04-EFT.WP.Core.Metrology v1.0 | Chapter 3 — Reference Conditions and Nondimensionalization

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Chapter 3 — Reference Conditions and Nondimensionalization

I. Objectives and Scope

• Establish mandatory conventions and recording rules for reference conditions RefCond, and define the model interface and verifiable properties of environmental corrections corr_env(•; RefCond).
• Fix the nondimensionalization convention t0 def= L0 / c_ref, and provide unified nondim / re_dim rules for fields, derivatives, integrals, and path quantities, kept consistent with the two arrival-time conventions.
• Outputs: extended postulates P90-11…P90-18; metrology workflow Mx-2 (reference conditions and nondimensionalization); implementation use cases bound to I40 3 and I40 7.

II. Reference Conditions and Recording Postulates (P90-11…P90-13)

• P90-11 Explicit-RefCond: every metrological quantity affected by environment must explicitly carry reference conditions RefCond = { p_ref, Temp_ref, humidity_ref } (or a superset). Implicit “standard state” is forbidden.
• P90-12 Traceable-Metadata: each RefCond component must include units and uncertainty: p_ref (unit="[Pa]", u), Temp_ref (unit="[K]", u), humidity_ref (unit="[1]", u).
• P90-13 Affine-Unit-Guard: any unit with a zero-point offset (e.g., degC) must be converted to its SI affine form before registry: Temp[K] = Temp[degC] + 273.15.

III. Environmental Correction Models and Properties (P90-14…P90-15)

1. P90-14 corr_env-Def: define environmental correction as a function family
   x_corr = corr_env(x_raw; RefCond, model, args), with model ∈ { linear, affine, Arrhenius, polynomial, piecewise }.
2. P90-15 Monotonicity-and-Dim: corr_env must preserve dimensions and be identity at the reference point: corr_env(x; RefCond_ref) = x. If monotonicity is declared, the sign of ∂ x_corr / ∂ Temp_ref or ∂ x_corr / ∂ p_ref must satisfy the model’s constraints.
3. Typical instances
   • Temperature drift (linearized): x_corr = x_raw * ( 1 + alpha_T * ( Temp_ref - Temp0 ) ), with dim(alpha_T) = "[Temp]^-1".
   • Pressure correction (affine): x_corr = a_p * x_raw + b_p * ( p_ref - p0 ), where dim(a_p) = "[1]", dim(b_p) = dim(x) * [Pa]^-1.
   • Arrhenius form: x_corr = x_raw * exp( -Ea / ( k_B * Temp_ref ) ), with a dimensionless argument.

IV. Canonical Scales and Convention (P90-16)

• bar_x = x / L0, bar_t = t / t0, bar_T = Temp / T0.
• Path measure: d bar_ell = d ell / L0, hence ∫_gamma ( n_eff / c_ref ) d ell = t0 * ∫_gamma ( n_eff ) d bar_ell.

V. Nondimensionalization and Re-Dimensionalization Postulates (P90-17…P90-18)

1. P90-17 Nondim-Operators: for any field f(x,t), define bar_f(x,t) def= f / f0 so that dim(bar_f) = "[1]". Derivatives and gradients obey:
   ∂ f / ∂ t = ( f0 / t0 ) * ( ∂ bar_f / ∂ bar_t ), ∂ f / ∂ x = ( f0 / L0 ) * ( ∂ bar_f / ∂ bar_x ).
2. P90-18 Integral-Measure: integral rules:
   • Time: ∫ f d t = f0 * t0 * ∫ bar_f d bar_t
   • Path: ∫_gamma f d ell = f0 * L0 * ∫_gamma bar_f d bar_ell
   • Volume: ∫ f d V = f0 * L0^3 * ∫ bar_f d bar_V
3. Consistency of the two arrival-time conventions under the canonical mapping
   • Constant-factored: T_arr = ( 1 / c_ref ) * ( ∫_gamma n_eff d ell ) = t0 * ∫_gamma n_eff d bar_ell.
   • General: T_arr = ( ∫_gamma ( n_eff / c_ref ) d ell ) = t0 * ∫_gamma n_eff d bar_ell.
   • Therefore bar_T_arr = T_arr / t0 = ∫_gamma n_eff d bar_ell.

VI. Dimensionless Groups and Indicators (Pi sets)

1. Construction steps
   • List all quantities and dimensions in the measurement equation; select baseline scales L0, t0, T0, ....
   • Replace the time scale via t0 = L0 / c_ref to form an initial set of dimensionless variables.
   • Generate a minimal basis of Pi groups and verify check_dim passes; ensure all statistical arguments are dimensionless.
2. Arrival-time example
   • Pi_1 = T_arr * c_ref / L_gamma, with L_gamma = ∫_gamma 1 d ell, reduces to Pi_1 = avg_gamma[n_eff].
   • If n_eff(Temp) exhibits temperature dependence, define Pi_2 = ( Temp_ref / T0 ) or Pi_2 = ( Temp_ref - Temp0 ) / T0 to characterize thermo-induced drift.

VII. Reference Conditions and Nondimensionalization Workflow (Mx-2)

• Fix RefCond fields and units: p_ref [Pa], Temp_ref [K], humidity_ref [1], recording uncertainties u(•).
• Choose baseline scales: specify L0 and c_ref, compute t0 = L0 / c_ref; define T0 if thermal processes are involved.
• Select and fit corr_env model: declare model and args; verify identity/monotonicity/convexity in the neighborhood of RefCond.
• Execute nondimensionalization: call nondim(•, L0, t0, T0) for inputs, parameters, and intermediates; transform derivatives/integrals per P90-17…P90-18.
• Validate: run check_dim(expr); ensure bar_• arguments enter log/exp/trig; verify statistical operators preserve dimensions.
• Re-dimensionalize and report: re_dim(bar_y, L0, t0, T0), append RefCond and the uncertainty budget, adhering to “combine first, then round.”

VIII. Implementation Binding and Examples (I40 3, I40 7)

1. Reference condition registration
   set_refcond(name="STD", p_ref=101325.0, Temp_ref=293.15, humidity_ref=0.45)。
2. Environmental correction call
   x_corr = corr_env(x_raw, ref="STD", model="linear", args={ "alpha_T": 2.0e-3, "Temp0": 293.15 })。
3. Nondimensionalization and re-dimensionalization
   • bar_t = nondim(t, L0=1.0, t0=L0/c_ref)；t = re_dim(bar_t, L0, t0)。
   • Equivalence check for line integrals: verify T_arr / t0 == integrate( n_eff, d_bar_ell )。
4. Constraints and checks
   • check_dim rejects log(Temp[K]); accepts log( Temp / T0 ).
   • convert(v, "degC", "K") performs only the affine mapping; apply corr_env or nondim thereafter.

IX. Typical Checklists (mixed pass/fail)

• bar_x = x / L0, bar_t = t / t0, d bar_ell = d ell / L0 (pass)。
• log( x ) with dim(x) != "[1]" (reject); rewrite log( x / x0 )。
• T_arr = ( ∫_gamma ( n_eff / c_ref ) d ell ) consistent with bar_T_arr = ∫_gamma n_eff d bar_ell (pass)。
• corr_env(x; RefCond) without declared model and args (reject)。
• Temp_ref stored as degC (reject); convert to K and record the affine offset。
• avg_t[f; Δt] nondimensionalized as avg_{bar_t}[bar_f; bar_Δt] (pass), where bar_Δt = Δt / t0。
• Pi_1 = T_arr * c_ref / L_gamma is dimensionless (pass)。

X. Cross-Volume Anchors and Reuse

• With Core.Parameters: in priors and likelihood construction, any argument entering log/exp/logit/softplus should be nondimensionalized via nondim; J and Fisher(theta) are recommended in the bar_theta space.
• With Core.Equations: maintain consistency of the two arrival-time conventions in bar_• form: bar_T_arr = ∫_gamma n_eff d bar_ell。
• With this volume’s Chapter 2: run Mx-1 prior to Mx-2 to ensure dimensional consistency before environmental correction and nondimensionalization.