31-EFT.WP.BH.TensionWall v1.0 | Chapter 5 — Wall Construction & Parameterization

Home ／ Docs-Technical WhitePaper ／ 31-EFT.WP.BH.TensionWall v1.0

Chapter 5 — Wall Construction & Parameterization

I. One-Sentence Aim
Provide an executable parameterization and construction workflow for the Tension Wall Sigma_TW, yielding a unified expression and calibration convention for the profile W(r), thickness Delta_w, center surface r_H, and strength measure Xi_TW(r). Ensure seamless coupling with Phi_T, n_eff, and both arrival-time gauges.

II. Scope & Non-Goals

• Covered: the geometric and physical parameter set TWProfile; model families for the wall profile; non-spherical extensions; smoothing and regularization; switching criteria between thin- and thick-wall regimes; calibration/inversion steps; interface and logging norms.
• Not covered: re-stating Chapter 3 minimal equations or Chapter 4 coordinate construction; implementing Chapter 8 matching or energy audits; device-level hardware solutions.

III. Minimal Terms & Symbols

• Geometry & interface: Sigma_TW, r_H, Delta_w, r, outward normal n_vec, tangent t_hat.
• Profile & strength: W(r) (normalized profile, 0 → 1), Xi_TW(r) = | dW/dr |.
• Potential & index: Phi_T(x,t), grad_Phi_T(x,t), n_eff(x,t,f).
• Gauges & paths: T_arr, gamma(ell), d ell, c_ref.
• Energy triplet: R_TW, T_trans, A_sigma, satisfying R_TW + T_trans + A_sigma = 1.

IV. Design Principles & Constraints (building on P40-*)

• Monotonicity & bounds: W(r) is monotone across the transition zone and flat on both sides; 0 ≤ W ≤ 1.
• Scale separation: default Delta_w << r_H. If violated, switch to thick-wall volume integration and record explicitly.
• Regularity & integrability: W(r) and dW/dr are piecewise continuous; combinations with Phi_T = G(T_fil) remain integrable along the path.
• Lower bounds & feasibility: n_eff ≥ 1; under both gauges, T_arr respects lower bounds.
• Auditability: all parameterizations and transforms must be written into the contract and logs, including units and coordinate mappings.

V. Parametric Model Families (S40-25 … S40-29)

1. S40-25 Normalized hyperbolic transition
   • W(r) = 0.5 * ( 1 + tanh( ( r − r_H ) / sigma_w ) )
   • sigma_w ≈ Delta_w / 2, Xi_TW(r) = | dW/dr | = 0.5 * sech^2( ( r − r_H ) / sigma_w ) / sigma_w
2. S40-26 Logistic transition
   • W(r) = 1 / ( 1 + exp( − ( r − r_H ) / sigma_w ) )
   • Equivalent smoothness to S40-25; convenient when fitting asymmetric tails in Delta_w.
3. S40-27 Piecewise monotone splines (C^1 or C^2)
   • With nodes { r_i } and values { W_i }, enforce 0 = W_0 < … < W_N = 1 and nonnegative first differences.
   • Use monotone splines or piecewise cubic Hermite to avoid ringing; Xi_TW follows from spline derivatives.
4. S40-28 Exponential tail correction
   When observations indicate a slow outer approach, augment W(r) ← W(r) + alpha * exp( − ( r − r_H ) / lambda ), then clip to [0,1].
5. S40-29 Non-spherical expansion
   • r_H(theta, phi) = r0 * [ 1 + ∑_{l=2}^{L} ∑_{m=−l}^{l} a_{lm} · Y_{lm}(theta, phi) ]
   • Apply the same approach to Delta_w(theta, phi); control |a_{lm}| to prevent multiple crossings and self-intersections.

VI. TWProfile Object & Fields

• Core fields: r_H or r_H(theta,phi), Delta_w or Delta_w(theta,phi), model ∈ {tanh, logistic, spline, exp_tail}, sigma_w, spline nodes/coefficients, alpha, lambda (if tail term used).
• Derived fields: routine handle for Xi_TW(r); grid proxy or implicit function F(x)=0 for Sigma_TW; geometric hash hash(Sigma_TW).
• Constraint fields: eta_w (thin-wall threshold), bounds (legal parameter ranges), audit_meta (generation method and seed).

VII. Smoothing & Regularization (S40-30 … S40-32)

1. S40-30 Convolutional smoothing
   • W_epsilon(r) = ∫ K_epsilon( r − s ) · W(s) ds, with ∫ K_epsilon = 1.
   • Typical kernel: Gaussian; use epsilon ≤ Delta_w / 3 to avoid over-smoothing.
2. S40-31 Tikhonov / TV regularization
   • J[W] = ∫ ( (dW/dr)^2 + lambda · W^2 ) dr or J[W] = ∫ | dW/dr | dr.
   • Set lambda via cross-validation or independent benchmarks.
3. S40-32 Physical clamping
   Enforce W ∈ [0,1], Xi_TW ≥ 0 during construction/evaluation; log trigger rates.

VIII. Non-Sphericity & Geometric Consistency

• Single-crossing guarantee: along any radial ray, W(r) transitions once to avoid segmentation ambiguity.
• Normal continuity: compute n_vec via the normalized cross-product of surface tangents from S(theta,phi); C^1 continuity stabilizes the interface normal.
• Step size & error: reduce path step size in regions with large |dW/dr| to keep segmentation endpoint errors within tolerance.

IX. Calibration & Inversion Flows (M40-7 … M40-12)

• M40-7 Priors
  Choose a model family and initial values { r0, Delta_w, sigma_w, alpha, lambda, a_{lm} }, plus physical bounds and soft constraints.
• M40-8 Benchmark path & band design
  Use multiple frequencies on the same path to isolate the path term; use multi-path, multi-angle data to identify non-spherical coefficients.
• M40-9 Objective
  min_theta ∑ ( ( T_arr_obs − T_arr_mod(theta) ) / u_c )^2 + R(theta ), where theta includes TWProfile parameters.
• M40-10 Iteration & linearization
  Use first-order sensitivities ∂T_arr/∂theta (see Chapter 3) with LM or BFGS; validate the linear regime via MC when necessary.
• M40-11 Consistency checks
  Compare thin- vs thick-wall pipelines; assess two-gauge consistency eta_T; ensure R_TW + T_trans + A_sigma = 1.
• M40-12 Archival
  Persist TWProfile, contract, logs, hashes, and falsification samples; record key values eta_w, epsilon, lambda.

X. Thin-/Thick-Wall Switching & Validation (S40-33 … S40-35)

1. S40-33 Switch criteria
   • If Delta_w / r_H ≤ eta_w, use thin-wall: represent wall contribution via Delta_T_sigma.
   • Otherwise, thick-wall: integrate directly within the wall ∫_{layer} ( n_eff / c_ref ) d ell.
2. S40-34 Consistency test
   Near the threshold eta_w, require | T_arr^{thick} − (T_arr^{thin} + Delta_T_sigma) | ≤ tau_switch.
3. S40-35 Sensitivity alert
   If ∂T_arr/∂Delta_w or ∂T_arr/∂r_H dominates, flag geometry–physics coupling degeneracy and expand the set of paths and frequencies.

XI. Implementation Bindings & Interfaces (I40- interop)*

• build_tension_wall_profile( M_bh, a_bh, params ) -> TWProfile
  params includes model, r0, Delta_w, sigma_w, a_{lm}, alpha, lambda, eta_w.
• apply_TW_matching( Phi_T, TWProfile ) -> Phi_T_matched
  Generate side limits of Phi_T using W(r) and the selected interface type.
• estimate_neff_TW( Phi_T, grad_Phi_T, rho, f, TWProfile ) -> n_eff
  Incorporate H_TW( W, Xi_TW, f ), maintaining n_eff ≥ 1.
• detect_TW_intersections( gamma, TWProfile ) -> { ell_i }
  Provide precision and convergence logs.
• simulate_multipath_TW( n_eff, gamma, TWProfile, mode ) -> { T_arr_i, weights }
  Generate echo orders and weights, consistent with R_TW, T_trans, A_sigma.
• All interfaces must perform dimensional and unit checks at ingress and emit hash(TWProfile) and key thresholds.

XII. Risks & Systematic-Error Safeguards

• Parameter degeneracy: { r_H, Delta_w } may correlate with { u0, u1 }; break degeneracy via multi-path, multi-band design.
• Non-spherical leakage: high-order a_{lm} are prone to fit noise; cap expansion order and enforce stronger regularization.
• Over-smoothing: too-large epsilon depresses Xi_TW, underestimating Delta_T_sigma.
• Crossing jitter: segmentation endpoint errors bias T_arr; tighten crossing solvers and fix interpolation order.
• Clamping & saturation: log trigger rates for W ∈ [0,1] and n_eff ∈ [1, n_max]; exceedances must trigger review.

XIII. Cross-References

• EFT.WP.BH.TensionWall v1.0 Chapter 3 (Minimal Equations & Structural Model), Chapter 4 (Geometry & Coordinates), Chapter 6 (Near-Wall Propagation Gauges), Chapter 8 (Interface Matching)
• EFT.WP.Propagation.TensionPotential v1.0 Chapters 4–6 (Phi_T, n_eff, paths & two gauges)
• EFT.WP.Core.Metrology v1.0 M05-, M10- (calibration & traceability)
• EFT.WP.Core.Equations v1.1 S06-* (notation & operators)

XIV. Deliverables

• TWProfile specification and example roster including model, r_H, Delta_w, sigma_w, a_{lm}, alpha, lambda, eta_w.
• Smoothing/regularization configuration records: epsilon, lambda, and sensitivity curves.
• Switching & consistency report template: thin–thick differences, two-gauge consistency eta_T, and energy-consistency audit fields.