31-EFT.WP.BH.TensionWall v1.0 | Chapter 2 — Axioms & Applicability

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Chapter 2 — Axioms & Applicability

• I. One-Sentence Aim
  State the minimal axioms P40-* and applicability boundaries for the Tension Wall Sigma_TW within the coherence window. Make explicit the hard constraints—energy consistency, arrival-time lower bounds, and convergence—so that wall profiling, arrival-time computation, and interface implementation rest on a unified, falsifiable foundation.
• II. Scope & Non-Goals
• Covered: wall existence and scale separation; axiom numbering and constraints; default boundaries and energy consistency; applicable scales and neglected terms; call anchors for the two arrival-time gauges; minimal flows for metrology and implementation.
• Not covered: concrete construction and proofs of the wall profile W(r) (see Chapter 3); device-level hardware details; astrophysical observation surveys.
• III. Minimal Terms & Symbols
• Geometry & interface: Sigma_TW (wall interface set), r_H (characteristic radius), Delta_w (wall thickness), outward normal n_vec, tangent t_hat.
• Fields & potentials: T_fil(x,t), Phi_T(x,t), grad_Phi_T(x,t).
• Propagation quantities: n_eff(x,t,f), c_ref, T_arr, path gamma(ell) and line element d ell.
• Interface energy triplet: R_TW, T_trans, A_sigma, satisfying R_TW + T_trans + A_sigma = 1.
• Naming isolation: strictly distinguish T_fil from T_trans; strictly distinguish n (number density) from n_eff (effective index).
• IV. Axioms & Constraints P40-*
• P40-1 Wall existence & measurability
  In the strong-gradient region near r_H, there exists a measurable wall layer Sigma_TW whose propagation impact can be modeled as an interface response confined within finite thickness Delta_w.
• P40-2 Scale separation
  Within the coherence window, Delta_w << r_H. When this condition fails, switch to a non-thin-wall model and state it explicitly in the report.
• P40-3 Propagation lower bounds & feasible domain
  The effective index satisfies n_eff(x,t,f) ≥ 1. Arrival-time lower bounds follow; any construction yielding n_eff < 1 is infeasible.
• P40-4 Energy consistency
  On the wall, reflection, transmission, and loss obey R_TW + T_trans + A_sigma = 1; all three are dimensionless and measurable within the band.
• P40-5 Potential & gauge
  On a simply connected domain, Phi_T = G(T_fil) exists, with gauge fixed by Phi_T(x_ref,t_ref) = 0. If observables depend only on grad_Phi_T, they are invariant under Phi_T → Phi_T + const.
• P40-6 Path measurability & convergence
  The path gamma(ell) is piecewise differentiable and n_eff( gamma(ell), t, f ) is piecewise continuous, ensuring the convergence of ∫ n_eff d ell or ∫ (n_eff/c_ref) d ell and enabling implementable discretizations.
• P40-7 Bandwise separability
  Within the target bandwidth, the decomposition n_eff = n_common(x,t) + n_path(x,t,f) has residuals below a threshold that is included in the uncertainty budget.
• P40-8 Interface matching types
  The wall admits three matching types: continuous; potential jump with C_sigma = Phi_T^+ − Phi_T^-; and flux jump with J_sigma = dot( grad_Phi_T^+ − grad_Phi_T^- , n_vec ). Types must be measurable and archived.
• P40-9 Composability of multi-path
  Reflections may induce multiple back-and-forth paths. Arrival time is the weighted sum across segments/events, with weights determined by R_TW, T_trans, A_sigma and geometry.
• V. Default Boundary Conditions & Consistency Constraints
• Outer boundary (choose one and record):
  Dirichlet: Phi_T → 0 at far field;
  Neumann: dot( grad_Phi_T , n_vec ) = 0;
  or Robin: alpha · Phi_T + beta · dot( grad_Phi_T , n_vec ) = g(x,t).
• Call anchors for the two gauges
• Constant factored out: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
• General gauge: T_arr = ( ∫ ( n_eff / c_ref ) d ell )
• Lower-bound consistency: from n_eff ≥ 1, obtain T_arr ≥ L_path / c_ref (equivalently embedded in the integrand for the general gauge).
• Energy consistency: every segmentation and correction must satisfy R_TW + T_trans + A_sigma = 1.
• VI. Applicable Scales & Neglected Terms
• Coherence window: spatial ell_coh and temporal tau_coh. Within the window, treat dynamics as quasi-steady; faster-than-tau_coh bursts are recorded as pulse-correction terms.
• Thin-wall approximation: enable zero-thickness corrections when Delta_w / r_H ≤ eta_w (threshold specified in the metrology contract); otherwise perform explicit thick-wall integration.
• Out-of-band high frequencies: treat out-of-band leakage as a systematic term in u_sys and list it explicitly in the report.
• Anisotropy: if dot( grad_Phi_T , t_hat ) is significant, enable the directional extension and provide parameterization in Chapter 5.
• VII. Minimal Equations & Calls S40-*
• S40-1 Wall-term coupling (schematic):
  n_eff = F( Phi_T, grad_Phi_T, rho, f ) + H_TW( W(r), … ), where W(r) is the wall profile (see Chapter 3).
• S40-2 Segmented arrival time (segmentation at the wall)
• Constant factored out: T_arr = ( 1 / c_ref ) * ∑i ∫{gamma_i} n_eff d ell
• General gauge: T_arr = ∑i ∫{gamma_i} ( n_eff / c_ref ) d ell
• S40-3 Zero-thickness correction (optional):
  Delta_T_sigma ≈ k_sigma · H( crossing ), where k_sigma is calibrated and H is a crossing indicator.
• S40-4 Differential isolation of the path term:
  Delta_T_arr(f1,f2) = ( 1 / c_ref ) * ∫ [ n_path(f1) − n_path(f2) ] d ell (or the corresponding general-gauge form).
• VIII. Metrology & Calibration Flows M40-*
• M40-1 c_ref calibration: use a benchmark path gamma_ref with T_arr_ref to calibrate c_ref; report u_stat(c_ref) and u_sys(c_ref).
• M40-2 Wall identification & classification: detect Sigma_TW; classify as continuous / potential-jump / flux-jump and record uncertainties.
• M40-3 Jump-parameter calibration: estimate C_sigma, J_sigma, and their stability and environmental dependence.
• M40-4 Reflection/transmission estimation: with multi-path and multi-band measurements, produce in-band curves and clamping intervals for R_TW, T_trans, A_sigma.
• M40-5 Two-gauge consistency: compute eta_T = | T_arr^{const} − T_arr^{gen} |; if out of spec, revisit c_ref and the n_eff decomposition.
• M40-6 Archival & audit: seal contracts, logs, hashes, and entrances for falsification samples.
• IX. Implementation Bindings & Interfaces I40-*
• declare_tw_contract( coords_spec, units_spec, gauge, boundary_config ) -> Contract
• build_tension_wall_profile( M_bh, a_bh, params ) -> TWProfile
• apply_TW_matching( Phi_T, TWProfile ) -> Phi_T_matched
• arrival_time_with_TW( n_eff, gamma, Sigma_TW, mode, c_ref ) -> T_arr
• estimate_RT_TW( data, TWProfile ) -> R_TW, T_trans, A_sigma
• check_dimension( expr ) -> DimReport ensuring dim(T_arr) = [T], dim(n_eff) = 1
• X. Cross-References
• 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 Chapters 4–8, S20-, M20-, I10-*
• XI. Verification & Falsification Lines
• Any stable path/band combination yielding n_eff < 1, or violating R_TW + T_trans + A_sigma = 1, constitutes a falsification sample.
• Persistent failure of two-gauge consistency (eta_T over threshold and irreparable after back-checks) is a falsification sample.
• When, under refinement to the asymptotic limit, zero-thickness corrections diverge from explicit thick-wall integrations beyond threshold, the thin-wall approximation is falsified.
• XII. Systematic-Error Safeguards
• Path & measure: reuse the same { gamma[k], Delta_ell[k] } for band differencing to avoid numerical contamination.
• Interface handling: integrate by segments with explicit endpoints; forbid cross-interface interpolation.
• Out-of-band leakage: treat out-of-band energy as part of u_sys; log leakage ratios.
• Clamping & saturation: enforce n_eff ∈ [1, n_max] and record trigger rates.
• Gauge & boundary: fix gauge and boundary_config consistently across experiments.
• XIII. Deliverables
• Axiom cards P40-1 … P40-9 with usage guidance.
• Dimension-check checklist for the two arrival-time gauges and segmented integration.
• Templates for wall identification & classification, reflection/transmission estimation records, and two-gauge consistency audits.