11-EFT.WP.Core.DrawingKinetics v1.0 | Appendix A Terminology and Unified Symbols

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Appendix A Terminology and Unified Symbols

I. Scope and Use

• Summarize the terms, symbols, and formulas used across this volume, unifying units and gauges so cross-references among chapters and calls from implementation bindings (I10-*) remain consistent.
• The default unit system is SI. Unless otherwise noted, x is the axial coordinate, t is time, and any path integral is parameterized as gamma(ell) with the explicit measure d ell.

II. Domains, Operators, and Paths

1. Variables and domains
   • x ∈ R: axial position (m); t ∈ R_+: time (s).
   • ell: path parameter along the streamline gamma(ell) (m); the measure is written as d ell.
   • Omega_x, Omega_t: spatial and temporal observation domains; a window.id is uniquely defined by its bounds t0, t1.
2. Common operators
   • Time derivative ( d/dt ) ( • ); path derivative ( d/dell ) ( • ).
   • Path integral ( ∫_{gamma(ell)} ( • ) d ell ); spectral integral ( ∫_{f_low}^{f_high} ( • ) d f ).
   • Dimensional audit check_dim( expr ); normalization error eps_norm; mass-conservation residual eps_mass.

III. Geometry and Kinematics

1. Stretch ratio and strain rate
   • lambda(x,t): stretch ratio, current length over reference length (dimensionless, lambda > 0).
   • s(x,t) = ( d/dt ) ( ln( lambda(x,t) ) ): Hencky strain rate (1/s).
2. Velocity and cross-section
   • v(x,t): drawing speed (m/s).
   • A(x,t): cross-sectional area (m^2); characteristic diameter d_char(x,t) = 2 * sqrt( A(x,t) / pi ) (m).
3. Path and measure
   gamma(ell): parameterization of the center streamline. Any integral is written as ( ∫_{gamma(ell)} ( • ) d ell ).

IV. Conservation and Flux

1. Material properties and line-form quantities
   • rho(x,t): volumetric density (kg/m^3).
   • rho_L(x,t) = rho(x,t) * A(x,t): line density (kg/m).
   • J(x,t) = rho_L(x,t) * v(x,t): mass flux (kg/s).
   • Q(x,t) = A(x,t) * v(x,t): volumetric flow rate (m^3/s).
2. Minimal conservation equation (S12-1)
   ( d/dt ) rho_L(x,t) + ( d/dell ) J(x,t) = 0.
3. Momentum and power (summary)
   Axial power density P_ax(x,t) = T_fil(x,t) * v(x,t) (W).

V. Constitutive Relations and Tension

1. Tension and parameters
   • T_fil(x,t): 1D axial tension (N).
   • K_el: elastic coefficient (N).
   • K_vis: viscous coefficient (N*s).
   • theta: optional dissipative or coupling weights (dimensionless).
2. Basic constitutive family (S12-3, exemplar)
   T_fil = K_el * ( lambda - 1 ) + K_vis * s + ... (extendable with Maxwell/Voigt/GLS terms and thermal couplings).
3. Collision mandate
   T_fil denotes tension only; T_trans denotes the transmission coefficient only—never mix.

VI. Spectral Analysis and Jitter

1. Definitions and gauges
   • S_xx(f): one-sided power spectral density of signal x(t) (units unit(x)^2/Hz).
   • Discrete window energy U_w = sum_n ( w[n]^2 ) (dimensionless).
   • Equivalent noise bandwidth ENBW = Fs * ( sum_n ( w[n]^2 ) ) / ( sum_n w[n] )^2 (Hz).
   • Discrete transform gauge
     S_xx(f_k) = ( 2 / ( Fs * U_w ) ) * | FFT( w[n] * x[n] )_k |^2.
   • Time–spectrum consistency
     var( x ) = ∫_{0}^{Fs/2} S_xx(f) df.
2. Typical target signals
   x ∈ { T_fil , v , lambda , s }. Report the window and ENBW explicitly in all outputs (Chapter 7).

VII. Time Base and Time of Arrival

1. Dual-gauge time base (Chapter 8)
   Linear mapping between internal monotonic time tau_mono and external timestamps ts
   ts = alpha + beta * tau_mono, where alpha (s) and beta (dimensionless) are published as timebase.{alpha,beta}.
2. Two arrival-time conventions (cross-volume unified)
   • T_arr = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff(ell,t) d ell ).
   • T_arr = ( ∫_{gamma(ell)} ( n_eff(ell,t) / c_ref ) d ell ).
   • Report the discrepancy in parallel delta_form = T_arr^{(1)} - T_arr^{(2)}, together with the path details and reference speed c_ref (m/s).
3. Reporting rule
   When T_arr is used, you must list the parameterization of gamma(ell), the measure d ell, alpha, beta, and delta_form (Chapter 8).

VIII. Dimensionless Groups and Characteristic Scales

1. Criterion numbers
   • Re = ( rho * v * d_char ) / mu: Reynolds number, with dynamic viscosity mu (Pa*s).
   • We = tau_rel * s: Weissenberg number, with material relaxation time tau_rel (s).
   • De = tau_rel * ( v / L_char ) = tau_rel * s_char: Deborah number; L_char is the characteristic length (m), s_char = v / L_char (1/s).
2. Characteristic quantities
   d_char(x,t) = 2 * sqrt( A(x,t) / pi ) (m); L_char is specified per case or segment and must be explicit in the manifest (Chapter 9).

IX. Parallelism, Execution, and Observability (Threads Interface Highlights)

1. Execution graph and timing
   • G = ( V , E ): execution graph; T_make(G): build/scheduling cost (s).
   • TS.*: observability and runtime metric family (aligned with Core.Threads), including TS.hb (causality heartbeat), TS.latency, TS.bp (back-pressure), TS.slo (SLO attainment).
2. Concurrency semantics
   Cross-thread hb causality must be preserved. If approximations or compensations are used, declare them in contracts and test matrices (Chapter 10).

X. Quality Gates, Errors, and Compliance Modules

1. Gates
   • gate.mass, gate.norm: thresholds on eps_mass and eps_norm, respectively.
   • gate.reprod.*: reproducibility thresholds—distance dist, spectral delta spec, and score delta score (Chapter 13).
2. Error quantities
   • eps_norm: normalization error (dimensionless).
   • eps_mass: mass-conservation residual (kg/s or dimensionless after normalization).
   • Spectral distance
     d_spec = ( ∫ | log( S_A(f) ) - log( S_B(f) ) | df ) / ( ∫ df ).
   • Reproducibility distance
     D_rep = max( D_T , D_v , D_lambda , D_s , D_A ), where
     D_T = rmse( T_A , T_B ) / T_ref, rmse(x,y) = sqrt( mean_t ( x(t) - y(t) )^2 ) (Chapter 13).
3. Compliance fingerprint
   fingerprint = H( code.digest || img.digest || data.digest || param.digest || H(alpha || beta) ); sig is the signing evidence (Chapter 13).

XI. Name Collisions and Legal Combinations

1. Collision list (mandatory)
   • T_fil vs T_trans: the former is mechanical tension (N); the latter is a transmission coefficient (dimensionless).
   • n vs n_eff: the former is a number density or count-like quantity; the latter is effective refractive index. Never mix.
2. Legal combinations and identities
   • rho_L = rho * A; J = rho_L * v = rho * A * v = rho * Q.
   • s = ( d/dt ) ( ln( lambda ) ); if lambda is differentiable, then s = ( 1 / lambda ) * ( d lambda / dt ).
   • Spectrum–variance consistency: var( x ) = ∫ S_xx(f) df (when spectral windowing is consistent).
   • The two T_arr conventions must be published in parallel with delta_form and the path gauge.
   • Any expression with division, integrals, or composite operators must be parenthesized and must explicitly state gamma(ell) and d ell.

XII. Symbol Quick Reference (grouped by topic)

• Geometry & kinematics
  lambda(x,t), s(x,t), v(x,t), A(x,t), gamma(ell), d ell, d_char(x,t).
• Conservation & flux
  rho(x,t), rho_L(x,t), J(x,t), Q(x,t), P_ax(x,t).
• Constitutive & tension
  T_fil(x,t), K_el, K_vis, theta.
• Spectrum & jitter
  S_xx(f), U_w, ENBW, Fs, var( x ).
• Time & arrival time
  tau_mono, ts, alpha, beta, c_ref, n_eff(x,t), T_arr, delta_form.
• Dimensionless groups
  Re, We, De, mu, tau_rel, L_char, s_char.
• Parallelism & compliance
  G, T_make(G), TS.*, gate.mass, gate.norm, gate.reprod.*, eps_norm, eps_mass, D_rep, d_spec, fingerprint, sig.

XIII. Citation Gauges and External Constraints (Digest)

• Time bases and measures follow Core.Sea and Core.Density. Concurrency semantics and TS.* metrics follow Core.Threads. Metrology and spectral methods follow Core.Metrology.
• Cross-volume citation format is unified: e.g., “see companion white paper ‘Energy Threads’ Chapter x S/P/M/I…”. This volume reserves the numbering ranges P11-* / S12-* / Mx-1* / I10-* and declares them per chapter.