31-EFT.WP.BH.TensionWall v1.0 | Chapter 1 — Terms & Preliminaries

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Chapter 1 — Terms & Preliminaries

• I. One-Sentence Aim
  Lock down the fixed terminology, symbols, unit/dimension checks, coordinate and path representations, and the two arrival-time gauges for the Tension Wall (Sigma_TW). This establishes a unified and reproducible notation system for subsequent wall-layer modeling, metrology, and implementation.
• II. Scope & Non-Goals
• Covered: the minimal term/symbol set for Sigma_TW, r_H, Delta_w, R_TW, T_trans, A_sigma; path representation and both arrival-time gauges; unit/dimension conventions; coordinates and interface representations; ambiguity-avoidance rules.
• Not covered: derivations of wall profiles and matching (see Chapters 3 and 8); device-level electrical/mechanical parameters; macroscopic statistical interpretations beyond the coherence window.
• III. Fixed Terms & English Equivalents (Minimal Set)
• Tension: T_fil(x,t) — the “tightness” state and response of the Energy Sea.
• Tension Potential: Phi_T(x,t) — a potential mapped from T_fil.
• Tension-potential gradient: grad_Phi_T(x,t).
• Tension Wall: Sigma_TW — the strong-gradient interface set surrounding a black-hole-like condensate.
• Wall thickness: Delta_w — effective thickness parameter of Sigma_TW.
• Characteristic radius: r_H — characteristic scale used to parameterize wall-layer location.
• Effective refractive index: n_eff(x,t,f) — the core quantity for propagation gauges; dim(n_eff) = 1, and require n_eff ≥ 1.
• Reference speed: c_ref — the reference constant/field in the arrival-time gauges.
• Arrival time: T_arr — two unified exemplars:
• Constant factored out: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
• General gauge: T_arr = ( ∫ ( n_eff / c_ref ) d ell )
• Interface energy triplet: R_TW (reflection), T_trans (transmission), A_sigma (loss); must satisfy R_TW + T_trans + A_sigma = 1.
• Path & measure: gamma(ell) with line element d ell; tangent t_hat(ell); L_path = ∫ d ell.
• IV. Symbols & Unit Conventions (inline always wrapped in backticks)
• Space & time: x, t; path parameter ell.
• Coordinates: default Cartesian; in the wall neighborhood use spherical r, theta, phi; at interfaces use outward unit normal n_vec and tangent t_hat.
• Potential & gradient: Phi_T(x,t), grad_Phi_T(x,t).
• Index & speeds: n_eff(x,t,f), c_ref, c_loc(x,t,f) = c_ref / n_eff(x,t,f).
• Interface & wall thickness: Sigma_TW, Delta_w, r_H; one-sided limits Phi_T^+ / Phi_T^-, n_eff^+ / n_eff^-.
• Units & dimensions (SI): length m, time s, speed m·s^-1, frequency Hz = s^-1; e.g., dim(c_ref) = [L][T^-1], dim(T_arr) = [T].
• Naming isolation: never mix T_fil with T_trans; never mix n (number density) with n_eff (effective index).
• V. Mathematical Notation & Ambiguity-Avoidance (TW edition)
• Explicit path integrals: ∫_gamma g(x) d ell = ∫_0^L g( gamma(ell) ) d ell; reparameterization does not change the value.
• Always parenthesize division/composite operators: a / (b + c), ∫ ( n_eff / c_ref ) d ell.
• Interface expressions: use one-sided limits and jumps, e.g., C_sigma = Phi_T^+ − Phi_T^-, J_sigma = dot( grad_Phi_T^+ − grad_Phi_T^- , n_vec ).
• Lower bound & feasible domain: always enforce n_eff ≥ 1; the arrival-time lower bound is
  T_arr ≥ L_path / c_ref (constant-factored semantics; equivalently embedded in the integrand for the general gauge).
• Gauge selection: if max |δc_ref/c_ref| ≤ eta_c, use the constant-factored gauge; otherwise use the general gauge and record c_ref(x,t,f) estimates and uncertainties.
• Band differencing: for same-path multi-band ΔT_arr(f1,f2), reuse the identical { gamma[k], Δell[k] } and the same step policy to isolate the path term.
• VI. Typical Dimension-Check Examples (TW edition)
• Arrival time (constant-factored):
  T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
  Check: ∫ n_eff d ell → [L]; multiply by 1/c_ref → [T].
• Arrival time (general gauge):
  T_arr = ( ∫ ( n_eff / c_ref ) d ell )
  Check: (n_eff/c_ref) → [T][L^-1]; times d ell → [T].
• Interface energy consistency: R_TW, T_trans, A_sigma are dimensionless and satisfy R_TW + T_trans + A_sigma = 1.
• Wall thickness & characteristic radius: dim(Delta_w) = [L], dim(r_H) = [L]; any dependence on Delta_w/r_H is dimensionless.
• VII. Coordinate Systems & Path Representation (near the wall)
• Coordinates: near the wall, prefer spherical r, theta, phi to describe geometry and profile W(r); in implementation, declare coords_spec in the Contract and provide the mapping to Cartesian.
• Paths: gamma(ell) may be given in spherical or Cartesian coordinates, but the units of Δell[k] must match those of c_ref; the interface crossing sequence { ell_i } is used for segmented integration.
• Normals/tangents: at Sigma_TW, compute n_vec and t_hat for directional terms (when the model includes dot( grad_Phi_T , t_hat )).
• VIII. Invoked Axioms & Consistency Constraints (calls)
• Invoke P40-* (Chapter 2): existence of the wall; scale separation Delta_w << r_H; n_eff ≥ 1; energy consistency and segmented convergence.
• Alignment with prior volumes: symbols and both gauges must match EFT.WP.Propagation.TensionPotential v1.0; dimensions, metrology, and error semantics align with the Core series.
• IX. Cross-References (volume + version + anchor)
• EFT.WP.Core.Terms v1.0 P10-*
• EFT.WP.Core.Tension v1.0 S12-*
• EFT.WP.Core.Sea v1.0 S08-*
• EFT.WP.Core.Equations v1.1 S06-*
• EFT.WP.Core.Metrology v1.0 M05-, M10-
• EFT.WP.Propagation.TensionPotential v1.0 S20-, M20-, Chapters 6 and 8
• X. Deliverables
• Minimal glossary of terms & symbols, reusable in each chapter’s “minimal set.”
• Dimension-check templates covering both arrival-time gauges, interface energy consistency, and the dimensionless ratio Delta_w/r_H.
• Path & interface representation checklist specifying generation and unit requirements for { gamma[k], Δell[k] }, { ell_i }, and n_vec/t_hat.