33-EFT.WP.Cosmo.EarlyObjects v1.0 | Chapter 8 — Interaction with the Layered Sea / Interfaces

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Chapter 8 — Interaction with the Layered Sea / Interfaces

I. One-Sentence Goal

Unify the three matching modes between an early object’s near field and the layered SeaProfile / environmental interfaces Sigma_env—continuous, potential jump, and flux jump—with a segmented propagation–correction convention. Define metrology and auditing of the energy triplet { R_env, T_trans, A_sigma }, the thin/thick consistency tau_switch, and connect everything end-to-end with the two arrival-time dialects T_arr / Delta_T_arr.

II. Scope & Non-Goals

• Covered: interface/layer types and side-limits; segmented integration and zero-thickness correction Delta_T_sigma; thick-layer volume integration and the equivalence criterion; estimation and auditing of the energy triplet; two-form consistency and lower bounds; stepwise flow (extended M70-7 … M70-8); interfaces and minimal logging set.
• Not covered: re-deriving the equations or spectroscopy in Chs. 3–6; replacing implementation details of LayeredSea Ch. 8; instrument and observatory pipelines.

III. Minimal Terms & Symbols

• Interfaces & layers: Sigma_env (set of environmental interfaces), SeaProfile (layered environment object), layer profiles W_k(chi), strengths Xi_k(chi), layer thickness Delta_k.
• Side-limits & jumps: Phi_T^+ / Phi_T^-, grad_Phi_T^+ / grad_Phi_T^-; C_sigma = Phi_T^+ − Phi_T^-, J_sigma = dot( grad_Phi_T^+ − grad_Phi_T^- , n_vec ).
• Propagation & path: n_eff(x,t,f) (dimensionless, n_eff ≥ 1), c_ref, arclength path gamma(ell), segments gamma_i, endpoints { ell_i }.
• Corrections & consistency: zero-thickness correction Delta_T_sigma, thin/thick consistency tau_switch; two-form consistency eta_T.
• Energy triplet: R_env, T_trans, A_sigma, with R_env + T_trans + A_sigma = 1.
• Naming isolation: T_fil (tension) ≠ T_trans (transmittance); n (number density) ≠ n_eff (effective refractive index).

IV. Three Matching Modes & Side-Limits (per LayeredSea Ch. 8)

• Continuous: Phi_T^+ = Phi_T^-, J_sigma = 0. If the constitutive map F is continuous, then n_eff^+ = n_eff^-; segmented integration suffices.
• Potential jump (jump_phi): C_sigma ≠ 0, J_sigma = 0. Side-limits may differ; under thin-layer conditions, the added delay is represented by Delta_T_sigma.
• Flux jump (jump_flux): C_sigma = 0, J_sigma ≠ 0. Introduce a normal-response term or interface coefficient in n_eff.
• Directional extension (optional): when directionality is significant, include b1_n • dot( grad_Phi_T , n_vec ) (dimensioning and auditing per Appendix A and Ch. 11).

V. Segmented Integration & Two-Form Control (enforced at layers/interfaces)

1. Segmentation rule: find { ell_i } such that gamma(ell_i) ∈ Sigma_env ∪ Sigma_sea; include endpoints explicitly in integrals; forbid cross-interface interpolation.
2. Two-form arrival time:
   • Constant-factored:
     T_arr = ( 1 / c_ref ) * ( ∑_i ∫_{gamma_i} n_eff d ell )
   • General form:
     T_arr = ∑_i ∫_{gamma_i} ( n_eff / c_ref ) d ell
3. Lower bound: with n_eff ≥ 1, T_arr ≥ L_path / c_ref (the general form enforces the same in the integrand).

VI. Thin/Thick Consistency & Zero-Thickness Correction

• Thin-layer condition: when Delta_k / L_char ≤ eta_w, use the event-type correction
  Delta_T_sigma ≈ k_sigma • H(crossing).
• Thick-layer volume integration:
  T_arr^{layer_k} = ∫_{layer_k} ( n_eff / c_ref ) d ell.
• Consistency quantification:
  tau_switch = | T_arr^{thick} − ( T_arr^{thin} + Delta_T_sigma ) |.
  Perform dual solves in the eta_w neighborhood and require tau_switch ≤ gate; if exceeded, revert to the thick-layer chain and re-audit endpoint tolerances and SeaProfile parameters.

VII. Metrology & Audit of the Energy Triplet (M70-7)

• Estimation: from incident/exit path pairs and power conservation, back-out R_env(f), T_trans(f), A_sigma(f); produce in-band curves with confidence bands.
• Audit: for each interface event, check the residual to R_env + T_trans + A_sigma = 1 over frequency; enforce n_eff^± ≥ 1 as a side-limit.
• Recording: write curves, residuals, methods, and thresholds into the Report/Log with coords_spec / units_spec / metric_spec.

VIII. Band Differential & Inter-Layer Consistency (aligned with Ch. 6)

1. Same-path differential:
   • Constant-factored:
     Delta_T_arr(f1,f2) = ( 1 / c_ref ) * ∫ ( n_path(f1) − n_path(f2) ) d ell
   • General form:
     Delta_T_arr(f1,f2) = ∫ ( ( n_path(f1) − n_path(f2) ) / c_ref ) d ell
2. Consistency requirement: the two bands must reuse the same { gamma[k], Δell[k] }, the same segmentation, and the same Delta_T_sigma configuration; out-of-band residuals go to u_sys, and the leakage ratio is logged.

IX. Stepwise Flow (M70-7 … M70-8 Extended)

• Interface detection & classification. Detect Sigma_env / Sigma_sea; classify as continuous / jump_phi / jump_flux / anisotropic; output { ell_i } with type labels.
• Segmentation & corrections. Build segments gamma_i from { ell_i } and perform segmented integration; invoke Delta_T_sigma for thin layers; execute volume integration for thick layers.
• Energy closure & side-limit audit. Output curves and residuals for R_env / T_trans / A_sigma; check n_eff^± ≥ 1.
• Two-form consistency. Compute T_arr^{const} and T_arr^{gen} in parallel; produce eta_T; if exceeded, revisit c_ref and n_eff decomposition.
• Thin/thick dual solve. In the eta_w neighborhood, compute tau_switch; if not met, fix to thick and revisit endpoint tolerances and SeaProfile.
• Archival. Record interface_marks:{ ell_i }, thresholds, methods, curves/residuals, and hash(SeaProfile / n_eff / gamma).

X. Interfaces & Implementation Bindings (aligned with template I70-*)

• detect_interfaces( gamma, Sigma_env or SeaProfile ) -> { ell_i, labels }
• apply_sea_matching( Phi_T, SeaProfile ) -> Phi_T_matched (reuse LayeredSea I/F)
• segment_integrals( n_eff, gamma, { ell_i }, mode ) -> { T_arr_i }
• interface_correction_sea( gamma, SeaProfile, params ) -> Delta_T_sigma
• estimate_energy_triplet( data, Sigma_env ) -> { R_env, T_trans, A_sigma }
• log_env_interface( meta, hashes, metrics ) -> Log
  Constraints: entry dimension checks; audits for T_arr ≥ L_path / c_ref and R_env + T_trans + A_sigma = 1; naming isolation is mandatory; logs include eta_T and tau_switch.

XI. Acceptance Criteria & Falsification Lines

• Accept if: segment endpoints are explicitly integrated with no cross-interface interpolation; two-form consistency eta_T within gate; tau_switch within gate; energy-closure residuals and side-limit lower bounds within gates; Delta_T_arr isolation valid in linear regime / specified order.
• Falsify if: any n_eff^± < 1; R_env + T_trans + A_sigma ≠ 1 persists; eta_T or tau_switch over gate with no resolvable back-trace; missing segments or inconsistent corrections.

XII. Traceability & Minimal Logging Set

• hash(SeaProfile), hash(n_eff), hash(gamma), Sigma_env labels and { ell_i } tolerances;
• mode, eps_T, eta_T, eta_c, eta_w, tau_switch, lower-bound margin T_arr − L_path/c_ref;
• Residual curves for R_env / T_trans / A_sigma, trigger stats for Delta_T_sigma, linear-regime checks and OOB leakage ratio for Delta_T_arr;
• coords_spec / units_spec / metric_spec, SolverCfg, RNG seed, falsification samples and replay handles.

XIII. Cross-References

• EFT.WP.Cosmo.EarlyObjects v1.0: Ch. 3 (minimal equations), Ch. 5 (coupling & growth laws), Ch. 6 (radiation & propagation), Ch. 7 (metrology), Ch. 9 (numerics).
• EFT.WP.Cosmo.LayeredSea v1.0: Chs. 6–8 (segmentation, matching, consistency).
• EFT.WP.Propagation.TensionPotential v1.0: two forms, differentials, and path conventions.

XIV. Deliverables

• Interface–layer coupling rules: actionable cards for continuous / jump_phi / jump_flux / anisotropic modes with unit/dimension checks.
• Segmentation & correction templates: { ell_i } detection, Delta_T_sigma / thick-layer integration call patterns, and logging fields.
• Audit dashboards: eta_T, tau_switch, energy-closure residuals, Delta_T_arr linear regime & OOB leakage, plus a falsification sample roster.