13-EFT.WP.Methods.SimStack v1.0 | Chapter 5: Time Bases, Synchronization, and Arrival-Time Calibration

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Chapter 5: Time Bases, Synchronization, and Arrival-Time Calibration

I. Scope & Objectives

• Establish unified conventions for time bases, synchronization mechanisms, and arrival-time calibration. Provide the mapping model from the internal time base tau_mono to the published time base ts, define parallel computation of the two T_arr formulations and the discrepancy report delta_form, and standardize the Mx-61 calibration flow with quality gates.
• Ensure, across devices, across layers (continuous kernel and discrete weaving), and across paths gamma(ell), that causality hb is upheld, units and measures are consistent, arrival-time computations are traceable, and published results are reproducible.

II. Terms & Symbols

1. Time bases and mapping
   • tau_mono: internal monotonic clock.
   • ts: externally published clock.
   • Linear mapping: ts_i(t) = alpha_i * tau_mono + beta_i + epsilon_i(t).
   • alpha_i: clock skew; beta_i: clock offset; epsilon_i(t): zero-mean jitter term.
2. Two formulations of arrival time
   • General form: T_arr(gamma) = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell ).
   • Pull-out-constant form: T_arr(gamma) = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell ).
   • Discrepancy: delta_form = | ( ∫ ( n_eff / c_ref ) d ell ) - ( 1 / c_ref ) * ( ∫ n_eff d ell ) |.
3. Path and measure
   • gamma(ell): piecewise-C1 path, ell ∈ [0, L_gamma], with line measure d ell.
   • L_gamma = ( ∫{gamma(ell)} 1 d ell ); avg_gamma[f] = ( 1 / L_gamma ) * ( ∫{gamma(ell)} f d ell ).
4. Observability and metrics
   • TS.time.offset, TS.time.skew, TS.time.jitter, TS.hb.violations.
   • Quality gates: eps_time_map, delta_form, eps_norm, eps_mass.

III. Postulates & Minimal Equations (P61-/S62-)

• P61-5 (Time-base coherence)
  Any cross-device timing comparison must be performed on tau_mono first, then mapped to ts for publication. Mapping parameters alpha/beta and their uncertainties must be recorded in the manifest.
• P61-6 (Two-form parallelism)
  Whenever arrival time is used for alignment or fitting, compute both T_arr forms in parallel and report delta_form, while recording the sources of gamma(ell), d ell, c_ref, and n_eff.
• P61-7 (Causality conservation)
  If hb(a,b) holds, no synchronization or compensation may alter that causal relation. Replay and compensation execute on tau_mono.
• S62-20 (Linear clock fitting)
  Given anchor pairs (tau_k, ts_k), estimate alpha_hat, beta_hat by weighted least squares minimizing
  Σ_k w_k * ( ts_k - ( alpha_hat * tau_k + beta_hat ) )^2.
  Residual RMS: r_rms = sqrt( Σ_k w_k * r_k^2 / Σ_k w_k ), where r_k = ts_k - ( alpha_hat * tau_k + beta_hat ).
• S62-21 (Two-form consistency metric)
  delta_form as defined above; report relative deviation
  delta_form_rel = delta_form / max( 1e-12 , T_arr ).
• S62-22 (Two-way measurement estimates)
  With bidirectional timestamps (t1A, t2B, t3B, t4A), approximate propagation delay and offset:
  d_prop ≈ ( ( t2B - t1A ) - ( t4A - t3B ) ) / 2,
  offset_AB ≈ ( ( t2B + t3B ) - ( t1A + t4A ) ) / 2.
  All times are recorded on their respective tau_mono; map to a common base before comparison.
• S62-23 (Steady-state jitter estimate)
  TS.time.jitter = std( epsilon_i(t) ) estimated over a sliding window; thresholds are scenario-dependent.

IV. Data & Manifest Conventions

1. Mapping and uncertainty
   • timebase.alpha, timebase.beta, timebase.r_rms, timebase.confidence.
   • ts_domain, tau_domain, window.size, window.step.
2. Arrival time and paths
   • T_arr.general, T_arr.factorized, delta_form, gamma.param, support.ell_range, measure.d_ell.
   • c_ref.source, n_eff.source, n_eff.context (e.g., temperature, frequency, tension).
3. Causality and synchronization
   sync.method ∈ [two_way, beacon, hybrid], sync.anchors, hb.violations, retry.count, comp.success.
4. Conservation and quality gates
   eps_time_map ≤ ε_time_gate, delta_form ≤ ε_form_gate; where applicable, also eps_norm ≤ ε_norm_gate, eps_mass ≤ ε_mass_gate.

V. Algorithms & Implementation Bindings (I60-*)

1. I60-4 calibrate_arrival_time(trace:any, paths:list, c_ref:float, medium:any) -> CalReport
   • Function
     1. Compute both T_arr forms and delta_form.
     2. Fit alpha/beta from anchor pairs and output r_rms with confidence intervals.
     3. If trace includes two-way probes, estimate d_prop and offset_AB.
   • Key steps
     1. Parse gamma(ell) and d ell, validate units and dimensions.
     2. Compute T_arr.general and T_arr.factorized.
     3. Assemble (tau_k, ts_k, w_k) and fit alpha/beta.
     4. Produce CalReport = {alpha, beta, r_rms, T_arr.*, delta_form, anchors, quality}.
2. I60-3 advance_dt with I60-2 exchange_packets
   • advance_dt records anchors at the end of each step and triggers conditional calibration.
   • exchange_packets records event causality and time-sync stamps at event boundaries, with idempotent deduplication.
3. I60-5 emit_metrics
   Periodically emits TS.time.* and TS.hb.violations, and persists snapshots of mapping parameters.

VI. Metrology Flow & Run Diagram (Mx-61)

1. Objective
   On startup, on drift excursions, and at window rotations, execute unified time and arrival-time calibration to guarantee cross-layer alignment and traceability.
2. Flow
   • Collect anchors: record a set of (tau_k, ts_k) on tau_mono, and gather path and medium context.
   • Arrival-time computation: for each path gamma, compute both T_arr forms.
   • Time-base fitting: estimate alpha/beta by weighted least squares, obtain r_rms and confidence bounds.
   • Consistency checks: verify delta_form, eps_time_map, and TS.time.jitter against gates.
   • Publish & persist: write timebase.*, T_arr.*, anchors, and evidence for passing gates.
   • Triggers: if any gate fails, shorten the window, raise anchor frequency, or switch the sync method.
3. Artifact
   Calibration report cal.json containing mapping parameters, two-form results, gate statuses, and replay directives.

VII. Verification & Test Matrix

1. Minimum required
   • Constant medium with static offset: expect delta_form ≈ 0, alpha ≈ constant, beta ≈ constant.
   • Segmented path: gamma = gamma_1 ⋃ gamma_2; verify T_arr additivity and stability of delta_form.
   • Two-way exchange: inject known d_prop and offset_AB, verify estimation error.
2. Boundary & extreme cases
   • High-jitter conditions: as var( epsilon_i(t) ) grows, confirm TS.time.jitter alerts and strategy switching.
   • Drift transitions: under piecewise changes in alpha, verify piecewise fitting and window adaptation.
   • Strong medium variation: rapidly varying n_eff causing divergence between T_arr forms, verify compensation and reporting.
3. Regression & thresholds
   With fixed anchors and conditions, compare Δalpha, Δbeta, Δr_rms, Δdelta_form, and TS.hb.violations.

VIII. Cross-References & Dependencies

• With the Continuous Kernel (Chapter 2)
  T_arr depends on explicit n_eff, c_ref, and gamma(ell) measures; passing eps_norm and eps_mass gates is a precondition for publication.
• With the Thread Network (Chapter 3)
  Anchor collection and event causality recording follow hb semantics and the idempotency contract; synchronization-induced retries require compensating transactions.
• With Coupled Advancement (Chapter 4)
  Trigger Mx-61 at strong-sync barriers and loose-sync window boundaries. Include both T_arr forms with delta_form in StepReport and window-alignment reports.

IX. Risks, Limitations & Open Questions

1. Risks
   • Omitting two-form computation and delta_form reporting makes arrival-time results incomparable.
   • Ignoring ts quantization and sampling delays biases alpha/beta estimates.
   • Poor anchor selection can undermine the representativeness of r_rms.
2. Limitations
   No specific time-sync protocol is mandated; only abstract methods and manifest requirements. Implementations must satisfy mapping and evidence persistence.
3. Open questions
   • Real-time estimation of nonlinear clock drift (temperature and load coupling).
   • Robust T_arr calibration and uncertainty propagation under strongly random path disturbances.

X. Deliverables & Versioning

1. Deliverables
   • Reference implementation and contract tests for I60-4.
   • Mx-61 run scripts and a cal.json template.
   • Dashboard configurations for TS.time.*, TS.hb.violations, delta_form, and gate statuses.
2. Versioning
   From v1.0, freeze field names and gate semantics. Add new statistics in a backward-compatible manner and include migration guidance.

XI. New Terms & Symbols (to memorize)

• Time bases & mapping: alpha_i, beta_i, epsilon_i(t), r_rms, eps_time_map.
• Arrival time & paths: T_arr.general, T_arr.factorized, delta_form, gamma(ell), d ell, L_gamma, avg_gamma[f].
• Synchronization & anchors: two_way, beacon, hybrid, anchors, offset_AB, d_prop.
• Metrics & gates: TS.time.offset, TS.time.skew, TS.time.jitter, TS.hb.violations, ε_time_gate, ε_form_gate.