25-EFT.WP.STG.Dynamics v1.0 | Chapter 6 — Transport and Conservation Laws (Network Flow / Continuity)

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Chapter 6 — Transport and Conservation Laws (Network Flow / Continuity)

One-Sentence Goal
On G = (V, E), establish a unified treatment of transport and continuity for conserved quantities (network flow / advection–diffusion / electro–hydraulic analogies), with computable and auditable continuous–discrete dual forms and enforceable contracts.

I. Scope and Objects

1. Objects
   • Conserved quantities: node density/inventory ρ(t) ∈ R^N, edge flux f(t) ∈ R^{|E|}, global amount Q(t) = ( Σ ρ ).
   • Drivers: sources–sinks s(t) ∈ R^N, potential φ(t) ∈ R^N, edge velocities v ∈ R^{|E|}, diffusion coefficients D ∈ R^{|E|}_{≥0}, capacities c ∈ R^{|E|}_{>0}.
   • Boundary conditions: Dirichlet / Neumann / Robin (node potential / flux / mixed), queueing / travel-time edges.
2. Inputs & outputs
   • Inputs: G, B, A, L, and { s, φ / v / D / c }, initial condition ρ(0), and boundary specs.
   • Outputs: { ρ(t_k), f(t_k) }, mass-balance reports, delta_form_cons, and contract pass rates.
3. Constraints
   • Explicit units: unit(ρ), unit(f) = unit(ρ)/[T], unit(s) = unit(ρ)/[T]; check_dim must pass.
   • Nonnegativity & capacity: ρ ≥ 0, | f_e | ≤ c_e if capacities are declared.

II. Terms and Variables

• Graph operators & geometry: incidence B ∈ R^{N×|E|} (column direction tail→head), graph gradient grad_G x = B^T x, divergence div_G f = B f, Laplacian L = B W B^T (W = diag(w_e)).
• Fields & parameters: ρ, f, φ, v, D, c; edge averages ρ̂ = edge_avg(ρ, method).
• Timebase & dimensions: settle on tau_mono, publish at ts; dual-form gap delta_form_cons.
• Optional: edge storage/inventory w_e(t) (pipelines / queues).

III. Axioms P706-*

• P706-1 (Continuity): for any subset S ⊆ V,
  d/dt ( Σ_{i∈S} ρ_i ) = Σ_{i∈S} s_i − Σ_{∂S} n•f; discretely dρ/dt = s − B f.
• P706-2 (Orientation & sign): column directions of B define positive flux; f_e > 0 points tail→head; persist orient.hash.
• P706-3 (Parallel dual forms): compute continuous-time integration and discrete updates in parallel and record discrepancies.
• P706-4 (Explicit measures): temporal integrals ( ∫_{t_k}^{t_{k+1}} • dτ ), spectral/edge sums ( Σ_{e∈E} • ) must be explicit.
• P706-5 (Nonnegativity & capacity): if ρ is a physical inventory, enforce ρ ≥ 0 and |f_e| ≤ c_e (or publish over-capacity handling).
• P706-6 (Units/dimensions): all fields entering equations declare unit(field) and dim(field) and pass check_dim( y − f(x) ).

IV. Minimal Equations S706-*

• S706-1 (Graph continuity):
  dρ/dt = s(t) − B f(t), with unit(dρ/dt) = unit(ρ)/[T].
• S706-2 (Constitutive law: potential-driven / Ohm–Darcy):
  f = − K ⊙ ( B^T φ ), where K = diag(k_e), k_e ≥ 0; equivalently B f = − L_K φ, L_K = B K B^T.
• S706-3 (Constitutive law: advection–diffusion):
  f = v ⊙ ρ̂ − D ⊙ ( B^T ρ ), where ρ̂ = edge_avg(ρ) (upwind/central, etc.).
• S706-4 (Capacity & projection):
  | f_e | ≤ c_e; numerically f_proj = clip(f, −c, c) or solve min_g || g − f ||_2 s.t. | g | ≤ c.
• S706-5 (Global conservation):
  with no boundary leakage and w ≡ 0, d/dt ( 1^T ρ ) = 1^T s;
  with edge storage w, d/dt ( 1^T ρ + 1^T w ) = 1^T s + boundary_flux.
• S706-6 (Discrete upwind / Godunov):
  ρ_{k+1} = ρ_k + Δt ( s_k − B f_up(ρ_k) ), CFL = max_e( | v_e | Δt / ℓ_e ) ≤ 1 (ℓ_e effective edge length; default 1).
• S706-7 (Semi-implicit diffusion):
  ρ_{k+1} = ρ_k + Δt ( s_k − B ( v ⊙ ρ̂_k ) ) + Δt ( B D B^T ) ρ_{k+1}.
• S706-8 (Dual-form gap):
  delta_form_cons = || ( ρ_{k+1} − ρ_k ) − ( ∫_{t_k}^{t_{k+1}} ( s − B f ) dτ ) ||_2.
• S706-9 (Network-flow optimization, optional):
  Static: min_f cost(f) s.t. B f + r = s, 0 ≤ f ≤ c;
  Dynamic: add time and inventories ρ, decoupling into sequential linear/convex programs.

V. Metrology Workflow M7-6 (Ready → Model/Estimate → Check → Persist)

1. Ready
   • Build B / Orient, declare units and RefCond; align time windows of { s, φ / v / D / c }.
   • Compute spectral/CFL references: λ_max(L_D), max | v_e |, ℓ_e.
2. Model / Estimate
   • Choose constitutive laws: { potential-driven, advection–diffusion, queue/travel-time }.
   • Estimate parameters { K or v / D }, edge-averaging policy and discretization { Euler, upwind, semi-implicit }; decide on capacity projection/optimization.
3. Checks
   • Mass balance: mb = ( ρ_{k+1} − ρ_k ) − Δt ( s_k − B f_k ); compute norms and percentiles.
   • Nonnegativity & capacity: min(ρ), viol_rate( |f| > c ).
   • Stability: CFL ≤ 1 or semi-implicit monotonicity evidence.
   • Dual forms: percentile of delta_form_cons.
   • Boundary: reconcile boundary flux and internal generation.
4. Persist / Publish
   manifest.stg.flow = { model, scheme, Δt, CFL, params: { K|v|D|c }, edge_avg, capacity_proj, mb_p95, neg_ρ_rate, cap_viol_rate, delta_form_p95, RefCond, method.hash }.

VI. Contracts & Assertions C70-6xx

• C70-601 (Mass balance): p95( || mb ||_1 / ( || ρ_{k+1} − ρ_k ||_1 + ε ) ) ≤ 1e−3.
• C70-602 (Nonnegativity): min(ρ) ≥ − ε_neg (suggest ε_neg = 1e−9 • scale(ρ)), with neg_ρ_rate = 0.
• C70-603 (Capacity compliance): cap_viol_rate ≤ 1e−4 or max_excess ≤ 1e−6 • median(c).
• C70-604 (Stability / CFL): explicit advection requires CFL ≤ 1; semi-implicit must show energy/residual monotonicity.
• C70-605 (Dual-form gate): delta_form_cons_p95 ≤ tol_cons (suggest tol_cons = 1e−3 • || ρ ||_1).
• C70-606 (Unit consistency): check_dim( dρ/dt − s + B f ) = "[0]" must pass.
• C70-607 (FIFO / travel-time, optional): queue edges satisfy FIFO and monotone travel time dτ/dt ≥ 0.

VII. Implementation Bindings I70-6*

• I70-61 build_incidence(G, orient) -> B
• I70-62 edge_avg(ρ, method) -> ρ̂ (method ∈ { upwind, central, arithmetic })
• I70-63 constitutive_flow(model, ρ, φ, v, D, K, params) -> f
• I70-64 advance_conservation(ρ, s, f, Δt, scheme) -> ρ_next (scheme ∈ { Euler, upwind, semi-implicit })
• I70-65 godunov_flux(ρ_left, ρ_right, v, F) -> f (F is the fundamental diagram / flux function)
• I70-66 project_capacity(f, c, mode) -> f_proj (mode ∈ { clip, qp })
• I70-67 mass_balance_report(ρ_k, ρ_k1, s_k, f_k, Δt, boundary) -> report
• I70-68 estimate_transport_params(data, G, model) -> { K|v|D|c }
• I70-69 check_flow_contracts(ds, rules) -> report

Invariants: non_decreasing(time); ρ ≥ − ε_neg; | f | ≤ c (if declared); check_dim passes; delta_form_cons ≤ tol_cons; methods/params/RefCond are traceable.

VIII. Cross-References

• Graph operators & spectra: Ch. 2; kernels & diffusion energy: Ch. 4.
• State/observation/noise & masks: Ch. 3 (for estimating s and boundaries).
• Coupling/synchronization and balancing conserved quantities: Ch. 5.
• Numerical stability & step strategies: Ch. 9; runtime panels & manifests: Ch. 14 & Appendix C.
• Physical-dimension and dual-form philosophy for arrival times: EFT.WP.Metrology.PathCorrection v1.0, Chs. 10/11.

IX. Quality & Risk Control

• SLIs/SLOs: mb_p95, delta_form_cons_p95, neg_ρ_rate, cap_viol_rate, CFL_max, runtime_per_step.
• Fallbacks: CFL > 1 → reduce Δt / switch to semi-implicit; ρ < 0 → upwind/limiters or capacity projection; frequent capacity violations → increase c or reroute/solve an optimization; mass-balance mismatch → audit boundaries/sources and B orientation.
• Audit: persist orient.hash, B.hash, parameter hashes & estimation evidence, contract pass rates, and slices of anomalous windows.

Summary
This chapter establishes a unified, computable framework for transport and conservation on networks: the continuity law dρ/dt = s − B f, constitutive laws, capacity/stability conditions, dual-form comparisons, and contracts. Key outputs: manifest.stg.flow.* (model/scheme/params/CFL/mass balance/dual-form gap/compliance modules) to support downstream identification, control, and runtime publication.