02-EFT.WP.Core.Equations v1.1 | Chapter 1 — Equation System and Numbering

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Chapter 1 — Equation System and Numbering

I. Chapter Goals and Scope

• Establish a unified numbering, naming, and citation scheme for the Sxx-? minimal equations; provide strong/weak-form templates and a precise convention for domain–boundary declarations; define implementation-binding constraints to I20-*.
• This chapter introduces no new symbols or units; all formulas, symbols, and definitions are written in English as plain text and wrapped in backticks.

II. Numbering Scheme and Naming Convention

1. Code structure: S<chapter two-digits>-<index>, e.g., S20-1, S40-3; index increases consecutively from 1, while allowing reserved gaps for future insertions.
2. Chapter mapping:
   • Chapter 2 Path & Arrival Time → S20-*
   • Chapter 3 Constitutive Laws & Mappings → S30-*
   • Chapter 4 Tension-Field Minimal Equations → S40-*
   • Chapter 5 Continuity & Transport → S50-*
   • Chapter 6 Boundary & Initial Conditions → S60-*
   • Chapter 7 Variational Forms & Weak Statements → S70-*
   • Chapter 8 Statistics & Coarse-Graining → S80-*
   • Chapter 9 Numerical Realization & Consistency → S90-*
3. Unified cross-volume citation: fixed format “see companion white paper Energy Threads, Chapter x, S/P/M/I…”.

III. Equation Categories and Metadata Fields

1. kind ∈ {algebraic, ODE, PDE, integrodifferential, variational, constraint}
2. Required metadata:
   • domain: Ω ⊂ R^d, time: (t0, t1], params: enumerate external parameters;
   • bc: ∂Ω = ∂Ω_D ∪ ∂Ω_N with associated data; ic: initial conditions (if applicable);
   • dependencies: referenced Sxx-?, Pxx-?, Ixx-?;
   • dim_check: dimensional-closure status (pass/fail);
   • weak_form: whether a weak form and trial spaces are provided;
   • notes: applicability and approximation level (approx).

Ω def= spatial domain; ∂Ω_D def= Dirichlet boundary; ∂Ω_N def= Neumann boundary; n(x) def= outward unit normal on ∂Ω.

IV. Strong and Weak Form Templates

• Strong form (conservation mother form, see S10-0):
  strong= ∂_t u(x,t) + div[ J(u, grad[u], x, t) ] = S_src(x,t) on Ω × (t0, t1]
• Boundary and initial data:
  u|_{∂Ω_D} = g_D(x,t)；n • J|_{∂Ω_N} = g_N(x,t)；u(x,t0) = u0(x)
• Weak form (explicit inner products and boundary flux):
  find u ∈ V : ∀ v ∈ V0, inner_V[ ∂_t u, v ] + inner_V[ J(u,grad[u]), grad[v] ] = inner_V[ S_src, v ] + inner_bdN[ g_N, v ]
• Trial/solution spaces:
  V def= { v | v satisfies bc on ∂Ω_D }；V0 def= { v | v = 0 on ∂Ω_D }
  inner_V[a,b] def= ∫_Ω ( a * b ) dV；inner_bdN[a,b] def= ∫_{∂Ω_N} ( a * b ) dS

V. Domain and Boundary Declaration Card

• domain: Ω, t ∈ (t0,t1], d ∈ {1,2,3}
• bc: ∂Ω_D with g_D(x,t)；∂Ω_N with g_N(x,t)
• ic: u(x,t0) = u0(x) (if applicable)
• measures: dV, dS; if a path integral is present, explicitly declare gamma(ell) and d ell

VI. Equation Card Template

• code: Sxx-?
• title: concise English title
• kind: PDE/ODE/…
• strong: one-line strong form
• weak: one-line weak form or N/A
• domain/bc/ic: as §V
• dependencies: ["S..", "P..", "I.."]
• dim_check: pass/fail
• notes: applicability, approx level

VII. Unified Authoring and Lint Rules

• Fractions, integrals, and composite operators must use parentheses, e.g., ( a / b ), ( ∫ ( n_eff / c_ref ) d ell ).
• For any path—arrival-time expression, gamma(ell) and d ell must be explicit, and the path length reported as L_gamma = ∫_gamma 1 d ell.
• Prohibited patterns: "∫ n d ell / c", omitted path/measure, bare "c", "T", "n".
• Dimensional closure: check_dim_equation(eqn) must return pass; the integrand ( n_eff / c_ref ) * d ell is dimensionless.
• Statistical windows must be declared: avg_t[f; Δt], avg_V[f; V=Ω], avg_gamma[f].

VIII. Chapter Postulates

• P11-1 (Unique Anchor): Every minimal equation must have a unique code Sxx-? and a stable title; once published, a code must not be reused.
• P11-2 (Paired Presentation): If a weak form exists, it must be presented alongside the strong form with explicit V, V0, and inner-product definitions.
• P11-3 (Explicit Domain–Boundary): A strong-form statement must declare Ω, ∂Ω_D, ∂Ω_N, and (t0,t1]; omission of any is non-compliant.
• P11-4 (Explicit Path): Any equation containing a path integral must explicitly write gamma(ell) and d ell, and state whether c_ref is constant.

IX. Example and Placeholder

• title: Conservation mother form
• kind: PDE
• strong: ∂_t u(x,t) + div[ J(u, grad[u], x, t) ] = S_src(x,t)
• weak: inner_V[ ∂_t u, v ] + inner_V[ J(u,grad[u]), grad[v] ] = inner_V[ S_src, v ] + inner_bdN[ g_N, v ]
• domain/bc/ic: Ω ⊂ R^d, u|_{∂Ω_D} = g_D, n•J|_{∂Ω_N} = g_N, u(x,t0) = u0(x)
• dependencies: []
• dim_check: pass
• notes: baseline template for the S50-* family

X. Implementation Binding: Minimal Registration and Validation Sequence

• register_equation(code="S10-0", eqn="∂_t u + div[J] = S_src", kind="PDE", anchors=["P11-1","P11-2"], depends=[]) -> IRef
• validate_equation(eqn="...", allowed={ "u","J","S_src","grad","div","inner_V","inner_bdN" }) -> bool
• check_dim_equation(eqn="...") -> "pass"
• (if path terms present) propagate_time(n_eff_path, ds, c_ref) for consistency regression
• export_equations("yaml") for archival export

XI. Quality and Release Checklist

• id: unique and consistent with the chapter map (e.g., this chapter uses S10-0 as the mother form).
• forms: strong/weak presented as a consistent pair; trial spaces and inner products fully defined.
• domain: Ω, ∂Ω_D, ∂Ω_N, (t0,t1] explicitly declared.
• paths: where path integrals occur, gamma(ell), d ell, L_gamma are explicit, and ( n_eff / c_ref ) has complete parentheses.
• dims: check_dim_equation passes; bare "c", "T", "n" are forbidden.
• stats: when using avg_*, windows or domains are declared.
• refs: dependencies and cross-volume references are complete and traceable.