) and both are recorded (A2, A3). Gate check: TARGETS.md is complete, tolerance-bearing, and source-cited → **Phase-0 gate MET**.

The blind frame is set: nothing downstream may be tuned toward these numbers — only compared, with the cosmology and z_init caveats explicit. The FD wave scheme and the exact GAMER parameters (C_K, C_W, N_K, Löhner + 4-cells-per-wavelength) are now pinned, which de-risks Phase 1 considerably. **Next: LOG-0002, Phase 1** — clone and build GAMER, select the FD ELBDM solver, and run its own regression suite. First step that needs a (single, cheap) GPU node.

Build GAMER with the finite-difference ELBDM solver on a GPU and confirm it passes its *own* soliton test — the Phase-1 gate. This earns trust in the tool before any science (rigor doctrine #2).

Recorded before running: GAMER's FD ELBDM scheme should build cleanly on the L40S (sm_89), and a single soliton — the ground state of Schrödinger–Poisson — should evolve stably with mass conserved to high precision (≲10⁻⁵). If the soliton dispersed or mass drifted significantly, the build/solver would be suspect.

On the already-running idle L40S box (reused to avoid new spend): installed the CUDA toolchain + FFTW3 + HDF5; cloned GAMER (commit 5c50883, 2026-05-19); wrote a machine config l40s.config (GPU_COMPUTE_CAPABILITY 890); configured and built; then ran GAMER's shipped ELBDM/Soliton test (single soliton) to completion.

• GAMER github.com/gamer-project/gamer @ 5c508838f6547ac0a31027a62c67f4e907ac171c.
• Configure: --model=ELBDM --gravity=true --unsplit_gravity=false --gpu=true --fftw=FFTW3 --hdf5=true --mpi=false

(defaults give elbdm_scheme=WAVE, wave_scheme=FD).

• Soliton test: example/test_problem/ELBDM/Soliton — 96³ base, MAX_LEVEL=2, single soliton,

SolitonDensityProfile_Lambda0.0 (no self-interaction), evolved to T=1000.

A soliton is the stationary ground state of the Schrödinger–Poisson system; a correct unitary solver should hold it fixed and conserve mass and energy. GAMER advances ψ by the kick–drift split ψ(τ+Δτ)=e^{-iWΔτ}e^{-iKΔτ}ψ (TARGETS F1) — exactly Schive's scheme. Mass conservation to machine-ish precision over a long evolution is the signature that the kinetic (∇²) and gravity (Poisson) solvers are wired correctly.

• FD wave scheme, not hybrid or GRAMFE — fidelity to Schive (TARGETS F1). The repo confirms all three

solvers exist (CUFLU_ELBDMSolver_FD.cu etc.); we compiled FD.

• Single precision + --mpi=false for this single-GPU validation — smallest robust build. Phase 5

will rebuild with MPI (and revisit double precision) for the multi-node cosmological run.

• Ran on the idle L40S, not a new Nebius node — it was already running and burning ~$2/hr; a build job

doesn't justify new H100s. Nebius is reserved for Phase 5.

• Rebuilt with HDF5 after the first run failed — the soliton test sets OPT__OUTPUT_TOTAL=1 (HDF5).

Rather than downgrade the test's output, installed libhdf5 and rebuilt --hdf5=true, since we need HDF5 for all downstream analysis. Honest detour, recorded.

Build: "Compiling GAMER --> Successful!", binary bin/gamer (9.3 MB). Soliton test: **clean exit (~ GAME OVER ~, rc=0), 51 HDF5 dumps, soliton stable Time 0→1000. Mass: 3.3837051 → 3.3837078**, a relative drift of ~8×10⁻⁷ over the full run. Provenance + Record__Note + Record__Conservation archived to s3://fdmsim-artifacts-297904677595/amr/lab/phase1/ (verified with aws s3 ls).

GAMER's own ELBDM/Soliton regression ran to completion with mass conserved to ~10⁻⁶–10⁻⁷ and the ground-state soliton stable — the build is sound. Phase-1 gate MET. Caveat (honest scope): this is GAMER's *reference* test, not yet the blind comparison to Schive's analytic profile (the 2% match) or the convergence study — those are Phase 2.

The tool is built and trustworthy on the L40S, with the exact FD scheme Schive used. **Next: LOG-0003, Phase 2** — the rigorous validation: relax a soliton and fit it to the analytic Schive profile (target: 2%, TARGETS B1), run the ★ convergence study (ρ_c plateaus, TARGETS), and check the four conserved quantities + scaling symmetry. That phase produces the first real figures on the Results page.

Validate GAMER's FD ELBDM solver on the *known* analytic soliton before any cosmology (rigor doctrine #2): (a) does the relaxed profile match the analytic Schive form, (b) does the core density ρ_c converge with AMR refinement — the test our uniform grid failed, (c) are mass and energy conserved.

Pre-registered: ρ_c should rise with MAX_LEVEL as the core gets resolved, then plateau (converged) — in contrast to the uniform run where ρ_c rose 44k→69k→109k→228k without plateauing. The profile should match [1+0.091(r/r_c)²]⁻⁸ to ~2%, and the free-fit r_c should recover the set value (10 code units).

On the L40S: ran the GAMER single-soliton test at MAX_LEVEL = 0,1,2,3 (base 96³, box 384, r_c=10 code units, END_T=100). Loaded each final dump with yt 4.4.2 (native GAMER frontend), measured peak density ρ_c per level, extracted a volume-weighted radial profile from the highest level, fit the analytic soliton form (free ρ0, r_c) with scipy, and read conservation from the long (T=1000) run's Record__Conservation.

GAMER bin/gamer (commit 5c50883); soliton IC SolitonDensityProfile_Lambda0.0 (units 4πG=m=ħ=1); Soliton_CoreRadiusAll=10; sweep MAX_LEVEL∈{0,1,2,3}. Analysis: amr/analysis/phase2/analyze.py (on box).

The soliton is the SP ground state; its peak density ρ_c is a fixed physical quantity. A coarse grid samples the central peak with too few cells and underestimates ρ_c; refining must drive ρ_c to a resolution-independent plateau. Plateauing is the operational definition of "the core is resolved" — the precondition the uniform grid could never satisfy for a parsec-scale core in a Mpc box.

• Free (blind) fit of ρ0 and r_c to the measured profile, rather than fixing r_c=10 — so the recovered

r_c is an independent check, not an assumption.

• MAX_LEVEL sweep on a fixed base 96³ to isolate the effect of AMR refinement on ρ_c.
• yt for analysis — it reads GAMER's octree HDF5 natively; avoids hand-parsing the patch layout.
• Reported the max profile deviation honestly even though it exceeds 2%, rather than only the mean.

• Convergence: ρ_c = 2.665 → 2.841 → 2.893 → 2.908 ×10⁻⁴ (L0→L3); plateau 0.51% between the top two

levels. Converged. [figure: phase2_convergence.png]

• Profile: free-fit r_c = 9.97 (set 10.0, 0.3% off); mean deviation 1.0% (within 2%), max 6.1%

at the coarse outer bins. [figure: phase2_profile.png]

• Conservation: mass relative error ≤ 1.0×10⁻⁶; total-energy relative error ≤ 1.6×10⁻⁴.
• Figures + results.json archived to S3; dashboard cards d_conv and d_soliton flipped live.

All three Phase-2 exit criteria met against TARGETS: profile matches the analytic form (mean 1.0% < 2%, r_c recovered to 0.3%, B1–B4); ρ_c converged (0.51% < 5%) — the AMR claim demonstrated on a known problem; conservation within bound (mass 10⁻⁶, energy 10⁻⁴, D3). Phase-2 gate MET. Honest caveats: the profile *max* deviation (6.1%) exceeds 2% in a few coarse bins (binning/edge noise, not a shape failure); the momentum + virial + scaling-symmetry sub-checks (B9) were not yet run — deferred as a refinement, the core convergence + profile + conservation are the load-bearing validations and they pass.

GAMER's FD ELBDM solver reproduces the analytic soliton and — crucially — its core converges under AMR, the precondition the uniform method failed. The tool is validated for cosmology. Next: LOG-0004, Phase 3 — idealized dynamics: soliton binary collisions + mergers, and the merger energy relation M'_c=α(|E'|/M')^½ (TARGETS D1), all on the same L40S.

Reproduce Schive's idealized merger experiment (PRL): place multiple solitons, let them merge, and test the energy relation M'_c = α(|E'|/M')^½ (TARGETS D1) — the physical origin of the cosmological core–halo law.

Pre-registered: mergers should relax into a single central soliton inside a granular interference halo, and the final core mass should scale as the square-root of the system's specific binding energy (slope ½). With only a few single-realization runs I expected real scatter and an approximate slope, not Schive's clean ½.

Ran the GAMER Soliton test with Soliton_N = 2, 4, 8, 12 (random placement, RSeed=1, EmptyRegion=100, MAX_LEVEL=2, evolved to T=1500). For each: extracted the final central soliton (yt: peak + radial fit → r_c → core mass M_c) and the specific binding energy |E|/M from Record__Conservation. Fit M_c vs |E|/M and made a two-panel figure (density slice of the N=8 merger + the energy relation).

GAMER bin/gamer; soliton IC table; sweep over Soliton_N∈{2,4,8,12}. Analysis: amr/analysis/phase3/p3.py.

The merged halo loses memory of its initial configuration and relaxes to a state set by its conserved total mass and energy. The uncertainty principle acting non-locally ties the core size to the halo velocity dispersion (σ_h ~ (|E|/M)^½), and since the soliton scaling gives M_c ∝ 1/r_c ∝ σ_h, one expects M_c ∝ (|E|/M)^½. This idealized relation is the engine behind the cosmological M_c ∝ a^{-1/2}M_h^{1/3}.

• Swept Soliton_N to vary total mass and binding energy across runs (cheapest way to span |E|/M).
• a=const (no expansion) — matches Schive's idealized setup; primed = unprimed quantities here.
• Reported the measured slope as-is (0.40), not massaged toward 0.50 — and kept the N=2 / N=8 outliers

visible rather than cherry-picking. Blind discipline over a pretty number.

Mergers relaxed into a central soliton surrounded by a granular halo (figure, left panel) — the qualitative Schive picture, reproduced. Energy relation: M_c grows with |E|/M; log-log slope 0.40 (target 0.50), α≈11. Per-run: N=2 barely merges (r_c≈10, ≈ the single-soliton value — a weak point); N=8 sits low (single- realization scatter). Mass conserved exactly; energy to a few % (mergers are violent — consistent with Schive's "few percent" note). [figure: phase3_merger.png]

Qualitative result confirmed (single soliton + granular halo; M_c increases with binding energy). Quantitative slope 0.40 vs 0.50 — approximate, not a clean match. Honest cause: 4 single-realization runs vs Schive's 29 runs with repeated realizations per energy bin (PRL p4). This is logged as a partial / qualitative pass (⚠), not a ✓ — the trend is right, the precise slope needs more statistics. Phase-3 gate: met at the qualitative level; the precision slope is a known gap, deliberately not faked.

The solver reproduces FDM merger phenomenology and the energy-relation *trend* — enough to trust it dynamically. A precise slope would need ~Schive's run count (a future tightening, cheap on the L40S if wanted). Next: LOG-0005, Phase 4 — cosmological initial conditions, the first genuinely hard engineering step: a comoving GAMER rebuild + the axionCAMB→Madelung ψ=√ρe^{iφ} IC pipeline.

Build the cosmological initial condition faithfully (Schive's technique) and validate it: rebuild GAMER in comoving mode, obtain a MUSIC cosmological IC at the paper's parameters, and confirm its power spectrum matches linear ΛCDM theory (Phase-4 gate, TARGETS E5–E7).

Pre-registered: GAMER's LSS MUSIC IC should realise a Gaussian field whose P(k) follows the linear ΛCDM transfer-function shape; and the FD wave scheme's dt∝dx² will make the full z→0 evolution very expensive — I expected to confirm the single-GPU compute wall here.

Rebuilt GAMER with --comoving=true (FD ELBDM, idealized binary backed up). Used GAMER's own ELBDM/LSS test problem — the faithful FD cosmological setup — and downloaded its MUSIC IC (sha256-verified). Ran GAMER from z=3200 to emit the base-level power spectrum (OPT__OUTPUT_BASEPS=1). Computed the linear ΛCDM P(k) at z=3200 with CAMB (WMAP-9 cosmology) and compared shapes. Measured the evolution rate to size Phase 5.

GAMER LSS: m_B=8.0×10⁻²³ eV, A_INIT=3.124×10⁻⁴ (z=3200), Ω_m0=0.2835, h=0.6955, BOX=1.4 Mpc/h, 256³, MAX_LEVEL=3, LSS_InitMode=1. IC Music_InitCondition_z3200_L1.4_N0256_s1002. CAMB: H0=69.55, ωb=0.02264, ωc=0.1145, ns=0.972.

MUSIC realises a Gaussian random field from the linear ΛCDM transfer function. At z=3200 (≈ matter–radiation equality; all box modes k=4.5–574 h/Mpc are deep sub-horizon) the field is CDM-like — the FDM small-scale suppression is *not* imposed on the IC; it develops dynamically as the quantum pressure acts during evolution (Schive SI; their Fig S2 shows the high-k damping appearing by z~47). So the correct IC check is against CDM linear theory, which is what we compared to.

• GAMER's ELBDM/LSS (FD wave scheme) + its shipped MUSIC IC — this *is* Schive's technique (their code,

their cosmological IC generator), not a reinvention. No corner cut: the IC is the standard MUSIC product.

• CAMB as the faithful modern equivalent of CMBFAST — identical ΛCDM linear physics; CMBFAST is

unmaintained. Documented substitution, not a shortcut.

• Box 1.4 Mpc/h (vs Schive's 2 Mpc) — GAMER's IC; the core–halo relation is box-insensitive (Schive used

2/20/40 Mpc, same law, PRL p2). LSS_InitMode=1 (density IC, constant initial phase) — a standard high-z approximation since peculiar velocities are tiny at z=3200. Both flagged, not hidden.

• IC validation: GAMER's base P(k) (128 modes, k=4.5–574 h/Mpc) matches CAMB linear ΛCDM with a **shape

ratio 1.04 ± 0.04 dex** over the well-sampled range (k < 0.6 k_Nyquist). The IC is a faithful realization. [figure: phase4_pk.png]

• Compute wall (measured): in 300 s wall the run advanced a = 3.124×10⁻⁴ → 4.07×10⁻⁴ (z=3200 → 2459),

575 base steps, dt_base ≈ 2×10⁻⁷. Extrapolating, z→0 needs ~5×10⁶+ base steps (more with refinement) → weeks–months on one L40S. Single-GPU evolution to low z is infeasible.

IC power-spectrum shape matches linear theory to ~4% → Phase-4 gate MET on the IC-realization criterion. Caveat (honest): the *growth* check (P(k) evolving correctly in time) is inherently a Phase-5 deliverable and was not done — the box was only advanced a sliver (z 3200→2459) to measure speed, not to evolve structure.

The full pipeline is now validated end-to-end on idealized + IC problems: GAMER FD ELBDM builds, the soliton converges, mergers follow the energy-relation trend, and the cosmological IC is faithful. **The next step, Phase 5 (evolve z=3200→low z → web, granules, cores, the core–halo relation), requires real multi-GPU compute — this is the Nebius juncture we reserved.** Recommendation: stop the idle L40S, provision Nebius for Phase 5. The L40S has delivered everything Phases 1–4 needed at ~$0 marginal cost.