19-EFT.WP.Methods.SynthData v1.0 | Chapter 2 — Axioms & Minimal Equations (Generation Baseline)

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Chapter 2 — Axioms & Minimal Equations (Generation Baseline)

I. Scope & Targets

1. Goals
   • Establish a unified generation baseline for bringing p_model(x; theta) close to p_data(x), with common conventions for divergence measures, time/path consistency, dimensional conservation, and privacy budgeting.
   • Define the necessary equations and threshold mappings required by the publication gate, serving as foundations for later implementation and evaluation chapters.
2. Inputs
   • Reference distribution & samples: D_real ~ p_data(x); schema & constraints: SRef, Rules; time & path anchors: tau_mono, ts, gamma(ell).
   • Budget & policy: Privacy = {eps, delta, accounting}; evaluation metrics: Metrics = {W1, KL, JS, MMD, FID, KID, covg}.
3. Outputs
   Baseline report {divergence, coverage, uncertainty, privacy}; threshold recommendations {tol_*}; contract mapping C40-2xx.

II. Terms & Variables

• Distributions & parameters: p_data(x), p_model(x; theta), theta, z ~ p(z).
• Divergences & kernels: KL, JS, W1, MMD(k), FID(mu,Sigma).
• Time & arrival: T_arr, c_ref, n_eff (effective refractive index—distinct from weight-based n_eff), gamma(ell), d ell.
• Uncertainty: u(x) (standard uncertainty), U = k * u_c (coverage/expanded uncertainty), alpha (confidence level).
• Privacy budget: DP(eps, delta), eps_total, delta_total.
• Coverage & representativeness: covg, supp(•), n_eff_weights = ( (∑ w)^2 ) / ( ∑ w^2 ).

III. Axioms P402- (Non-Negotiables for the Baseline)*

• P402-1 (Explicit Measures): Any integral or expectation must specify domain and measure, e.g., ( ∫_Omega p(x) dx ) = 1.
• P402-2 (Declared Divergences): Fidelity is defined only via explicit divergence/distance families—no visual heuristics as substitutes.
• P402-3 (Dimensional Conservation): Any physical quantity entering models or metrics must satisfy check_dim(expr); compute only after unit harmonization.
• P402-4 (Unified Time Base): Evaluate windows on tau_mono; publish on ts with offset/skew/J recorded.
• P402-5 (Dual Arrival-Time Forms): When propagation/paths are involved, compute both
  T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ) and T_arr = ( ∫ ( n_eff / c_ref ) d ell ), and publish delta_form.
• P402-6 (Explicit Privacy Budgeting): Before any synthetic release, provide DP(eps, delta) accounting and cumulative budgets eps_total, delta_total.
• P402-7 (Minimum Publication Criterion): pass = fidelity_ok ∧ privacy_ok ∧ contract_ok ∧ manifest_signed.
• P402-8 (Reproducibility): With identical SynthSpec, seed/rng, and version, D_syn must be statistically equivalent.
• P402-9 (Coverage Lower Bounds): Coverage covg for key strata/enumerations must meet policy minima.
• P402-10 (Steady State & Backpressure): Streaming generation on G=(V,E) satisfies stability and backpressure constraints (see Core.Threads v1.0).

IV. Minimal Equations S402- (Necessary Baseline Formulae)*

1. S402-1 (Fitting Objective)
   • theta* = argmin_theta D( p_model(•; theta ) || p_data(•) ), where D ∈ { KL, JS, W1, MMD }.
   • Constrained form: min_theta ( D + λ * R(theta) ), where R encodes rule/physical/geometry/referential-integrity penalties.
2. S402-2 (Wasserstein-1 Distance)
   W1(P,Q) = inf_{pi ∈ Π(P,Q)} ( ∫ c(x,y) d pi(x,y) ), with typical c(x,y)=||x-y||_1 or ||x-y||_2.
3. S402-3 (MMD)
   MMD^2(P,Q;k) = || μ_P - μ_Q ||_H^2 = E_{x,x'} k(x,x') - 2 E_{x,y} k(x,y) + E_{y,y'} k(y,y') (declare kernel k and bandwidth).
4. S402-4 (FID/KID in Image/Embedding Space)
   • FID = || mu_r - mu_s ||_2^2 + Tr( Sigma_r + Sigma_s - 2 * ( Sigma_r * Sigma_s )^{1/2} ).
   • KID uses a kernel unbiased estimator via multiple subsamples; declare feature-extraction protocol.
5. S402-5 (Coverage & Support Sets)
   Discrete approximation: covg = | supp(D_syn) ∩ supp(D_real) | / | supp(D_real) |; in continuous space, approximate via grids or kernel mass ratios.
6. S402-6 (Uncertainty Publication)
   • Expanded uncertainty: U = k * u_c, with k set by target coverage 1 - alpha; for bootstrap, publish percentile bands {q_{alpha/2}, q_{1-alpha/2}}.
   • Delta method (short form): var( g( hat{theta} ) ) ≈ ( ∇g )^T * cov( hat{theta} ) * ( ∇g ).
7. S402-7 (Privacy Accounting, Simple Composition)
   eps_total = ( ∑_{r=1}^R eps_r ), delta_total = ( ∑_{r=1}^R delta_r ); for advanced accounting, publish the accountant and its parameters.
8. S402-8 (Dual Arrival-Form Discrepancy)
   delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |, and assert delta_form ≤ tol_Tarr.
9. S402-9 (Time Mapping & Jitter)
   ts = map_tau_to_ts( tau_mono; offset, skew ); publish jitter bound J under manifest.synth.timing.
10. S402-10 (Weight-Effective Sample Size, if Reweighting Applied)
    n_eff_weights = ( (∑ w_i)^2 ) / ( ∑ w_i^2 ), and require W_norm = ( ∑ w_i ) / N ≈ 1.

V. Synthesis Flow M40-2 (Baseline Verification)

• Readiness
  Confirm SRef/Rules; units & dimensions pass check_dim; set Metrics and preliminary thresholds {tol_W1, tol_MMD, tol_FID, covg_min}.
• Training/Fitting
  Solve theta* = argmin_theta D + λR; record theta_ref and seed/rng.
• Path/Time Consistency
  Compute both T_arr forms and delta_form; run map_tau_to_ts and measure offset/skew/J.
• Metrics & Uncertainty
  Evaluate {W1, KL, JS, MMD, FID, KID, covg}; produce U or CI.
• Privacy Accounting
  Aggregate eps_total, delta_total; if over budget, roll back or retrain with down-sampling/noise.
• Contracts & Persistence
  Execute C40-2xx contracts; on pass, write manifest.synth and sign; otherwise, issue remediation guidance and rollback points.

VI. Contracts & Assertions C40-2xx (Generation Baseline)

• C40-201 (Divergence Thresholds): W1 ≤ tol_W1 ∧ MMD ≤ tol_MMD (or FID ≤ tol_FID for imaging).
• C40-202 (Coverage Lower Bound): covg ≥ covg_min, and per-stratum covg_group ≥ covg_min_group.
• C40-203 (Arrival Consistency): delta_form ≤ tol_Tarr.
• C40-204 (Time-Base Jitter): J ≤ J_max; |offset| ≤ off_max; |skew| ≤ skew_max.
• C40-205 (Privacy Budget): eps_total ≤ bud_eps ∧ delta_total ≤ bud_delta; disclose accounting method.
• C40-206 (Dimensional Conservation): check_dim(expr)=true; publish the unit table with the manifest.
• C40-207 (Effective Sample Size, if Reweighted): n_eff_weights / N ≥ r_min and W_norm ≈ 1.
• C40-208 (Reproducibility): repeat-run deviation |metric' - metric| ≤ tol_repro.

VII. Implementation Bindings I40- (Chapter Anchors)*

• I40-201 measure_fidelity(real, syn, metrics) -> report (implements W1/KL/JS/MMD/FID/KID/covg with CI or U)
• I40-202 privacy_accounting(steps, mechanism) -> {eps_total, delta_total}
• I40-203 propagate_uncertainty(estimates, method) -> {U or CI} where method ∈ {delta, bootstrap, bayes}
• I40-204 enforce_timepath_baseline(ds_syn, ref) -> {delta_form, offset, skew, J}
• I40-205 evaluate_weight_effective(w) -> {n_eff_weights, W_norm}
• I40-206 assert_synth_contract(report, rules) -> decision
• Invariants: sum(w)/N ≈ 1; eps_total within budget; delta_form ≤ tol_Tarr; unit/dimension checks pass; identical seed reproduces equivalent outputs.

VIII. Cross-References

• Cleaning: units/dimensions (Ch.4), timeline & synchronization (Ch.5), paths & arrival time (Ch.6), contracts & release (Ch.10).
• Imaging: metric conventions & embedding spaces (PSF→metric interpretation, Ch.5; quality & audit, Ch.14).
• Cross-Statistics: uncertainty propagation (Chs.4/5), multiplicity (Ch.6), drift & alignment (Ch.7).

IX. Quality Metrics & Risk Control

• Baseline panel fields: metrics.{W1,MMD,FID,KID,KL,JS,covg}, uncertainty.{U,CI}, timing.{offset,skew,J}, arrival.delta_form, privacy.{eps_total,delta_total}.
• Risk policy: when W1 or MMD exceed thresholds, prioritize reweighting/mapping or retraining; when eps_total nears budget, switch to low-fidelity mode or delay release; when delta_form breaches, mandate parameter review for path/medium.

Summary

• This chapter codifies the baseline’s non-negotiables P402-* and supplies the necessary equations S402-* for divergence, coverage, arrival time, time-base, privacy, and uncertainty.
• Through the verification flow M40-2 and contracts C40-2xx, it provides a unified gate and implementation anchors for subsequent generator families, conditional control, and release governance.