19-EFT.WP.Methods.SynthData v1.0 | Chapter 7 — Conditional & Controllable Generation (Prompt/CFG/Rules)

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Chapter 7 — Conditional & Controllable Generation (Prompt/CFG/Rules)

I. Scope & Targets

1. Goals
   • Define a unified convention for conditional and controllable generation: x ~ p_model(x | c; theta), where c may be a text prompt, structured conditions, numeric bounds, or a set of rules.
   • Specify cooperative mechanisms among CFG (classifier-free guidance), hard constraints, and soft penalties; achieve interpretable control without violating units/dimensions or physical constraints.
   • Bring condition alignment, rule satisfaction, and downstream utility under contracts and manifests: manifest.synth.cond.*.
2. Inputs
   Generation engine engine(theta) (see Chapter 5), condition set Cset, rules & constraints Rules = { g_j(x,c) ≤ 0 }, a reference distribution or panel ref, time-base and arrival-time conventions, SLOs and thresholds.
3. Outputs
   Conditional samples D_syn(c), alignment reports and acceptance rate acc_rate(c), control strengths and schedules w_cfg(t), contract evaluation report, and manifest.synth.cond.
4. Applicability
   Applies to tabular, time series, image/audio/text, and multimodal settings; when physical chains are involved, execute jointly with Chapter 6 (record T_arr, delta_form).

II. Terms & Symbols

• Conditions & control: c (condition/prompt), enc(c) (condition encoding), w_cfg ∈ R_+ (CFG strength), lambda_j ≥ 0 (Lagrange multipliers for rules), schedule w_cfg(t) (diffusion timestep schedule), policy (sampling & rules policy).
• Rules & acceptance: g_j(x,c) ≤ 0 (j-th hard constraint), m_acc ∈ {0,1} (accept indicator), acc_rate = ( ∑ m_acc ) / N.
• Alignment & utility: sim_embed(x,c) (condition–sample similarity, e.g., embedding cosine), util(x,c) (downstream utility score), penalty(x,c) (rule penalty).
• Time & arrival: tau_mono, ts, T_arr, gamma(ell), delta_form, offset/skew/J.
• Metrology & units: unit(•), dim(•), check_dim(expr).

*III. Axioms P407- **

• P407-1 (Explicit Conditions): The semantics, domain, and units of c must be explicit and persisted; implicit defaults are prohibited.
• P407-2 (Minimal Distortion): Control must preserve the statistical fidelity of p_model(x|c); any rule or post-processing should minimize distortion to p_model.
• P407-3 (Hard-Constraint Priority): Use hard constraints (reject or project) for safety, physics, and referential integrity; use soft penalties or resampling for the rest.
• P407-4 (Auditable Schedules): w_cfg(t) and related schedules (sampling temperature, truncation, etc.) must be functional and reproducible.
• P407-5 (Measurable Alignment): Evaluate condition–sample alignment with repeatable metrics sim_embed(x,c) and bring them into contracts.
• P407-6 (Units & Dimensions): Conditional control must not violate unit/dim; check_dim(expr) must pass.
• P407-7 (Time Base & Arrival): When time/path propagation is present, record both T_arr formulations and delta_form; evaluate windows on tau_mono.
• P407-8 (Fairness & Bias): Publish coverage and disparity metrics across important subgroups for p(•|c); provide debiasing or reweighting paths.
• P407-9 (Reproducibility & Trace): Persist seed/rng/enc(c)/w_cfg(t)/Rules in the manifest and sign it.
• P407-10 (Multimodal Consistency): Declare the shared embedding or alignment mapping for multimodal conditions to avoid cross-modal ambiguity.

*IV. Minimal Equations S407- **

1. S407-1 (Conditional Generation Base Form)
   x ~ p_model( x | c; theta )，目标最小化 D( p_model(x|c) || p_ref(x|c) )，D ∈ {W1, KL, MMD}。
2. S407-2 (Generic CFG Form)
   With guided fields s_cond(z,t) = s_theta(z,t|c) and s_uncond(z,t) = s_theta(z,t|∅), define
   s_guided(z,t) = s_uncond(z,t) + w_cfg(t) * ( s_cond(z,t) - s_uncond(z,t) )。
3. S407-3 (Soft Penalties with Lagrange Multipliers)
   min_theta E_{c} E_{x~p_theta(•|c)} [ L_fid(x,c) + ( ∑_j lambda_j * g_j^+(x,c) ) ]，其中 g_j^+(x,c) = max( 0, g_j(x,c) )。
4. S407-4 (Hard-Constraint Accept/Reject or Projection)
   • m_acc(x,c) = 1 若 ∀j, g_j(x,c) ≤ 0，否则 0；接受率 acc_rate = ( ∑ m_acc ) / N。
   • Constraint projection: x' = Pi_C(x) = argmin_{z ∈ C} d(z,x) with residual res_cons = d(x',x).
5. S407-5 (Alignment Thresholds & Utility)
   sim_embed(x,c) ≥ sim_min，util(x,c) ≥ util_min；不达标触发重采样或增益控制。
6. S407-6 (Sequential KL Regularization)
   max_theta E_{x~p_theta(•|c)}[ R(x,c) ] - beta * KL( p_theta(•|c) || p_ref(•|c) )，beta ≥ 0。
7. S407-7 (Dual Arrival Forms)
   T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ) 与 T_arr = ( ∫ ( n_eff / c_ref ) d ell )，
   delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |。

V. Metrology Flow M40-7 (Conditional & Controllable Loop)

• Ready
  Fix Cset, the encoding enc(c), alignment metric sim_embed, rules Rules, schedule w_cfg(t), thresholds sim_min/util_min, acc_min, SLOs, and manifest keys.
• Condition Encoding
  Parse c into enc(c) (text/structured/numeric); normalize units and run dimensional checks check_dim.
• Candidate Sampling
  Have the engine produce initial states z_t or x0; record seed/rng.
• Guidance & Stepping
  Compute s_cond and s_uncond, combine to s_guided, and step with w_cfg(t) to propose x.
• Rule Checking
  Evaluate g_j(x,c); for violations, either reject or project x ← Pi_C(x); record res_cons and m_acc.
• Alignment & Utility
  Measure sim_embed(x,c) and util(x,c); if below thresholds, adaptively raise w_cfg or resample.
• Reweighting & Debiasing (optional)
  Compute a map or weights w(i) so p_syn(•|c) aligns with ref; persist weights and n_eff.
• Time Base & Arrival (if applicable)
  Write tau_mono/ts, the two T_arr formulations with delta_form, and offset/skew/J.
• Persist & Publish
  Emit manifest.synth.cond including enc(c), w_cfg(t), Rules, thresholds, metrics, acc_rate, and signature; freeze release or roll back.

VI. Contracts & Assertions C40-7xx

• C40-701 (Valid Encoding): enc(c) lies within the registered domain; syntax and units are valid.
• C40-702 (Hard Constraints): P( g_j(x,c) ≤ 0 ) ≥ p_pass_min, or res_cons ≤ tol_cons.
• C40-703 (Alignment): sim_embed_p50 ≥ sim_min and sim_embed_p05 ≥ sim_floor.
• C40-704 (Acceptance Rate): acc_rate ≥ acc_min; if not, publish the retry rounds R used.
• C40-705 (Units & Dimensions): check_dim(expr)=true.
• C40-706 (Time/Arrival): non_decreasing(tau_mono), J ≤ J_max, delta_form ≤ tol_Tarr.
• C40-707 (Fairness & Coverage): For key subgroup g, | metric_g - metric_all | ≤ gap_max, or publish the reweighting map.
• C40-708 (SLO): latency_ms_p99 ≤ SLO_cond, oom_rate ≤ oom_max.
• C40-709 (Reproducibility): seed/rng/w_cfg(t)/Rules replayable; hash signatures consistent.

VII. Implementation Bindings I40-7*

• encode_condition(c, registry) -> enc(c)
• compose_guidance(engine, method, w_schedule) -> engine' (where method ∈ {CFG, classifier, energy})
• sample_conditional(engine', n, enc(c), seed) -> ds_syn
• evaluate_rules(ds_syn, Rules) -> {m_acc, res_cons, report}
• accept_or_project(ds_syn, Rules, projector) -> ds_syn'
• measure_alignment(ds_syn', enc(c), metric) -> {sim_stats, util_stats}
• rebalance_conditional(ref, ds_syn', method) -> {map|w}
• annotate_time_arrival(ds_syn', ref_path) -> ds_syn'' (write T_arr, delta_form, offset/skew/J)
• emit_conditional_manifest(artifacts) -> manifest.synth.cond
• Invariants: reproducible(seed); acc_rate ≥ acc_min; sim_embed_p50 ≥ sim_min; delta_form ≤ tol_Tarr; units/dimensions pass checks.

VIII. Cross-References

• This volume: Chapter 5 (CFG implementations in deep generators), Chapter 6 (physics/simulation & constraint projection), Chapter 12 (fidelity & utility evaluation), Chapter 13 (release & manifests).
• Methods.Cleaning v1.0: Chapter 10 (contracts & release freeze).
• Methods.CrossStats v1.0: Chapters 7/9/14 (distribution alignment, calibration transfer, statistical SLOs).
• Methods.Imaging v1.0: Chapter 13 (time/path-gated arrival-time consistency).

IX. Quality Metrics & Risk Control

1. Core SLIs
   sim_embed_p50/p05/p95, acc_rate, res_cons, n_eff, latency_ms_p99, oom_rate, fairness_gap, delta_form, J.
2. Common risks & mitigations
   • Overlarge CFG causing mode collapse → use increasing w_cfg(t) schedules and early stop; introduce KL regularization.
   • Overtight rules causing low acceptance → relax to soft penalties, switch to projection, or stage constraints.
   • Condition–sample mismatch → improve encoders or embeddings; apply adaptive sim_min.
   • Unit/dimension violations → run check_dim and range clamping in both encoding and post-processing.
   • Subgroup bias → enable reweighting or mapping alignment; publish gaps and corrective evidence.
   • Time/arrival drift → re-run annotate_time_arrival; audit delta_form and J.

Summary