25-EFT.WP.STG.Dynamics v1.0 | Chapter 11 — Multiscale Coarsening and Renormalization (Spectrum-Preserving / RG)

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Chapter 11 — Multiscale Coarsening and Renormalization (Spectrum-Preserving / RG)

One-Sentence Goal
Establish a unified coarsening framework on G = (V, E) that preserves spectra and supports renormalization (RG), providing computable mappings from operators/states to scaled dynamics parameters, and ensuring cross-scale consistency via dual-form comparisons and contracts.

I. Scope and Objects

1. Objects
   • Graph–operator: adjacency A, Laplacian L (or normalized Laplacian).
   • State–dynamics: ( d x / d t ) = f(x, t) = − κ L x + g(x, t; θ) or discrete x_{k+1} = Φ_{Δt}(x_k).
   • Multiscale mappings: restriction R: R^{|V|} → R^{|V_c|}, prolongation P: R^{|V_c|} → R^{|V|}.
2. Inputs
   Base { G, A, L }, target size |V_c| or ratio b, preservation goals (leading λ’s, heat kernel, effective resistance), dynamical conservation constraints, RefCond, and units.
3. Outputs
   Coarse graph G_c, coarse operator L_c, up/down maps { P, R }, parameter rescaling θ', cross-scale errors and delta_form_ms, and manifest.stg.ms.
4. Constraints & boundaries
   • unit(x) and unit(t) required; if mass/energy conservation or nonnegativity is declared, coarse operators and time stepping must preserve the discrete analogs.
   • Coarsen within connected components; merge or drop isolated nodes and record annotations.

II. Terms and Variables

• Spectral objects: σ(L) = { λ_i }, eigenvectors U, spectral distance d_spec; heat kernel H(t) = exp( − t L ); effective resistance R_eff(i, j).
• Multigrid / Galerkin: R (restriction), P (prolongation), L_c = R L P (Galerkin).
• Aggregation & matching: heavy-edge matching (HEM), aggregation (Agg), Kron reduction Kron(G, S).
• RG scaling: spatial factor b > 1, time rescale t' = b^{z} t, parameter map θ' = RG_b(θ).
• Dual-form gap: fine-simulate–downsample vs coarse-simulate discrepancy delta_form_ms.

III. Axioms P711-*

• P711-1 (Parallel dual forms) — For any stimulus/window, compute
  x̂_fine↓(t) = R * Solve(L, θ) and x̂_coarse(t) = Solve(L_c, θ'), and record
  delta_form_ms = ( ∫ || x̂_fine↓(t) − x̂_coarse(t) ||_2 dt ).
• P711-2 (Explicit domains/measures) — Spectral/heat-kernel errors use ( Σ_{i∈Ω_k} • ) or ( ∫_{t∈T_ref} • dt ); dynamics rely on ( ∫_t • dt ).
• P711-3 (Units & dimensions) — check_dim( d x / d t + κ L x − g ) = "[0]"; unit(P) = unitless, unit(R) = unitless.
• P711-4 (Conservation alignment) — If 1^T L = 0 (mass conservation) then require 1^T L_c = 0; if energy semidefinite x^T L x ≥ 0, maintain x_c^T L_c x_c ≥ 0.
• P711-5 (Spectral priorities) — Prioritize low-frequency eigenpairs (λ_1…λ_k and subspace angles) and Frobenius matching of heat kernels on T_ref.
• P711-6 (Mapping stability) — Require R P ≈ I_{|V_c|} and bounded || P R ||_2; cond(L_c) must not deteriorate beyond threshold.

IV. Minimal Equations S711-*

• S711-1 (Galerkin coarsening):
  L_c = R L P; with symmetric restriction R = P^T W (weight matrix W), L_c remains SPD when L is SPD.
• S711-2 (Kron reduction):
  For kept set S and eliminated set F = V \ S,
  L_c = L_{SS} − L_{SF} L_{FF}^{−1} L_{FS} (Schur complement), yielding graph G_c.
• S711-3 (Spectral distance):
  With target indices Ω_k,
  d_spec = ( Σ_{i∈Ω_k} | λ_i(L) − λ_i(L_c) | / max( ε, | λ_i(L) | ) ) / |Ω_k|;
  subspace angle Θ = || U_k U_k^T − P U_{c,k} U_{c,k}^T R ||_2.
• S711-4 (Heat-kernel matching):
  E_H = ( ∫_{t∈T_ref} || R e^{−t L} P − e^{−t L_c} ||_F^2 dt ); use to infer θ' or tune P, R.
• S711-5 (Parameter renormalization):
  For diffusion invariance e^{− κ t L} ≈ R^{†} e^{− κ' t' L_c} P^{†}, obtain
  κ' = κ * b^{ 2 − z } (if L approximates a 2nd-order operator), t' = b^{ z } t.
  For nonlinear g, match responses:
  θ' = argmin_{θ'} ( ∫ || R Φ_{Δt}(x; θ) − Φ'_{Δt'}( R x; θ' ) ||_2 dμ(x, t) ).
• S711-6 (Energy & conservation):
  Coarse energy E_c(t) = x_c^T L_c x_c ≈ x^T L x using x ≈ P x_c.
• S711-7 (Dual-form gap):
  delta_form_ms = ( ∫_{t0}^{t1} || R x_fine(t) − x_coarse(t) ||_2 dt ); window average \bar{δ} = delta_form_ms / ( t1 − t0 ).

V. Metrology Workflow M7-11 (Ready → Model/Estimate → Validate → Persist)

1. Ready
   • Choose coarsening strategy { HEM, Agg, Kron, AMG/Galerkin, Learning-based }; set weights for spectrum/heat-kernel/resistance; specify T_ref, Ω_k.
   • Declare physical constraints (mass/energy/nonnegativity) and runtime limits (|V_c|, memory/latency).
2. Model / Estimate
   • Build aggregates and R, P (e.g., R_{ij} = 1 / | Agg(j) | to cluster centers), or learn P = argmin E_H + α d_spec.
   • Form L_c (Galerkin or Kron); consistently construct G_c (edge weights/self-loops) to preserve metrics.
   • RG parameters: initialize κ' from b, z; refine via T_ref heat/pulse response to obtain θ'.
3. Validate
   • Frequency domain: d_spec, subspace angle Θ, heat-kernel error E_H, effective-resistance error E_R.
   • Time domain: for canonical stimuli (impulse, step, white noise) compute delta_form_ms, energy-curve gaps, conservation-closure errors.
   • Numerics: cond(L_c), ρ( exp( − κ' Δt' L_c ) ), stability regions.
4. Persist / Publish
   manifest.stg.ms = { method, R_hash, P_hash, L_c_hash, G_c_meta, targets: { Ω_k, T_ref }, errors: { d_spec, Θ, E_H, E_R }, dynamics: { κ', θ', z, b }, deltas: { \bar{δ}, δ_p95 }, constraints, RefCond, units }.

VI. Contracts & Assertions C70-11xx

• C70-1101 (Spectral preservation): d_spec ≤ tol_spec and Θ ≤ tol_subspace (suggest tol_spec ≤ 0.1, tol_subspace ≤ 0.2).
• C70-1102 (Heat-kernel consistency): E_H / | T_ref | ≤ tol_heat.
• C70-1103 (Dynamics dual forms): delta_form_ms_p95 ≤ tol_ms (suggest tol_ms = 5 • ( atol + rtol • || x || )).
• C70-1104 (Conservation / nonnegativity): if declared, ensure 1^T L_c = 0 and min(x_c) ≥ − ε_neg over the full window.
• C70-1105 (Conditioning bound): cond(L_c) ≤ γ • cond(L) (suggest γ ≤ 5); otherwise do not publish.
• C70-1106 (Topology consistency): preserve the number of connected components; annotate isolated-node handling with tags.isolated.
• C70-1107 (Unit consistency): check_dim(L_c) = "[1]", check_dim(κ') = "[1/T]" prior to release.

VII. Implementation Bindings I70-11*

• I70-111 build_aggregates(G, target_size, method) -> { Agg, R, P }
• I70-112 kron_reduce(L, keep_set) -> L_c
• I70-113 galerkin_coarsen(L, R, P, mode) -> L_c (mode ∈ { std, normalized })
• I70-114 rg_rescale_params(θ, b, z, criteria) -> θ'
• I70-115 match_heat_kernel(L, L_c, R, P, T_ref) -> { κ', tune_log }
• I70-116 simulate_multiscale(fine_model, coarse_model, R, P, stimuli) -> { traj_fine↓, traj_coarse, delta_form_ms }
• I70-117 eval_spectral_metrics(L, L_c, Ω_k) -> { d_spec, Θ }
• I70-118 eval_resistance_metrics(G, G_c, pairs) -> E_R
• I70-119 emit_ms_manifest(results, policy) -> manifest.stg.ms

Invariants: R P ≈ I; L_c SPD/semidefinite consistent with the prototype; delta_form_ms ≤ tol_ms; traceable RefCond / method / hash.

VIII. Cross-References

• Graph operators & kernels: Ch. 4 (diffusion/wave/filter kernels & spectral approximations).
• Numerical integration & stability: Ch. 9 (stable stepping and events on the coarse model).
• Causality & intervention: Ch. 10 (cross-scale consistency of do-effects).
• Runtime & panels: Ch. 14 (multi-resolution caching and adaptive switching).
• Manifests & contracts: Appendices C and B.

IX. Quality & Risk Control

1. SLIs/SLOs: d_spec, Θ, E_H / | T_ref |, E_R, delta_form_ms_p95, cond(L_c), runtime_speedup.
2. Fallbacks
   • Poor spectral/heat-kernel match → enlarge aggregates, switch Kron → Galerkin, or learn P, R.
   • Large dynamics deviation → tune z / κ', expand T_ref, add nonlinear closure (g_c = R g(P•) regression).
   • Conditioning worsens → spectral correction L_c ← L_c + ε I or rebuild aggregation.
   • Conservation broken → adjust diagonal to enforce row-sum zero or project onto the conservation subspace.
3. Audit: persist Agg / R / P, L_c / G_c hashes, spectral & heat-kernel curves, dual-form residual time series, and dashboard snapshots.

Summary
This chapter provides Galerkin / Kron / learning-based coarsening routes, spectral–heat-kernel–resistance consistency metrics, and RG parameter scaling. With C70-11xx contracts and manifest.stg.ms.*, multiscale models become simultaneously deployable, auditable, and performant across frequency and time domains.