32-EFT.WP.Cosmo.LayeredSea v1.0 | Chapter 4 — Geometry & Coordinate Choice

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Chapter 4 — Geometry & Coordinate Choice

• I. One-Sentence Objective
  One-sentence goal: Without relying on any specific cosmological solution, standardize the geometric description and coordinate conventions for a layered Energy Sea by providing the mapping from conformal-time/comoving-radius to physical arclength d ell, the arclength parameterization of the path gamma(ell), rules for detecting intersections with layer interfaces Sigma_sea and for piecewise segmentation, together with normal/tangent construction and step-size/error control — thereby supplying a stable geometric foundation for propagation and arrival-time calculations.
• II. Scope & Non-Goals
• Coverage: coordinate and metric conventions; the chi → physical arclength mapping; arclength parameterization and reparameterization invariance; computation of the interface normal n_vec and path tangent t_hat; intersection detection { ell_i } with Sigma_sea and segmentation; adaptive step sizing driven by Xi_k(chi); data-contracts and implementation anchors.
• Non-Goals: no derivation of concrete cosmological metric equations; no device-level surveying schemes; no replacement for solver details and performance tuning in Chapter 9.
• III. Minimal Terms & Symbols
• Coordinates & Metric: eta (conformal time), chi (comoving radial), a(eta) (scale factor); the line element is declared in the Contract.
• Layered Sea & Interfaces: SeaProfile, layer profile W_k(chi), layer strength Xi_k(chi) = | dW_k/dchi |, thickness Delta_k, interface set Sigma_sea.
• Path & Measure: gamma(ell) (arclength-parameterized path), ell ∈ [0, L], line element d ell, piecewise segments gamma_i.
• Direction Vectors: unit tangent t_hat(ell) = d gamma / d ell / norm( d gamma / d ell ); outward normal n_vec defined on the interface.
• Constraints: dim(d ell) = [L]; do not conflate T_fil with T_trans, nor n with n_eff.
• IV. Mapping from Coordinates & Metric to Arclength (chi → d ell)
• Line-Element Declaration (at the data-contract layer)
  The Contract must provide the metric and the mapping from comoving coordinates to physical arclength via metric_spec. At minimum, the physical arclength along the path element is
  d ell = a(eta) · norm( d x_comov ), where d x_comov is composed of variations in chi and angular variables.
• Pure radial segment: with no angular variation, d ell = a(eta) · d chi.
• With angular variation: under the chosen spatial curvature, write
  d ell = a(eta) · sqrt( d chi^2 + S_k(chi)^2 dΩ^2 ),
  with S_k(chi) and the curvature sign declared explicitly in the Contract.
• Units & Consistency: any coordinate–metric combination must ensure dim(d ell) = [L]. Unify coords_spec and units_spec at entry, and persist a DimReport in logs.
• V. Arclength Parameterization & Reparameterization Invariance
• Arclength parameterization: construct gamma(ell) such that norm( d gamma / d ell ) = 1, ensuring well-conditioned coupling of integrals and integrands.
• Reparameterization invariance: for a strictly monotone, differentiable change of variables sigma = h(ell),
  ∫_0^L g( gamma(ell) ) d ell = ∫_{h(0)}^{h(L)} g( gamma( h^{-1}(sigma) ) ) d sigma.
• Implementation note: a captured trajectory gamma_raw(s) must be normalized by reparameterize_by_arclength( gamma_raw ) -> gamma(ell) to avoid systematic errors from nonuniform sampling.
• VI. Intersection Detection with Sigma_sea & Segmentation
• Intersection definition: ell_i satisfies gamma(ell_i) ∈ Sigma_sea.
• Numerical detection workflow
• Sign-change localization: on the discrete { gamma[k] }, for each layer construct a discriminator (example) F_k(chi) = W_k(chi) − 0.5, or use an explicit interface implicit form G_k(chi) = 0, and search intervals exhibiting sign changes.
• Refinement & root-finding: refine each interval to tolerance tol_x by bisection/secant/Newton to obtain ell_i.
• Segment construction: partition the path by { ell_i } into gamma_i = gamma|_[ell_i, ell_{i+1}], each segment lying on a single one-sided limit.
• Hard rules: include { ell_i } explicitly in integrals; forbid interpolation across an interface; record intersections and tolerances in the interface_marks log.
• VII. Computing n_vec and t_hat
• Tangent: t_hat(ell) = d gamma / d ell / norm( d gamma / d ell ) (already unit if arclength parameterized).
• Normal (surface parameterization): if the interface is a parametric surface S(θ, φ) with physical embedding x = X(θ, φ), then
  n_vec = normalize( ∂X/∂θ × ∂X/∂φ ).
  If the interface is given implicitly by G(x) = 0, then
  n_vec = normalize( grad G(x) ).
• Usage: directional extensions (if enabled) require dot( grad_Phi_T , t_hat ) and dot( grad_Phi_T , n_vec ), which enter n_eff via S60-3 in Chapter 3.
• VIII. Adaptive Step Size & Error Control (Coupled to Xi_k(chi))
• Three triggers for step reduction
• Geometric-curvature threshold: when norm( d^2 gamma / d ell^2 ) ≥ tau_geom.
• Medium-variation threshold: when | d n_eff / d ell | ≥ tau_medium.
• Layer-strength threshold: when Xi_k(chi) ≥ tau_sea (steep interfaces).
• Local error estimate: within a segment, use a two-order quadrature difference as a local error proxy; aggregate global error by root-sum-of-squares and enforce
  | T_arr^{(fine)} − T_arr^{(coarse)} | ≤ eps_T.
• Consistency checks: enforce symmetric sampling near Sigma_sea to avoid endpoint bias; audit thin/thick dual-computation with tau_switch (see Chapters 6 and 9).
• IX. Data Contracts & Recording (Minimal Implementation Set)
• Minimum Contract fields
• coords_spec (e.g., Comoving-Spherical), units_spec (SI), metric_spec (metric and chi → d ell mapping), mode ∈ { constant, general };
• Tolerances & thresholds: eps_T, tau_geom, tau_medium, tau_sea, eta_w, tau_switch;
• Hashes: hash(SeaProfile), hash(gamma); logs: interface_marks:{ ell_i }, DimReport.
• Consistency hooks: at entry, run check_dimension to ensure dim(d ell) = [L], dim(T_arr) = [T]; audit T_arr ≥ L_path / c_ref and energy consistency R_sea + T_trans + A_sigma = 1.
• X. Implementation Bindings & Interface Anchors (Aligned with the Template Family)
• I.Path.Capture → recommended: capture_path(raw_track, coord_spec) -> { gamma[k], Δell[k], t_hat[k] }.
• I.Path.Segment → detect_sea_intersections(gamma, SeaProfile) -> { ell_i, layer_id };
  segment_integrals(n_eff, gamma, { ell_i }, mode) -> { T_arr_i }.
• I.Interface.ApplyMatching → apply_sea_matching(Phi_T, SeaProfile) -> Phi_T_matched (for one-sided consistency).
• I.Report.Log → record coords_spec/units_spec/metric_spec, thresholds and endpoint tolerances, DimReport, and hash(*).
• XI. Pass Criteria & Falsification Lines (Chapter Level)
• Pass Criteria
• Clear metric→arclength mapping with dimensional checks passed;
• Path arclength parameterized; endpoints { ell_i } explicitly included; no cross-interface interpolation;
• Three-threshold triggering effective; eps_T satisfied;
• Logs include metric_spec, interface_marks, DimReport.
• Falsification Lines
• dim(d ell) ≠ [L] or T_arr < L_path / c_ref;
• Missing intersections or endpoints not included; any interpolation across an interface;
• No step reduction in strong-layer regions leading to loss of convergence, or tau_switch above threshold.
• XII. Cross-References
• EFT.WP.Cosmo.LayeredSea v1.0: Ch. 3 (Layer Profiles & Minimal Equations), Ch. 6 (Propagation & Arrival Time), Ch. 8 (Interface Matching), Ch. 9 (Modeling & Numerical Realization).
• EFT.WP.Propagation.TensionPotential v1.0: path and the two formulations; differential isolation.
• EFT.WP.Core.Equations v1.1: operator and notation consistency; EFT.WP.Core.Metrology v1.0: units and traceability.
• XIII. Deliverables
• Geometry–Coordinate Contract Template: coords_spec, units_spec, metric_spec, threshold set, and hash fields.
• Path–Segmentation Implementation Checklist: arclength parameterization, intersection solving, endpoint inclusion, tolerances, and logging rules.
• Normal/Tangent Computation Card: numerical routines and audit points for n_vec and t_hat.