Editing Openai/6961e230-053c-8013-9840-af60b05b3856 (section)

=== Assistant: Top. ===
Top. Hieronder krijg je precies wat reviewers (en engineers) willen zien: een parameterblok + pseudo-code integratiestap die η-coherentie, Casimir-regularisatie en AC/DC-governance expliciet afdwingt. Dit is “copy-paste klaar” voor Methods / Numerical Implementation.

==== QEF-MTOYE Stabiliteitsparameters ====

Numerical Stability & Coherence Control (η, Casimir, AC/DC)

===== Parameterblok (default/prototype waarden) =====

Deze defaults zijn conservatief (stabiliteit > agressieve resolutie) en bedoeld als startpunt; je kalibreert ze op je grid, resolutie en AGN-injectieschema.

Coherentieband
* ηmin⁡=0.980\eta_{\min} = 0.980ηmin=0.980
* ηmax⁡=0.99995\eta_{\max} = 0.99995ηmax=0.99995  (boven dit riskeer je “numerieke lock-in”)
* η0=0.990\eta_0 = 0.990η0=0.990 (referentie voor tijdstapregeling)

AC/DC-balans
* χmax⁡=0.35\chi_{\max} = 0.35χmax=0.35  (AC mag pieken, maar niet domineren)
* χmin⁡=0.05\chi_{\min} = 0.05χmin=0.05  (onder dit = over-smoothing risico)

Casimir-regularisatie
* κCas=10−3 Hhydro,core\kappa_{\mathrm{Cas}} = 10^{-3}\, \mathcal{H}_{\mathrm{hydro,core}}κCas=10−3Hhydro,core (geschaald op lokale bulk-energiedichtheid; dimensionloos in code via normalisatie)
* p=2p = 2p=2 (coherentie-gevoeligheid; 2 is “veilig”)
* ℓ=max⁡(Δx,λmfp)\ell = \max(\Delta x, \lambda_{\mathrm{mfp}})ℓ=max(Δx,λmfp) (grid-schaal of fysische mean free path proxy)
* f(ℓ)=(ℓ/ℓ0)−4f(\ell)= (\ell/\ell_0)^{-4}f(ℓ)=(ℓ/ℓ0)−4 met ℓ0=Δx\ell_0 = \Delta xℓ0=Δx (sterke demping op kleine schalen, mild op grote)

Tijdstapregeling (stiffness-aware)
* Δt0=ΔtCFL\Delta t_0 = \Delta t_{\mathrm{CFL}}Δt0=ΔtCFL
* q=1.5q = 1.5q=1.5 (η-stiffness exponent)
* r=1.0r = 1.0r=1.0 (AC/DC exponent)
* Δtmin⁡=10−4ΔtCFL\Delta t_{\min} = 10^{-4}\Delta t_{\mathrm{CFL}}Δtmin=10−4ΔtCFL
* Δtmax⁡=ΔtCFL\Delta t_{\max} = \Delta t_{\mathrm{CFL}}Δtmax=ΔtCFL

Demping / filters (alleen wanneer nodig)
* Coherence-damping gain: gη=0.15g_{\eta} = 0.15gη=0.15 (alleen activeren bij η boven η_max of runaway indicator)
* Anti-smoothing gain: gχ=0.10g_{\chi} = 0.10gχ=0.10 (als χ < χ_min, demping terugschroeven)

Runaway guards
* Espike=5σE_{\mathrm{spike}} = 5\sigmaEspike=5σ boven lokale energiedichtheid-fluctuatie (sliding window)
* Max. Casimir aandeel: ∣HCas∣≤0.05 Hhydro|\mathcal{H}_{\mathrm{Cas}}| \le 0.05\,\mathcal{H}_{\mathrm{hydro}}∣HCas∣≤0.05Hhydro

==== Pseudo-code: QEF-stabiele integratiestap (1 tijdstap) ====

<syntaxhighlight lang="text">Given state at time n:
  Hydro fields: ρ_g, v, P, T, ...
  Coherence state: Ψ
  AGN sources: {AGN_i} with injection schedules
  Grid scales: Δx, Δt_CFL
  Parameters: η_min, η_max, χ_min, χ_max, κ_Cas, p, q, r, ...

STEP 0 — Precompute baseline timestep
  dt0 = dt_CFL

STEP 1 — Update / predict coherence state Ψ (predictor)
  Ψ* = CoherencePredictor(Ψ, hydro_fields, AGN_sources)

STEP 2 — Compute coherence invariant η
  num  = |∫_V Ψ*(x) dV|^2
  denom= ∫_V |Ψ*(x)|^2 dV
  η    = clamp(num / denom, 0, 1)

STEP 3 — Compute AC/DC energy partition and ratio χ
  E_DC = EnergySlow(hydro_fields, gravity_potential, smoothed_terms)
  E_AC = EnergyFast(shocks, AGN_injection_rate, wave_terms, residuals)
  χ    = ||E_AC|| / max(||E_DC||, ε)

STEP 4 — Stiffness-aware timestep control (dt_η and dt_χ)
  dt_η = dt0 * ((1 - η) / (1 - η0))^q          // smaller dt when η → 1
  dt_χ = dt0 * (χ_max / max(χ, ε))^r           // smaller dt when χ is large
  dt   = clamp(min(dt0, dt_η, dt_χ), dt_min, dt_max)

STEP 5 — Compute Casimir regularization term (local, scale-aware)
  ℓ     = max(Δx, mfp_proxy(hydro_fields))
  fℓ    = (ℓ / ℓ0)^(-4)
  H_Cas = - κ_Cas '' fℓ '' η^p

  // Guard: Casimir must not dominate bulk hydro energy
  H_Cas = clamp(H_Cas, -0.05''H_hydro_local, +0.05''H_hydro_local)

STEP 6 — Apply governance loop: coherence damping / anti-smoothing
  if η > η_max or DetectRunawayEnergySpike(hydro_fields, E_spike):
       Ψ''   = Ψ'' * (1 - g_η)                   // mild coherence damping
       ApplySpectralFilter(E_AC, strength=g_η) // tame stiff oscillations
  if χ < χ_min:
       ReduceDamping(strength=g_χ)             // protect tSZ-relevant structure

STEP 7 — Main update: hydro + AGN + QEF terms (corrector)
  hydro_fields^{n+1} =
       HydroSolverStep(hydro_fields^n, dt,
                       source_terms = {gravity, AGN_injection(dt), H_Cas, QEF_coh(Ψ*)})

STEP 8 — Coherence corrector (synchronize Ψ with updated hydro fields)
  Ψ^{n+1} = CoherenceCorrector(Ψ*, hydro_fields^{n+1}, dt)

STEP 9 — Diagnostics & acceptance criteria
  Recompute η^{n+1}, χ^{n+1}
  if η^{n+1} < η_min:
       Flag: "Decoherence drop" (increase resolution or adjust AGN cadence)
  if χ^{n+1} > χ_max and persists N steps:
       Flag: "AC runaway risk" (tighten dt or κ_Cas slightly)
  Log: dt, η, χ, max|H_Cas|/H_hydro, tSZ_proxy(Compton-y)

Return state at n+1

</syntaxhighlight>

==== Mini “Reproducibility Note” (voor in je paper) ====

Je kunt er één compacte zin onder zetten zoals:

: “We enforce numerical stability using an explicit coherence invariant η, an AC/DC energy governance loop (χ), and a scale-aware Casimir regularization term that prevents coherent-regime runaway while preserving resolved tSZ-relevant thermodynamics.”

Als je dit nóg sterker wilt positioneren voor protoclusters zoals SPT2349–56, kun je één extra output-metric toevoegen die reviewers herkennen: een Compton-y proxy uit je sim (zelfs als je geen volledige radiative transfer doet). Dat maakt de brug van “QEF stabiliteit” naar “observeerbaar tSZ-signaal” keihard.